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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 !
( ) (