Advanced Methods of Structural Analysis

14.2 Free Vibrations of Systems with Finite Number Degrees of Freedom: Force Method 529

Frequencies of the free vibration in increasing order

!^21 D

1

 1 mı 0

D

768

31:5563

EI

ml^3

D24:337

EI

ml^3

!! 1 D4:9333

r

EI

ml^3

;

!^22 D

1

 2 mı 0

D

768

2:0

EI

ml^3

D 384

EI

ml^3

!! 2 D19:5959

r

EI

ml^3

;

!^23 D

1

 3 mı 0

D

768

0:44365

EI

ml^3

D1731:09

EI

ml^3

!! 3 D41:6064

r

EI

ml^3

:

(d)

For each i-th eigenvalue, the set of equation (b) for calculation of amplitudes is

.9i/A 1 C11A 2 C7A 3 D0;

11A 1 C.16i/A 2 C11A 3 D0;

7A 1 C11A 2 C.9i/A 3 D0:

(e)

Equations (e) divide byA 1 .Let 2 DA 2 =A 1 ; 3 DA 3 =A 1.

Equations for modes become

.9i/C112iC73iD0;

11 C.16i/2iC113iD0;

7 C112iC.9i/3iD0:

(f)

AssumingA 1 D 1 we can calculate 2 and 3 for each calculated eigenvalue.

For their calculation we can consider set ofanytwo equations.

1.Eigenvalue 1 D31:5563

.931:5563/C11 2 C7 3 D0;

11 C.1631:5563/  2 C11 3 D0:

Solution of these equations is 2 D 1:4142;  3 D 1:0. Therefore, the first

(principal) mode is defined asy11;y 21 D

p

2 y 11 ;y 31 Dy 11. If we assume that

y 11 D 1 , then the eigenvector® 1 which corresponds to the frequency! 1 is

® 1 D

1:0

p

21

̆T

.

Corresponding mode shape vibration is shown in Fig.14.10c.

 1 D31:5563

! 1 D4:9333

s

EI

ml^3

Note, that substitution of 2 and 3 into third equation (f) leads to the identity.
2.Eigenvalue 2 D2:0. In this case

.92:0/C11 2 C7 3 D0;

11 C.162:0/  2 C11 3 D0: