Advanced Methods of Structural Analysis

14.3 Free Vibrations of Systems with Finite Number Degrees of Freedom 537

If eigenvalue is denoted as

D

m!^2

r 0

D

m!^2 h^2

24i

;

then the system (a) may be rewritten as

.1/ A 1 A 2 D0;

A 1 C2.1/ A 2 A 3 D0;

A 2 C2.1/ A 3 D0:

(b)

Eigenvalue equation is

DD

2

4

1   10

12.1/  1

0 12.1/

3

(^5) D 0
4.1/^3 3.1/D 0 or.1/
h
4.1/^2  1
i
D 0
The eigenvalues in increasing order are
 1 D 1 
p
3
2
; 2 D1;  3 D 1 C
p
3
2
:
Corresponding frequencies of free vibration (eigenfrequencies)
!^21 D 24
1 
p
3
2
!
EI
mh^3
;! 22 D 24
EI
mh^3
;!^23 D 24
1 C
p
3
2
!
EI
mh^3
:
Mode shapes vibration. Now we need to consider the system (b) for each calcu-
lated eigenvalue. If denote 2 DA 2 =A 1 and 3 DA 3 =A 1 , then system (b) may be
rewritten as
.1/ 2 D0;
 1 C2.1/  2  3 D0;
 2 C2.1/  3 D0:
(c)
This system should be solved with respect to 2 and 3 for each eigenvalue.
First mode
! 1
 1 D 1 
p
3
2
!
: