liatlonal Bureau or btaiidara?

Library. N.W. Bldg

JAN 2

NBS MONOGRAPH 67

Methods for the

Dynamic Calibration

U.S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
THE NATIONAL BUREAU OF STANDARDS

Functions and Activities

The functions of the National Bureau of Standards are set forth in the Act of Congress,

March 3, 1901, as amended by Congress in Public Law 619, 1950. These include the development and maintenance of the national standards of measurement and the provisioh of means

and methods for making measurements consistent with these standards; the determination of

physical constants and properties of materials; the development of methods and instruments

for testing materials, devices, and structures; advisory services to government agencies on

scientific and technical problems; invention and development of devices to serve special needs

of the Government; and the development of standard practices, codes, and specifications. The

work includes basic and applied research, development, engineering, instrumentation, testing,

evaluation, calibration services, and various consultation and information services. Research

projects are also performed for other government agencies when the work relates to and supplements the basic program of the Bureau or when the Bureau's unique competence is required.

The scope of activities is suggested by the listing of divisions and sections on the inside of the

back cover. Publications

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publications or in the journals of professional and scientific societies. The Bureau itself publishes three periodicals available from the Government Printing Office: The Journal of Research,

published in four separate sections, presents complete scientific and technical papers; the Technical News Bulletin presents summary and preliminary reports on work in progress; and CRPL

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and the Supplement to National Bureau of Standards Circular 460, July 1947 to June 1957

($1.50), and Miscellaneous Publication 240, July 1957 to June 1960 (includes Titles of Papers

Published in Outside Journals 1950 to 1959) ($2.25); available from the Superintendent of

Documents, Government Printing Office, Washington D.C. 20402
UNITED STATES DEPARTMENT OF COMMERCE • Luther H. Hodges, Secretary

NATIONAL BUREAU OF STANDARDS • A. V. Astin, Director

Methods for the Dynamic Calibration

of Pressure Transducers

J. L. Schweppe, L. C. Eichberger, D. F. Muster,

E. L. Michaels, and G. F. Paskusz

Prepared by Dresser Electronics, Southwestern

Industrial Electronics Division, under contract with the National Bureau of Standards.

National Bureau of Standards Monograph 67

Issued December 12, 1963

For sale by the Superintendent of Documents, U.S. Government Printing Office Washington, D.C., 20402 - Price 60 cents
Library of Congress Catalog Card Number: 63-60069

II
Contents

Foreword

Preface

List of symbols

Chapter 1. Introduction

1. General

1.1. Statement of objectives

1.2. Background

2. Pressure transducers

2.1. Tjrpes of pressure transducers

2.2. Physical characteristics of transducers2.3. Mechanical models

2.4. Characteristic differential equations

3. Transducer calibration and analysis

3.1. Methods of analysis . 3.2. Experimental calibration methods

3.3. Relationship of calibration to analysis _ 4. Fourier methods and spectral analysis

4.1. Fourier series and line spectra

4.2. Fourier transform and continuous spec- trum

4.3. Laplace transform and continuous

spectrum

5. References Chapter 2. Analytic methods for linear transducers1. Input, output, and transfer function relations.

1.1. Direct input-output relation in the time domain

1.2. Transformation from time to fre- quency domain

1.3. Transfer function

1.4. Input from transfer function and re- sponse record

1.5. Phase-plane analysis

2. Periodic input functions

2.1. Sine function

2.2. Square wave function

2.3. Rectangular pulse train

3. Aperiodic input functions

3.1. Rectangular pulse function

3.2. Step function

4. Phase-plane method

4.1. Introduction

4.2. The phase-plane (phase space)

4.3. The phase-plane method applied to

linear systems

5. References Chapter 3. Approximate methods of linear trans- ducer analysis

1. General

2. Approximation of periodic functions

2.1. Harmonic analysis

3. Approximation of aperiodic functions

3.1. Staircase function

3.2. Straight-line segments

3.3. Trapezoidal method

3.4.

''^ approximation

3.5. Number series transformation

3.6. Pseudo-rectangular pulse approximation

4. Instrumental aids

4.1. Henderson's mechanical harmonic analyzer

V 4.2. Montgomery's optical harmonic anavi lyzer 45

''ii 4.3. Photoelectric Fourier transformer 48

1 4.4. Electronic analyzer with magnetic

1 transient storage 50

1 5. References 51 Chapter 4. Analysis of nonlinear transducers 53

^ 1. General 53

2. Physical aspects of nonlinearity 54

3. Methods of analysis (nonlinear systems) 56

4. The concept and application of the describing-

^ function method 57

^ 5. Bilinear approximation method for determin-

^ ing the transient response of a nonlinear

® system qq

^ 6. The phase-plane method 63

^ 6.1. Method of isoclines 63

^ 6.2. Phase-plane-delta method 64

6.3. Displacement-time plots from phase

^ paths 65

6.4. Other phase-plane methods 66

^ 7. References _ 66

9

^

^ Chapter 5. Simple aperiodic-function generators 69

1. General 69

1.1. Types of simple aperiodic-function genj

^ erators 69

1.2. Place in pressure transducer calibration- 69

j2 1-3. Range of operation 69

j2 2. Dropping ball 69

2.1. Description 69

^2 2.2. Theory 69

^2 2.3. System design 70

j2 2.4. Evaluation of test data 70

13 3. Quick-opening devices 70

16 3.1. Description of burst-diaphragm device. 70

17 3.2. Description of Eisele's device 71 19 3.3. Description of NBS device 71 19 3.4. Evaluation of test data 74

24 4. Explosive devices 74

25 4.1. Description of JPL bomb 74

25 4.2. Description of NOL bomb 74

25 4.3. Evaluation of test data 74

5. References 74

26 Chapter 6. Shock tube methods 75

28 1. General 75

1.1. The shock tube as a step-function generator 75

31 1.2. Information obtained from shock tube

^2 calibration 75 32

2. Description 75

2.1. Description of components 75

2.2. Qualitative description of shock tube

phenomena 75

3. Shock tube theory 76

2^ 3.1. General 76

40 3.2. Contact surface velocity 77

3.3. Shock wave pressure 77 43 3.4. The reflected shock wave 78 44 3.5. The rarefaction wave 78

3.6. Pressure limitations 78

44 3.7. Real shock tube behavior 79

III
Chapter 6. —Continued Page

4. Design for pressure-gage testing 80

4.1. General requirments 80

4.2. Tube dimensions 80

4.3. Burst-diaphragm selection 80

4.4. Gas supply and control system 80

4.5. Shock velocity measurement 81

4.6. Transducer output records 81

5. Evaluation of test data 82

5.1. Amplitude characteristic of response

function 82

5.2. Predominant frequency or frequencies- 82

5.3. Logarithmic decrement or damping

characteristic 83

5.4. Determination of the tranducer system

transfer function 83

6. References 83 Chapter 7. Periodic-function generators 85

1. General 85

1.1. Types of periodic-function generators.. 85

1.2. Place in pressure transducer calibra- tion 85

2. Acoustical-shock generator 86

2.1. Description 86

2.2. Theory 86

2.3. Evaluation of test data 86

3. Rotating-valve generator 87

3.1. Description 87

3.2. Evaluation of test data 87

Page

4. Sirens gg

4.1. Description gg

4.2. Evaluation of test data gg

5. Piston-in-cylinder steady-state generators 89

5.1. General gg

5.2. Theory 89

5.3. Equipment go

5.4. Evaluation of test data go

6. Electrical and mechanical exciters 90

6.1. General go

6.2. Piezo-electric exciters 91

6.3. Electrodynamic vibration machine 92

6.4. Low-frequency pneumatic sinusoid generator 92

6.5. Electro-magnetic methods for calibrat- ing some pressure transducers 92

7. References g3

Chapter 8. The electronic compensator gs

1. General g5

2. The principle of the compensator 95

3. Compensator circuit 96

4. Calibration and operation of the compensator. 97

4.1. Calibration 97

4.2. Setting the compensator constants 98

5. Frequency response 98

6. Limitations 102

7. References 102

IV
Foreword

Accurate dynamic measurements of pressure are a necessity in the design

and development of modern rocket engines. Thus, with the growth of missile

and space vehicle programs, such measurements have become increasingly

important. To insure their accuracy, precise calibration methods must be

employed. As pressure changes in rocket engines may exceed 200 psi at fre- quencies above lOkc/s. the measurement and calibration requirements of the

dynamic pressure transducer are extremely rigorous.

This publication is designed to assist the practicing engineer who is faced

with the problem of making dynamic measurements of such rapidly changing

pressures. It provides a single reference source in which he may find, for a

particular pressure transducer, the appropriate mathematical model, the mathematical and instrumental methods of analysis, the methods of calibration,

and the specific methods for evaluation of test data from each method of

calibration.

A. V. AsTiN, Director.
Preface

This Monograph is published as part of a continuing program on telemetering transducers

which has been conducted in the Mechanical Instruments Section of the National Bureau of Standards under the sponsorship of the Bureau of Naval Weapons; Aeronautical Systems

Division, U.S. Air Force; White Sands Missile Range, U.S. Army; and the National Aeronautics and Space Administration. Publication of the Monograph was conceived while

Edward C. Lloyd was Chief of the Section, and under his direction Dr. Frederick F. Liu of Dresser Dynamics prepared a report entitled "The Dynamic Calibration of Transducers."

Dr. Liu's report, which was completed in 1959, is a major source of the material included in

this Monograph.

The final manuscript was prepared by Dresser Electronics, Southwestern Industrial

Electronics Division, Houston, Texas under contract with the National Bureau of Standards.

The text was written by Drs. J. L. Schweppe, L. C. Eichberger, D. F. Muster, E. L. Michaels,

and G. F. Paskusz of the University of Houston. Dr. Knut Seeber and the late Dr. H. E.

Hollman of Dresser Electronics, SIE Division made major contributions. Raymond O. Smith,

Pauls. Lederer, and Dr. Hansjorg Oser of the National Bureau of Standards and Dr. Henry

L. Mason, formerly of the National Bureau of Standards and now of the Veterans Administration, assisted materially with critical review, comments, and suggestions.

In order to calibrate and use dynamic pressure transducers, the practicing engineer needs

to be familiar with (1) the characteristic differential equations and their solutions, (2) the

methods of analyzing pairs of input and output functions to determine the transfer function

and the frequency response curve, (3) the methods of generating precise input functions, and

(4) the specific methods for using precise experimental measurements to determine the dynamic

characteristics of a particular pressure transducer. The first two of these topics are covered

in chapters 1 through 4, and the last two are covered in chapters 5 through 7. Chapters 1 through 4 include an introduction to pressure transducer calibration, the

characteristic differential equation, analytical methods of analysis for both linear and nonlinear

transducers, and approximate methods of analysis for both linear and nonlinear transducers.

Chapters 5 through 7 cover the description of many types of input function generators, the

theory of calibration with each generator, the design of the calibration system, and the specific

methods for evaluating the calibration data. Chapter 8 is devoted to a discussion of the

applications and limitations of the electronic compensator, a device designed to compute

the input function directly from the transducer output.

Arnold Wexler, Chiej,

Mechanical Instruments Section.

VI
Symbol Concept

List of Symbols

Symbol Concept

OnA

bbnB

CnC

d

D

exp E

En

f

5

9G

hH

ZH

Im

J

jk

K

K„

IL

arbitrary constant speed of sound

Fourier coefficient of cos wco< arbitrary constant

area, cross-sectional function

sin X maximum amplitude of ^

coefficients arbitrary constant

Fourier coefficient of sin tmI

arbitrary constant

function

arbitrary constant damping constant

Fourier coefficient, eq (1.10) arbitrary constant

fuhction diameter

arbitrary constant

function base of Napierian or natural logarithm

(e = 2.718 . . .) function

voltage or electromotive force

e'' =exp a

error nonlinear differential equation

function

function frequency

input function amplitude of the input spectrum

force Fourier transform of /

Laplace transform of /

maximum value of /

Fourier transform operator

inverse Fourier transform operator

modified input function Fourier transform of g function

function amplitude of the transfer function (describing function) function phase angle of the describing function

function imaginary part of a complex number

function square root of minus one amplitude factor

scale factor spring constant numerator of partial fraction arbitrary constant bulk modulus

length

dimension of length

inductance

function

X

Xm

M

nN

PiP

9

QiQ

TR

Re

Rp

sS

u

MlU

VV

X

XX

Laplace transform operator

inverse Laplace transform operator mass

arbitrary constant Mach number

function nth quantity or number in general

dimensionless number

modulus of inverse equivalent transfer function function

pressure

velocity

generalized coordinate

pressure function point located at coordinates {qk,qk) transducer output

generalized coordinate

maximum value of q root of auxiliary equation

function gas constant

electrical resistance

real part of a complex number

ramp function complex variable <r+jaj excitation parameter

function phase space time

dimension of time

operator

period

temperature

total transmitted light

triangular pulse function

velocity gas velocity into which the shock moves

gas velocity out of which the shock moves

shock velocity unit step function

unit pulse function

velocity component along x-axis (x) function volume

maximum value of x coordinate along x-axis function instantaneous displacement

approximate solution or value amplitude of the response spectrum

function Fourier transform of x maximum displacement

response function Re (Z(s)), eq (3.73) coordinate along j/-axis function Im{X{s)), eq (3.74) vn
Symbol Concept Superscript Concept

Z

za

yr

sA

e

Poa

Subscript Concept

CO

arbitrary variable (n) coordinate along z-axis response function + (plus) arbitrary constant approximate solution or value

angle

attenuation constant —(bar) output phase angle

logarithmic decrement • (dot) angle

phase angle of the transfer function

nonlinearity coefficient * attenuation constant

ratio of specific heats

function logarithmic decrement

function

finite difference arbitrary constant phase angle

damping ratio c angle

wavelength ^

absolute viscosity

phase angle * function A; mass density m

radius w maximum value of p angle o

real part of s = (r+jw V

normalized time (=co<) ^

time constant time increment used in trapezoidal method s

(fig. 3.4c) <S input phase angle t force (c/|a;|) function T

angular velocity (circular frequency) 0 undamped natural frequency

nth derivative of a quantity (the nth deriv- ative of X is denoted Z<"*)

limit of the function when the independent

variable approaches the point to be evaluated from the right (/(O approaches the limit /(0+) as I approaches zero from the right) mean value (written over, not after, the

principal symbol to which it applies) derivative with respect to time (number of dots over primary symbol indicates number

of times derivative is taken)

approximation

average

critical upper limit damped

effective ith quantity or number in general

fcth quantity or number in general

maximum

fundamental

nth quantity or number in general

output

pulse

rectangular

ramp

steady-state or particular solution step

transient or complementary solution

transition point

triangular

(zero) : initial condition peak value

NOTES: 1. This list of symbols is based on American

Standards Letter Symbols for Rocket

Propulsion, ASA YlO.14-1959. Some symbols have been added from other sources.

2. Vectors are indicated by an arrow over the

principal symbol. The amplitude of a vector is represented by the principal

symbol without the arrow.

3. In general functions of time are indicated by lower case letters, functions of frequency

by capitals. The same letter is used so, when a given function such as /(<) is trans- formed to the frequency domain it be- comes F(co).

VIII
1. INTRODUCTION

L. C. Eichberger ' and J. L. Schweppe ^

1. General

1.1. Statement of Objectives

The purpose of this pubHcation is to provide

assistance to the practicing engineer who is faced

with the problem of making dynamic measure- ments of rapidly changing pressures. This chapter

and the chapters which follow cover the mathematical models for transducers, the methods of analyzing pairs of input and output functions to determine the transfer function and the frequency

response curve, the methods of generating precise

input functions, and the specific methods for using precise experimental measurements to determine the dynamic characteristics of a pressure

transducer.

The specific objectives of Chaper 1 are to introduce (1) the types and characteristics of pressure transducers, and (2) the concepts of calibration and analysis.

1.2. Background

The missile and space vehicle programs have

brought about a need for precise measurements of rapidly changing pressures. Along with this need for high precision and response, there is an

associated need for improved calibration techniques. As a result, both experimental and

analytical methods have evolved rapidly.

For some time the frequency-response concept

has been established in the electronic and servo

fields. It is therefore a natiiral development for

this concept to be applied to pressure transducers,

which are components in electronic systems.

This development has been accelerated by the

increasing application of transform methods such

as Fourier and Laplace. The frequency-response

concept and the use of mathematical methods are introduced in this chapter.

2. Pressure Transducers

A pressure transducer is an electromechanical

device through which an input pressure signal is converted to an output electrical signal. In most pressure transducers the pressure force causes

the displacement of a spring. This displacement

produces a change in some electrical property

which in turn is measured by an appropriate

electrical system. The electrical property of the

transducer is most commonly the resistance of a potentiometer, the resistance of a bonded or unbonded strain gage, the capacitance between

two plates, the inductance of a coil, or the piezoelectric property of a crystal. Depending on the

physical combination of parts, any one of these

gages may either approach a simple oscillator

having one natural frequency, or it may have a combination of frequencies. And it may have a

linear or nonlinear response to the input signal.

Some good general references on pressure trans- ducers are Hernandez [1] ^, Borden and MayoWells [2], Lion [3], and Roberts [4].

2.1. Types of Pressure Transducers

The potentiometric transducer utilizes a potentiometer circuit in which the slider location is determined by the magnitude of the pressure force,

' Assistant Professor of Mechanical Engineering, The University of Houston; Technical Staff, Houston Engineering Research Corporation.

2 Professor of Mechanical Engineering, The Universitj' of Houston; Presi- dent, Houston Engineering Research Corporation.

3 Figures in brackets indicate the literature references on page 9.

figure l.l. A multiplying linkage is used between

the slider and the force-summing member to minimize the required movement of the latter part.

The many moving parts make friction, vibration,

and inertia substantial problems; and, although

these transducers are carefuUy designed to reduce

the effects of friction and vibration, they do not

respond well to high rates of pressure change.

However, because a high-output a-c or d-c signal

may be obtained without use of an amplifier, the

potentiometric transducer is widely used for measuring static and low-rate-of-change pressures. The strain gage transducer system measures the

pressure through its effect on a bonded or un- bonded strained element in the transducer. In

the example shown in figure L2, the strain gages

are bonded to a cylindrical strain tube which is compressed when pressure is applied. A flush catenary diaphragm separates the transducer components from the pressure region and transmits

the pressure to the strain tube. Two strain gages

are bonded, one longitudinally and one circumferentially, to the outside of the tube to form the

two active arms of a Wheatstone bridge. Two

inactive precision resistors and a precision potentiometer are added externally to complete the

bridge. The unbonded strain gage transducer

utilizes the gage directly as the strained element.

That is, no element comparable to the strain tube

of figure 1.2 is used. For further details see Hernandez [1].

1
INCREASING PRESSURE , (D ®

PRESSURE DIAPHRAGM

PIVOT POINT

MOVABLE CONTACT

RESISTANCE ELEMENT

Figure 1.1. Potentiometric pressure transducer.

(a) Photograph and typical dimensions (Reproduced by courtesy of Fairchild Controls Corporation)

(b) Schematic diagram

Figure 1.3 shows one example of the unbonded

strain gage transducer. Bonded and unbonded

gages are available from a number of manufacturers and both may be used to measure either

static or dynamic pressures.

Still another strain gage system measures pres- sure through its effect on the resistivity of a semiconductor crystal. In this system the crystal

functions both as the strained element and the

measuring element. The crystal is mounted

between a fixed plane and a diaphragm which

separates the components from the pressure region.

The piezoresistive transducer has a gage factor of the order of magnitude of 150 compared to 2 to 4

for a conventional strain gage [5]. The capacitive transducer system measures the

pressure through its effect on the electrical capaci- tance of a movable-plate condenser. In one system the movable-plate condenser is a part of a very-high-frequency circuit. As the capacitance

changes, so does the impedance. Since for a fixed

position of the movable plate the impedance is constant, the capacitive transducer system can be

used for either static or dynamic pressure measurements. It is important to note that the cooxial

cable which connects the transducer with the

external circuit is a part of the circuit whose impedance change is an analog of pressure. Therefore

Figure 1.2. Bonded strain gage pressure transducer.

(Reproduced by courtesy of Norwood Controls Unit of Detroit Controls Corporation)

(a) Photograph with enlarged view of disassembled pickup

(b) Cross section of catenary diaphragm pressure transducer

electrical characteristics of the coaxial cable are

critical. A photograph, a dimensional drawing,

and a cross-sectional view .of a typical capacitive

transducer are shown in figure 1.4. The inductive transducer measures the pressure

by its effect on the inductance of a coil or on the

inductance ratio of a pair of coils. In a typical

instrument the pressure force moves a diaphragm

and thereby changes the magnetic coupling

between the coils, figure 1.5. The inductive

transducer has the advantages that it can be used

for static or dynamic measurements and that it has a high output and high signal-to-noise ratio. But it is influenced by stray magnetic fields and

it has a low useful frequency range in the order

of 500 c/s. The piezoelectric transducer system measures

the pressure through its effect on a piezoelectric

material. When the crystal is distorted by the

pressure force, an electric charge is generated.

The amount of this charge, which is a function of

the pressure force, is measured with an electrometer. Since the electric charge leaks off, the

piezoelectric transducer is inherently a transient

device and cannot be used for static measurements.

A photograph and a cross-sectional view are shown

in figure 1.6.
PRESSURE INPUT FEEDBACK AMPUFB1

J]

Ir 1

frDC OUTPUT DEMODULATOR

Figure 1.3. Unbonded strain gage pressure transducer.

(Reproduced by courtesy of Transducer Division, Consolidated Electro- dynamics.)

(a) Photograph

(b) Typical dimensions

(c) Wiring diagram

Figure 1.5. Inductive pressure transducer.

(Reproduced by courtesy of Astromies Division of Mitchell Camera

Corporation.)

(a) Internal view and typical dimensions

(b) Circuit diagram

INNEFi DIAPHRAGM

WATER PASSAGE

-COIL ASSEMBLY

- INSULATOR ASSEMBLY

COPPER GASKET

STACKED CRYSTAL

CRYSTAL UNIT CLAMP NUT

THIN DIAPHRAGM

Figure 1.4. Typical capacitive transducer.

(a) Photograph and typical dimensions (Reproduced by courtesy of Photocon Research Products.)

(b) Cross-sectional view

Figure 1.6. Piezoelectric pressure transducer.

(a) Photograph and tvpical dimensions (Reproduced by courtesy of Kistler Instruments Corporation.)

(b) Cross section

3
2.2. Physical Characteristics of Transducers

The above descriptions of several types of transducers show the similarities and the differ- ences in physical characteristics. First, each

transducer must have certain major parts. These

include an active or sensing element, a body to hold the active element in place in the wall of the

pressure region, a diaphragm or other device to separate the components from the pressure region

and to transmit the pressure force to the sensing

element, and an electrical means of removing the

output signal from the transducer. The signal

output may be through a simple transmission line or a rather complicated electronic device such as a carrier bridge.

A simple or "ideal" transducer would have a

linear response to the input pressure signal, and

it would have a single degree of freedom. That is, the oscillatory movement would be limited to a

direction parallel to the line of action of the applied

force. Also, the "ideal" transducer would deflect

little when the input pressure is applied, but would

generate a substantial output signal.

A real transducer may have a response which

approaches the ideal linear response to the input

signal, but it will have more than one mode of

oscillation. The number of additional modes and

their importance depend on the particular trans- ducer. In the analysis which follows, it will be

assumed that the pressure input is applied directly

to the diaphragm. That is, cavity effects will not

be included. Also, it will be assumed that the

///////////////

c

f(t)

Figure 1.7. Single-degree-of-freedom vibrating system.

(a) Mechanical model

(b) Free-body diagram

(c) Electrical force-voltage analog

body of the transducer is rigid—or that it has a natural frequency so far above the operating

frequency that it need not be considered.

For each transducer, then, two parts are considered in the synthesis of the mechanical model.

They are the sensing element and the diaphragm

and, in many cases, the electrical equipment

required to complete the energy transformation to the voltage or current analog. Each of these

parts will have one or more modes of oscillation,

with associated spring constants and damping

factors. Mechanical models for several typical

transducers are synthesized in the next section.

It should be noted that such mechanical models

do not always completely describe the transducer,

but that often one or more additional electrical

modes of oscillation need to be indicated. The

mathematical treatment need not distinguish

between mechanical and electrical resonances.

2.3. Mechanical Models

A mechanical model idealizes a vibrating system.

The model consists of at least one inertial mass,

a spring, a viscous resistance, and an external

driving or exciting force. For the ideal transducer

the spring force and the viscous resistance are

linear, i.e., the spring force is directly proportional

to the change in length and the viscous resistance

is directly proportional to the velocity. Nonlinearity results if the spring force or the viscous

resistance, or both, are nonlinear. The arrangement and number of components establish the

number of degrees of freedom for the model, i.e., the minimum number of coordinates necessary to specify the configuration of the vibrating system

at any time [6, 7]. A single-degree-of-freedom vibrating system can be idealized by the mechanical model shown in figure 1.7a. The characteristics of the capacitive

and piezoelectric transducers are approximated by

this model. For these transducers the response,

or output, is normally dependent upon one of two

masses: the mass of the sensing element, or the mass of the diaphragm. In the capacitive trans- ducer the moving mass in the sensing element

(air) is negligible compared to the mass of the

diaphragm (active, or movable plate of the capacitor). In the piezoelectric transducer the mass of

the diaphragm normally is negligible compared

to the mass of the sensing element.

A two-degree-of-freedom vibrating system can

be idealized by the mechanical model shown in

figure 1.8a. Lederer and Smith of the National

Bureau of Standards have shown that this model

is a fair approximation for the strain gage trans- ducer shown in figure 1.2. They considered the mass of the diaphragm to be negligible and the mass of the strain-generating tube to respond in two modes, a longitudinal mode and a radial mode.

In general, the synthesis of a mechanical model

must be preceded by a detailed study of the

arrangement, size, and orientation of the physical

components of a transducer. The validity of the

4
synthesis is verified by comparing the response of

the model to that of the transducer for a given

input, or external driving function.

2.4. Characteristic Differential Equations

Once the mechanical model has been established,

the characteristic differential equation which describes the motion of the system is obtained by

application of d'Alembert's principle. For the

single-degree-of-freedom vibrating system, figure

1.7b, summing the forces gives

///////////////

111

mx-\-cx-\-kx =f(t)

.

(1.1)

The damping constant, c, is the number of units

of resistive force per unit velocity of motion; the

spring constant, k, is the nimiber of units of force

required to stretch the spring a unit length; and

J(t) is the external driving force, a function of time.

Application of d'Alembert's principle to the

two-degree-of-freedom vibrating system, figure

1.8b, yields

<

t(t)

111

miXi

CiXi +kiXi— C2 (iz —Xi) —^2 (iCz —Xi) =0

"^2*2 +C2(X2 —Xi)+k2(X2 —x{) =f(t)

.

(1.2)

The number subscripts distinguish between like elements.

In general the nimaber of such characteristic

second-order differential equations of motion will agree with the number of degrees of freedom of the

vibrating system.

Figure 1.8. Two-degree-of-freedom vibrating system.

(a) Mechanical model

(b) Free-body diagram

(c) Electrical force-voltage analog

3. Transducer Calibration and Analysis

Calibration is the establishment of a known

relation or transfer function between the input or driving function and the output or response function. This transfer function exists only if the

transducer is describable by a linear differential

equation. For a simple static calibration the

transfer function is the ratio of output to input.

For a dynamic calibration the transfer function is normally a complex function of frequency in which

are included certain time constants. This function may be found in one of two ways: (1) If the

transducer system can be described by a characteristic differential equation, the transfer function

may be obtained by analytical solution of the

equation. With this solution the response for a given input, or the input which will produce a given response, can be computed. Or (2) if the

characteristic differential equation is not known,

the transfer function can be obtained from knowledge of a pair of associated input and output functions. From this transfer function, frequencyresponse curves can be computed, and these curves can be used to determine the response for a given

input, or the input which will produce a given

response.

In the following, the reader is introduced to the

analytical solution of the characteristic differential

equations, experimental calibration methods, and

the relationship of calibration to analysis.

3.1. Methods of Analysis

Methods used for solving differential equations

of the motion of a vibrating system are dependent on the type of these differential equations. We

shall deal here exclusively with homogeneous and

nonhomogeneous linear, second-order, differential

equations with constant coefficients. Several methods are available to find solutions

of these equations. The oldest method, which we

shall refer to as the classical method, consists in

finding one particular solution of the nonhomogeneous equation and adding to it the general solution

of the homogeneous equation. The former solution is also called the steady-state solution, whereas

the latter is the transient solution, which is found,

for example, by substituting exp{rt) into the homogeneous equation. Calling the solutions of the

resulting equation rj and Vo, we obtain the general

solution of the homogeneous equation as

5
Ci exp(rif) + C2 exp(r2t), with arbitrary constants

Ci and C2. They are determined from the initial conditions.

The operational methods are Laplace and

Fourier transformations, which are dealt with in

section 4 of this chapter. Both transient and

steady-state solutions are obtainable by either of these methods.

Other methods of analysis make use of digital or analog computers. On the digital computer the

solution is determined on a discrete set of values of

time. Because of the cost of programming highspeed electronic computers, these machines are economical only if many differential equations of

the same type have to be solved, or if high accuracy

is desired which is not obtainable otherwise.

However, since automatic programming is now

available for almost any computer, programming

costs have been reduced considerably.

The analog computer simulates the original dif- ferential equation, the dependent variable usually

being a voltage which is made to satisfy the given

differential equation and the initial conditions.

It is an extremely versatile device which can also be made to simulate many types of mathematical

nonlinearities or graphic relationships not expressible in analytic form, or even to permit the ina

b

Figure 1.9. Associated input and output functions from

shock tube calibration of a pressure transducer.

(a) Input function (b) Output function

elusion of actual transducer components in the

simulated system. The output of the analog

computer may be a continuous curve, e.g., an

inked trace or a display on a cathode ray tube, or

it may be digital through the use of analog-todigital converters. Because the parameters and

variables of a problem are uniquely evidenced by

knob settings or dial readings during the solution,

the rapid scanning of parameter influence is possible. This puts the analog computer clearly

ahead of any other method if some quantitative

results with moderate accuracy are desired.

Purely mathematical methods for other types of

differential equations exist, but their discussion is outside the scope of this treatise.

3.2. Experimental Calibration Methods

In general, the imposed input for dynamic cali- bration may be either a nonperiodic or a periodic

function. In either case the dynamic properties

may be expressed as a plot of the instrument's

response to the input. The nonperiodic function

may be an impact of short duration which is quickly released, or it may be a step function

which changes the pressure level from one specific

value to another in a very short tune. The

dropping ball is an example of the impact-type,

nonperiodic function generator. Step-function

generators include quick-opening devices, explosive

devices, and shock tubes. Figure 1.9 shows the

response of a typical gage to a pressure step

generated in a shock tube in the laboratory at the

National Bureau of Standards.

The ideal periodic-function generator produces

pure sine waves of controlled frequency and amplitude, and the frequency response cm-ves are determined directly as they are for electronic

circuits. Satisfactory sinusoidal calibration os microphones, a form of pressm-e transducer, if carried out in air over a wide frequency range

but at very low amplitudes. Since accurately

known sinusoidal input functions in the form of pressure cannot be generated in a gas at appreciable ampUtudes [8, 9], other waveforms describable

as the sum of a number of sinusoids are used when

tests must be conducted at pressxu-es up to 200

psia and at frequencies up to 30 kc/s. Periodic

waveforms used for transducer calibration include

repetitive impulses, square waves, sawtooth waves,

and the like. Periodic-function generators include

acoustical shock generators, rotating valves, shens,

piston-in-cylinder devices, and electrical and

mechanical oscillators. Figure 1.10 shows the

response of a typical pressiu-e transducer to a sine wave generated by a hydrauUc oscillator [10].

3.3. Relationship of Calibration to Analysis

Even though it may be possible to describe the

characteristics of a real transducer by a simple

mechanical model and the associated differential

equation, the spring constants and damping coefficients still must be determined experimentally.

6
Figure 1.10. Associated input and output functions from calibration of a pressure transducer with a hydraulic oscillator.

(Reproduced from [10] with permission from J. Sci. Instr., published by The Institutes of Physics and The Physical Society.)

(a) Input function (b) Response function And, when it is impossible or impractical to describe the characteristics of the transducer

analytically, the transfer function must be obtained from a knowledge of a pair of associated

input and output functions which must be imposed

and measured experimentally.

These experimental input and output functions

are recorded as a function of time, or are expressed

in the time domain. In order to obtain the transfer

function, which is expressed in the frequency

domain, the input and output functions must be

transformed into the frequency domain. Therefore, the method chosen for analyzing the experimental input and output functions must include

this transformation. Fourier and Laplace analysis

fulfill this requirement and, at the same time, are more efficient than methods which operate in the

tune domain. Accordingly, Fourier and Laplace

methods are used exclusively in this work.

4. Fourier Methods and Spectral Analysis

4.1. Fourier Series and Line Spectra

Any periodic function i{z) with period 27r can be expanded into a Fourier series of the form

/(a;)=^+X) {<^n COS nz +hn sin nz), (1.3) ^ n=l

where the Fourier coefficients a„ and are given

by

1 o„=- f{z) COS nzdz (1.4)

IT J -r

1 f"" b„=- f{z) sin nz dz. (1.5)

The validity of the Fourier series oij{z), eq (1.3), as used in engineering applications, is covered by

the Dirichlet conditions. Dirichlet proved that,

if j{z) is finite in a given interval and has a finite number of maxima and minima in one period, the

Fourier series of /(s) is convergent, and its sum is equal to/(2), if /(e) is continuous at the point z. If j{z) is discontinuous at the point z, then the

Fourier series will converge to the average value

of f{z) at tliis point. Immediately near and on

both sides of the discontinuity, the Fourier series overshoots the function /(s). The amount of the

overshoot is about 9 percent of the jump, so the

Fourier series approximation is not very satisfactory for points very near to the discontinuity [11]. In most engineering appfications, a function of time is given, and eqs (1.3), (1.4), and (1.5) must

be modified. If T is the period in time of the

given function /(O, then z= (2Tr/ T) t==ut, since

z is the argument of the sine and cosine function

and must be an angle. The circular frequency, w, wiU always have the value of 2ir/T. Therefore

y(i)=^+Xi (an cos nu:t +bn sin nwt), (1.6)

where, for 71= (0, 1, 2, . . .)

T/2 j{t) cos n(j}t dt (1.7)

-TI2

6„=4 J{t) smnwtdt. (1.8)

Equation (1.6) can also be expressed in an

alternate form:

/(^)=?+Z: c„ cos (nco^-^J (1.9)

n = l where , c„=v^7+^ (1-10)

and

^=tan-' {bja„). (1.11)
0 To T

1.905 ATp

1.652 ATp

Figure 1.11. Characteristic spectral plots.

(a) Harmonic amplitude line spectrum

(b) Eelatlve frequency distribution for a single rectangular pulse

Still another alternate form of the Fourier series

is the complex exponential form which is used to introduce the Fourier integral and Fourier trans- form. This form is obtained by substituting the

exponential equivalents of the sine and cosine

terms into eq (1.3). Expanding, collecting like terms, and writing the results in a symmetrical

form yields

where

re = — 00

1 rT/2

-t J -T/2

(1.12)

(1.13)

Spectral analysis has evolved from Fourier

analysis solutions [12]. A plot of c„ or from

the Fourier series representation of f{t), eq (1.9), as ordinate and nco as abscissa, consists of discrete

vertical lines and so is called a line spectrum.

Specifically, the c„ and ^„ versus nco plots are, respectively, the harmonic-amphtude spectrum

and the phase-angle spectrum. In general, plots

of this type are called by that quantity which is plotted against ?7co, and are a part of the family

referred to as Fourier series spectra (see fig. 1.11a).

For a more detailed discussion on Fourier series see Pipes [13].

4.2. Fourier Transform and Continuous

Spectrum

The Fourier series is adequate to accomplish

the expansion of any periodic function satisfying

the Dirichlet conditions [11]. However, in many

problems encountered, the given function of time

is aperiodic and such a function cannot be handled

directly by the Fourier series. However, in the

limit, as the fundamental period T becomes

infinite, the series passes into an integral form

[14]. The resulting integral is called the Fourier

integral and is expressed as

fit)=j e'^^'"d<rj f(t)e-^^^'"dt (1.14)

where o- is a real variable. This representation

of J(t) is valid, provided that in every finite interval /(i) satisfies the Dirichlet conditions and

J

00 \J(t) \dt exists.

- CO Equation (1.14) may be written in a slightly

different form by the introduction of u=2Ta as a new variable. With this,

f{t)=^j\^-'do^j^J{t)e-^"'dt. (1.15)

Then, eq (1.15) becomes

y(^)=^ J

V(c.)€^-"'c?co (1.16)

where

(1.17)

Equations (1.16) and (1.17) constitute what is known as a Fourier transform pair; F{w) is called

the Fourier transform of j{t) and, conversely, j{t)

is called the inverse Fourier transform of F{o})

:

(1.18)

(1.19)

The Fourier transformation of eq (1.18) trans- forms a function of t (in the time domain) into a function of co (in the frequency domain). The

inverse Fourier transformation of eq (1.19) trans- forms a function of w (in the frequency domain)

into a function of t (in the time domain)

.

4.3. Laplace Transform and Continuous

Spectrum

If a function is identically zero before some

time, say t=0, then it may be represented by the

Fourier transform pair, eqs (1.16) and (1.17), if the

lower limit of F(co) is changed from — oo to 0 [16].

8
/(i)==l- J *°F(co)e^'-'(/co (1.16)

F(c^)=j^ fit)e-^-'dt. (1.20)

Equations (1.16) and (1.20) are knoAvn as the

unilateral Foiirier transform.

In contrast to the line spectra concept from the

Fourier series anah'sis of periodic functions,

Fourier integral analysis of aperiodic functions

yields a continuous amplitude spectrum. This

transform, when applied to a single rectangular

pulse, yields the spectrum of figure 1.11b.

Equations (1.16) and (1.20) are meaningless if the integral F{u)) does not exist. A sufficient condition for the existence of F(co) is that f{t) be

absolutely integrable, i.e., that

j: f(t) \dt exists. There are cases where f(t) is not absolutely integrable, but still represents a physically well-behaved

stimulus. Examples are the step function and

sin ut. For this reason there is a need for an

extended definition of the transforms which jaelds

F(a)) whenever that integral exists, and also gives

a meaningful answer for some other cases of physical interest.

If for t tending towards infinity, f(t) remains

finite or tends to infinity at a rate less rapid than

then j^^lM^'"'. dt exists for aU cr>6.

Under these conditions the transforms (1.16) and

(1.20) can be considered for the function g{t) =

fit)e~''\ where g{t) is the modified function and

f{t) is the function of actual interest. Applying

the unilateral Fourier transform

27r J_

6{w)e^"'dc^

g(t)e"=M=~ 0{o>)e''+'''''dw

J -co Jo

In these equations let s=(7-\-jo: and ds=j du.

Since s is a function of u, 6(co) can be replaced bv

^(s)=J^ J(t)e-"dt. (1.22)

Equations (1.21) and (1.22) are the Laplace

transform pair. The function F(.s) is knowm as the Laplace transform of/(<), and the integral for

f{t) is known as the complex inversion integral:

Fis) = X[f{t)]

f{t) = £-\F{s)].

(1.23)

(1-24)

The Laplace transform, therefore, is a special case

of the unilateral Fourier transform, expressed

symbolically as

xuit)]=m(t)e-''].

That is, the Laplace transform is identical with

the unilateral Fourier transform of the same

function multiplied by a convergence factor. Applications of both of these transforms are discussed

in chapters 2 and 3.

5. References

[1] J. S. Hernandez, Introduction to Transducers for Instrumentation, Stratham Instruments, Inc. (Los

Angeles, Calif.).

[2] P. A. Borden and W. J. Mayo-WeUs, Telemetering

Systems, Reinhold Publ. Corp. (New York, 1959).

[3] K. S. Lion, Instrumentation in Scientific Research

Electrical Input Transducers, McGraw-Hill Book

Co., Inc. (New York, N.f., 1959).

[4] H. C. Roberts, Mechanical Measurements bj^ Electrical Methods, Instruments Publ. Co., Inc. (Pitts- burgh, Pa., 1951).

[5] G. Landwehr and K. F. Zobel, A new device for measlu-ing quickly changing high pressure (in German),

Z. Instrumentenk. 65, No. 11, 220 (1957).

[6] N. O. Myklestad, Fundamentals of Vibration Analysis, McGraw-HiU Book Co., Inc. (New York, N.Y.,

1956).

[7] D. K. Cheng, Analysis of Linear Systems, AddisonWesley Publ. Co.,' Inc. (Reading, Mass., 1959).

[8] Ralph A. Bowersox, Calibration of high-frequencyresponse pressure transducers, ISA J. 5, 98 (1958).

[9] C. R. TaUman, Transducer frequency response evaluation for rocket instability research, ARS J. 29,

119-122 (1959).

[10] A. W. Melville, Hydraulic oscillator for the dynamic

calibration of pressure recording systems, J. Sci. Instr. 36, 422 (1959).

[11] T. von Karman and M. A. Biot, Mathematical Methods in Engineering, p. 335, McGraw-Hill Book Co.,

Inc. (New York, N.Y., 1940).

[12] E. Weber, Linear Transient Analj^sis, Vol. 1, John

Wiley & Sons, Inc. (New York, 1954).

[13] L. A. Pipes, Applied Mathematics for Engineers and

Physicists, I\IcGraw-Hill Book Co., Inc. (New York,

N.Y., 1958).

[14] C. R. Wj'lie, Jr., Advanced Engineering ^Mathematics,

McGraw-HQl Book Co., Inc. (New York, N.Y.,

1951).

[15] R. V. Churchill, Operational Mathematics, McGrawHiU Book Co., Inc. (New York, N.Y., 195S).

[16] M. F. Gardner and J. L. Barnes, Transients in Linear

Systems, John Wiley & Sons, Inc. (New York,

N.Y., 1942).

694-211 0-63—

2

9

2. Analytic Methods for Linear Transducers

L. C. Eichberger '

In this chapter the reader is intioduced to the procedural steps of analytical analysis.

These steps are applied to a hnear tiansducer which, for simplicity, is assumed to be a singledegree-of-freedom system represented by the mechanical model shown in figure 1.7a. The

characteristic differential equation of motion foi this system is given by eq (1.1). Response

functions for the system are obtained for a given input function by both the classical and

the operational methods of analysis. The periodic (sine, square wave, and rectangular pulse)

and aperiodic (rectangular pulse and step) functions are the input functions considered. These functions represent the idealized inputs used in experimental dynamic calibration,

as discussed in the later chapters of this work.

1. Input, Output, and Transfer Function Relations

1.1. Direct Input-Output Relation in the

Time Domain

It is assumed that the characteristic differential

equation of motion for the transducer is Imown,

and that the input function / (periodic or aperiodic) can be expressed analytically as a function

of time, J{t). Then the general solution, or re- sponse function, x{t), of the differential equation

of motion can be obtained through the application

of the classical method of analysis. The response

function consists of a complementary function or

transient solution, and a particular integral or steady-state solution. This operation is illustrated

systematically in figure 2.1 by the path ABC in the time domain. The differential equation in

this operation acts as a transfer function. In the actual calculation of a response function

as indicated above, it will be found that the classi- cal method of analysis is more adaptable to the

periodic class of input functions than the aperiodic

class. Most periodic functions encountered in transducer analysis will satisfy the Dirichlet conditions [1] ^ and, therefore, can be approximated

by a Fourier series. Standard solutions are readily

available for the sine and cosine terms contained

in the series, and by the principle of superposition

the particular integral is readily obtained.

Aperiodic functions are readily treated by the

Laplace transformation. This method transforms

a given function in the time domain to one in the

domain of the complex variable s=a-\-jco, where

0) is a real frequency. When an aperiodic input

function and the associated differential equation

are known analytically, then the entire equation

is transformed to the frequency domain as an algebraic equation. The Laplace transform of the

solution is obtained from this equation and the

> Assistant Professor of Mechanical Engineering, The University of Houston; Technical StaflE, Houston Engineering Research Corporation.

' Figures in brackets indicate the literature references on p. 28.

inverse transform of this expression yields the

required response function.

On the other hand, if the response function has

been expressed analytically and the characteristic

differential equation of motion is known, the input

function can in theory be determined by substituting the response function into the differential

equation. In fact, only the particular integral

part of the response function need be used, since

r"

_j

-I

Time Domoin Frequency Domain 1

INPUT FUNCTION

INPUT SPECTRUM

AyPLITUOE PHASE ANGLE

PHASE ANGLE

THANSFEfi FUNCTION

AMPLITUDE XM

PHASE ANGLE

(Dl RESPONSE SPECTRuy

I I J Figure 2. 1 . Routes of analysis.

11
the complementary function is the solution of the

homogeneous part of the differential equation,

i.e., when/(0 is zero. This op(-ration is illustrated

in figure 2.1 by the path CBA.

1.2. Transformation From Time to Frequency

Domain

With reference to figure 2.1, both the input

function, f{t), and the response function, x{t), can be transformed from the time to frequency

domain, respectively indicated by the paths from

A io F and C to Z>. The transformed input

function and response function are denoted as the

input and response spectrum, respectively. Both

spectra consist of two distinct parts, an amplitude

and a phase angle. The input and response

functions are related by a transfer function which

also consists of an amplitude and a phase angle.

If the input function is periodic and satisfies the

Dirichlet conditions, then it can be represented

by a Fourier series. When the Fourier series is expressed in the form shown in eq (1.9), the

amplitude F{u)) and the phase angle ip{u>) are given by the combined Fourier coefficient c„ and

the phase angle (Pn, respectively.

The response of a linear system to a periodic

input function is also periodic, and therefore can

be represented by a Fourier series similar in form

to that shown in eq (1.9), or

CO x(0=^o+S COS {nwt—an).

n = l The amplitude X{o}) of the response spectrum is given by X„ and its phase angle a(w) by a„.

If the input function /(i) is periodic or aperiodic

and expressed analytically, then the input

spectrum can be obtained directly from the

Laplace transform providing that /(^) =0 for ^<0

and the integral

J*

\f{t)e''''\dt exists. Since the

Laplace transform of the input function is a complex number in the frequency domain, it provides both the amplitude and phase angle of the input spectrum. Likewise, the Laplace trans- form of the response function provides both the

amplitude and the phase angle of the response

spectrum.

It should be noted that the phase angles are an essential part of the frequency domain description for both periodic and aperiodic functions. In

the time domain, any delay of an aperiodic

function is described by the shift factor, e.g., mt-T,), p. 22.

1.3. Transfer Function

The transfer function plays an important role in the analysis of a linear transducer. It allows

the characteristics of the transducer to be described

in the absence of the characteristic differential

equation of motion. The transfer function is determined from the analytical expressions of the

input and response spectrum. This requires that an accurate record of the response function for

a known input function be made available, so that the response function can be expressed

analytically by the methods given in chapter 3. Such a procedure for obtaining a response record

for a known input function is included in the

dynamic calibration of a transducer.

The transfer function is described by two separate characteristics, figure 2.1, which are defined

as follows: the amplitude of the transfer function

Hiw)=X{o:)/F{o:) (2.1)

where X{w) and F(u) represent the amplitudes of

the response and input spectra respectively, and

the phase-angle of the transfer function

/3(a;)=a(co)-^(co), (2.2)

where, respectively, a(co) and ^(w) are the phase

angles of the response and input spectra.

Among aperiodic functions, the pulse of infinitely short duration, but of unit area, has

particular significance. The Laplace transform

of the response to this input is the transfer function

itself.

1.4. Input From Transfer Function and

Response Record

If the transfer function for a transducer is known analytically, and a response record from

the transducer for an unknown input is also

known, then it is possible to determine an ex- pression for the input function by operational

methods. The stepwise procedure is described

below:

(a) Express the response record as a function

of time, x{t).

(b) For a periodic ftmction evaluate the amplitude and the phase-angle characteristics of the

response spectrum by the Fourier series. For

an aperiodic function, take the Laplace transform

of the response function, which is the frequency

response characteristic in amplitude and phaseangle. This is step C to D, figure 2.1.

(c) Evaluate the input spectriom by eqs (2.1)

and (2.2). The amplitude and phase-angle transfer functions are known for the transducer, and

the response spectrum was obtained in step (b).

This step is represented by path DEF in figure 2.1

.

(d) Finally, obtain the input function either

by the Fourier series representation, i.e., by

matching c„ and <Pn in eq (1.9), for periodic

fimctions; or by the inverse Laplace transform

for either periodic or aperiodic functions. This

operation is illustrated in figure 2.1 by step F to A.

1.5. Phase-Plane Analysis

In the course of studying dynamical expressions

which are invariant under canonical transformations, Poincare [2, 3] used the concept of a phase

space, a Cartesian space formed of coordinates

qt and pt (i=l, 2, ... n) in which the complete

dynamical specification of a mechanical system

is given by a point. Other investigators have

12
since used the phase plane (a two-dimensional

phase space) to study the behavior of linear and

nonlinear, damped and undamped dynamical

systems. The most notable among the early

papers on the graphical phase-plane method for the determination of transient response are, perhaps, those due to Lamoen [4, 5]. More

recently the graphical phase-plane method was

apparently rediscovered independently by Fuchs

[6], Braun [7], and Rojansky [8]. Bishop [9] has

published a comprehensive survey and Andronow

and Chaikin [10], Minorsky [11], and Kryloff and

Bogoliuboff [12] have included lengthy discussions

of phase-plane techniques in their books on

nonlinear mechanics.

In addition, Truxal [13] and Murphy [14] have

shown the application of the phase-plane method

to the analysis of servomechanisms, particularly

those characterized by nonlinear behavior. Jacobsen and A3a;e [15] use the method extensively in their recent book on vibrations. The current

literature includes numerous applications to specific problems in the fields of dynamics, the design

of the circuits in electronic instruments and

servomechanisms, and transducer systems. These

include contributions by Klotter [16, 17], Magnus

[18], Gibson [19], Cosgriff [20], Bass [21], Ergin

[22], Stout [23], Jacobsen [24], and Liu [2.5].

2. Periodic Input Functions

In this section, known periodic input functions

are imposed on a hypothetical transducer for the

pm-pose of illustrating the analytical methods of

analysis presented in the previous section. The

order of presentation is preserved and, for simplicity, a single-degree-of-freedom transducer is chosen

for analysis. The mechanical model for this

transducer is shown in figure 1.7a and its characteristic differential equation is given by eq (1.1).

The inputs imposed on the transducer are the

f (t)

sine, square wave, and rectangular pulse functions.

The analysis for the sine function is presented in

detaU, whereas only the results and pertinent

details are given for the square wave and rectangular pulse function.

2.1. Sine Function

a. Direct Input-Output Relation in the Time Domain

The sine input function is shown in figure 2.2a,

f (f)

f (f)

I

I T Tp + T

kf(t)

F

dFigure 2.2. Periodic and aperiodic input functions.

(a) Sine function

(b) Square wave function

(c) Rectangular pulse train

(d) Step function

13
and is expressed by

f{t) =F sin wt (2.3)

where w is the circular frequency in radians per

unit time.

The characteristic differential equation of motion for a single-degree-of-freedom transducer,

eq (1.1), becomes

mx-\-cx-\-kx=F sin ut. (2.4)

The transient solution (complementary function)

is obtained from the homogenous part of this

equation, i.e.,

mx-\-cx-\-kx—0. (2.5)

The standard solution to a differential equation

of this type is

Xt=Cie'i'-\-C2e'2* (2.6)

where Ci and C2 are arbitrary constants and Vi and r2 are the two roots of the auxiliary equation

Solution of eq (2.7) gives two roots

ri = —c/2m +Vc74 ^ —^/

w

(2.7)

(2.8)

(2.9)

Since m, c, and k are always positive, the type

of root obtained is dependent upon the evaluation

of the radical. If the radical in eqs (2.8) and

(2.9) is positive, the motion given by eq (2.6)

is of a gradually subsiding nature. If the radical

is negative, the roots are complex numbers and

the motion given by eq (2.6) is of a fluctuating nature. The limiting case is when the radical

is equal to zero, for which c^=4:km. This value

of c is called the critical damping Cc and is given by

Cc=2-\Jkm=2'mo}n (2.10)

where co„=V^7m is the natural circular frequency.

The dimensionless ratio c/Cc is called the relative

damping ratio f and is given by

(2.11)

When f>l, the damping is called supercritical,

and when f<l, the damping is called subcritical.

Since for practical transducers f<l, it is more

convenient to Mo-ite eqs (2.8) and (2.9) in the form

Then, eq (2.6) becomes for f<l

By introducing new arbitrary constants A and B,

defined by

C,=AI2+Bl2j

a=A/2-B/2j,

and, using the exponential equivalents for the

sine and cosine functions

sm Xcos x=

2i

eq (2.12) becomes

x,=e-(''n'{A cosVl-fW+5 sinVl-fW},

(KD

which may be written more conveniently in the

form

x,=Xoe-^'^n' cos {Vl-rW-^ol (KD- (2.13)

The constants Xq and <po can be evaluated at the

initial conditions.

The stea-dy-state solution (particular integral)

is obtained by using the complete differential eq

(2.4). For input functions which can be ex- pressed in terms of sine and cosine functions, the

steady-state solution is taken to be of the form

X5=Csin coi+Z? cos o^t, (2.14)

where C and D are arbitrary constants. Successive differentiation of eq (2.14) gives

Xs=^Cw cos cot—Do) sin wt (2.15)

Xs=—Cui^ sin wt—DoP' cos wt. (2.16)

Substituting eqs (2.14), (2.15), and (2.16) into

eq (2.4) and collecting like terms yields

mw^)Csin wi + (t— ww^)D cos coi —cwD sin wt-j-cwC cos o:t=Fsin wt. (2.17)

Equating coefficients of like terms in eq (2.17)

gives

{k-mw')C-cwD=F (2.18)

cwC+{k-mw^)D=0. (2.19)

Solving eqs (2.18) and (2.19) simultaneously

14
yields

C=

D-

(/t-mo;2)2+c2-.2

and eq (2.14) becomes

[{k—mo}^) sin co^— cwcos ut].

After normalizing the bracketed expression to give

sin a and cos a, this may be written more convenietitly in the form

where

md

Xs=X cos (cat—a)

X

(2.20)

(2.21)

V(Ar-mw=')2+cV

a=arc tan (^^^)- (2.22)

Therefore, the required response function (general

solution) is given by

or

X=-X'o«~^'*'"' cos (Vl--pw„<—<^o)+^ cos (w<— a)

(2.23)

where JY'o and <po are constants of integration which

can be evaluated at the initial conditions; and

X and a are given by eqs (2.21) and (2.22).

To obtain the input function when the characteristic differential equation of motion and the

response function are known analytically, substitute the steady-state solution, eq (2.20), into

the original differential equation of motion, eq

(1.1), and solve for /(<).

b. Transformation From Time to Frequency Domain

The transformation of the input and response

functions from the time to the frequency domain

requires that these functions be expressed as functions of time. Let us find the spectrum for the input function Fsin oit. Since this is a periodic

function of time, it can be represented by a Fourier

series of the form given in eq (1.9), i.e.

i^sin (d =Ci cos {wt— (pi) (2.24)

Expandhig the right hand side of eq (2.24) by the

function for the sum of angles yields

F sin u}t =Ci cos oit cos (^i+Ci sin cat sin <pi, and the equation is satisfied if Ci =F and <pi=Tr/2.

Therefore, the input spectrum for F sin is

and

Fico)=c„=F

<p(w)=^„ =7r/2

(2.25)

(2.26)

where F(u) and (p(ui) are the amplitude and phase

angle of the input spectrum, respectively, and we

see that both are constant for any value of w. The spectrum for the response function given

by eq (2.23) is evaluated by the same procedure

used for the input spectrum. However, to evaluate the response spectrum, only the steady-state

solution of the response function need be considered. The steady-state solution of eq (2.23) is a

periodic function of time and has a Fourier series representation of the form given in eq (1.9). Therefore,

.X^ cos (it}t — a)=Ci COS (wt —(pi). By inspection X=Ci and a=<pi. Then, the

response spectrum is

and

X{u:)=c„=X,

where X and a are given by eqs (2.21) and (2.22)

respectively. Therefore the ampHtude of the

response function becomes

X(a,) = F

V(A:-maj2)2+cV

and the phase angle

a(w) =arc tan /k—mui^\

\ —ceo /

(2.27)

(2.28)

c. Transfer Function

The transfer function defined by eqs (2.1) and

(2.2) is obtained from the input and response

spectra. Making use of the input spectra of eqs

(2.25) and (2.26), and the response spectra of eqs (2.27) and (2.28), eqs (2.1) and (2.2) become

and

V(yt-/nco^)2+cV

(2.29)

^(co) -arc tan (^3^')-7r/2 (2. 30)

where H{w) and 0(a), respectively, are the amplitude and phase-angle of the transfer function.

d. Input From Transfer Function and Response Record

Determining an expression for the input function

when the transfer function and the response record

are known requires that the response record be

expressible as a function of time. From the pre- vious results the spectrimi for the response function, eq (2.23), is given by eqs (2.27) and (2.28).

15
The transfer function is given by eqs (2.29) and

(2.30). Equations (2.1) and (2.2) yield

and

F{<^) =X{<^)IH{o=)=F=c,

(p(w) =a(co) —)3(co) =7r/2 =^i.

Since a periodic response must be the result of a

periodic input, the input function can be repre- sented by a Fourier series of the form of eq (1.9),

j{t)=Ci cos {o)t—<Pi)

where

cos (ojf— (pi) = cos (wf —7r/2) =sin cot.

Therefore

f{t) =F sin uf.

This agrees with eq (2.3).

2.2. Square Wave Function

The square wave function, shown in figure 2.2b,

is represented in the time domain for one complete

period by

f{t) = -F for -T/2<:t<0 (2.31)

f{t)=F for 0<t<T/2. (2.32)

The given square wave, being periodic and piecewise continuous, can be approximated by a Fourier

series. Since the waveform is an odd function of

t, no cosine terms can be present in the series, i.e., a„ in eq (1.6) must be zero. Therefore, it is only

necessary to evaluate b„, the Fourier coefficient of

sin ncjot. By eq (1.8)

F) sin nut dt

T/2rT/2 AW

+2/T Fsmnwtdt=—^{l-cosn(oT/2]-

Jo ncol

Hence, the Fourier series representation for the

square wave function described bv eqs (2.31) and

(2.32) is /(0=-^S - ( 1—cos^^ ) sin nw^. (2.33)

The circular frequency w is equal to 2-k\T and

cos W7r=l for n even; therefore eq (2.33) becomes

AF 1 ^(0=— S - sin wcof, for w odd. (2.34)

a. Direct Input-Output Relation in the Time Domain

Substituting the Fourier series representation

for the square wave driving function, eq (2.34),

the differential equation of motion, eq (1.1),

becomes

A.F 1 mx-\-czAr^^=— S - sin ncoi, for n odd. (2.35) K n

The transient solution of eq (2.35) is identical to the solution of eq (2.5), and is given by eq (2.13).

The steady-state solution is obtained by assuming

the solution to take the form of

a;,,=Xl (C« sin ncoi+Z>,, cos moO- (2.36)

The procedure for evaluating the constants C„

and Dn is the same as that presented in section

2.1 for the sine function. This procedure yields

n _ \^F{k—n^m(jp') » , Cn— TTi 2 ON 2 I s~o s^' lor n odd; ^ 4Fnc(jo

" mr[{k—n'^mu^y-{-n'c^w^

Therefore, eq (2.36) becomes

4:F^ 1

for n odd.

r,, 2 2\2 I 2 2 21 [{k—n^mo}^) sinwa;^

—ncoi cos no}t], for n odd,

which may be written more conveniently in the

form

Xs=Xl cos (r-wi— a„), for n odd (2.37)

where

and

4F

k—rt^mw^^

a„=^arc tan /k—rrmw^X

\ —new )

(2.38)

(2.39)

Therefore, the response function of eq (2.35) is x=X^e cos (Vl— Wti^— «Po) -fy^ Xn cos (na}t—a„), for n odd. (2.40)

The constants Xo and <po are evaluated from a

set of initial conditions, and Xn and a„ are given

by eqs (2.38) and (2.39) respectively.

To obtain the input function when the characteristic differential equation of motion and the

response function are known analytically, substi- tute the steady-state solution of eq (2.40) into the

original differential equation of motion, eq (1.1),

and solve iorfit).

b. Transformation From Time to Frequency Domain

As shown for the sine function, section 2.1, the

input spectrum of a periodic function is obtained

directly when c„ and (p„ of eq (1.9) are known for the input function. Therefore, rewriting eq (2.34)

16
with 03=2ttIT and sin «,co<=cos {nwt—Tr/2) yields

fit) =^-^cos (no)t—^' for 7?, odd. (2.41)

no^T

Comparing eq (2.41) to eq (1.9), shows that

and

,=c„, for n odd

Therefore, the amplitude of the input spectrum is 8F

F(u))=Cn= — forw odd

and the phase angle is

<p{ui)=(Pn

IT

(2.42)

(2.43)

Similarly, the response spectrum can be evaluated when c„ and (p„ of eq (1.9) are known for the

response function. These quantities are readily

obtained by equating eq (2.37) to eq (1.9), where

by inspection

and

a„=(p„.

Therefore, the amplitude of the response spectrum

is given by eq (2.38) with cc=2w/T, or

X(co)=c„=

8F

-> for n odd

(2.44)

and the phase angle of the response spectrum is given by eq (2.39), or

(J^ ^2 ^2\

)' for n odd. (2.45)

—new J

c. Transfer Function

The amplitude of the transfer function H(iS) is obtained by substituting the amplitude of the

input spectrum 7^(co) and the response spectrum

X(w) into eq (2.1). These quantities are given,

respectively, by eqs (2.42) and (2.44). Therefore,

the amplitude transfer function, eq (2.1), becomes

The phase angle of the transfer function ;8(co) is defined by eq (2.2), where ^„ and a„ are given by

eqs (2.43) and (2.45), respectively. Therefore,

eq (2.2) becomes

^(ca))=arc tan 0^—^ raw with «, odd. (2, \ —nwc / 2

47)

d. Input From Transfer Function and Response Record

The input function can be described by a thorough knowledge of the input spectrum. The

input spectrum is obtainable from eqs (2.1) and

(2.2). Using the response spectrum described by

eqs (2.44) and (2.45), and the transfer function

given by eqs (2.46) and (2.47), the amplitude of

the input spectrum is %F F{w) =^^^' for n odd

and the phase angle of the input spectrum is

<p{(i))=ir/2.

Therefore, since a periodic response is the result

of a periodic input, the input function can be

represented by a Fourier series of the form similar

to that of eq (1.9), where

and

Cn =Fiw)

The input function then becomes

/(0=S ^j;^^^^ (ncof—7r/2), for n odd

or ^F 1 i{t)= — y2, - sin noit, for n odd

which agrees with eq (2.34).

2.3. Rectangular Pulse Train

The train of rectangular pulses to be analyzed is shown in figure 2.2c, with period T taken to lie between — T/2 and T/2. Therefore, one complete

period can be represented in the time domain by

f{t) =0 for -r/2<K0 (2.48)

M-FiovO<t<T, (2.49)

/(0=0 for T,<t<T/2. (2.50)

The Fourier series representation for this

pulse train is obtainable by the evaluation of an and 6„ given by eqs (1.7) and (1.8), respectively. These equations yield

2F

nwT

sin nwT„

Hiw) = l/^/ik-n'mw')^+nW,ioTnodd. (2.46) and

bn=-^^ (cos nuTp—l).

(2.51)

(2.52)

For n=0, eq (2.51) becomes indeterminate.

However, application of I'Hospital's rule jdelds

2FTr,

(2.53)

17
Therefore, the Fourier series, eq (1.6), for the

period rectangular pulse train described by eqs

(2.48), (2.49), and (2.50) yields

FT 2F °° ri

fit) =—Fi^^—7r S - sin nwTp cos nut

1 oil n = l —- (cosncoTp— 1) sinnojfl n J

which may be written more conveniently in the

form

J{t)=-7r+TjCnCOsino>t-cp„) (2.54)

where

c«=^^^Vl-cos noiTp (2.55)

and

tPn—QXC tan ( —-. TfT-^ )• (2.56)

Note that the average value FTj,l T is really part

of the spectrum, and may be viewed as an "amplitude" at zero frequency.

a. Direct Input-Output Relation in the Time Domain

The differential equation of motion, eq (1.1), for the given rectangular pulse train input is

CD mx-\-cx-^kx=FTj,IT-\-Y^ c„ cos {rMt—<Pn) (2.57)

n=\

where c„ and <p„ are given by eqs (2.55) and (2.56),

respectively.

The transient solution of eq (2.57) is given by

eq (2.13). The steady-state solution consists of two distinct parts: (a) a constant term corresponding to zero frequency, and (b) an oscillatory term with

nco>0. For convenience, the solution is broken

down into two problems, each considered to be

independent of the other. The first problem is the solution of

mx+cx-^kx=FTplT, (2.58)

for which the solution is a constant, and by

inspection is

XA.. =-

FTr,

The second problem is the solution of

(2.59)

Here again assume the solution of eq (2.61) to take the form of eq (2.36). Applying the same

procedure as presented in section 2.1 for the sine wave to evaluate C„ and Dn yields

p _2F[ncco sin nwTp— (k—n^mw'^) (cos nuTp—l)] "~ nwT[{k-n'ma>'y-\-nW]

(2.62)

and^ _2F[{k—n^mo:'^) sin ncoTp+ncwjcos nwTp—l)]

nwT[{k-n'mcoy+nW]

(2.63)

Therefore, eq (2.36) becomes

X]

'

{ [new sin nuiT >o oiTt^xn[{k-n^mo}''Y+nW] — [k—n^mu') {cos nuTp—l)] sin nut

-\- [{k—n^mu^) sin noiTp

-\-ncw(cos nuiTp—l)] cos nut}.

This expression may be written more conveniently

in the form

a;.]„„>o=Z; X, cos {nut-ard, (2.64)

where

2V2^

^ nuT V(^-i

-COS nuT^

n^m,u^y-\-n^c^(j}^

(2.65)

and

a„=arc tan

^vcoi sin nuTp— (k—n^mu^) (cos nuTp—1)

({k—n^moo'^) sin nwTp-'rncuicos nuTp

(2.66)

By the principle of superposition, the steady

solution for eq (2.61) is

or FT X, COS {nut-aj (2.67) kT ' t^i

where Xn and a„ are given, respectively, by

eqs (2.65) and (2.66). Therefore, the response

function of eq (2.61) is mx+cx+kx=^ Cn cos {nut—^n), (2.60) x^X^e cos (Vl—f^^n^—<Po)-f FTp

kT

where c„ and tpn are given by eqs (2.55) and (2.56),

respectively. Rewriting eq (2.60) more conveniently

2i^ " ri

mx^cx-\-kx=—pf, y] - sin nuTr, cos nut uT t^i \_n —^ (cos ncoTp— 1) sin nut^ (2.61)

+S cos (moi-aj (2.68)

n = l

where X^ and are evaluated from the knowledge of initial conditions; and, Xn and a„ are given by eqs (2.65) and (2.66), respectively.

To obtain the input function when the characteristic differential equation of motion and response

18
function are known analytically, substitute the

steady-state response given by eq (2.67) into

the original differential equation of motion,

eq (1.1), and solve for Jit).

b. Transformation From Time to Frequency Domain

The input spectrum is completely described

by eq (2.54). The amplitude of the input spec- trum F(u) is obtained by adding the constant

amplitude FTJT to the oscillatory amplitude

c„ as given by eq (2.55). The phase angle is equal to cpn and is given by eq (2.56).

Similarly, the response spectrum is determined

by eq (2.67). The amplitude of the response

spectrum -X'(co) is equal to FTp/kT plus X„, where

Xn is given by eq (2.65); and the phase angle

a(u)) is equal to a„ and is given by eq (2.66).

c. Transfer Function

The transfer function for the system is given

by eqs (2.1) and (2.2). Since the amplitudes

of the input and response spectra both consist of a constant and oscillatory part, then the amplitude

of the transfer function will also contain these same characteristics. The amplitude H(ui) of the transfer fimction is obtained by dividing the

constant amplitude FTp/kT and the oscillatory

amplitude of the response spectriun, eq (2.67),

respectively, by FTp/T and c„ of the input

spectrum, eq (2.54). This may be stated in

equation form as

a(w), eq (2.66), as indicated by eq (2.2). /3(aj)=arc tan /ncu) sin nuTp—(k—n^mio^) (cos no}Tp—l) \

Xik—n^moi^) sin tiw Tp-fnew (cos nuTp—l)

)

-arc tan /1-cosn.rA

\ sm noiTp J

d. Input From Transfer Function and Response Record

The input spectrum can be found from F(u) =

X(w)/H{u) and <p(w) =a(co) —/3(w), eqs (2.1) and

(2.2) respectively. Using a transfer function

expressed by eqs (2.69) and (2.70), and a steadystate response characteristic expressed by eqs

(2.67) and (2.66), these relations become, re- spectively,

and

7-,, s FTp , 2-J2F n =- F{u>)=^+^-^ Vl-cosnwTp

1 noil

(2.71)

<p(w) = arc tan { —-. )• (2.72) \ sm nwTp /

Equations (2.71) and (2.72) are, respectively,

the amplitude and phase angle of the input

spectrum. Therefore, the input function can be

represented by a Fourier series of the form of eq

(1.9), since the response function is periodic.

Hence

J{t) =FT

or H{oi)=r+ ^ k ^j{k-n'mu,')'+nW

C„ cos {no}t—<fin) 1 n=l

(2.73)

(2.69)

The phase angle )3(w) of the transfer fimction is obtained by subtracting v'(w), eq (2.56), from

where c„ and are given by eqs (2.71) and (2.72),

respectively.

Equation (2.73) agrees with the original Fourier

series representation of the periodic rectangular

piilse train, eq (2.54). The constant term FTp/T

is readily obtained from a static calibration of the

system.

3. Aperiodic Input Functions

In this section aperiodic input functions are treated in a manner similar to that given the

periodic functions in the previous section. A

single degree-of-freedom transducer which can be

represented by the mechanical model shown in figure 1.7a and eq (1.1) is chosen for analysis.

The inputs imposed on the transducer are the

rectangular pulse and step functions. The analysis illustrates the analytical methods presented

in section 1 of this chapter. For the rectangular

pulse fimction the analysis is presented in detail. For the step function only the results and pertinent details are given.

3.1. Rectangular Pulse Function

a. Direct Input-Output Relation in the Time Domain

The aperiodic rectangular pulse function consists of a single phase such as the first pulse shown

in figure 2.2c and is expressed as a fimction of time

by

/(O =0for«0 (2.74)

f{t)^F (or 0<t<Tp (2.75)

M=0 for t>Tp. (2.76)

Aperiodic functions are not representable by a Fourier series and, therefore, are best treated by

the Laplace transformation, eq (1.22), which

represents time functions in terms of the complex

variable s^a+ju. Both the aperiodic input

function and the characteristic differential equation of motion are transformed to the frequency

domain.

The Laplace transform for the rectangular

pulse function described by eqs (2.74), (2.75), and

(2.76) is obtained by eq (1.22).

19
F(s)=j' f(t)e-"dt

F(s) =-- Fe-"dt=-- {e-'''p-l). (2.77)

^ Jo ^

Equation (2.77) represents the Laplace transform

of the rectangular pulse function which is often

written in the form

£U(t)]=^(l-e-^^p). (2.78)

o

The characteristic differential equation of motion for the sytem under analysis is given by eq

(1.1), and is

mx+cx->rkx=f{t)

where m, c, and k are constants. Applying the

Laplace transformation to both sides of this

equation yields

X[mx+cx-\-kx]=£[J{t)]. (2.79)

Equation (2.79) can be rewritten if we recognize

that the Laplace transform of the sum of two functions is equal to the sum of the transforms of the

individual functions, and that the Laplace trans- form of a constant times a function is the constant

times the transform of the function [26, p. 161].

m £[x] +c £[x] +kX[x] = Xifit)].

Applying the differentiation theorem [27, p. 175]

and substituting eq (2.78) into the equation yields

m {s'X[x]-sx{0+) -x{0+) } +c{sX[x]-x{0+)

}

+kX[x]=- (l-e-'^'j,). (2.80)

The initial displacement of the sensing element

of the transducer is usually zero. However, if not zero, it will have a constant displacement

which can be eliminated for the purpose of analysis

by a shift in the coordinate system. The initial velocity of the sensing element is likewise usually

zero. In this case, a;(0+)=i;(0+)=0, and eq

(2.80) becomes

ms'X[x]+csX[x]+kX[x]=- (l-e-'^'T'). (2.81)

5

Solving eq (2.81) for X[x]

X[x]= F{l-e-''^p)

s{ms^-\-cs-\-k)

(2.82)

The desired solution to the characteristic differential equation of motion is obtained by taking the

inverse transform of eq (2.82), i.e..

:=X ^X[x]=X ^ 1^

F{1- ,-sT, s(ms^+cs-\-k) (2.83)

The inverse transform of most linear systems is determined by expanding into partial fractions the

function upon which X~^ operates. The method

of partial fractions enables us to utUize tables of Laplace transform pairs already evaluated for our convenience. Therefore, for the purpose of

applying the method of partial fractions, eq

(2.83) may be written more conveniently in the

form

F p.

m

(2.84)

The form of the partial fraction best suited to handle the two functions enclosed by the brackets

in eq (2.84) is given by

A{s)_\

B{s)

=1 r

b \_s—Si -S2 s—s^

(2.85)

which is restricted to functions that are rational

algebraic fractions with denominators of a higher

degree than the numerators. In eq (2.85), 6 is a

constant, and Si, S2, . . ., Sk, . . , are the roots of

B{s)=0. Equation (2.85) applies when the roots

of 5(s)=0 are all distinct, i.e., no two roots are

equal. The numerators of the partial fractions,

Ki, K2, . . ., Kk, . . . K„ are determined by

K, {(.-..) ^}^^^. (2.86)

Therefore, the partial fraction for the first term

enclosed by the brackets in eq (2.84) is obtained

as follows by inspection:

A{s) =l B^ \ m mJ

The roots of B{s) =Q are

si=0

V4m2

S3-- c '2m V 4m^ m

(2.87)

(2.88)

(2.89)

The roots defined by eqs (2.88) and (2.89) are

identical to the roots ri and r2 defined by eqs (2.8)

and (2.9), respectively. Therefore, eqs (2.88)

20
and (2.89) may be written in the same form as Vi and

S2=-fco„+i«„Vl-f' (2.90)

S3=-fa;„-iw„Vl-f'. (2.91)

Once the roots of B(s)=0 are established, the numerator of the partial fractions can be evaluated by eq (2.86). Therefore, for k=l eq (2.86) becomes:

\ m m/_

bm

However, k/m=o}„^; therefore Ki =b/con^. For k=2, eq (2.86) becomes

s— ( —fco„+ico„Vl K2=b n 1

= 6 r L__ 1

Ls[s+(fco„+iwnVl^-F)]J»-»-f"n+;'-„Vl^

b K2

f2 2ico„Vw'(-f"«+i".Vw^)

2co„n(i-f2)+jfVi-n

And, similarly, for k=3, eq (2.86) yields

Hence, the partial fraction expansion of eq (2.87) becomes

I =1- (I) =L

V m m/

s-(-rw„+ico„VT-F)]

1 2a;„^[(l-n-ifVl-n[s-(-fa>n-ia'«Vl-n]

(2.92)

Taking the inverse transform of both sides yields

1

\ m mj

where the constants Ki, K2, and K3 and the roots S2 and S3 are reintroduced for simpKcity. By

utilizing the summation and multiplication theorems [27, p. 161] this equation may be written as

X1 b IsJ 6 LS-S2J b is-ssj

(2.93)

\ m m/

Referring to a table of Laplace transform pairs, the inverse transforms for eq (2.93) can be shown

21
to be

Therefore, eq (2.93) becomes

K,_K2 ... K.

' b h

Introducing the expressions for K^, K2, K3, S2, and S3, this equation becomes

J_ exp [(-fa;„+ia;„Vl-f)<]

2a,„^[(l-n+ifVW']

exp [{-^oin-jo}„^|l-^^)t]

2a;„^[(l-n-jfVW'] }

which reduces to

1

\ m m/

+-^|^sinVw"w)]-- (2.94)

The second term enclosed by the brackets in eq

(2.84) is identical to the first term multiplied by

g-'^p. For such a case the shifting theorem

[27, p. 166] apphes and

j{t-T,)U{t-T,) = £-\e-^''vF{s)].

This expression, where U{t— Tp) is defined as a

unit step occurring at Tp, shows that multiplication by a factor e~'^p simply amounts to a shift

in the independent variable from t to t— Tp.

The procedure used in evaluating the inverse

transform of the second term within the brackets

of eq (2.84)

is as follows: first perform

1

_ \ m m/_

(2.95)

and then apply the shift in t, noting that j{t— Tp)

will be zero for 0<^<rp. The inverse transform

of eq (2.95) is given by eq (2.94); therefore,

applying the shift in t yields

x-r ^

\ m mJ = ^2^1-exp ([-fco„(«-r,)]

[cosVl-fV(«-rp) +

sm

where

and

VwV(«-?^p)])j-t/(^-r,) (2.

C7(f-Tp) =0for t<Jp

Uit-Tp) =0 for >Tp.

96)

Substituting eqs (2.94) and (2.96) mto eq (2.84),

the complete solution to the characteristic

differential equation of motion, or response

function, is x=-^2 i 1 —exp (— fw„0 fcosVl —

f sm Vr3p„^^J_(^l_exp [_fco„(i-T,)]

[

cosVl-r'w„(f-T,)-f

sm^Jl-^'wn{t-Tp)'J)U{t-Tp)y• (2.97)

To obtain the input function when the characteristic differential equation of motion and the

response function are known analytically, substitute the response function, eq (2.97), into the

original differential equation of motion, eq (1.1),

and solve for J(t). In this analysis keep in mind

the properties of 17(1— Tp).

b. Transformation From Time to Frequency Domain

The transformation of an aperiodic function

from the time to the frequency domain is ac- complished by use of the Laplace transform, with

s=jw. For the input function described by eqs

(2.74), (2.73), and (2.76) the transformation has

already been performed and is given by eq (2.77).

For s=jw, eq (2.77) becomes

F(j.)=j^ il-e -Jo>T, (2.98)

where Fijw) is denoted as the complex input

22
spectrum for real frequencies w and can be simplified as foUovps:

2F fexp 0-cor,/2)-exp {-jo:Tj2y

03

[exp (-jo:Tj2)]

=(^sm ^) exp i-jo:TJ2). (2.99)

This complex number, eq (2.99), consists of two

parts: (a) an absolute value or modulus, which

is the amplitude of the input spectrum; and (b) an argument (the coefficient of j in the exponent),

which is the phase angle. These are given,

respectively, by

F{w)=mod Fijw) =

and —2F . coT, sm —

2

<p{a>)=a.TgF{j<,) =-'^-

(2.100)

(2.101)

Note that here wTp 9^ 2t because Tp is not the

period of a periodic function; therefore, co is not

fixed. Taking the Laplace transform of the response

function, eq (2.97), results in eq (2.82). For

s=jo3, eq (2.82) becomes

X{jo3) = X[x]=. F{1 , — juiT jw[{k—mo}^)+jcoi] (2.102)

where X(jw) is denoted as the complex response

spectrum. Equation (2.102) may be written in a more convenient form by noting that

[1-exp (-jwTp)]

=i^(ja,)=(^sin ^) [exp {-j<^Tpl2)]

and

(k—mw^)-\-jca}

=V{k— mco^)2+cVexp j |^arc tan ^

^_^^^2^J J"

Therefore, eq (2.102) becomes

2F

sm

X{jo>) =

V(ii:-mw2)24-cV exp[iarctan(^3^,)]

or 2F

sm

X{jo:) =

(2.103)

The amplitude of the response spectrum is given by

-X'(co)=mod X{jo)) =

and the phase angle by

2F— . coTp sm —2

w 2 V(^-ma,2)2-fcV

(2.104)

a(co)=arg Z(ico) =-^-arc tan (y-^A

(2.105)

c. Transfer Function

The transfer function for the system is defined

by the relationships expressed by eqs (2.1) and

(2.2), i.e., division of the moduli and subtraction

of the arguments. The transfer function may

also be expressed in complex number form as follows

:

where by eqs (2.99) and (2.103) the complex

transfer function becomes

V(/:-mw2)2+cV

X{exp[-iarctan(^-^)]} (2.107)

with

fl^(a;)=mod Hijoi)- (2.108) V(F^W?+cV

/3(w)=arg fl'(jw) =—arc tan (j^ir^)' (2-109)

and

d. Input From Transfer Function and Response Record

The input function is obtained through the

knowledge of the response record and the transfer

function for the system. A response record such

as eq (2.97) has been shown to be Laplace trans- formable and can be expressed in the frequency

domain by the substitution of s=jco into the trans- form. This substitution results in the complex

response spectrum X(jco) given by eq (2.103). The

complex transfer function H(Jo}) has been derived in the previous section and is given by eq (2.107).

Therefore, by eq (2.106) the complex input spec- trum F(ju) becomes

ui ^0.,=(?fsi„f)exp(=MV)

Equation (2.110) is identical to eq (2.99) which was shown to be derivable from eq (2.98) in

23
section c, or

Taking the inverse Laplace transform of this

expression, which is readily obtainable from a

table of Laplace transform pairs and the shifting

theorem, the input function is

f(f.)=F[l-U{t-T,)] (2.111)

where Uit— Tp) has the properties previously

defined. Equation (2.111) is in full agreement

with the input described bv eqs (2.74), (2.75),

and (2.76).

The inverse Laplace transform in eq (2.116) is identical to that given in eq (2.94). Therefore,

the response function, eq (2.116) becomes

+Vf=p ^nty^. (2.117)

When the characteristic differential equation of

motion and the response function are expressed

analytically, the input function is readily obtained

either directly by substituting the response function eq (2.117) into eq (1.1), or by the Laplace

transforms of these expressions.

•3.2. Step Function

a. Direct Input-Output Relation in the Time Domain

The step function to be analyzed is shown in

figure 2. 2d and is expressed as a function of time

by

j{t) =QioYt<0 (2.112)

j{t) =Fiox 0<^< 00. (2.113)

The Laplace transformation for the step function described above is obtained through eq (1.22).

Therefore,

f{t)e-"dt

or F

X[f{t)]=F{s)=rFe-''dt=- (2.114)

Jo s

where s is a complex number equal to cr+i".

The characteristic differential equation of motion

for the system analyzed is given by eq (1.1).

Applying the Laplace transform operator to this

equation, the form subsequently arrived at is equivalent to eq (2.81) except for the right hand

side of eq (2.81):

where

ms'^X[x\+csX[x\ +kX[x\

X[x\=

s{ms'^-\-cs-\-k)

(2.115)

Rewriting eq (2.115) in a more convenient form,

and taking the inverse Laplace transform, yields

the response function

m

1

\ m m/

(2.116)

b. Transformation From Time to Frequency Domain

The Laplace transform of the input function is given by eq (2.114). Fors=jco, eq (2.114) becomes

(2.118)

where F{jw) is the complex input spectrum.

Equation (2.118) may be written in a more convenient form as F -Ja; (2.119)

with the amplitude of the input spectrum given by

i^(co)=mod F{jw)=^

and the phase angle given by

(2.120)

<p(a,)=argi^(ico) = --- (2.121)

The Laplace transform of the response function,

eq (2.117), results in the complex response spectrum, eq (2.115). For s=jw, eq (2.115), becomes

F

(2.122)

Equation (2.122) can be written in a more

convenient form by recognizing that

and

(k—mo}^)-\- jc(j3

-F{jo:)-

F -il

=V(^-mco2)^+cV^exp [i arc tan (^i^a)]" ^

24
Therefore, eq (2.122) can be written as

F

03 V(il:-Wa;2)2+cV

r X (exp| -J g+arc tan (^I^.)]})

with

F

X{w)=mod X{jo3)

V(^-wco2)2+cV

and

(2.123)

(2.124)

Q;(w) =arg X(ia)) =-^-arc tan (^z?^)'

(2.125)

and

/3(w)=arg fl'(jco) = -arc tan „ \ (2. 128)

Note that the transfer function is a characteristic

of the linear transducer system and is not dependent

upon the type of excitation or input to the system.

This in part is substantiated by the fact that eq

(2.126) is identical to eq (2.107), where the

transfer function relations were derived for two

different inputs.

d. Input From Transfer Function and Response

Record

The input function is directly determined from

the complex input spectrum F{ju), which can be

obtained from eq (2.106). Tliis requires that the

complex response function X{ju) and transfer

function Hiju) be knowm analytically. For

X(jo)) and H{ju) given by eqs (2.122) and (2.126),

respectively, the complex input spectrum is

c. Transfer Function Fijo^yThe complex transfer function HijcS) is given

by the division of F(jo:) into Xijui), where these

quantities are given by eqs (2.119) and (2.123),

respectively. This division yields

ff(i«) =

1 V(^-mco2)2+cV

X{exp[-iarctan(^:^,)]} (2.126)

with

H{w)=mod Hijo3) - 1 V(A:-mco^)=^+cV

(2.127)

By a table of Laplace transform pairs, the inverse

transform of F(jui)=F{s) is M=F

The interpretation of this result is

Jit) =0 for t<0

m=FioTt>0,

which is based on the property of a function which

is Laplace transformable. These resiilts agree

with, the input function described by eqs (2.112)

and (2.113).

4. Phase-Plane Method

4.1. Introduction

We introduce the concept of the phase-plane

method here, in a section concerned with linear

transducers, only to relate it to the more common

methods by which the beha\aor of linear djmamical

systems is characterized. In chapter 4, the apphcation of the phase-plane method in characterizing the behavior of nonlinear transducers is dis- cussed in detail. For linear systems (where the

principle of superposition is vahd), the classical,

direct methods of analysis are more commonly

used and the existence of a meaningfid transfer

function precludes the necessity for using the

phase-plane method. For nonhnear systems the same transfer function does not exist; however,

the time response of certain nonhnear systems can

be determined by phase-plane techniques. Other

methods for approximating the response of nonlinear transducer systems are described in chapter

4. Here we confine ourselves to an introductory

exposition of the phase-plane method as it apphes

to linear systems.

4.2. The Phase Plane (Phase Space)

The phase plane is a two-dimensional phase

space, the coordinates of which are related in that

the one is the time derivative of the other. In the analysis of transducer S5'stems, the appropriate phase-plane coordinates are the transducer

output and its first time derivative, say, q and q, respectively.

The general concept of a phase space is well

known in physics and perhaps best known for

its application to kinetic theory of gases. In such an application of the general concept, the

disturbed response of a dynamical system with n degrees of freedom is characterized for time t by

the set of 2n coordinates comprising the positional

coordinates qi{i=l, 2, . . . n) and their velocities

694-211 0-63 —

3

25
q^(i=l^ 2, . . . n). The qt, qi may be considered

as the coordinates of a space S of 2n dimensions

called a phase space. At time t, for each state of the system there exists a point P with coordinates

q^)- As t is permitted to vary, the point P

describes a curve called a path or trajectory, which

characterizes a history of the system. An infinite number of such paths exists, each determined

uniquely by specification of a single point of the

path. For a given system the totality of such

paths in the (g;, 2i) space is the phase portrait of the system. Since each of the paths is determined

uniquely by specifying a single point, it can be

inferred that there is one and only one path

through each point of the phase space.

4.3. The Phase-Plane Method Applied to

Linear Systems

Let us consider the viscously damped, singledegree-of-freedom system described by the linear,

second-order equation of motion

g +2fo;„2 +co„2g=0, (2.129)

where cj ^ is the undamped natiiral frequency

(=V^7m). The state of the system at any time

is fixed by the values of q and q; for example, if q{Q) and q{Q) are known, the solution for all time

is determined. This dependence of future

states of the system on the initial conditions can be shown graphically in the phase plane, the {q, q)

plane. The phase portrait for the linear system

of eq (2.129) with relative damping ratio f=0.5

is shown in figure 2.3. Let us say that Pq at q'o, go describes the initial state of the system. Then

P], P2 • • • describe successive states as t progresses. Note that time appears in the phase portrait only implicitly as a parameter changing

value along any path.

Clearly, there is only one path through each

point in the phase plane, since the solution of eq

(2.129) is determined uniquely by specifying both

q and g at a given instant of time. Thus, the

1.0 \

; 0.5 \ \

^

J

\°

1.0

1 a.o

1 -1.0

qO.5

H.O

hi

-1.5 ^^^^

Figure 2.3. Phase portrait of linear viscously damped

system (f = 0.5). (Reproduced from [35] with permission from Clarendon Press)

phase portrait for the linear system is a family of

noncrossing paths describing the system behavior

after all possible initial conditions.

There are three methods for constructing the

phase portrait : by direct solution of the differential

equation, by reducing the order of the original

differential equation and solving for g as a function

of q, and by plotting the isoclines corresponding

to various slopes of the phase paths. The three

methods are described below in terms of eq (2.129)

with f =0, the equation of an undamped system,

q + o}„\ =0. (2.130)

a. Direct Solution

The solution of eq (2.130) is given by

q{t)=^Q sin {w„t+<p) (2.131)

and the velocity by

q{t) = Qu„ cos {o:„t+(p). (2.1321

By squaring and adding eqs (2.131) and (2.132),

time is eliminated as an explicit factor and we

obtain

(2.133)

which describes the phase portrait of the system.

The paths are ellipses with axes of magnitude Q

and Q(j)„, or, normalizing the velocity, the phaseplane coordinates are q, q/un, and the corresponding phase paths are circles of radius Q. In either

case, the initial conditions specify a particular

ellipse (or circle) which characterizes the dynamic

behavior of the system.

The same procedure, that is, direct solution of the differential equation, can be used infrequently

in the analysis of nonlinear systems. However,

this is discussed in chapter 4 where the phase-plane

method is used to describe the behavior of nonlinear transducers.

b. Solution of the Equation for g as a Function of q

In certain systems, it is simpler to pass directly,

without integration, from differential eq (2.130) to representation on the phase plane. The procedure by which this can be accomplished is, first, to replace the initial second-order equation by

two equivalent, first-order equations.

dq

dp

dt'

(2.134)

But eq (2.134) can be written as the following

first-order equation

dp

dq =—

P

(2.135)

26
It can be seen that integration of eq (2.135) will jdeld

2'+^=QS (2.136)

a solution (phase portrait) identical to that given

by eq (2.133).

c. Method of Isoclines An approximation of the phase portrait can be

constructed by studying the slopes of the phase

paths. This approximate method is, perhaps,

most useful in the analysis of nonlinear systems

or of linear systems where integration of the

differential equation is difficvdt. The method of

isoclines offers little advantage whenever, as in our illustrative case, separation of variables per- mits easy integration of the equation.

For the imdamped linear system it is clear that

the slopes of the totality of paths on the phase

plane are given by eq (2.135). A locus of constant slope values is termed an isocline. The

isocline corresponding to dp/dq=a can be found

from the equation

2 3

p

(2.137)

The family of isoclines determined by eq (2.137)

are straight lines passing through the origin of

the phase plane with slope — wja or, if the velocity

p is normalized, with slope — 1/a.

The path from any given point (initial state)

in the phase plane can be constructed in the

following manner. Point Po in figure 2.4 lies on

the isocline corresponding to a=1.00. The motion of the path away from Po is clockwise with

reference to the origin. The isocline adjacent to that through Pq in a clockwise direction is that

for a= — 1.50. From Pq a directed line segment

of slope —1.25 (the average of —1.00 and —1.50)

is drawn to intersect the a = — 1.50 isocline at

{)oint Pi. From Pi the process is repeated; a new

ine segment of slope —1.75 is drawn to intersect

the a= —2.00 isocline at point Pz. As the process

is repeated an approximation of the phase path

can be sketched by joining successive line segments.

The exactness of the approximation to the phase

path is dependent upon the number of isoclines

used in its construction: the greater the number,

the more exact the approximation. Inherent in

this lack of exactness lies the greatest disadvantage

of the method. An important aspect of the phase

portrait for nonlinear systems is the existence of closed paths. In some cases, with even a rela- tively dense array of isoclines upon which to base

the construction, it is difficult to sketch accurately

the phase path in the region of the origin. Is the

path closed or is it, in fact, approaching the origin

slowly with each circuit around the origin? As a

practical consideration this deficiency is usually more distm-bing than serious.

p

/ y'P y a--i.»o ^°J\*^ a-- 1.00

Q--Z.0O >^

a-- 1.00 /

Phoio Poih - Po Pt

Figure 2.4. Construction of phase path from isoclines.

d. Application of the Phase-Plane Method to a Linear,

Viscously Damped System by Means of Oblique

Coordinates

If it is assumed that the initial conditions for the motion described by eq (2.129) are qiO)=qo

and i{0)=po, then the solution of (2.129) for f<l is

g=e-f cos con^/l —^H ^Po+^ sin c.„VW^^)> (2.138)

which may be rewritten as

where

and

2=e-r«n<^ cos (co^VW'^-^), (2.139)

(2.140)

tan v'=

\w„vi—

By eq (2.140), the normalized velocity then

becomes

l=_e-ro,„' S sin (w„^^T^H-<t,+^), (2.141)

where the new phase angle ^ is defined by

tan ^= (2.142)

27
Figure 2.5. Oblique-coordinate phase plane used for analysis of viscously damped systems.

Earlier, in the undamped case, time was eliminated

as an explicit parameter by squaring and adding

the expressions for q and g/co„. Here, if the phase

coordinates as given by eqs (2.139) and (2.141) are squared and added, the presence of the new phase

angle ^ unduly complicates the resultant expression. However, the phase paths can be simplified

if we use an oblique phase-plane coordinate system

[15, 34].

Let q and q/un (the transducer output signal

and its first normalized time derivative) be plotted on the oblique coordinates of figure 2.5. The

magnitude of the radius vector p is

=q'+iq/cCny+2q(qM sin ^ (2.143)

By using eqs (2.139) and (2.141) in (2.143) we

obtain

p=e-r"n<^ cos ^

=Poe-^-', (2.144)

the equation of a logarithmic spiral in polar

coordinates, which must be projected onto the

oblique axes of figure 2.5 to give q and g/co„. It was stated earher that the time does not

appear expHcitly in the phase-plane representation

of the motion of a system; however, time is given

implicitly by the angular location of the phase

points representing successive states of the system.

In eq (2.139) the angle in the trigonometric argument is expressed in terms of the angiilar velocity

oj^^l— f2 From this it can be concluded that

the angular velocity of the phase point on the

path will have this value. Consequently, the

implied time for an angular movement of 6 radians

in the phase plane will be j== and, upon

substituting this in (2.144), the explicit equation

of the spiral phase path in the oblique coordinate

plane becomes

p=Pog-ft«f. (2.145)

Clearly, for each value of there exists a particular spiral.

While the construction of exponential spirals can be tedious, it presents no other problem.

For values of f <0.5, Jacobsen [15, p. 204] suggests

several approximate methods of construction.

5. References

[1] T. von Karman and M. A. Biot, Mathematical

Methods in Engineering, p. 335, McGraw-Hill

Book Co., Inc. (New York, N.Y., 1940).

[2] H. Poincar^, On the contours determined from a dif- ferential equation (in French), vol. I, Oeuvres de Henri Poincar^, Gauthier-Villars (Paris, 1928).

[3] H. Poincar^, New methods of Celestial Mechanics (in French), vol. I, Gauthier-Villars (Paris, 1892).

[4] J. Lamoen, On the dynamic entreaty of wind on tall structures (in French), Mem. Acad. Roy. Belgique,

Classe Sci. 12, 1437 (1932).

[5] J. Lamoen, Graphical study of the vibrations of singledegree-of-freedom systems (in French), Rev. Universelle Mines [8] 11, No. F, 213 (May 1935).

[6] H. 0. Fuchs, Spiral diagrams to solve vibration damping problems, Prod. Eng., 294-296 (Aug. 1936).

[7] E. Braun, On the graphical solution of the differential equation for forced vibration, considering an arbi- trary law for damping, restoring force and driving

force (in German), Ing. Archiv 8, No. 3, 198-202

(1937).

[8] V. Rojansky, Gyrograms for simple harmonic systems

subjected to external forces, J. Appl. Phys. 19, No.

3, 297-301 (Mar. 1948).

[9] R. E. D. Bishop, On the graphical solution of transient vibration problems, Proc. IME (London) 168, No.

10, 299-322 (1954).

[10] A. A. Andronov and C. E. Chaikin, Theory of Oscillations, Princeton Univ. Press (Princeton, N.J., 1949).

[11] N. Minorsky, Introduction to Non-Linear Mechanics,

Edwards Brothers (Ann Arbor, Mich., 1947); also Non-Linear Oscillations, D. Van Nostrand Co., Inc. (Princeton, N. J., 1962).

[12] N. KrylofT and N. BogoliubofF, Introduction to NonLinear Mechanics (Kiev, USSR, 1937); trans. S. Lefschetz, Princeton Univ. Press (Princeton, N.J., 1943, 1947).

[13] J. G. Truxal, Automatic Feedback Control Svstem

Synthesis, McGraw-Hill Book Co., Inc. (New York,

N.Y., 1955).

[14] G. J. Murphy, Control Engineering, D. Van Nostrand

Co., Inc. (Princeton, N.J., 1959).

28
[26] D. K. Cheng, Analysis of Linear Systenns, AdflLson- Wesley Pub. Co., Inc. (Reading, Mas.s. 1959).

[27] R. V. Churchill, Operational Mathematics, McGrawHill Book Co., Inc. (New York, N.Y., 195S).

[28] J. N. MacDuff and J. R. Curreri, Vibration Control,

McGraw-Hill Book Co., Inc. (New York, N.Y.,

1958).

[29] A. Bronwell, Advanced Mathematics in Physics and

Engineering, McGraw-Hill Book Co., Inc. (New

York, N.Y., 1953).

[30] W. R. LePage, Complex Variables and the Laplace

Transform for Engineers, McGraw-Hill Book Co.,

Inc. (New York, N.Y., 1961).

[31] C. R. Wylie, Jr., Advanced Engineering Mathematics,

McGraw-Hill Book Co., Inc. (New York, N.Y.,

1951).

[32] L. A. Pipes, Applied Mathematics for Engineers and

Physicists, McGraw-Hill Book Co., Inc. (New

York, N.Y., 1958).

[33] N. O. Myklestad, Fundamentals of Vibration Analysis,

McGraw-Hill Book Co., Inc. (New York, N.Y.,

1956).

[34] Fliigge-hotz, Irmgard, Discontinuous Automatic Control, p. 24, Princeton Univ. Press (Princeton, N.J., 1953).

[35] N. W. McLachlan, Ordinary Non-Linear Differential Equations in Engineering and Physical Sciences,

Clarendon Press (Oxford, England, 1956).

29

[15] L. S. Jacobsen and R. S. Ayre, Engineering Vibrations, p. 203, McGraw-Hill Book Co., Inc. (New

York, N.Y., 1958).

[16] K. Klotter, How to obtain describing functions for non-linear feedback systems. Trans. ASME 79,

509-512 (1957).

[17] K. Klotter, Steady-state oscillation in non-linear multiloop circuits. Trans. IRE, 13-18 (1954).

[18] K. Magnus, On a method for the analysis of nonlinear vibration and control systems (in German),

VDI Forschungsheft [13] 21, 1-32 (1955).

[19] J. E. Gibson, Non-linear system design. Control Eng.,

69-75 (Oct. 1951).

[20] R. L. Cosgriff , Non-Linear Control Systems, McGrawHill Book Co., Inc. (New York, N.Y., 1958).

[21] R. W. Bass, Proc. Symp. Non-Linear Circuit Analysis,

Polytechnic Inst. Brooklyn, vol. 6, 1956.

[22] E. I. Ergin, Transient response of a non-linear spring- mass system, J. Appl. Mech. 33, 635-641 (1956).

[23] T. M. Stout, Basic methods for non-linear control system analysis, Trans. ASME 79, 497-508 (1957).

[24] L. S. Jacobsen, On the general method of solving

second-order, ordinary differential equations by

phase-plane displacements, J. Appl. Mech. 19, 543

(1952).

[25] F. F. Liu, Recent advances in dynamic pressure measurement techniques, J. ARS 38, 83-85, 128-132

(1958).

3. Approximate Methods of Linear Transducer Analysis

L. C. Eichberger '

1. General

In chapter 2 it was assumed that the characteristic differential equation of motion of the

transducer system is known. Corresponding re- sponse functions were then obtained analytically

from the differential equation for a number of given input functions by classical or operational

mathematics. This procedure of analysis is the

exception rather than the rule. In general, the

characteristic differential equation of motion is not known. What is known is the response of the transducer system to (1) a known input

fimction and (2) the response of the same trans- ducer system to an unknown input function. The

response function will usually be in the form of a record—either a strip-chart record from an oscillo- graph or a photographic record from an oscillo- scope. The response function will be referred to the time domain since its graphical representation

is a function of time. The graphical form of the

response function brings about a need for a method or methods by which the response function can either be expressed analytically as a function

of time or transformed directly to the frequency

domain. This chapter will be devoted to the

various methods available for fulfilling this need.

Once the response function has been expressed as a function of either time or frequency, the transfer function and the input function can be obtained

by following the routes of analysis shown in figure 2.1 and illustrated in detail in chapter 2. The major part of the analysis in this chapter is based on the evaluation of the Fourier integral eq

(1.17). The integral, as it stands, is cumbersome

to compute, and the solution to some very ele- mentary situations often turns out to be a tedious

and time-consuming task. To relieve this tedium,

different approximations and/or computing aids

are introduced to simplify the evaluation. Some

of the approximations considered are harmonic

analysis, staircase function, straight-line segment,

trapezoidal, sin a;/x, number series transformation,

and the pseudo-rectangular pulse. The computing aids considered are Henderson's analyzer,

Montgomery's optical Fourier analyzer, photoelectric Fourier transformer, and an electronic

analyzer with magnetic transient storage used by

Lederer and Smith.

Before discussing the approximations in detail

let us examine the Fourier integrals, eqs (1.16)

' Assistant Professor of Mechanical Engineering, The University of Houston; Technical Staff, Houston Engineering Research Corporation.

and (1.17), more closely. For convenience, let us rewrite eqs (1.16) and (1.17) as

x{t)=^ r ^We'-'dw (3.1)

Xijw)= x{t)e-^'''dt, (3.2)

tJ — CO

where x{t) is the response function and X{jw) is its direct Fourier transform, the complex frequency

response of the system.

Let us assume that the response function x(t) is identically zero for t<CO. This assumption for all practical purposes is true since it implies that the

input function to the system is also identically zero for f<0. The acceptance of this assumption

allows us to use either the real or imaginary part

of Xijo}) and thus eliminates the exponential form

of the transform [1].^ This can be easily shown

[2] by rewriting eq (3.2) with Xijo))=Xi{io) +

jX2{(j}) and e~^"'=cos wt—j sin wi. Therefore,

eq (3.2) becomes

J

CO x{t) cos ut dt

- CD /» 00 —j I x{t) sin wtdt

from which

^

CD Xi{co)= j x{t) cos wt dt

X2(o:)= I x{t)smwtdt.

Assuming that x(t) is an even function, then Xiiu)

is even and determines the even part of x{t) in eq (3.1), wliile X2{o}) is odd and vanishes. Or

assume x{t) to have even and odd parts, then,

since x{t) is zero for i<0, the even and odd parts

must cancel. It follows that they are identical

for i>0. Therefore, one-half of either part or one-fourth of the total can be used to compute

x{t) for positive t. On the basis of this argument,

2 Figures in brackets indicate the literature references on p. 51.

31
eqs (3.1) and (3.2) may be rewritten as

2 x{t)=- X^ico) cos cot do) (3.3)

TT Jo

Xi(u)=\ xit)coswtdt, (3.4)

where -X'i(co) is a real function of a real variable.

2. Approximation of Periodic Functions

2.1. Harmonic Analysis

Harmonic analysis enables us to represent any

known periodic phenomenon by an empirical

function. The periodic phenomenon, the response

fmiction, can always be approximated by the

Fourier series of the form

/(i)=Oo+<^i cos o)t-ra2 cos 2o:t-\- . . . +a„ cos nut

-\-bi sin sin 2wt-\- . . . -\-b„ sinnwt. (3.5)

This function is periodic and has a period of

2ir. If values of the dependent variable f{t) are known for certain equidistant values of the independent variable wt, then the unknown coefficients

tto, ai, a2, . . ., an, bi, b2 , bn can be easily foimd.

Values of f(t) are readily obtainable from the

record of the response function. This record may

be in the form of numerical data, a graph, or a photograph. Once these values have been obtained the analysis for evaluating the coefficients

is as follows: for illustrative purposes, let us assume that the function is periodic with period

2t as shown in figure 3.1, and that/(^) is known

for six equidistant values of cot (or ^=3 in fig. 3.1),

i.e.,

at m

0 60 120 180 240 300

/O /l /2 /a /4 /5

Equation (3.5), for the 6-ordinate system,

becomes

/(^)=ao+ai cos (d^a2 cos 2wt-\-az cos 3a)f

-\-bi sin U-^b2 sin 2wt. (3.6)

f(i)

to t| tik-i Iji,

FiGUHE 3. 1. Periodic response function.

Substituting the corresponding set of values for

jit) and oit iuto eq (3.6), gives

/o =«0 +«l+02+a3

(3.7)

y3=(Jo—cti+ct2—03

J I 1 1 V3 7 V3 , .75=«o+iai— ^«2—^3— "2" —2 ^

To solve these equations for the a's and &'s apply

the following rule: To find (or b„), multiply

each equation by the coefficient of o„ (or 6„) of

that equation and add the results [3]. Applying

this rule to eqs (3.7), yields

6ao=/o+/l+/2+/3+/4+/5

3ai =/o +i/i ~~ hji —j%—hfi +i/s

3^2 =/o—5/1 i/2 4"./3 hfi 1/5

6a3=/o-yi+/2-/3+/4-/5

36.=f/.+f/2-f/4-|/.

OA V3 , _V3 r , V3 /• V3 302=y/i —2 -/2+ yJ-f

(3.8)

The values of the a's and b's could be obtained

directly from these equations without too much

diflfiiculty. However, if the present intervals were

halved, this would constitute a 12-ordinate system

which would yield 12 equations similar to those

shown above with some containing at least 12 terms. For such a system the evaluation of the a's and b's would be a tedious process. Therefore, let us show for the six-ordinate system how an addition

and subtraction scheme can be developed to aid in the evaluation of the a's and b's. Suppose the terms in eq (3.8) are regrouped as foUows:

32
6ao= (/0+/3) + (/l + (/2+/5)

3a, = (/o -/a) +K/i -/J -1(/2 -/s)

302= (/0+/3) -K./. +/4) -K/2+/5)

6a3= (/0-/3) - Ui-J') - ih-h)

362=f (/l+/4)-^ (/2+/5).

/o-/3=A

/2-/5=A.

Let us now put

/o+/3 ='S'o

/2+/5='S2 Then these equations become

6ao =5*0 + (S*!

3a.=Do+§A-iD2

3a2=<S'o —^(Si

Qa3=Do-Di-D2

362=

Finally, substitute

Sfj—Uo

Si-\-S2=U,

Do=Ro

VS Q, V3 Q

Then the equations for finding the coefficients in

the Fourier series are

a2=UUo-hU^)

(3.9)

The results of the addition and subtraction scheme

could be tabulated as follows:

/o /l /2 So Si Do D,

/a /) /s D2

Sum ^0 'S'l ^2 C/o ?7i Ro R,

DiflF. Do Z?i D2 Fi P,.

A reliable check on the computed a's and h's, is obtained by a study of eqs (3.7) which yields

/o=ao+«i+a2+a3

to be used as a check on the a's, and

/l-/6=V3(il +62)

for a check on the 6's.

It should be noted that, since the terms of the

Fourier series are additive, it is necessary that the

coefficients all be computed to the same number

of decimal places.

A system for any number of Fourier coefficients can be developed in the same manner. To

expedite this procedure, it can be shown [4] that

eqs (3.8) can be written directly from the following

(with reference to fig. 3.1)

:

Y 2k-l

2k-l a;=T Z) /t cos

I

2fc-l

IJTk

-j=l,2, . . .,k-l.

V3

1

Tables 3.1 and 3.2 give systems for determining

12 and 24 Fourier coefficients, respectively. For

other systems the reader is referred to iSIacDuff

and Curreri [4], 4S-ordinate system; Running [5], 6-, 8-, 10-, 16- 20-, etc., ordinate systems ; and Pollak

[6], 3- through 40-ordjnate systems, inclusive.

Since periodic functions ha^nng a period different

than 2w can be reduced to the form of eq (3.5) by

a linear substitution of the independent variable

[7], the procedures outlined in this section are also

valid for functions having periods other than 27r.

33
Table 3.1. 12-ordinate system {12 Fourier coefficients) Table 3.2. 24-ordinaie system (24 Fourier coefficients)

/o fi

J* u U

/6 /.. /lO h fs h

Si S, Sz Si s.

Do Dz Di

So

CI Si

Cf Cj iJi

S2 s, Ds Di

Uo Ui u. Ri R,

Vo Vi Pi

Ui Pi

Ut P,

Sum L H

Diff. M G

oo^VM+L)

ai=vj(^Do + ^Vi+V2V^

a2=}i{Vo+V2M)

az=V6{Do-V2)

ai=V6{Uo- V2L)

V^(^Do-'^Vi+V2V^

a,= Vi2(Vo-M)

bi=vj(^Dz+V2Ri+^R2^

V3(?

as

hz=Vi{Ri-Dz)

bi- 2/3H'' 12

5= vj(^D.Z+V2Rl-^R2

Check formulas (check results for a's and 6's)

2a==/o

(61 + 65) +263+V3(62+64) = A.

/o fi J ^ J A Ji Jof. ff. j~ fa f« 78 ^9 J 10 7ii

/.2 /23 /2I /2O /l9 /ifi fii /16 /l5 fu fn

Sum Si S2 S3 Si s, S^ S7 Si Si Sio Sii Diff. Di Dz Di Di Di Di Di Dt Dm ^11

So Si S2 Sz Si s,

s. Sii Sio s. Ss Sr Sum Uo Ui U2 Uz Ui u,

DifiF. y 0 V,y 1 y 2 y 3 V' i

Di D2 Dz Di Ds

Dii Dio D, D, D7

Sum Ri R2 Rz Ri R,

DifiF. Pi P2 Pz Pi P5

Uo Ui u. Pi P2 Li Gi Uz u, Ui P4 Ps L2 G2 Sum u Li u Hi H2 C J

Diff. Mo Ml M2 Gi G2 E N

^ =sin 15° = 0.25881 90 B=cosl5° = 0.9659258

ao=H4(Lo+C)

a,=H2 (^Do + BVi+^ ^2+-^ ^3+ V2Vi+AV,^

a2= /12 ( Vo+Y Mi+

az= V12 [^Do +^{Vi-Vz- V,) - F4]

ai=Vi2{Mo+V2E)

ai==Vi2 (Do + AVi-^ Vi-^Vz+VzVi+BV^

ai=Vn(yo-M2)

a,= Vi2 {Do-AVi-^ V2^^Vz^\Wi-BV^

ai=Vx2(Lo-V2C)

09= V12 [^0-;^ {Vi-Vz-v,) - F4]

a,o= V12 (^0-^ M,+

aii = V12 {Do-BVi^^ F2-;^ ^3+^2^4 -AV^

ai2=V2i{Mo-E)

34
Table 3.2. 24-ordinate system (S4 Fourier coefficients) —Continued

6i ' 24

65=^^2 (^BR,+ V2R2-y^ ^'--y Ri + ARs + D,^

611= (^fti- V3

V2R2 +-7^ «3-if

V2

bt=Vl2(Hi-P3)

b^=Vl2 (^BRi-V2R2-^ ^^4+^/1:5- a)

V2

V2 " 2

Check formulas (check results for a's and b's): 2a=/o

2^ (6, + 611) + (62 + 6,0 ) + V2 (63 + 69) + VS (64 + fcs) + 2^(65 + 67)+ 26, = Z),.

3. Approximation for Aperiodic Functions

3.1. Staircase Function

Let us assume a response function as shown in

figiu'e 3.2a. This graphical record may be either a strip-chart record or a photographic record.

From this record it is required that the frequency

characteristics of the system be determined.

Since analytical evaluation of eq (3.4) generally

proves rather difficult, it would help us httle to

find a mathematical expression for x{t). Therefore, an approximate integration by graphical means seems appropriate. To begin with, let us

diflPerentiate eq (3.1) with respect to t and eq (3.2)

with respect to jw.

X (t)

x(n)

x(N)

i(<)=^

J_"

X{jw)e^-'{jw)do:={jc^)x{t) (3. 10) •(t)

and

X{jw)=j' xit)e-^'''i-t)dt^-tX{jo)). (3.11)

Substituting the results of eq (3.10) and (3.11),

respectively, into eqs (3.2) and (3.1) gives

{ju)X(joi)=f x(t)e-^''' dt (3.12)

*/ — 00

-txit)=:^ j'^ X(jo^)e^"'dw. (3.13)

Equations (3.12) and (3.13) show that differentiation in the time domain corresppnds to multiphcation by joi in the frequency domain, while

differentiation in the frequency domain corresponds

to multiphcation by —t in the time domain.

In a manner analogous to the derivation of eqs

(3.3) and (3.4), with an assumption of fictitious odd and even components in x{t) for t<^0, a similar

0 T

b

2T 3T 4T 5T 6T 7T

•(t)

t 1 I t t . 0 T

C

2T 3T 4T 5T 67 TT

Figure 3.2. Application of the staircase function approximation.

(a) Response function

(b) Staircase function approximation of the response function

(o) First derivative of the staircase function approximation

35
set of equations can be derived for eqs (3.12) and

(3.13); these are

x{t) =-^S Xi(co) sin coi c^w (3.14)

TTf Jo

Xiiw) = -- ( x{t)smo:tdt. (3.15)

<^ Jo Let us consider eq (3.15) and return to the

original problem of evaluating the frequency

characteristics from figure 3.2a. Eq (3.15) re- quires that the slope of the original function x{t)

be known. This is accompHshed by approximating

x(t) by a staircase function as shown in figure 3.2b. The derivative of this function is the impulse

distribution shown in figure 3.2c. Let the impulses

(inclusive of their algebraic signs) be denoted by

tti, a2, . . ., and their time of occurrence by

ti, ti, . . ., h, then eq (3.15) yields

Xi{w) = —- {ai sin wti-\-a2 sin co^2+ • • • sin oit^^

CO (3.16)

where k is the number of impulses.

The only error committed in the above analysis

is that due to the staircase approximation of x{t).

The impulses shown in figure 3.2c are true impulses, so eq (3.16) is good for all values of the

frequency w. By making a less precise staircase

approximation (which need not, incidentally,

involve uniform increments) an expression for

x{t) can be obtained with fewer terms than before.

However, this is done at the expense of acciu-acy.

To obtain fewer terms in x{t) without sacrificing

accuracy it is necessary to take further advantage

of the differentiation characteristics of eqs (3.1)

and (3.2). This is discussed in more detail in the

following section on straight-line segments.

3.2. Straight- Line Segments

Before introducing the straight-line segment

approximation, let us examine more closely the

successive differentiation of eqs (3.10) and (3.11)

with respect to t and jco, respectively. Therefore

x{t) = {jo:)x{t)

and

X{jo>) = -tX{joi)

which by eqs (3.10) and (3.11) become

x{t) = {ji»yx{t) (3.17)

and

X{jo>)=^{-tyX{jc.). (3.18)

Substituting eqs (3.17) and (3.18) into eqs (3.2)

and (3.1), respectively, yields

ijo}yX{jw)= r xiDe-^^'dt (3.19)

and

(-0'a;(0=^J_" X(jw)e^-'dc^. (3.20)

Comparing eq (3.20) with eqs (3.1) and (3.10) and

eq (3.19) with eqs (3.2) and (3.11), the following

general equations evolve:

(-i)"a;(i)=^J X^"\jo:)e^-'dw (3.21)

and

(»«X(iw)= x^^\t)e-^'''dt (3.22)

where the nih. derivative of X{joi) and x{t) is denoted, respectively, by X'"'(jco) and a;<">(i). The corresponding real part of the time function,

eq (3.21) is

n 2( 1)2 f"° x{t)= X<"'(jw) cos o)t do3, for n even

irt Jo

(3.23)

and

n+l x{t)= ^''~^}„ ' r"x("'(ico)sincofcZf, for rt odd.

Jo

(3.24)

Similarly, the real part of the frequency function,

eq (3.22), is

n Xi{<x))=-—^ r a;'"'(^) cos wt da, for n even w Jo

(3.25)

and

n+l

X^{o:) = ^~^l

^ f x^^^t) sin oit dt, for v odd. ^ Jo

(3.26)

Let us now consider approximating x(t) by straightfine segments as shown in figiu^e 3.3a. Then its

first derivative becomes a step approximation,

figure 3.3b, and its second derivative a sum of impulses, figure 3.3c. The expression for Xi{o:) is given by eq (3.25) for n=2, therefore

Xi{a)=-K (di cos coii+02 cos 0)^2+ • • • -hd'k cos uU)

CO (3.27)

where ai, a2, • • . and <i, ^2, • • ., are defined in

the same manner as in eq (3.16), and k is now the

number of impulses in the presentation of x(t). The basic difference between the staircase function

approximation and the straight-line segment

approximation can be readily seen by comparing

figures 3.2c and 3.3c. Thus, the latter approximation results in fewer impulses, or an expression

for x(t) involving fewer terms. Therefore, for

the same number of impulses as in the staircase

function approximation, the straight-line segment

approximation to xit) is better.

36
X *(t)

Figure 3.3. Application of the straight line segment

approximation.

(a) Straight line segment approximation of the response function shown

In fig. 3.2(a)

(b) First derivative of the straight line segment approximation

(c) Second derivative of the straight line segment approximation

The same line of reasoning can be extended to higher order approximation such as approximating

the response function by a second-degree curve. The first derivative of such an approximation is a straight-fine segment approximation, while its second derivative is a staircase function approximation. The third derivative is a sum of impulses.

Thus, the expression for xit) is readily obtained

by the appropriate form of eq (3.26). There is, however, a great deal of difficulty experienced in implementing this approximation and those of

higher order. For this reason, higher order

approximations will not be considered further.

For a detailed discussion of higher order approximations the reader is directed to the work of Guillemin [1].

3.3. Trapezoidal Method

In section 1 , the determination of the amplitude

of the frequency characteristic for the stated

problem is reduced to the approximate evaluation

of eq (3.4), or

^i(co) x{t) COS oit dt

X •(!)

Figure 3.4. Application of the trapezoidal method of approximation.

(a) Trapezoidal approximation of the response function

(b) Approximating trapezoids

(c) Symbol description for trapezoid

where

Xi(co)=Ke [Xijo:)].

Let us assume the response fimction to be that

shown in figure 3.2a and let it be approximated

by the straight-line segments shown in figure 3.3a. For convenience figure 3.3a is repeated as figure

3.4a. It can be seen that the curve xit) can be

approximated piecewise by several trapezoidal

boundaries as shown in figure 3.4a. Therefore

x{t)^

i = l (3.28)

where the x*i (t) denotes the ith trapezoidal boundary segment of n boundaries shown in figure 3.4b. Substituting eq (3.28) into eq (3.4) yields

^i(w)=l] f

x*i{t) cos wtdt. (3.29)

! = l Jo

Thus, the frequency response is the sum of iute37
grals of the form

/• 00 Xi{w)i=j x*i{t) cos ut dt (3.30)

where, with reference to figure 3.4c,

' a;o«= const 0<f<(^j— r<) ti+Ti-t

2t,

Lo

(3.31)

Substituting eq (3.31) into eq (3.30) yields

Xi{u)— I Xoi cos ut dt

Jo

dt

\ cos wt dt- r- COS dt-^ r- t cos (jit dt.

Integrating gives

A''i(co)i=— sin H ^ sin ut

0 ZTio;

ti+Ti

t sin ut cos co^

Xpi J 2r< sin ui{ti-Ti) + {ti+Ti)

X [sin w(<i+r<)] — (ii+Ti) sin aj(«i-7<) — (<(+Ti) sin c<;(«(+rt) + (fi— T<)

or

X [sin 03{ti—Ti)] [cos u{ti+Ti)

03—COS u{ti—Ti)]

Xi{w)i=—-~-2 [cos w(ti + Ti)—COS u{ti—Ti)]

}

(3.32)

where

cos w(^i+r<)— cos o}{ti—Ti) =—2 sin uti sin un

Therefore, eq (3.32) becomes

^i(w)i=--% sin uti sin wtj

Ti03

or

/ N / * \ /sin 03ti\ /sin orA

A,(a))<=-(Xoi<<) ( 7— )

( )• \ 0}ti / \ WTi /

Therefore,

XT / \ / ^ s /sin cofA /sin ojTjX „„

j=i \ uti / \ a)T< /

The evaluation of eq (3.33) does not present any

difficulties, especially if tables of the function ^

are available. For tables of ?HL5 ^j^g reader

X X

is referred to the work by Solodovnikov [8], where

five-place tables appear in Appendix 1 for arguments of ranging from 0 to 10 in increments of 0.01

from 10 to 20 in increments of 0.1, and from 20 to 100 in increments of 1.

3.4. §HL? Approximation

Sin X In this section will be used to approximate

*c the response function x{t). Shannon [9] noted

that the spectrum for almost aU response functions

had an upper limit, say w^. Shannon also proved

that a function of this type could be synthesized

sin X

exactly by a sum of functions of the type

Synthesis of a response curve by

sm x

fvmctions is shown in figure 3.5. For piirposes of clarity only

sin X

portions of tlie —-— curves are shown. The

response function can be expressed as x*(0=S4.TM^(t2l), (3.34)

„=0 Ucit —riT)

where A„, the amplitude of sm X

is equal to the

ordinate of the response function at t=nT, and t

is the time spacing, which must be smaller than

or equal to tt/uc- Substituting eq (3.34) into eq (3.2) yields

CD /• CD

n=0 J — <B

sin Wcit — nr)

Ucit —nr)

e-^"' dt, (3.35)

where Xijw) is the complex frequency characteristic of the system. To eliminate the exponential form of the transform, we evaluate the

real part of eq (3.35). The Re [X{ju)] is

n=0 We J-c

sin Ucit—riT)

(t-nr)

cos wtdt. (3.36)

Hence, the frequency response is the sum of integrals of the form

sin (j3c{t—nT)

{t-nr) cos (j}t dt. (3.37)

38
X (t) t)

1

/

1

1 / L

r

\

\

\\—

/ \

'\

,r-\ \ -1

I

\

-\

\

\

1

I —(

\

/

I

I

\ /

\

1 —+r

\ 1

\

1

1 /

V

I i \(v—r-

^

Figure 3.5. sm X approximation of the response curve shown in figure 3.2. (Reproduced from [11] with pennission from Proc. IRE)

Evaluation of eq (3.37) involves a number of trigonometric manipulations. The substitution

of sin Wet cos TicocT —sin nwcT cos cocf=sin adt —ut)

into eq (3.37) gives

A, C" r(sin wj cos nwcT— sin nwcx cos Uct)

J — L

X [cos ujt] dt

An C f" si

\^ J-oo {t —nr)

J"

cos Wet cos wt dt

Tl ^ {t-nr)

{t-nr)

sin cos (Jit

]

dt

(3.38)

However, sin coscoi=^ sin (aj+coc)^— | sin

[{(>}— (j3c)]t and cos Uct cos wt=\ cos (o+coj^ cos [(w— coc)]t Therefore, eq (3.38) becomes

^i(w)»=^ "l^cosww.T

[^J

sin (w+wj^

(i— 711

sin (w —coj^ dt

X

-rJ-oD {t —nr

|~JCOS (co +wji

sm no>cT

^ p

cos (co-..)^ ^ny

(^ —Tlr) J_„ (^—nr) Jj

At this point let t —nT =z, thus dt=dz. The

limits of — 00 to 00 remain unchanged for this substitution. Also let w—Wc=a, and w+Wc II ^•

Then

/^^ r f sin /3(2+wr)(/2 Ai(w)n=2^ cosnojcrj

J

! ^—

f" sin a{z-\-nT)dz~\

~J-„ ^ J-8in7»a,.r

^l^J"

COS /3(2+7tr)(j2

^ J°°

COS a(2+nr)(j2

j

Expanding the sine and cosine functions in the

above equation for the sum and difference of two

angles, and factoring out the non-integrable

terms, Xi (w) „ becomes

Xi{u>)n=^^ cos nwcT l^cos n^T ^

sin j32 dz

+sin n^T

/» CO

•/ — e

-sm nar

cos /32 dz

-cos TJar

»y — CD

sin a2 (Z2

J°°

C0Sa2^"]_

-0=2 J

vyf o f" cos (82 C?2 XI cos 7i/3r J ^

r " cos q:2 dp +COS Tiar

sm nucT

—sin nj3r

sin TkxT

- C

sin /32 (^2

(3.39)

The improper integrals in eq (3.39) are now in a form such that known properties of improper

integrals may be introduced to aid in their evaluaCOS cx

tion. For example, integrands of the form —^—

where c is a constant, are odd functions which

have a Cauchy Principal Value of

cos cx dx

-0- (3.40)

sm cx

Also, integrands of the form are even functions and

X

sin CX

dx. (3.41)

The latter form of the improper integral is found

in standard integral tables [10], which gives

j;

sm cx X

TT—}

2

dx='' —TT

)

2

c>0

c=0

c<0. (3.42)

In eq (3.39) put x=z and c=a=^u —Uc, where

We is the upper limit on the frequency. Therefore,

39
a<CO and eq (3.41) yields 3.5. Number Series Transformation

sm az

(3.43)

Also, c =^=w-\-03c\ therefore (3>0 and eq (3.41)

gives

sin

dz=ir. (3.44)

Substituting eqs (3.40), (3.43), and (3.44) into

eq (3.39), eq (3.39) becomes

tA

Xi{w)n=Tr^ {cos 7lC0^r(cOS 7?-/3t+COS nar)

+sin mO(,T(sin n^T—sm nar) } . (3.45)

Reintroducing a =o) —Wc and j8=co +Wc in^to eq

(3.45), expanding the sine and cosine for the

sum and difference of two angles, multiplying

and removing common factors, eq (3.45) fuially

becomes

Xi{w)n=- cos nccT.

The frequency response for the system is

(3.46)

Xi(co) = = COS WCOT,

(^c ra=0 forr<— (3.47)

This equation relates the frequency response

to the measured points Aq, Ai, A2, ... of the

response curve. In laying out the response function in equal

intervals of r at which the ^'s are to be measured,

the first point (n=0) should not be more than one interval to the right of the first variation

in the curve, figure 3.6. Also, the intervals

should continue to the point where the curve

ceases to vary.

In a problem similar to the above, Samulon [11] has developed an expression which considers

measuring the first difference of the A's, An—

An-x rather than An. He also developed a number of tables and nomographs to facilitate numerical evaluation. The original tables and

nomographs are for 0.05 fi sec-intervals; however,

these can be scaled to fit most problems.

x(t)

Figure 3.6. Interval layout of response curve.

The number series transformation method was

first introduced by Tustin [12] in 1947. Since

that time Madwed [13] has been a principal contributor in this area. The fundamental ideabehind the number series transformation method

is the concept of representing or approximating a function /(O by a number series. For our purpose

fit) will hereafter be denoted as the response function x(t). Therefore, x{t) is the dependent variable, displacement, and t is the independent variable, time. Since the independent variable is time, the number series is called a time series. To

best present this method let us illustrate it in the

form of an example. Again let us take the re- sponse function, shown in figure 3.2a, and approximate it in turn by a triangular, rectangular, step,

and ramp time series transformation. The tri- angular time series transformation is presented in detail whereas only the results and pertinent details are given for the rectangular, step, and ramp time

series transformation.

»'<t)

(n-i)T (n*OT

« (t) I

(n-()T

x(t)

Figure 3.7. Triangular time series transformation.

(a) Unit triangular pulse function

(b) Triangular pulse function

(p) Triangular 40
a. Triangular Time Series Transformation

Any continuous arbitrary function xit) can be

approximated by a set of triangular pulse functions,

all of which have the same width, as shown in

figure 3.7c. When x{t) is approximated in this manner for analytical purposes, then by definition

this approximation for x{t) is called triangular time

series transformation. For symbol simplicity all time series transformations or approximations of

xif) will be denoted by x*{t). With reference to figure 3.7c, it is easily recognized that when a line

is drawn parallel to the ordinate at point A on

the abscissa, it cuts two adjacent pulses at B and

C. The distance AD is laid off equal to the sum

of AB and AC, and thus establishes a point on the

approximate curve x*{t). It can be shown that

repeating this .process for an infinite number of

points in a given time interval t results in a chord

approximation of x{t) in that interval. Therefore,

the sum of all the triangular pulse functions from

^=0 to i= 00 gives the chord approximation of x{t)ovx*{t).

Before developing an analytical expression for

x*{t), let us first define two basic quantities used

in the triangular time series transformation. The

first quantity to be defined is the unit triangular

pulse function rising at time t={n—\)T, shown in figure 3.7a, and denoted by Urit—nT). By

definition the area enclosed between the unit

triangular pulse function and the time axis is equal

to unity. Therefore, with a base dimension of 2r,

the altitude of the unit triangular pulse function

must be equal to 1/r. The properties of the unit

triangular pulse function then become

0

t-{n-l)T

1T{n+l)T-t

Lo

for K(w-1)t

for {71— l)T<t<nT

for t—TiT

for nT<t< (n+l)^"

for ty{n+l)r.

(3.48)

The second quantity to be defined is the general

triangular pulse function, shown in figure 3.7b,

and denoted by T. By definition the triangular

pulse function is equal to the area enclosed

between the triangular pulse function and the

time axis, multiplied by the unit triangular pulse

function. Therefore, the nth triangular pulse

function at x{nT) is T{nT) = Tx{nr)UT{t-nT). (3.49)

It is now possible to write an analytical expression

for the triangular time series transformation, since

the response function can be approximated by

summing the triangular pulse function, eq (3.49),

from n=0 to n= 00 . Therefore

x*{t) = TY^x{nT)UT{t-nT),

«=0

(3.50)

where Urit—nT) is given by eq (3.48). Having

once establislied tlie approximation to tlio response

function x*{l), tiiis approxiinatioii can tiien be

substituted into the unilateral Fourier transform

eq (1.20), thus achieving the desired transformation of the response function from the lime to the

I'requency domain.

b. Rectangular Time Series Transformation

A unit rectangular pulse function, denoted by

Unit —riT), is shown in figure 3.8a. Since, by

definition, the area enclosed by the unit rectangular pulse function and the time axis must be equal

to unity, it can be seen that the height of the

rectangle must be 1/t. Therefore, the properties

UR(t-nT)

R(nT)

x(t)

T (n-i)T nr (n+i)T t

FiGUBE 3.8. Rectangular time series transformation.

(a) Unit rectangular pulse function

(b) Rectangular pulse function

(c) Rectangular pulse function approximation of the response function

694-211 0-63—

4

41
of the unit rectangular pulse function are

' 0 for t<nT

UR{t—nT) = ^ 1/t fornr<i<(w+l)r

,0 for ^>('/i+l)r. (3.51)

The general rectangular pulse function, shown

in figure 3.8b, is denoted by R and is defined as the area under the pulse function multiplied by the

unit pulse function. That is.

R{nT) = Tx{nT)URit —nr). (3.52)

Since the ordinate xinr) is not uniquely determined

for the rectangular pulse function approximation,

and since rxinr) is the area of the nth rectangular

pulse, an approximation to the area under the curve x(t) in the interval between ut and {n-\- 1)t is

Tx{nT)=A{nT)

(re+l)T x{t)dt, (3.53)

x(t) URoU-riT)

1(0

0 (n-i)T nr t

x(nT-T) /

1 1 » 1

x(t)

? T (n-i)T nr (n+i)T

Figure 3.9. Ramp time series transformation.

(a) Unit ramp function

(b) Ramp function

(c) Ramp function approximation of the response function

where Ainr) is the area under the nth rectangular

pulse, then xinr) is defined to be

X(7lr)=^^^«- xit)dt. (3.54)

J TIT Therefore, eq. (3.52) becomes

R(nT)=AinT)URit-nT) (3.55)

and the rectangular time series transformation is given by

x*{t)=j: A(nT)UR(t-nT),

n=0

(3.56)

where Unif—nT) and^(nr) are given, respectively,

by eqs (3.51) and (3.53). The rectangular pulse

function approximation is shown in figure 3.8c.

c. Ramp Time Series Transformation

Figure 3.9a shows a unit ramp fvmction which is denoted by Uitp{t—nT). By definition the height

of a unit ramp function is equal to unity. Therefore, the unit ramp function properties become

TJRp{t—nT)= ->

0

t—in-l)T

Li

for K(«-l)r

for {n—l)T<t<nT

iort>nT. (3.57)

The general ramp function, denoted by Bp and

shown in figure 3.9b, can be defined as foUows:

Rpinr) = {xinr) —x{nT— t) ]URp{t—nT). (3.58)

Therefore, the ramp time series transformation is expressed as

^

x*(i)=S {x{nT)-x{nT-T)]URj,{t-nT) (3.59)

where Uupit—riT) is defined by eq (3.57).

d. Step Time Series Transformation

The unit step function, shown in figure 3.10a, is denoted by Usit—nr) and has the following properties TT /, N r 0 for t<nT Us{t-nr) =

^^ fori>nr. (3.60)

The general step function is denoted by S and is shown in figure 3.10b. The expression for the nth

step function is S{nT)=[x{nT)-x{nT-r)]Usit-nr). (3.61)

3 From the definition of a unit triangular pulse function, it follows that a unit ramp function for ra = Ocan be expressedasasummation of the triangular pulse function, that Is

CD URp(t)=r Y. UT(t-nr).

n-O

Also the triangular pulse and ramp function times series transformation

results in a straight-line approximation of the original function.

42
Again the ordinates x{nT) and x(nr— r) are not

uniquely determined for the step function approximation. However, let us define them, with refer- ence to figure S.fOc, in a manner analogous to the

evaluation of the ordinate x{nT) for the rectangular

pulse function approximation (i.e., xinr), defined

by eq (3.54), repeated here for clarity):

x(wr)=^^^=^- x{t)dt (3.54)

and

x(nr— r) =

Ajnr) _l

r T Jn

A{nT—T) 1 J'nr

(.n-l)T x(t)dt. (3.62)

Therefore, eq (3.61) becomes

S(nT)=- {A{nT)-A{nT-T)]Us{t-nT). (3.63) T Thus the step-time series transformation can be

expressed as

*

a;*(0=-Z; {A{nT)-A(nT-T)}Us{t-nr) (3.64)

T n=0 where A(nT) and Usit—riT) are defined by eqs

(3.53) and (3.60), respectively.

3,6. Pseudo-Rectangular Pulse Approximation

Let us consider the unilateral Fourier integral

as given by eq (1.20) and rewritten in a notation

that complies with the response, i.e.,

X(ico)=J^° x(Oe-^"' dt, (3.65)

where X(ju) denotes the complex response spec- trum of the response function x(t). Bowersox and Carlson [14] applied the fundamental theorem of calculus to eq (3.65) and

obtained

n = l (3.66)

where d=wAt. Let us now consider a response

function x{t) which reaches a steady-state condition, such as the response shown in figure 3.2a. Then eq (3.66) becomes

X{jui)=MSj2 x{n)e-^'''+Mx{N) S e'

(3.67)

* From the definition of a unit rectangular pulse function, it follows that a

unit step function for n=0 can be expressed as a summation of the rectangular

pulse function, that is

00 Usit)=T 2 Uait-UT).

n-O

Also, that the rectangular pulse and step function time series transformation

results in a step curve approximation of the original function.

where x(N) is the amplitude of the response function at time t—NAt, the time for which the re- sponse function first reaches the steady-state condition. The summation part of the last term in eq (3.67) is recognized as the geometric series and

can be written as

n=N+]

Therefore, eq (3.67) becomes

(3.68)

Schweppe [15] defined Xijw) to be

X(jo:) = iX+jY)e~ht (3.69)

for which the amplitude of the response spectrum

x(t)

x(t)

s(nT)

x(nT-T)

.z_

x(t) A(nD

x(t)^

/

//

r - -

(n-OT nT (n»i)T

Figure 3.10. -S<ep iimt series transformation.

fa) Unit step function

(b) Step function

(c) Step function approximation of the response function

43
is given by TV X=X) a;(ri) cos nd

n=l

x(N)

X(cc)=modXijw) =-^X'+Y' (3.70)

and its phase angle by

a(co)=arg X(jw)=arc tan (3-71)

Rewriting eq (3.68) to include the definition of

X{ju>), as given by eq (3.69), gives

_jl c N

(X+jY)e ^At=At < Sa;(w)e-^"''

+^M^y (3.72,

Equation (3.72) can be separated into real and

imaginary parts [14]

N

F=—2 x{n) sin nd71=1 x(N)

(^sin Nd cot 1+ cos Nd^

(3.73)

(cos Nd cot |—sin A^6'^-

(3.74)

Therefore, given a response function x{t) which

reaches a steady-state condition, the amplitude

of the response spectrum is given by eq (3.70),

and its phase by eq (3.71), where X and Y are given respectively by eqs (3.73) and (3.74), and

e=coAt. The fundamsntal theorem of calculus is based on approximating the area under a curve by ele- mental rectangular areas. The error involved in

such an approximation is directly related to the

width of the elemental area, as, in this case, the

time increment. A better approximation to the

problem could be found in the adoption of the

trapezoidal approximation or Simpson's rule [16].

4. Instrumental Aids

4.1. Henderson's Mechanical Harmonic

Analyzer

The need to remove the tedium from numerical

analysis is ever present. Henderson [17] developed the mechanical harmonic analyzer shown in

figure 3.11 to fill this need. The analyzer is synthesized on the fact that any periodic function

J{t) can be represented by a Fourier series of the

form given by eq (1.6), in which the coefficients are determined by the integral forms of eqs (1.7) and

(1.8). Evaluation of these integrals for a given

periodic function /(i) is known as harmonic analysis, and is the sole job of the analyzer. The

analyzer shown schematically in figure 3.12

essentially does just that. That is, on tracking a curve over one complete period, the displacement

of the resolver plate gives a vector that represents

the amplitude and phase of the derivative of a

harmonic present in the curve. The fact that

the instrument gives the derivative of the sinusoidal components, rather than the components

themselves, is no disadvantage when analyzing

periodic functions, as will be shown later. With reference to figure 3.12, a brief description

of the operation of the analyzer is as follows: the curve to be analyzed is placed on table a, which

is moved from right to left either by a motor or by the handwheel labeled "traverse." As the

table moves it drives a variable-speed gear, b,

Figure 3.11. Henderson's mechanical harmonic analyzer.

(Reproduced from [17] with permission from Engineering)

Figure 3.12. Schematic of Henderson's mechanical

harmonic analyzer.

(Reproduced from [17] with permission from Engineering)

44
which causes the friction wheel, c, to rotate about

its vertical axis, perpendicular to the resolver

plate, d. The friction wheel makes contact with,

and drives, the resolver plate, which is a flat plate

free to move in the horizontal plane, but unable

to rotate; its movement is recorded by a rigidly

attached stylus. The friction wheel is driven

about its horizontal axis by a flexible wire coupled

to the cursor, which is operated manually by the

handwheel labeled "ordinate." The rotary motion of the wheel is proportional to the change in

the ordinate of the c\u"ve.

Before the start of analysis, the analyzer must

first be initialized, that is, the friction wheel is alined with d—Q, as indicated in figure 3.12, when

the cursor is at the origin of the curve. Then the

variable-speed gear is adjusted to give n (the

harmonic of the curve to be determined) revolutions to the friction wheel as the cursor is moved

through one period. The analyzer is now ready

to determine the nth harmonic of the curve. Let us consider the motion of the resolver plate

when the curve has been moved an amount 6, and then incremented by dd. The corresponding

change in the ordinate is dh, hence, the friction

wheel, which is at an angle of nd to its original

direction, will drive the resolver plate a dh in the

direction of nd. The constant a is denoted as the

amplification factor. Then the displacements in the X and y directions become

and

dx=a ^ sin nd dd

dd

dy=a ^ cos nd dd.

If the curve is tracked continuously throughout one period, the final coordinates of the locus

traced by the resolver plate are given by

and

'-dh

dd

C'-dh

sin nd dd

cos nd dd

(3.75)

(3.76)

dh^d^

dd dd

where ^=-^ [/(<?)]• Substituting ut=d into eq

(1.6), ^ becomes

y=—naa„ j sin nd cos nddd-\-nab„ I cos' riSdd

Jo Jo

for which it can be readily shown tliat

sin^ nd dd= cos^ nd dd= tt

r2ir f

sin^ nd dd=

Jo Jo

Jo

Therefore,

sin nd cos nd dd=0.

x= —Trnaa„

y^irnabn

or

irna

y

irna

(3.78)

(3.79)

Thus, the resultant vector of the locus traced by

the resolver plate has rectangular components

that are proportional to the Fourier coefficients a„ and 6„. Therefore, the resultant vector repre- sents the amplitude and phase of the nth harmonic

in the derivative of the tracked ctu-ve. As mentioned earlier, the analyzer gives the

derivatives of the sinusoidal components rather

than the components themselves. Let us now

look at this more closely. To aid us in this in- terpretation let us recognize that taking the derivative of a sinusoidal quantity results in another

sinusoidal quantity n times larger and 90 degrees

in advance of the first. That is 4i(cos nd)

clu = —nsmnd=ncos^d-{-^- Thus, by interchanging the designation of the reference axes and by dividing the projections of the resultant vector of the traced locus by the harmonic order,

the true components present in the tracked curve

are obtained. The equations for the coefficients a„ and bn, eqs (3.78) and (3.79), respectively, must

be rewritten to comply with this change as follows:

and

dn--

bny

(3.80)

(3.81)

Oi sm d—2a2 sm 26—. . .-na^ sm nd

+bi cos 0+262 cos 2d-\-. . .+r26„ cos nd. (3.77)

Substituting eq (3.77) into eqs (3.75) and (3.76)

gives

'•J.

x=—naan I mi'nBdd+nabn ^ sm nd cos nddd

Let us further illustrate this point by an example.

Figure 3.13a shows the waveform to be analyzed

and figure 3.13b shows the resiilts of the analysis

for the first two harmonics. The projections of the resultant vector, shown as a dotted line ex- tending from point 0 to A in figure 3.13b, on the

X and y axes corresponding to the components

present in the derivative cui've. Thus, by eqs

(3.78) and (3.79) the fundamental terms in the

derivative of the waveform are O.39sin0 and

45
0.714 cos 6. The analytical results for the fundamental terms of this same waveform are O.7sin0

and 0.405 cos d. Comparison of the fundamental

terms shows that the coefficients should be interchanged for agreement. Therefore, eqs (3.80)

and (3.81) will yield the correct results, since

they were written purposely to incorporate this

interchange.

The analyzer can also be used to obtain an

approximate evaluation of Fourier integrals which

have the form

J*

x{t) cos nwt dt, where x{t) is an

aperiodic convergent function, since the displacement of the resolver plate represents the value of

the integral, point by point, as the integration

proceeds. It can also be shown, from the locus

traced by the resolver plate after a sufficiently long

period of integration, that the remaining portion

of the integral is of negligible magnitude. Thus,

the frequency response can be determined by

analysis of the response function.

This analyzer can determine harmonics up to the 20th in curves of 27 in. period and 4 in. peakto-peak amplitude. Analysis can be made on

curves with smaller periods, however, at a proportional loss of harmonic range.

0

Figure 3.13. Analysis of the 1st and 2d harmonic of a tri- angular wave by Henderson's mechanical harmonic

analyzer.

(a) Triangular wave

(b) Solution locus for triangular wave

(Reproduced from [17] with permission from Engineering)

4.2. Montgomery's Optical Harmonic

Analyzer

Instead of evaluating the coefficients of the

Fourier series for a periodic function by a mechanical method, Montgomery [18] utilizes the principles

of optics to accomplish this feat. This method

requires that the function to be analyzed be

represented on a photographic film by one of two

methods: (1) the variable area record, shown in

figure 3.14a where the part between the curve and

the t-axis is transparent, the remainder is opaque;

and (2) the variable density record, shown in figure 3.14b. With either type of representation

of a function x{t), the amount of light transmitted

through a narrow vertical strip of width dt is proportional to x{t)dt. If two or more such

records are superimposed, the light transmitted

through all of them will be proportional to the

product of the recorded functions, provided

that only one of the records is of the variable area type.

To determine the Fourier coefficients a„ and bn, proceed as follows: consider the two records shown

in figure 3.14, where xit) is a variable area record

and cosnwt is a variable density record. Here

again the only requirement is that both shoiild

not be variable area records. Therefore, by superimposing one film on the other, the amount of

light transmitted through both of them between

the limits zero to 2t is J

x(t)cosn(jotdt, and

hence proportional to a^. The coefficient 6„ is obtained in the same manner, i.e., by replacing

the cos not film by a sinncoi film. This can be

accomplished very simply by moving the cos nwt

in

b) VARIABLE DENSITY RECORD Figure 3.14. Representation of x(t) and cos nwt on film. {Reproduced from [18] with permission from the Bell System Technical

Journal.) 46
film one-quarter wavelength along the axis. Also,

if the cosine film is moved a whole wavelength

along the axis, the transmitted light will go

through a maximum. This maximum value is proportional to c„ where Cn='^(in-\-hn, and the

position of the cosine along the axis at which it occurs is the phase angle ^„=arc tan —

Since x{t) and cos nw^, in general, will have both

negative and positive values they cannot be

directly represented by the transmission of light.

Therefore, to eliminate this difficulty a constant

is added to each function which results in a constant in the measured amplitude.

To illustrate this analytically, let us consider

the optical transmission of the film on which x{t)

is recorded to be expressed as

A-^x{t) (3.82)

where is a constant large enough to make the

expression positive for all values of t. Similarly,

the transmission of the cosine film is 5J1+M„ cos (3.83)

where 6 denotes the position of the cosine film

along the i-axis, M„ is a constant less than unity,

known as the modulation of the record, and 5„ is a constant which is the average optical trans- mission of the film. Therefore, the total transmitted light when these

two films are superimposed on each other is given

by

T=JJ"5J^+x(0][l+M„ cos (nwt-9)]dt

--AB„ dt+AB„Mn cos {nwt—e)dt

Jo Jo

+Bn x(t)dt+B„M„ x{t) cos {nut-e)dt.

Jo Jo

(3.84)

Integrating each term in eq (3.84) except the

last integral yields

T=2TAB^+Bng{t) r+5„M„

/•2ir J.

'x(t) cos {nut—e)dt, (3.85)

where g{t) = J'x(t)dt. The last integral can be

evaluated by a sequence of mathematical manipulations and previously derived expressions. Let us

first rewrite the integral in the form

j;

x{t) cos[{no3t—<p„) — {e—<pj]dt.

Expanding the cosine function for the difference

of two angles gives

COS {d—(p„)j x{t) cos {no}t—<pn)dt-{-?>\n {6—iPn)

xj^J* x{t) sin (nw^—^„)(/iJ.

Again expanding the cosine term in the integrand

yields

cos {d—(pj j^cos <PnJ x(t) cosnoit (Z^+sin (p,^ X

J*

x{t) sin nwt ^

J+sin (d—(p„) |^cos (p„

X-^J* x{t) sinnut dt"^ — sinipn^ x{t) cosnut dt~^

The integrals are now recognized as a„ and 6„, eqs (1.7) and (1.8) respectively, the coefficients of

the Fourier series for a function x{t) whose period

is 27r. Therefore,

TT cos {e —(Pn)[an COS <Pn +b„ siu (p„] +ir sin {e—'Pn)[bn COS o„ sin (p„]. From eq (1.11) it can be shown that cos v?„ =

ajc„ and sin ip„=b„lcn; therefore

x{t) cos {nwt—6)dt=TvCn cos {6—<Pn)

and eq (3.85) becomes

r=27r5„(^+C)+x5„M„c„cos (e-^J (3.86)

where C=9(t)

2tt

; c„ is the resultant of the Fourier

series coefficient a„ and 6„, defined by eq (1.10);

and (p„ is the phase angle defined by eq (1.11).

To obtain a„, take the difference in T for 5 =0

and TT ; therefore

T{Qi)-T{-^)

(3.87)

Similarly, to obtain 6„, take the difference in T

for 0=7r/2 and 37r/2; thus

hn'- 27r5„M„

As previously discussed, the maximum value of T occurs at 0 =<p„ thus establishing ip„. This can

also be seen by referring to eq (3. 86). The

minimum Tocciu-s at 0 =<p„ +7r; thus the difl'erence between maximum and minimum values of T

gives

-^max

1*mlg 27rB„Mn

(3.SS)

47
Figure 3.15. Montgomery's optical harmonic analyzer.

(Reproduced from [18] with permission from the Bell System Technical

Journal)

LIGHT ^—^ I CELL SOURCE A B Figure 3.16. Schematic of optical system in optical har- monic analyzer.

(Reproduced from [18] with permission from the Bell System Technical

Journal)

By making 5„ and. M„ approximately constant for

all screens, the coefficients for either form of the

Fourier series are directly proportional to the

change in transmitted light for specified pairs of

position of the cosine film. The process which the analyzer is required to carry out consists of superimposing the function

to be analyzed on a cosine film and measuring the

variation in the transmitted light when the cosine

film is moved along the ^-axis. This is repeated

with a different cosine film for each harmonic to be measured. Such an analyzer is shown in

figure 3.15 and schematic of its optical system in

figure 3.16. With reference to figure 3.16, the

film containing x{t) is placed in a holder at slot A and illuminated by passing light from an in- candescent lamp through a condenser lens. An

enlarged image of x{t) is formed at 5 on a window

bounded by two knife edges 0.750 in. apart.

The portion of x{t) to be analyzed must be adjusted

so as to just fill the window. The cosine films

slide in a track directly behind the window, and

receive the image of x{t). The transmitted light

is collected by another lens and brought to a

photocell. The photocell output is coupled di- rectly to an oscillograph which records the variations in the form of a strip chart record. Each

succeeding cosine film is brought into play and

passed across the window at B automatically, by

a series of cams and levers. The analysis is fully automatic except for adjustments at the

beginning of each analysis.

The analyzer shown in figure 3.15 takes recordsof x{t) between Me and in. in length and no

higher than its length, and is capable of analyzing

the first 30 harmonics.

4*3. Photoelectric Fourier Transformer

The photoelectric Foiu-ier transformer is an

optical analyzer which produces the Fourier

transform of a given function automatically and

instantaneously in the form of a graph on the screen of an oscilloscope. Furth and Pringle [19] based their analyzer on a principle similar to that

used by Montgomery [18], but extended it for continuous Fourier analysis, i.e., Fourier integration. The analysis is based on the light trans- mitted through a closely spaced fringe pattern

rotating in front of a narrow slit, beyond which is the mask with the pattern to be analyzed. The

light reaching a photocell can be expressed by an

equation similar to eq (3.84), or T=AB^ Cdt+AB^Mn f

cos {nwt-e)dt

+B, rx{t)dt+BnM„ Cxit) cos inut-e)dt, (3.89)

J a J a where the only difference is in the range of integration. Let us consider only one standard fringe

pattern, say cos ut; therefore n=l and integration of eq (3.89) gives

T=AB,Mig{o^)+B,MM<^) +C (3.90)

g{o>)=- [sin (a;6-0)-sin (wa-0)] (3.91)

CO

hiw)= r x{t) cos {wt-e)dt (3.92)

J a C^AB,{h-a)+B,{\{t)dt. (3.93)

If (0 is now made to extend and shrink periodically

in time between co =0 and an upper limit w^, the

spacing of the standard fringe pattern along the

(-axis (defined by X =27r/w), wiU vary periodically in time between X,=2ir/co, and infinity. Thus, the

time dependence of the transmitted Ught will be re- presented by eq (3.90) with the Umits of minus co, and

plus CO,. If ^=0 (the given function is positive

everywhere) or if b=—a and d=±ir/2, then eq

(3.90) represents the Fourier transform of x{t).

The constant C in eq (3.90) is considered insignificant [19]. The transmitted Ught which is directly proportional to the Fourier transform of x{t) is now con- centrated on a photoelectric cell. The photoelectric cell produces a current proportional to the intensity of light received, and therefore pro48
Figure 3.17. Schematic of the photoelectric Fourier transformer.

(Reproduced from [19] with permission from The Philosophical Magazine)

(a) optical and recording system

(b) Plate drive and generator system portional to the Fourier transform of x{t). The

current is then amplified and fed into an oscillo- scope where it is made to deflect the beam in the

vertical direction. At the same time a harmonic

time base which is synchronized to the variation

of CO is fed into the oscilloscope to drive the beam

in the horizontal direction proportional to w. A schematic of the analyzer is shown in figure

3.17a. Light from an incandescent lamp D is passed through a diffuser E and a condenser lens F. The light is then passed through a small

glass plate A, bearing a photographic pattern

consisting of 20 fringes having a harmonic variation in optical transmission. The plate is mounted

in a journal bearing B, so that it can be rotated

with a constant angular velocity about the axis of the optical system. The light then passes through

lens G which produces a reduced image on a fine adjustable vertical slit at H. The slit at H is arranged to make the axis of rotation pass exactly

through the center of the slit. In this way a symmetrical distribution of light intensity is obtained along the slit as plate A is rotated. The

width of the slit must be small compared to X^. The variation in the spacing of the light distribution, as plate A is rotated at constant angular

veloctiy w, is given by

(3.94)

-1-0 -l-l -7-«-5 -4-3-?-l A X 5 4 5 6 7 B 9 HI

1-0 «-«-7-«-5-< S-M 1 I S * S « 7 • f l«

sm (lit

Figure 3.18. Typical photoelectric Fourier transformer

results with associated analytical solutions.

(Reproduced from [19] with permission from The Philosophical Magazine.)

(a) Photoelectric Fourier transformer result

(b) Analytical solution

49
Figure 3.19. Electronic and magnetic ana

Lederer and Smith.

jzer used by

Therefore, the angular frequency w varies in time

according to

27r 2x . co=—=— sm coi A he or w=Wc sin Zit. (3.95)

From the slit at H, the light passes through a

cylindrical lens / (with its longitudinal axis of

symmetry perpendicular to the slit). The cylindrical lens rotates the vertical image at the slit H

into an enlarged horizontal image at J. The image

at J consists of a 3X3 in. fringe pattern which

contracts and expands in time according to eqs

(3.83) and (3.95). At J the record of the function

to be analyzed is introduced, i.e., x{t). The optical

arrangement is completed by the condensing lens

K and the photoelectric cell L.

The remaining pieces of equipment shown in

figm"e 3.17a are the milliammeter A^, the potentiometer M, amplifier 0 and Z, oscilloscope P, and

a generator. The motor Q rotates plate A and

indirectly rotates plate S of the generator. The

generator consists of core T, magnets V, and iron

pole pieces W. The purpose of the generator is to produce a voltage which is synchronized with

the angular rotation of plate A to drive the beam

of the oscilloscope in the horizontal direction.

Typical results from this analyzer are shown in

figure 3.18a. In figure 3.18b are the analytical

solutions worked out for the fimction analyzed.

The analytical solutions are scaled to the identical

scale used by the analyzer; therefore a direct

comparison can be made.

4.4. Electronic Analyzer With Magnetic

Transient Storage

Lederer and Smith of the National Bureau of

Standards use a magnetic drum recorder to store the transient response of the pressm-e transducer

and to repetitively play it back to a commercially

available electronic analyzer, figure 3.19. The

resonating frequencies may be viewed on the

screen of the analyzer as shown in figure 3.20. The recorder has a variable recording time of 1 to 6 msec and a frequency range of 1 to 100 kc/s.

The analyzer will detect any resonance of amplitude not less than 1 percent of that of the cali- brated resonance nearest in frequency through a frequency range of 1 to 100 kc/s, provided that

the resonant frequencies are separated by at least

1 kc/s. In general, the greater the frequency

separation, the lower the amplitude ratio at which

resonances can be detected. At frequency separations of less than 1 kc/s resonance can be detected

if the amplitude is high enough.

Figure 3.20. Frequency analysis from analyzer used by Lederer and Smith.

50
Information from this system is limited to the

amphtude characteristic. No information regarding phase is obtained. Use of the information is based on the assumption that the transducer sys- tem has a sufficiently low damping constant to

make possible construction of the frequency re- sponse curve. Thus it is possible to determine to what frequency the transducer can be used with

acceptable fidelity.

5. References

[1] E. A. Guillemin, Computational techniques which

simplify the correlation between steady-state and

transient response of filters and other networks,

Proc. Nat. Electron. Conf., 1953, 9, 513-532 (1954).

[2] A. Sommerfeld, Partial Differential Equations of Physics (in German), W. Klemm (Wiesbaden,

1947).

[3] J. B. Scarborough, Numerical Mathematical Analysis,

The Johns Hopkins Press (Baltimore, 1955).

[4] J. N. MacDuff and J. R. Curreri, Vibration Control,

McGraw-HiU Book Co., Inc. (New York, N.Y.,

1958).

[5] T. R. Running, Empirical Formulas, John Wiley &

Sons, Inc. (New York, N.Y., 1917).

[6] L. W. Pollak, Multiplication Table for Harmonic

Analysis (in German), J. A. Barth (Leipzig, 1926).

[7] L. A. Pipes, Applied Alathematics for Engineers and

Physicists, McGraw-Hill Book Co., Inc. (New

York, N.Y., 1958).

[8] V. V. Solodovnikov, Introduction to the Statistical Dynamics of Automatic Control Systems, Dover

Publ., Inc. (New York, N.Y., I960)."

[9] C. Shannon, Communications in the presence of

noise, Proc. IRE 37, 10-21 (Jan. 1949).

[10] B. O. Peirce and R. M. Foster, A Short Table of

Integrals, Ginn and Co. (Boston, ^lass., 1956).

[11] H. A. Samulon, Spectrum analysis of transient re- sponse curves, Proc. IRE 39, No. 2, 175-186 (Feb.

1951).

[12] A. Tustin, A method of analyzing the behavior of

linear systems in terms of time series, J.I.E.E., Proc. Conv. Autom. Regulation and Servomechanisms, London, 94, Part II-A, No. 1, p. 130 (Mav

1947).

[13] A. Madwed, Numerical analysis by the number series transformation method, Ch. 27, Instrument Engineering, Vol. II, by C. S. Draper, W. McKay, and

S. Lees, McGraw-Hill Book Co., Inc. (New" York,

N.Y., 1953).

[14] R. B. Bowersox and J. Carlson, Digital-computer cal- culation of transducer frequency response from its response to a step function. Progress Report No.

20-331, Jet Propulsion Laboratory (July 26, 1957).

[15] J. L. Schweppe, Calibration of pressure transducers with aperiodic input-function generators, Paper

46.1.62 presented at the 17th Ann. ISA Instr.- Automation Conf. (Oct. 1962).

[16] R. W. Hamming, Numerical Methods for Scientists and Engineers, McGraw-Hill Book Co., Inc. (New

York, N.Y., 1962).

[17] J. G. Henderson, Mechanical harmonic analyzer and

some applications to servo systems, Eng. 173, 52-

53, 68-70 (Jan. 1952).

[18] H. C. Montgomery, An optical harmonic analyzer,

Bell System Tech. J. 17, 406-415 (July 1938); U.S. patent 2098326.

[19] R. Furth and R. W. Pringle, A photoelectric Fourier

transformer, Phil. Mag. [7], 37, No. 264, 1-13 (Jan.

1946).

[20] G. F. Floyd, Method of approximating the transient response from the frequency response, in Principles

of Servo Mechanism hy G. S. Brown and D. P. Campbell, Ch. 11, John Wiley & Sons, Inc. (New

York, N.Y., 1948).

[21] L. C. Ludbrook, Step to frequency response trans-

. forms for linear servo systems. Electron. Eng. 27, No. 311, 27-30 (Jan. 19"54). [22] F. H. Raven, Automatic Control Engineering,

McGraw-Hill Book Co., Inc. (New York, N.Y.

,

1961).

[23] A. V. Bedford and G. L. Fredenall, Analysis, synthesis and evaluation of the transient response of

television apparatus, Proc. IRE 30, No. 10, 440-

457 (Oct. 1942).

[24] R. Bowersox, Calibration of high frequency response

pressure transducers, ISA J. 5, No. 11, 9S-103

(Nov. 1958).

51

4. Analysis of Nonlinear Transducers

D. Muster

1. General

Physical nonlinear systems can be discussed only

in terms of mathematical models that are characterized by differential equations amenable to some

form of solution over restricted regions of the

motion parameter(s), or for specific mathematical

descriptions of the damping and/or restoring force terms. In addition, little can be done by analysis

to predict a response function for systems which

cannot be represented by one- or possibly twodegree-of-freedom models.

For damped linear systems, if Xi(f) is a solution

of mx-\-cx-\-kx=Fi{t), and Xiit) is a solution of mx-\-cx-\-kx=F2{t), then x=Xx-]rX2 is a solution of mx +cx-\-kx=Fi{t)^F2{i). This fundamental fact, a direct consequence of the linearity of the differ- ential equation, is called the principle oj superposition. It is important to note explicitly that

the principle does not hold for nonlinear differential equations. It follows as well that the concepts

of free and forced oscillation, classical normal-mode

analysis and resonance, all intimately related to the principle of superposition, have real meaning

only for linear systems. As a consequence of these limiting conditions, any form of analysis con- ducted to evaluate specific characteristics of nonlinear systems is restricted stringently. The

investigator is confronted immediately with the

decision: which optimum mathematical model,

among those amenable to useful solution, wUl provide an adequate characterization of the nonlinear

system's behavior?

The decision on an optimum mathematical

model raises several specific questions. Can the

system be adeq[uately described by a single-degreeof-freedom system? Is the nonlinearity of the

damping (restoring) force sufficiently small that

it can be represented adequately by viscous damping (a linear spring)? Is the region of interest

sufficiently narrow that a linear approximation

wUl be adequate? Can a solution be obtained by

considering the system characteristics as being

piece-wise linear (say, bilinear)? In summary,

every attempt is made to represent the system by

linear damping and restoring force characteristics

or, if this is not possible, to minimize the magnitude and subsequent effect of the nonlinearity.

Fortunately, the response of most pressure

transducers (as well as other electromechanical

sensing devices) can be represented adequately by

solutions of linear differential equations. Here

' Professor of Mechanical Engineering, The University of Houston.

we will discuss some of the methods of analysis for those transducer systems with sufficiently great

nonlinear restoring force or damping characteristics, that linear analysis is not adequate, but not

so great that solution is a practical impossibility.

We wUl restrict the discussion to nonlinearities of the type and relative magnitude likely to be

encountered in pressure transducer systems.

In this chapter, we discuss first the physical and

analytical aspects of nonlinearity. In the main,

this discussion will focus on the physical parameters of damping and stiffness and the manner of

their representation in the anatysis of transducer

systems. The remainder of the text is concerned

with analytical methods as they are applied to nonlinear systems. Particular attention is drawn

to those methods and solutions of systems where

the nonlinearity is confined to the restoring force

term onl}' and, later, to those where it is confined

to the damping term only. Among the methods

of analysis, the describing-function method, the

bilinear approximation and the phase-plane

method are discussed in some detail. At the end of the chapter, there is a combined

bibliography and reference list. The literature sources cited in the chapter text are listed first: uncited sources used for background information

and general reference in preparing the chapter are

listed last. Among the general references are

several to which we draw the reader's special

attention: the translations of the books by

Andronov and Chaikin [1],^ Minorsky [2J, and

Kryloff and Bogoliuboff [3], the vibrations text

by"^ Jacobsen and Ayre [4], that on noidinear

analysis by Cunningham [5], and those concerned

with the design of control-systems by Truxal [6] and Murphy [7]. In addition, there is a review

by Ku [8] of recent advances in analytical methods

for studying nonlinear control problems. The

extensive bibliography in the Ku paper is oriented

primarily towards the theory of nonlinear control

systems; however, man^^ references to general

analytical methods are included. Analysts in- terested in other nonlinear sj-stems wUl find them

useful. Obviously, the adequacy of the results obtamed

with any method of analysis is intimately related

to the values assigned to the damping and stiffness parameters in the governing differential equation.

There are no standard accepted procedures for determining these values. Usually, the stiffness

2 Figures in brackets indicate the literature references on p. 66.

53
of a mechanical system is determined from a

static test and the assumption is made that

dynamic effects are either negligible or behave in

accordance with a known relationship. The

magnitude and form of the damping characteristic are usually assumed to be those associated with

either viscous or coulomb damping. For a damped, one-degree-of-freedom system, the differ- ential equation for free vibration can be integrated

completely in only two cases; that is, when the

damping force is proportional to the first power of the velocity, or is a constant value with a sign the same as that of the velocity—the coulomb friction

type. In case the damping force is proportional

to the square of the velocity, a first integral can be

obtained and from this a relationship between

two successive maximum displacements can be

established [9]. The inverse problem, that of

determining the damping parameter from a set of observed values, has been solved in general

terms only for systems in which viscous or coulomb

damping occur separately or jointly [10]. The

case of combined viscous-coulomb damping has

been studied by Den Hartog [11] and applied to the frequency response of vibration instruments

[12].For more general damping laws, when the

damping force is proportional to either any power

of the velocity or sums of such terms (or powers

of the displacement or displacement amplitudes),

neither the direct problem of finding an expression

for the attenuation nor the inverse problem of

determining the damping law from observed values

has been solved in general terms. The reason for

this has been mentioned earlier; the governing

nonlinear differential equations are such that they

do not admit of closed form solutions. However,

if the damping forces are sufficiently weak, an

approximate method due to Kryloff and Bogoliuboff [3] leads to a closed-form description of the

attenuation of free damped vibrations for a variety

of damping laws. Klotter [13] has shown that

from the expressions for the attenuation it is possible to deduce, from a set of observed values, the

damping law which governs the behavior of a system. In [13], damping laws are deduced for cases where the damping forces are proportional

to a single power of the velocity, displacement or displacement amplitude, or to the sum of two such

terms. In summary, the relationships that govern tha

stiffness and damping parameters can be deduced

from observations for a limited class of cases

(usually, for one-degree-of-freedom systems characterized by relatively weak damping forces).

For other cases, the assigned values are obtained

through trial and error or through intuition.

2. Physical Aspects of Nonlinearity

One of the most desirable characteristics of a transducer is for it to respond in a linear manner

over its useful dynamic range. In general, trans- ducers are designed so as not to exhibit appreciable

nonlinearity; for example, in the case of static

calibration, there is no significant departure from

linearity between input and output functions, or, under dynamic conditions, the transducer does not

Figure 4.1. Hardening, softening, and linear springs.

exhibit such phenomena as jumps and sub- or super-harmonic resonances. In the presence of

significant nonlinearities in damping or stiffness,

the principle of superposition does not hold, with

the consequences enumerated earlier. In figure 4.1, softening (sub-linear) and hardening (supra-linear) stiffness ciu"ves are superposed on that for a linear spring. The nonlinear character of these curves can often be approximated

by the addition of a cubic term, that is, the restor- ing force of a transducer may be given by

k(x±^^7?), (4.1)

where the relative magnitude of the nonlinearity

is given by the coefficient ^ (with dimension

L~'^) and the hardening or softening influence of

the cubic term is indicated by the plus or minus

sign, respectively. Consider the motion of an undamped transducer with sinusoidal forcing and

a hardening stiffness characteristic given by eq

(4.1) . If the transducer system can be represented

by a one-degree-of-freedom idealization, the equation characterizing its motion is mx+^(a;±i8V)= F cos o^t, (4.2)

the weU-known Duffing equation [4]. The steadystate response of a system governed by eq (4.2) is shown in figure 4.2 for a hardening stiffness characteristic and in figure 4.3 for a softening

characteristic. After [4], the response is given

in terms of the dimensionless excitation parameter

54
LC=o

Figure 4.2. Response curves for Duffing-equation model

with hardening spring, no damping (jump phenomena)

.

(Reproduced from [4] with permission from McGraw-Hill Book Co., Inc.)

C-O.l

Figure 4.4. Same as for figure 4-^, but with viscous damping

(f = 0.i) ; no jump for S =OJ.

(Reproduced from [4] with permission from McGraw-Hill Book Co., Inc.)

t»0

Figure 4.3. Response curves for Duffing-equation model

with softening spring, no damping (jump phenomena)

.

2 «S^u,n2 3

Figure 4.5. Same as for figure 4-3, but with viscous damping

(f = 0.7); jump phenomena exists for S =0.1.

(Reproduced from [4] with permission from McGraw-HlU Book Co., Inc.) (Reproduced from [4] with permission from McGraw-HiU Book Co., Inc.)

S (where S=(-^l3F/k)). ^he ordinate is the di- mensionless parameter VSMiSi where x denotes an approximate solution of Buffing's equation

obtained by, say, the Ritz averaging method [4]; the abscissa is w^/co„^ (where co„^=k/m) . The cross- hatched area in each figure indicates the areas of unstable motion and the vertical lines represent

the jump phenomenon, in which the displacement

changes suddenly from a lower to a higher value

with no change in applied force.

The quantitative effect of hardening stiffness (that is, systems in which )3->0) is to bend the

S=0 curve of figure 4.2 toward larger values of

co/w„, whUe for softening-stiffness s.vstems (fig. 4.3), for which |8^<0, the S=0 curve bends toward

values of w/a)„<l. The greater the value of |)3-|, the more marked the bend in the response curves. The S=0 curves referred to above represent the

free vibration case of the Duffing-equation model.

It can be shown [4] that the natural frequency and

55
the displacement amplitude are related by

Wd=w„Vl±i^. (4.3)

The response functions of actual physical systems

exhibiting hardening or softening stiffness characteristics will vary in detail from those of figure 4.2 and 4.3 but not in general appearance. The

tendency for the response to lean towards larger

values of aj/w„ is associated with a hardening

characteristic, the tendency to bend towards values

of w/w„<l with a softening characteristic. The

existence of regions of instability (jump phenomena) are characteristic of the presence of either

hardening or softening.

The effect of nonlinear damping on the response

of a mechanical system is more subtle. Whereas

the stiffness characteristic determines the basic

shape of the response function (that is, no bend

for the linear case and bent response curves for

the nonlinear cases), a nonlinear damping characteristic only modifies the shape and extent of the

domains of instability (if they exist) and makes

the resonance amplitudes finite. In systems with

linear restoring force and with damping proportional to, say, the nth power of the velocity, it can be shown [14] that the steady-state response can be approximated closely by assuming that the

actual damping term can be replaced by an equivalent viscous term. The criterion of equivalence

3. Methods of Analysis

By comparing the response curves of the nonlinear cases above with those for the linear cases discussed in chapter 2, it can be shown [4] that

nonlinear behavior makes it difficult (if not practically impossible) to evaluate the dynamic characteristics of some transducers. For transducers

with relatively weak nonlinearities, the response

function can be approximated by that for a linear

system with characteristics which are first-order

approximations of those of the actual system.

An alternate approach is to assume that the

behavior of a system is linear in the neighborhood

of an operating point or between two points on

the curves describing the system parameters [15]. In the remainder of this chapter, the nonlinearities of stiffness and damping are assumed to be

sufficiently great that it is necessary to recognize

the system as being nonlinear. Within this framework, we will discuss the methods available for the analysis of such systems as they actually are. We will divide these methods into three major

categories

1. Perturbation, iteration, and variational methods.

2. Linearization methods (describing functions).

3. Phase-plane methods (including the method

of isoclines and other graphical methods).

The mathematical methods of the first category

are most useful in determining the approximate

steady-state response of nonlinear systems and,

in many cases, cannot be used to determine tranis that of equal dissipative work per quarter cycle

of the motion of the system.

As an example of the influence of damping upon

response curves, let us consider a viscously damped

Duffing-equation model. The governing differential equation of motion is F

x-{-2^WnXi-Wn\x±l3h^)=— cos ait, (4.4)

where the damping ratio is f =c/c<. =c/2mw„ and c is the

damping coefficient. Response curves analogous

to those of figure 4.2 and 4.3 are shown in figures

4.4 and 4.5, respectively. It can be seen that the

damping has made the resonance amplitudes

finite and that the maximum amplitudes of the

system are but very slightly larger than the

response amplitudes. The two domains of stable

motion in figures 4.2 and 4.3 are seen to merge in the damped response plot of figures 4.4 and 4.5,

since the backbone of the plot (the S=0 curve)

is no longer a boundary. As before, the unstable

domain is bounded by the locus of the vertical

tangents to the families of constant-excitation S

curves. In the case of damping proportional to the nth

power of the velocity, the general configuration of

the plots in figures 4.4 and 4.5 remains the same,

but the crossings of the resonance-point loci with

the *S'=0 curves change quantitatively.

(Nonlinear Systems)

sient response. For the latter reason, and because

of.the sometimes formidable algebraic computation

associated with their use, they enjoy, perhaps, the

least popularity among analysts concerned with

calibration problems. As computers and analogto-digital conversion devices become more readily

available, mathematical computation procedures

will probably find wider acceptance.

The perturbation method was developed by

Poincare [16] and Lindstedt [17], primarily for

application to astronomical problems. Somewhat

later, Ritz [18] and then Galerkin [19] developed

equivalent approximate methods for solving steadystate responses of linear as well as nonlinear problems. Duncan [20, 21] and others used the method

to treat problems concerning the statics and dynamics of elastic bodies. Prior to the early 1950's,

when Klotter called attention to the interdependence and equivalence of the two forms in which

the method was known, it had been used only

infrequently to solve dynamical problems (other

than eigenvalue problems). Since that time

(usuallv under the name of the Ritz averaging

method), it has been used extensively [4, 22, 23,

24, 25]. In particular, the method can be applied

to an oscillatory system described by differential

equations of the type

E„(g„, q„, q„, t)=0, (4.5)

for example, eqs (4.2) and (4.4).

56
The exact solution would yield the coordinates

g„(<) characterizing the response of the system at any time t . In the absence of a theory or method

for obtaining the exact solution of eq (4.5) and

hence of (4.2), (4.4), . . ., the Ritz averaging

method wUl yield an approximate steady-state

solution by the following procediu'e:

1. Each of the coordinates g„ is replaced by an assumed approximate solution of the form

:

Sn(0=2:^..*'z(0

(n=l, 2, 3, . . .), (i = l, 2, 3, . . .) (4.6)

where the A^i are a set of unknown coefficients

and the functions tptit) are assumed functions

expected to yield the best average solution over

a period.

2. Then the assiuned solutions are introduced

into eq (4.5) and integrations of the following

type performed over the period indicated

:

E„{q„, g„, g„, t)<pi{t)dt=0

(71=1,2, 3,...),(i =l, 2,3, ...). (4.7)

A set of {mn) algebraic equations is obtained

from which the coefficients A^i can be determined.

Examples in which the method is applied appear

in the literature and in later sections of this

chapter.

As might be expected, the usefulness of per- turbation, iterative, and variational procedures

has been enhanced by the development of elec- tronic computers and programing techniques.

Bowersox and Carlson [26] and Hylkema and

Bowersox [27] have investigated the digitalcomputer calculation of transducer frequency

response from its response to a step function.

The recent development of electronic devices

which can convert the analog output of a trans- ducer to a digital form acceptable to a computer

suggests new possibilities for more effective use

of mathematical computing procedures. At

present, however, there is no simply applied,

effective method of analysis available to all engineers, rather than only to specialists with

access to extensive electronic computer facilities. Among the linearization methods of the second

category, we have selected the "describingfunction" method [28] as, perhaps, the most

practical method for application to determine the

response of nonlinear transducers. The approximation involved in the method is similar to that

used earlier by Kjyloff and Bogoliuboff [3]. Independent development of essentially the same method as Kryloff-Bogoliuboff was made by

Goldfarb [29], Tustin [30], and Oppelt [31]. A second method discussed later is that due to Ergin [32]. He has showm that for even substantial nonlinearities in restoring force, a bilinear

approximation can be used with satisfactory

results. The third category covers phase-plane methods.

For a general development of the method (including the method of isoclines) and for applications of it to linear systems, we refer the reader to section

4 of chapter 2. Here we will restrict the discussion to nonlinear systems. Transient disturbances

as well as free vibrations of such systems can be

treated hy phase-plane methods, but the steadystate forced vibrations of any system are not

readih' dealt with by these methods unless the

periodic excitation functions are of the square- wave type.

4. The Concept and Application of the Describing- Function IVIethod [6, 28]

In linear systems the concept of frequency

response, in the form of a (direct or inverse)

transfer function, is used to describe the dynamic

behavior of transducers. The concept of such

a transfer function is not valid for transducer

systems with substantial nonlinearity. Kochenburger [28] has proposed a nonlinear counterpart

to the transfer function, a so-called "describing

function" based on the frequency response. The

method has been used extensively and extended

in the recent past, primarily bv Klotter [33, 34,

35], West [36], Grief [37], Johnson [38], Tou and

Schultheiss [39], and Truxal [6]. The describing-function method of analysis is based on three assumptions [28]:

1. There is only one nonlinear element in the

system.

2. The output of the nonlinear element depends

only on the present value and past history

of the input. No time-varj'ing characteristics are included in the nonUnear element.

3. If the input of the nonlinear element is a sinusoidal function, only the fundamental

component of the element output is considered.

The implications of these assumptions are dis- cussed in more detail b\- Truxal [6]. In order to demonstrate the use of the method,

let us consider a transducer system consisting of a mass m, a coulomb damper (that is, a dry

friction device), and a diaphragm of negligible

stiffness. The governing differential equation for the sinusoidaUy forced motion of the system is [5]

mx+^,=F cos i^t (4.8)

where ip is the magnitude of the force needed to overcome the static friction of the damper. The

sign of this force is the same as the sign of the

velocity.

694-211 0-63 —

5

57
The exact steady-state solution of eq (4.8) can be

determined by a piecewise linear solution of two

equations, with the applicable equation depending

upon the sign of x. The equations, with their

regions of applicability, are x>0: /3<oji<7r+/3

x=— I cos ut—-^ ) m\ r /

x<0:7r+|8<«<<27r+/3

x=|(cosa,f+|,)

^[sina,«+|,(co<-^-/3)] (4.9b)

where sin ^= —k(pI2F. A plot of the variation of

force and velocity is shown in figure 4.6, where the

arbitrary choice has been made, cp/F=1/t, so that

/3=— 30°. The velocity vanishes at values of wt determined by the phase angle 13. The latter is negative and tends to zero for values of <p, which

are negligibly small with respect to F. The upper

limit of real values for /3 is <p/F=2/Tr. If W^>2/7r,

the velocity does not vary smoothly through zero, but so-called dead bands exist in which there is no

motion. In fact, by examining eq (4.9a), it can

be seen that, if tp/F'^l, no steady-state motion is possible.

The fundamental component of the velocity

expressed by eq (4.9a) can be found by Fourier

analysis. The amplitudes of the sine and cosine

terms of the fundamental will be found to be,

respectively,

iT+fe sin (S) mo \

ai=— cosA (4.10) mw\7r /

FORCE / *

/la. / / 2 /

Figure 4.6. Variations in force and velocity for system gov- erned by m'x + <px/\ X I = F cos cot(*?/F= 1/n, ,3 = - 30°)

.'

(Reproduced from [5] with permission from McGraw-Hill Book Co., Inc.)

The describing function we seek is defined as the

ratio

where

Thus,

^-F

V=ai cos o}t-\-bi sin wt.

(4.11)

1%*'''^ ^ '-^+(l+^sin sin 0,^]

(4.12)

where

sin /3= —

2F

The amplitude of the describing function is

\H\-.

with phase

ZH=—9,vc tan

arc tan [^(^^+sin ^ sec isj (4.13b)

In theory, the relations given in eqs (4.13) are valid for <pIF< 2/ir. In practice, a describing function is meaningful only so long as the variations

are reasonably close approximations of simple

harmonic functions. This requirement limits ipJF to small values. In table 4.1, there is a comparison

between velocity values obtained by the exact

analysis of eq (4.9a) and the describing function

of eq (4.12). As in figure 4.6, the ratio of friction

force to driving force is chosen to be (p/F=l/Tr,

from which it follows that j8=— 30°.

Table 4.1. Comparison of exact solution to describing function

wt xmuir/F Hmuir

0 1. 05 1. 09

tr/6 2. 10 2. 21

7r/4 2. 49 2. 54

7r/3 2. 71 2. 71

ir/2 2. 62 2. 51

27r/3 1. 66 1. 62

3ir/4 0. 92 1. 00

57r/6 0 0

The values of mo:\H\ and ZH from eqs (4.13) are plotted in figure 4.7. It can be shown that similar

describing functions can be obtained by the Ritz

averaging method [5] and also by assuming that

the coulomb friction force can be replaced by an

equivalent damping force [5]. In summary, the describing-function concept as developed by Kochenburger [28] embodies the

idea that, in a nonlinear element, a steady-state

sinusoidal input will produce a periodic but nonsinusoidal output. It is assumed that adequate

58
mw|H|

0.1 0.2 0.3 0.4 0.5 0.6

1.0

0.5

0

-30

-60

-90

(DEG) Figure 4.7. Magnitude and phase of describing-function for system governed by eq {A. 8).

(Reproduced from [5] with permission from McGraw-Hill Book Co., Inc.)

results can be obtained by using only the fundamental component of this output.

Klotter [33] called attention to the fact that

the cases which had been treated (until 1957) were those where the relationship between input

and output can be given by a nonlinear expression of the variables themselves as contrasted to other cases in which the nonlinear element is characterized by differential relationships between

the variables. In the literature, cases can be cited

where the nonlinearity is caused by dead bands

[40, 41], saturation [40], limiting [42], linkages [43], contactors [28], and coulomb friction [44]. (For

the last situation see the illustrative problem dis- cussed above, eq (4.8) et seq.) In [33] Klotter

presents a method by which describing functions

for nonlinear elements whose behavior is described

by nonlinear differential equations can be obtained

without solving the differential equations. The

fundamentals of the method are explained in [22] and [24]; here we will focus our efforts on demonstrating its use. Let

E=M{z)-¥y{t)=Q (4.14)

denote the nonlinear differential equation of the

element imder consideration with y(t)=F cos wt standing for the input and z for the unknown output. After [22], we will treat the second-order

differential equation

£=2+2ra;,^(i) + a)„y(2)-co„2ircos cot =0, (4.15)

which contains two nonlinear terms g{z) and/(2).

For simpUcity of presentation, let us assume that

g{z) and/(2) are odd functions of their respective

arguments.^ As an example, we will choose the

parameters of eq (4.15) to be those of a Duffingequation model with viscous damping (see eq (4.4) et seq.), that is, 9{z) =z

J{z) =z+0'^ (4.16)

£:=2-}-2fco„iW(3+/3V)-co„2iPcos «e=0. (4.17)

We seek an approximation z to the output function z{t) of the form (assumption 3)

z=Z cos (w< —e). (4.18)

with the parameters Z and e chosen in such a way

that they represent "best" values.

Because the function z{t) of eq (4.18) cannot

satisfy the differential equation of eq (4.17) at every instant, we will satisfy it in some weighted

average. In [22] and [24], it is shown that appropriate weighting functions are cos w< and sin ut. By the Ritz averaging method then, the integrals

X

E[z{<t)] cos a dff=0

E[2{a)] sin ff d<T=0

(4.19)

and

' In [22] the treatment is outlined also for the case of even functions.

From these two equations, we wiU determine the

parameters Z and e in the approximating function

of eq (4.18). The steps for accomplishing this are:Step 1. From the two functions /( 2) and g{z),

eq (4.16), we derive two new functions 4>{Z) and

r(Z, w) by performing the following integrations:

4 1 f''^ 0(Z)=- \ j{Z cos a) cos <r da

or equivalently V (4.20a)

4 1 T'^^ =- I /(Z sin a) sin a da

TT ZJo J

4 11 C'^ T{Z, 0))= ^ g{Zo} sin <r) sin a da

or equivalently =

4 11 f''^

I g{Z(j} cos a) cos a da

TT W„ ZJo J

(4.20b)

Equations (4.20) are equivalent to the following

procedure: Introduce Z cos ut and Zw sin wt for

z and i, respectively; replace the functions/ and g

of these arguments by functions of multiple arguments (2a)/ Scof, . . .) and retain the fundamental

terms only.

Step 2. Using the functions <t> and r of eqs

(4.20), we obtain the amplitude Z and the phase

e from the two equations (which foUow from eq

59
(4.19))

and

and

2f

tan € (4.23b)

tan c= 2fr

-(0

(4.21b)

In terms of the parameters of eq (4.17), the

functions 0 and T are

14-|/32Z=^-(^)

Equations (4.23) give both the modulus

F

|A7"|=^ and the argument e of the "inverse equivalent transfer function" * -(0 (4.24)

CO r=—

(4.22)

by which eqs (4.21) become

This (complex) transfer function can be plotted

as a family of curves, either with I3Z, as a parameter of the family and w/w^ varying along the

individual curves, or vice versa. In summary, an equivalent transfer function

(describing function) for the output of a viscously

damped, Dviffing-equation model has been derived.

The expression for it is given by eq (4.24) and

F

the modulus |A7^|=^ and argument e by eqs (4.23).

5. Bilinear Approximation Method for Determining the Transient Response of a

Nonlinear System [32]

A line-segment approximation method by which

the transient response of a nonlinear transducer can be determined has been developed by Ergin

[32]. In particular, the method is applicable to systems with a nonlinear restoring force characteristic. Ergin shows that for many problems

involving even large nonlinearities, two line seg- ments are sufficient to yield an adequate approximate solution. At least for the examples

chosen by Ergin, the bilinear approximation gives more accurate results than the classical perturbation methods of Poincare [16], Linstedt [17],

and Kryloff and Bogoliuboff [3], and it is faster

than graphical and numerical methods. It may

also be adapted to an iterative procedure, should

it be desirable to improve the accuracy of th^

method.

There are other allied problems in which the

effect of a nonlinear restoring-force function is represented by linear or piecewise linear functions

of the displacement; for example, simple piecewise linear problems are considered by Hansen

and Chenea [45], and others are discussed in some

detail by Flugge-Lotz [46]. More recently, Mahalingham [47] developed a one-term approximate

solution for the amplitudes of a single-degree-offreedom system with nonlinear (nonsymmetrical)

spring characteristics. The method of [47] is similar to that of Martienssen [48], but the

construction uses a modified frequency function

in place of the actual spring characteristic, the

function being so chosen that it yields the correct frequency of free vibration. A system with a tri- linear restoring-force function (that is, one which

is piecewise linear by joining successively an initial linear section, a softening section, and a hardening

section) has been studied by Dost and Atkinson

[49] by means of an electronic differential analyzer.

In addition, we include the following literature

references to similar problems: Atkinson and

Heflinger [50], Katz [51], Evaldson, Ayre and

Jacobsen [52], Ludeke [53], Wylie [54], Jacobsen

and Jespersen [55], Rauscher [56], and Burgess

[57].In most of the cases cited above, a solution is obtained for systems in which a nonlinear re- storing-force function is piecewise linear, either

bi- or trUinear. It is worth noting explicitly that

Ergin [32] solved a somewhat different problem;

namely, that of

(a) approximating the nonlinear restoring-force

function of a system by a bUinear approximation and then

(b) finding a solution such that the meansquare error between the actual nonlinear

restoring-force function and the assumed

bilinear approximation is a minimum.

It is a well-known principle that any function can be approximated by a number of straight-line

segments. Each line can be determined by its slope and a point through which it passes; this joint can be picked to be the transition point

where two segments meet. The slopes and transition points can be determined so that the meansquare error between the function and the series of line segments is minimized within a range of variables. The application of this concept to the

solution of nonlinear differential equations is a

direct one. Any nonlinearity, which is a function

* In an attempt to emphasize the analogy to the corresponding linear concept, Klotter [33] chooses to replace the term "describing function" by

"equivalent transfer function" with qualifying adjectives, such as inverse,

direct, etc., as the case may be. For comparison, see eq (4.11).

60
of the independent variable only, can be described by a number of line segments chosen so as

to approximate the nonlinear function as closely

as possible. The nonlinear problem, then, re- duces to the solution of the same number of linear

equations as there are line segments, with the

proper matching of displacement and velocity at the transition points. The determination of the

optimum slopes and transition point locations for

line segments requires the solution of (2w— 1) nonlinear simultaneous algebraic equations.

There are certain simplifications which make

this method of linear approximations feasible. Most functions encountered in practice can be approximated satisfactorily by only two line seg- ments. In addition, it is possible to choose the

slope of the first line segment as the specified slope

of the function at the origin without significant

loss of accuracy. This simplifies the problem to the determination of the slope of the second line segment and the location of the transition point;

the choice of both being such that the mean square

error is minimized.

In [32] it is shown that, for any nonlinearity

which can be expressed as a power of the independent variable, the second slope and the location

of the transition point are simple functions of the

maximum displacement amplitude. It is also shown that for small nonlinearities, the solution in many cases is insensitive to the precise choice of the maximum deflection used to calculate the

parameters of the bilinear approximation.

In what follows we will state the basic relations

upon which the method is based and apply it to the undamped Duffing-equation model of eq (4.2).

Consider the original nonlinear differential

equation which governs the motion of a transducer

system to be of the form [32]

(4.25)

We assume that eq (4.25) can be approximated

by the two linear equations:

x+gi{x)=j{t) for |x|<x,

x+g2{x)=j{t) for \x\yxt

(4.26)

(4.27)

where gi(x) and g2{x) are linear functions of x, and Xi is the "transition amplitude" where the

linear restoring-force changes from gi{x) to g^ix).

As mentioned earlier, gi{x) is taken equal to

kix, where ki is the slope of the nonlinear restoringforce function at the origin. Then,

gi{x)=kix

g2{x)=k2X-\r{k1— ^2)a;«}''

(4.28)

where ^2 and are as yet undetermined constants- The form of gi{x) and g2{x) require that the

piecewise-linear approximation is continuous at x^x^.

The parameters ^2 and x« are to be determined

such that the mean square error {EY between

the nonlinear restoring-force function and the

bilinear approximation of it is minimized;

{EY=^i [(H{x)-g{x)]'dx Xm \^J{)

+£"'[g2{x)-g{x)Ydxy (4.29)

where gi{x) and ^2(x) are defined by eq (4.28);

g{x)=ki{x)-\-h(x) ; and x^ is the maximum dis- placement amplitude of the motion.

The conditions that (Ey has an extremum

are [58] ^(^''-0

(4.30)

and

dx.

bk.

=0. (4.31)

Since gi{x) and g{x) are not functions of x,, and

9iixt)=g2{xt), then eq (4.30) reduces to

Jx, f

[g2(x) —gix)]dx=0.

Similarly, eq (4.31) reduces to fJx, [g2{x)-g{x)]^^dx=^0

but

C)g2(x)

d^2

dk2

—X Xt,

(4.32)

(4.33)

which when used in eq (4.33) and recalling (4.32)

yields

fJx,

[g2{x)—g(x)]x dx=0. (4.34)

By appropriate substitution of ^2(x) and </(x), eqs (4.32) and (4.34) can be simplified to

ki—kx-

{Xm —X )dx (4.35)

and

k2—ki =

(Xm —X()^(2x„ +Xi) Jx,

' h(x)x dx, (4.36)

from which the desired values of x, and k2 can be

found.

For the case where h{x) is of the form

and

/i(x)=€X" (n>l)

Xt —OiXmt

(4.37)

(4.38)

where a is a constant of proportionahty, the

simultaneous solution of eqs (4.35) and (4.36) is

61
relatively easy.^ We see that they then become,

respectively,

and

^2—^1=

2ex^-Hl-a"^'')

" (l-a)^(n+l)

(l-a)^(a+2)(7i+2)

(4.39)

(4.40)

Solving for the constant a in the latter two

equations, we obtain

(1+2=

3(n+l)(l-a"+'')

(n+2)(l-a"+')

Since a<l and w>l then 7- -j-^-

(4.41) yields ^ n-\

n+2

(4.41)

1 and eq

(4.42)

where n is the power of the nonlinear restoringforce term (4.37). In turn, with this value of a, the slope of the second line segment can be computed from eq (4.39) and from eq (4.38).

Let us apply [32] the results of the analysis

above to the transient response of an undamped

Duffing-equation model to the half-sine pulse

excitation of the form

i+w„^(a;+/3V)=sin co^ for 0<Kr (4.43)

=0 for tyr (4.44)

where r, the pulse length, is ir/co. For t=ir/w, the

solutions of eqs (4.43) and (4.44) must be equal.

Since an exact solution of homogeneous eq (4.44)

exists [59], our main interest lies in finding an

approximation to eq (4.43) such that the motion

during the piilse can be obtained.

For convenience in computing the results, let us

assvune the initial conditions are at ^=0:

and

x(0)=i;(0)=0

)=2

Then, eqs (4.43) and (4.44) become

,3_ ,

sin 2t for 0<K?

0 for t>l- (4.45)

For a cubic nonlinearity, the transition amplitude

X( is, from eqs (4.38) and (4.42)

Xt=OAx^ (4.46)

and the second slope ^2, k2=h + l.ZbxJ=l + \.ZbxJ. (4.47)

The parameter Xm is computed as the maximum

« Note that the Duffing-equation type of restoring-force function, eq (4.1) is Included.

amplitude of the linear solution of eq (4.45) ^; hence

x-|-x=sin2i,

the solution of which for the same initial conditions is x=|(2 sin sin 2t)

and, in the range 0<^<7r/2, x„,=0.667. Thus,

X, =0.267 and A:2=1.60. (4.48)

The two bilinear equations to be solved are

xi+xi=sin 2t; Xi(0)=Xi(0)=0 for |x|<x, (4.49)

X2+I.6X2— 0.16=sin 2t; X2(f,)=Xi(i,); X2{tt)=Xx{tt)

(4.50)

where tt is the transition time, the time which it takes Xi to reach x«. The solution of eq (4.49) is Xi=\{2 sin i—sin 2t),

from which the transition parameters are Xi(i,)=x,=0.267 '

1.011 (radians) ^- (4.51)

x(^,)=0.646

The solution of eq (4.50) which satisfies the conditions X2(^,) =0.267 and X2(^) =0.646 is

X2=0.1+0.582 sin (Vl^i -0.084) -^^ sin 2t.

(4.52)

The displacement and velocity at the end of the

pulse (t—ir/2) are, respectively

X2 (^0=0.651 and X2 (^^=0.592.

Comparable values obtained by niunerical itera- tion to four-place accuracy are

0.6518 and x (|)=0.6001,

that is, the approximate results obtained by solving the two bilinear eqs (4.49) are within 0.5

percent in displacement and 1.5 percent in velocity

of a numerical solution of eq (4.45) when the

nonlinear force at the maximum deflection point

is about 60 percent of that for the linear model.

In summary, a method is shown for approximating a nonlinear restoring-force function by a

particular bilinear approximation for which the

slope and location of the transition point have

been selected so that the mean square error between the original function and the approximation

is minimized.

^ In [32] this is shown to be a reasonable assumption.

62
6. The Phase-Plane Method [4, 5]

A general exposition of the phase-plane method

(including the method of isoclines) is given in section 4 of chapter 2. Here we will restrict the

discussion to three central points:

(a) the method of isoclines applied to a nonlinear system,

(b) the phase-plane-delta method as applied to a nonlinear system, and

(c) the determination of displacement-time

plots from phase portraits.

As mentioned earlier, a dynamical system with n degrees of freedom depends on n positional

coordinates q^t. The state of the system at any

instant of time depends upon the values of qt, and

the velocities qi. Thus, a phase-space of 2n

dimensions is associated with the behavior of the

dynamical system. To each state of the system

there corresponds the point Q with coordinates

qi) in the phase space; as t varies, the point Q

describes a curve called a path or trajectory, which

describes the history of the system. The totality

of aU paths is called the phase portrait of the

system. It represents all the possible histories

of the system, any one of which is determined

by specifying a single state ^o- In a geometric

sense, this means there is one and only one path

through each point of the phase space.

6.1. Method of Isoclines [1]

It is upon this last concept that the method of

isoclines is based. In chapter 2, the method is applied to linear systems. Here we will recall only the essentials of the method and apply it to a nonlinear system.

Only certain equations are amenable to solution

by the isocline method, namely, those which can

be reduced to a single first-order equation. An

example is x+/(i, x)=0. (4.53)

Equations (4.2), (4.4), (4.8), (4.43), and (4.45) in

homogeneous form are all examples of eq (4.53), an autonomous equation.^ It may be reduced to a first-order equation by introducing the new

variable v=x; then,

.._ . _(dv\ /dx\_vdv

\dx/\dtj dx'

and eq (4.53) becomes

This is a first-order differential equation in the

independent variable x and hence can be treated

by the isocline method. It is convenient in some

cases to normalize the velocity coordinate, that is,

' An autonomous equation is one in which the independent variable (here time t) appears In the derivatives only.

to specify the coordinates of the phase plane as

(x, v), where v=x/w„.

As an application of the isocline method to the

solution of a nonlinear diflFerential equation, let us

consider [59] a system whose motion is governed

by the van der Pol equation

x-€(l-x2)x-|-x=0, (4.55)

the equation for an oscillatory system with variable

damping. In our illustrative ^ problem, let £ =0.1. In eq (4.55), let us introduce the variable v=x,

then

dv__ {l—x^)v—x

dx V The algebraic equation for an isocline curve is then

X

^ e(l—x^) —

m'

where m is a specified value of dv/dx, the slope of a solution curve (phase path). Isoclines found

from this relationship with indicated values of m

are plotted in figure 4.8. Because the equation is nonlinear, the isoclines are curves and not straight

lines as in figure 2.4 of chapter 2. An infinity of

isoclines come together at the origin, which is a

singularity." For e=0.1, the isoclines have a rather simple geometrical shape, suggestive of figure 2.4. Line segments with appropriate slope

are located along the isoclines of figure 4.8. For a

precise construction of the phase trajectories, more

isoclines and line segments are needed than are shown here.

A most interesting property of the solution for the van der Pol equation is that there is a particular closed solution curve that is ultimately

achieved, regardless of the initial conditions.

This closed curve represents a steady-state periodic

oscillation, determined only by the properties of

the equation itself, and independent of the way

the oscillation is started. It is a phenomenon

characteristic of oscillating systems with nonlinear

damping and cannot occur in a linear system. A

more intensive investigation of the phenomenon '° shows that, if the initial point is inside this closed

curve, the corresponding phase path (solution

curve) spirals outward. If the initial point is outside the curve, the phase path spirals inward. In

either case, ultimately the phase paths approach

this closed curve, called a limit cycle by Poincare

[60]. An important property of this limit cycle

for eq (4.55) is that the maximum magnitude of x

is always close to two units, regardless of the

value of

^ The relative magnitude of t has slgniflcant effect upon the configuration

of the phase paths which characterize the behavior of systems governed by

eq (4.55). For a study of the effect of varying e and the concept of limit cycles, see [9], p. 148 et seci., and [5], p. 35 et seq.

" For a system whose motion Is characterized by an autonomous equation

(such as equation (4.53)), the points at which both i and r vanish are called singular points. For a discussion of such points, see [1], p. 9.

i» For example, see [59], p. 148 et seq.

II See [59], figures 58 and 59; see [5], figures 3.4 and 3.5.

63
€=0.1

Figure 4.8. Phase path for equation x — e{l —x^) x-\-x = 0

for e = 0.1. The closed curve Cls the limit cycle. Isoclines are given for constant m

(Reproduced from [59] with permission from The Clarendon Press, Oxford)

6.2. Phase-Plane Delta Method [61]

In starting a solution by the isocline method,

the entire plane must be filled with line segments

which fix the slope of the phase paths. If only a

single such trajectory is sought, only a few of

these line segments are actually used. Considerable simplification would result if only information

related directly to the desired solution curve were

used. Jacobsen [61] has proposed just such a method. It is an extension and refinement of the

earlier work by Lamoen [62] and Fuchs [63]. In

[61], Jacobsen generalizes and consciously formulates the method as an efficient working tool. Let us assume that either inherently or by a

suitable division it is possible to reduce the equation governing the motion of a system to the form

x+H{x, X, t)=0 (4.56)

without specifying any particular kind of function

H. Then, after choosing an arbitrary constant k

(of dimension T~^), eq (4.56) may be written as x+k%(x, X, t)=0 (4.56a)

with h of the same dimension as x. If we introduce the phase-plane coordinates x and y=x/k, we

may replace eq (4.56a) by

x=ky

y = —kh

(4.57)

from which we may obtain the differential equation of the trajectories in the phase plane.

dx (4.58)

In the phase-plane, the tangent of the path through

P, figiu-e 4.9, will be perpendicular to line QP. A

graphical procedure, leading to an approximation

of the trajectory by a sequence of arclike seg- ments, can be based on this geometric relationship.

It is apparent that if k (as used here) is the same

as the quantity tj„ (used in the chapter 2 discussion of the phase-plane), the coordinates y and v will coincide. The length h (fig. 4.9) may be

divided into the two components x and S, from

which it follows that

dv x+5

dx (4.59)

where v=x/w„ and d=8{v, x, t) =h(v, x, t) —x. Equation (4.59) is similar to eq (4.54) used in the dis- cussion of the isocline method. For 5 constant,

the variables of eq (4.59) are separable and can be

integrated to yield

v''+{x+by : constant =p^, (4.60)

the equation of a circle of radius p with its center

at (x, ») = (— 5, 0). Thus, for a suitably small

increment, the trajectory is the arc of a circle having these properties. In reality 5 is not a

constant. This implies that the true centers of

curvature of the path segments are located some- where on the line PQ (fig. 4.9), but not necessarily at Q.

P(X,V)

'2 Note the nonautonomous nature of eq (4.56) in contrast to eq (4.53) et seq. Figure 4.9. Graphical representation of segment of phase

path at P(x, v).

64
Let us apply the delta method to a Duffingequation model: let the motion of such a system

be governed by [61] x +25(l +0.1x2)x=0 (4.61)

with initial conditions at i=0: 0. Then,

putting the equation in the form of eq (4.56a),

where Wn=25, h—x-\-8=x-\-0.lj(^, and (in terms

of the phase-plane coordinates)

dv X +0.1x3

dx

In figure 4.10, a plot of the solution curve for eq

(4.61) is shown.

Construction of the approximation to the cm-ve

is shown on the figure. The initial point is x(0)=3, »(0)=0. As the first step, it is assumed

that X decreases to the value 2.8, or Ax*'' = —0.2, as shown. The average of x during this interval

is Xav'"=2.9, for which the value of 5 can be read

directly as 5av*"=2.4, approximately. Thus, the

center for the first circular arc is located at x=-2.4 and v=0. The radius is (3+2.4) =5.4,

and with this radius an arc is drawn to the point

where x=2.8, indicated in the figure by the small

dot marked x(0) + Ax'''. By continuing in this manner, the entire solution curve can be buUt up as a sequence of circular

arcs. Dots along the curve of figure 4.10 indicate

Figure 4.10. Phase path for x+25(l + 0.1x2)x = 0 constructed by delta method.

Dots on phase path indicate junction point on arcs (Reproduced from [5] with permission from McQraw-Hill Book Co., Inc.)

the junction points of successive arcs used in obtaining the phase path.

While the delta method is most immediately

applicable to equations with oscillatory solutions

(that is, closed-curve phase-paths), it can be

applied to other types of equations as well. Chu

and Abramson [71] have applied it to systems with

several degrees of freedom.

In summary, the phase-plane-delta method rele- gates all the terms of an ordinary second-order

differential equation into the type form

x +co„2(x+5)=0. (4.62)

Using the graphical procedure described above

and the parameters of eq (4.62), an approximation to the sought-for solution curve is plotted in terms of the phase-plane coordinates (x, v). The

accuracy of the graphical procedure depends

upon judicious choice of thi size of the x increments. Test cases in [61] involving elliptic

integrals, Bessel and Neumann functions, show

good accm-acy for increments of practicable

size. For reasons of acciu^acy, the method is not

recommended for solutions involving long durations; for example, as in Mathieu-type equations.

In the case of forced-motion systems, eq (4.56)

may be written as mx+H{x, X, t)=j{t).

Rearrangement into the type-form (eq (4.62))

yields

from which

i +a;„='[x+5-^/(o]=0,

5e«=5-^[./(i)].

If the external driving term x*{t) is a forced

motion of the "ground" end of the spring element

of a spring-mass system, then eq (4.56) may be

written as mx+/:[x-x*(0+5]=0.

If the relative displacement of the mass is denoted

by y~x—x*{t), then we have

from which

y+o^n'\y+^+^2 x*(o]=o

5eM =5+— x*(<).

Solutions for systems of these types are treated

in [61] for several step-function excitations.

6.3. Displacement-Time Plots From Phase

Paths "

Phase-plane trajectories involve time implicitly

but it can be determined explicitly and the dis13 See [59], chapter 7.

, j • >« A graphical integration and also a geometrical method are given m

[6, p. 6231.

65
placement plotted as a function of time. One

method requires step-by-step integration and can

be performed as follows

For small increments Ax and A^, the average

velocity is _J_ Ax

C0„ Al

A small increment Ax can be measured on the

phase path and the corresponding v^y determined.

The increment in time needed to traverse the

increment Ax is then

Atsw„A^=—

In figure 4.11a, a portion of the phase path of

figure 4.10 is shown. We recall that the initial conditions for this case are, at i=0, x=3, x=v=Q.

Increments (negative here) in x are assigned as Ax*^\ Ax*^', Ax'^' . . ., and corresponding average velocity values are ^av^ \ ^av ? ^av • * •? (also negative). The corresponding increments in r( =w„Ai) are then detremined from eq (4.63) and

are positive. Individual points obtained in this

way are plotted in figure 4.1 lb. As in any numerical integration procedure, the choice of the magnitude for Ax is a compromise. In general, Ax

should be chosen sufficiently small so that the

corresponding changes in x and v are also small.

6.4. Other Phase-Plane Methods

A review of six general types of recent analytic

7. Re

[1] A. Andronov and C. E. Chaikin, Theory of Oscillations, Princeton Univ. Press (Princeton, N.J., 1949).

[2] N. Minorsky, Introduction to Non-Linear Mechanics,

Edwards Brothers (Ann Arbor, Michigan, 1947)

"Non-linear Oscillations", revised ed., D. Van

Nostrand Co., Inc. (Princeton, N.J., 1962).

[3] N. Kryloff and N. Bogoliuboff, Introduction to nonlinear mechanics (trans, and ed. S. Lefschetz, "Ann. Math. Studies", No. 11), Princeton Univ.

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[4] L. S. Jacobsen and R. S. Ayre, Engineering Vibrations,

p. 286 et seq., McGraw-Hill Book Co., Inc. (New

York, N.Y., 1958).

[5] W. J. Cunningham, Introduction to Non-linear

Analysis, p. 208, McGraw-Hill Book Co., Inc. (New

York, N.Y., 1958).

[6] J. G. Truxal, Automatic Feedback Control Synthesis,

McGraw-Hill Book Co., Inc. (New York, N.Y.,

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[8] Y. H. Ku, Theory of non-linear control, J. Franklin

Inst. 271, 108-144 (1961).

[9] W. E. Milne, Damped vibrations, Univ. Oregon Publ.

2, No. 2 (Aug. 1923).

[10] K. F. Rubbert, On vibrations with combined damping,

Ing. Arch. 17, No. 3, 165-166 (1949).

[11] J. P. Den Hartog, Forced vibrations with combined

coulomb and viscous friction, Tratis. ASME 53,

107-115 (1931).

[12] T. A. Perls and E. S. Sherrard, Frequency response of second-order systems with combined coulomb and

viscous damping, J. Research NBS 57, No. 1 (July

1956) RP 2693.

Figure 4.11. Numerical integration methoa for obtaining

displacement-time plot. techniques and new transforms (particularly, as they apply to nonlinear control systems) appears

in [8]. In addition, Liu [64] has developed an

electronic system which plots directly on an

oscilloscope tube-face the phase path for a specified

motion of a system (say, the response of a pressure

transducer to a pulse excitation). Timing marks

for constant At are marked on the phase path by

an internal circuit. The inherent advantages of

such a direct display of test information are obvious. However, detailed information con- cerning the stiffness and damping relations which

govern the motion must be obtained by additional

tests.

[22] K. Klotter, Non-linear vibration problems treated by

the averaging method of W. Ritz, pp. 125-131,

Proc. 1st U.S. Natal Cong. Appl. Mech., ASME

(1951).

[23] C. P. Atkinson and C. P. Bourne, The solution of DufRng's equation for a softening spring system

using the Ritz-Galerkin method with a three-term

[13] K. Klotter, The attenuation of damped free vibrations and the derivation of the damping law from recorded

data, p. 85, Proc. 2d U.S. Nat'l Cong. Appl. Mech.,

ASME (1954).

[14] L. S. Jacobsen, Steady forced vibrations as influenced by damping. Trans. ASME 52, 169 (1930).

[15] J. E. Gibson, Non-hnear system design, Control Eng.,

69-75 (Oct. 1951).

[16] H. Poincar^, New Methods of Celestial Mechanics (in French), vols. I and II, Gauthier-Villars (Paris, 1892, 1893).

[17] A. Lindstedt, Differential equations of perturbation

theory (in German), Mem. Acad. Imp. Sci. St. Petersburg 31, No. 4 (1883).

[18] W. Ritz, On a new method of solution of certain variation problems of mathematical physics (in German), Crelles J. f.d. reine w. ang. Math. 135, 1-61 (1909).

[19] B. G. Galerkin, Series solution of some problems of

elastic equilibrium of rods and plates, Vestnik

Inzhenerov Petrograd, No. 19, 897 (1915).

[20] W. J. Duncan, Galerkin's method in mechanics and

differential equations. Aero. Research Com. (London), R&M No. 1798 (Aug. 1937).

[21] W. J. Duncan, The principles of the Galerkin method.

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(Sept. 1938).

66
approximation, pp. 71-77, Proc. 3d U.S. Nat'l Cong. Appl. Mech. ASME (1958).

[24] K. Klotter, Steady-state vibrations in systems having

arbitrary restoring and damping forces, pp. 234-258,

Proc. Symp. Non-linear Circuit Analysis 2, Brooklyn Poly. Inst. (1953).

[25] K. Klotter and P. R. Cobb, On the use of nonsinusoidal approximating functions for non-linear

oscillation problems, J. Appl. Mech. [E] 27, No. 3, 579-583 (1960).

[26] R. B. Bowersox and J. Carlson, Digital-computer

calculation of transducer frequency response from

its response to a step function, Progr. Rept. 20-331,

Jet Propulsion Laboratory (July 1957).

[27] C. G. Hylkema and R. B. Bowersox, Experimental

and mathematical techniques for determining the dynamic response of pressure gages. Memo No.

20-68, Jet Propulsion Laboratory (Oct. 1953).

[28] R. J. Kochenburger, A frequency response method

for analyzing and synthesizing contactor servo- mechanisms. Trans. AIEE 69, Pt. 1, 270 (1950).

[29] L. C. Goldfarb, On some non-linear phenomena in regulatory systems (in Russian), Avtomat. i Tele- mekh. 8, No. 5, 349-383 (1947).

[30] A. Tustin, The effects of backlash and of speeddependent friction on the stability of closed-cycle control systems, J. Inst. Elec. Engr. (London),

Pt. 94, II-A, 152-160 (1957).

[31] W. Oppelt, On the ortho-curve method as applied to an example with friction (in German), Z. Ver. deut. Ing. (Berlin) 90, 179-183 (1948).

[32] E. I. Ergin, Transient response of a non-linear system,

J. Appl. Mecb., 23, No. 4, 635-641 (1956).

[33] K. Klotter, How to obtain describing functions for non-linear feedback systems. Trans. ASME 79,

509-512 (1957).

[34] K. Klotter, An extension of the conventional concept

of the describing function, pp. 151-162, Proc. Symp. Non-linear Circuit Analysis 6, Brooklyn

Polytechnic Institute (1957).

[35] K. Klotter, On the Use of Replacing the Transient

Response Function of the Solution of Non-linear

Feedback Systems (in German), Fachtagung

Regelungestechnik (Heidelberg, 1956).

[36] J. C. West, The Use of Frequency Response Analysis

in Non-linear Control Systems, Fachtagung Regelungetechnik (Heidelberg, 1956).

[37] H. D. Grief, Describing function method of servo- mechanism analysis applied to most commonly

encountered nonlinearities. Trans. AIEE 73, Pt. II, 243-248 (1953).

[38] E. C. Johnson, Sinusoidal techniques applied to nonlinear feedback systems, pp. 258-273, Proc. Symp.

Nonlinear Circuit Analysis, Vol. 2, Brooklyn Polytechnic Institute (1953).

[39] J. Tou and P. M. Schultheiss, Static and sliding

friction in feedback systems, J. Appl. Phys. 24,

1210-1217 (1953).

[40] H. Chestnut, Approximate frequency-response methods for representing saturation and dead band,

Trans. ASME 76, 1345-1364 (1954).

[41] C. H. Thomas, Stability characteristics of closed- loop systems with dead band. Trans. ASME 76,

1365-1382 (1954).

[42] R. J. Kochenburger, Limiting in feedback control systems. Trans. AIEE 72, Pt. 2, 180-194 (July

1953).

[43] E. C. Johnson, Sinusoidal analysis of feedback-control systems containing nonlinear elements. Trans. AIEE 71, Pt. 2, 169 -181 (July 1952).

[44] V. B. Haas, Jr., Coulomb friction in feedback control systems. Trans. AIEE 72, Pt. 2, 119-126 (1953).

[45] H. M. Hansen and P. F. Chenea, Mechanics of Vibration, John Wiley & Sons, Inc. (New York,

N.Y., 1952).

[46] I. Flugge-Lotz, Discontinuous Automatic Control,

Princeton Univ. Press (Princeton, N.J., 1953).

[47] S. Mahalingham, Forced vibrations of systems with

nonlinear, nonsymmetrical characteristics, J. AddI

Mech. 24, No. 3, 43.'-)-439 (1957).

[48] O. Martienssen, On a new response phenomena in alternating current circuits (in German), Phvs. Z

II, 448 (1910).

[49] M. Dost and C. P. Atkinson, Electronic comput(-r

stimulation of a system with a trilinear restoring

function, pp. 133-140, Proc. 3d U.S. Nat'l. ConK

Appl. Mech., ASME (1958).

[50] C. P. Atkinson and I. Heflinger, Subharmonic and

superharmonic oscillations of a bilinear vibrating

system, J. Franklin Inst. 262, No. 3, 185-190

(Sept. 1956).

[51] A. M. Katz, Biharmonic vibrations of a dissipative nonlinear system activated or kept activated by a harmonic disturbing force, Appl. Math, and Mech.

Inst. Mech. Acad. Sci., U.S.S.R., 18, 425-444 (1954)

.

[52] R. L. Evaldson, R. S. Ayre, and L. S. Jacobsen,

Response of an elastically nonlinear system to transient disturbances, J. Franklin Inst. 248, 548-

562 (1949).

[53] C. A. Ludeke, An experimental investigation of forced

vibrations in mechanical systems having a non- hnear restoring force, J. Appl. Phys. 17, 603-609

(1946).

[54] C. R. Wylie, On the forced vibration of non-linear

springs, J. Franklin Inst. 236, 273-284 (1943).

[55] L. S. Jacobsen and H. J. Jespersen, Steady forced vibration of a single mass system with a non-linear

restoring force, J. Franklin Inst. 220, 467-496

(1935).

[56] M. Rauscher, Steady oscillations of systems with nonlinear and unsymmetrical elasticity, Trans. ASME

60, A169-A177 (Dec. 1938).

[57] J. C. Burgess, Harmonic, superharmonic and sub- harmonic responses for single degree of freedom

systems of the Duffing type. Tech. Rept. No. 27, Div. Eng. Mech., Stanford Univ. (Sept. 1954).

[58] R. Courant, Differential and Integral Calculus, vol. 1, p. 160 et seq, Interscience Publ., Inc. (New York,

N.Y., 1937).

[59] N. W. McLachlan, Ordinary Non-linear Differential Equations in Engineering and Physical Sciences,

p. 24 et seq, Oxford Univ. Press (London, 1950).

[60] J. H. Poincar6, On the curves defined by a differential equation (in French), J. Math. 8, 251 (1882).

[61] L. S. Jacobsen, On a general method of solving secondorder ordinary differential equations by phaseplane displacements, J. Appl. Mech. 19, No. 4, 543-553 (1952).

[62] J. H. Lamoen, Graphic study of vibrations of systems

with one degree of freedom (in French), Rev. Univ.

Mines [8] 11, No. 7, 213 (May 1935).

[63] H. O. Fuchs, Spiral diagrams to solve vibration problems. Prod. Eng. 271, 108-144 (1961).

[64] F. F. Liu, Recent advances in dynamic pressure meas- urement techniques, J. ARS, Jet Propulsion 28, 83-

85, 128-132 (1958).

[65] R. E. D. Bishop, On the graphical solution of transient vibration problems, Proc. IME (London) 168, No.

10, 299-322 (1954).

[66] F. J. Beutler and E. O. Gilbert, Applying phase-plane

techniques to a non-linear system, ISA J. 8, No. 7, 47-50 (1961).

[67] R. D. Mindlin, Dynamics of package cushioning. Bell System Tech. J. 24, 353-461 (1945).

[68] J. J. Stoker, Non-linear Vibrations in Mechanical and

Electrical Systems, Interscience Publ., Inc. (New

York, N.Y., 1950).

[69] M. A. Sadowsky, Nonlinear strings, J. Franklin Inst. 240, 469-476 (1945).

[70] N. Minorsky, Modern non-linear trends in engineering, Appl. Mech. Rev. 4, No. 5, 266-267 (1951).

[71] H. N. Abramson and W. H. Chu, Application of the

generalized phase-plane 5 method to multi-degree- of-freedom vibrating svstems, J. Appl. Mech. 29,

Series E, No. 3, 580-582 (1962).

67

5. Simple Aperiodic-Function Generators

J. L. Schweppe '

1. General

All methods of transducer calibration require

that a known input be applied and that the out- put be measured precisely. In this and the fol- lowing chapters, methods of producing the required

input and methods of evaluating the measured

response will be discussed. This chapter covers

three types of simple aperiodic-function generators: the dropping ball, the quick-opening device,

and the explosive device.

1.1. Types of Simple Aperiodic- Function

Generators

Perhaps the most simple, though not the most

satisfactory, aperiodic-function generator is the

dropping ball. The transducer is set in motion

by bouncing a ball bearing on the diaphragm, an

action which approaches a true impulse.

Quick-opening devices include both quickopening valve and burst-diaphragm devices.

The pressure change produced approaches a step-function with the direction of the step either

positive or negative.

Finally, a pressure step or pulse can be produced by an explosion. This is an adaptation of the closed bomb which has been used traditionally

to determine thermodynamic properties and to determine the rate of reaction of fuels and oxidants.

1.2. Place in Pressure Transducer Calibration

The dropping ball was perhaps the first effective

aperiodic generator which could be used for the

determination of the natural frequency of trans- ducers [1].^ The method has a number of limitations which are discussed in section 2 of this chapter. Because of these limitations the dropping ball is not used extensively, although it can

be used at higher frequencies than any of the

other simple devices.

The quick-opening devices are used to provide

precise pressure steps for amplitude calibration. In general, they are too slow for effective fre- quency calibration.

The explosive devices do not provide precise

pressure steps. Therefore, calibration requires

comparison with a gage which has been calibrated

by some other method. For this reason explosive

devices are not used extensively.

1.3. Range of Operation

The range of operation for each of the devices

discussed in this chapter is given in table 5.1. The amplitude ranges are those reported by the

laboratory or manufacturing concern where the

device was built or used. The rise time of a pressure step of given amplitude (represented by

the response of a "fast" transducer to the pressure

step) is also that reported by the laboratory.

The frequency ranges show the highest frequency

mode of oscillation of the transducer which can be excited by a given aperiodic generator. These

are approximations based on the assumption that

the rise time for maximum swing of the transducer

must not exceed one-fourth of the period of oscilla- tion of the transducer.

Table 5.1. Range of Simple Aperiodic-Function Generators

Device

Dropping ball Quick-opening JPL burst-diaphragm..

Eisele's valve NBS valve Explosive JPL , NOL

Amplitude

range to

psia Not applicable. 200-

3,000-.

50,000.

Time for pres- sure step

Rise

psta

200

1,250 23,500

700 700 300

40,000. 40,000 400X10'

Rise time

250 200X103

25

Fre- quency

range to

cIs 100,000

1,000

10, 000

8, 000

2.1. Description

2. Dropping Ball

transducer system's response to the impact is

The first simple aperiodic -function generator

to be considered here is the dropping ball. This

generator in its most simple form consists of a small ball which is dropped onto the sensing end

of the transducer [1, 2, 3], figure 5.1. Ordinary

ball bearings have good elastic properties and are customarily used. The ball is released by cutting

off the current to an electromagnet which holds

the ball directly above the transducer. The

1 Professor of Mechanical Engineering, The University of Houston; Presi- dent, Houston Engineering Research Corporation.

2 Figures in brackets indicate the literature references on p. 75.

displayed on an oscilloscope. The oscilloscope

sweep may be triggered by the transient voltage

generated across the electromagnet when the current is cut off, or by a nucros\\atch or photocell

actuated by the ball dropped on the transducer or by a second ball dropped at the same time,

figure 5.1a.

2.2. Theory

The dropping ball transmits its kinetic energy

to the transducer during the very short time the

two masses are in contact. The ball then bounces

away and the transducer is left to vibrate at its

69
TRANSMITTER TO-SWEEP UNIT

VAAAA/

TO

OSCILLOGRAPH

Figure 5.1. Drop-ping hall calibrator.

o

(0zu

V)o1- >

q: o UJ UJ z

1- UJ

1-

SMI

PING z

<

q: :op 1- K

O

UJz

UJ

MASS- RATIO DROPPING BALL iviMj>a rcMiiu, SENSOR

Figure 5.2. Effect of mass-ratio on vibration energy of transducer

own natiiral frequency. If the ball and the sersor were perfectl}^ elastic, and if the sensor

deflection were zero, all of the kinetic energy of

the dropping ball would be retained by the ball when it bounced away. This condition is approached as the mass of the dropping ball approaches zero. As the mass of the ball is increased

from zero, the amount of energy transmitted to the sensor Increases, reaches a maximum, and

then decreases again [2], figure 5.2. The fraction of the dropping energy transmitted

to any given pressure transducer is affected by a number of factors. Some of these factors are the

elasticities of the ball and the sensor, the relation

between the force applied to the sensor and deflection of the sensor, and the relation between the

rate of application of force and rate of deflection.

2.3. System Design

The design of a dropping-ball system for cali- brating pressure transducers must take into

account the factors which are discussed in section

2.2. Further consideration must be given to the

selection of the dropping ball, the height of the

drop, the oscilloscope sweep speed, and the

triggering time. Since the proper choice of these

design variables is generally different for each

transducer to be tested, the testing system should

allow for adjustments in these variables.

2.4. Evaluation of Test Data

The product of a dropping-ball calibration is an

amplitude versus time plot of the transducer

output. Four factors limit the value of this out- put function: (1) The forcing function cannot be

determined precisely. (2) The response during

the finite period when ball and transducer are in contact is a function of the joint mass. (3) Thin

diaphragm transducers are sensitive to noise.

And (4) the dropping ball exerts a point-concentrated force on the transducer and, depending on

the configuration of the transducer, the response

may not be the same as the response to a uniformly

distributed pressure force. It is, however, usually

possible to determine the frequency of the pre- dominant mode and to treat the transducer as if this were the only mode of oscillation. The

dropping-ball method is not recommended for

amplitude response determinations. This method

is useful for determining the transducer's pre- dominant mode of oscillation in the range of 2 to 100 kc/s.

3. Quick-Opening Devices

In general, the quick-opening device utilizes two pressure chambers—a small gage chamber

and a relatively large reservoir. Initially the two

chambers are at different pressures and the pres- sure step is generated by opening quickly the pas- sage between the two chambers. Since either

chamber may be at the higher pressure initially,

the pressure step may either rise or fall. Hylkema

[4] states that, for the JPL (Jet Propulsion Laboratory) device, the pressure-fall (or negativegoing step) arrangement develops a closer approach to the ideal step function. While this

is generally true lor quick-opening devices, it is important to recognize that all gages do not have

the same character of response to negative- and

positive-going steps. With the negative-going

step the transducer oscillates about a median

which is near the zero-pressure point. As a

result it is possible for the diaphragm to lose contact with the force-summing component, in which case the recorded transducer response differs from that for oscillation about a median at a

significant positive pressure point.

70
3.1. Description of Burst-Diaphragm Device

One of the simplest quick-opening devices for pressure gage calibration utilizes a thin plastic

diaphragm as the separator between the two

pressure chambers. Because of the bursting

characteristics of the diaphragm, a pressure fall in the gage chamber produces a sharper pressure

step than a pressure rise. Hylkema and Bowersox [4] of the Jet Propulsion

Laboratory have reported the construction details

of one typical burst-diaphragm device, figure 5.3a. The JPL device has a 200-in.^ reservoir chamber

and a very small gage chamber separated by a

thin plastic diaphragm. Either chamber can be

pressurized with an inert gas. The diaphragm

is punctured by a 1/4-in. knife operated by a

solenoid. A microswitch on the knife assembly

triggers the sweep of the monitoring oscilloscope. In the JPL device, both chambers are lined with

felt to eliminate pressure oscillations due to

reflected pressure waves. The JPIj calibrator can operate at pressures

up to 200 psi. The time to reach a steady pres- sure is about 250 ^sec for a 200-psi change. This

device can be used to shock a transducer into

b FtGURE 5.3. Burst-diaphragm device.

(a) Cross section of JPL device (Reproduced from [4] with permission from Proc. ISA.)

(b) Wiancko device (Reproduced by courtesy of WIANCO Engineering Co.)

oscillation when the predominate resonance fre- quency is below about 1000 c/s. At higher resonance frequencies the device is limited to amplitude calibration.

Burst diaphragm devices (fig. 5.3b) are available

commercially and custom models are being used

in a number of laboratories. P. M. Aronson of the Naval Ordnance Laboratory, White Oak, Md.,

has built a device which operates at pressures up

to 1000 psi.

3.2. Description of Eisele's Device

In some quick-opening devices a valve is used

in place of a burst-diaphragm. The general

principle of operation is the same. In Eisele's

device [3] the reservoir is pressured with compressed air, the pressure is adjusted to the desired

level with a needle-valve, and then a quickopening hand valve is opened to admit the highpressure air to the gage chamber, figure 5.4. In principle Eisele's device can be operated at any pressure for which equipment can be designed

and at which gas is available. The particular

device described here can be operated at pressures

up to 3000 psi. The pressure rise rate is much

lower than for the burst diaphragm device,

6.2X10^ versus 8X10^ psi per second. Application is limited to amplitude calibration.

3.3. Description of NBS Device

D. P. Johnson of the National Bureau of

Standards has developed a quick-loading, liquidmedium pressure calibrator which is now in service in the Mechanical Instruments Section of the Bureau (figs. 5.5, 5.6, and 5.7). In this device

the two chambers are separated by a special

quick-opening valve. The gage is mounted in the smaller chamber, and the pressure step is obtained by opening the valve between the two

chambers. In Johnson's device the main reser- voir is about 1000 times that of the gage chamber

and its length is about 40 times as great.

The operating procedure is as follows: (1) Open the bleeder. (2) Open the poppet by opening

and closing the quick-acting dump valve. (3) Fill the system by pimiping. (4) Close the

bleeder. (5) Close the poppet by increasing the

differential pressure between the piston and the

main reservoir several psi (pump-to-piston valve

open and pump to main reservoir valve closed)

.

(6) Open pump to main reservoir valve and pump

to desired pressure behind piston and in main

reservoir. (7) Open bleeder. If some main reservoir pressure is lost, repump. During pumping

leave pump to main reservoir and pump to piston

valves open. (8) Close pump to main reservoir

and pump to piston valves. (9) Close bleeder.

(10) Dump pressure behind piston by operation

of quick opening valve.

Note that, as long as the poppet valve is closed,

the reservoir pressure exerts a force which tends

to hold the valve head in the closed position. As

the actuating force is increased, the tension in the

71
AIR BOTTLE

CONNECTION

AIR RELEASE

VALVE

Figure 5.4. Eisele's quick-opening device. Figure 5.5. NBS quick-loading calibrator. (Reproduced from [3] with permission from Archiv fur Technlsches Messen.)

-TRANSDUCER ADAPTOR -BLEEDER VAIVE

^^^^^^^

Figure 5.6. Principal parts of NBS quick-loading calibrator.

Figure 5.7. Flow diagram for liquid-medium pressure-step generator.

72
10 m SEC

Figure 5.9. JPL closed bomb.

(Reproduced from JPL Memorandum No. 20-168 with permission from Jet Propulsion Laboratory)

BOLTS

LID

EXHAUST

RING

DIAPHRAGM FIRING- LEAD

SEAL

POWDER CHARGE

FIRING CHAMBER

BASEPLATE

TEST

VEHICLE

PRESSURE

GAUGE

Figure 5.10. NOL closed bomb.

(Reproduced from 5 with permission from Trans. ASME.)

eopMILLISECONDS

Figure 5.8. Transducer response to NBS calibrator.

694-211 0-63—

6

Figure 5.11. Typical dynamic pressure pulse in XOL

closed bomb.

(Reproduced from 5 with permission from Trans. ASME.)

73
valve stem increases almost to the breaking point.

An expansion wave travels down the stem moving

the poppet first in the direction of tighter closure,

then in the opposite direction. The elastic force

in the stem snaps the valve open at a rapid rate. In Johnson's device the hydrostatic pressure in the reservoir is measured by its effect on the resist- ance of a manganin wire. The gage sensitivity is estimated to be 0.5 psi at 30,000 psi with a cali- bration repeatability of ±0.1 percent. The

calibration accuracy of the device is limited by

two inherent errors. (1) Closure of the bleeder

valve (step 8) reduces the volume of the gage

cavity, thus raising the pressure to something

above 0 psig. (2) It takes 5 to 10 msec for the

piston to complete its travel. The volume of the

main reservoir is increased by the travel of the

piston so that the gage experiences a transient

pressure somewhat higher than the steady final pressure. Both errors appear to be independent

of pressure, so that for a given configuration of the

device there is a pressure below which the error is not acceptable.

The time required for the pressure in the gage

chamber to reach the main reservoir pressure

depends on the gage cavity size, the gage cavity

aperture diameter and length, the viscosity of the

liquid, and the poppet valve stem stress. The

damping restriction shown in figure 5.6 is designed

to utilize the viscosity of the liquid to damp the

oscillations of the poppet. With the most rapid

pressure rise, the pressure step causes the gage

cavity to ring at a frequency of about 10 kcs

This ringing frequency limits the use of the liquidmedium calibrator for testing the periodic re- sponse of a gage in a hquid. By increasing the

time for the pressure rise to about 2 msec, a very

nearly monotonic step can be produced for testing

the nonperiodic transient errors. Typical response curves are shown in figure 5.8.

3.4. Evaluation of Test Data

AH of the quick-opening devices provide a

positive means for amphtude calibration within

the ranges of pressure specified for each device.

Evaluation of the amplitude data simply involves

a determination of the output-input ratio as a function of the input magnitude.

None of these devices is particularly suited for the determination of frequency response. Since

each is a step generator the rise time must be short

in comparison with the period of the main resonant

frequency of the transducer, if the transducer is to be shock excited. This requirement limits the

usefulness of the quick-opening devices to a fre- quency range of about 0 to 10,000 c/s. The

frequency response data which are valid can be

evaluated by the methods described in chapters

2-and 3.

4. Explosive Devices

4.1. Description of JPL Bomb

Among the closed-bomb pressure generators

which have been described in the literature is a simple calibration device developed at Jet Propulsion Laboratory [4]. This bomb was designed by

L. R. Williamson and was described by C. G.

Hylkema and R. D. Bowersox (fig. 5.9). The

outer structure is a heavy-walled steel pipe nipple,

about 6 in. long, fitted with screwed pipe caps. Aluminum liners are used to protect the outer

case from exploded fragments and to vary the

internal volume. The pressure is generated by

exploding a dynamite cap within the bomb, and

the peak pressure is adjusted by varying the

internal volume. Since blasting caps are not

perfectly uniform, successive shots may not produce the same pressure. This makes it necessary

to use a calibrated comparison gage to determine

the actual pressures attained.

Although bombs can be designed for very high

pressures, the JPL bomb is limited to a maximum

pressure of 700 psi. Rise time, using blasting

caps, is about 300 usee. The average rise rate of

2.3 X 10® psi per second is comparable to that of the

quick-opening devices.

4.2. Description of NOL Bomb

Mickevicz [5] of the Naval Ordnance Laboratory has described a device which operates at much higher pressures (fig. 5.10). Mickevicz'

dynamic pressiu-e pulse generator is capable of

generating reproducible, pressm-e-time pulses in

the order of 40,000 psi. The pressm-e is generated

by igniting a propellent powder within the bomb,

and the pulse is produced by releasing the pressure

through a rupture disk when it reaches a pre- determined value. The pressure transducer is mounted in the side of the firing chamber.

A typical djmamic pressure plot is shown in

figure 5.11. The pulse rise rate is dependent on

the powder grain size, and the decay rate is dependent on the escape aperture size. Rise

rates of 14X10" psi per second have been attained,

corresponding to 40,000 psi in 4,000 fisec. Dynamic calibration is accomplished by comparing

the response of the imknown gage with that of a

calibrated high-frequency response gage.

4.3. Evaluation of Test Data

The explosive devices produce somewhat more

rapid pressure rise rates than the quick-opening

devices. However, they are not adequate for determining frequency response above about 8,000

c/s.Since precise pressure steps are not produced,

amplitude calibration can be determined only by

comparison with a gage which has been calibrated

by some other method. Further, any determination of frequency response also involves the use

74
of a previously calibrated gage. Evaluation then

requires a determination of the input function

through use of the transfer function and the out- put of the calibrated gage, followed by evaluation

of the transfer function for the gage being tested. The methods used are described in chapters 2 and 3. The explosive device provides only a secondary method for calibration.

5. References

[1] W. Gohlke, Quartz pressure measuring apparatus for high natural frequencies (in German) , VDI-Forschungsheft 407 (1941).

[2] W. Gohlke, Introduction to Piezoelectric Measuring

Techniques (in German), 2d ed.. Academic Publ. Co. Geest & Partig K.-G. (Leipzig, 1959).

[3] E. Eisele, Calibration of indicators II. Dynamic calibration, Archiv. Tech. Messen J-137-7 (May 1951).

[4] C. G. Hylkema and R. D. Bowersox, Mathematical

and experimental techniques for the determination

of dynamic performance of pressure gages, Proo. ISA 8, 115 (1953).

[5] E. J. Mickevicz, Techniques and equipment for generation of dynamic high pressures, Trans. ASME 75,

325-327 (1953).

75

6. Shock Tube Methods

J. L. Schweppe '

1. General

1.1. The Shock Tube as a Step- Function

Generator

In chapter 5 a number of simple aperiodic function generators are described and limitations in performance are pointed out. All of these simple

generators produce pressiu-e steps or pulses with

relatively slow rates of pressure change or with

uncertain magnitude or both. The rates of pres- sure change are such that dynamic calibration

with these devices is limited to transducers with

predominant modes of oscillation at frequencies

below 10,000 c/s. Since calibration is required

for transducers having predominant modes of

oscillation in the range of 50,000 to 100,000 c/s or

higher, the step-function generator must be capable of changing the pressure from one known level

to another in a sufficiently short time to shock

excite these high-frequency transducers.

The shock tube meets the requirement for a

very short rise time between pressure levels—as short as 10~^ sec for a positive pressure step [1].^ With proper design the pressure can be held steady

for 4 or 5 ^sec after the step change in a shock

tube of moderate length. Step amplitudes are

easily controlled from a few psi to about 600 psi

in a shock tube of relatively simple construction.

1.2. Information Obtained From Shock Tube

Calibration

The direct product of the shock tube calibration

procedure is a time-base record of the response of the transducer to the step input. This record is obtained by photographing the trace made by the

transducer output on an oscilloscope (figure 1.9). Methods described in chapter 2, section 3.2 may

be used to determine the frequency response from

the transient response to a step-function input.

Numerical methods also may be used effectively

with the aid of a digital computer [2].

2. Description

2.1. Description of Components

The shock tube consists of two elongated chambers, usually of constant cross section, separated

by a burst diaphragm. Initially the gas pressure

in one chamber is higher than in the other. When

the diaphragm ruptures the expansion of the highpressure gas into the low-pressure chamber generates a shock wave which travels faster than the

expanding gas. In addition to the compression

and expansion chambers and the diaphragmpunctiu-ing mechanism, the equipment includes a gas supply and regulating system, static pressure

and temperature-measiu'ing devices, a shock ve- locity-measuring system, and a transducer output

recording system. See figure 6.1.

GAS SUPPLY AND CONTROL

SYSTEM

STATIC PRESSURE

AND TEVPERATURE

MEASURING SYSTEM

TRANSDUCER

ELECTRONIC SYSTEM

SHOCK VELOCITY hEASURING SYSTEM

OSCILLOSCOPE

INDICATOR

Figure 6.1. Shock tube schematic.

' Professor of Mechanical Engineering, The University of Houston; Presi- dent, Houston Engineering Research Corporation.

2.2. Qualitative Description of Shock Tube

Phenomena [3]

Initially, the gas in the compression chamber

is at a higher pressure than the gas in the expansion chamber, figure 6.2a. When the diaphragm

bursts, the contact sm-face between the gas originally in the expansion chamber and that originally

in the compression chamber moves into the expansion chamber. The pressure and the velocity of the medium on either side of the contact surface

must be the same, but the temperature and density differ. A short time after the diaphragm

bursts the idealized waves appear as shown in figure 6.2b. The leading edge of the rarefaction wave moves

into the compression chamber at the velocity of sound for the undisturbed medium while the trail- ing edge moves in the same direction at a lower

velocity. The reduced velocity of the trailing

edge results from the fact that it moves in a me- dium which is at a reduced temperature and it moves against the velocity of the stream. The

rarefaction wave will, in time, reach the end of the compression chamber where it will be reflected

as a rarefaction, figure 6.2c. After reflection the wave ^vill move in the same direction as the shock

2 Figures in brackets indicate the literature references on p. 85.

77
RARCFACTKM

RAREFACTKM

I CONTACT SURFACE

I CONTACT SURFACE

I "7 foMfTACT SURFACE

RAflEFACTHM I

I CONTACT SURfWZ

RAREFACTKM

CONTACT SURFACE

Figure 6.2. Pressures and waves in shock tube.

(a) Before diaphragm Is burst

(b) After burst and before reflection of rarefaction

(c) After refiection of rarefaction

(d) Reflected rarefaction overtakes shock wave

(e) Shock wave reflected by closed end of tube

(f) Rarefaction wave reflected by open end of tube wave at the velocity of sound in the medium pius

the particle (contact surface) velocity. If the

tube is long enough the rarefaction wave will over- take the shock wave, figure 6. 2d. Under this cir- cumstance the strength of the shock drops off. For a shorter tube, the shock wave reaches the

end and is reflected before the rarefaction wave

overtakes it. If the end of the tube is open the

reflection is a rarefaction and if the tube is closed

the reflection is a shock wave of more than twice

the magnitude of the incident pressure step, figures

6.2e and 6.2f.

3. Shock Tube Theory

3.1. General

The theory of shock waves is covered thoroughly

in a number of references [3, 4, .5, 6, 7, 8, 9, and

10]. The general equations for shock and expansion waves may be used to analyze the flow conditions in a shock tube. In figure 6.3 the region

of flow ahead of the shock in the expansion chamber is designated (1), the shock front is represented by a line of slope 1/C7, and the contact

REGION (9)

REGION (8)

REGION ( I

DISTANCE, FEET Figure 6.3. Distance/time plot of events.

surface by a line of slope 1/u. The region between the shock front and the contact surface is designated (2), the region in front of the head of

the rarefaction wave is designated (3), and the

region between the contact surface and the tail of the rarefaction wave is designated (4)

Let us consider right-traveling shocks as in

figure 6.3. The velocity of the shock relative to the gas into which it moves is U—Ui, and relative to the gas out of which it moves is U—U2. The

flow through a shock discontinuity may be treated

as though the flow were steady at each instant of

time and the relations for steady-flow normal

shocks may be applied if the coordinate system moves with the shock [6, p. 995]. The following

basic equations are taken from Shapiro [6], chapter 25, where detailed derivations are presented.

The plus sign refers to a right-traveling and the

minus sign to a left-traveling shock wave. (See

List of Symbols.)

Translation particle velocity (Shapiro, eq 25.20,

p. 1001):

U2—U1 ±2

7+

5 [Uzzii ai 1. ±2 r 1 T

iL «i u-u,] 7+1 mJ

(6.1)

Shock wave pressure (Shapiro, eq 25.23, p. 1002)

(6.2)

78
Relation between particle velocity and velocity of sound (Shapiro, eq 24.20, p. 945):

1 j-lnl /-U2—

2 V a, /

(6.3)

Contact surface velocity (Shapiro, eq 25.32c. p. 1008)

:

P2

U2 Pi

-1

(6.4)

Relation between reflected and incident shock

(Shapiro, eq 25.35, p. 1020):

7+1 VPi/^

(6.5)

Relation between incident and reflected pressure

step (Shapiro, eq 25.36, p. 1021):

2y

k+l

P2-P1 Pi , 7-1 (6.6)

P2 ' 7+1

Relation between temperature and shock velocity

(Shapiro, eq 25.5, p. 995)

:

)(^

2^—] Mi^-1

(7+1)^

(6.7) Ml'

2(7-1)

3.2. Contact Surface Velocity

For a shock front which moves toward the

right into a region at rest, Ui —0 and the positive

sign is selected in eq (6.1). The resulting relation

is not closely checked by experiment, but it is useful for planning experiments and for designing

shock tubes.

U2

di 7+

In eq (6.8), 7 refers to the ratio of specific heats

for the gas in region (1). For air, 7=7/5 and eq

(6.8) becomes

Since in a helium-to-air tube the shock front also moves into air in region (1), eq (6.9) applies.

3.3. Shock Wave Pressure

Equation (6.2) relates the shock velocity to the

shock wave pressure. In the derivation of this

equation it is assumed that the perfect gas laws

apply, that the specific heats are constant, and

that flow is adiabatic. For Ui=0

7+

27

7+1

(M,2- -1).

(6.10)

A second equation, which relates the compression-expansion pressure ratio to the shock- wave pressure, is developed as follows: Write eq

(6.3) for a rarefaction wave moving left into region

(3) at the velocity of the region (4) front. In this case 7 is the ratio of specific lieats (assumed constant) for the gas in the compression chamber and

^3=0. Also assume that the isentropic relation

between a and p applies. That is.

27 P

P,

2t,

1 \aj

Then

ffla 73—1 \ aJ 73—1 L \P3/

But Ui=U2 and P4=P2. Therefore

(6.11)

U2_ 1 03 73- Now combine the above equation with eq (6.4)

and solve for P1/P3. The result is

P3 P

^i(73-l) (p-^) 7,-1

(6.12)

If the same gas is present at the same temperature

in both the compression and expansion chambers,

73=7i, a3=ai, and

27

Pi

P3"

Pi

"P2

7-1

27

El

Pi

1 7-1

(6.13)

For air in both chambers, eqs (6.10) and (6.13)

reduce to (^-l)=^(Mi^-l) (6.14)

P

El

Pi

(6.15)

79
For helium-to-air, eq (6.14) applies and eq (6.12)

reduces to

(6.16)

If the temperatures in the two chambers are as- sumed to be the same, and the velocity of sound is computed assuming perfect gases

Pi

Pz_ Pi

(6.17)

3.4. The Reflected Shock Wave

The pressure ratio for the reflected-to-incident

shock wave is given by eq (6.6). In both air-to-air

and helium-tp-air tubes the equation is the same. P5-P2 IP.

QPi+P2 (6.18) or P..-P

P2-P

Combining eq (6.19) with eq (6.14) yields a more convenient relation

The relative Mach number of the reflected

shock may be determined by writing eq (6.1) for the left-traveling reflected shock with 1/5=0.

Thus

U2 MJ (6.21)

Since this left-traveling shock moves into region

(2), its absolute velocity to the left is reduced by

the amount of the right-traveling particle velocity

in region (2). U5=U2—Msa2. (6.22)

The velocity of sound in region (2) is given by

eq (6.21), and the Mach number M5 by eq (6.2)

written for the shock between regions (5) and (2)

\P2 J 27

fl.

Then

(7+l) + (7-l) Ps

-U2

(6.23)

(6.24)

The negative sign indicates that the reflected

shock moves left.

3.5. The Rarefaction Wave

When the diaphragm bursts a rarefaction wave

travels to the left into the expansion chamber,

region (3), as shown on figure 6.3. The rarefaction wave tip travels at the speed of sound, as, in the

negative direction until it is reflected by the closed

end of the expansion chamber. It then travels in the positive direction; first at a changing rate through the expansion fan, then at a constant rate

until it reaches a point where it overtakes the contact surface or meets the reflected shock, or both.

Tha diu-ation of the pressure step in the shock

tube depends on the time required for the head of the rarefaction wave to meet the reflected shock.

Wright [7] and others have described how to determine the position of the rarefaction wave tip as it moves through the expansion fan. Passage through the expansion fan slows the

rarefaction wave tip slightly, and hence shifts

the region (4) distance-time cm"ve to the left. If this shift is neglected and the constant velocity

into region (4) is used to compute the position of

the rarefaction tip, the result is an estimate of the

duration of the pressm-e step which tends to be

shorter than the actual step. Design based on

the above assumption is therefore conservative.

It is recommended that design be based on a negative rarefaction wave tip velocity of 0.3 and

a positive rarefaction wave tip velocity of

dx

di =U^-\-a^. (6.25)

3.6. Pressure Limitations

Consider eq (6.12), which relates the initial compression and expansion chamber pressures to the size of the pressure step. If P1/P3 approaches

zero, P1/P2 becomes very small. But Pi cannot be

zero in a real shock tube. Therefore we can

divide both sides of eq (6.12) by Pi and rearrange.

P3

«i(73— 1)

27_3 73-1

(6.26)

Now P3 may become very large, but since P2

must always have a finite value, the ratio P2/P3 can approach zero as a limit. At this limit P2 reaches a maximum value. Solving eq (6.26) atP2/P3=0,

\Pl /max KaJ

7i(7i +l) (73-1)^

+--^A/(-y(^y+4- (6-27)

ai 73— 1 V \73— 1/

Inserting eq (6.27) into eq (6.10) and solving

80
for Ml,

For the air-to-air shock tube, eqs (6.27) and (6.28)

reduce to

The maximum pressure ratio P2/P1 for an air-to-air

shock tube is 44.1 and the maximum Mach number is 6.16. Higher pressure ratios and Mach numbers may

be obtained by using properly selected gases in the compression and expansion chambers. Higher

ratios also may be obtained by increasing the

energy ratio by heating the compression chamber

or cooling the expansion chamber, or both.

Figure 6.4 shows the solutions of eqs (6.15) and

(6.17) for air-to-air and helium-to-air shock tubes

with the same initial temperatures in the compression and expansion chambers.

Extremely high compression-to-expansion pres- sure ratios are required to approach the limiting

maximum pressure step. For example, to reach

a shock strength of P2/A = 10 a compression-to-expansion ratio of about 600 is required for an air-toair tube. The same shock strength can be obtained

in a helium-to-air tube with a compression-toexpansion ratio of 43. Table 6.1 below shows the

effect of compression and expansion chamber pres- sure on shock amplitude for both air-to-air and

helium-to-air tubes.

Table 6.1

Shock tube Pi Pi Pi- Pi

Alr-to-air 500 7.6 41.5 34 600 15.0 66 SI 600 30.0 104 74

1, 000 16.0 83 68 Hellum-to-alr 600 15.0 132 117

3.7. Real Shock Tube Behavior

Experimental measurements indicate that val- ues of shock strength computed from eq (6.12) are higher than those actually attained. The differ- ence between the theoretical and experimental re- sults can be explained by reviewing the assumptions made in the derivation of eq (6.12). The fol- lowing assumptions were made : (1) the perfect gas

laws apply; (2) flow is adiabatic; (3) the isentropic

relation between a and p applies; (4) the specific

heats are constant; (5) the diaphragm burst is instantaneous; and (6) viscous forces are negligible. The first three assumptions are somewhat

more realistic than the last three. Glass and Patterson [9] have discussed the conditions under

which variations of specific heat are of importance.

We will concentrate here on a discussion of the

influence of diaphragm burst and viscous forces. The diaphragm is, of course, a necessary component of the shock tube system. It separates the

high-pressure gas in the compression chamber

from the relatively low-pressure gas in the expansion chamber. It is stretched nearly to the bursting point by the high pressure of the expansion

chamber and is caused to burst by the action of a knife or punch. Two factors related to the diaphragm cause the real shock strength to be less than theoretical. First, energy is required to separate and fold back a metal diaphragm or to

81
accelerate the particles of a friable diaphragm to the flow velocity of the gas. And second, the

diaphragm burst delays the formation of a one- dimensional shock wave. For each shock a cer- tain distance must be traveled before the wave is fully developed.

In their paper on experimental aspects of shock wave attenuation, Glass and Martin [10] discuss

both the growth and development of the shock wave and the effect of viscous forces. As the wave moves down the tube it tends to decelerate

due to the growth of the boundary layer. At the same time the contact surface accelerates.

Although real shock tube behavior is not ideal, in a well-designed tub the attenuation of the wave is small.

4. Design for Pressure- Gage Testing

4.1. General Requirements

Satisfactory pressure-gage calibration with a step-function generator requires generation of a

step of reasonable duration and known amplitude

over the necessary pressure range. To accomplish this with the shock tube, the compression and

expansion chamber lengths must be properly

proportioned, the tube must be designed for

minimum wave distortion, minimum mechanical

vibration, and minimum secondary shock or

noise; and the shock velocity must be precisely

measured.

4.2. Tube Dimensions

The expansion chamber must be at least long

enough for a plane wave to develop. Although

there is no theoretical relation for determination

of this length, experience shows that a length of ten times the largest dimension of the cross section is sufficient [11]. The size of the cross

section therefore affects the minimum length of

the expansion chamber and hence the length of

the tube.

The vertical or largest dimension of the cross section must be large enough for the test purpose.

Wolfe [12] chose a 1% in. square section for pres- sure gage testing because a very few pressure

transducers have diameters greater than 1 in. If one-dimensional investigations are to be made (or

if optical timing devices are to be used) the

width or smallest dimension should be as small

as possible, but large enough that the shock wave

will deflect light rays for distinct schlieren pictures.

Bitondo and Lobb [11] found that a 2-in. width is adequate for this purpose (they did not determine

the minimum adequate width).

The compression chamber must be long enough

to keep the rarefaction wave from overtaking the

shock wave before it passes through the test

section. The time required for this process, and

hence the best compression chamber length, depends on the initial pressure ratio between the

two chambers and the gases initially present in the

chambers. For side gage location the maximum

constant-pressure interval is achieved when the

compression-expansion chamber length ratio is such that the reflected rarefaction reaches the gage

port simultaneously with the reflected shock. For

end gage location the maximum constant-pressure interval occiu's when the compression-expansion chamber length ratio is such that the rarefaction tip, the contact surface, and the reflected

shock reach the same point in the tube at the same instant [12]. Since most pressure transducers exhibit some

response to mechanical vibration, it is desirable

to reduce the mechanical vibration to a minimum.

This can best be accomplished by making the

tube massive and rigid. The tube at the NBS

used for pressure gage testing weighs about 1500

lb, and has a minimum wall thickness of %-in.

For satisfactory performance a shock tube must

be built within reasonable dimensional tolerances.

It is generally agreed that the inside surface should

be machined but that surface grinding is an unnecessary refinement. Bitondo and Lobb [11]

point out that waviness in the longitudinal direction is especially to be avoided and they suggest

a manufacturing tolerance of 0.035 in. in a 12-in.

length. If the tube bends the shock wave will not remain perpendicular to the wall as it travels

down the tube.

4.3. Burst-Diaphragm Selection

In section 3 of this chapter theoretical relations were developed based on the assumption that the

diaphragm bursts instantaneously and completely.

Real diaphragms are rated by the degree to which

they satisfy this assumption.

Glass and Patterson [9] have reported that, for low-pressure operation, Red Zip cellophane has

the best ruptiire properties and leaves the smallest

obstruction. Glass and Patterson and others have

also reported good results with cellulose acetate

film and with Monsanto's Vuepak F. All of the

plastic materials give best results when they are loaded near the biu-sting point before the rupturing

mechanism is activated.

For high pressures, metal diaphragms must be

used. Copper or aluminum sheets have been used

successfully and particularly good results have

been reported for hard or half-hard brass shim

stock. Annealed shim stock gives poor results.

4.4. Gas Supply and Control System

The gas supply and control system indicated in

figm-e 6.1 includes the source of gas or gases, the

pressvire-regulation system, the loading system,

and the diaphragm-punctiu-e system.

The gas source is usually purchased in cylinders

for use in the laboratory, although a compressor

may supply air for an air-to-air tube. In either

82
SHUTOFF —IX

HG^ COMPRESSION —^) PUNCTURE ly SUPPLY ^SUPPLY —CXJ—'ENT X-O LOAD SOLENOID

SUPPLY —00- VENT r

-LO«D SOLENOID

EXPANSION "JVi/ PRESSURE

FiGUBE 6.5. Gas supply and control system.

case the gas is stored in high-pressure cyUnders

prior to use in the tube, and in either case moisture

free gas must be provided. On the way to the

shock tube the gas passes through a regulator

which is set at a pressure somewhat above the

maximum desired in the shock tube chamber,

figure 6.5. The puncture pressure is set some 250

psia above the desired compression chamber

pressiu'e. The load and vent switches are operated

manually to adjust the pressures in the compression and expansion chambers. The compression

chamber is loaded first to avoid distortion of the

diaphragm. When the desired pressures have

been attained and pressures and temperatm-es have been recorded, the tube is fired by actuating

the puncture solenoid.

4.5. Shock-Velocity Measurement

Shock wave velocities are measured by determining the time required for the wave to traverse the measured distance between two points. Because of the high velocity of the wave, the instrument used to detect the passing of the wave must

have a rapid response and the instrument used to measure the time interval must be capable of

counting in very small increments. Methods of

detecting the shock wave and determining the

velocity are discussed by Cady and Bleakney in section C of Physical Measurements in Gas

Dynamics and Combustion [13]. Cady describes

ion-tracer techniques, use of luminous particles,

electric discharge anemometry, and acoustic Mach

meters, and Bleakney describes the light-screen

techniques. The latter method is perhaps used

most commonly to measure shock wave velocity. Pressure transducers are also used to detect the

passage of the shock wave. The light-screen technique is used in one of two

types: schlieren or reflection. In the schlieren

system for detecting the passage of the shock wave,

the camera or viewing screen is replaced by a f)hotocell. The reflection method uses the shock

ront as a reflecting surface to bend the light rays

along a path through three staggered knife edges.

The two light-screen methods are equally accurate,

but the reflection method has the advantage of

simplicity and cheapness.

The light-screen methods permit defection of

the arrival of a shock wave within 1 usee. In

general, detection with a pressure transducer is somewhat less accurate.

4.6. Transducer Output Records

The calibration system must include eq^uipment

for recording the transducer output m some

permanent form, either on photographic film or in punched or printed form, or both. The trans- ducer system includes the electronic circuits which

are directly associated with a given transducer.

Examples are the piezoelectric transducer and

electrometer, the capacitance-type transducer and

oscillator-detector, and the strain gage transducer

and Wheatstone bridge.

Additional electronic equipment must be in- cluded as a part of the calibration system. Triggering and time delay circuits must be provided to start the recording equipment a few microseconds

before the shock wave reaches the transducer.

And appropriate recording equipment must be

available, figure 6.6. The input to the trigger circuit may be the

output from one of the detectors in the shockvelocity measuring system. This trigger-pulse

is fed to an adjustable time-delay generator which

puts out a pulse to the recording equipment a few

microseconds before the shock wave hits the trans- ducer which is being tested. The output from the

transducer system also goes to the recording

system.

The simplest recorder is an oscilloscope fitted with a camera. The scope is set to give a single

sweep when triggered, and by holding the camera

shutter open during the sweep, a photographic

record can be obtained. This record can be superimposed on a voltage-calibrated grid if the grid is lighted during the time the photograph is being

exposed. The photographic record can then be

analyzed to determine the response spectrum of

the transducer system.

The photographic output record can be con- verted to numerical data by manual graphical

means, or it can be scanned by an electronic

device which "reads" the curve at fixed time in- crements and records the numerical values in printed or punched-card form. The conversion

of the transducer system output directly to a numerical output would be more satisfactory, but

z:

TRIGGER AND

TIME DELAY

CIRCUITS OUTPUT

RECORDING EQUIPMENT

TRANSDUCER

ELECTRONIC

CIRCUITS

Figure 6.6. Transducer output recording system.

83
it is difficult to provide analog-to-digital equipment

which can operate sufficiently fast. A third means of recording the transducer

system output was described in chapter 3, section

4. This is a magnetic drum recording which can

be played over and over again into a frequency

analyzer.

5. Evaluation of Test Data

The transducer system is evaluated on the basis

of the recorded response to the step-function

input. From this recorded response the amplitude

characteristic of the response can be compared

to the input step, the predominant frequency

or frequencies and the logarithmic decrement

or damping c];iaracteristic can be determined,

and the amplitude and phase characteristics of

the complex transfer function can be computed.

The general background and the relation of cali- bration to analysis are covered in chapter 1, section 3, and the methods for determining the transfer function are presented in chapters 2 and 3. Here we concentrate on the application of methods

described elsewhere.

5.1. Amplitude Characteristic of Response

Function

Before shock tube tests are made the transducer

system should be calibrated with a static calibration device if possible. For transducer systems

which include piezoelectric transducers, for example, leakage prevents accurate calibration by

strictly static means. In such cases, pseudo-static

or short-time-period static calibration replaces

static calibration. The result of either of the

9

2

0

n

0 100 200 300 400 500

INPUT PflESSUBE. psi Figure 6.7. Static calibration of transducer system.

Figure 6.8. Response of transducer system to step input.

500

400

100

INPUT-STEP AMPLITUDE , pti Figure 6.9. Response amplitude versus input-step

amplitude.

above calibrations is transducer system response

in terms of an electrical output as a function of

pressure input. A typical calibration curve is shown in figm-e 6.7. Figure 6.8 shows an idealized response to a stepfunction input. For the air-to-air tube the step

amplitude is computed from eq (6.14) for side

mounted gages and from eq (6.20) for end-mounted

gages. The step amplitude may also be evaluated

from the transducer response—either by a graphical averaging process or by arithmetic average of numerical values taken at equal time increments.

Since the transducer response is in electrical units,

it must be converted to pressiire for comparison

with the input pressure step. This conversion is made with the aid of the static calibration. A

typical plot of response amplitude versus inputstep amplitude is given in figure 6.9.

5.2. Predominant Frequency or Frequencies

When the predominant frequency of oscillation

of the transducer is far removed from other

84
oscillating frequencies and/or the magnitude of the

predominant oscillations is much greater than that

of the other modes of oscillation, the transducer

response can be approximated by a single-degreeof-freedom model. In this case it is a simple

matter to scale the transducer response record

and count the number of oscillations in a given

period of time. The computation of the damped

frequency is then straightforward.

If more than one major frequency exists (that is, if the transducer has two or more degrees of

freedom), determination of these frequencies is not so simple, and a more sophisticated method of analysis is required. One such method is dis- cussed below in section 5.4; other methods are discussed in chapter 3.

5.3. Logarithmic Decrement or Damping

Characteristic

In a typical transducer system, actual damping

is only a smaU fraction of the critical damping.

This is the case for the system illustrated in figure

6.8 and we assume that the systems we are con- cerned with here are all of this "underdamped"

category. For underdamped, single-degree-offreedom systems the logarithm of the ratio of the

magnitudes of successive maximum (or minimum)

displacements is constant. The following expression is known as the logarithmic decrement [14, p. 205]:

=ln (^). (6.31)

\yn+2/

The logarithmic decrement is related to the frac- tion of critical damping by the equation

Cc V5'+47r2

(6.32)

Knowledge of the damping ratio and the damped

frequency can be used to compute the undamped

natural frequency, w„.

(6.33)

Performance of the transducer system may then

be predicted from standardized curves of magnification ratio and phase angle versus w/c<j„, for a

single-degree-of-freedom system [14, pp. 208-9]. For more complex sj^stems, more sophisticated methods of analysis are required. One such method is discussed in the following section.

5.4. Determination of the Transducer System

Transfer Function

Approximation of the response function may be

carried out by any one of the methods described

in chapter 3. As an example, use the method of

section 3.6 based on closely spaced coordinates.

It is necessary to express the step input-function

in the complex plane in a form compatilile with

the approximation of the response function. For

a step input, the Laplace transform may be shown

[2] to be

Fis) =

j:

Fe-"dt = --AtF (n-icot|) (6.34)

Equation (6.34) represents a complex vector of

amplitude F{(jS)=FmI2 sin {612) and phase angle

^(w) = (-9O°-0/2).

The amplitude characteristic of the transfer

function is given by eq (2.1). For a frequency

(jo=d/At, the transfer function is

2 sin

The phase characteristic |3(w) of the transfer

function is given by eq (2.2).

i3(co)=a(aj)-^(a>) =tan-i|^(^^^+90°J (6.36)

a(a;)=tan-' |)-

where

We have shown here one method for determining

the amplitude and phase characteristics of the

complex transfer function for a transducer sj-stem.

The response function was approximated by

numerical methods and the transfer function

might be computed with the aid of a digital

computer. The methods of analysis are described more completely in chapter 2 and the methods

of approximation are given in chapter 3.

6. References

[1] G. R. Cowan and D. F. Hornig, The experimental

determination of the thickness of a shock front in a gas, J. Chem. Phys. 18, 1008-1018 (Aug. 1950).

[2] R. B. Bowersox and J. Carlson, Digital-computer

calculation of transducer frequency response from

its response to a step function, Prog. Rept. No.

20-331, Jet Propulsion Laboratory (July 26, 1957).

[3] W. Bleakney, D. K. Weiner, and C. H. Fletcher,

The shock tube as a facility for inyestigation in

fluid dynamics, Rev. Sci. Instr. 20, 807 (1949).

[4] R. Courant and K. O. Friedricks, Supersonic Flow

and Shock Waves, Interscience Publ., Inc. (New

York, N.Y., 1948).

[5] A. H. Shapiro, The Dynamics and Thermodynamics

of Compressible Fluid Flow, vol. I, The Ronald

Press Co. (New York, N.Y., 1953).

[6] A. H. Shapiro, The Dynamics and Thermodynamics

of Compressible Fluid Flow, vol. 11, The Ronald

Press Co. (New York, N.Y., 1954).

[7] J. K. Wright, Shock Tubes, John Wiley & Sons, Inc. (New York, N.Y., 1961).

85
[8] L. D. Landau and E. M. Lifschitz, Fluid Mechanics

(transl. from the Russian by J. B. Svkes and

W. H. Reid), Addison-Wesley Publ. Go., Inc. (Reading, Mass., 1959).

[9] I. I. Glass and G. N. Patterson, A theoretical and

experimental study of shock tube flow, J. Aeronautical Sci. 32, 73-100 (Feb. 1955).

[10] I. I. Glass and W. A. M; rtin, Experimental and

theoretical aspects of shock wave attenuation,

J. Appl. Phys. 36, 113-120 (1955).

[11] D. Bitondo and R. K. Lobb, The design and construction of a shock tube, Toronto University

Institute of Aerophysics Rept. No. 3, pp. 48 (1950).

[12] A. E. Wolfe, Shock tube for gage performance studies,

Rept. No. 20-87, Jet Propulsion Laboratory

(May 1955).

[13] R. W. Ladenburg, B. Lewis, R. N. Pease, and H. S. Taylor, Physical Measurements in Gas Dynamics

and Combustion, vol. IX, High Speed Aerodynamics and Jet Propulsion, Princeton Univ. Press

(Princeton, N.J., 1954).

[14] C. R. Wylie, Jr., Advanced Engineering Mathematics,

McGraw-Hill Book Co., Inc. (New York, N.Y.,

1960).

86
7. Periodic-Function Generators

D. F. Muster

1. General

1.1. Types of Periodic- Function Generators

In chapters 2 and 4, respectively, the response

of linear and nonlinear pressure transducers to periodic input functions is discussed. It is demonstrated that (even in theory) th^re are inherent

advantages to calibrating certain transducers

with steady-state (repetitive or periodic) inputs

rather than step inputs. The relative value of

these advantages will be discussed with reference

to each of the periodic-function generators mentioned below.

For convenience, let us restrict this initial discussion to linear transducers; the effect of nonlinearities (particularly, of stiffness) is dis- cussed separately. As sensed and displayed by

a linear transducer system, the shape, magnitude

and time of occurrence of an arbitrary forcing

function are modified. In effect, the transducer

system filters the input and the displayed response

is modified accordingly. If the forcing fimction

is a sinusoid, the response of a linear transducer

wiU also be a sinusoid; thus, the shape of the

input and response are the same. However, the

magnitude and phase of the response may be

markedly different from that of the input. As

was mentioned earlier, this fact places sinusoidal

forcing fimctions in a unique position in the

dynamic calibration of all categories of trans- ducers. Further, it gives experimentalists a feel for the physical significance of Fourier analysis

and the concept of frequency spectra. If any

input can be represented as the sum of a number

of sinusoids, then it follows that the output of a linear transducer can be represented by these same sinusoids modified only in amplitude and

phase. Each is sensed and displayed by the

transducer as if it were present alone (the principle of superposition). The response function is the sum of the output sinusoids.

It is shown in chapter 2 that from a knowledge

of the steady-state characteristic of a system the

response waveform for any specified input wave- form can be computed. For example, if the specified input includes components at frequencies in a given range and it is fcaown that the transducer

system is linear in the same frequency range, then

the following relationship will hold

:

X{(^)=kF{<^)€-^'>', (7.1)

where X{ui) and F{io) are the output and input

spectra, respectively, k an amplitude factor, and

' Professor ol Mechanical Engineering, The University of Houston.

f 11)

r

-T+Tp

C

Tp T T*Tp 2T

Figure 7.1. Periodic input functions.

(a) Sine function

(b) Square-wave functions

(c) Pulse function

(f the phase shift, all in the given frequency range.

The specific periodic functions of primary use

in the dynamic calibration of pressure transducers

are sinusoids, square-wave, and impulse functions

(fig. 7.1). In consequence, in this chapter we will be concerned primarily with generators capable

of producing pressure changes which approximate

closety one of these wave forms. These include

acoustical shock generators, rotating valve and

piston-in-cylinder devices, sirens, and electrical,

mechanical, and electro-mechanical exciters.

1.2. Place in Pressure Transducer Calibration

The descriptive phrase "frequency response"

implies that the response under consideration

(pressure, acceleration, velocity, etc.) is a function

of frequency. Two examples are the measured

87
output of an electronic amplifier in response to

sinusoidal inputs of known frequency from an

oscillator, and the oscillating electrical output of a microphone in response to known sinusoidal

changes in pressure level. In the first case, it is relatively easy to generate sinusoidal signals up

to frequencies in the megacycle per second range. In the second case, it is diflftcult to generate a

relatively distortion-free sinusoidal sound field at high amplitudes and/or high frequencies. The

latter is in conflict with the requirements for shock wave and rocket-engine research, where the

ambient acoustic environment may be characterized by amplitudes of greater than one atmosphere ( 188 db re 0.0002 microbar) and frequency

components up to 30 kc/s. To date, sinusoidal

pressure oscillations at these amplitudes and

frequencies cannot be produced in a gaseous

medium. At lower amplitudes and/or frequencies

calibrating devices have been developed and are described in a later section.

The lack of an acoustic high-amplitude, highfrequency sinusoidal calibration device is not due

to a lack of ingenuity on the part of experimentalists. It is due to certain fundamental physical

limitations associated with the problem of attempting to force the motion of gases in a sinusoidal manner over a wide range of frequencies. If the

amplitude is kept constant, as the forcing frequency is increased the wave form degenerates from a sinusoid to a saw tooth. This is due to thermodynamic causes, particularly to shock wave interaction. As a result, sinusoidal acoustic exciters

in pressure transducer calibration are limited to pressure levels and frequencies considerably lower

in scale than those associated with most rocketengine research.

In some cases, input fimctions to a pressure

transducer in the range of frequencies and/or

amplitudes of interest can be more easUy generated

as a square wave or repetitive impulse. The

superposition principle will still apply, if the

transducer system is linear, and the frequency

response for either input will be the same as for

the sinusoid, although the method of interpreting

the results is different. There are two principal limitations on calibration with impulse generators. The impulse must

be of short duration compared with duration of

the finest detail of the output transient which is to be reproduced accurately; that is, the impxilse

spectrum must be flat over the entire frequency

range of the device under test. In turn, this

implies that, in order to achieve a significant

response at some frequencies the impulse must be

so great in amplitude that the transducer system

under test is often forced out of its linear range at other frequencies.

The second disadvantage is that, when testing

wide-band devices, low-frequency eflfects are difii- cult to observe. The one-test advantages of

impulse methods are offset by the incapability of

transducer systems to react in the same way to all frequencies and, as a consequence, low-frequency

cutoff limits their effectiveness.

In summary, in the low- and medium-pressure

range, the available methods of dynamic calibration include the use of repetitive acoustical-shock

generators, sirens, rotating-valve and electromechanical exciters. Piezoelectric devices are

useful in the higher frequency and pressure-level

ranges ; strictly mechanical devices are most useful

in the lower frequency and, most often, the lower

pressure-level ranges.

2. Acoustical- Shock Generator

2.1. Description 2.2. Theory

The acoustical-shock generator is well adapted

to the dynamic calibration of microphones and

other low-pressure transducers. It is characterized by the following features: low power

requirement, suitability for automatic operation,

and the high-energy, short-duration acoustic

pulses it generates are accurately reproducible. In principle, the operation of the device is similar

to suddenly bursting an inflated paper bag. A

piston in a cylinder is forced against a compression

spring and held in this retracted position. The

open end of the cylinder is covered by a paper strip.

When the piston is released, the air ahead of it is compressed adiabatically until the paper strip

breaks at a certain critical pressure. The piston

is again retracted. A fresh strip of paper is drawn

over the end of the cylinder and the process

repeated. The device is motor-driven and the

length of time between impulses is controlled

through a gear train. In a typical generator, the

pressure at bursting is about 50 psi, the time

between pulses from 3 to 30 sees.

The shock wave associated with each burst is propagated in an essentially spherical manner. The value of the energy in the pressure wave can be computed by integrating the square of the

pressure-level function from t=0 to t-^ oo . To the

first order, the pressure wave may be assumed to have the shape of a saw-tooth, whose base represents the duration of the wave (say, 0.4 msec). At

a radius of 1 m and for a pressure of 200 dynes/cm^,

the energy content of the shock wave is approximately 0.25 watt-sec and the power of the acoustic source is about 600 acoustic watts.

2.3. Evaluation of Test Data

Oscillograms of the pressure pulses from an

acoustical-shock generator (as measured at a given

radius and various angles with the generator at the origin) show that the shock wave propagates

spherically. The main pulse is essentially triangular and a typical value of its duration is, say, 250 jusec. Whereas the amplitude of the main

88
pulse may be almost 100 db (re 0.0002 microbar),

the amplitude of succeeding pulses is considerably

less (down 20 db or more)

.

A typical frequency spectrum is essentially flat from about 1000 to 8000 c/s. This is roughly two

octaves below and one octave above the center frequency of 4000 c/s, which corresponds to the

duration of the main pulse. At higher frequencies,

the diameter of the generator opening is not small

compared to the acoustic wavelengths and the

pressure waves associated with these frequencies

are not propagated spherically. As a consequence,

each frequency spectrum will display directional

characteristics and the measured spectrum is no

longer flat. In addition to the gradual decrease in level asso- ciated with the radiation characteristics of the

source, there are resonances and anti -resonances observed in the measured spectra. These are due

to the presence of a short tube just in front of the

generator cylinder and along an extension of its axis. At the moment of burst the cylinder plus

tube acts as a quarter-wave resonator. At the end

of the piston stroke, the tube alone acts in the same

manner. In effect, this affects the frequency

spectra by strengthening certain frequency components and weakening others, causing relative

peaks and nulls in the measured pressure levels. The reproducible quality of the pulses from an

acoustical-shock generator have been well established. A generator was placed in an anechoic

chamber and the relative maximum sound-pressure

levels per octave were measured. The results of many tests were analyzed in this way and the peroctave sound pressure levels compared. The

deviations are less than ± 0.5 db for all octaves

from 100 to 6400 c/s; thus, the reproducible quality

of the pulses produced by the generator was established. Above 6400 c/s, the deviation was as much as ±3 db.

3. Rotating-Valve Generator

3.1. Description

It is often desirable in the calibration of pressure

transducers that the forcing function be of the

form of a series of repetitive, precisely controlled

pressure pulses. A mechanical device which generates such a forcing function (but which is necessarily limited to low frequencies) is reported

by Hermann and Stiefelmeyer [1] ^ (and attributed

to Staiger and Mohilo) . A schematic representation of the device is shown in figure 7.2a and of a

similar device developed by Eisele [2] in figure 7.3a. Pressure fluctuations of the kind in figure 7.2b

[1] or 7.3b [2] are possible by suitably arranging

the rotating valves and the sources of high-pressure

gas. The levels of the pressure in either figure

7.2b or 7.3b can be preset precisely by means of pressure gages. By adjusting the speed of the

driving motor, a series of valves are opened and

closed sequentially at the desired repetition rates. The transducer is subjected to a sequence of pre- determined pressure changes, the duration of which are determined by the speed of the rotation.

3.2. Evaluation of Test Data

At higher repetition rates, Eisele [2] has found

that the pressure steps are distorted by resonance

effects which can be attributed to the inertia of the gas column in the valve system (fig. 7.3b).

As a consequence, the usefulness of the device as a

calibrator is limited to frequencies below where

these effects are observed.

Normally, the square pressure pulses generated

by the rotating-valve device are not truly shock

waves. Unlike the short-term, high-intensity

pulses generated by the acoustical-shock generator,

the pulses from the rotating-valve generator are

of longer duration (say, about 100 msec as compared to 250 /isec) and lesser intensity (say, about

2 Figures in brackets indicate the literature reference on p. 95.

1. ROTITING VALVE

2. PRESSURE INLET PORT

3. EXHAUST PORT

4. VALVE HOUSMO

5. TRANSDUCER

Figure 7.2. Rotating ralve pulse-function generator.

(Reproduced from [1] with permissioa from Franzis-Vcrlag)

(a) Constructional features of pressure calibrator

(b) Representation of oscillograph trace for single-step callDratlon

694-211 0-63—

7

89
"I- •<\() COMPRESSED AIR FLASKS

b) ROTATING GATE VALVE

c) TRANSDUCER

a d) MOTOR

b MOTOR SPEED 390rpm'--'760rpni

Figure 7.3. Rotating-valve staircase-function generator.

(Reproduced from [2] with permission from R. Oldenbourg Verlag)

(a) Schematic of pressure calibrator

(b) Calibration of curves derived from apparatus shown in (a) at two

different motor speeds

10 to 15 psi per step as compared to 200 psi).

A major disadvantage of the rotating-valve

generator is the presence of the relatively long

tube between the transducer under test and the

rotating-valve mechanism. In effect, it introduces

a half-wave resonator into the calibration system

which limits the effectiveness of the generator at high frequencies.

4. Sirens [3, 4, 5, 6, 7] 4.1. Description

The siren-tuned-cavity oscillator is a useful

device for generating periodic (but not necessarily

sinusoidal) pressure waves for the calibration of microphones and other low- and medium-pressure

transducers. The essential elements of the generator are a cylindrical chamber with an axial

orifice in one end and a revolving disk or flanged

wheel with a number of equally spaced holes

HGH-PRESSURE AIR

a \PERFORATED

DISK

FiGUBE 7.4. Siren-tuned-cavity generator.

a) Rotating disk

b) Timed cavity

(c) Typical waveform

arranged around its periphery. The wheel is positioned such that the holes in its periphery are

close to the axial hole in the cyhnder and, as the wheel rotates, the flow of air from the cylinder

is interrupted. A schematic representation of

this arrangement is shown in figure 7.4 with a sketch of the waveform produced by such a

generator.

As the name "siren-timed-cavity oscillator"

implies, two adjustments are necessary in order

to operate the generator effectively. For each

rotational speed of the siren (which, with the

number of holes, determines the frequency of the

pressm-e waves), the length of the cyhnder must

be adjusted so that it will act as a half-wave

resonator; thus, the speed of the wheel determines

the basic pulse length (frequency) and the tuning

adjustment reinforces the pressure wave and

determines its amplitude at the transducer

location (fig. 7.4).

For purposes of calibration, the value of pres- sure at the cavity-transducer interface is measured

with a previously calibrated transducer, or it is computed using methods described by Oberst [8] When measured pressures are used as a standard,

the caUbrated gage must be an instrument with a uniform frequency response up to frequencies

several times higher than the test frequency.

Oberst [8] shows that the maximum pressure

magnitude at the transducer, figure 7.4, is given by

\Po\=Pem.

Here Pe is static pressure (as measured with

a Bourdon gage) at the end of the tube where the

opening is located (called excess pressure by

Oberst), and I is the tube length. The term

/3(=8MiTVVPav<i^) is the attenuation constant per

unit tube length for a plane wave of frequency /in

90
a medium of mean equilibrium density p^^. In

the expression for /3, a is the velocity of sound.

The actual damping for waves in air is small

for moderate frequencies (ji3=0.24X 10~* for /= 1000/2 TT in air), but increases rapidly with

increasing frequency.

4.2. Evaluation of Test Data

The saw-tooth shape of the pressure wave at the pressure transducer can be approximated by

the series p n 1 P= — rsin koit,

where Po is the peak value of pressure. Thus,

although the basic pulse length is established by

speed of rotation of the siren wheel, each of the

pulses consists of many harmonic components

whose respective amplitudes are an inverse function of frequency.

The saw-tooth shape is due to tlie nature of the

reinforcement making any pulse steeper-fronted

as the wave propagates. Even quasi-sinusoidal

pulses are distorted and originally steep-fronted

pulses (such as can be generated by a siren)

become close approximations of the saw-tooth

shape.

It has been shown that a saw-tooth wave will develop within the tunnel cavity when the pres- sure pulse generated at the orifice is greater than

10 percent of the pressure of the air flowing in the

cavity. Peak pressures of about 30 psi (equivalent to over 190 db re 0.0002 niicrobar) are possible. Basic repetition rates from 50 to 1000 pulses

per second can be achieved without difficulty.

5. Piston-in-Cylinder Steady- State Generators

5.1. General

A generator which can produce a steady-state

(rather than repetitive-pulse) forcing function

(sinusoidal at low amplitudes, distorted sinusoids

at high amphtudes) has many advantages, as has

been mentioned in this chapter and earlier. In

particular, nonlinear effects occur when an attempt

is made to generate pressure waves characterized

by both high frequency and high amphtude.

Thus, low-frequency, low-amphtude pressure waves can be sinusoidal, whereas high-frequency,

high-amplitude waves are not sinusoidal and, in many cases, may exhibit shock characteristics.

The obvious way to pi'oduce a sinusoidal pres- sure variation is by means of a piston-in-cyMnder

device. Pressure changes produced in this manner

can be reinforced by resonance effects at fre- quencies related to the geometry of the cyhnder

and the physical properties of the fluid. There is an extensive literature on the distortion of pressure waves due to nonlinearities in these properties

under dynamic conditions [9, 10]. In particular, Schmidt [9] and Betchov [11] have

studied the formation and behavior of oscillating

shock waves which are generated by the coales- cence of compression waves.

5.2. Theory

A theory for the forced motion of gases in tubes

is developed by Betchov [11]. It is assumed that

the caUbration device is a long tube, closed at one

end with a piston at the other (fig. 7.5). The

motion of the piston is sinusoidal with circular

frequency aj( =27r/). The gas in the tube is assumed to behave as an ideal gas {P=pRT). In

this one-dimensional theory, for small motions

(i.e., u, the pressure wave velocity is small), a

linear solution is a close approximation of the

actual motion. As in all idealizations of dynamical systems which do not account for friction in its various forms, infinite amplitudes are predicted

at the resonant frequencies for which the length

of the tube is a multiple of the half-wavelength

^=j; a is the velocity of sound in air.^

In an actual piston-in-cyUnder device, the

motions of the gas are not necessarily small (and

tlie Unear solution is not valid). In turn, the

x-x,cosa>t

Figure 7.5. Pixton-in-cylinder model.

91
shape of the pressure wave is determined by the

piston motion, the gas-dynamics and acoustic

factors associated with the phenomena, frictional

effects at the lateral boundaries and inherent in the

gas and, when the motion of the gas is forced into

the nonlinear domain, shock fronts. The shock fronts, whether strong or weak, tend

to distort the shape of the pressure wave and form

oscillating shock waves in the tube length.

Energy is dissipated in the formation and continued motion of the shock waves. It is suflficient to raise the gas temperature slightly, but not so much that the resonant frequencies are affected

appreciably. Losses (watts dissipated per unit

piston area) may, in fact, exceed the average

power delivered.

From the observations of Schmidt and others

it is clear that, at resonance, oscillating shock waves are always present in the gas column of a

piston-in-cylinder device. Even at nonresonant

frequencies, the motion of the gas is sufficiently

complex that it can safely be assumed the amplitude and phase of the gas will not agree with that

of the piston motion. The nonlinear and cumulative effects of friction can be seen in the trans- formation of an initially shockless and almostsinusoidal motion into one characterized by shock waves and a distorted pressure-wave shape.

Some pressure-wave generators use a liquid as the active medium instead of gas. The principle

differences between liquid- and gas-piston-incylinder devices are associated with the incompressibility of liquids. The dynamics of long, liquidfilled tubes have been studied by several investigators [12, 13, 14, 15]. Their studies have

included resonance effects, coupled vibration,

cavitation, and the softening effect of entrapped

air.A basic factor in each of these studies is the bulk

modulus (or its inverse, compressibility) of liquids.

AP

It is defined as the ratio — .tt<t7 where AP is the AV/V

increment of pressure applied to the volume of

liquid V to cause a decrease in volume—

Bulk modulus has the dimensions of pressure and

depends upon both pressure and temperature

The stiffness of a liquid-filled, piston-in-cylinder

device can be computed from the equation

6. Electrical and Mechanical

6.1. General

The source of force (or motion) in many pres- sure-transducer calibration systems is an electrical,

mechanical or electromechanical vibration exciter

capable of delivering sinusoidal excitation to the

transducer over a wide range of frequencies.

Among the most useful of such generators are the

following types: piezoelectric, mechanical directdrive and force-reaction, hydraulic, pneumatic,

where A and I are the cross section and length,

respectively, and Kn the bulk modulus. Typical

values of K„ for several liquids are

Liquid Km

Water (68 °F)

psi

3. 5 X 105 Methyl alcohol (68 °F) 1. 76X 106 Carbon tetrachloride 1. 63X 106 Kerosene _ _ . 1. 91

106

5.3. Equipment

The basic elements of a piston-in-cylinder cali- brator consist of a piston in a liquid-filled cylinder

to which the transducer is connected. The cylinder can be pressurized and the piston is connected

to a sinusoidal-motion driving mechanism. In a

recent report, Cobb [16] describes such a device

for which the driving force is furnished by one or two electromagnetic exciters, each capable of delivering up to 2500 lb-force at frequencies up to 2000 c/s. A plastic diaphragm seal is fastened

to the end of cylinder with the piston acting

against the side of the diaphragm away from the

liquid. This arrangement permits separate application of the static and dynamic components of pressure. Cobb suggests two systems: one in which the

cylinder is mounted rigidly to a stiff structural

frame and the piston to the exciter, and a second

in which both the piston and cylinder are secured

to the armatures of separate exciters arranged to operate out of phase. The second arrangement

performed in a more satisfactory manner.

5.4. Evaluation of Test Data

Cobb [16] reports the two-exciter calibrator sys- tem is capable of supplying ±150 psi with a static

pressure of 500 psi in the cylinder and at frequencies between 10 and 1000 c/s. Above 500 c/s there

is a tendency to distortion at the larger pressurevariation amplitudes. A major difficulty in operating the device is the softening effect of entrapped

gas on cavity stiffness. This tends to lower the

frequency at which the first resonance occurs and

is the principal reason for limiting the effective

range of the calibrator to 500 c/s. Exciters [17, 18, 19]

and electrodynamic. They cover the frequency

spectrum from almost-zero c/s to 10,000 c/s and

can deliver force amplitudes as high as several

thousands pounds. No one machine can span

these ranges, but from this group, machines can

be selected which cover large portions of them.

If a range of calibration frequencies from 5 to 10,000 c/s is desired, the electrodynamic-type

machine most nearly satisfies the requirements of the ideal calibrator. For high-frequency, high92
force applications, the piezoelectric type is most

useful, despite its limitation of being capable of producing only very small displacements. The

latter disadvantage makes it difficult to use as a low frequency source. In addition, the motion of

the exciter must be measured by interferometer

techniques or external displacement-measuring

sensors such as the capacitance pickup. Mechanical direct-drive and force-reaction types are used

only for low-frequency (less than 100 c/s) applications. Hydraulic and pneumatic exciters are used

least of all as sources of vibration, but (as we have

mentioned earlier in this chapter) they are used

directly in pressure-transducer calibration. When

used as vibration exciters their limitations (sensitivity to distortion, low-force output at high

frequencies) outweigh their advantages (force

transmitted can be calibrated under static conditions, the generated force can be interrupted

abruptly). Conversely, their direct use (that is, when the transducer is an integral part of the

fluid chamber) is well established and is reported

earlier in this chapter.

A vibration generator which is to be used in calibration work should provide:

1. Distortion-free sinusoidal forces (or motions).

2. Stepless variation of frequency and output

within specified limits, easily adjustable

during operation.

3. Horizontal as well as vertical motion, if required.

4. Features permitting the use of absolute cali- bration methods when calibrating the

calibrator.

5. True rectilinear motion normal to the main

axis of the exiciter without the presence of other motions.

6.2. Piezoelectric Exciters

The piezoelectric vibration machine uses a stack of piezoelectric elements as a source of mechanical vibration. The piezoelectric elements

are cut from crystals having piezoelectric properties or are made from ferroelectric ceramics.

When the elements are subjected to mechanical

pressure, electrical charges proportional to the

pressure are generated. If the element is compressed between two electrically connected metal

plates applied to opposite faces of the element,

and a metal band is placed around the middle of

the element between the band and the end plates, an electromotive force of considerable magnitude

will be developed between the band and the end

plates, upon small changes of pressure between

the end plates. The piezoelectric effect is re- versible. A variable voltage applied between the

plate and the band will cause the element to expand and contract mechanically in synchronism

with the varying applied voltage [20, 21].

Three types of materials are used in transducers

and exciters which make use of this effect. These

are Rochelle salt, ammonium dihydrogen phosohate (ADP), and barium titanate (BaTiOs). In

either case, a strain across the element results

PIEZOELECTRIC STACK

LIQUID

PRESSURE TRANSDUCER Figure 7.6. Piezoelectric exciter for a pressure-transducer

calibration system

when a voltage is applied. The direction of the

strain depends upon the orientation of the crystallographic axes of the original crystal relative to the crystal element, or in the direction of polarization of the ceramic element, and on the location

of the electrodes. The strain in the piezoelectric

stacks of elements (as shown in fig. 7.6) alternately causes an increase and decrease in the thickness

of the elements, thereby imparting motion to the

liquid-loaded plate of the calibrating chamber.

These exciters are usable in the range from

1,000 to 20,000 c/s, the span of the range depending upon the configuration and size of the particular unit; the peak-to-peak displacements are 0.001

in. or less. As a consequence of the latter,

mounting the piezoelectric stack is important.

A massive backup base holds fixed the end of the

stack away from the liquid. In order to terminate

the end of the stack in an essentially infinite mechanical impedance, the base material is pre- ferably highly damped and the base itself made as massive as possible.

When a voltage is applied to the electrodes of

the piezoelectric stack, the magnitude of the deformation produced is dependent upon the fre- quency of the applied voltage and the frequenc}'

response of the electromechanical system. If the mass of the load carried by the piezoelectric

element in combination with the stiffness of the

element gives a resonant frequency well above the

highest operating frequency, the system is spring

controlled, providing a displacement proportional

to the voltage and independent of frequency.

For a liquid loading, which is essentially mass-like

and produces a resonant frequency in or near the

operating frequencies, a typical resonance effect

is observed.

The principal features of the piezoelectric

exciter are:

1. The design can be made compact and rigid,

providing good high-frequency (\\-ith highamplitude) operation.

2. Normally, only small-displacement (and, as a result, relative^ high-frequency) operation is possible.

3. The electrical load presented to the poAver source is highly reactive and is of high

impedance. Unless it is possible to use many piezoelectric elements (connected

electrically in parallel but mechanically in

series) several Idlovolts may be required

in order to produce the requisite output.

93
6.3. Electrodynamic Vibration Machine

Electrodynamic exciters are used in pressure

transducer calibration in one of two ways: as sources of sinusoidal motion for diaphragms, or in

piston-in-cylinder calibrators such as that reported

by Cobb [16]. In the former case, a pressure trans- ducer is placed in the end of a liquid-filled chamber

at the opposite end of which is the driven diaphragm. The latter example is discussed in sec- tion 5 of this chapter.

The electrodynamic exciter derives its name

from the method of force generation: the force

causing the motion of the table (to which the load

is attached) is produced electrodynamically by the

interaction between an alternating current flow in

the driver coil and the intense magnetic field which

cuts the coil (fig. 7.7).

A wide range of frequencies is possible, with a properly selected electric power source, from 0

to 20,000 c/s. Small, special-purpose machines

have been made with the first axial resonance mode

above 26,000 c/s, giving a resonance-free, flat response to 10,000 c/s [17]. Frequency and amplitude are easily controlled by adjusting the powersupply frequency and voltage. In the case of an

electronic power supply, the vibration machine

load is not reflected back to the oscillator signal source, giving zero frequency regulation at any

load.

Rated force (continuous sinusoidal peak value)

varies according to manufacturer and unit design,

but the smallest self-contained units deliver approximately 1 lb, the largest, about 25,000 lb. For most transducer calibration applications,

machines rated between 200 and 2500 lb are used.

Pure sinusoidal table motion can be generated

at all frequencies and amplitudes. Inherently, the

table acceleration is the result of a generated force

proportional to the driving current. If the electric power supply generates pure sinusoidal voltages and currents, the waveform of the table' motion will be sinusoidal and background noise will not be present. Operation with table waveform

distortion of less than 10 percent through a dis- placement range of 10,000 to 1 is common. Random vibration, as well as sinusoidal vibration, or a combination of both, can be generated

by supplying an appropriate input voltage.

ELECTROMAGNET

Figure 7.7. Electrodynamic vibration machine.

ECCENTRIC BUSHING

b Figure 7.8. Displacement adjustment method.

(a) ffi=6. Displacement=6—a=0

(b) a?^6. Displacement=2c (PEAK-TO-PEAK)

6.4. Low- Frequency Pneumatic Sinusoid

Generator

For low-trequency, low-amplitude calibration, a pneumatic generator has been developed [22]. The

generator consists of an electric motor operating

through a variable-speed reducer to drive a cam

which positions the input crank of displacementbalance, pneumatic-pressure transmitter. In fig- ure 7.8 is a method of providing stepless adjustability of crank offset between zero and full dis- placement. The boundary surfaces between shaft

eccentric 1 and eccentric bushing 2 and between 2 and the connecting rod are circular cylinders. For

a fixed-displacement setting, the eccentric 1 and

the eccentric bushing 2 remain fixed relative to each other but rotate about the rotation center of

the driving shaft. The displacement adjustment

is made by rotating eccentric bushing 2 relative to

shaft eccentric 1 [17]. Nominally, the pressure transmitter has an out- put range of from 5 to 25 psi, but different ranges

can be obtained by adjusting the operating stroke

at the input crank.

6.5. Electromagnetic Methods for Calibrating

Some Pressure Transducers

Transducers which use diaphragm^ of magnetic

materials can be calibrated by a direct-excitation

technique. E. J. Diehl and H. Visser of the in- ternal combustion laboratory of the Technische

Hogeschool in Delft are reported to have used an

iron core (solenoid) placed coaxially in close proximity to the diaphragm (fig. 7.9). The oscillating

motion of the diaphragm is sensed by a capaci- tance pickup.

The damping present in the diaphragm assembly

can be sensed and measured by recording the decay of the diaphragm vibration on a level recorder.

From these decay-rate measurements, the logarithmic decrement can be computed directly.

Very low-pressure, acoustical transducers can

94
mm

7^

COIL

mm

FiGUBE 7.9. Electromagnetic excitation of a transducer diaphragm.

be calibrated by fastening them directly to the

table of an electrodynaraic vibration machine [23]. By exposing the pressm-e-sensitive diaphragm to the atmosphere, the oscillatory motion of the trans- ducer produces a corresponding pressure variation on the sensitive element. The absolute movement

of the diaphragm can be sensed and measured

optically (at low frequencies) or by a capacitance

pickup. The frequency range over which this method has been used is reported to be 3000 c/s

under normal condit'ons and, with special equipment,

8000 c/s [23]. Pressme levels are low (usually no more than 50 to 60 db or, say, 0.1 dyne/cm^).

7. References

[1] p. Hermann and G. Stiefelmayer, Contribution to the electrical measurement of pressure changes (in German), Elektronik, No. 4, 94-98 (1956).

[2] E. Eisele, Calibration of gages, II dynamic calibra- tion (in German), Arch. Tech. Messen, J137-7

(May 1951).

[3] C. H. AUen and B. G. Watters, Siren design procedure,

J. ASA 31, 842 (1959).

[4] C. H. Allen and B. G. Watters, Siren for producing

controlled wave forms with amplitude modulation,

J. ASA 31, 463 (1959).

[6] C. H. Allen and B. G. Watters, Siren for producing

controlled wave forms with high intensities, J. ASA 31, 177 (1959).

[6] R. Clark Jones, A fifty-horsepower siren, J. ASA 18, 371

(1946).

[7] C. H. Allen and I. Rudnick, A powerful high-frequency siren, J. ASA 19, 857 (1947).

[8] H. Oberst, A method for producing extremely strong

standing sound waves in air (in German), Akust. Z. 5, 27-38 (1940). English translation by L. L. Beranek, J. ASA 12, 308-309 (1940).

[9] E. Schmidt, Large amplitude oscillation of gas columns

in piping (in German), Z. VDI 79, No. 22, 671-673

(June 1935).

[10] C. Mayer-Schuchard, Oscillation of air columns with

large amplitude (in German), VDI Forsch. 376

[B], 1, 13-22 (Jan.-Feb. 1936).

[11] R. Betchov, Non-hnear oscillation of a gas column,

Phys. Fluids 1, 205 (1958).

[12] R. W. Boyle and D. Froman, Vibrations of liquids

in cyUndrical tubes, Nature 126, No. 3181, 602

(Oct. 18, 1930).

[13] R. D. Fay, Waves in a liquid-filled cylinder, J. ASA

24, 459 (1952).

[l4] G. S. Field, Longitudinal and radial vibrations in liquids contained in cylindrical tubes, Can. J. Research 5, 131 (1931).

[15] Y. C. Lee and C. C. Miesse, On the forced vibration

of a tank of hquid. Shock & Vib. Bull. No. 22, USNRL, 125 (July 1955).

[16] A. W. Cobb, Design of a dynamic calibrator for pressure transducers, MS Thesis, UCLA Engrg.

Dept., UCLA Engrg. Lib. (1956).

[17] C. O. Harris and C. E. Crede, Shock and Vibration

Handbook, Vol. I, Ch. 16; Vol. II, Ch. 25, McGrawHiU Book Co., Inc. (New York, N.Y., 1961).

[18] L. Beranek, Acoustic Measurements, John Wiley &

Sons (New York, N.Y., 1949).

[19] E. Rule and T. A. Perls, Hand-held calibrator for pressure-measuring systems, J. ASA 32, 535 (1960).

[20] W. P. Mason, Piezo Electric Crystals and Their

Applications to Ultrasonics, Appendix D, Van

Nostrand Co. (Patterson, N.J., 1950).

[21] W. G. Cady, Piezoelectricity, McGraw-Hill Book

Co., Inc. (New York, N.Y., 1946).

[22] D. P. Eckman and J. C. Moise, A pneumatic sine- wave generator for process control study, Proc. ISA 7, 13 (1952).

[23] W. Erler, Measurement methods for the calibration

of structure-borne vibration and sound-measuring

instruments (in German), Hochfrequency Technik

und Elektroakustik 68, 18 (May 1959).

95

8. The Electronic Compensator

E, L. Michaels ' and G. F. Paskusz

1. General

The electronic compensator is an electronic

device designed to provide a transfer function

which is the inverse of the transfer function of a given transducer. When an electronic compensator, whose transfer function is exactly the inverse

of a given transducer transfer function, is incorporated into that transducer system, the result is that the input or driving function waveform and

the output waveform are exactly alike. It was pointed out in chapter 2, section 1.4, that

knowledge of a transducer's transfer function and

of the system response is sufficient for the determination of the system driving function. Thus,

if a pressure transducer's output and transfer

characteristics are known, its input can be obtained. That the operations necessary for this determination may often be laborious must be

obvious from the preceding chapters. Moreover,

the results are valid only for output ranges severely

limited by the transducer frequency response.

It is therefore generally desirable to extend the

usable frequency range of a transducer system and

to reduce the labor involved in the determination

of the driving function. The electronic compensator discussed in this chapter is designed to perform both of these functions.

A general scheme for the representation of trans- ducer systems was developed in chapter 2, and is in part repeated in figm-e 8.1. In this scheme the effect of the transducer on

the system is represented by a "transfer function."

Basically, this transfer function is a description of

the input-output relationship, and so it may be an

equation, a set of equations, or one or more

characteristic curves. Equation (1.1), characteristic of the transducers described in this report, is reproduced here:

where

wx+cx

kx =j{t)

,

/(i)= driving function, a function of time

a;=response of system, a function of time

m, c, ^=physical constants previously defined.

The transfer function is an operator implicit in

the equation. The equation may be written in

operator form as

2. The Principle of the Compensator

by 1/T, the equation may be rewritten as [T-^x=M. (8.2)

The response of the system may be obtained by

the inverse operation. The inverse operation,

[T], is defined by eq (8.3) as [r][T-i]x=x=ir]/(0, (8.3)

where [T] is the operation which, when applied

to [T-^x, will yield x. The operator [T] in the time domain thus represents all the main operations involved in solving the differential equation.

Equation (8.3) relates the known output of

the system, x, to the driving force /(/) which is to be determined. The solution to this equation

may be obtained by the same scheme as that

employed for the solution of the transducer eq

(8.2), i.e., by the use of the inverse operation.

Thus

[T-'][T}f{t)=fit)MT-']x. (8.4)

(1.1)

(8.1)

where the expression in the brackets is considered

as operating on x. If this operator is designated

INPUT

FUNCTION

fit)

DIFFERENTIAL EQUATION "TRANSFER FUNCTION- RESPONSE

FUNCTION

x(t)

The electronic compensator, by performing

the operation implied in eq (8.4), thus generates a system output (an output voltage) which is analogous tof{t), the driving fimction.

The transducer system, including the compensator, may thus be described b}- the diagram of

figure 8.2.

Figure 8.1. A general transducer system.

1 C^'airman of Electrical Engineering, The University of Houston.

2 Associate Professor of Electrical Engineering, The University of Houston. Figure 8.2. The transducer system with compensation.

97
3. Compensator Circuit

The electronic compensator is an electronic

circuit whose transfer function is the inverse of

the transfer function of the transducer. The

compensator is resigned to perform the operation

[T"'], i.e., to solve the differential eq (1.1). It

is convenient for the discussion of compensator

operation to modify this equation by using some

of the conventional substitutions discussed in

chapter 2. Thus, in terms of the undamped

natural frequency, a;„, and the damping ratio,

f, eq (1.1) may be written in the form

±x+^x+x=-^-^=git), (8.5)

where g{t) differs from j{t) only by the constant

scale factor h. This is an ordinary, second-order,

linear differential equation with constant coeffi- cients. Equations of that type are of the form

Ax+Bx+Cx=g{t), (8.6)

where both x and g{t) are functions of time and A,

B, and C are constants. Solution of this equation

by electronic analog computer is simple and may

be accomplished by use of an analog computer

scheme similar to the one shown in figure 8.3,

where d/dt represents a differentiator; A, —B, and

C represent constant multipliers and 2 represents

a summation amplifier.

Here the transducer output, x, is differentiated

twice to yield x and x, respectively. Then x is multiplied by C, x hy B, x by A, and the three

terms Ax, Bx, and Cx are summed, thus resulting

in the computer output g(t).

A slightly more detailed diagram of the elec- tronic compensator is shown in figure 8.4 [1].^ Two amplifiers are used per stage to provide

negative signals corresponding to the variables

X, X, and x, for summation, and for final inversion

after summing. Amplifiers labeled CF are cathode followers; those numbered 1, 2, 4, and 6 are phase inverters or sign changers; and those

numbered 3 and 5 are differentiators. Amplifier

6 is also an adder or summing amplifier as well as a phase inverter. Constant multiplication is achieved by means of potentiometers. A is adjusted by means of the ganged potentiometer

while B is adjusted by the single potentiometer

shown. C is unity in this case and thus needs no

adjustment.

OUTPUT

9(1)

FiGUEE 8.3. Analog computer schematic for solution of eq {8.6.).

a

Figure 8.4. Analog computer schematic diagram of electronic compensator

' Figures In brackets Indicate the literature references on p. 104.

98
4. Calibration and Operation of the Compensator

The calibration and operation of the electronic

compensator involve (1) determination of the

constants in eq (8.6) by static or dynamic calibration methods, and (2) setting the compensator

constants.

4.1. Calibration

Static calibration methods may yield results

with an error no greater than 0.01 percent for

quartz crystals. Piezoceramic transducers, which

are more sensitive than quartz crystals, have a much lower internal resistance. The time constant of the piezoceramic transducer is thus re- duced and the calibration accuracy at any given

reading speed is much reduced below that of the

quartz crystal. It then becomes mandatory to use ceramic calibration by periodic, step, or impulse function excitation.

A comparison of eqs (8.5) and ('8.6) shows that

the constants w„, f, and k specify completely the

constants needed for the compensator eq (8.6).

The natural frequency without damping (w„) maybe derived from the damped frequency (co^) if the damping ratio f is known. The damping

ratio, in turn, may be obtained from the logarithmic decrement in the following way.

It was shown in chapter 2 that the response of a slightly damped system to step function excitation is of the form

x=Ae~"' sin w<, (8.7)

which is a damped sinusoid of the form shown in figm-e 8.5.

At successive peaks, e.g., and x,,, sin w^wl

and therefore

X3«^e-«'3 (8.8)

and the ratio between two successive peaks is

(8.9)

where ar is called the logarithmic decrement.

The time of one period, r, is related to the fre- quency by

J=\- (8.10)

Also, 27r/=co, and c/2m=a (see chapter 2), so

that, from eq (2.11),

Thus, the damping ratio may be determined

from a dynamic calibration. From the damping

ratio and the equation

a; =co„V(l-n (8.12)

the damped natural frequency w can be found

[3,4].

The remaining system constant, k, may be

determined statically.

X

Figure 8.5. Exponentially decaying oscillations.

99
4.2. Setting the Compensator Constants

To set the constant A into the compensator, a sine-wave generator of frequency co„ (the un- damped natural frequency of the transducer) is connected to the input terminals, the B potentiometer is set to zero, and the A potentiometer is adjusted until a small output is obtained.

Under these conditions the signal is x=Vi sin ct>„t. (8.13)

Substitution of this function into eq (8.5) results

in the output

g(t)=-M^± (_^„2y^ sin o:J)

+— (co^Fi COS co„0+^i sin co„i. (8.14)

This simplifies to

g{t)^2^ViCosoiJ. (8.15)

If the B potentiometer is now set to zero,

(fi=2i/co;,), the output must be zero.

This sets the potentiometer A. The B potentiometer can now easily be set by adjustment until

the output amplitude is 2fFi.

5. Frequency Response

Ideally, amplitude and phase response of trans- ducer and compensator are the inverse of each

other, resulting in a flat overall system characteristic. The characteristics of such an ideal sys- tem are shown in figure 8.6. In any real system the frequency characteristics

and sensitivity of the amplifiers, and the noise

generated in the amplifiers and in the transducer,

will naturally adjust the capability of the overall

system. This is particularly important at the

high frequency end, where the transducer response

falls off and the compensator response increases

as shown in figure 8.6a, thus tending to exaggerate

high-frequency noise.

Real transducer characteristics were discussed

in chapter 1. The similarity between real and

ideal compensator characteristics may be judged

from figure 8.7, which shows the characteristics of an early model compensator [1]. The characteristics were obtained by setting the compensator's

a control for the natural frequency of the trans7' /

/

/ k PHASE RE OF COMPE

SPONSE

NSATOR

;

/

/

1- /4-FREOUENCY / RESPONSE OF

' COMPENSATOR

/V

OVERALL FREQUENCY

AND PHASE RESPONSE

OF COMPENSATOR SYSTE / /

\

LOG FR ;quency \

\ ^* FREQUENCY \ RESPONSE C \ TRANSDUCEJi

F

\

\

\

* PHASE RESi

1^ OF TRANSD

>ONSE

JCER

\\ \\

\ \ \

+ 140

+1 20

+ 100

+eo

+ 60

+ 40 w

(J <20g

0.4 05 0£ 4 9 6

Figure 8.6. Frequency response of ideal system.

100
fn- TRANSDUCER'S NATURAL

FREQUENCY

n* 12 Kc fn

'

fn-30 Kc H—-]P 1

RESPONSE

100 Kc

)F COMPENSATOR C'O

\ \ J vv

\ \ \ \

NOTE: 00TTEO LINE AflE THEORETICAL R EXTENSIONS

10 Kc 100 Kc LOG FREQUENCY

R -W\AFigure 8.8. Electrical analog transducer.

Figure 8.7. Attenuation characteristic of early model

compensator.

tmemf(t)yDRIVING \

afunction/

/transduce r\

v response )

Q X* /3x* X = liU.

k (OUTPUT)

Figure 8.9. Functional representation of simulated transducer and compensator

system.

ducer used, and adjusting the input to obtain

constant output. The plotted attenuation is then

the inverse of the compensator gain.

Compensator characteristic determination can

be made somewhat more independent of trans- ducer characteristics by replacing the physical

transducer b}^ an analogous electrical cu'cuit such

as that shown in figure 8.8. (We say "somewhat" because lumped electrical elements are known not to be entirely independent of frequency

either)

.

Signals present at various parts of this pseudosystem excited b}^ an electronic pulse of less than

0.1 microsecond rise time are shown in figure

8.9.

Figure 8.10 shows a three-Msec pulse, the pseudotransducer output, and the reconstructed dri%-ing

function (compensator output) [1]. Expansion of the fii'st part of figure 8.10 in

figxu-e 8.11 makes apparent the )2=A(sec rise and

decay times, and the 0.3-Msec delay of the compensator output.

An improved version of the compensator is capable of 20-Msec rise times.

The effect of the compensator on recordable

system output is graphically demonstrated by

figures 8.12 and SJ3. Figure 8.12 shows the

response of a capacitance-type transducer to a complex waveform produced by a shock tube.

101
Figure 8.11. Expansion of first part of figure 8.10.

1. f{t) (driving function)

2. I (transducer response)

Figure 8.12. Transducer response without compensator.

102
Figure 8.13. Transducer response with compensator.

Figure 8.14. Response of transducer to shock wave.

Figure 8.15. Response of figure 8.14 using compensator.

103
Figure 8.16. Response of figure 8.15 with expanded sweep.

Figure 8.13 shows the same response with compensator.

A similar improvement for a shock-excited trans- ducer is shown in figures 8.14, 8.15 and 8.16.

6. Limitations

The limitations of the compensator are of two

kinds: fu-st, the compensator is essentially limited

in the high-frequency region, where the compensator performance deviates appreciably from

the desired characteristic. (See fig. 8.7.) A

second, and possibly more severe, limitation lies in the basic principle of the compensator operation: namely, the compensator will faithfully

reproduce the system input, provided that the transducer is adequately described by an ordinary

linear differential equation of the second order and

with constant coefficients. And so nonlinearities or other deviations from

these ideal transducer characteristics wUl result in erroneous system output even at frequencies

which would yield readable results which could

be accurately related to system input functions

without use of the compensator.

Without the electronic compensator, any vibration frequencies that may be present in the trans- ducer system will appear generally in distorted

form in the output, the amount of distortion

depending on the nature of the amplitude and

phase characteristic of the transducer at those

frequencies. With the compensator in the system,

however, any vibration frequencies that may be

present will appear unaltered in amplitude and

phase in the output, provided that the vibration

frequencies occur in the frequency range over which compensation is effective. If the vibration

frequencies occur outside the range over which

compensation is effective, they will be altered in

amplitude and phase depending on the nature of

the overall transfer function of the system at those frequencies.

7. References

[1] F. F. Liu, and T. W. Berwin, Extending transducer

transient response by electronic compensation for high speed physical measurements, Rev. Sci. Instr.

29, 1, 14-22 (Jan. 1958).

[2] J. P. Den Hartog, Mechanical Vibration, McGrawHill Book Co., Inc., 3d ed. (New York, N.Y.,

p. 37).

[3] Gordon J. Murphy, Control Engineering, p. 89, D. Van Nostrand Co., Inc. (Princeton, N.J., 1959).

[4] C. S. Draper, Walter McKay, and Sidney Lees, In- strument Engineering, p. 249, vol. 11, McGrawHill Book Co., Inc., (New York, N.Y., 1953).

104

U.S. GOVERNMENT PRINTING OFFICE: 1963 O 694-211
THE NATIONAL BUREAU OF STANDARDS

The scope of activities of the National Bureau of Standards at its major laboratories in Washington, D.C., and Boulder,

Colorado, is suggested in the following listing of the divisions and sections engaged in technical work. In general, each

section carries out specialized research, development, and engineering in the field indicated by its title. A brief description of the activities, and of the resultant publications, appears on the inside of the front cover. WASHINGTON, D.C.

Electricity. Resistance and Reactance. Electrochemistry. Electrical Instruments. Magnetic Measurements. Dielectrics. High Voltage. Absolute Electrical Measurements.

Metrology. Photometry and Colorimetry. Refractometry. Photographic Research. Length. Engineering Metrology.

Mass and Volume.

Heat. Temperature Physics. Heat Measurements. Cryogenic Physics. Equation of State. Statistical Physics.

Radiation Physics. X-ray. Radioactivity. Radiation Theory. High Energy Radiation. Radiological Equipment.

Nucleonic Instrumentation. Neutron Physics.

Analytical and Inorganic Chemistry. Pure Substances. Spectrochemistry. Solution Chemistry. Standard Reference

Materials. Applied Analytical Research. Crystal Chemistry.

Mechanics. Sound. Pressure and Vacuum. Fluid Mechanics. Engineering Mechanics. Rheology. Combustion

Controls. Polymers. Macromolecules: Synthesis and Structure. Polymer Chemistry. Polymer Physics. Polymer Characterization. Polymer Evaluation and Testing. Applied Polymer Standards and Research. Dental Research. Metallurgy. Engineering Metallurgy. Metal Reactions. Metal Physics. Electrolysis and Metal Deposition.

Inorganic Solids. Engineering Ceramics. Glass. Solid State Chemistry. Crystal Growth. Physical Properties.

Crystallography.

Building Research. Structural Engineering. Fire Research. Mechanical Systems. Organic Building Materials. Codes and Safety Standards. Heat Transfer. Inorganic Building Materials. Metallic Building Materials. Applied Mathematics. Numerical Analysis. Computation. Statistical Engineering. Mathematical Physics. Operations Research. Data Prbcessing Systems. Components and Techniques. Computer Technology. Measurements Automation. Engineering Applications. Systems Analysis.

Atomic Physics. Spectroscopy. Infrared Spectroscopy. Far Ultraviolet Physics. Solid State Physics. Electron

Physics. Atomic Physics. Plasma Spectroscopy.

Instrumentation. Engineering Electronics. Electron Devices. Electronic Instrumentation. Mechanical Instruments.

Basic Instrumentation.

Physical Chemistry. Thermochemistry. Surface Chemistry. Organic Chemistry. Molecular Spectroscopy. Ele- mentary Processes. Mass Spectrometry. Photochemistry and Radiation Chemistry.

Office of Weights and Measures. BOULDER, COLO.

CRYOGENIC ENGINEERING LABORATORY

Cryogenic Processes. Cryogenic Properties of Solids. Cryogenic Technical Services. Properties of Cryogenic Fluids. CENTRAL RADIO PROPAGATION LABORATORY

Ionosphere Research and Propagation. Low Frequency and Very Low Frequency Research. Ionosphere Research.

Prediction. Services. Sun-Earth Relationships. Field Engineering. Radio Warning Services. Vertical Soundings

Research. Troposphere and Space Telecommunications. Data Reduction Instrumentation. Radio Noise. Tropospheric Measurements. Tropospheric Analysis. Spectrum Utilization Research. Radio-Meteorology. Lower Atmosphere Physics.

Radio Systems. Applied Electromagnetic Theory. High Frequency and Very High Frequency Research. Frequency

Utilization. Modulation Research. Antenna Research. Radiodetermination.

Upper Atmosphere and Space Physics. Upper Atmosphere and Plasma Physics. High Latitude Ionosphere Physics.

Ionosphere and Exosphere Scatter. Airglow and Aurora. Ionospheric Radio Astronomy.

RADIO STANDARDS LABORATORY

Radio Standards Physics. Frequency and Time Disseminations. Radio and Microwave Materials. Atomic Frequency

and Time-Interval Standards. Radio Plasma. Microwave Physics.

Radio Standards Engineering. High Frequency Electrical Standards. High Frequency Calibration Services. High

Frequency Impedance Standards. Microwave Calibration Services. Microwave Circuit Standards. Low Frequency

Calibration Services. Joint Institute for Laboratory Astrophysics-NBS Group (Univ. of Colo.).