Page 210 - The Combined Finite-Discrete Element Method

DYNAMICS OF IRREGULAR DISCRETE ELEMENTS SUBJECT 193

M x

t
(a)

hM x

0 h 2h 3h 4h 5h
t
(b)
Figure 5.5 Approximation of external load. (a) Continuous load, (b) equivalent series of impulse
loads.

5.6.9 Change in angular velocity during a single time step
The assumption that the change of angular momentum is instantaneous implies that the
motion of the discrete element during the time step is free of external moments. In other
words, the angular momentum at the beginning of the time step (before the discrete
element has changed its spatial orientation, but after the external impulse load has been
taken into account) is equal to the angular momentum at the end of the time step, i.e.

t+h H = t H (5.76)
The angular momentum at time t + h is a function of the angular velocity at time t + h,i.e.

 
t+h H ˜x
t+h H =  t+h H ˜y  (5.77)
t+h H ˜z
 
    ˜ ˜ ˜  
0 0 t+h i x t+h j x t+h k x
t+h i ˜x t+h j ˜x t+h k ˜x I x t+h ω ˜x
˜
˜

=  t+h i ˜y t+h j ˜y t+h k ˜y   0 I y 0   ˜ t+h j y t+h k y   t+h ω ˜y 
 t+h i y
0 0
˜
˜
˜
+ht i ˜z t+h j ˜z t+h k ˜z I z t+h i z t+h j z t+h k z t+h ω ˜z
Equations (5.76) and (5.77), when combined, yield angular velocity at time t + h
 
t+h ω ˜x
(5.78)
t+h ω ˜y
 
t+h ω ˜z
  
    −1 ˜ ˜ ˜  
t+h i ˜x t+h j ˜x t+h k ˜x I x 0 0 t+h i x t+h j x t+h k x t H ˜x
  ˜ ˜ ˜ 
 t+h i ˜y t+h j ˜y t+h k ˜y   0 I y 0   t+h i y t+h j y t+h k y   t H ˜y 
= 
t+h i ˜z t+h j ˜z t+h k ˜z 0 0 I z ˜ ˜ ˜ t H ˜z
t+h i z t+h j z t+h k z