Page 205 - The Combined Finite-Discrete Element Method

188 TEMPORAL DISCRETISATION

5.2.5 The inertia of the discrete element
The inertia of the discrete element in translational motion is defined by the mass of the
discrete element

m = ρdV (5.47)
vol
where ρ is mass per unit volume of the discrete element.
The inertia of the discrete element in rotational motion is defined by the inertia tensor:

 
Ixx Ixy Ixz
I =  Iyx Iyy Iyz  (5.48)
Izx Izy Izz

where the moments of inertia are given by

2
2
Ixx = (y + z )ρ dV (5.49)
vol

2
2
Iyy = (z + x )ρ dV (5.50)
vol

2
2
Izz = (x + y )ρ dV (5.51)
vol
while the products of inertia are

Ixy = Iyx =− (xy)ρ dV (5.52)
vol

Iyz = Izy =− (yz)ρ dV (5.53)
vol

Ixz = Izx =− (xz)ρ dV (5.54)
vol
As the axes of the element frame are assumed to coincide with principal axes of inertia,
the products of inertia are zero, and the inertia tensor is represented by a diagonal matrix:
   
Ixx 0 0 Ix 0 0
I =  0 Iyy 0  =  0 Iy 0  (5.55)
0 0 Izz 0 0 Iz

5.2.6 Governing equation of motion
In the inertial reference frame, the motion of the discrete element is governed by the
Euler equations:
dL dv
F = = m (5.56)
dt dt
dH
M = (5.57)
dt