Transcript Problems eigenvalue sensitivity - UF-MAE

Sensitivity of Eigenproblems
• Review of properties of vibration and
buckling modes. What is nice about them?
• Sensitivities of eigenvalues are really
cheap!
• Sensitivities of eigevectors.
– Why bother getting them?
– Think of where you want your car to have the
least vibrations
The eigenproblem
• Common notation for vibration and buckling
Ku   Mu  0
u Wu  1
T
• For vibration M is mass matrix, for buckling it
is geometric stiffness matrix.
• Usually W=M
• u is vibration or buckling mode, and  is the
square of the frequency of buckling load
• What are the properties of K and M?
• What do we know about the eigenvalues and
eigenvectors?
Derivatives of eigenvalues
• Differentiate:
du d 
dM
 dK

Mu   

dx dx
dx
 dx
du
dW
2uT W
 uT
u
dx
dx
 K  M 
• Pre multiply by uT:
dM
 dK
u 

d
dx
dx
 
dx
uT Mu
T
• What is the physical meaning?
• Why is it cheap to calculate?

u


u

Problems eigenvalue sensitivity
• How you would apply the physical interpretation
of the derivatives of eigenvalues to raising or
lowering the frequency of a cantilever beam?
• Check this by using the beam in the semianalytical problem, assuming that it has a crosssection of 4.5”x2”, and is made of steel with
density of 0.3 lb/in3. Compare the effect of
halving the height of the first and last of the 10
elements. Check the frequency of the original
beam against a formula from a textbook or web.
Eigenvector derivatives
• Collecting equations
K  M
 uT W

dM  
 du    dK



 u 
 Mu   dx    dx
dx 

  

0   d   
dW
0.5uT
u 
 dx  

dx
• Difficult to solve because top-left matrix is
singular. Why is it?
• Textbook explains Nelson’s method, which
uses intermediate step of setting one
components of the eigenvector to 1.
Spring-mass example
• Fig. 7.3.1
• Stiffness and mass matrices (all springs
and masses initially equal to one.
1  k
K 
 1
1
2 
1 0 
M 

0 1 
• Solution of eigenproblem
1  1 2  3
For u Mu  1
T
1 1
u

2 1
For um  1
1
u 
1
Derivative w.r.t k
• Derivatives of matrices
1 0 
K'

0
0


0 0 
M '

0
0


• Derivative of eigenvalue
uT  K '  M ' u
 '    ' 
 uT K ' u  0.5
T
u Mu
2
• See in textbook derivative of eigenvector
u' 
2 1
 
8 1
• Do those pass sanity checks?
Eigenvectors are not always unique
• When can we expect two different vibration modes with
the same frequency?
• Why does optimization with frequency constraints likely
to lead to repeated eigenvalues?
• Vibration modes are orthogonal when eigenvalues are
distinct, but any combination of modes corresponding
to the same frequency is also a vibration mode!
Ku1   Mu1  0
Ku2   Mu2  0
hence: K ( u1   u2 )   M ( u1   u2 )  0
Example 7.3.2
• Problem definition
and solution
2  y x 
K 
W M I

2
 x
y
1,2  2   x 2  0.25 y 2
2
• Eigenvectors for x=0
1
u1   
0
• Eigenvectors for y=0
1
u1   
1
0
u2   
1
1
u2   
1
•At x=y=0 eigenvalues are the same and
eigenvectors are discontinuous
Eigenvalues for example 7.3.2
3.5
.
3
2.5
mu1
mu2
2
1.5
1
0.5
-0.8
-0.6
-0.4
-0.2
0
0.2
x (y=0)
0.4
0.6
0.8
1
1.2
Deriviatives of repeated
eigenvalues
• Assume m repeated eigenvectors
m
u   qi ui  Uq
substitute into
i 1
 K  M 
du d 
dM
 dK

Mu   

dx dx
dx
 dx
d 

A

Bq  0

dx 


T
u
and
multiply
by
U


dM 
 dK
T
A UT 

U
B

U
MU

dx 
 dx
• To find eigenvalue derivatives need to solve a
second eigenvalue problem!
Calculation of derivatives w.r.t x
• At x=y=0 any vector is an eigenvector.
1 0 
U 

0 1 
yielding
1
q1   
1
• Similarly get
0 1 
1 0 
Then A  
B


1
0


0 1
  
  

1



  1

x

x

1

2
1
q2   
1
  

 1
 y 1
1 
1
q  
0 
  

 0
 y 2
0 
2
q  
1 
Why are these derivatives of limited
value
• What happens if we try to use them for
dy=2dx=2dt?
d   dt  2dt


Cannot be obtained from d  
dx 
dy
x
y
Problems (optional)
1.
2.
3.
Explain in 50 words or less why
derivatives of vibration frequencies are
relatively cheaper than derivatives of
stresses
When eigenvalues coalesce, they are not
differentiable even though we can still use
Nelson’s method to calculate derivatives.
How can you reconcile the two
statements?
Why is the accuracy of lower frequencies
(and their derivatives) better than that of
higher frequencies?
Source: Smithsonian
Institution
Number: 2004-57325