Page 284 - Finite Element Modeling and Simulations with ANSYS Workbench

Structural Vibration and Dynamics 269

We have the following equation for the free vibration (EVP):

2 v
   
0
 
[K −ω 2 M ]   =  
0
θ 
  2    
where

6
EI  12 − L  ρAL 156 −22  L
K = 3  2  , M =   2
6
L  − L 4 L  420  −22 L 4 L 
The equation for determining the natural frequencies is
12 − 156λ − 6 + 22 λ
L
L
2
− 6 + 22 λ 4L 2 − 4 λ = 0
L
L
L

2
4
in which λ= ωρAL / 420 EI.
Solving the EVP, we obtain:
1
2 v

 EI  2    1  
.
ω= 3 533   ρAL 4   ,   =  138 L  ,
.
1
θ 
 
 
2
1   1  1 
 EI  2    
v 2
.
ω 2 = 34 81   ρAL 4   ,   =  762 
.
2
 θ 2 2     L  
The exact solutions of the first two natural frequencies for this problem are
1 1
 EI  2  EI  2
ω= 3 516  4  , ω= 22 03.  4 
.
2
1
 ρAL   ρAL 
We can see that for the FEA solution with one beam element, mode 1 is calculated
much more accurately than mode 2. More elements are needed in order to compute
mode 2 more accurately. The first three mode shapes of the cantilever beam is shown in
the insert above.
8.3.1 Modal Equations
Use the normal modes (modal matrices), we can transform the coupled system of dynamic
equations to uncoupled system of equations or modal equations.
We have:
2 M ]u = , 0 i = 1, 2, ..., n (8.21)
[K −ω i i
where the normal modes u i satisfy:
 uKu j = 0,
T

i
 uMu j = 0, for i ≠ j
T
  i