MECA029 Mechanical Vibrations

Gatan Kerschen

Space Structures & Systems Lab (S3L)

MECA029 Mechanical Vibrations

Lecture 3: Damped Vibrations of

n-DOF Systems

3

Outline

Preliminary Remarks

4

Next Thursday

First exercise session with Maxime Peeters

The session starts at 9.30pm

5

Quiz

Results and answers

6

Matlab GUI

Example available on the S3L web site

Help Matlab help Demos Matlab Creating GUI

7

Further Reading

Available on the S3L web site

Modal analysis in a nutshell

Structural design using mode shapes

Difference between real and complex modes

8

Additional Movie

Glider flutter

9

Outline

Previous Lectures

10

Previous Lectures

L1: Analytical dynamics of discrete systems

L2: Undamped vibrations of n-DOF systems (L2.1, L2.2)

L3: Damped vibrations of n-DOF systems (L3.1, L3.2)

L4: Continuous systems: bars, beams, plates (L4.1,L4.2)

L5: Approximation of continuous systems by Rayleigh-Ritz and finite element methods (L5.1, L5.2)

L6: Solution methods for the eigenvalue problem

L7: Direct time-integration methods (L7.1, L7.2)

L8: Introduction to nonlinear dynamics (L8.1, L8.2)

11

Previous Lectures

Engineering structure

int 02 2 1

1

( )( )n

exts r rs

rs s s s s

V V TT T Td DQ t q gdt q q q q t q=

+ = +

&& & &

L1

12

Previous Lectures

Engineering structure

ODEs( ) ( ) ( )t t t+ =Mq Kq p&&

L2 (linearization)

int 02 2 1

1

( )( )n

exts r rs

rs s s s s

V V TT T Td DQ t q gdt q q q q t q=

+ = +

&& & &

L1

13

Previous Lectures

Engineering structure

Normal modes, FRFs, time series

ODEs( ) ( ) ( )t t t+ =Mq Kq p&&

L2 (linearization)

L2

int 02 2 1

1

( )( )n

exts r rs

rs s s s s

V V TT T Td DQ t q gdt q q q q t q=

+ = +

&& & &

L1

14

Usefulness of Normal Modes

Clear physical meaning:

Structural deformation at resonance

Synchronous vibration of the structure

Important mathematical properties:

Orthogonality

Decoupling of the equations of motion

Constant system characteristics for linear systems

15

Usefulness of Normal Modes

16

Normal Mode Computation

)(txq =

Natural frequencies, resonant frequencies,

eigenfrequencies

( )2( ) ( )r r =K M x 0

0+ =Mq Kq&& &

( )2det =K M 0

Normal modes, eigenmodes

( ) cos( ) sin( )r r r r rt t t = +Normal mode

modulation, normal coordinates

17

Usefulness of FRFs

Constant system characteristics for linear systems

Convenient way of locating natural frequencies

Convenient way of assessing the potential danger of a structural resonance (damped systems)

18

FRF Computation

( ) cos( )t t=q x

cos( )t=Mq+Kq s&&

( )2 ( ) -1x = K - M s = H s

Frequency response function (FRF)

19

Link Between Structural Features

Engineering structure

Modal parameters (natural frequencies, damping

ratios and mode shapes)

FRFs

Time series (displacements, velocities,

accelerations)

Link ? Link ?

20

FRF vs. Modes

( )2 2 21 11 T Tm n m( i ) ( i ) ( s ) ( s )

ii s s s( )

= =

= +

u u x x

H

Mathematical relationship between modes and FRFs (useful for experimental modal analysis)

21

FRF vs. Time Series

( ) ( ) ( ) ( ) ( ) ( )t t t = =q h p Q H P

22

Modes vs. Time Series

( ) ( ) ( ) ( ) 01 1

( ) ( )( ) ( ) 0

1 1

( ) cos

sin

m n mT T

i i s s si s

Tm n ms sT

i i si s s

t t

t t

= =

= =

= +

+ +

q u u x x M q

x xu u M q&

Modal superposition Free response case

( ) ( )0 00 0given ,+ =

= =

M q K q 0q q q q&&

& &

23

Modes vs. Time Series

Mode displacement method Forced response case

( ) ( )0 00 0( t )

,+ =

= =

M q K q pq q q qgiven&&

& &

( ) ( )( ) ( )0

1

1( ) sinTn ts s

ss s s

t t d =

= x x g

q

24

Modes vs. Time Series

( ) ( )0 00 0( t )

,+ =

= =

M q K q pq q q qgiven&&

& &

( )( ) ( ) ( ) ( )1 201 1

( ) ( )sin ( )T Tk kts s s s

ss ss s s s

t t d t

= =

= +

x x x x

q p K p

Interest of modal approach: truncation !

Mode acceleration method Forced response case

25

Link: A Cantilever Beam Example

Force: harmonic excitation with increasing frequency

(chirp)

0 5 10 15 20 25 30 35 400

50

100

150

200

250

300

350

400

450

500

Time (s)

Freq

uenc

y (H

z)

Mode number

1

2

3

Frequency (Hz)

23.66

148.28

415.30

Response: measured acceleration

26

Digression: Chirp in Matlab

yyy=chirp([0:0.01:10],1,20,5);plot([0:0.01:10],yyy)

0 2 4 6 8 10-1

-0.5

0

0.5

1

Time (s)

Cos

ine

with

var

ying

exc

itatio

n

27

Link: A Cantilever Beam Example

0 5 10 15 20 25 30 35 40-400

-200

0

200

400

0 5 10 15 20 25 30 35 400

200

400

600

0 50 100 150 200 250 300 350 400 450 500

-50

0

50

FRF

Frequency (Hz)

Time (s)

Time (s)

Excitation frequency

(Hz)

Acceleration

ChirpExcitation

28

Link: A Cantilever Beam Example

Coordinate along the beam

Mod

e 1

Coordinate along the beam

Mod

e 2

Coordinate along the beam

Mod

e 3

Modes

FRFs

Time series

29

Today

L1: Analytical dynamics of discrete systems

L2: Undamped vibrations of n-DOF systems (L2.1, L2.2)

L3: Damped vibrations of n-DOF systems (L3.1, L3.2)

L4: Continuous systems: bars, beams, plates (L4.1,L4.2)

L5: Approximation of continuous systems by Rayleigh-Ritz and finite element methods (L5.1, L5.2)

L6: Solution methods for the eigenvalue problem

L7: Direct time-integration methods (L7.1, L7.2)

L8: Introduction to nonlinear dynamics (L8.1, L8.2)

30

Today

Engineering structure

Normal modes, FRFs, time series

L3

int 02 2 1

1

( )( )n

exts r rs

rs s s s s

V V TT T Td DQ t q gdt q q q q t q=

+ = +

&& & &

L1

( ) ( ) ( ) ( )t t t t+ + =Mq Cq Kq p&& & ODEs

L2 (linearization)

31

Today

1. Modal damping assumption

2. Forced harmonic response Force

appropriation testing

3. State-space formulation

32

Damping ?

Damping is any effect that tends to reduce the oscillation amplitude

0 50 100 150-1

-0.5

0

0.5

1

Time (s)

Dis

plac

emen

t (m

)

UndampedDamped

1320 1325 1330 1335 1340 1345 1350 1355 1360

-10

-5

0

5

10

Time (s)

Dis

plac

emen

t (m

)

Lightly dampedModerately damped

Free response of a 1DOF system Forced response of a 1DOF system

33

Damping In Practice

All real-life engineering structures are damped

Damping is either deliberately engendered or inherent to a system

There exists different kinds of structural damping

Viscous damping

Hysteretic damping

Friction

34

Damping In Practice

35

Damping In This Course

Focus on viscous damping here

0TD = q Cq& &

( ) ( ) ( ) ( )t t t t+ + =Mq Cq Kq p&& &

C symmetric

Damping plays a crucial role on the concept of phase between oscillators

36

Normal Equations

( ) ( ) ( ) 0t t t+ + =Mq Cq Kq&& & ( )1

( ) ( )n

s ss

t t=

=q x

1112

1 1 1

21

...0 0

( ) ... ... ... ( ) 0 ... 0 ( ) 00 0...

n

n nn n

n n

t t t

+ + =

&& &

Coupled normal equations !

37

Damping Distribution

( ) ( ) 0 for T

rs r s r s = x CxDamping has usually a different distribution from that of stiffness and inertia

Stator blades (Techspace Aero)

38

Modal Approach of Limited Interest ?

In principle, all normal equations are coupled

Strong damping

YES ! NO !

In practice, the modal coupling may be weak

Modal damping assumption (weak damping)

39

Alternative ?

Direct integration of the equations of motion (L7) using, e.g., Newmarks algorithm

Robust: damping may be arbitrary

But computationally expensive !

( ) (