115
714

 

THS

THE EFFECT GE VAMATEON OF THE-

PHYSECAL COESTAhfiTS G? A THREE

MASS SYS?EM LENSES THE" RESONANT
FREQUENCEES

Thais far {he Dagmar 36 M. S.
WCMGAN STATE CGiLEGE
Zigurd: jmés Levefisfséns
19547

'I‘HESIS

This is to certify that the
thesis entitled
THE EFTECT OF‘ VARIATION OF THE PHYSICAL CONSTANTS OF A
THREE MASS SYSTEM UPON TPE RESONANT FREQUENCIES

presented by

ZIGURDS J. LEVFNSTEINS

has been accepted towards fulfillment
of the requirements for

MASTER OFW degree in APPLIED MECHANICS

Major professor
DateDeQembgr 6, 125’;

0.169

THE EFFECT OF VARIATION OF THE PHYSICAL
CONSTANTS OF A THREE MASS SYSTEM UPON
THE RESONANT FREQUENCIES

by

Zigurds Juris Egyensteins

A THESIS
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 of

MASTER OF SCIENCE

Department of Applied Mechanics
1951+

ACKNOWLEDGEMENT

The author wishes to express his sincere thanks to
Dr. Charles 0. Harris, under whose lofty inspiration,
constant supervision, and unfailing interest this

investigation was undertaken and to whom the results

are herewith dedicated.

THE EFFECT OF VARIATION OF THE PHYSICAL
CONSTANTS OF A THREE MASS SYSTEM UPON
THE RESONANT FREQUENCIES

by

Zigurds Juris Levensteins

AN ABSTRACT

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 of

MASTER OF SCIENCE

Department of Applied Mechanics
Year 195%

Approved

._.—r._...- ‘RA

Zigurds Juris Levensteins

The object of this study was to investigate the rela-
tionship between the resonant frequencies and the physical
constants in a three mass system. In particular, it was
desired to see the effect that variation of the physical
constants of the system has upon the distribution of the
resonant frequencies.

Two methods of analysis were used. First the sixth
degree algebraic resonance equation was solved by a direct
method, obtaining expressions for the three resonant
frequencies in a trigonometric form as functions of the
physical constants of the system. Second, by an inverse
method of approach, relationships between the resonant
' frequencies and the physical constants of the system were
derived in an algebraic form.

It was concluded from the direct method of solution
that the expressions were well suited for the computation
of the resonant frequencies in any given three mass system.
Also, the results obtained by the direct method of analysis
shdhd that it is possible to have only two distinct reso-
nant frequencies in a three mass system, provided that the
physical constants of the system have a particular rela-
tionship. However, this relationship is so complicated
that it appears to be difficult to devise a physical system
in which the masses and springs fulfill the required rela-
tionship.

Zigurds Juris Levensteins

For the purpose of determining the effect that varia-
tion of the physical constants of the system has upon the
distribution of the resonant frequencies, it was found
that the inverse method of solution was simpler than the
trigonometric form as found by the direct method.

Some numerical calculations were made with the inverse
method solution. By varying one physical constant of the
system at a time and keeping all others unchanged, the
effect that each constant has on the distribution of the

resonant frequencies was determined.

TABLE OF CONTENTS
PAGE

Object .............................. ..... ......... 1
Introduction ................................ ......
The Steady State Metion of a Three Mass System ....

Resonant Frequen01es .OIOOOOOOOOOOOOOO0.0.0.0000...

00mm»:

Trigonometric Form of Solution ....................
Conditions for Two Resonant Frequencies

to be Equal .................................... 11
Distribution of Resonant Frequencies .............. 13
Numerical Examples ................................ 16
Conclusions ....................................... 18

Bibliography OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 19

92129.2
The object of this study was to investigate the reso-

nance conditions of a vibrating three mass system and deter-
mine the relationship between the resonant frequencies and
the physical constants of the system. In particular, it was
desired to see the effect that variation of the physical
constants of the system have upon the distribution of the

resonant frequencies.

Introduction

Vibration is a repetitive motion. It can develop on
every occasion where there is an elastic body and a repeti-
tive force acting on it. Therefore, it is a very important
problem in engineering.

To describe completely the vibrating system at any
instant, a certain number of independent coordinates is
necessary. This number of independent coordinates is called
the number of degrees of freedom of the system. There are
systems with a single degree of freedom as well as systems
with infinitely many degrees of freedom. The simplest vi-
brating system is with one degree of freedom and is compar-
atively easy to analyze, but as the number of degrees of
freedom increases the problem becomes exceedingly difficult.
A system with three degrees of freedom was chosen for this
analysis, so let us turn to the investigation of that par-

ticular type.

‘Ihg Steady Stggg Motion 9; a Th;gg-Mgg§ System

Figure one shows a system with three degrees of freedom.
It consists of three masses, M1, M2, and M3 and three springs
with spring constants k” k; , and In“, respectively. It is
assumed that the springs have negligible mass. Furthermore,
all friction in the system (causing damping of the vibrations)
is neglected. A harmonic driving force mmtslne caused
by a rotating unbalance is applied to the mass M1. Thus vi-
bration of the system will occur. It seems reasonable to
assume the frequency of the vibration to be the same as the
frequency of the driving force. This assumption is supported
by experimental results and theory.

Figure la shows the configuration of the system in
static equilibrium, and it is assumed that the vibration will
be about this configuration. Only the vertical vibrations of
the system will be considered in this analysis; thus in this
case the independent coordinates necessary to describe the
system are the three vertical displacements.'9¢., 96;, and

9¢,of the masses M1, M2, and M3, respectively. Only steady
state vibrations of the system will be considered, neglecting
the transient vibrations. A

Figure 1b shows the configuration of the system when it

is vibrating in the steady state. Also Figure lb shows all

forces acting on the masses M1, M2, and M3 at any time'* .

By applying Newton's Second Law of Motion to the system three

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simultaneous second order differential equations are obtained.

They are:

M153. + 1999-761) 4- k,('x..- x,) = mrm‘sinw, (1)
M1521" k, (9". " x1.) "' kz'xn.‘ O , (2)

M36t5—k,(x,-x,)-o . (3)

Where it is assumed that the frequency of the driving force
""135an is constant, so that 9303+ and 9 can

be replaced by art .
For steady state vibrations, the solutions of equations

(1), (2), and (3) can be assumed in the forms:

x..= X,sinwf , (tr)
xf-‘i nginmt, ( 5)
9": X’sinwt . (6)

Where X1, X2, and X3 are the amplitudes of motion of M1, M2,
and M3, respectively. Substition of equations (H), (5), and
(6) in equations (1), (2), and (3) yields,

(kf’ k’-utM.)X. - k'xz " k3x3= mrw" 3
~k,X. + (k.+ k,-<3'M,_)X._= o , <7)

"" k; X. "' (ks " th1)X3= O 3

which is a system of three simultaneous algebraic equations
that can be solved for X1, X2, and X3.
Solving the system by determinants and introducing the

notations,
I t t k a
«ta-=12, Cow—k"; w.=—'Ma
I t 3
M, , 9: Ms

the amplitudes of motion are,

x3315 (5;) )0" (wlgfig'fihg' H65]

(8)

 

x1: WSW-Silas; (9,
D

 

a (a? [A' (at (at a $1

X33; . D

(10)

D is the coefficient determinant of the system of equations
(7), and is,

I.

D. 04%,) J [mi—:3;1(ai-(sa‘][(aa‘-)+v-(ai1
amp“ -v[-;= a} at -(-$-.>‘J

 

 

(ll)

Resonant EIeguencies

Whenever the denomination D is equal to zero the ampli-
tudes of motion X1, X2, and X3 will become infinite. The
condition under which an amplitude tends to increase without
limit is called a resonant condition. It is very important
to know for what values of the driving frequency on the
amplitudes become infinite in any given vibrating system,

because at those frequencies an extremely violent shaking

of the system will result. In some problems this condition
is desired, in others it is to be avoided. To determine the
values of as for which resonance occurs, the coefficient
determinant D must be equated to zero. Thus an algebraic

equation of sixth degree in'gga is obtained,

 

6 It a,
a» w w __ ._
(ag1'A(m,1+B(zs,1 C-O- <12)
The coefficients A, B, and C are functions of the physical

constants of the vibrating system, namely, 031, 002, (.03,
ft, and I, . They are:

A=('+#1%;1*('3‘:1* v+I . <13)

‘ w .‘
§1+F(%, + VS“ 1. (11+)

IS

ashes-:1) 334a)?

8

I. I.
3.316%: . (15)

An equation like (12) must have six roots, but only three

of them are different numerically; that is, in absolute value

there are only three roots. This is true because equation

(12) can be written,

{—Aaf-o-sa-c-o , (16)

by letting, '-
1(E§:)IISK .
Thus it can be seen that a system with three degrees of free-
dom has three distinct values of the driving frequency for
which resonance occurs. Since the coefficients of the reso-
nance equation (12) are functions of the physical constants
of the system, the location of the resonant points can be
varied by changing the masses and spring constants in the
system. I
Now assume that the roots of equation (16) are 5"1’
A2, and 9s 3. Then the following relationship between

the coefficients A, B, and C and the roots A1, $2,

and 9&3 must be true,
A: A. +AL+75$ , (l7)
B=7M31+Atzt+atxt ) (l8)

c=7\u7\z7‘s ' (19)

,4.
4

From equation (19) it can be seen that 2-3-3'30 can never be
a resonant point, since that would require constant C to be
zero, but for the system to be of three degrees of freedom C

must be different from zero as can be seen from equation (15).

Igigonometric Eorm.g£ Solution
A trigonometric form of solution of equation (16) will

now be attempted. First, to eliminate the quadratic term
let,

A=z+£§ -
Then,
23+3Hz+G=0. (20)
where, H: 35 ‘Al ’
' TE) (21)
and,

G__AB-ZA::_Z'IC ’
- 21

and A, B, and C are as previously given by equations (13),

(22)

(1%), and (15). From physical considerations it can be
assumed that the roots 7s 1, A2, and 9x3 of equation
(16) and therefore also' fill, .252, and 153 of equation
(20) are distinct positive real quantities. Hence the dis-
criminant 42; of equation (20) must be positive.

The discriminant of a reduced cubic equation is,
I.
A=-Z'1(G +4H’) .

1- 5
Therefore, €3‘*‘LF44{(> and since G2 cannot be negative,
H must be, from which it follows that

A">55 .

This is one condition that must be fulfilled in order for a
system with three degrees of freedom to have three resonant
points. It can be proven that the solution of equation (20)

can be given in a trigonometric form* as,

MI(‘2?W' +2.1“!
z= 2.\/-H C05[ *5 vfi] 3 (231

 

where kao,l,2. . Substitution of equations (21) and
(22) into equation (23) yields,

 

1 1
(aAe-zA-z'IcNA-ss ]
ONCOSE' ' ch -35 +ka

k.o,u,z .

 

*Conkwright, N. B. Introduction to the filheorz 26.
Equations. New York: Ginn and Company, 19 1.

10

A
Noting that A=E+fl5 , this expression gives the three
desired roots of equation (16),

3
$2» ~2A-21c2th- as

 

 

 

 

9. 5%- A"-56 cos 1 3
+ g 3 (21+)
o,[_(fl-ZK-21€2‘\/At-bb 3+2
m=§ 555 c05{ 2. g“ r
+55- <25)
a"m[_caAs-2A‘-z1c)\IN-SD 4"
' aft—3'; ZOE-351‘
?\gprir l‘fl' C:H5 31

Thus it can be seen that the location of the resonant points
represented above by equations (2h), (25), and (26) are de-
pendent on all the physical constants of the system ‘0 1,
m2, and (03, ft , and 9’ , but the functional relationship
between them is extremely complicated. Since the cosine
funtion cannot be larger than plus one or smaller than minus
one, for any given system the maximum and minimum values of

resonant frequencies are,

11

 

 

A+
m“('é's1n:+5 AI,” ’ (27)
min(€,—Q ;%-% A"- 35‘ . (28)

Conditions £2; T g Resonant Frequencieg £9 be Equal

Now let us look at the possibility of making the roots
of the resonance equation equal. Assume that the roots of
equation (20) are 21 and 32- 2.3. The latter can be
true if and only if the discriminant 1:5 is equal to zero,
that is

6241-3. 0 . (29)

In equation (23) let

¢IOVCCOJ- am 3

2
but H3 '“- ~%-; if there are repeated roots,

 

therefore,

(baarccos- G sarccostl,
20‘s

12

depending on whether G is positive or negative. If G>O ,

¢=arccos -| ; and «ts-11' . If 640 ;
¢=GPCCOS+l ; and (P80 . Therefore, if there
are repeated roots of the resonance equation (16), they must
be,
9\,=—% At-SB +% a
.._L 1-... A (30)
we +1;- :

semi-(#535 +% . <31)

If 6<O . For equations (30) to be possible in a

physical system,

2. 1. A
"?5' /\W'EMBl'TT§;)’()',

13

I.
A
or E5}? '5: . For equations (31) to be possible in a

physical system,
I 't (AK
3 A‘SB+5>O’

or E5=>() ,*which is always true, since all quantities in

equation (1%) are positive.

Distribution g;_B§sonant frequencies
To see the effect that variation of the physical constants

of the system have upon the distribution of the resonant fre-
quencies, as given by equations (2%), (2S), and (26), differ-
ent values of 001, (02, Q3, ,4. , and I, must be substi-
tuted in the equations (13), (1%), and (15) to obtain the
quantities A, B, and G. Then in turn the known A, B, and C
must be substituted in equations (2%), (25), and (26); and
the variation of 7N1, $2, and $3 noted. However, due
to the complexity of the expressions for the resonant fre-
quencies, and also quantities A, B, and C, the substitution
and computation would be a very tedious andlengthy process.
Therefore, let us try another method of approach.

Equations (13), (1%), and (15) give the quantities A, B,
and C (they are coefficients of the resonance equation (16))
in terms of the physical constants of the system. Equations
(17), (18), and (19) give the relationship that must exist

between the roots of the resonance equation 9‘1, $2, and

1h

9K3 and A, B, and C. Therefore, the right hand sides of
equations (13), (1%), and (15) may be equated to the right
hand sides of equations (17), (18), and (19), respectively,
because the left hand sides of the above equations are also

equal, respectively. This gives:

7s,+7\,_+ 7x3=(l+-,!;)(%j-+ 3:14-04- | (32)

 

“schemes -(""r~)(" mg)+(‘a’;)+(a,3¢$—5

0' “3'1. aJa'L
+7r(a's,)+"(‘5.) , (33>
.....,=(%;a§ . a)

In this system of three simultaneous algebraic equations,

eight quantities are involved, namely, 75 1, $2, 7N3,

cal, :02, 003, It , and P . By assuming five of them,

the other three can be uniquely determined. Suppose we assume

values for $1, 002, O0 3, ’4. , and 9 and solve for 00 l,
9K2, and $3. From equation (31+),

%)=u $.7V19\

(”1)

' (35)

15

Substituting this expression in equations (32) and (33) and

solving for 2 3 yields,

mm:- "9'” [@‘j—AJ - <36)
(I’N)i(|+fi)%3-I 41%?

Substitution of equation (36) into equation (35) gives the
.923.
unknown (‘95 ,

(a 1'_ N-P- ”[95:31:- 3"] p , (37)

w’ —(":;l('+fi)J L531} 71"

It can be seen from equation (37) that an interesting rela-

tionship is obtained if 5N.1 is chosen to be unity; then,

1 't
6' =’£[|— 93-")1 . (38)
no t ’
Recalling that 12%) , man means that

con-w,- Hence, if the frequency of the driving force
is equal to the natural frequency of the mass M3 and spring
F(3 considered alone, then the equation (38) holds true.
Substituting in this equation the previously defined quan-
tities of C01, C02, and It we get,

16

I.
k.--k.,+ M103, . <39)
as 1‘
Knowing 23;) from equation (37) the quantities A,

B, and C may be computed. Then by knowing A, B, and C the

other two resonant frequencies may be found from the equation,

 

M-(m-Auflm—Af-‘i-i
2-

(40)

where using the plus sign before the radical gives one root
and the minus sign the other.

Thus this inverse method of approach gives eXpressions
that are simpler for variation and computation of the para-

meters.

Numegical Examples
Now let us apply the inverse method developed in the

latter part of the previous section to obtain results in
graphical form. ‘1

First, parameters 9 and (a) were chosen as
variables and the resonant frequency ratio 7\, was
held constant at 1.1. The parameter t, was chosen to
vary from 0.01 to 0.1 and (%:)t was given the values
0.01, 0.02, 0.03, and 0.0%, successively. Figure 2

shows the results obtained for the variation of the reso-

nant frequency ratios; and Figure 3 the variation of

l7

frequency, and spring characteristics of the mass Ml. Figure
2 shows that the middle resonant frequency ratio is quite

sensitive to variations in t, , but its variation is negli-
'1

 

act. VODL
gible with respect to (“’3 , at least up to‘@ 20.04- .

0n the other hand, the lowest resonant frequency ratio seems

to have no variation with respect to I, , but it increases

1.
with increasing values of (agi' . Figure 3 shows that the
I.
parameter 94) , along with the spring constant k1, is
9

greatly- influenced by variations in I, , but the variation of
@553 has negligible effect on both of them, at least in the
range of values used.

Second, for two different values of the parameter I)

(9:0.l and 1’20.06) , the parameter [It was made
to vary from 1.5 to 2.5. The effect of this variation on the
resonant frequency ratios is shown in Figure h. For the
lowest resonant frequency ratio, the magnitudes for it,
obtained by the two different values of the parameter I’
were so nearly equal that it was impossible to draw two
separate curves. Figure 4 shows that the variation of the
resonant frequency ratios with parameter II. is very slight.
This is particularly so for the lowest and highest resonant
frequency ratios. If variation of the resonant frequency
ratios with the parameter /l is to be avoided, it is evident
from Figure H that a choice of higher values of the parameter

I, is advantageous .

usroOOo

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18

Conclusions
From this study of a vibrating three mass system it may

be concluded:

I. A direct method of solution of the sixth degree
algebraic resonance equation, where the solution is expressed
in a trigonometric form, offers good possibilities for com-
puting the resonant frequencies of any given vibrating system
with three degrees of freedom.

2. It appears to be possible to have a vibrating system
with three degrees of freedom that has only two distinct reso-
nant frequencies; that is, the resonance equation may have
repeated roots. However, it seems difficult to devise such
a system, because of the extremely complicated relationships
that the physical constants of the system have to satisfy.

3. An inverse method of solution gives simpler relation-
ships between the resonant frequencies and the physical con-
stants of the system than the trigonometric form. The ex-
pressions obtained by the inverse method are well suited for
determining the effect that variations of the physical con-
stants have upon the distribution of the resonant frequencies.

H. A notable simplification in the relationships between
the resonant frequencies and the physical constants of the
system is obtained by making the frequency of the driving
force equal to the natural frequency of mass M3 and spring

k3 alone .

l9

BIBLIOGRAPHY

Conkwright, N. B. Introduction to the TheOry g§_§guation .
New York: Ginn and Company, 195T.

Thomson, W. T. Mechanical Vibrations. New York: Prentice-
Hall, Incorporated, 1958.

Timoshenko, S. Vibration Eroblems in Engineering.
New York: D. Van Nostrand Company, Incorporated, 1937.

Timoshenko, S., and D. H. Young. Advanced Dynamic .
New York: Mc Graw-Hill Book Company, Incorporated, l9h8.

 

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