Advanced Methods of Structural Analysis

14.2 Free Vibrations of Systems with Finite Number Degrees of Freedom: Force Method 523

If a structure has ndegrees of freedom, then the modal matrix ˆ D
' 1 ' 2 ::: 'n

̆

.

Example 14.1.Design diagram of the frame is shown in Fig.14.8a. Find eigenfre-
quincies and mode shape vibration.

P 1 =1

1 ⋅l P^2 =1

M 1 M 2

1 ⋅h

l

h

m

EI

q 1

q 2

2 EI

a

1.1328

m

1.0

w 1 -first mode

0.8828

b

m 1.0

w 2 -second mode

Fig. 14.8 (a) Design diagram of the frame and unit states; (b) Mode shapes of vibration

Solution.The system has two degrees of freedom. Generalized coordinate areq 1
andq 2. We need to apply unit forces in direction ofq 1 andq 2 , and construct the
bending momens deagram. Unit displacements are

ı 11 D

M 1 M 1

EI

D

1

2 EI



1

2

 1 ll

2

3

 1 lC

1

EI

 1 lh 1 lD

l^3

6 EI

C

l^2 h

EI

I

ı 22 D

M 2 M 2

EI

D

1

EI



1

2

 1 hh

2

3

1 hD

h^3

3 EI

I

ı 12 Dı 21 D

M 1 M 2

EI

D

1

EI



1

2

 1 hh 1 lD

h^2 l

2 EI

Let hD2l and ı 0 Dl^3 =6EI. In this case ı 11 D13ı 0 Iı 22 D16ı 0 I
ı 12 Dı 21 D12ı 0.
Equation for calculation of amplitudes (14.4)

13ı 0 m!^2  1



A 1 C12ı 0 m!^2 A 2 D0;

12ı 0 m!^2 A 1 C



16ı 0 m!^2  1



A 2 D0:

(a)