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Quadratic Eigenvalue Problem July
Recen t Dev elopmen ts in
Quadratic Eigen v alue Problems
F ran
coise Tisseur
Departmen t of Mathematics
Univ ersit y of Manc hester
ftisseur
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man
ac
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Quadratic Eigenvalue Problem July
Generalized Eigenproblems
Standard eigen v alue problem
Ax
x
Generalized eigen v alue problem
GEP
Ax
B x
Quadratic eigen v alue problem
QEP
Q
is an n
n matrix p olynomial of degree
Q
M
C
K
The QEP is to solv e
Q
x
y Q
x
y
x
y
righ t and left eigen v ectors corresp onding to
the eigen v alue
Quadratic Eigenvalue Problem July
Vibration Analysis of Structural Systems
Undamp ed structural eigenproblem in engineering
Mx
Kx
In practice systems are sub ject to damping
Mx
C x
Kx
Prop ortional or Ra yleigh damping
C
M
K
Damp ed QEP transformed to n decoupled eq
Mo dern structures lead to non
classically damp ed
eigen v alue problems
Mx
C x
Kx
More generally
M q
t
C
G
q
t
K
D
q
t
f
t
Quadratic Eigenvalue Problem July
Problems Considered
Matrix
Eigen v alue Eigen v ector
M nonsingular
n
nite
s
M singular
nite
s
M
C
K
s are real or x righ t ei
v ec of
e in pairs
x
righ t ei
v ec
of
M
C
K
s are real or x righ t ei
v ec of
e in pairs
x left ei
v ec
of
M
C
K
Re
M
C
K
s are real
n linearly ind
M
C
K
gap bet w een n ei
v ecs corresp
larg
small
to n largest
s
M
K Hermitian
s purely x righ t ei
v ec of
M
imaginary or
x left ei
v ec
C
e in pairs of
M
K
real
s are purely
T
C
C imaginary
M
C
K
min
x Cx
x Mx
x Kx
jj x jj
Quadratic Eigenvalue Problem July