02a eigenvalue analysis

8/10/2019 02a Eigenvalue Analysis

1/5

d.kuhl, institute of mechanics and dynamics, university of kassel 2 eigenvalue analysis2.1 generalized evp

point of departure - homogenoeus equation of motion (without damping)M u + K u = 0

homogeneous mean with vanishing right hand side r(t) = 0

second order differential equation

exponential ansatz for solving differential equationu(t) = e

t

imaginary number =1 velocities and accelerations by time derivation of ansatz

u(t) = et u(t) =2 et

inducing ansatz and time derivatives in equationequation of motionK e

t M2 et =h

K 2 Mi

et

= 0

yields withet

= 0 generalized eigenvalue problem of dynamicshK2 M

i = 0

represents the eigen angular frequencies

are eigenvectors or eigenforms of dynamic system

generalized eigenvalue problem 1


2/5

d.kuhl, institute of mechanics and dynamics, university of kassel 2 eigenvalue analysis2.1 generalized evp

generalized eigenvalue problem of dynamics

hK 2 Mi = 0

non-trivial solutions forK 2 M

= 0

defines a characteristic polynomial in terms of eigen angular frequency2 NEQ zero points of characteristic polynomial are eigenvalues 2i fori[1, NEQ] eigen angular frequencies i =q2i eigen frequencies fi = i2 oscillation periods Ti = 1fi =

2 i

eigenforms or eigenvectors defined by solutions of homogeneous equation

hK 2i Mi i = 0

linear system of equations can be multiplied by an arbitrary scalar [K2i M]i =0all vectors i are eigenvectors of eigenvalue problem

is chosen such thati= 1

eigenvalues and eigenvectors 2


3/5

d.kuhl, institute of mechanics and dynamics, university of kassel 2 eigenvalue analysis2.2 specialized evp generalized / standard eigenvalue problem of dynamicsh

K 2 Mi

= 0h

K 2 Ii = 0

native transformation generalized into standard eigenvalue problem

generalized eigenvalue problem pre-multiplied by inverse mass matrix M1hM

1K2 M1M

i =

hM

1K 2 I

i = 0

identical eigenvalues i and eigenvectors i as generalized eigenvalue problem because of inverse mass matrix theoretically possible but not practicable

transformation using Choleskyfactorization of mass matrix Choleskyfactorization of positive definite mass matrixM= L L

T

generalized evp, pre-multiplyed by L1, substituting eigenvectors = LTL1

K 2 L1 L LT = L1 K LT 2 L1 L LT LT = 0

standard eigenvalue problemhK 2 I

i = 0 K = L

1K L

T

transformations= L

T = L

T

generalized vs standard eigenvalue problem 12


4/5

d.kuhl, institute of mechanics and dynamics, university of kassel 2 eigenvalue analysis2.2 specialized evp

homogeneousequation of motion (free vibration)

M u + K u = 0

generalized eigenvalue problem (evp) specialized eigen value problem (evp)

trial solution and derivatives Choleskysplit of mass matrix

u = et

u = 2 et

u = 2 etM= L LT

insert in equation of motion insert in generalized eigen value problem2 M et + K et = 0 K |{z}LT

2L LT | {z }

= K 2L = 0

reformulation reformulation (pre-multiplying by L1)

K

2 M

= 0

hL1

K LT

2 I

i = 0

eigen values and eigen vectors eigen values and eigen vectors

i, i = LT

i i, i, i= 1, , NEQmodal matrix

=h

1 2 NEQi

eigen value analysis 13


5/5

d.kuhl, institute of mechanics and dynamics, university of kassel 2 eigenvalue analysis2.2 specialized evp

Cholesky factorization of symmetric positive definite matrix MM= L L

T

Lis a lower triangular matrix

L=

2664

L11 0 0L21 L22 0... ... . . . 0LNEQ1 LNEQ2 LNEQNEQ

3775

algorithmloop over components i[1, NEQ]

next component i

loop over components j [i+ 1, NEQ]

next component j

diagonal components Lii = [MiiPi1

k=1 L2ik]

1/2

all other components Lji = [MjiPi1k=1 Ljk Lik]/Lii

Choleskyfactorization of mass matrix 14

02a eigenvalue analysis

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