Page 203 - The Combined Finite-Discrete Element Method

186 TEMPORAL DISCRETISATION

rotate) together with the discrete element. In the following text, this frame is referred to
as the ‘element frame’ or ‘moving frame’. The axes of the Cartesian coordinate system
associated with the element frame are assumed to coincide with the principal axes of
inertia of the discrete element. The orientation of the element frame is defined by a triad
of unit vectors that are orthogonal to each other, (Figure 5.3).
(i, j, k) (5.37)

5.2.2 Kinematics of the discrete element in general motion

The general motion of the discrete element in 3D is described by the translation superposed
by rotation about the centre of mass. Translation of the discrete element is described by
the position r of its centre of mass, (Figure 5.3). From the position of the centre of mass,
the velocity v of the centre of mass is obtained:

 
dr v ˜x
˜ ˜
˜ ˜
˜ ˜
˜
˜
˜
v = = (v · i)i + (v · j)j + (v · k)k = v ˜x i + v ˜y j + v ˜z k =  v ˜y  (5.38)
dt
v ˜z
It should be noted that in equation (5.38), both the position and velocity of the centre of
mass are given in the inertial frame.
A change in orientation of the discrete element occurs due to the presence of angu-
lar velocity. In the combined finite-discrete element method, it is convenient to express
angular velocity ω in the inertial frame:
 
ω ˜x
˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜
ω = (ω · i)i + (ω · j)j + (ω · k)k = ω ˜x i + ω ˜y j + ω ˜z k =  ω ˜y  (5.39)
ω z
5.2.3 Spatial orientation of the discrete element

Representation of the spatial orientation of the discrete element using approaches such as
Euler or Cardan angles of different kinds is difficult to implement in the context of the
combined finite-discrete element method, because of the presence of singularities associ-
ated with specific values of these angles. To avoid these singularities, in the combined
finite-discrete element method, spatial orientation of the discrete element is described by
the orientation of the element frame, i.e. by the triad of unit vectors
(i, j, k) (5.40)

The components of these unit vectors are expressed in the inertial frame of reference

˜
i = i ˜x i + i ˜y j + i ˜z k (5.41)
˜
˜
˜
˜
j = j ˜x i + j ˜y j + j ˜z k
˜
˜
k = k ˜x i + k ˜y j + k ˜z k
˜
˜