Advanced Methods of Structural Analysis

524 14 Dynamics of Elastic Systems

Let

D

1

ı 0 m!^2

D

6 EI

m!^2 l^3

:

In this case equation (a) may be rewritten

.13/ A 1 C12A 2 D0;

12A 1 C.16/ A 2 D0:

(b)

Frequency equation becomes

DD



13 12

12 16 

D.13/ .16/ 144 D0:

Roots indescending orderare 1 D26:593I 2 D2:4066

Eigenfrequenciesin increasing orderare

! 1 D

s

6 EI

 1 ml^3

D0:4750

r

EI

ml^3

;! 2 D

s

6 EI

 2 ml^3

D1:5789

r

EI

ml^3

:

Mode shape vibration may be determined on the basis of equations (b).

For first mode. 1 D26:593/ratio of amplitudes are

A 2

A 1

D

13 

12

D

13 26:593

12

D1:1328;

A 2

A 1

D

12

16 

D

12

16 26:593

D1:1328:

Assume thatA 1 D 1 , so the first eigenvector®becomes

'D

' 11 ' 21

̆T

D

1 1:1328

̆T

For second mode. 2 D2:4066/ratio of amplitudes are

A 2

A 1

D

13 

12

D

13 2:4066

12

D0:8828

A 2

A 1

D

12

16 

D

12

16 2:4066

D0:8828

The modal matrixˆis then defined as

ˆD



11

1:1328 0:8828

:

Corresponding mode shapes of vibration are shown in Fig.14.8b.