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This is to certify that the
thesis entitled "

THE NATURE OF THE FORCE FUNCTION
IN SELF-EXCITED VIBRATIONS

presented by

WILLIAM J. RUB!

has been accepted towards fulfillment
of the requirements for

MASTER'S d in CIVIL ENGINEERING

MOM

Major professor

March 13, 1951

 

 

’b-

 

. m: ‘v'W-¢l:=_'h m. , -_- .

Jo-

THE NATURE OF THE FORCE FUNCTION

IN SELF-EXCITED VIBRL'I‘ICNS

By

William John__§u_by

A {IBESIS
Submitted to the'School of Graduate Studies of Michigan
State College of Agriculture and Applied Science
in partial fulfillment of the requirements

for the degree or

MASTER OF SCIENCE

Department of Civil Engineering

1951

THESIS

Acknowledgement

The writer wishes to express his gratitude for the timely
suggestions, mending inspiration, and unfailing interest given

him by Dr. Charles 0. Harris during the preparation of this paper.

- ' .9
an'lf‘a

II.

III.

V.

TABLE OF CCNTENTS

Statement of Problem -

Introduction - - - - - .-

Oonversion of Data - ...... .. .. .. - -

Preliminary Data -

Idealized Case -

Development of Function
Comments and Conclusions

Bibliography - .. - - - -

1. Statement of Problem

The object of this paper is to determine the nature of the driving
force causing self-excited vibrations. It has been assumed, that the

curve, showing amplitude plotted against time, is available.
II. Introduction

.L self-excited vibration is a phenomenon in which the force
furnishing the energy to cause oscilation is dependent upon the motion
of the system.under consideration, in contrast to the usual forced
vibration which does not depend on the motion.

In general, this subject has two natural groupings:

(1) systems where this form.of vibration is necessary for
the unit to carry on its intended function, and,

(2) systems where a self-excited vibration inhibits the
intended action.

Some typical systems where self-excited vibrations may occur are:
vibration of transmission lines coated with sleet, airplane-wing
flutter, nosing of locomotives, and some cases of hunting of generators.
All bowed string or blown musical.instruments, the doorbell, the
automobile horn, power-driven tuning fork, and many toys function
because of self-excited vibrations.

The characteristic, common to all systmms subject to this type

of vibration, is instability. A.motion is said to be unstable when

the driving force tends to increase the amplitude of the existing
motion. A.system.is not necessarily unstable for all frequencies
or conditions. There are usually certain ranges of frequencies for
which the motion is stable and other ranges for which the motion is
unstable, for any given system.

It is desirable to control the self-excited vibration in many
systems. There are two approaches leading to the control of this
form.of vibration. They are:

(1) the elimination of the instability of the unit; and,
(2) the reduction of the vibration by introducing suffi-
cient damping in the existing system.
Both of these cannot, in general, be applied to all systems. When
it is possible to eliminate the instability and retain the useful-
ness of a system, the problem is usually quite easily solved. If
the second expedient is to be utilized, it would be a distinct
advantage to have some expression for the force to be reduced by

damping.
III. Conversion of Data

Preliminary data. The information which was assumed to be
available, at the outset of the problem, was the curve showing
amplitude plotted against time. This curve varies for each par-
ticular case, however, the general shape varies only slightly. A
typical build-up curve, shown in Figure I, has time plotted as

abscissa and amplitude as the ordinate.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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Since the exciting force in a self-excited vibration depends
on the motion, it would take an indefinite length of time for the
vibration to start and reach, say, 10 per cent of its maximum value.
It is because of this characteristic that some random.disturbance
must cause the initial motion of the system. As can be seen from
Figure I, it takes an infinite length of time for the amplitude to
reach a maximum, however, 90 per cent of this maximum.can be reach-
ed in a short period of time. The time lapse between 10 per cent
and 90 per cent of the maximum.amplitude will vary from a few sec-
onds to an hour or more, depending on the system.under consideration.

Idealiged case, For the work in this paper, the portion of
the build-up curve between 10 per cent and 90 per cent of the maxi-
mum.amplitude was used due to the uncertainty of both extremities.
The origin was placed on the curve, mid-way between the ordinates
representing 10 per cent and 90 per cent of maximum amplitude re-
spectively, as shown in Figure II.

t In order to apply a general development to a particular case,
it is convenient to use expressions of a non-dimensional nature as
much as possible throughout the deve10pment. It is with this in
nfind that the choice of abscissa and ordinate in Figure III was
made. The assumption has been made that this ideal curve is symp

metrical with respect to the origin.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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_

 

 

 

I'V. Development of Function

Two assumptions are needed before proceeding. They are:
(1’ that the natural damping force of the system during
the period under consideration is small compared
to the driving force and thus can be neglected; and,
(2) that the motion is of a sinusoidal nature.
With these assumptions it seems reasonable to assume a sine

series to represent a curve of the shape shown in Figure II. Thus,

a::bnsin9%§, l.

where a = amplitude, hn : coefficient of the series, t : time,
T : total time interval under consideration. From the fact that

CT 1
a : DA and, t : -é- , equation 1 gives, .

-}. m
D-‘ansin . 2.

It can be readily shom,z that for a system undergoing a peri-
odic motion, the change in energy per cycle, U, is,
U =

a -.12£a:. 30

NM

2
2

where a1 and a2 are the amplitudes at the beginning and end of the

cycle, respectively, and k is a constant depending on the physical
properties of the system.3 Again using a : DA, equation 5 can be

resriten as ,

U 2
_ 2
b- D2 - D1 ° 4e

By assuming that the system will require a considerable number

of cycles to build up, the following approximation can be made,
- ’ 52.11
a - a1 / A w 0 50

Here, A, is the slope of the build-up curve, half way between suc-
cessive points of maxinmm displacement, and g! represents the time
for one cycle. To change equation 5 to non-dimensional form, it

I
is necessary to determine A . First,

d-

Then,sincea: .gfi-3A.°=-—.§%=%S

I

A:% 31%. 6.

Nov letting the slope, 23%, on the non-dimensional curve in Figure III

be equal to A , equation 6 becomes,

X899. Baa

Substituting for X , its value from equation 6a and a : DA, in

equation 5 gives,

. MYA
”2-91“ m“ '7'

When the right hand side of equation 4 is factored and the value

of D1, from equation '7, substituted in the second factor, it gives,
2U - _ -
E? - (D2 ,l D1) (D2 131) - (D2 ,4 D1) (.3019). 8.

Further, D2 ,l D1 : 2D, where D is the ordinate half way between D2

and D1. This is also the point where the slope A is to be evalu-

ated. Therefore, from equation 8,

U3Me 9e
w'I'

the two variables, D and A , can each be expressed as a series,
1 °° nwc
D a . sin --—
I Z ha 8 v

:Q-l‘ 21!. 3112
A (50". :bn 2°” 2 ‘

Using these series in equation 9 gives,

U:§a“—'§:hnain nyé-EZhnncos 1:122. 10.

The energy transfered per cycle,4 through the driving force F,

to the system under consideration is ,

«as.
U: Ira-at, 11.

where y is displacement as shown in Figure IV, and can be expressed

as, y = a sinwt . 12.

Using the expression ,

D
ll

3 ha sin %§ gives,

'4
ll

[ibn sin 3-3;] sinwt . 12a.

The derivative of y with respect to t yields,

§§.w[iq, sin N13] 00-th [ibnE-gcos 91%] sinwt .

Rewriting equation 11, using the above expression for 5% gives,

U :7: {w [inn sin 31?] ooswt ,l [inn 2;! cos 9?] sinwt dt . 13.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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12

To convert equation 13 to non-dimensional form, use is made of t = 9%,

dt 3 § do; and the resulting expression is,

d

p
1%: [inn Iin Dig—9] «WE—:3} [2% 231'. cos 5&9] Iin“-’-§9- g-dc . 14.
| I

The two expressions for U can now be equated; however, if the

derivative of each is taken with respect to C, the integral will be

eliminated. This gives,
31-5- [(2% n cos 52—0);- Zhn sin— n? 2% n3 sin ‘2']
=1'w[ (Elan sin —-)coI“’-— To;(ann ;cos'n-gg Iin %]§. 15.
Solving equation 15 for F gives,

me 2 " nIIc " nIc
Zfi‘k (21:11 11 cos T)“ E'bn sin—é— Zen n2 sin -2-

Mif.(23:51s11135-:;-(-:-)cos—1‘ (:bnn cos n—‘é'--c)sin“fm]

F:—

HOT

Going back to equation 2 which states,

NH

.e
2%, 8111212,
I

The % can be placed within the summation, thus,

13

When evaluating the series constants, by harmonic analysis, either
bn or b; can be round, depending on whether it is desirable to use
dimensional or non-dimensional form.

If the bn in equation 16 is replaced by b]; A it becomes,

1 212 [(2125 n °°’ mg")2 2511‘" '1” we Tin b. ”2 ’1” me]

3 168..

n «E [2:32 (2b; .myé)ms%T2/(:bnn°°'m§g).mu%é]

 

 

This is the expression for the non-dimensional force F'.

 

1. See Figure III for definitions of C, D, and A.
2. See any complete textbook on vibrations.

3. Essentially the familiar spring constant used in theory or
fib‘tiOflSe

4. See any complete textbook on vibrations.

14

V. Comments and Conclusions

The non-dimensional force function F', expressed in equation

16a, depends upon the frequency of vibration,aJ, the build-up time,

T, and the various constants which are known or can easily be found

for a particular system. The general shape of this function has

been plotted in Figure V, using only the terms for n = l.

The conclusions are as folloss:

1.

2.

3.

4.

5.

6.

The shape of the curve for the non-dimensional force func-
tion seems to be reasonable from the solutions of analogous
problems.

The function is periodic.

The frequency of the force is the same as that of the
motion of the system.and its higher integral harmonics.
When the displacement of the system, y, is a.minimum, the
force function is also a minimum.in magnitude and vice versa.
The expression 16a for F' could be used to advantage in

the calculation of a damping unit for a system.subject to
self-excited vibrations.

Since there is nothing in the function representing the
condition of the source of the energy increase, it could

be assumed to influence the force function through the fre-
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16

Bibliography

1. Baker, J. G. "Self-Induged'Vibrations”. Trans.
A. s. M. E. Vol. 55, mes-z, 19:33.

2. Baker, J. G., and 8. J. Mikina. ”go Cglglaticn
of Dampers for Systems Subject to Self-Indug§g_
‘Vibration‘. Trans. A. S. M. E. 1-121, 1936.

5. Don Hartog, 3. P. ”Transmission Line Vibratign_
Due to Elect". Trans. A. I. E. E. Vol. 51, p. 1074,
Dec. 1932.

 

 

 

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