PUBLICATIONS

Allin3

I l^^^f United States Department of Commerce

I ^1 I^^P I National Institute of Standards and Technology

NIST BUILDING SCIENCE SERIES 168

Periodic and Chaotic Motions of a

Modified Stoker Column:

Experimental and Numerical Results
NATIONAL INSTITUTE OF STANDARDS &

TECHNOLOGY

Research Information Center

Gaithersbiirg, MD 20899

DATE DUE

-

Demco, Inc. 3S-293
NIST BUILDING SCIENCE SERIES 168 /

Periodic and Chaotic Motions of a

Modified Stoker Column:

Experimental and Numerical Results

Graham R. Cook

Emil Simiu

Center for Building Technology

National Institute of Standards and Technology

Gaithersburg, MD 20899

Issued February 1990

U.S. DEPARTMENT OF COMMERCE, Robert A. Mosbacher, Secretary

NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY

John W. Lyons, Director
National Institute of Standards and Technology Building Science Series 168

Natl. Inst. Stand. Technol. Bldg. Sci. Ser. 168, 44 pages (Feb. 1990)

CODEN: NBSSES

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1990

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, DC 20402-9325
ABSTRACT

Records are presented of typical measured motions of a modified forced Stoker

column, including periodic motion around a stable fixed point, periodic snapthrough motion, and chaotic motion. Characterizations of the recorded chaotic

motion include: the autocorrelation function; the spectral density plot;

capacity dimensions; and the Lyapounov exponent. Two sets of numerical

simulations were performed, in which the same spring stiffnesses (measured under

static conditions) and the same dissipative forces (based on the viscous damping

model) were used. The first set, in which the device was modeled as a multidegree of freedom system to account for the distributed mass and stiffness

of the springs, yielded chaotic motions qualitatively similar to those recorded

in the laboratory. No chaotic motions could be obtained from the second set,

which did not reflect the fact that the spring properties are distributed and

in which the device was therefore modeled as a one degree of freedom system. To

the writers' knowledge this is the first reported instance of an experimental

structural system with continuous members for which a qualitatively successful

simulation of the chaotic motion appears to necessitate the inclusion in the

model of additional degrees of freedom to account for the effects of continuity.

Keywords: buckling; chaotic motion; dynamical systems; nonlinear ity ; structural

dynamics; structural engineering; vibrations.
ACKNOWLEDGMENTS

Partial support for this work was provided by the Minerals Management Service

(MMS) , U.S. Department of the Interior. Charles E. Smith of the Technology

Assessment and Research Branch, MMS, was Project Research Manager. The

experimental device was built by Frank Rankin of the Center for Building

Technology, NIST. The fractal dimension calculations were performed on the

massively parallel processor of the National Aeronautics and Space

Administration (NASA) by Judith E. Devaney of the Center for Computational and

Applied Mathematics, NIST.

iv
PREFACE

The nonlinear dynamic behavior of structural systems can be of concern in the

design of both conventional and novel types of structures. For conventional

structures, nonlinearities are principally due to inelastic material behavior

in the vicinity of ultimate loads, and nonlinear dynamic analyses can be helpful

in efforts to achieve realistic estimates of actual safety margins. Novel types

of structures exhibiting nonlinear behavior include compliant offshore

platforms, structures designed to be compliant under the effect of strong

seismic loads, large orbiting structural networks, flexible robot arms, and

other novel flexible or long- span structures. (Some of these structures may

require passive or active controls.) In these cases nonlinearities are due to one or more of the following factors: large deformations, materials and/or

joints with nonlinear behavior, and nonlinear loading.

The design of these new types of structures requires more than the ability to

perform nonlinear dynamic response calculations corresponding to specified

parameters and initial conditions. What is needed in addition is a qualitative

understanding of the nonlinear behavior for the system at hand. Features of

interest include:

(a) bifurcations, that is, changes in the character of the motion due to changes

in the values of the system parameters (such as mass, damping, stiffness, wind

or current speed, strength of ground motion, amplitudes and frequencies of

excitation induced by waves or by control system actuators)

(b) the possible existence for any given

stable oscillatory forms (attractors)

initial conditions (basin of attraction)

set of parameters of two or more steady

each with its corresponding set of

(c) jumps from one oscillatory form to another that can occur in bi- or multistable systems;

(d) the possible existence of motions that are sensitive to initial conditions

and therefore cannot, in practice, be predicted numerically for sufficiently

long times, since this would require virtually error- free initial conditions and

computational algorithms (these motions are referred to as chaotic).

Examples of bifurcations in structural dynamics problems include: the onset of

the critical flutter speed (corresponding to a Hopf bifurcation, also referred

to by some mathematicians as the Poincare -Andronov bifurcation, i.e., a bifurcation from a fixed point to periodic motion) ; and bifurcations from

periodic or quasiperiodic to chaotic motion for certain types of offshore

structures [1-3], for single forced or coupled autonomous galloping oscillators

[4] , or for forced circular cylinders excited by vortices shed in their wake

[5].

As pointed out in recent studies of nonlinear and chaotic motions of compliant

offshore structures reported in [1-3], the design of nonlinear structures can

be grossly inadequate or even disastrous if the analyst fails to account for the

diversity of possible types of responses that may correspond to (a) various sets

of system parameters, or (b) a given set of system parameters and various sets

V
of initial conditions. Even before the advent of compliant platforms such

failures have occurred: the design of the old Tacoma Narrows bridge, which

collapsed in 1940, is one example. In this case the possibility of an aeroelastic instability, that is, of a Hopf (Poincare -Andronov) bifurcation

apparently did not even occur to its designers.

The solution of nonlinear structural engineering problems such as those

mentioned earlier requires approaches that depend in a fundamental way upon the

quality of physical modeling. For example, in spite of the availability of the

requisite mathematical tools that had previously been developed for the analysis

of airfoil flutter, aeronautical experts -- including von Karman -- were unable

to calculate the critical flutter speed for the old Tacoma Narrows bridge until

it was recognized three decades later that aerodynamic lift forces and moments

derived for airfoils are inapplicable to bluff shapes [6]. However, intertwined

with questions of physical modeling are mathematical questions that can be of

great difficulty, so that the engineer needs whatever nonlinear mathematics

relevant to his problem is available (and usually more) . These questions become

more difficult when the nonlinearities are strong, as is the case when they are induced by hydrodynamic -- as opposed to aerodynamic -- effects.

The assimilation and stimulation by structural engineers of mathematical

developments applicable to previously unknown types of structures goes far back

in the history of structural engineering. The development by Euler of the theory

of elastic stability occurred in time to help designers of steel and cast iron

columns avoid the need for costly lessons through disastrous structural

failures. We have already mentioned the old Tacoma Narrows bridge, whose

designers were less fortunate, since no aeroelastic theory applicable to bridges

was available to them. .

In view of forthcoming developments in the design of structures with nonlinear

behavior, including structures with passive or active controls, it appears

reasonable to create a body of structural engineering knowledge drawing on recent progress -- and with the potential for stimulating further progress -- in nonlinear dynamical systems theory. For this reason, the Center for Building

Technology and the Center for Computational and Applied Mathematics of the

National Engineering Laboratory, National Institute of Standards and Technology,

are engaged in a cooperative effort the ultimate goal of which is to facilitate

the application by structural engineers of methods developed within the

framework of nonlinear dynamical systems theory. The first phase of this effort

is aimed at studying the experimental, numerical and analytic behavior of two

structural systems chosen for their relative simplicity and for their capability

to exhibit chaotic motions for certain ranges of system parameters. The writers

believe that these systems offer a useful opportunity for familiarization with

existing tools used in dynamical systems theory, and the potential for gaining

useful insights into the behavior of real systems susceptible of experiencing

chaotic motion. One type of system, briefly described in [4], is fluidelastic

in nature and consists of galloping oscillators, either single and harmonically

forced, or coupled and autonomous. This system is of potential interest in power

line engineering and in ocean engineering applications involving cables with

marine growth. A second type of system, consisting of a structural member

buckled under compression and subjected to periodic forcing, is described in the

present report.

vi
LIST OF SYMBOLS

a = propagation velocity

A = amplitude of forcing arm motion

c = viscous damping coefficient

F = force

F-^, F2 = forces defined in figure 4.1

g. Si- &2' &92' Sli = dissipative forces

^li' ^2i ^ dissipative forces

k = spring constant

^1' ^2 stiffness of left and right horizontal spring, respectively

K^^ = total stiffness of torsional springs

K2 = total stiffness of vertical springs

2 = half of total column length

L = spring length

m = mass assumed to be concentrated at column midpoint (fig. 2.1)

M = sum of mass m and tributary masses of horizontal springs

^hl ' "'h2 total mass of left and right horizontal spring, respectively

m^ = total mass of vertical spring

"^1 ' "^2 ^ mass assumed to be concentrated at upper and lower column support

M]^ = sum of mass m-^ and tributary mass of vertical spring

M2 = sum of mass m2 and tributary mass of vertical spring

N = capacity dimension

n^ = number of segments into which horizontal springs are divided

n^ = number of segments into which vertical springs are divided

Pq = total tension in vertical springs corresponding to x = 0

^li' ^2i ^ vertical spring forces (figs. 4,1 and 4.2)

^lo' ^2o ^ tensions in left and right horizontal springs corresponding to zero displacements of mass m and of the forcing arms

^li' ^2i ^ horizontal spring forces (figs. 4.1 and 4.3)

t = time

T = torsional moment (fig. 4.1)

tQ = ratio Z/t

tp = disturbance propagation time

^11' ^21 ^ forces defined in figure 4.1 w = nondimensional displacement in Duffing equation

X = displacement of mass m

Xq = ratio x/w

^li' ^2i ^ displacements of lumped masses of left and right horizontal spring

segments (fig. 4.2)

y-^^ = displacements of lumped masses of vertical spring (fig. 4.3)

Zj^ = auxiliary coordinates

-a, /3 = coefficients of linear and cubic term, respectively, in eq (4.34)

7 = Lyapounov exponent

5(x) = displacement of mass m]^ and of mass m2

£ = strain

d = angle defined by figure 4.1 and eq (4.12)

p = mass per unit length

r = nondimensionalized time w = frequency forcing arms motion

Q = nondimensionalized frequency of forcing arms motion

' denotes differentiation with respect to time

vii

CONTENTS

Page

Abstract iii

Acknowledgments iv

Preface v List of symbols vii

1. Introduction 1

2. Description of Experimental Device 2 3. Experimental Results 3 4. Analytical Modeling 5 4.1 Modeling of Spring Masses and Stiffnesses 5 4.2 Modeling of Dissipative Forces 6 4.3 Equations of Motion 6 4.3.1 Equations of Motion of Multidegree of Freedom System .. 7 4.3.2 Single Degree of Freedom System 8 4.3.3 Duffing Equation 9 5. Numerical Simulations 10

5.1 Numerical Simulation of Observed Chaotic Motion 10

5.2 Effect of Simplified Representation of the Device as a One

Degree of Freedom System 11

6. Summary and Conclusions 12

7. References 12

List of Figures

Figure 2.1. Schematic of Forced Modified Stoker Column 14

Figure 2.2. Drawing of Forced Modified Stoker Column 15

Figure 2.3. Photograph of Forced Modified Stoker Column 16

Figure 3.1a. Time History of Displacement x 17

Figure 3.1b. Time History of Displacement x 18

Figure 3.1c. Time History of Displacement x 19

Figure 3.2. Time History of Displacement x 20

Figure 3.3. Phase Plane Diagram 21

Figure 3.4. Intersections of Orbit x, x' , t with Planes t=27rn/w

(n = 1,2,...) 22

Figure 3.5. Mapping of Successive Local Maxima of x, x^^ versus .• 23

Figure 3.6. Spectral Density Function of x 24

Figure 3.7. Autocorrelation Function of x 25

Figure 3.8. Snap- through Oscillations 26

Figure 3.9. Spectral Density of Snap-through Oscillatory Motion 27

Figure 3.10. Oscillations Around a Stable Fixed Point of the Unforced

Motion 28

Figure 3.11. Spectral Density of Oscillatory Motion Around Fixed

Point 29

Figure 4.1. Free Body Diagrams 30

Figure 4.2. Lumped Masses Representing Vertical Springs (n^ =4) 31

Figure 4.3. Lumped Masses Representing Horizontal Springs (n-^ = 4) ... 31

Figure 4.4. Dependence of Potential Function on Displacement |x|

for One Degree of Freedom System 32

Figure 5.1. Time History of Simulated Chaotic Motion 33

Figure 5.2. Stroboscopic Poincare Plot of Simulated Chaotic Motion ... 34

Figure 5.3. Spectral Density Plot of Simulated Chaotic Motion 35

ix

1. INTRODUCTION

The purpose of this report is twofold. First, it describes an experimental

device designed to demonstrate periodic and chaotic behavior of a buckled

column subjected to harmonic forcing. Second, it attempts to reproduce the

behavior of the system numerically using relatively simple analytical models

comparable to those that might be used in engineering practice. Comparisons

between models with various degrees of complexity are made, which provide some insight into issues related to analytical modeling for structures that can experience chaotic behavior. The work presented here can be viewed as the first

phase of a broader experimental, analytical, and numerical investigation into

the behavior of the forced buckled column, which would explore additional

issues of engineering interest. The conditions under which noise can change

dynamic behavior qualitatively are one such issue. Another issue is the

occurrence of chaotic behavior when the excitation, rather than being periodic,

is due to the superposition of two harmonics (a topic investigated theoretically

for certain simple systems in [7]), or to a relatively large number of harmonics

that mimic a stochastic process such as excitation by waves.

The experimental device was designed to exhibit clearly the physical mechanisms

underlying its excitation and response and -- to within unavoidable noise

effects -- the deterministic nature of those mechanisms. Its predecessors are

a pioneering magnetoelastic analog [8] and an electrical analog [9] of a buckled

column. However, our device is strictly mechanical and actually consists of a physical column subjected to compressive loads that cause it to be buckled at

all times.

In view of the similarity of the device of [8], it was judged that the device

has the capability of exhibiting chaotic behavior, in spite of its structurally

more complex nature, which will be examined subsequently.

A Stoker column is an ideally frictionless buckled column with constant

compressive load and with mass and bending stiffness concentrated at its

midpoint [10, p. 54]. A Stoker column with energy dissipation, forced

periodically at its midpoint in a direction normal to the load, can exhibit

chaotic behavior for certain parameter ranges [8,11]. Practical difficulties

associated with the constancy of the compressive load appear to have prevented

so far the construction of such a device. The interesting motions associated

with the forced, dissipative Stoker column are due to the presence of a homoclinic point at the origin of the phase plane diagram for the corresponding

unforced, Hamiltonian (nondissipative) system. However, the compressive load

need not be constant for such motions to occur. The dissipative forced Stoker

column was therefore modified as follows. The compression forces are supplied

by prestressed springs connected to the column ends. Since the elongations of

those springs vary during the forced motion of the column, so do the compression

forces. In addition, for reasons of construction, the modified column also has

concentrated masses at its ends. Finally, the springs have considerable mass,

so that the system being considered is continuous, i.e., it has an infinite

number of degrees of freedom. The device so constructed is referred to here as the modified Stoker column.

1
Section 2 of the report contains a description of the experimental device.

Section 3 presents descriptions of column motions, including time histories,

phase plane diagrams, Poincare plots, spectral density functions, and

autocorrelation functions. Estimates of fractal (capacity) dimensions and of

Lyapounov exponents characterizing the motion are also included. Section 4

briefly discusses analytical models used for the modified Stoker column and

includes the derivation of corresponding equations of motion. Section 5 includes

results of numerical simulations based on various analytical models and

discusses their adequacy in light of the experimental results. Section 6 consists of a summary and conclusions.

2. DESCRIPTION OF EXPERIMENTAL DEVICE

A schematic of the device is shown in figure 2.1. The column is assumed to have

three concentrated masses; a mass m at its midpoint, a mass m-^ at its top end,

and a mass m2 at its bottom end. Torsional springs at the midpoint supply the

total concentrated bending stiffness K-^. The column is tied at its ends by

vertical springs with total mass m^ and total stiffness K2 . The column ends and

the masses attached to them are free to slide vertically but are constrained in

the horizontal direction. The mass m is free to slide horizontally but is

constrained in the vertical direction. This ensures the symmetry of the column

motion with respect to the horizontal line passing through the column midpoint.

The mass m is tied by two springs, with masses m-^-^ and m|-^2 and stiffnesses k]^ and k2 , to two rigid arms forced by a motor to move harmonically (with

amplitude A and circular frequency w) and in phase in the horizontal direction.

A drawing and a photograph of the device are shown in figures 2.2 and 2.3.

Each column consists of a pair of identical members. Each member is equipped

with a vertical spring and with two torsional springs. This arrangement,

similar to that of a tied arch discussed in [12, p. 55], allows the occurrence

of snap- through oscillations. Friction between the moving mass ra and its

horizontal guides was reduced by providing the mass with roller bearing guide

wheels. The columns are connected at their ends to horizontal axles sliding

within vertical guides. To reduce friction the vertical guides are lined with

teflon, and the ends of the axles are provided with teflon pads. The connections

of the ends of the columns to the axles was done through roller bearings. The

vertical springs are connected to the bottom axles by S-hooks positioned onto

the axles by clip rings. The top ends of the vertical springs are suspended from

the lower eye of turnbuckles whose upper eye rests on the axles. The turnbuckles

allow the adjustment of the pretension in the springs. The rigid arm to which

mass m is connected by the horizontal springs is driven at the desired frequency

by a low inertia servomotor with tachometer feedback. Measurements indicated

that the frequency varies in time by as much as 1% of its mean value. The

amplitude of the forced rigid arm motion can also be adjusted as needed within

the range 0.02 to 0.035 m. Displacements of the mass m are measured using a single turn precision potentiometer with a ball bearing mounted shaft and a linearity of 0.05%.

All springs were manufactured from chromium

for its superior fatigue performance. Both

behave linearly under static testing (i.e..

silicon steel, which was vertical and horizontal

low loading rate) except

selected

springs near the
origin of the force-displacement diagram. The undeformed length of the vertical

springs was 0.18 m; their outside diameter 0.01667 m; the wire diameter was 0.002324 m; and the inside diameter of the hooks was 0.0096 m. The total mass

of the two vertical springs, m^, was 0.234 kg. The undeformed length of the

horizontal springs was 0.1825 m; the outside diameter was 0.01429 m; the wire

diameter 0.0015875 m; and the inside diameter of the hook was 0.0873 m. The mass

of each horizontal spring, mj^]^ = m]-j2 was 0.07 kg. Within the linear range, the

spring stiffnesses were K2 = 2,640 N/m for the vertical springs and k]^ = k2 = 264 N/m for the horizontal springs.

The springs are pretensioned so as to be under tension and linear at all times.

The spring extensions corresponding to x = 0 were 30 mm for the vertical

springs. For the horizontal springs, when the displacements of the forcing rigid

arms were equal to zero, the spring extensions corresponding to x = 0 were 80 mm

The torsional springs, as mounted on the vertical shaft located at the midheight

of the columns, are linear under static testing over the range of angular

deformations 0 - 0.336 radians (|x| < 0.066 m) , their total stiffness within

this range being K]^ = 10.9 Nm/rad. For angular deformations larger than 0.336

radians (|x| > 0.066 m) the torsional springs behave nonlinearly, and

measurements under static testing yielded the approximate relation

Ki(x) = (27015.3 |x| - 1772.1) Nm/rad (|x| > 0.066 m) . (2.1)

The design of the experimental device makes it possible to change the springs,

the masses, and the amplitude and circular frequency of the horizontal forcing

motion. The total length of the column, however, is fixed (2-? = 0.4 m) .

3. EXPERIMENTAL RESULTS

The properties of the system for which the experimental results presented in

this section were obtained were:

Z = 0.200 m m = 0.490 kg

m^^ = 0.155 kg

m2 = 0.128 kg

A = 0.02425 m

The forcing frequency was varied as described below. The spring properties were

those given in section 2. Mass m as given above includes, in addition to the mass attached to the column midpoint, the tributary mass of the columns

themselves and the total mass of the torsional springs (0.084 kg). Masses m^^ and

m2 consist of the tributary mass of the columns themselves, the spring

attachments and the mass of the column axles.

We show in figures 3.1a, 3.1b, and 3.1c a 300 s time history (x, t) of steady

state motion corresponding to the properties just listed. For comparison with

subsequent figures we also show at an expanded scale in figure 3.2 the time

3
history for the first 40 s of figure 3.1a. The forcing frequency for the motions

of figures 3.1 and 3.2 was to = 37.315 rad/s

Figure 3.3 shows the phase plane diagram (x, x') corresponding to figure 3.2.

Figure 3.4 shows the points obtained by intersecting the trajectory (x, x' , t)

with planes t = 27rn/cj (n = 1 , 2, . . .), where w is the nominal forcing frequency.

(The velocities x' were estimated from the recorded time series x.) This diagram

would be a stroboscopic Poincare plot if the nominal and actual forcing

frequency were the same. As indicated in section 2, however, the actual

frequency varies by as much as 1% of its mean value. The plot of figure 3.4 is

therefore affected by errors associated with this variation.

Figure 3.5 shows a different Poincare plot which is not susceptible of

experiencing such errrors . It was obtained by using the surface of section x' = 0 and therefore represents the mapping of the local maxima of the

displacement, > versus their immediate antecedents, the local maxima x^ . If

the motion were periodic or quasiperiodic , this plot would consist,

respectively, of a finite number of points or of a dense set of points on a closed curve. The plot of figure 3.5 appears, however, to be a fractal object.

This was confirmed by calculations of its capacity dimension [13, 14] , which was estimated to be 1.25. The estimate was based on about 2000 points from figure

3.5 (corresponding to a record approximately 300 s long). It was performed by

Judith E. Devaney of the Center for Computational and Applied Mathematics, NIST,

using the procedure developed by Sullivan and Hunt in [15].

The spectral density of the displacement is represented in figure 3.6. The

broadband components of the spectrum are noted. To a broadband spectrum there

corresponds a decaying autocorrelation function, as seen in our case in figure

3.7. The decay and the vanishing (to within noise effects) of the

autocorrelation function reflects the unpredictability of future motions given

a known time history. Thus, the broadband character of the spectrum is an indication that the motion is chaotic.

For the three-dimensional attractor of the system the capacity dimension was estimated to be 2.3. This estimate was based on the three-dimensional timedependent vector obtained by embedding the time series of the response, x(t),

in a three-dimensional space as follows. The first coordinate of the vector was

x(t) itself. The second and the third coordinates were x(t -i- r) and x(t -i- 2t),

respectively, where the delay time t = 0.6425 s is the time lag corresponding

to the first zero of the autocorrelation function shown in figure 3.7 (see,

e.g., [13]). The estimate was performed for a first set of 65,000 points

obtained in this fashion, and corresponding to the first half of 300 s long

record. The estimated capacity dimension based on the second half of that record

was also 2.3. These estimates were also performed by J . E. Devaney using the

procedure of [15].

The Lyapounov exponent was estimated using the procedure described in [16]. The

computer program for implementing the procedure was made available to the

writers by Professor A. Wolf. For embedding dimensions D = 2 and 3 we obtained

estimates of the Lyapounov exponents corresponding to input parameters

appropriate to our problem (SCALMIN = 0.002 m, SCALMAX = 0.01 m, EVOLV = 500,

4
see [16, p. 313]). The estimated values were 7 - 2.99 bits/s for D = 2, and 7 = 3.14 bits/s for D = 3. For higher embedding dimensions noise effects appeared

to affect the results significantly, although the estimated Lyapounov

coefficients remained positive and of the same order of magnitude up to D = 7

.

The forcing frequency was then changed from w =- 37.315 rad/s to w = 22.702

rad/s , while all other parameters were left unchanged. A periodic snap- through

motion was recorded, as shown in figure 3.8. Figure 3.9 shows the corresponding

spectral density function. The presence of noise is clearly reflected in figure

3,9, but its level is considerably lower than that of the broadband components

of the chaotic motion shown in figure 3.6.

Finally, we show in figure 3.10 the time history of periodic motion about a stable fixed point of the unforced column. This occurred for a forcing frequency

(j) = 22.800 rad/s, all other system parameters being unchanged. It is noteworthy

from an engineering standpoint that the behavior of the system is radically

different from that of figure 3.8 even though the respective frequencies differ

by less than 0.5%. The spectral density of the motion of figure 3.10 is shown

in figure 3.11.

4. ANALYTICAL MODELING

4 . 1 Modeling of Spring Masses and Stiffnesses

The system under consideration is continuous, i.e., it has an infinite number

of degrees of freedom. This is due to the relatively significant mass of the

horizontal and vertical springs.

For a linearly elastic member in tension the propagation velocity for a disturbance is a = [ ( F/e )/p ] '-/^

, where F = force acting on member, e = strain

induced by force F, and p = mass of member per unit length. For a linear spring

with total mass and length L to which there corresponds a spring constant k,

F = k, e = 1/L, p = mg/L, and a = (k/mg) •'/^L. The time required for the

propagation of a disturbance from one end of a spring to the other is tp = L/a

,

and the length, spring constant and mass are approximately 0.18 m, 2640 N/m, and

0.234 kg for the vertical springs, and 0.18 m, 265 N/ra, and 0.07 kg for each of

the horizontal springs. Therefore, the propagation times are about 0.01 s and

0.016 s, respectively. These times should be compared with a quarter of the

forcing period (i.e., the time required for the forcing arm displacement to go

from zero to its maximum value), which for the chaotic oscillations is 27r/37.315 = 0.042 s, that is, only a few times larger than the propagation times. This

suggests that the continuity of the system due to the masses of the horizontal

and vertical springs may be a significant factor in the modeling of the

experimental device.

An additional modeling issue is whether the spring stiffnesses as measured under

static conditions are applicable to dynamic conditions. Given the constraints

of the project dynamic testing could not be performed, and it was assumed in the

simulations that the spring stiffnesses under static and dynamic conditions are

the same. However, we believe that the modeling of the system might have been

5
improved if the spring stiffnesses had been carefully measured under dynamic

conditions, particularly for the as -mounted torsional springs.

4 . 2 Modeling of Dissipative Forces

Energy dissipation is due to both internal friction in the springs and to

friction between moving parts of the device, including in particular the

friction of the torsional springs against the shaft onto which they are mounted.

The measurement of the dissipative forces was beyond the scope of this work.

Some of these forces are nonlinear, and our observations indicated that this is

in particular the case for the dissipative forces associated with the torsional

spring forces |x| > 0.066 m. Nevertheless, in simulating the motions we used

the simplified assumption that the total damping is proportional to the velocity

x' of the column midpoint, the factor of proportionality being a constant

viscous damping coefficient. We also carried out simulations in which we assumed that the viscous damping coefficient was constant for |x| < 0.066 m and

increased linearly for |x| > 0.066 m. The results of our simulations will be

commented upon in light of the limitations of these simplified models.

4 . 3 Equations of Motion

The equations of motion can be derived by considering the free body diagram for

masses M, M]_ and M2 , where M is the sum of mass m and the tributary masses of

the horizontal springs, and M;|^ (M2) is the sum of mass m]^ (m2) and the tributary

mass of the vertical spring. The motion of masses m-^ and m2 is constrained;

denoting their displacements by 5(x) we have (fig. 2.1):

5(x) = i - (^2 . x2)l/2 . (4.1)

Let the differentiation with respect to the time t be denoted by '. It follows

from figure 4.1 that

= Pll(x) + Ml 5" (x) + gi (4.2)

V21 = P2i(x) + M2 5"(x) + g2 (4.3)

T

+ I sin^ - F]^ i cos^ - = ^ (^•^)

2

T

- + V21 i sin^ - F2 a cose - = ^ (^-5)

2

M x' ' + g - F]^ - F2 - Rii + R21 = 0 (4.6)

where g, g-^ , g2 , g^2 denote the dissipative forces (moments). For the

reader's convenience we note that

5'(x) = xx'/('2^ - x2)l/2 (4.7)

5"(x) = [1/(^2 - x2)l/2] [xx" + x'2 + x2x'2/(i2 . ^2)] . (4.8)

6
4.3.1 Equations of Motion of Multidegree of Freedom System

To account for the forces of inertia associated with the mass of the springs, we divide the springs into segments, each having a lumped mass and stiffness.

If we divide the vertical springs into equal segments, the mass and the

stiffness of each segment will be m^/n^ and n^K2 , respectively, where m^ and

K2 are, respectively, the mass and the stiffness of the original undivided

spring. A tributary mass m^/ (2u^) will be added to the top mass mi as well as to the bottom mass m2 . Similar statements hold for the horizontal springs, which

are each divided into n^ segments. We will assume for convenience that n^ and

n^ are even.

The system of eqs (4.1) through (4.6) must then be complemented by the equations

of motion of the spring segments. Owing to symmetry, ?2X (^) = Pii(x). and we need to write only the equations of motion of the upper n^/2 - 1 lumped masses

of the vertical springs (fig. 4.2):

yii" + gli - Pli + Pli+1 = 0 (i = 1, 2, ..,n^/2 - 1) . (4.9)

For the horizontal spring segments (fig. 4.3):

""hi

"h

TMh2

"h

^li"' + hii + Rii - Rii+i = 0 (i = 1, 2, • • - % 1) (4

^2i"' + h2i - R2i + R2i+1 = 0 (i = 1, 2, •

• 1) (A

Sli' ^li' ^2i denote the dissipative forces . The forces in the springs

Pll = Po - n^K2 [Six) + Yii] (4

Pli = Po + i^vK2 (y2i-i - yii) (i = 2 3 • ,nv/2 1) (4

Rll = + n^ki(x;Ll - ^) (4

Rii = Rio + n^ki(xii - xi^.i) (i = 2,3, . • ,nh - 1) (4

Rii = + n^k]^(Asin(wt) - xn.i) (i = (4

R21 = R20 - nhk2(x2i - x) (4

R2i = R20 - nhk2(x2i - X2i.i) (i = 2,3,. • '"h

- 1) (4

R2i = R20 - n^k2(Asin(wt) - ^21-1^ (i = "h - 1) (4

where is the total initial tension in the vertical springs (corresponding to

X = 0) and R^o' P-2o initial tensions in the left and right horizontal
spring, respectively (corresponding to zero displacements of mass m and of the

forcing arms) . Finally we have

T(t) = 2 Ki e(t) (4.20)

6 = sin-l(x/i) . (4.21)

We consider, as an example, the particular case k]^ = k2 , m^^.

''"h2 > ^lo ^ ^2o

(to which there correspond antisymmetric motions of the two horizontal springs)

and n^ = 4, n^ = 4 . Denoting x = Z]^, x^^ = Z3, X2 = Z5 , X3 = zy, Yi = zg, and

assuming that the dissipative force g = c x' while the other dissipative forces

are zero (i.e., assuming that the dissipation is entirely concentrated in the

term c x' ), the equations of motion can be written as

zi' = Z2 (4.22)

m + %i/4 + {mi + m2 + ni^/4) 7.-^ / - zi^)

n n (mi + m2 + nVy,/4) 2^ z^ zi

^ ^

- zi^)2

2Po zi - 2 (i + Z9) (4K2) zi - 2Ki sin-l(zi/^)]

' ~~~2":";^2;i72

- 2[4ki - 2(4K2)] zi + 2 (4ki) Z3 ) (4.23)

Z3' = Z4 (^-24)

Z4' = 4ki (zi + Z5 - 2z3)/(mh/4) (4.25)

Z5' = zg (4.26)

Z6' = 4ki (Z3 + Z7 - 2z5)/(mh/4) (4.27)

Z7'= zg (4.28)

zg' = 4k]^ [Z5 + A sin (wt) - 2z7]/(m|^/4) (4.29)

zg' = zio (4.30)

zio' = - 4K2{[^ - (^2 . 2^2)1/2] + 2 Z9}/(m^/4) . (4.31)

4.3.2 Single Degree of Freedom System

If instead of n.^. = n^ = 4 we have n.^ = nj^ = 1, then

8
1 x' ' +

m + + (mi + m2 + m^/2)x2/(i2 . ^2)

{cx' +

(mi + m2 + xx'2

(^2 . x2)2

- 2 (Pq - 2iK2)x - Ki sin'l(x/i)

(i2 . x2)l/2

+ 2(k]^ - 2K2) X - 2kxA sin(wt)} = 0 . (4.32)

Equation 4.32 was verified by deriving it independently from the Lagrangian of

the system. The potential from which the restoring force of eq (4.32) is derived

is represented as a function of the absolute value of the displacement x in

figure 4.4. This double-well potential is similar to that characterizing the

Duffing equation [17, p. 82]. It may therefore be expected that to eq (4.32)

there can correspond forced motions similar qualitatively to those typical of

the Duffing equation with forcing [17, p. 84], that is (1) periodic motions

around one of the stable fixed points of the unforced Hamiltonian counterpart

of the system, as shown (to within effects of noise) in figure 3.10, (2)

periodic motions around the unstable fixed point, as in figure 3.8 (periodic

snap-through motions), and (3) motions that are neither periodic nor quasiperiodic (chaotic motions) , characterized by successions of irregular

oscillations around the three fixed points, as in figure 3.1.

We note that the potential function corresponding to the multidegree of freedom

system cannot be conveniently plotted, owing to its dependence upon a large

number of generalized coordinates.

4.3.3 Duffing Equation

If in eq (4.32) the masses mj^ , m2 , and m^ are zero, then

c (Pq - 2K2i) X - Ki sin"^(x/i) ki - 2K2

X" + x' - 2 + 2 X

M M(^2 . x2)l/2 M

2ki = A sin(cjt) (4.33)

M

where M = m + m-^i/1. If, as has been done for the classical Stoker column [10,

p. 54], eq (4.33) is expanded in a Taylor series and terms of order five and

higher are neglected, we obtain

9
c 1 x" + MMx'+ (2ki + 2 ^2

= A sin(wt) . (4. 34)

M

In eq (4.34) we denote the coefficient of x' by r, the coefficient of x by -a, the coefficient of x"^ by ^, and the coefficient of sin(wt) by Fq . The parameters

k]^ , K^^ , K2 , Pq , and Jl are chosen so that a > 0 and ^ > 0. By

nondimensionalizing eq (4.34) so that x = x^w, t = t^r , where t^ and Xq satisfy

the relations at^^ = 1/2, ^x^^t^^ = 1/2, and by using the notations rt^ = 7,

F^t^^/xQ = f, and wt^ = Q, we obtain the standard form of the Duffing equation

1 1 w' ' + 7w' w + w^ = f sin(Q7-) . (4.35)

2 2

Useful explorations of the motions corresponding to various regions of the

parameter space for the Duffing equation are reported in [11]. For this reason

the Duffing equation was used as a guide in preliminary attempts to obtain

chaotic motions of the one-degree of freedom system represented by eq (4.32).

We note, however, that unlike eq (4.32), eq (4.35) cannot be viewed as an approximation to the experimental device since masses m2^ , m2 , and m^^ are negligibly small.

4Ki _ . 2 ) X + ( + 2 ) X- M 3r

5. NUMERICAL SIMULATIONS

Numerical simulations were performed for two purposes. First, an attempt was made to reproduce by numerical simulation the chaotic motion obtained

experimentally. Second, for a given set of system parameters, simulations were performed with a view to ascertaining whether two models, one representing a simplified version of the other, yielded motions that belonged to the same of

the three types of response listed at the end of section 4.3.2. Motions

belonging to the same type of response will be referred to here as qualitatively

similar

5 . 1 Numerical Simulation of Observed Chaotic Motion

It was suggested in section 4.1 that the mass of the springs appears to be a significant modeling factor. For this reason we modeled the motion of the

experimental device by dividing the vertical and each of the horizontal springs

into several segments having the respective masses lumped at their centers. We

used eqs (4.22) through (4.31), which correspond to the division of each of the

tension springs into four segments, and in which the dissipative force was asssumed to be proportional to x' , the factor of proportionality being a constant c. The torsional stiffness was modeled by eq (2.1), and the various

parameters of the system had the values given in sections 2 and 3. Although we did not attempt to perform exhaustive numerical studies, we ascertained that

chaotic motions were obtained for a fairly broad range of values c. For c = 3.0,

10
which would correspond in a linear system with unit mass and stiffness 2k-^ to

a damping ratio of about 7%, we obtained the time history of figure 5.1. While

the motions of figures 3.2 and 5.1 are similar qualitatively, they are clearly

different quantitatively. We note that quantitative differences between

experimental and numerically simulated motions were also pointed out for a similar oscillator in [17, p. 84]. Changing the constant c, or assuming that c was constant for |x| < 0.066 and increased linearly for |x| > 0.066 until it

reached at |x| = 0.075 a value n times larger than for small |x| (n = 2,

3,..., 10), did not result in a substantially improved reproduction of the

experimental motion. The stroboscopic Poincare plot and the spectral density

function corresponding to figure 5.1 are shown in figures 5.2 and 5.3,

respectively. Note that the maximum velocities are considerably higher for the

simulated than for the observed motion (fig. 5.2 versus fig. 3.4).

Detailed modeling and measurements of the dissipative forces and of the spring

stiffnesses under dynamic conditions -- particularly for the torsional springs

-- were beyond the scope of this project. Had such modeling and measurements

been performed, and had the springs been divided for simulation purposes into more that four segments, it is likely that the observed chaotic motion would

have been reproduced more closely. Nevertheless, we note that in spite of the

imperfect modeling of the dissipative forces and possibly of spring stiffnesses

as well, the modeling of the device by a system with 10 degrees of freedom

(reduced owing to symmetries and antisymmetries to a system with 5 degrees of

freedom) was capable of reproducing qualitatively the observed chaotic motion.

5 . 2 Effect of Simplified Representation of the Device as a One Degree of Freedom

System

We also attempted to reproduce qualitatively the observed chaotic motion by

modeling the device as a system with three masses. As shown in section 4, owing

to the geometrical constraints of the device, if the spring masses are neglected, the system with masses m, m]^ and m2 (to which tributary spring masses

are added) has one degree of freedom. We therefore simulated the motion of the

device by using eq (4.32), with the same dissipative forces and stiffnesses as in the simulations of the corresponding multidegree of freedom system (sec.

5.1). In none of the tens of cases for which chaotic motion was obtained for the

multidegree of freedom system did the corresponding one degree of freedom system

result in chaotic motion. Additional tens of simulations for the one degree of

freedom system, in which the parameters defining the assumed dissipative forces were varied from simulation to simulation while all other parameters were kept

equal to their counterparts in the experimental device, also failed in all cases

to yield chaotic motion.

It may be that, had we used a model that reproduced more closely the actual

dissipative forces (and possibly also the dynamic spring forces), chaotic motion

could have been obtained even by modeling the device as a one degree of freedom

system. Nevertheless, our results allow us to draw the interesting conclusion

that motions of the multidegree of freedom systems represented by the

simulations of section 5.1 cannot be reproduced even qualitatively if those

systems are modeled as one degree of freedom systems. This result differs from

that obtained for the oscillator discussed in [17, p. 84], for which the

approximation of a continuous beam by a one degree of freedom mass did not alter

11
the observed behavior qualitatively. The effect of the additional degrees of

freedom of the system is significant in our case owing to the relative

magnitudes of relevant parameters, in particular the spring masses and

stiffnesses, masses m, m-^ and m2 , and the forcing frequency u>

.

6. SUMMARY AND CONCLUSIONS

Following a description of the modified Stoker column as constructed at the

Center for Building Technology, National Institute of Standards and Technology,

we presented records of typical motions of that device as measured in the

laboratory, including periodic motion around a stable fixed point, periodic

snap-through motion, and chaotic motion. Characterizations of the recorded

chaotic motion included: the autocorrelation function, which decays and then

vanishes (to within noise effects) after a 0.74 s lag, thus indicating that

knowledge of past history of the motion cannot be used to make deterministic

inferences on future behavior; the spectral density plot, which exhibits

relatively substantial broadband components, as is expected for a time history

with decaying autocorrelation; the capacity dimensions of a Poincare plot of the

strange attractor of the motion (N = 1.25) and of the attractor constructed by

embedding the time history in a three-dimensional phase space (N = 2.3); and the

Lyapounov exponent, whose estimated value was approximately 3.00.

The stiffness measurements for the springs used in the construction of the

device were performed under static conditions, and no measurements were performed of the dissipative forces occurring during the motion. Nevertheless,

numerical simulations based on an analytical model in which the spring

stiffnesses measured under static conditions were used and the damping was assumed to be viscous yielded chaotic motions qualitatively similar to those

recorded in the laboratory. That model entailed the representation of the device

as a multidegree of freedom system to account for the fact that, owing to the

spring properties, times required for disturbances to travel from one end of a spring to the other are significant in relation to the forcing period of the

system. On the other hand, we were unsuccessful in our attempts to simulate the

chaotic motion when representing the device as a one degree of freedom system

(i.e., by assuming that disturbances are propagated through the springs

instantaneously) . To our knowledge this is the first instance of an experimental

structural system with continuous members for which a qualitatively successful

simulation of the chaotic motion appears to require the inclusion in the model

of additional degrees of freedom to account for the effects of continuity.

A videotape of the chaotic motions of the device is available upon request from

the authors

.

7. REFERENCES

[1] J. M. T. Thompson, "Complex Dynamics of Compliant Offshore Structures,"

Proceedings . Royal Society of London, A 387 . pp. 407-427, 1983.

[2] J. M. T. Thompson, A. R. Bokaian, R. Ghaffari, "Subharmonic and Chaotic

Motions of Compliant Offshore Structures and Articulated Mooring Towers,"

Journal of Energy Resources Technology . ASME , Vol. 106, pp. 191-198, June 1984.

12
[3] J. M. T. Thompson and H. B. Stewart, Nonlinear Dynamics and Chaos . John

Wiley and Sons, New York, 1986.

[4] E. Simiu and G. R. Cook, "Periodic, Quasiperiodic and Chaotic Motions of

Forced and Autonomous Galloping Oscillators," Proceedings of the 8th Colloquium

on Industrial Aerodynamics, Department of Aeronautics, Fachhochschule Aachen,

Aachen, W. Germany, Sep. 4-7, 1989, Part 1, pp. 167-178.

[5] D. J. Olinger and K. R. Sreenivasan, "Nonlinear Dynamics of the Wake of an Oscillating Cylinder," Physical Review Letters . Vol. 60, No. 9, pp. 2798-2801,

1988.

[6] R. H. Scanlan and J. J. Tomko , "Airfoil and Bridge Deck Flutter

Derivatives," Journal of the Engineering Mechanics Division . ASCE, Vol. 97, pp.

1717-1737, 1971.

[7] S. Wiggins, Global Bifurcations and Chaos . Springer-Verlag , New York, 1988.

[8] F. C. Moon and P. J. Holmes, "A Magnetoelastic Strange Attractor," Journal

of Sound and Vibration . Vol. 69, No. 2, pp. 285-296, 1979.

[9] H. Clemens and J. Wauer, "Durchschlagen vorgekruemmter Staebe unter

harmonisch pulsierender Belastung,", Proceedings of the 8th International

Conference on Nonlinear Oscillations, Institute of Thermomechanics , Czechoslovak

Academy of Sciences, Prague, 1978, pp. 191-197.

[10] J. J. Stoker, Nonlinear Vibrations . Interscience Publishers Inc , New York,

1950.

[11] E. H. Dowell and C. Pezeshki, "On the Understanding of Chaos in Duffing'

s

Equation Including a Comparison with Experiment," Journal of Applied Mechanics . ASME, Vol. 53, March 1986, pp. 5-9.

[12] J. M. T. Thompson, Instabilities and Catastrophes in Science and

Engineering . John Wiley and Sons, New York, 1982.

[13] P. Berge, Y. Pomeau, and C. Vidal, Order within Chaos . John Wiley and Sons,

New York, 1984.

[14] F. C. Moon, Chaotic Vibrations . John Wiley and Sons, New York, 1987.

[15] F. Sullivan and F. Hunt, "How to Estimate Capacity Dimension," Nuclear

Physics B . (Proc. Suppl.), Vol. 5B, 1988, pp. 125-128.

[16] A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano, "Determining

Lyapunov Exponents from a Time Series," Physica D . 1985, pp. 285-317.

[17] J. Guckenheimer and P. Holmes, Nonlinear Oscillations. Dynamical Systems,

and Bifurcations of Vector Fields . Springer-Verlag, New York, 1983.

13
Fig. 2.1 Schematic of Forced Modified Stoker Column.

14
F^g- 2.2 Drawing of Forced Modified ^iied Sto. Stoker Column.

15
16


19
20
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21
(s/uuuu) A;!00|8A

22
r

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29
Fig. 4.1 Free Body Diagrams.

30
yii

1

yi2

1

Pi 2

P12

P2I = Pi 1

Fig. 4.2 Lumped Masses Representing Vertical Springs (n„ = 4)

Asinwt X12 X11 X X21 X22 X23 Asinwt

| ><(j)><:(!)><(!)><:(b><:(b>c(b> > < | Rl4 Ri3 R12 R11 R21 R22 R23 R24

Fig. 4.3 Lumped Masses Representing Horizontal Springs (n^ = 4)

31
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34


NIST-114A U.S. DEPARTMENT OF COMMERCE

(REV. 3-89) NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY

BIBLIOGRAPHIC DATA SHEET

1. PUBLICATION OR REPORT NUMBER

NIST/BSS-158

2. PERFORMING ORQANIZATION REPORT NUMBER

3. PUBUCATION DATE February 1990

4. TITLE AND SUBTITLE PERIODIC AND CHAOTIC MOTIONS OF A MODIFIED STOKER COLUMN:

EXPERIMENTAL AND NUMERICAL RESULTS

5. AUTHOR(S)

Graham R. Cook and Emil Simiu

6. PERFORMING ORGANIZATION (IF JOINT OR OTHER THAN NIST, SEE INSTRUCTIONS)

U.S. DEPARTMENT OF COMMERCE

NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY

GAITHERSBURQ, MD 20699

7. CONTRACT/GRANT NUMBER

8. TYPE OF REPORT AND PERIOD COVERED

Fi nal

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (STREET, CITY, STATE, ZIP) National Institute of Standards and Technology (NIST)

Gaithersburg , MD 20899

10. SUPPLEMENTARY NOTES

DOCUMENT DESCRIBES A COMPUTER PROGRAM; SF-185, FIPS SOFTWARE SUMMARY, IS ATTACHED.

11. ABSTRACT (A 200-WORD OR LESS FACTUAL SUMMARY OF MOST SIGNIFICANT INFORMATION. IF DOCUMENT INCLUDES A SIGNIFICANT BIBUOQRAPHY OR

LITERATURE SURVEY, MENTION IT HERE.)

Records are presented of typical measured motion of a modified forced Stoker column,

including periodic motion around a stable fixed point, periodic snap-through motion, and

chaotic motion. Characterizations of the recorded chaotic motion include: the autocorrelation function; the spectral density plot; capacity dimensions; and the Lyapounov

exponent. Two sets of numerical simulations were performed, in which the same spring

stiffnesses (measured under static conditions) and the same dissipative forces (based on the viscous damping model) were used. The first set, in which the device was modeled

as a multidegree of freedom system to account for the distributed mass and stiffness of

the springs, yielded chaotic motions qualitatively similar to those recorded in the

laboratory. No chaotic motions could be obtained from the second set, which did not

reflect the fact that the spring properties are distributed and in which the device

was therefore modeled as a one degree of freedom system. To the writers' knowledge,

this is the first reported instance of an experimental structural system with continuous

members for which a qualitatively successful simulation of the chaotic motion appears to

necessitate the inclusion in the model of additional degrees of freedom to account for

the effects of continuity.

12. KEY WORDS (6 TO 12 ENTRIES; ALPHABETICAL ORDER; CAPITALIZE ONLY PROPER NAMES; AND SEPARATE KEY WORDS BY SEMICOLONS)

Buckling; chaotic motion; dynamical systems; nonlinearity; structural dynamics;

structural engineering; vibrations

13. AVAILABILITY T UNLIMITED FOR OFFICIAL DISTRIBUTION. DO NOT RELEASE TO NATIONAL TECHNICAL INFORMATION SERVICE (NTIS). ORDER FROM SUPERINTENDENT OF DOCUMENTS, U.S. GOVERNMENT PRINTING OFFICE, WASHINGTON, DC 20402. ORDER FROM NATIONAL TECHNICAL INFORMATION SERVICE (NTIS), SPRINGFIELD, VA 22161.

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