Dynamic Responses - Mechanics of Structures: Variational and Computational Methods - page 621

STEP 8. Compute Physical Displacement Responses by Modal Superposition
The response in terms of V is found using

q i

= = ( ) φ i

V Φq t (10.90)

STEP 9. Compute Other Responses
Responses other than displacements, e.g., accelerations and elastic forces, are frequently
of interest in design. Accelerations are computed by taking derivatives of the displacement
responses. Thus,

φ i q i

V

( ) = ( )

t t (10.91)

In some cases q i ( t ) can be obtained readily from Eq. (10.84).

EXAMPLE 10.9 Forced Response of Two-DOF System
Co mp u te the response of the system of Fig. 10.5. A suddenly applied horizontal force
P 2

P 0 is imposed on mass 2 (see Fig. 10.7).
Follow the procedure outlined in this section.
Step 1. Set up the governing equations for the free vibration
m 0
0 m

=

u 1
u 2

2 k u 1

u 2

− k

+ = 0

− k 2 k

(1)

V

M + KV = 0

with the initial conditions

) = V V ( 0 ( 0 ) = 0 (2)

796266 √ k

Step 2. Fr om Example 10.4, it is already known that ω = 0 . / m and ω =

538188 √ k

1 2

1 . m and the mode shapes are found in Eq. (5) of Example 10.4.

From Eq. (10.61a)

/

366025] m 0
0 m

1

. 000

φ 1 M φ 1 =

M 1 = [1 . 000 1 .

.

1 366025

φ 2 M φ 2 =

= 4 . 732049 m, M 2 = 1 . 267949 m (3)

FIGURE 10.7
Loading for Example 10.9.