Dynamic Responses - Mechanics of Structures: Variational and Computational Methods - page 623

FIGURE 10.8
Loading for Examples 10.10, 10.11, and 10.12.

FIGURE 10.9
Displacement of the frame of Example 10.10 at node c in the X direction.

EXAMPLE 10.10 Transient Response of a Frame
Calculate the response U Xc of the frame of Fig. 10.2 due to the applied force of Fig. 10.8
at node b in the X direction. Assume 2% modal damping

(ζ = 0 . 02 ) for all modes of the

i

structure, and that the structure is initially at rest.
The solution procedure for this problem follows the steps outlined in this section. The
reduced stiffness matrix and consistent mass matrix, in which the degrees associated with
the rotational DOFs are condensed out, are used, and the natural frequencies are found
in Example 10.7. Use the modal superposition method to find the transient response. This
leads to the uncoupled equations of motion of the form of Eq. (10.84) and the associated
initial conditions. After solving these equations, Eq. (10.90) is used to compute the physical
response. The response U Xc

( )

t of the structure is shown in Fig. 10.9.