Page 289 - Finite Element Modeling and Simulations with ANSYS Workbench

274 Finite Element Modeling and Simulation with ANSYS Workbench

where

 1 1
 A = () 2 M + 2∆t C,
∆
t



 F() t = f −  K − 2 M u −  1 M − 1  

 k   () 2   k   () 2 2∆t C u k −1
∆t
∆t


We compute u from u and u , which are known from the previous time step. The
k−1
k+1
k
0
solution procedure is repeated or marching from tt 1, , … tt k 1, + , … , until the specified max-
k
imum time is reached. This method is unstable if Δt is too large.
Newmark Method:
We use the following approximations:
t ∆ () 2
u k +
u k+ ≈ u k + ∆ [( 1 − β ) u k + β → ( u k+ = )
t
2
2
u k+1 ],
1
1
2 (8.34)
≈ + t∆ [(1 ) +γ u k ]
u k+1 ≈ u k −γ u k +1
where β and γ are chosen constants. These lead to the following equation:
Au k+ = F t() (8.35)
1
where
γ 1
A = K + C + M,
β∆t β ∆t() 2
CM uuu k )
F t () = f ( f +k 1 , ,,γβ ∆t,, , k , k ,

This method is unconditionally stable if
1
2β≥ γ≥
2
For example, we can use γ= 12/ , β = 14, which gives the constant average acceleration
/
method.
Direct methods can be expensive, because of the need to compute A , repeatedly for
−1
each time step if nonuniform time steps are used.
8.5.2 Modal Method

In this method, we first do the transformation of the dynamic equations using the modal
matrix before the time marching:

m
∑
u = u ii zt() = Φz , (8.36)
= i 1
2
i
i
i z +ξ ω ii z + ω ii z = p t(), i = 1, 2, … m.,