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Dr.Dr. Ing.Ing.ValentinaValentina SalomoniSalomoni
Dipartimento di Costruzioni e TrasportiDipartimento di Costruzioni e Trasport i
UniversitUniversitdegli Studi di Padovadegli Studi di Padova
DINAMICA DELLE STRUTTUREDINAMICA DELLE STRUTTURELAUREA SPECIALISTICALAUREA SPECIALISTICA -- INGEGNERIA CIVILEINGEGNERIA CIVILE
Introduzione alla soluzione di problemi agliIntroduzione alla soluzione di problemi agli autovaloriautovalori
nellnell analisi dinamica dei continuianalisi dinamica dei continui
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Eigenvalue problems
1) Standard problem (eigenvalues of the stiffness matrix)
Eigenvalues
Eigensolution
K matrix can be positive definite of semi-definite (rigid body
motions).
1. Introduction
StructuralStructuraldynamicsdynamics
=K
n ...0 21
K =
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Eigenvalue problems
2) Generalised eigenproblem
Eigenvalues
Eigenpairs
Eigensolution
1. Introduction
StructuralStructuraldynamicsdynamics
MK =
),(...,),,(),,( 2
2
2
1
2
21 nn
222 ...021 n
M
K
=
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Eigenvalue problems
3) Linearised buckling analysis
Eigenvalues
Eigenpairs
Eigensolution
1. Introduction
StructuralStructuraldynamicsdynamics
KK =G
),(...,),,(),,( 2211 nn
K
K =G
n ...0 21
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Linear properties
1) Connection to static problems
Posing
We have to solve the static problem
2) Eigenvectors are defined as directions (not uniqueness)
2. Properties of eigenvectors
StructuralStructuraldynamicsdynamics
iii MK =
iwith == URUK
RM =ii
)()( iii MK =
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Linear properties
3) M-orthonormalisation
4) K-orthonormalisation (as consequence)
5) Multiple eigenvalues (and corresponding eigenvectors): with
multiplicity m
Eigenvectors are not unique, while eigenspace is unique
2. Properties of eigenvectors
StructuralStructuraldynamicsdynamics
ijj
T
i =M
ijijTi =K
miii ++ === 11 ...
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Linear properties
6) Conditions for eigenvectors
7) In buckling analysis they are
2. Properties of eigenvectors
StructuralStructuraldynamicsdynamics
IMT =
KT =
IKT =
K GT =
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3. Characteristic polynomials
StructuralStructuraldynamicsdynamics
The eigenvalues of the generalised problem are the roots of
the characteristic polynomial
or
In other terms:
If I has multeplicitym, the last m elements of D matrix arezero.
)det()( MK =p
0)( = ii MK
( ) ii
n
ii
dp1
det)(=
== TLDL
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3.1 Separation of eigenvalues
StructuralStructuraldynamicsdynamics
Sturm sequence property holds for the characteristic
polynomials of the constraint associated problems.
Eigenproblem of the rth associated constraint problem of the
generalised eigenproblem (obtained deleting last r rows and
columns from K and M)
Characteristic polynomial
)()()()()( rrrrr MK =
( ) ( ))()()()()( det rrrrrp MK =
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3.1 Separation of eigenvalues
StructuralStructuraldynamicsdynamics
The eigenvalues of the (r+1)st associated constraint problem
separate those of the rth constraint problem, i.e.
Consequence:
If we factorise the matrix
The number of n g tiv elements in D is equal to the number
of eigenvalues smaller than
)()1(
1
)(
1
)1(
2
)(
2
)1(
1
)(
1 ... r
rn
r
rn
r
rn
rrrr
+
++
TLDLMK
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3.1 Separation of eigenvalues
StructuralStructuraldynamicsdynamics
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3.2 Shifting
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Used to accelerate eigenvalues calculation (and eliminate
rigid body modes).
Resolve the generalised eigenproblem performing a shiftingon K-matrix by calculating
Then we consider the eigenproblem
That it can be rewritten as
Comparing this with the generalised eigenproblem
we have (due to uniqueness)
MK =
MKK =
MK )( += MK =
ii += ii =
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3.1 Effect of zero mass
StructuralStructuraldynamicsdynamics
If we have r zero diagonal elements in the mass (lumped)matrix M, then the generalised eigenproblem can be rewrittenas
were = -1. Hence the eigenpair
is a non-trivial solution of the generalised eigenproblem:
[ ] 0.00...010...00 == iT
i
KM =
( ) ( )kii e,, =
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3.2 Standard form of the G.E.P.
StructuralStructuraldynamicsdynamics
Hyp.: M positive definite (mii > 0, i=1,,n if lumped; orbanded if consistent)
Transform M using the following decomposition
and substitute in the G.E. equation
Premultiplying l.h.s. and r.h.s. by S-1, we have
with
TSSK =
TSSM=
~~~
=K
TT SSKSK == ~~ 1 and
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3.2 Standard form of the G.E.P.
StructuralStructuraldynamicsdynamics
Choices for S - matrix
1) Cholesky factorisation
that gives
2) Spectral decomposition
that gives
T
MMLLM ~~
=
MLS ~=
DRS=
T2RDRM=
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3.2 Standard form of the G.E.P.
StructuralStructuraldynamicsdynamics
Cholesky factorisation is more efficient than Spectraldecomposition, because less operations are involved, butthe latter can be more accurate.
If M is ill-conditioned the transformation process of theG.E.P. to S.E.P. il also ill-conditioned.
its better to use Spectral decomposition thanCholesky factorisation. (Es. 10.9)
Its also possible to use a factorisation of K-matrix, that is
usually better conditioned than M, but the transformed M-matrix becomes always full.
Other reasons to transform G.E.P. into S.E.P.
i) properties of solution of S.E. hold also for solution of G.E.
ii) Sturm sequence properties of S.E. valid for G.E.
http://lezioni%20di%20dinamica%20%28corsi%20a.a.%202004%202005%202006%202007%29/92.%20Esercizi%20cap.%2010/Es%20Bathe%2010.8%20e%2010.9.xlshttp://lezioni%20di%20dinamica%20%28corsi%20a.a.%202004%202005%202006%202007%29/92.%20Esercizi%20cap.%2010/Es%20Bathe%2010.8%20e%2010.9.xls7/25/2019 Dinamica Lineare FEM 3
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3.3 Approximate solution techniques
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Purpose: calculate the lowest eigenvalues and correspondingeigenvectors of the G.E.P. (with large order of system)
Techniques:
1) Static condensation2) Rayleigh-Ritz analysis
3) Component mode synthesis
4) Lanczos method
The same techniques will be used as first iteration in thesubspace iteration algorithm (used to solve large eigenvalueproblems).
MK =
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3.3.1 Static condensation
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Static condensation applies if structural masses are lumpedin at some specific d.o.f. without affecting eigefrequencesand eigenvectors accuracy.
Condensed masses can be eliminated using the followingpartitioning technique:
From the second row it can be found:
0=+ cccaca KK
=
c
aaa
c
a
ccca
acaa
000M
KKKK
acac cc KK 1
=
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3.3.1 Static condensation
StructuralStructuraldynamicsdynamics
The following reduced problem can be found
with
A standard technique is often used to solve the above G.E.P.
An analogy exists with Gauss elimination in static analysis of
massless degrees of freedom, for which the followinginterpretation of the load vector R is given
caccacaaa KKKKK 1=
aaaa MK =
=
0
aa MR
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3.3.1 Static condensation
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The equivalent static problem becomes
hence, i.e. Cholesky factorisation of Ka matrix can beperformed.
The accuracy of the technique for resolving the G.E.P. bymeans of the approximation given by the reducedeigenproblem depends on the specific mass lumping chosen.
However the bandwidth of Ka is in general increased withrespect to K, and the computational effort is higher, even if
only the smallest eigenvalues are needed to be calculated forthe reduced E.P. and this can lead to go back to the originalG.E.P.
In summary, the success of the procedure to be adopteddepends on a large extent on the experience of the analyst indistributing the masses appropriately. (Example: tower bell).
aaa RK =
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3.3.2 Rayleigh Ritz analysis
StructuralStructuraldynamicsdynamics
Assume that K and M are positive defined (i > 0, i=1,n) for agiven GEP.
The Rayleigh minimum principle states that
with Rayleigh quotient
and the following upper and lower bounds
M
KT
T
=)(
)(min1 =
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3.3.2 Rayleigh Ritz analysis
StructuralStructuraldynamicsdynamics
Ritz analysis: consider a set of vectors linearcombinations of Ritz basis vectors with a typical
vector equal to
andxi the Ritz coordinates.
In Rayleigh-Ritz analysis we aim to determine the specificvectors that best approximate the required eigenvectors.
This is obtained using Rayleigh minimum principle. Starting
from the Rayleigh quotient
m
k
mxx
kxx
q
j
ijj
q
i
i
q
j
ijj
q
i
i
~
~
~
~
)(
1 1
1 1 ==
= =
= =
i
q
i
ix =
=1
qii ,...,1, =
j
T
i
j
T
i
m
kand
M
K
=
=~
~
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3.3.2 Rayleigh Ritz analysis
StructuralStructuraldynamicsdynamics
The necessary condition for a minimum of is given by
in thexivariables. Being
the condition for the minimum is
2
11
~
~~2~~2
)(
m
mxkkxm
x
q
jijj
q
jijj
i
==
=
)(
qixi ,...,1,0/)( ==
qiforxmk j
q
j
ijij ,...,10)~~(
1
===
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3.3.2 Rayleigh Ritz analysis
StructuralStructuraldynamicsdynamics
In matrix form we have the eigenproblem
with K and M (q x q) matrices and x the vector of Ritzcoordinates
The eigensolution is given byq
eigenvalues
approximating
and q eigenvectors
used to evaluate
approximating
eigenvectors
[ ]qT
xxx ...21=x
xMxK
~~
=
[ ]
[ ]q
q
qqT
q
q
Tq
T
xxx
xxx
xxx
...
...
...
...
21
22
2
2
12
11
2
1
11
=
=
=
x
x
xq,...,1
q,...,1
q,...,1
q,...,1
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3.3.2 Rayleigh Ritz analysis
StructuralStructuraldynamicsdynamics
The approximation to eigenvectors are given by
and the approximated eigenvalues are upper bounds to thereal eigenvalues
The solution found is good if the vectors i span a subspaceV
q
close to the least dominant subspace of K and M spannedby .
Ritz analysis is a very general tool and also StaticCondensation can be regarded as a particular case of the
same method of analysis.
nqq ...,,, 332211
qix j
q
j
i
ji ,...,11 == =
q ,...,1
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3.3.3 Component Mode Synthesis
StructuralStructuraldynamicsdynamics
Also CMS is a particular case of RA (as well as SC). Again, theresults are good as far as the Ritz basis vectors are closeenough to the real eigenvectors.
The basic idea of CMS is the following: Suppose to have a large structure and to have partitioned that in
substructures
Calculate eigeinpairs separately for each substructure (withsuitable boundary conditions).
Use approximate values of eigenpairs to calculate the frequencesand modal shapes of the whole structures (Modal Synthesis).
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3.3.3 Component Mode Synthesis
StructuralStructuraldynamicsdynamics
Matrices of component structures (substructures)
Matrices of the whole structure
Assume the lowest eigenvalues/vectors of each componentstructure have been calculaled:
....
,...
=
=
M
II
I
M
II
I
M
M
M
M
K
K
K
K
.,...,,;,...,, MIIIMIII MMMKKK
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3.3.3 Component Mode Synthesis
StructuralStructuraldynamicsdynamics
In a component mode synthesis, approximated eigenpairs canbe obtained by perfoming a RR analysis with the followingloads at the r.h.s.
=
......0
...........
..I00
...00
...0I0
...00
R
M
IIIII,
II
III,
I
MMMMM
IIIIIIIIII
IIIII
MK
...
MK
MK
=
=
=
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3.3.4 Lanczos Method
StructuralStructuraldynamicsdynamics
Used for finding Eigenpairs of tridigonalized matrices.
The method starts with the transformation of the GEP in aSEP form.
An arbitrary starting vector x is used for this purpose and it isnormalised with respect to M matrixto obtain a second vectorx
1
.
Then a sequence of vectors xi
, i = 1,q is generated by means
of an algorithm, producing a matrixX for which
where Tq is tridiagonal of order q. We solve now
The eigenvalues of Tn are reciprocals of the eigenvalues ofthe given GEP and
q
1TTM)X(MKX =
~1~
=nT
~X=
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Examples
Blast doors
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