Advanced Methods of Structural Analysis

548 14 Dynamics of Elastic Systems

14.7.A massless beam carried body of massMand moment of inertiaJxwith
respect tox-axis (Fig.P14.7); radius of gyration isDJx=M l^2. Calculate the
frequencies of the free vibration and find the modal matrix.

M

Jx

EI

l

y

z

x

r

Fig. P14.7

Ans.! 12 D

3 EI

 1 Ml^3

,! 22 D

3 EI

 2 Ml^3

,1;2D

1 C3 ̇

p

1 C3C9^2

2

14.8.A massless beam carried two lumped mass (Fig.P14.8). Calculate the fre-
quencies of free vibrations and find the modal matrix. (Hint: For construction of
bending moment diagram in unit states use the influence lines).

M 1 =M

EI

l/ 3 l/ 3

l l

M 2 =2M

y 1 y^2

Fig. P14.8

Ans.! 1 D6:379

r

EI

l^3 M

,! 2 D11:047

r

EI

l^3 M

,ˆD



11

1:4142 0:3462

14.9.A uniform circular rod is clamped at pointBand carrying the lumped mass
Mat free end as shown in Fig.P14.9. Calculate the frequencies of free vibrations,
and find the modal matrix and show the shapes of vibrations.

B

M

R

R

Fig. P14.9