Advanced Methods of Structural Analysis

530 14 Dynamics of Elastic Systems

Solution is 2 D 0:0;  3 D1:0, and second eigenvector becomes® 2 D

1:0 0:0 1

̆T

; this mode shape vibration is shown in Fig.14.10d.

 2 D2:0;

! 2 D19:5959

s

EI

ml^3

:

3.Eigenvalue 3 D0:44365. In this case

.90:44365/C11 2 C7 3 D0;

11 C.160:44365/  2 C11 3 D0:

Solution 2 D1:4142;  3 D1:0, and Eigenvector® 3 D

1:0

p

21

̆T

.

Third mode shape vibration is shown in Fig.14.10e.

 3 D0:44365;

! 3 D41:6064

s

EI

ml^3

:

We can see that a number of the nodal points of the mode of shape vibration one

less than the number of the mode.

The modal matrix is defined as

ˆD

® 1 ® 2 ® 3

̆

D

2

4

' 11 ' 12 ' 13

' 21 ' 22 ' 23

' 31 ' 32 ' 33

3

(^5) D
2
4
p11 1
20:0
p
2
1  11
3
(^5) ; (g)
where thei-th andk-th indexes at'mean the number of mass and number of
frequency, respectively.

14.3 Free Vibrations of Systems with Finite Number Degrees

of Freedom: Displacement Method

Now we will consider a dynamical analysis of the structures with finite number de-

grees of freedom using the concept of unitreactions. For several types of structures,

displacement method is more preferable than the force method.

14.3.1 Differential Equations of Free Vibration in Reactions

In essence, this method consists in expressing the forces of inertia as function ofunit

reactions. According to the displacement method, we need to introduce additional