Page 285 - Finite Element Modeling and Simulations with ANSYS Workbench

270 Finite Element Modeling and Simulation with ANSYS Workbench

and

T
  uMu i = 1,
i

 T 2 for i = 12,, …, n
i
i
  uKu i =ω ,
Form the modal matrix:
=[ uu 2 u ] (8.22)
Φ (nn× ) 1 n
We can verify that:

 ω 1 2 0 0 
 0 2 
T
Φ K Φ = Ω=  ω 2  ( Spectral matrix)

 0  (8.23)
 
2
  0 0 ω  
n
Φ Φ MΦ = I.
T

Transformation for the displacement vector:
u = z 11 u + + z n u = Φ z (8.24)
u + z 2
2
n
where
zt () 
1
 
 zt ()

z =  2 
 
zt

 n () 
are called the principal coordinates.
Substitute Equation 8.24 into the dynamic Equation 8.8 and obtain:
Φ
Φ
Mz + Cz + Kz = f()
Φ
t

Premultiply this result by Φ , and apply Equation 8.23:
T
z + z + Ω z = p() (8.25)
t
C ϕ
where C φ =α +βΩ if proportional damping is applied, and p =Φ T f t().
I
If we employ modal damping:
ξω 1
2 1 0 0 
 0 
ξω 2
C φ =  2 2 
 
 
ξω n
 0 2 n 