II 3l Bureau ci

SCONBS TECHNICAL NOTE

487

Considerations in Computing

the Useful Frequency Range

of Piezoelectric Accelerometers

**t OF

A- W ^ 4»

z\

cSafi

^fAU Of '

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U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards
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UNITED STATES DEPARTMENT OF COMMERCE

Maurice H. Stans, Secretary

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

NBS TECHNICAL NOTE 487

ISSUED JULY 1969

Nat. Bur. Stand. (U.S.), Tech. Note 487, 20 pages (June 1969)

CODEN: NBTNA

Considerations in Computing the Useful Frequency Range

of Piezoelectric Accelerometers

Nathan Newman

Institute for Applied Technology

National Bureau of Standards

Washington, D.C. 20234

NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide a means for making available scientific data that are of transient or limited interest. Technical Notes may be

listed or referred to in the open literature.

For sale by the Superintendent of Documents, Government Printing Office

Washington, D.C, 20402 - Price 30 cents
Contents

Page

1. Introduction 1

2. Computations for Resonant Frequencies 2

2.1. Mechanical Drive 2

2.2. Electrical Drive 4

3. Resonant Frequency Solutions for Mechanical vs. Electrical Drive 6

4. Some Limiting or Special Cases 7

5. Summary and Conclusion 8

List of Tables

Table 1: Lowest Relative Resonant Frequency, oo/co , for Mechanical

Drive

Table 2: Lowest Relative Resonant Frequency, co/co and the Percent

Deviation of the Resonant Frequency, PD of RF, for Electrical

Drive

Table 3: Parameter Sets, (a,b,r), Corresponding to 2% and 10% Levels

of Percent Deviation of Resonant Frequency , PD of RF

List of Figures

Figure 1: A Typical Configuration of a Piezoelectric Accelerometer

Mounted on an Aircraft Panel

Figure 2: Lumped Parameter Model of a Mounted Accelerometer

Figure 3: Mechanical Drive: Applied Force and Displacements

Figure 4: Electrical Drive: Applied Force and Displacements

ii
Considerations in Computing the Useful Frequency Range

of Piezoelectric Accelerometers

Nathan Newman

This paper analyzes two lumped-parameter models for

computing the usable frequency range of piezoelectric ac- celerometers. The analyses indicate why application of an electrical excitation to the piezoelectric element of a mounted pickup does not, in general, give the same result as application of a mechanical acceleration to the structure on which the pickup is mounted. Tabular results of the computations for various sets of parameters indicate those cases for which the electrical drive will give resonant frequency

values within 2% of those for the pickup mounted on a vibrating structure. For these parameter sets, the electrical

drive can be used as a reliable substitute.

Key words: Electrical excitation; lowest resonant

frequency; mechanical excitation; piezoelectric

accelerometer ; usable frequency range.

1 . Introduction

Piezoelectric accelerometers are often favored for measuring structural vibration because of their small size and mass, independence of

external bias or excitation, and wide range of usable frequencies. Fig- ure 1 shows a typical configuration, mounted on an aircraft panel. The

small size generally means that the effect of the accelerometer on the

structure's vibrational characteristics will be minimal. On the other

hand, as will be shown, the structure very often does affect the vibrational characteristics of the attached accelerometer.

The sensitivity of an accelerometer (often called a pickup) is defined as the ratio of the output signal (voltage in this paper) to the

acceleration of its base. The major resonant frequency of the accelerometer is the lowest frequency for which the sensitivity has a maximum.

The usable frequency range of the accelerometer is generally taken as that region in which the sensitivity does not change significantly

(viz. 5%) from the value found near 100 Hz when calibrated on a con- ventional shaker table. However, the upper limit of the range for which

a particular accelerometer is usable when mounted on a particular

structure is lower than that determined by the resonance of the
accelerometer alone, when the mass of the accelerometer affects the

motion of the structure. The properties of principal interest are the

stiffness of attachment to, and the effective mass of the structure.

For most applications, an accelerometer is not usable if the motion of

the structure with the pickup attached is significantly different from

the motion it would have without the pickup.

This paper analyzes two lumped-parameter models which represent ex- perimental techniques often used to determine the resonant frequency of

an accelerometer. The two experimental techniques analyzed are: the

application of a sinusoidal acceleration of constant amplitude to the

structure on which the pickup is mounted, and the application of a sinusoidal voltage of constant amplitude to the piezoelectric element. The

results of the two techniques differ, because when a constant voltage is

applied to the piezoelectric element, the motions of all the other elements of the mechanical system are determined by the mechanical parameters of the system, whereas application of a constant acceleration to

the mounting structure imposes an overriding influence from outside the

pickup which limits, to the other elements, the effect of these parameters. In both cases, the upper limit on the usable range of the

pickup is set by the lowest resonance of the "pickup-structure" system

under the driving conditions imposed.

2. Computations for Resonant Frequencies

The method of computation used to find the resonant frequencies is

very considerably simplified by the use of lumped-constant models. The

distributed masses are treated as though concentrated at single points

and are designated mi, m 2 , and m 3 in Figure 2. Their interconnections

are assumed to be massless and are represented by the spring constants

k and k . The accelerometer is represented by a "seismic mass", mi,

linked to its base, m2 , by a spring, k a

. The accelerometer is attached

to the test structure, 1113, by a spring, k

s

.

2.1. Mechanical Drive

Suppose a sinusoidally varying mechanical force, F cos cot, is imposed on m3, from outside the system (Figure 3), so that its position,

X3, is A3 cos cot. The resultant motions of mi and m 2 (i.e. x x and x 2 ) are now considered as the frequency of the drive force is varied. After

transient effects die away, the equations describing the motion of mi

and m 2 under the dynamic forces are:

mi xi + k a

(xi - x 2 ) =0 (1)

m 2 x 2 + k

a

(x 2 - xi) + k

s (x 2 - x 3 ) = (2)

The resonance sought is the lowest value of co at which a maximum of

(xi - x 2 ) occurs. When the system is in dynamic equilibrium and reso- nance is approached from below, the motions will be at the drive frequency and in phase.
Xj = Ai cos o)t (3)

X2 = A 2 cos cot (4)

xi = - Aiw cos cot (3a)

x

- 2 = - A200 2 cos cot (4a)

By sutstitution in (1) and (2) , we obtain

(k - mico 2

) Ai - k_ A 2 = (5)

- k a A x + (k

s + k a

- m 2 co 2

) A 2 = k s A 3 (6)

Resonance occurs when Ai - A 2 (and therefore Ai and A 2 individually)

has a maximum value. There is no problem due to phase considerations

because resonance is approached from below and, with no damping, Xi and

x 2 can be considered to be in phase with the motion of the driving element i.e. with X3. The solution for Ai and A 2 each has the determinant

of its coefficients in the denominator. Thus maximum values of k\ and

A 2 [and consequently (xi - x 2 )] occur when this determinant vanishes.

An equation in the resonant frequency co (explicitly in co 2

) results:

(k a

- m x co 2

) (k a

+ k

s

- m 2 co 2

) - k

a

= (7)

,2 Rewritten as a quadratic in co , k k

w" - [k„ (~ + —) + k s —] oo 2 + -2-5. = (8)

At this point, calculations are simplified by introducing non-dimensional parameters a, b, and r where

a = m 2 /mi

b = m3/mi (b is not used in this

section)

r = k s /k a

In addition, if we use

? - 1 co Q = k

a/mi (or co = /k a/mi)

we obtain equations (8a) and (8b)

A ri ! . *\ ,.2 ,,2 , r u

co H - (1 + - + -) co; co z + - co^ = (8a)

a a a

* C^ 2 - (1 -!• Ir^T + T-O (8b)

2 a a 2 a % wo

3
Note that the use of the non-dimensional relative frequency , oo/oo , as well as the non-dimensional parameters a and r, make Equation (8b) non- dimensional. Conforming with the verbal description of co/oo , a and b

may be referred to as relative masses and r, as a relative spring

constant .

The relative resonant frequency is given by:

o£ _ [i + -kci +

, r)i V [1 + t (1 + r)]2

%

a

v

_

J IL a

v V

1 a (9)

Although the solution (9) is explicitly for oo 2 /u)q, it will be referred

to as the solution for the relative resonance, uyco , as well without

specifically stating, each time, that the square root process is involved. Only the negative sign is used for the radical because the

lower resonance is sought. Since the motion of the structure m 3 is

forced, the value of m 3 does not affect the resonance and thus b does

not appear in (9)

.

Earli er in this section co Q was introduced as an abbreviated symbol

for /k /m . The choice of this symbol was not fortuitous. The

quantity v/ k

& /m

1 is generally accepted as the undamped natural frequency

for a single mass driven by a single spring whose other end attaches to

a relatively very large mass. Throughout the paper, u) , is used as a reference frequency so that division of co by co Q gives the non- dimensional quantity, relative resonant frequency.

The use of relative parameters has a twofold purpose. The initial

benefit is the simplification of the mathematical manipulations in the

solution of a differential equation. However, its more significant

worth perhaps lies in the potential of scaling one set of solutions of

co/(jO qinto many. An example of the scaling procedure is given in the

section on "limiting and special cases".

2.2. Electrical Drive

The case of electrical drive is shown in Figure 4, with the applied

force and displacements indicated. A sinusoidal voltage applied to the

piezoelectric element of the pickup generates a force, F cos cot, across

k

a

. The equations of motion expressing forces acting on the three

masses are:

m i x i + ^a ( x i " x 2) = F cos wt CIO)

m

2 x 2 + ^a *- x 2 " x i) + ^s ( x 2 ~ X 3^ = ~ F cos cot t 11 )

m 3 x 3 + k

s (x

3 - x 2

) = (12)
At dynamic equilibrium below resonance the motions and accelerations

are:

Xi = Bi cos cot (13)

x 2 = B2 cos cot (14)

x 3 = B3 cos cot (15)

and

Xi = -Bico cos cot (13a)

x 2 = -B2W2 cos cot (14a)

x 3 = -B3C0 2 cos cot (15a)

By substitution in the equations of motion,

(k - mico 2

) Bi - k B 2 = F (16)

ci 3.

k Bi + (k + k - m 2 co 2

) B 2 - k B 3 = -F (17)

a a s

^ ; ^ s

c v. ,

k B 2 + (k - m 3 co 2

) B 3 = (18)

and resonances occur when the determinant of the coefficients vanishes

Setting the determinant equal to zero, we get:

(k - mico 2

) (k + k - m2 co 2

) (k - m 3 co 2

) - k

2 (k - m 3

co 2

) d do o do

- k

2 (k - mi co 2

) = (19)

o d

Performing the indicated multiplication, and removing the factor co 2 which corresponds to non-oscillatory motion

co

1* - [k (— + —)+ k (—+—)] co 2 + k k (—— + —— + ——

)

L a 'mi m 2 s m2 m 3^ J a s

v mim 2 m 2 m 3 m 3 mi

= (20)

Converting to relative parameters, as was done in equation (8) and

solving:

n 1,;' -; r, / ri l, n

, r,

2 4r 4r,, 1. [1 t ¥

(^r) + g] -y [1 + -CUT) + F

] - - - ^-(1+-)

This is a general solution for co 2 /cOq and includes the previous solution,

equation (9), as a special case for which b is very large, (i.e. m 3 » mi)

.
3. Resonant Frequency Solutions for Mechanical vs. Electrical Drive

The equations (9) and (21) lead to solutions for co 2 /Wq (and therefrom oo/o) ) which are in general different, but not necessarily significantly so. The relative resonant frequency co/u) , directly converts to

the resonant frequency, go, for any given co = /k /mi

.

In order to distinguish between the values of co/co , they will be

identified by co(a,r)/co , the relative resonant frequency for the mechanical drive and by co(a,b,r)/coQ

, the relative resonant frequency for the

electrical drive. Abbreviated descriptive designations are RRF-MD and

RRF-ED, respectively. The latter, which is a function of "b", as well

as of "a" and M r", indicates the presence of a structure which responds

to the electrical drive rather than to other forces imposed on it.

Ultimately, we would like to express the departure of the RRF-ED

from the RRF-MD as a percentage of the RRF-MD. This is the percent deviation of the relative resonant frequency using the electrical rather

than the mechanical drive. This is designated in Table 2 of PD of RF,

(see Equation 23) . In terms of the two values of U)/a) ,

w(a,b,r)/oo - oo(a,r)/u)

(jo(a,r)/a)

Since oo is constant for any particular set (a,r) , it follows that

PD of RRF = - r-7 100% (22)

0)(a,r)/o)

PD of RRF = ^Cajb.r) - o)(a,r)

% = pD f Rp ^ co(a,r)

The expression in the middle is thus the percent deviation of the reso- nant frequency , of electrical relative to mechanical drive.

Table 1 applies to the case of mechanical drive, as computed from

Equation (9). It lists co(a,r)/co for a number of representative pairs

of values (a,r). It will be found that the value of r = °° makes w/oo = \

whatever the value of a. This corresponds to the accelerometer base ni2 rigidly attached to a very large mass 1113, so that mi and k

a are the only

resonance-determining parameters.

The relative resonant frequency for an electrical drive, co(a,b,r)/oo

is given in Table 2, as determined by calculations using Equation (21).

Because of the additional parameter, b, Table 2 is much larger than Table

1 and is presented in four parts. The final column, headed PD of RF,

gives a numerical indication of how far apart the results of the two

computations are, for any particular parameter set, (a,b,r).

The parameter sets, (a,b,r), for which results are given in Table 2,

correspond to a rather wide range of mass and spring constant ratios.
However, the PD of RF do not fall into a readily predictable pattern, and

from Table 2 it is not easy to determine the resonant frequencies and

parameters (a,b,r) that combine to give a particular value of PD of RF.

Table 3 attempts to partially fill this gap. Two levels of PD (percent

deviation) were arbitrarily selected; the one 2% (or less), tolerable,

the other 10% (or more), intolerable. Herein, "tolerable" means that

results could be accepted without any resonant frequency correction being

required; "intolerable" means that results (PD of RF) point to a significant difference between co(a,b,r) and co(a,r). Thus the electrical drive

can make the accelerometer appear to have a higher resonant frequency.

4. Some Limiting or Special Cases

Equation (21) provides the general solution for the relative reso- nant frequency, co/co . Unless 1113 >> mi, co = co(a,b,r); if 1113 is infinite,

co = co(a,r). Some interesting examples arise for certain specific values

of one or more of the parameters.

If k

s = 0, there is no attachment to the structure and we have the

familiar case of the unmounted accelerometer. If the base is driven

mechanical ly, co = co Q or co/co n

= 1 . If the drive is electrical Cu(a) =

co /l + 1/a or co(a)/co = /l + 1/a. In terms of the fundamental masses,

co(a)/coQ

= /l + mi/m2. The percent deviation of resonant frequency (PD of

RF) is to a first approximation (l/2a) 100% [using the equivalent of

Equations (22) and (23)].

In a similar manner, if the accelerometer base is rigidly fastened

to a structure, 1112 and 1113 combine and co/co = 1 for the mechanical drive

and co(a,b)/co = /l + l/(a+b). In terms of the fundamental masses,

co(a,b)/co = /l + mi/(m2 + ma). Again to a first approximation PD of RF

is [1/2 (a+b)] 100%. If the mass ratio, "a" or "b", (or "a" for the un- mounted accelerometer) is large (i.e. >> 1), then the PD of RF becomes

small and both test methods give essentially the same results. Of

course, if "a" or "a+b" for the respective examples are sufficiently

small, the arbitrary limit of 10% for PD of RF will be exceeded and the

resonant frequency difference will be "intolerable".

An illustrative example, sometimes used in texts, considers three

equal masses (a = b = 1) interconnected by two like springs (r = 1)

.

Treating this problem by the two methods, we find that direct substitution in Equation (9) results in co 2 (a,r)/coQ = (3-/~5)/2 so that co(a,r)/co = (/5-l)/2 (or 0.618). Substitution in Equation (21) gives co2 (a,b,r)/coQ = 1 so that co(a,b,r)/w = 1. PD of RF = [(1-. 618)/. 618] 100% or 61.8%, which

is quite "intolerable". However, these relative parameters are not likely to be encountered in practical situations, and the example is merely

intended to show how large a spread of resonant frequency determinations

one might find (without an unreasonable search for wild parameter

values) . This particular example corresponds to the first row in Table

2 and uses the first row in Table 1.
Scaling is a procedure by which one solution set, co(a,b,r)/co , can be cascaded into many. Although oo/oo remains fixed during the process,

oo itself can change.

Let a and g be two non-zero but not necessarily unequal constants.

Now let

m' = a m

m' = a m

2 2 m

1 = a m (24)

3 3 v J

k ;

- B \

k

s

= 6 k

s Note that the system constants (m[, m^ , m'3

, k^, k' )

, leave the parameter

set, (a,b,r), unchanged. Accordingly, oo/oo is unchanged even though the

specific system constants were changed, allowing that both a and 6 were

not unity. If 3 and a are equal, neither oo/co nor oo is changed. However, if 6 is different from a, then go , and therefore oo, are each

changed by the factor /3/ou

5. Summary and Conclusion

Laboratory experience has demonstrated that the lowest resonant

frequency of a mounted accelerometer, as found by applying a sinusoidal

acceleration to the structure on which it is mounted, often differs from

the resonance found by applying a sinusoidal voltage to its piezoelectric

element. Significant differences do exist, and the test method used appears to be responsible. Magnitudes of differences are tabulated in

terms of the system parameters. Lumped constant models, while not exact

representations of the physical system, do permit an approximate solution

of the problem.

Two specific physical models (mechanical and electrical drive) have

general solutions given in Equations (9) and (21), respectively. Using

various parameter sets (m 2 /m 1 and k

g /k )

, Equation (9) gives the relative resonant frequency for the mechanical drive. Equation (21) does

the same for the electrical drive. Results for the former are given in

Table 1; those for the latter in Table 2. Table 2 also shows (in the

last column) the important quantity, PD of RF. This column, in essence,

summarizes the tolerability of substituting the electrical for the me- chanical drive method.

The solutions to Equations (9) and (21) give relative resonant frequency rather than resonant frequency itself. The conversion to co involves only direct multiplication by/ka/m1

. It is most fortunate that

the percent deviation is the same for the resonant frequency as for the
0—'

Vrv

Vx

GJL\ 'l*.i&^< ,

.! I

>

> lf 1
CD

6DCo

fi5

8.5 H i-r *•! *> K . W OO

<U SO x 8T a 5 p,

a- s . C3 „ CO u q o s I*. . A « o

J3 3 O

s> £ u o ^ a ^ Z°S« 2 o ^

»- 5 ^ es§s © H « w 2

<5« s o

4>

g |i . a a nib

iS 1 o 5

a-

*-
relative resonant frequency and the "PD of RF" is used as the column

loading in Tables 2 and 3.

Table 3 selects values of PD of RF from Table 2 and regroups them.

A 2% (or less) PD of RF is listed as "tolerable" and a 10% (or more) PD

of RF is listed as "intolerable" and must not be ignored.

Analysis indicates that for the mechanical drive (except for

k

s >> k &) for which co = co Q

) co < a> . The situation is different for the

electrical drive. It develops that for

k

s /k a

< m 3 /m 1 w < w ;

k

s /k

&

= m 3 /m 1 co = co Q

;

k-s/ k

a

> m 3/m i w =

°°o •

All three relationships apply regardless of the value of mj/mj. It is

interesting to note that when k

g /k = m 3 /m 1

, co = co no matter what the

value of m2 /m 1

.

In using the tables, the uncertainty of both oo/oo and PD of RF as computed depends on the adequacy of the lumped parameter representation

and on the accuracy with which the values of the parameters can be as- signed. However, the analysis and tabulations in this paper will permit

for improved confidence in the resonant frequency found by the use of an electrical drive.

Grateful acknowledgment is tendered to Seymour Edelman for his

constructive and clarifying suggestions during the preparation of this

paper.
Table 1 - Lowest Relative Resonant Frequency,* oj/oj , for Mechanical Drive

1

1

1

1

1

2

22

2

2

4

4

4

4

16

16

16

16

16 °° 1.000

(see Fig. 3 for configuration)

= k

s

/ k

a

co/w

1 .618

2 .765

4 .874

16 .968

oo 1.000

1 .541

2 .707

4 .848

8 .929

00 1.000

1 .437

2 .600

4 .781

16 .962

00 1.000

1 .243

4 .481

16 .877

64 .925

*The lowest resonant frequency, co, is equal to con

• —. Since

u) Q

= /k /m 1} its value is not specified, oo,

10
Table 2.1 - Lowest Relative Resonant Frequency, co/oo , and the

Percent Deviation of the Resonant Frequency, PD of RF,

for Electrical Drive

(see Fig. 4 for configuration)

b = m 3 /m 1 a = nu/mj

1 1 24

8

16

24

8

16

124

8

16

14

16

64

1 1

r = k

s /k a

co/u) pd of :

1 1.000 61.8'

.848 37.2

.745 20.5

.683 10.5

.648 4.8

.618 0.0

2 1.126 47.2

1.000 30.7

.902 17.9

.837 9.4

.840 5.1

.765 0.0

4 1.179 34.9

1.083 23.9

1.000 14.4

.944 8.0

.908 3.9

.874 0.0

16 1.214 25.4

1.078 11.4

1.000 3.3

.976 0.8

.968 0.0

oo 1.224 22.4

1.154 15.4

1.095 9.5

1.054 5.4

1.000 0.0

any 1.414

11
Table 2.2 - Lowest Relative Resonant Frequency, oo/co Q

, and the

Percent Deviation of the Resonant Frequency, PD of RF,

for Electrical Drive

b = nig/mj a = n^/mj

2 2 4

24

8

16

24

8

16

= Vk

a

co/go PD of RF

1 .833 54.0%

.707 30.7

.631 16.7

.541 0.0

2 1.000 41.4

.890 25.8

.811 14.7

.707 0.0

4 1.073 26.5

1.000 17.9

.939 10.7

.848 0.0

8 1.098 18.2

1.045 11.5

1.000 7.6

.969 4.3

.929 0.0

00 1.118 11.8

1.080 8.0

1.049 4.9

1.022 2.2

1.000 0.0

any 1.224

12
Table 2.3 - Lowest Relative Resonant Frequency, co/co , and the

Percent Deviation of the Resonant Frequency, PD of RF,

for Electrical Drive

b = mg/mj a = m 2 /m 1

4 4

8

16

4

8

16

16

80

4

8

16

96

= Vk

a

w/co PD of RF

1 .651 49.0%

.555 27.1

.437 0.0

2 .931 55.2

.753 25.5

.682 13.7

.600 0.0

4 1.000 28.1

.927 18.2

.866 10.9

.781 0.0

16 1.033 7.4

1.012 5.2

1.000 4.0

.970 0.8

.962 0.0

00 1.061 6.1

1.041 4.1

1.025 2.5

1.005 0.5

1.000 0.0

any 1.11;

13
Table 2.4 - Lowest Relative Resonant Frequency, w/o) , and the

Percent Deviation of the Resonant Frequency, PD of RF,

for Electrical Drive

b = m /nij a = m2 /m 1

16 16

32

64

16 16

64

16 16

64

16 16

64

16 16

64

r = k

s /k

a

00/O) o PD of RF

1 .347 42.9%

.300 23.5

.272 12.0

.243 0.0

4 .688 43.0

.521 8.3

.481 0.0

16 1.000 14.0

.943 7.5

.877 0.0

64 1.082 17.0

1.000 8.1

.925 0.0

oo 1.015 1.5

1.006 0.6

1.000 0.0

16 any 1.031

14
Table 3 - Parameter Sets, (a,b,r), Corresponding to 2% and 10%

Levels of Percent Deviation of Resonant Frequency, PD of RF

PD of RF = 2% PD of RF = 10%

r = k

s /k a

a = n^/nij b = nig/n^ r = k

s Aa

a = n^/nij b = m 3 /m 1

16 16 375 16

12 130

8 64

4 36

2 30

1 26

16 420

8 225

4 125

2 75

1 50

16 34

12 9

10 4

8 3

4 3.4

2 3.7

1 4.6

16 80

8 43

4 24

2 14

1 9

16 425 1 16 81

15
mi

k

a

k

seismic mass

piezoelectric crystal compliance

accelerometer base mass

attachment spring constant - combination of mounting stud and

compliance of attachment region

1113 - attachment region mass

Figure 1: A Typical Configuration of a Piezoelectric Accelerometer

Mounted on an Aircraft Panel
m 1 - seismic mass

m 2

- accelerometer base

m 3 - attachment region

of structure

attachment spring constant,

seismic mass to base

attachment spring constant,

accelerometer base to

structure

nn

^TVm

kg. q n hs

L

rrry

Figure 2: Lumped Parameter Model of a Mounted Accelerometer

Xi

TTTT

Xt

F«» <*/t.

•X» = A,<oS wt

TTTT

Figure 3: Mechanical Drive: Applied Force and Displacements

FcoS *>t

Xx

nn

^i -^WB1-

X3

m\

k± ^ru

Figure 4: Electrical Drive: Applied Force and Displacements

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