Advanced Methods of Structural Analysis

14.2 Free Vibrations of Systems with Finite Number Degrees of Freedom: Force Method 525

Example 14.2.Design diagram of structure containing two hinged-end members is
shown in Fig.14.9a. Modulus of elasticityEand area of cross sectionAare constant
for both members;landl

p
3 are length of the members, ̨D 60 ı;ˇD 30 ı.Find
eigenfrequencies, modal matrix and present the mode shapes.

m

E,A,l l 3

y 1

y 2

a b

P 1 =1

S 1 = 32 S 2 = 12

First state

P 2 =1

S 1 = 12 S 2 =−^32

Second state

a

3

1.0

First mode

1 3

1.0

Second mode

b

Fig. 14.9 (a) Design diagram of the structure and unit states; (b) Mode shapes vibrations

Solution.The structure has two degrees of freedom. The first and second unit states
and corresponding internal forcesfor each member are shown in Fig.14.9a.

Equations (14.4) for unknown amplitudes



mı 11!^2  1



A 1 Cmı 12!^2 A 2 D0;

mı 21!^2 A 1 C



mı 22!^2  1



A 2 D0:

(a)

Unit displacements

ı 11 D

PRS 1 S 1

EAdsD

1

EA

p

3

2 

p

3

2 lC

1

2 

1

2 l

p

3



D4EAl



3 C

p

3



;

ı 22 D

PRS 2 S 2

EAdsD

1

EA



1

2 

1

2 lC

p

3

2 

p

3

2 l

p

3



D4EAl



1 C 3

p

3



;

ı 12 Dı 21 D

PRS 1 S 2

EAdsD

1

EA

p

3

2 

1

2 l

1

2 

p

3

2 l

p

3



D4EAl

p

3  3



:

Let us denoteı 0 Dl=4EA; D1=mı 0!^2 ,thenı 11 Dı 0 .3C

p
3/; ı 22 D
ı 0 .1C 3

p

3/; ı 12 Dı 21 Dı 0.

p

3 3/and equation (a) becomes



3 C

p

3 



A 1 C

p

3  3



A 2 D 0

p

3  3



A 1 C



1 C 3

p

3 



A 2 D 0

or

.4:7320/ A 1 1:2679A 2 D 0

1:2679A 1 C6:1961A 2 D0:

(b)