Page 215 - The Combined Finite-Discrete Element Method

198 TEMPORAL DISCRETISATION

The z-axis is one of symmetry. The inertia tensor of the element and initial angular
velocity at time t = 0 are as follows:

     
0 0 J 0 0 2I 0 0
I x
I c =  0 I y 0  =  0 J 0  =  0 2I 0  (5.94)
0 0 I z 0 0 I 0 0 I
   
ω xo 0
ω o =  ω y o  =  1  (5.95)
ω zo 100

The analytical solution for this problem is obtained using the Euler equations:
       
M x I x ˙ω x + (I z − I y )ω z ω y I ˙ω x − (I − J)ω z ω y 0
0
 M y  =  I y ˙ω y + (I x − I z )ω x ω z  =  I ˙ω y − (I − J)ω x ω z  =   (5.96)
M z I z ˙ω z + (I y − I x )ω y ω x J ˙ω z 0
The third equation yields
(5.97)
ω z = constant = ω z o
Differentiation of the second equation and substitution into the second equation yields

J 2I

2 2
¨ ω y + p ω y = 0; p = − 1 ω z o = − 1 100 = 100 (5.98)
I I
A similar equation is obtained for ω x , which after substitution of the initial condi-
tions, yields

cos pt = cos 100t (5.99)
ω y = ω y o
sin pt =− sin 100t (5.100)
ω x =−ω y o
The same problem has been solved using direct integration. A comparison of the analytical
and numerical result for the time step h = 0.0005 seconds is given in Figure 5.7. The
analytical and numerical results for this time step are almost indistinguishable.
A similar comparison of the numerical and analytical results for ω y , using time step
h = 0.001 seconds, is shown in Figure 5.8. Again, very good agreement between the
numerical and analytical results is obtained.
In Figure 5.9 a comparison of analytical results ω y and numerical results obtained
using time step h = 0.005 seconds is shown. The results show relatively good agreement
during the first 0.05 seconds. Afterwards, the difference in results is increasing due to the
numerical result lagging behind the analytical results.
Application of the Munjiza direct integration scheme in a problem comprising irregular
discrete elements is shown in Figure 5.10, where numerical simulation of the motion of
two very irregular discrete elements inside a rigid box is presented. The discrete elements
represent pebbles subject to initial velocity and acceleration of gravity towards the bottom
of the box. Initially, the pebbles move at different velocities, impact against each other
and bounce apart in mid-air (Figure 5.10), while at the same time falling towards the