Determining eigenvalues of free vibration of beams carrying concentrated mas

Amba-Rao, C. L.

English

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WYLE LABORATORIES - RESEARCH STAFF 
Report WR 65-34 
ON THE VIBRATION OF BEAMS OR RODS 
CARRYING A N  ARBITRARY NUMBER OF 
CONCENTRATED M A S S E S  I 
by 
C.L. Amba-Rao 
This report was completed under Contract NAS8-5384 
Prepared by 
C .L . Amba-Rao Approved by 4. A4z+r!@ R .W. White 
Approved by 62G5$=q 
Director of Research \ 
i 
SUMMARY 
An easy solution i s  given to the problem of the free vibrations of simply supported 
beams carrying an arbitrary number of concentrated masses i n  an arbitrary way. 
Finite Fourier sine transforms and. the modification of the beam to a variable den- 
sity beam by the introduction of Dirac S - functions are utilized i n  the solution 
of the problem. The simplified frequency equations corresponding to (i) a single 
concentrated mass in  the middle of the beam and (ii) two equally spaced identi- 
cal concentrated masses, are given. For the latter case, some of the roots of the 
frequency equation are numerically evaluated. 
#&no 1 
TABLE OF CONTENTS 
I 
Section 
SUMMARY 
TABLE OF CONTENTS 
LIST OF SYMBOLS 
1 .O INTRODUCTION 
2.0 FORMULATION OF THE PROBLEM 
2.1 Special Cases 
2.2 Concluding Remarks 
, 3 .O REFERENCES 
Page -
.. 
I I  
... 
I l l  
IV 
1 
2 
8 
10 
13 
M 
El 
Y 
P 
A 
P 
+ 
LIST OF SYMBOLS 
Mass added to beam 
Flexural rigidity of the beam 
Transverse deflection of the beam 
Constant mass- density of the beam material 
Area of cross-section of the beam 
Natural frequency of the beam 
Mode shape of beam 
I 
1 .O INTRODUCTION 
The problem of the vibration of beams carrying CI concentrated mass i n  the middle has 
evinced considerable interest Very iecently, Yu Chen, reference (1) applying the 
Laplace transform technique, originally developd by Thornson, reference (2) for 
static beam problems, solved this problem. A review of literature indicates that the 
problem of v?bra,tion of beam, currying a concentrated mass i n  the middle has been 
solved by Kbrmin and Blot reference (3) ami Hoppmann, reference (4) by dif- 
ferent methods. In references 1 , 3, and 4, attention was focused on symmetrical 
modes of vibration only, because of the special position of mass on the beam. 
In the present note, the authors presented an alternate method of determining the 
eigenvalues of the free vibration equation of a simply supported beam with a finite 
number of discrete masses. The partial differential equation for free vibrations of 
a bar with a number of discrete masses, which can be expressed by using Dirac b - 
functions as a variable mass density beam, i s  ini t ial ly reduced to an ordinary differen- 
t ia l  equation by standard methods of separation of variables. The ordinary differential 
equation in space variables, with i t s  boundary conditions, i s  solved by an application 
of finite Fourier sine transforms. The infinite set of eigenvalues i s  obtained as the 
roots of a transcendental equation, If a l l  masses are taken to be zero, we get the 
classical solution. 
1 
, -  
2.0 FORMULATION OF THE PROBLEM 
Let us consider the free vibrations of a simply supported beam carrying an arbitrary 
number of discrete masses M M2,,,M on the beam. In this anolysis , it i s  
1 '  q 
I assumed that E 1, the flexural rigidity of the beam, i s  constant, even though the 
beam carries a set of masses. The equation of motion, with damping neglected, 
can be written as follows: 
where 
E I 
Y 
P 
A 
M. 
flexural rigidity of the beam, 
= y (x,t ) i s  the transverse deflection of the beam, 
=: constant mass - density of the beam material. 
= area of cross-sectionof the beam. 
= i mass locatedat x = x. 
b - function i s  defined by the following relation 
th 
I I 
and where the 
S ( x - x . )  d x  = 1 J' * I 
-a0 
The boundary conditions are: 
where a i s  the span of the beam. For a bar vibrating with amplitude 
we can write for a single harmonic: 
q ( x  ) , 
Substituting (4) in  (11, we have 
d 4 *  -: [ P A + A M i  6 ( x - x i )  II, = O  
d x  1 E l  t = l  4 
and the boundary conditions 
Equations (5) and (6) constitute an eigenvalue problem, the solution of which can 
be obtained eastly by an appltcation of  finite stne transformation, Let us denote (5) 
Taking finite sine transforms of the differential equation (1) and using the transforms 
of the boundary conditions in (6), we have 
n n x .  
= 0 (8) I 
4 4  2 
.[ - k 4 ]  -p 2 Mi ( x )  1 sin E l  i= l  a 
where 
E l  
Therefore: 
~ 
E l  (f$ - k4) 
The inverse of she transform 1s given by reference (5) 
Applying (10) to (9) one obtalns 
- k  
4 
a 
Summation of Series 
Equation (1 1) may be written as follows: 
4 
where 
n n x. n n x  
I 
= 2 sin a sin a 
s *  04 
4 
4 4  
4 
I 
n n  
a 
- k  n = l  
We can evaluate the sum of the infinite series of (13) as follows, reference (6) . 
We use the known series 
cos n I 1 n cosh a ( n  - z )  
(a 1 + -  - - - -  
2 a  sinh n a 
2 2 a  
2 
n = l  n + a  
and similarly 
- 1 1 cos a ( n - z  ) - - - 2 n - a  2 a  2 a  sin II G n = l  
to obtain the following relation 
1 - 1 - 1 cos a ( n - z )  + cosh a (n - z) n = l  n - a  2 a  -.[ 4 a  sin n a sinh n a 
a k  which holds for 0 < z < 2 r  and where a = -
n 
5 
n n  - cos - n n  ( X + X i )  
a cos -( x  - x i  1 
a 
4 4 
n - a  
4 
a 
2 n  4 n - 1  
si (XI = 
a 
- -   
4 k3 
and 
for x c x. 1 I sinh k ( a  - x .  ) sinh k x I sin k ( a  - x. ) sin k x I - 
sin k a sinh k a 
for x > x. 1 I sin k xi sin k ( a  - x ) slnh k X. sinh k ( a  - x  ) I - - -  
sln k a sinh k a 
 
or we can write (14) in a compact form as 
sinh k ( a  -x.) sinh k x a [ sinh k ( a  - x i  ) sin k x I - 
s. (XI = - I
4 k 3  1 sin k a sinh k a 
1 - U ( x - x i )  sin k ( x - x i )  - sinh k ( x  - x i  ) 
where U (x) i s  the unit step function. 
If x = x. then (12) reduces to 
I 
6 
where 
I 
sin k ( a  o x .  ) sin k x. sinh k ( a  ox.)  sinh k x. 
I 1 -  I I 
sin k a sinh k a - ‘ii 
(1 7) 
I 2 1 - a 77- 
1 Thus (15) can now be written as 
0 
For a non-trivial solution of the set of homogeneous equations in (18) the determinant 
of the coefficients must vanish, Hence 
where 
2 
E l a  
2p Y =  
2.1 SPecial Cases 
(a) Suppose all masses are identically zero 
M = O  Ml = M2 - ----------- 
q 
Then, from (19) 
S i ;  (Nosumon 1 ),-boo i = 1 , 2 ,  0 . 0 0 .  q 
which implies 
sin k a j  Q) 
The consecutive roots of (20) are 
k a  = n n  n =  1,2,  4 . 0 0  00 
which agrees with the classical result. 
(b) One mats a t  the middle of the beam, i .e. 
i = 1; a = 21; and x = I . 1 
Then, 
sinh k x - sin k x 
4E I k3 cosh k I cos k I 
1 =  
8 
I 
Equation (22) simplifies to 
I 
1 2 M p  ( tank1 = tanh k l )  - 4 E l k 3  = 0 
I 
Equation (23) agrees with equation (12) of reference 1. 
(c) Solution of the frequency determinant for two identical masses equally 
spaced . 
In the case of 2 equal masses, equally spaced on a simply supported beam, the 
determinantal equation (1 9) reduces to: 
, 
= 2 1  0 x2 where a = 3 I x1 = I 
I 
Equation (24) reduces, after lengthy computations to: 
1 1 
U 
+ 8 X u [sinh 2 u - sin 2 u  + 2 (sinh 2 u  cos 2 u - cosh 2 u sin 2 u )  
[ 4 (sinh u sin u - sinh 2 u sin 2 u ) + 3 (cosh 2 u - cos 2 u ) 
sinh 3 u sin 3 u 
sinh u sin u 
= o *  + 8 X 2  
where X = &  = %  and u = k l  . M M 
9 
I '  
The first 1 1  roots of the above transcendental equation are obtained on an IBM 
1620 digital computer. Table 1 
first 1 1  modes of vibration, for different values of X ; ( the ratio of the mass of 
the beam to the attached mass ) . 
shows thevalues of kl / corresponding to the 
The values of  kl corresponding to the limiting 
n n  
case of A- oo are obtained from the classical theory as 3where n i s  an 
integer. The fundamental and the lower frequencies are highly dependent on the value 
of  A ,  while the effect of X i s  negligible on the higher frequencies. 
Every third frequency i s  a constant and equals n n where n i s  an integer, One 
can easily obtain the following relations 
where 
and f = frequency i n  cycles a 
Table 2 shows the distribution of for different values of X . The lowest two 
valuesof p ,  namely, 0.174 and O0664, corresponding to values of X = 0.001 
coincide with the frequencies of a massless beam, 
two degrees of freedom, ( 0,174, 0,675 ) 
( X J  0 ) treated as a system with 
2.2 Concluding Remarks 
For a beam whose boundary conditions at x 0 and x = 2 I are d * /  dx = 0 
3 3 
and d e /  dx = 0, finite cosine transforms can be used. It may be noted, that for 
such a b&m, without any masses, the classical frequency equation is'the same as that of 
a simply supported beam, namely sin k a = 0. The advantage of sine and cosine 
transforms over Laplace transforms i s  that the inverse In  the former is a converging infinite 
series, which normally can be summed. ( a = 2 I ). 
TABLE 1 
10. 1.023 
5. 1.001 
2. 0.946 
1. 0.880 
0.5 0.7% 
0.001 0.186 
7n 
3 
-a0 
10. 7.186 
5. 7.093 
2. 6.948 
1, 6.854 
0.5 6.790 
0.001 6.708 
2 n  
3 
- n 
2.045 
2.000 
1.889 
1.753 I t  
1.579 
0.366 
II 
I I  
I I  
I t  
I t  
8n 3 n  
3 
-
8.201 II 
8 .@75 
7.858 
7.703 II 
7.587 
7.430 
II 
I t  
I 1  
I t  
4n 
3 
- 
4.098 
4.030 
3.899 
3.790 
3.699 
3.557 
10 TI 
3 
10.282 
10,177 
10.036 
9,957 
9.907 
9.849 
5 n  2 n  
3 
-
5.119 
5 . 027 
4.840 I t  
4.676 
4.534 
4.298 
I t  
II 
I t  
I t  
I 1  
1 1  n 
3 
-
11.289 
11.142 
10.920 
10.781 
10.688 
10.572 
1 1  
I I  
0.053 
0.071 
0.100 
0.123 
0.143 
0.174 - 
1.987 
2.81 0 
4.443 
6.283 
8.086 
19 3 6 9  
0.21 1 
0.285 
0.402 
0.489 
0.561 
0.664 - 
2.600 
3.581 
5.433 
7.477 
10.377 
22.648 
TABLE 2 
10.0 
5.0 
2 .o 
1 .o 
c 0.5 
0.001 
10.0 
5.0 
2.0 
1 .o 
0.5 
0 .OOl 
0.497 
0.702 
1.111 
1.571 
2.221 
4.967 
3.385 
4.641 
6,949 
9.444 
12.956 
27.784 
0.845 
1.156 
1.711 
2.286 
3.080 
6.369 
4.468 
6.319 
9.991 
14.180 
19.983 
44.683 
1.319 
1.799 
2 -636 
3.480 
4.627 
9.297 
5.321 6.414 
7.372 8 3 3 6  
11.335 13.420 
15.779 18 -499 
22.091 25.712 
48.819 56.251 
i 2  
i 1 
I 
I I 
I 
I 
t 
I 
I 
I 
I , I 
1. 
2. 
3. 
40 
5. 
6. 
RE FERE NCES 
Yu Chen, I' On the Vibration of Beams or Rods Carring a Concentrated 
Mass 'I, JOURNAL OF APPLIED MECHANICS Volume 30, No. 2, 
1963, pp. 310 - 311 . 
Thomson, W. T ., I' Laplace Transformation", Prentice-Hall Inc o ,  
New York, 1950, pp. 78 -89. 
Karman To, and Blot, M. A., 'I Mathematical Methods in Engineering", 
McGraw-Hill Rook Company, Inc,, New York, New York. 1940. 
pp, 290 - 294. 
I I  
Hoppmann, W, H. II, 'I Forced Lateral Vibration of Beam Carrying A 
Concentrated Mass", JOURNAL OF APPLIED MECHANICS Volume 
74, 1952, pp. 301 - 307. 
Sneddon, I .  N ., I' Fourier Transforms", McGraw-Hi11 Book Company, 
Inc., New York, New York. 1951, pp. 74, 
Tlmoshenko, S . and WoinowskyXrieger, S ., " Theory of Plates and Shells", 
McGraw-Hill Book Company, Inc., New York, New York. 1959, pp, 141 - 142. 
13 


On the vibration of beams or rods carrying an arbitrary number of concentrated masses

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