Dynamic Responses - Mechanics of Structures: Variational and Computational Methods - page 646

10.20 Find the first three natural frequencies of a fixed-pinned beam of length L . Use both
consistent and lumped mass matrices. Compare your results with exact frequencies.
Discuss means of convergence to the exact results.

Answer: Exact Frequencies:

ω 1 = 4182 EI

15 . /ρ L 2 rad / s ,

9651 EI

L 2

ω = 49 . /ρ

2

2461 EI

.
10.21 Consider the axial motion of a uniform rod fixed at the left end and free at the
right end. The cross-sectional area is A and the length L . Use the stiffness matrix
for extensions of Chapter 4 and a consistent mass matrix. Compare the frequencies
obtained using one element and a two element model with the exact solution.

ω = 104 . /ρ L 2

3

Answer:

C

C E ω i = /γ L 2 Mode Exact 1 Element 2 Elements

γ = mass/volume 1 1.571 1.732 1.610

2 4.712 5.628

10.22 For the transverse motion of a clamped-free uniform beam, compare the first two
natural frequencies obtained using one, two, and three element models with the exact
solution. Use the stiffness matrix for beam bending from Chapter 4 and a consistent
mass matrix.

C 2 EI

Answer: ω i = /ρ L 4

C

Mode Exact 1 Element 2 Elements 3 Elements

1 1.8751 1.880 1.8754 1.8751

2 4.6941 5.900 4.7130 4.7041

Transient Responses
10.23 Use the modal superposition method to compute the longitudinal response of a bar
fixed at the left end and free at the right end, subject t o the sudden application of a
uniformly distributed longitudinal force of intensity p x (force/length).

Answer:

1

∞

16 L 2 p x
π

1
i 3

sin i

π x

cos i

ct
2 L

π

u ( x, t ) = −

3 c 2

ρ L

i = 1 , 3 , 5

EA

c = /ρ

10.24 Find the mode shapes of the frame in Problem 10.1. Verify their orthogonality con-
ditions and normalize the mode shapes.
10.25 Calculate the X direction displacement of point c of the frame in Fig. P10.1 using
the modal superposition method. The applied force is shown in Fig. 10.8, and acts at
point c in the X direction. Assume 2% modal damping for all modes of the structure
and that the structure is initially at rest.

= .

Answer: Max X direction displacement at node c 0 0188 in.