Page 206 - The Combined Finite-Discrete Element Method

DYNAMICS OF IRREGULAR DISCRETE ELEMENTS SUBJECT 189

where v is the velocity of the centre of mass, H is moment of momentum about the centre
of mass, F is the resultant force acting at the centre of mass, and M is resultant moment
about the centre of mass.
Equation (5.56) governs translational motion of the discrete element, and is best
resolved in the inertial reference frame, as the velocity components of the mass centre
are given in the inertial reference frame.
Equation (5.57) governs the rotational motion about the centre of mass of the discrete
element. The discrete element is rigid, and therefore the moment of momentum is express-
ible in terms of the inertia properties of the discrete element and angular velocity. Thus,
it is simply referred to as the ‘angular momentum’. The angular momentum about the
centre of mass is given as follows:

H = (p × (ω × p))ρdV = (H · i)i + (H · j)j + (H · k)k (5.58)
vol
= H x i + H y j + H z k
     
0 0
H x I x ω x
=  H y  =  0 I y 0   ω y 
H z 0 0 I z ω z
The time derivative of angular momentum in equation (5.57) is taken in the inertial
reference frame, while the angular momentum in equation (5.58) is expressed in the
element reference frame, which is fixed to the body and rotates at angular velocity ω
relative to the inertial reference frame. Thus, the time derivative of angular momentum
is as follows:

dH
˙
= H (5.59)
dt
˙
˙
˙
= H x i + H y j + H z k + ω × H
= I x ˙ω x i + I y ˙ω y j + I z ˙ω z k
+ (I z − I y )ω z ω y i + (I x − I z )ω x ω z j + (I y − I x )ω y ω x k
 
I x ˙ω x + (I z − I y )ω z ω y
=  I y ˙ω y + (I x − I z )ω x ω z 
I z ˙ω z + (I y − I x )ω y ω x
After substitution into equation (5.58), the Euler equations governing rotational motion
of the discrete element are obtained:
   
M x I x ˙ω x + (I z − I y )ω z ω y
 M y  =  I y ˙ω y + (I x − I z )ω x ω z  (5.60)
M z I z ˙ω z + (I y − I x )ω y ω x
Combined finite-discrete element systems usually comprise very large numbers of discrete
elements, all interacting with each other. Contact forces between discrete elements change
rapidly with the motion of the discrete elements. For such a general motion of a particular
discrete element, a closed form solution to the Euler equations is not available. It follows