Advanced Methods of Structural Analysis

546 14 Dynamics of Elastic Systems

Fundamental data for one-span uniform beams with classical boundary condi-
tions are presented in Table A.26. This table contains the frequency equation, and
the first, second and third eigenvalues. For the nodal points of the mode shapes of a
free vibration, the origin is placed on the left end of the beam.

Problems.......................................................................

14.1.Statically determinate beam carried one lumped mass. Determine the fre-
quency of free vibration.

M

b

l a

c

M

l

2 EI

l

M EI

a

EI, l

a b

Fig. P14.1

Ans. (a)!D

r

3lEI

a^2 b^2 M

;(c)!D

r

2 EI

3l^3 M

.

14.2.Statically indeterminate beamcarried one lumped mass (Fig.P14.2). Deter-
mine the frequency of free vibration.

M

b a

EI M

l/ 2

EI

l/ 2

ab

Fig. P14.2

Ans. (a)!D

s

12 EI

a^2 .3bC4a/ M

;(b)!D

r

768 EI

7l^3 M

.

14.3.Symmetrical frame with absolutely rigid cross bar of total massMis shown
in Fig.P14.3. Find the frequency of free horizontal vibration.

EI=∞, M

h EI

Fig. P14.3

Ans.!D

r

24 EI

h^3 M