mechanical vibrations for the eigenvalue problem solution ... · pdf filethe aim of the...

Mec

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ions

Chapter 6

Solution Methods

for the Eigenvalue Problem

2

xMxK 2ω=

Equations of dynamic equilibrium eigenvalue problem

The eigensolutions of this problem are written in the following order:

⎪⎩

⎪⎨⎧ ≤≤≤≤

)()2()1(

222

21

,,,0

n

n

xxxωωω

Introduction

3Criteria for selecting the solution method

• Number of degrees of freedom in the system (n)

Reduction methodsn > 25000Class V

Inverse iteration method, subspace method, Lanczos’ method2500 ≤ n ≤ 25000Class IV

Band character of K and M250 ≤ n ≤ 2500Class III

Jacobi’s method, power algorithm10 ≤ n ≤ 250Class II

Development of the characteristic equation1 ≤ n ≤ 10Class I

4

• Required frequency spectrum

• Ability to separate close eigenvalues

• Rate of convergence

• Computational cost

• Automatic extraction of rigid-body modes

• Handling of coupled problems

• Use within the substructuring context

Interested readers may refer to Géradin’s book.

Criteria for selecting the solution method

5Reduction and substructuring methods

The reduction and substructuring methods are often used in industry for two reasons:

1. As only the low frequency range is of interest for mechanical design purposes, it is advantageous to reduce from the start theeigenvalue problem to a smaller dimension.

2. In the context of large projects, the analysis is divided into several parts (often performed by distinct teams). A separate model is constructed for each part of the system and will be used to reconstruct the whole original model. That is what is called substructuring techniques.

6

Automated Transfer Vehicle

Courtesy of EADS Space Transportation

282 000 DOF

Stator of a turbojet engine

Courtesy of Techspace Aero

1 278 000 DOF

Industrial examples

7

Initial problem

Reduction and substructuring methods

n n×K n n×M

n ~ 105, 106

Reduced problem

m m×K m m×M

m ~ 102, 103

Transformation matrix?

8Reduction and substructuring methods

The general principle of a method for reducing the size of an eigenvalue problem

consists of building a subspace R of dimension n x m (m < n) so that the solution can be written in the form

xMxK 2ω=

)1()()1( ×××

=mmnn

yRx

9

Variational problem

022

1022

1 22=⎟⎟

⎠

⎞⎜⎜⎝

⎛−⇔=⎟⎟

⎠

⎞⎜⎜⎝

⎛− yMyyKyxMxxKx TTTT ωδωδ

yMyK 2ω=

Reduced stiffness and mass matrices

RMRMRKRK TT == and

Reduction and substructuring methods

( )max max 0T Vδ − =

10Static condensation (Guyan-Irons reduction)

The aim of the Guyan’s condensation method is to obtain an eigenvalue problem of reduced size without altering too much the low eigenfrequencyspectrum of the initial problem.

For this purpose, the degrees of freedom are partitioned into nR dynamic

(retained) coordinates (with nR << n) and nC condensed coordinates.

RR RC

CR CC

⎡ ⎤= ⎢ ⎥⎣ ⎦

K KK

K KRR RC

CR CC

⎡ ⎤= ⎢ ⎥⎣ ⎦

M MM

M MR

C

⎧ ⎫= ⎨ ⎬⎩ ⎭

xxx

The dynamic behaviour of the structure will be described by the retained coordinates only.

11Static condensation (Guyan-Irons reduction)

xMxK 2ω=The equation can be put in the form =K x F

where 2ω=F M x is the vector of inertia forces.

RR RC R R

CR CC C C

⎡ ⎤ ⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬⎢ ⎥⎣ ⎦ ⎩ ⎭ ⎩ ⎭

K K x FK K x F

with FC ≈ 0

The inertia forces FC may be neglected if the masses affected to the condensed degrees of freedom are equal to zero or negligible.

If it is the case, one finds

1C CC CR R

−= −x K K x

12

Thus we can define the transformation matrix R

11RR

R RC CC CC RR CCR

− −⎧ ⎫⎧ ⎫= = = =⎨ ⎬ ⎨ ⎬

−⎩ ⎭

⎡ ⎤⎢ ⎥−⎣ ⎦⎩ ⎭

xxx

Ix x

x K K xR

K K

Static condensation (Guyan-Irons reduction)

CRCCRCRRT

RR KKKKRKRK 1−−==

CRCCCCCCRC

CRCCRCCRCCRCRRT

RR

KKMKK

MKKKKMMRMRM11

11

−−

−−

+

−−==

It follows that the reduced stiffness and mass matrices are given by

13

Remarks

• The validity of the Guyan’s reduction method depends on the

extent to which the vector of inertia forces FC is negligible.

• It can be shown that static condensation always leads to an excess approximation to the eigenvalue spectrum.

• In computational practice, the reduction matrix

Static condensation (Guyan-Irons reduction)

CC CR CR= −K R K

1CRCC CR

−⎡ ⎤ ⎡ ⎤= =⎢ ⎥ ⎢ ⎥− ⎣ ⎦⎣ ⎦

I IR

RK K

is computed by solving the static problem (with nR second members)

14Example: the beam clamped at both ends

Two finite element model Three finite element model

Comparison of the results (no reduction)

1

1 2

2 3

w2Ψ2

w2 w3

1

1 22 4

Ψ2 33

Ψ3

3 8033 956.967202

14 62021 405-3

39 94484 537-4

500.6504.67516.921

exact3 elements

4 DOF2 elements

2 DOFMode n°

42r

m LE I

ω

15Example: the beam clamped at both ends

Three finite element modelw2 w3

1

1 22 4

Ψ2 33

Ψ3

Condensation of the rotational degrees of freedom

38033957400267202

500.6504.7506.5516.91

exact3 elements

4 DOF3 elements

2 DOF2 elements

2 DOFMode n°

42r

m LE I

ω

16

Beam clamped at both ends with 100 finite elements (= 396 DOF)

Example: the beam clamped at both ends

Reduction of the rotational degrees of freedom 198 DOF

0 10 20 30 40 50 60 70 80 90 1000

2

4

6

8

10

12

14x 109

0 1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12x 1054

2r

m LE I

ω Close-up

Mode n°

42r

m LE I

ω

Mode n°

The relative error on the first 10th modes is less than 5 10-5 % !

17

x exact

o Guyan (25 DOF)

FE (13 elements, 26 DOF)


Guyan reduction: 396 DOF 198 DOF 25 DOF

42r

m LE I

ω

Mode n°

0 1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

14x 105

Close-up

0 1 20

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

8 94

5

6

7

8

9

10

11x 10 5

Close-up

18

Relative error %

…………

7.934 1058.196 1057.964 1059

3803.53804.23803.62

10

1

Mode n°

1.184 1061.239 1061.191 106

500.6500.6500.6

exactFE 26 DOF

Guyan25 DOF0 1 2 3 4 5 6 7 8 9 10

0

1

2

3

4

5

o Guyan (25 DOF)

FE (13 elements, 26 DOF)


Guyan reduction: 396 DOF 198 DOF 25 DOF

Mode n°

19Substructuring methods

Let us consider a substructure which is connected to the rest of the system by a set of boundary degrees of freedom q2

internal DOF q1

boundary DOF q2

The internal degrees of freedom q1 are free.

The substructure is described by its stiffness and mass matrices K and M

20

( ) gqMK =− 2ω

applied force amplitudes

and the impedance matrix is defined as

( ) ( )2 2 1 2ω ω ω−= − =Z K M H

The dynamic equilibrium equation of the substructure writes

Concept of mechanical impedance

internal DOF q1

boundary DOF q2

21Concept of mechanical impedance

⎭⎬⎫

⎩⎨⎧=

⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡

22

12

222

21

212

211 0

)()()()(

gqq

ZZZZ

ωωωω

External loads and/or boundary reactions

internal DOF q1

boundary DOF q2

Since the internal degrees of freedom q1 are not loaded, we may write

From the first equation, we can eliminate the internal degrees of freedom.


122 21 11 12

*22

−= −Z Z Z ZZ

Reduced impedance matrix

11 11 12 2

−= −q Z Z q

So we deduce the relationship

( )* 222 2 2ω =Z q g

where

One notes that admits as poles the zeros of Z11 which corresponds to the eigenfrequencies of the subsystem with its

boundary degrees of freedom q2 fixed.

*22Z


Let us consider the subsystem clamped on its boundary

internal DOF q1

boundary DOF q2

The eigensolutions of the subsystem

( )211 11 0ω− =K M x

are numbered in the following order

( ) ( )

2 21

1

0 n

n

ω ω⎧ < ≤ ≤⎪⎨⎪⎩ x x

24

( )( ) ( )1

* 122 22 21 11 12

1 1 1 122 21 11 12 21 11 12 21 11 11 11 12

2 221 21 ( ) ( ) 21 21

4 24

21

2

( )

TTni i i i

i ii

ω ω

ω ω ω

ω

ω

−

− − − −

=

= −

− − − +

− −−

−∑

Z K K K K

M M K K K K M K K M K K

K M x x K M

Concept of mechanical impedance

Based on the spectral expansion of , it can be shown that the reduced impedance matrix takes the form

111−Z

where the terms of orders 0, 1 and 2 in ω 2 have been isolated.

What do we recognize in this equation?


The first two terms corresponds to a static condensation of the substructure on its boundary (Guyan’s reduction method).

122 21 11 12

1 1 1 122 21 11 12 21 11 12 21 11 11 11 12

RR

RR

−

− − − −

= −

= − − +

K K K K K

M M M K K K K M K K M K K

( ) ( )1

*22

2 221 21 ( ) ( ) 21 214

4 2 2

2

1 ( )

R

TTni i i

RR

i

i

R

i i

ω ωω

ω

ω

ω ω=

=

− −−

−

−∑

KZ

K M x x K M

MGuyan’s reduction method

The last term represents a correction term that can be exploited to improve Guyan’s reduction method.

26Craig and Bampton’s method

( ) ( )1

*22

2 221 21 ( ) ( ) 21 214

4 2 2

2

1 ( )

R

TTni i i

RR

i

i

R

i i

ω ωω

ω

ω

ω ω=

=

− −−

−

−∑

KZ

K M x x K M

MGuyan’s reduction method

The term of order ω 4 represents the contribution of the subsystem eigenmodes in clamped boundary configuration.

27

So the dynamic behaviour of a substructure is fully described by:• the static boundary modes resulting from the static condensation,• the subsystem eigenmodes in clamped boundary configuration.

Craig and Bampton’s method

28

where the Guyan’s reduction matrix has been complemented by the set of n internal vibration modes obtained by solving


Accordingly, it means that the following transformation may be applied to the initial degrees of freedom

1

0

CC CR

R

I−

⎡ ⎤ ⎧ ⎫= ⎨ ⎬⎢ ⎥⎭⎣ ⎦− ⎩

I xx

yΦK KnR boundary DOF

nI intensity parameters (nI = n – nR)

( )211 11 0ω− =K M x

29

1

0

CC CR m−

⎡ ⎤= ⎢ ⎥

−⎣ ⎦

IR

K K Φ


In practice, only a certain number m < nI of internal vibration modes are kept:

( ) ( )1m m⎡ ⎤= ⎣ ⎦Φ x x

They can be selected according to the intensity of the associated boundary reactions

( )2( ) ( )RC i RC i RC iω−K M x K x

This yields the final reduction matrix of dimension n x (nR + m)

30

2 andRR RR Rm

mRm

⎡ ⎤ ⎡ ⎤= =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎣ ⎦⎣ ⎦

K 0 M MK M

M I0 Ω

Working the reduced stiffness and mass matrices explicit gives

( )

1

1 1 1 1

1

RR RR RC CC CR

RR RR RC CC CR RC CC CR RC CC CC CC CRT T

mR m CR CC CC CR Rm

−

− − − −

−

= −

= − − +

= − =

K K K K K

M M M K K K K M K K M K K

M Φ M M K K M

with


In the finite element context, matrices and constitute a so-called superelement.

K M

31

Beam clamped at both ends with 100 finite elements (= 396 DOF)


Guyan’s reduction method 80 DOF

Craig-Bampton’s substructuring method 80 DOF + 2 internal modes

80 DOF + 5 internal modes

Superelement40 elements (80 DOF)

32

Relative error %

x exact (FE with 396 DOF)

o Guyan (80 DOF)

Craig-Bampton (2 modes)

Δ Craig-Bampton (5 modes)

42r

m LE I

ω

Mode n°

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5x 107


Close-up

0 1 2 3 4 5 6 7 8 9 100

200

400

600

800

1000

1200

1400

1600

1800

0 1 2 3 4 50

10

20

30

40

50

60

70

80

90

100

Mode n°

33Example (Courtesy of Techspace Aero)

Casing

789 072 DOF

Stator

1 278 000 DOF

Rotor

113 176 DOF

Analysis of the radial freeplay

34After reduction: validation in [0-300 Hz]

Casing

2 812 DOF

Craig-Bampton

Rotor

85 DOF

Craig-Bampton

Analysis of the radial freeplay

Stator

7 014 DOF

Guyan

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