Page 204 - The Combined Finite-Discrete Element Method

DYNAMICS OF IRREGULAR DISCRETE ELEMENTS SUBJECT 187

In a similar way,

˜
˜
˜
i = i x i + i y j + i z k (5.42)
˜
˜
˜ j = j x i + j y j + j z k
˜
˜
k = k x i + k y j + k z k
˜
˜
˜
˜
5.2.4 Transformation matrices
Representation of the spatial orientation by using equation (5.41) also provides an efficient
way of transforming vector components from one reference frame into another (much used
in contact procedures). Any particular vector written in the element frame is easily written
in the inertial frame:

a = a x i + a y j + a z k (5.43)
˜
˜
˜
= a x (i ˜x i + i ˜y j + i ˜z k)
˜
˜
˜
+ a y (j ˜x i + j ˜y j + j ˜z k)
˜
˜
˜
+ a z (k ˜x i + k ˜y j + k ˜z k)
˜
˜
˜
= a ˜x i + a ˜y j + a ˜z k
The vector components in the inertial frame are therefore calculated from the vector
components in the element frame using the following transformation:
     
a ˜x i ˜x j ˜x k ˜x a x
 a ˜y  =  i ˜y j ˜y k ˜y   a y  (5.44)
a ˜z i ˜z j ˜z k ˜z a z
The components of the same vector in the element frame can be calculated from the
components given in the inertial frame of reference as follows:

 
      −1  
˜ ˜ ˜
a x i x j x k x a ˜x i ˜x j ˜x k ˜x a ˜x
 ˜ ˜ ˜ 
 a y   a ˜y  =  i ˜y j ˜y k ˜y   a ˜y  (5.45)
=  i y j y k y 
a z ˜ ˜ ˜ a ˜z i ˜z j ˜z k ˜z a ˜z
i z j z k z
As the triad of unit vectors are orthogonal to each other, these transformation matrices
are orthogonal, and
 
˜ ˜ ˜   −1   T
i x j x k x i ˜x j ˜x k ˜x i ˜x j ˜x k ˜x
˜ ˜ ˜ (5.46)
 
 i y j y k y  =  i ˜y j ˜y k ˜y  =  i ˜y j ˜y k ˜y 
˜ ˜ ˜ i ˜z j ˜z k ˜z i ˜z j ˜z k ˜z
i z j z k z