Dynamic Responses - Mechanics of Structures: Variational and Computational Methods - page 634

proportional to the mass matrix, 0.000625 M . The external force is applied at node b in
the X direction, with the time-dependence shown in Fig. 10.8. The central difference and
Wilson θ methods are used to calculate the transient response of the beam.

Central Difference Method

Use Eq. (10.100) to calculate the response. From Eq. (10.109), the critical time step is

2 2

3686

10 − 3 t crit =

max =

032 =

0 . 5426 ×

ω .

where ω

max is taken to be

ω

6 in Eq. (1) of Example 10.7. Different time steps

( t =

10 − 3 sec) are used in the calculation. The displacement of node c in the
X direction is shown in Figs. 10.12 and 10.13. It can be seen that for the time step 0

10 − 3 , 0 . 5 × t = 0 . 6 ×

10 − 3 ,

. 5 ×

10 − 3 , the integration
is very unstable. The accumulated error makes the absolute value of the results extremely
large.

the integration is stable up to 0.012 sec, but for the time step 0 . 6 ×

Wilson θ Method

The calculations use Eq. (10.124). Employ the same time steps as used above for the central
difference method, as well as 10 − 2 The displacement of node c in the X direction

is shown in Figs. 10.12, 10.13 and 10.14. The results are stable.

t = 0 . 1 × .

FIGURE 10.12
Comparison of the central difference method with the Wilson

10 − 3

θ method ( t = 0 . 5 × sec).