Advanced Methods of Structural Analysis

526 14 Dynamics of Elastic Systems

Frequency equation

DD



4:7320 1:2679

1:2679 6:1961

D 0

Roots of frequency equation and corresponding eigenfrequencies are

 1 D6:9280!! 12 D

1

 1 mı 0

D

4EA

6:9280ml

D0:5774

EA

ml

;

 2 D4:0002!!^22 D

1

 2 mı 0

D0:9999

EA

ml

Š

EA

ml

:

Mode shape vibration may be determined on the base equation (b).

For first mode. 1 D6:9280/ratio of amplitudes are

A 2

A 1



3 C

p

3  1

p

3  3

D

4:73206:9280

1:2679

D1:73D

p

3;

A 2

A 1



p

3  3

1 C 3

p

3  1

D

1:2679

6:19616:9280

D

p

3:

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

'D

' 11 ' 21

̆T

D

1 

p

3

̆T

For second mode. 2 D4:0002/ratio of amplitudes are

A 2

A 1

D

4:73204:0002

1:2679

D0:577D

1

p

3

;

A 2

A 1

D

1:2679

6:19614:0002

D0:577D

1

p

3

:

The modal matrixˆis then defined as

ˆD



11



p

31=

p

3

:

Corresponding mode shapes of vibration are shown in Fig.14.9b.

Example 14.3.The beam in Fig.14.10a carries three equal concentrated masses
mi. The length of the beam islD4a, and flexural stiffness beamEI. The mass of
the beam is neglected. It is necessary to find eigenvalues and mode shape vibrations.

Solution.The beam has three degrees of freedom. The bending moment diagrams
caused by unit inertial forces are shown in Fig.14.10b.