Page 212 - The Combined Finite-Discrete Element Method

DYNAMICS OF IRREGULAR DISCRETE ELEMENTS SUBJECT 195

the total angle of rotation:

h
2 ψ = 2 ω (5.83)
2
&
2 2 2
2 ψ = 2 ψ + 2 ψ + 2 ψ ˜ z
˜ x
˜ y
and the second approximation of the intermediate spatial orientation:

' (
( 2 ψ · t i) ( 2 ψ · t i)
2
t+h/2 i = 2 2 ψ + t i − 2 2 ψ cos( 2 ψ) (5.84)
2 ψ 2 ψ
1
+ ( 2 ψ × t i) sin( 2 ψ)
2 ψ
' (
( 2 ψ · t j) ( 1 ψ · t j)
2
t+h/2 j = 2 2 ψ + t j − 2 2 ψ cos( 2 ψ)
2 ψ 2 ψ
1
+ ( 2 ψ × t j) sin( 2 ψ)
2 ψ
' (
( 2 ψ · t k) ( 2 ψ · t k)
2
t+h/2 k = 2 2 ψ + t k − 2 2 ψ cos( 1 ψ)
2 ψ 2 ψ
1
+ ( 2 ψ × t k) sin( 2 ψ)
2 ψ

Step 3: calculate the third approximation of the average angular velocity:

 
3 ω ˜x
3 ω =  3 ω ˜y  =
3 ω ˜z
   
2 i 2 2   −1 2 ˜ 2 ˜ 2 ˜
j
j
k
t+h/2 ˜x t+h/2 j ˜x t+h/2 k ˜x I x 0 0 t+h/2 x t+h/2 x t+h/2 x
2 2 2
k
i
j
 i i   2 ˜ 2 ˜ 2 ˜ 
 t+h/2 ˜y t+h/2 ˜y t+h/2 k ˜y   0 I y 0   t+h/2 y t+h/2 y t+h/2 y 
2 2 2 0 0 2 ˜ 2 ˜ 2 ˜
i
i
k
j
t+h/2 ˜z t+h/2 j ˜z t+h/2 k ˜z I z t+h/2 z t+h/2 z t+h/2 z
   
    ˜ ˜ ˜    
t i ˜x t j ˜x t k ˜x I x 0 0 t i x t j x t k x t ω ˜x t M ˜x h
 0 0  ˜ ˜ ˜  
 t i ˜y j ˜y t k ˜y   I y   t i y t j y  t ω ˜y  +  
 t k y  t M ˜y h 
˜
˜
˜
t i ˜z t j ˜z t k ˜z 0 0 I z t i z t j z t k z t ω ˜z t M ˜z h
(5.85)
the total angle of rotation:
3 ψ = h 3 ω
&
2 2 2
3 ψ = 3 ψ + 3 ψ + 3 ψ ˜ z (5.86)
˜ x
˜ y