Page 213 - The Combined Finite-Discrete Element Method

196 TEMPORAL DISCRETISATION

and the third approximation of the intermediate spatial orientation:

' (
( 3 ψ · t i) ( 3 ψ · t i)
3
t+h i = 2 3 ψ + t i − 2 3 ψ cos( 3 ψ) (5.87)
3 ψ 3 ψ
1
+ ( 3 ψ × t i) sin( 3 ψ)
3 ψ
' (
( 3 ψ · t j) ( 1 ψ · t j)
3
t+h j = 2 3 ψ + t j − 2 3 ψ cos( 3 ψ)
3 ψ 3 ψ
1
+ ( 3 ψ × t j) sin( 3 ψ)
3 ψ
' (
( 3 ψ · t k) ( 3 ψ · t k)
3
t+h k = 2 3 ψ + t k − 2 3 ψ cos( 3 ψ)
3 ψ 3 ψ
1
+ ( 2 ψ × t k) sin( 3 ψ)
3 ψ

Step 4: calculate the fourth approximation of the average angular velocity:

 
4 ω ˜x
4 ω =  4 ω ˜y 
4 ω ˜z
  
j
k
i
j
k
i
 3 3 3    −1 3 ˜ 3 ˜ 3 ˜
t+h ˜x t+h ˜x t+h ˜x I x 0 0 t+h x t+h x t+h x
3 3 3
   0  3 ˜ 3 ˜ 3 ˜ 
t+h ˜y t+h ˜y t+h ˜y  t+h y t+h y t+h y  (5.88)
 i j k I y 0  i j k
= 
3 i 3 j 3 k 0 0 3 ˜ 3 ˜ 3 ˜
i
j
k
t+h ˜z t+h ˜z t+h ˜z I z t+h z t+h z t+h z
   
    ˜ ˜ ˜    
0 0 t i x t j x t k x t M ˜x h
t i ˜x t j ˜x t k ˜x I x t ω ˜x
  ˜ ˜ ˜  
t i ˜y j ˜y t k ˜y I y  t i y t j y t ω ˜y
   0 0    +  
 t k y  t M ˜y h 
0 0 t M ˜z h
˜
˜
˜
t i ˜z t j ˜z t k ˜z I z t i z t j z t k z t ω ˜z
Step 5: calculate the average angular velocity:
         
ω ˜x 1 ω ˜x 2 ω ˜x 3 ω ˜x 4 ω ˜x
1
ω =  ω ˜y  =  ω ˜y  + 2  ω ˜y  + 2  ω ˜y  +  ω ˜y  (5.89)
3
1
4
2
6
ω ˜z 1 ω ˜z 2 ω ˜z 3 ω ˜z 4 ω ˜z
Step 6: calculate the total angle of rotation:
ψ = hω
&
2 2 2
ψ = ψ + ψ + ψ (5.90)
˜ x ˜ y ˜ z