111

Dr C S ManoharDepartment of Civil Engineering

Professor of Structural EngineeringIndian Institute of ScienceBangalore 560 012 India

manohar@civil.iisc.ernet.in

Vibration analyis of discrete MDOF systems

2

Preliminaries

Nature of equations of motionInput - output relations in time domainInput - output relations in frequency domainForced vibration analysis using modal expansion

3

A rigid bar supported on two springs

1 1 1 2 22

O : Elastic centre O : Centre of gravity

k L k L

Two DOF-s1 Translation1 Rotation

1 2 1 2l l L L L

4

1 1 2 2

2 2 2 1 1 1

1 2 1 1 2 22 2

1 1 2 2 1 1 2 2

0

0

00

0

my k y l k y l

I k y l l k y l l

k k k l k lm y yk l k l k l k lI

is diagonal and is non-diagonalM KStatic coupling

5

1 22 2

1 1 2 2

00

0k km me z z

k L k Lme m

is non-diagonal and is diagonalM KInertial coupling

6

1 1 2 22

1 2 2

0m ml k k k Lx x

ml m k L k L

is non-diagonal and is non-diagonalM KStatic and inertial coupling

7

Remarks

•Equations of motion for MDOF systems are generally coupled

•Coupling between co-ordinates is manifest in the form of structural matrices being nondiagonal

•Coupling is not an intrinsic property of a vibrating system. It is dependent upon the choice of the coordinate system. This choice itself is arbitrary.

•Equations of motion are not unique. They depend upon the choice of coordinate system.

8

Remarks (continued)

•The best choice of coordinate system is the one in which the coupling is absent. That is, the structural matrices are all diagonal.

•These coordinates are called the natural coordinates for the system. Determination of these coordinates for a given system constitutes a major theme in structural dynamics. Theory of ODEs and linear algebra help us.

9

u t u t

1m

2m

3m

1k

2k

3k

1z t

2z t

3z t

A building frame under support motion

10

1 1 1 1 2 1 2 1 1 2 1 2

2 2 2 2 1 3 2 3 2 2 1 3 2 3

3 3 3 3 2 3 3 2

0

0

0

m z c z u c z z k z u k z z

m z c z z c z z k z z k z z

m z c z z k z z

11

1 1

2 2

3 3

1 1 1 1 2 1 2 1 1 2 1 2 1

2 2 2 2 1 3 2 3 2 2 1 3 2 3 2

3 3 3 3 2 3 3 2 3

x z ux z ux z u

m x c x c x x k x k x x m u

m x c x x c x x k x x k x x m u

m x c x x k x x m u

1 1 1 2 2 1 1 2 2 1

2 2 2 2 3 3 2 2 2 3 3 2

3 3 3 3 3 3 3 3

0 0 0 00 00 0 0 0

m x c c c x k k k xm x c c c c x k k k k x

m x c c x k k x

1

2

3

0 0 1 0 0 1

0 0 1

mm u

m

12

gx t gy t

A frame with asymetric plan under multi-componentsupport motions

13

x t

y t

t

O

O

14

15

16

1 11 4 2 3

1 2 3 4

1 14 3 1 2

1 2 3 4

1 2 3

4

( ) ( )2 2 2

( ) ( )2 2 2

( / 2)( / 2),

s s s s sS

s s s

s s s s sS

s s s

a a

s s S s s s

b b b b bm m m m mx

tb d m m m m

d d d d dm m m m my

tb d m m m mm m m A hm A h m tb d

17

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hIE

kh

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2212

223

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sssssss

sss

ssaassscolumnsslab

rhArhAyx

hAyxyxyx

hA

dy

bxdtbdb

dtbI

hAhAdtbmmm

18

0)()()()(

)()()()(

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0)()()()(

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xyxcxyxkxyxcxyxk

xyxcxyxkxyxcxyxkxm

)(tfKuuCuM

x tu t y t

t

19

0 00 ; 0, and , in general, are non-diagonal

Equations are coupledSuppose we introduce a new set of dependentvariables ( ) using the transformation

( )where is a tra

MX CX KX F t

X X X XM C K

Z tX t TZ t

T n n

nsformaiton matrix, to beselected.

How to uncouple equations of motion?

20

0 00 ; 0

( )

( )

( )

( )

, ,& structural matrices in the new coordinate system.( ) force vector in the new coord

t t t t

MX CX KX F t

X X X X

X t TZ t

MTZ t CTZ t KTZ t F t

T MTZ t T CTZ t T KTZ t T F t

MZ t CZ t KZ t F t

M C KF t

inate system

Can we select such that , ,& are all ?If yes, equation for ( ) would then represent a set of uncoupled equations and hence can be solved easily.

T M C KZ t

QuestionDIAGONAL

21

How to select to achieve this?TConsider the seemingly unrelated problem ofundamped free vibration analysis

0Seek a special solution to this set of equations in whichall points on the sturcutre oscillate harmonically at thesa

MX KX

2

2

me frequency.That is

exp ; 1,2, ,

or, exp where is a 1 vector.

exp & exp

exp 0

k kx t r i t k n

X t R i t R n

X t i R i t X t R i t

MR KR i t

22

22

2

exp 0

0

This is a algebraic eigenvalue problem.

; is positive semi-definite is positive definite

Eigensolutions would be real valuedand eigenvalues w

t t

MR KR i t

RM KR

KR MR

K K M MKM

Note

ould be non-negative.

23

2

2

12

12 2

12

12

0

Let exist.

0

0 0

If exists, =0 is the solution.

Condition for existence of nontrivial solution is that

should not exist

KR MR

K M R

K M

K M K M R

IR R

K M R

K M

2

2 2 21 2

1 2

.

0

This is called the characteristic equaiton.This leads to the characteristic values

and associated eigenvectors, , , .

n

n

K M

R R R

24

2

2

2

Consider - th and -th eigenpairs. (1)

(2)

(1)

(3)

(2)

r r r

s s sts

t ts r r s r

tr

tr s s

r sKR MR

KR MR

R

R KR R MR

R

R KR

Orthogonaility property of eigenvectors

2

2

2

2 2

(4)Transpose both sides of equation (4)

Since & , we get (5)

Substract (3) and (5)

0

tr s

t t t ts r s s r

t t

t ts r s s r

tr s s r

R MR

R K R R M R

K K M MR KR R MR

R MR

0

0

ts rts r

R MR r s

R KR r s

2

Normalization1ts s

ts s s

R MR

R KR

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21 ( )

2 2 21 2

Introduce

Diag

n n n

n

R R R

t

t

M IK

Orthogonality relations

Select T

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0 0

2

0 ; 0

( )

( )

( )

( )

; 1,2, ,

How about initial conditions?(0) 0

(0) 0 0

0 (0) & 0 (0)

t t t

r r r r

t t

t t

MX KX F t

X X X X

X t Z t

M Z t K Z t F t

M Z t K Z t F t

IZ t Z t F t

z z f t r n

X Z

MX M Z Z

Z MX Z MX

ConsiderUndampedForcedVibrationAnalysis

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0

1

1 0

0 10 cos sin sin

0 10 cos sin sin

tr

r r r r r rr r

n

k kr rr

tnr

kr r r r r rr r r

zz t z t t t f d

X t Z t

x t z t

zz t t t f d

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0 00 ; 0

( )

( )

( )

( )

If is not a diagonal matrix, theequations

t t t t

t

t

MX CX KX F t

X X X X

X t Z t

M Z t C Z t K Z t F t

M Z t C Z t K Z t F t

IZ t C Z t Z t F t

C

How about damped forced response analysis?

of motion would still remain coupled.

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If the damping matrix is such that is a diagonal matrix, then equaitons would

get uncoupled.Such matrices are called classical damping matrices.

Rayleigh's proport

t

CC

C

Classical damping models

Exampleional damping matrix

t t t

C M K

C M KI

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2

2

[ ]

[ ] [ ]

2 2

T T

T T

i

n n

nn

n

C M KC M KI K

I Diag

c

31

100

101

102

103

10-4

10-3

10-2

10-1

100

frequency rad/s

eta

mass and stiffness proportional

stiffness proportionalmass proportional

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2

0

( )

0 (0) & 0 (0)

2 ; 1,2, ,

with 0 & 0 specified.

exp cos sin

1 exp

t

t t

r r r r r r r

r r

r r r r dr r dr

t

r r rdr

IZ t C Z t Z t F t

Z MX Z MX

z z z f t r n

z z

z t t a t b t

t f d

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0

1

1 0

1exp cos sin exp

1exp cos sin exp

1,2, ,

t

r r r r dr r dr r r rdr

n

k kr rr

tn

kr r r r dr r dr r r rr dr

z t t a t b t t f d

X t Z t

x t z t

t a t b t t f d

k n

34

u t u t

1m

2m

3m

1k

2k

3k

1z t

2z t

3z t

Example 1

35

36

0.024 0.009 0.007 t

37

1 2 3

38

39

gx t gy t

Example 2

40

Physical properties of the frame members

41 0.02 0.02 0.01 t

42

1 2 3

Mode shapes

43

44

Summary

• Normal modes of vibration of a structure are special undamped free vibration solutions such that all points of the structure oscillate harmonically at the same frequency with the ratio of displacements at any two points being independent of time.

• Thus, for a structure vibrating in one of its modes, the phase difference between oscillations at any two points is either 0 or π.

• The frequencies at which normal mode oscillations are possible are called the natural frequencies.

45

Summary (Continued)

• Modal matrix is orthogonal to mass and stiffness matrices. This helps is diagonalising the mass and stiffness matrices.

• Undamped normal modes, in conjunction with proportional damping models, simplify vibration analysis procedures considerably.