Dr. Peter Avitabile
Presentation Topics
Intent
Modal Overview
Modal Analysis & Controls Laboratory
University of Massachusetts Lowell
SDOF Theory
MDOF Theory
Measurement Definitions
IMAC 19
Young Engineer Program
Excitation Considerations
MPE Concepts
TUTORIAL:
Linear Algebra
Basics of Modal Analysis
Dr. Peter Avitabile
peter_avitabile@uml.edu
Friday, February 07, 2003
1
Intent of Young Engineer Program
The intent of the Young Engineer Program is to expose the new
or young engineer to some of the basic concepts and ideas
concerning analytical and experimental modal analysis.
It is NOT intended to be a detailed treatment of this material.
Rather it is intended to prepare one for some of the in-depth
papers presented at IMAC so that the novice has some
appreciation of the detailed material presented in these papers.
This presentation is intended to identify the basic methodology
and techniques currently employed in this field and to expose
one to the typical modal jargon used in the field.
Intent of Young Engineer Program
1
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Experimental Modal Analysis
A Simple Non-Mathematical Presentation
Dr. Peter Avitabile
Mechanical Engineering - UMASS Lowell
Could you explain
and how is it
used for solving
dynamic problems?
modal analysis
Illustration by Mike Avitabile
Experimental Modal Analysis
Illustration by Mike Avitabile
1
Illustration by Mike Avitabile
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Analytical Modal Analysis
Equation of motion
[M n ]{&x& n } + [C n ]{x& n } + [K n ]{x n } = {Fn ( t )}
Eigensolution
Experimental Modal Analysis
[[K n ] − λ[M n ]]{x n } = {0}
2
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Could you explain modal analysis for me ?
Simple time-frequency response relationship
RESPONSE
increasing rate of oscillation
FORCE
time
frequency
Experimental Modal Analysis
3
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Could you explain modal analysis for me ?
Sine Dwell to Obtain Mode Shape Characteristics
MODE3
MODE 1
MODE 2
Experimental Modal Analysis
MODE 4
4
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Just what are the measurements called FRFs ?
A simple inputoutput problem
8
5
2
Magnitude
Real
8
1
2
3
0
-3
8
-7
6
1.0000
Phase
Experimental Modal Analysis
-1.0000
Imaginary
5
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Just what are the measurements called FRFs ?
Response at point 3
due to an input at point 3
1
2
3
1
1
2
3
2
3
1
h32
2
1
2
3
3
1
2
Drive
Point
FRF
Experimental Modal Analysis
3
h33
h31
6
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Why is only one row/column of FRFs needed ?
The third row of the FRF matrix - mode 1
The peak amplitude of the imaginary part of the
FRF is a simple method to determine the mode shape
of the system
Experimental Modal Analysis
7
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Why is only one row/column of FRFs needed ?
The second row of the FRF matrix is similar
The peak amplitude of the imaginary part of the
FRF is a simple method to determine the mode shape
of the system
Experimental Modal Analysis
8
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Why is only one row/column of FRFs needed ?
Any row or
column can be
used to extract
mode shapes
- as long as it is
not the node of
a mode !
?
Experimental Modal Analysis
?
9
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
More measurements better defines the shape
MODE # 1
MODE # 2
MODE # 3
DOF # 1
DOF #2
DOF # 3
Experimental Modal Analysis
10
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
What’s the difference between shaker and impact ?
1
h 13
1
2
2
3
1
3
2
1
3
2
h 23
3
h 33
h 31
h 32
h 33
Theoretically - - Experimental Modal Analysis
11
NOTHING ! ! !
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
What measurements do I actually make ?
ANALOG SIGNALS
OUTPUT
INPUT
ANTIALIASING FILTERS
Actual time signals
Analog anti-alias filter
AUTORANGE ANALYZER
ADC DIGITIZES SIGNALS
OUTPUT
INPUT
APPLY WINDOWS
INPUT
OUTPUT
COMPUTE FFT
LINEAR SPECTRA
LINEAR
OUTPUT
SPECTRUM
LINEAR
INPUT
SPECTRUM
Digitized time signals
Windowed time signals
Compute FFT of signal
AVERAGING OF SAMPLES
COMPUTATION OF AVERAGED
INPUT/OUTPUT/CROSS POWER SPECTRA
INPUT
POWER
SPECTRUM
OUTPUT
POWER
SPECTRUM
CROSS
POWER
SPECTRUM
COMPUTATION OF FRF AND COHERENCE
FREQUENCY RESPONSE FUNCTION
Experimental Modal Analysis
Average auto/cross spectra
Compute FRF and Coherence
COHERENCE FUNCTION
12
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
What’s most important in impact testing ?
Hammers and Tips
40
COHERENCE
dB Mag
FRF
INPUT POWER SPECTRUM
-60
0Hz
800Hz
COHERENCE
40
FRF
dB Mag
INPUT POWER SPECTRUM
-60
0Hz
Experimental Modal Analysis
200Hz
13
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
What’s most important in impact testing ?
Leakage and Windows
ACTUAL TIME SIGNAL
SAMPLED SIGNAL
WINDOW WEIGHTING
WINDOWED TIME SIGNAL
Experimental Modal Analysis
14
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
What’s most important in shaker testing ?
AUTORANGING
AVERAGING WITH WINDOW
1
2
AUTORANGING
4
3
AVERAGING
1
AUTORANGING
Random
with
Hanning
2
3
4
Burst
Random
AVERAGING
Different
excitation
techniques are
available for
obtaining good
measurements
Sine
Chirp
1
2
3
4
Experimental Modal Analysis
15
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
How do I get mode shapes from the FRFs ?
MODE 2
2
1
4
3
6
MODE 1
5
2
4
1
3
6
5
Experimental Modal Analysis
16
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
How do I get mode shapes from the FRFs ?
The FRF is made up
from each individual
mode contribution
which is determined
from the
a ij1
ζ
1
a ij2
a ij3
frequency,
ζ
2
ζ3
ω
1
Experimental Modal Analysis
ω2
damping,
residue
ω3
17
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
How do I get mode shapes from the FRFs ?
MDOF
SDOF
The task for the modal test engineer is to
determine the parameters that make up the pieces
of the frequency response function
Experimental Modal Analysis
18
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
How do I get mode shapes from the FRFs ?
HOW MANY POINTS ???
RESIDUAL
EFFECTS
Mathematical
routines help to
determine the
basic parameters
that make up
the FRF
RESIDUAL
EFFECTS
HOW MANY MODES ???
Experimental Modal Analysis
19
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
What is operating data ?
Why and How Do Structures Vibrate?
INPUT TIME FORCE
f(t)
y(t)
FFT
IFT
INPUT SPECTRUM
f(j ω)
Experimental Modal Analysis
OUTPUT SPECTRUM
h(j ω)
20
y(j ω)
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
What is operating data ?
If I make measurements on a structure at an
operating frequency, sometimes I get some
deformation shapes that look pretty funky .
Maybe they are just noise?
Is that possible ???
Experimental Modal Analysis
21
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
What is operating data ?
But if I make a measurement at an operating
frequency and its close to a mode, I can easily
see what appears to be one of the modes
MODE 1 CONTRIBUTION
Experimental Modal Analysis
MODE 2 CONTRIBUTION
22
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
What is operating data ?
And if I make a measurement at an operating
frequency and its close to another mode, I can
easily see what appears to be one of the modes
Experimental Modal Analysis
23
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
What is operating data ?
I think I just answered my own question !!!
I think I’m starting to understand this now !!!
Experimental Modal Analysis
24
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
What is operating data ?
The modes of the structure act like filters
which amplify and attenuate input excitations
on a frequency basis
OUTPUT SPECTRUM
y(j ω)
f(j ω )
INPUT SPECTRUM
Experimental Modal Analysis
25
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
So what good is modal analysis ?
EXPERIMENTAL
MODAL
TESTING
FINITE
ELEMENT
MODELING
MODAL
PARAMETER
ESTIMATION
PERFORM
EIGEN
SOLUTION
MASS
DEVELOP
MODAL
MODEL
Repeat
until
desired
characteristics
are
obtained
RIB
STIFFNER
SPRING
STRUCTURAL
CHANGES
REQUIRED
Yes
No
DONE
DASHPOT
USE SDM
TO EVALUATE
STRUCTURAL
CHANGES
STRUCTURAL
DYNAMIC
MODIFICATIONS
Experimental Modal Analysis
26
The dynamic
model can be
used for studies
to determine the
effect of
structural
changes of the
mass, damping
and stiffness
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
So what good is modal analysis ?
Simulation, Prediction, Correlation, … to name a few
FREQUENCY
RESPONSE
MEASUREMENTS
FINITE
ELEMENT
MODEL
CORRECTIONS
PARAMETER
ESTIMATION
EIGENVALUE
SOLVER
MODAL
PARAMETERS
MODEL
VALIDATION
MODAL
PARAMETERS
SYNTHESIS
OF A
DYNAMIC MODAL MODEL
MASS, DAMPING,
STIFFNESS CHANGES
FORCED
RESPONSE
SIMULATION
STRUCTURAL
DYNAMICS
MODIFICATION
MODIFIED
MODAL
DATA
Experimental Modal Analysis
REAL WORLD
FORCES
STRUCTURAL
RESPONSE
27
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Single Degree of Freedom Overview
f(t)
x(t)
m
c
k
100
T = 2 π/ω n
ζ=0.1%
ζ=1%
X1
ζ=2%
X2
ζ=5%
10
ζ=10%
ζ=20%
0
1
-90
ζ=20%
ω/ω n
ζ=10%
t1
ζ=5%
ζ=2%
1
h (s) =
ms 2 + cs + k
ζ=1%
ζ=0.1%
-180
ω/ω n
SDOF Overview
t2
1
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
SDOF Definitions
Assumptions
• lumped mass
x(t)
• stiffness proportional
to displacement
• damping proportional to
velocity
• linear time invariant
m
k
c
• 2nd order differential
equations
SDOF Overview
2
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
SDOF Equations
Equation of Motion
d2x
dx
m 2 + c + kx = f ( t )
dt
dt
or
m &x& + cx& + kx = f ( t )
Characteristic Equation
ms 2 + cs + k = 0
Roots or poles of the characteristic equation
s1, 2
SDOF Overview
2
c
=−
±
2m
 c  + k


m
 2m 
3
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
SDOF Definitions
Poles expressed as
s1, 2 = − ζω n ±
jω
(ζωn )2 −ωn 2 = − σ ± jωd
POLE
Damping Factor
σ = ζωn
Natural Frequency
ωn = k
% Critical Damping
ζ= c
m
ζωn
cc
Critical Damping
c c = 2mωn
Damped Natural
Frequency
ωd = ωn 1−ζ 2
SDOF Overview
ωd
4
CONJUGATE
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
σ
SDOF - Harmonic Excitation
100
0
ζ=0.1%
ζ=1%
ζ=2%
ζ=5%
10
ζ=10%
-90
ζ=20%
ζ=20%
ζ=10%
ζ=5%
1
ζ=2%
ζ=1%
ζ=0.1%
-180
ω/ωn
ω/ω n
x
1
=
δst (1 − β 2 )2 +(2ζβ )2
SDOF Overview
 2ζβ 
φ=tan −1
2
1 − β 
5
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
SDOF - Damping Approximations
MAG
T = 2 π/ω n
X1
X2
0.707
MAG
t1
ω1
ωn
ω2
ωn
1
=
Q=
2ζ ω2 − ω1
SDOF Overview
t2
x1
δ = ln ≈ 2πζ
x2
6
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
SDOF - Laplace Domain
Equation of Motion in Laplace Domain
(ms 2 +cs+k)x (s) = f (s)
with
b(s ) = (ms 2 +cs+k)
System Characteristic Equation
b(s) x (s) = f (s)
x (s) = b −1 (s)f (s) = h (s)f (s)
and
System Transfer Function
1
h (s) =
ms 2 + cs + k
SDOF Overview
7
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
SDOF - Transfer Function
Polynomial Form
1
h (s) =
ms 2 + cs + k
Pole-Zero Form
1/ m
h (s) =
(s − p1 )(s − p1* )
Partial Fraction Form
a1
a1*
+
h (s) =
(s − p1 ) (s − p1* )
Exponential Form
1 −ζωt
h(t) =
e sin ωd t
mωd
SDOF Overview
8
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
SDOF - Transfer Function & Residues
Residue
a1 =
h (s)(s − p1 )
s→p1
1
=
2 jmωd
related to
mode shapes
Source: Vibrant Technology
SDOF Overview
9
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
SDOF - Frequency Response Function
h ( jω) = h (s)
SDOF Overview
s = jω
a1
a1*
=
+
( jω − p1 ) ( jω − p1* )
10
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
SDOF - Frequency Response Function
Coincident-Quadrature Plot
Bode Plot
0.707 MAG
Nyquist Plot
SDOF Overview
11
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
SDOF - Frequency Response Function
DYNAMIC COMPLIANCE
DISPLACEMENT / FORCE
MOBILITY
VELOCITY / FORCE
INERTANCE
ACCELERATION / FORCE
DYNAMIC STIFFNESS
FORCE / DISPLACEMENT
MECHANICAL IMPEDANCE
FORCE / VELOCITY
DYNAMIC MASS
SDOF Overview
FORCE / ACCELERATION
12
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Multiple Degree of Freedom Overview
[B(s )]−1 = [H(s )] = Adj[B(s )] = [A(s )]
det[B(s )] det[B(s )]
f1
p1
k1
f2
p2
c1
f3
p3
m2
m1
m3
k2
c2
k3
c3
R1
D1
MODE 1
MODE 2
MODE 3
R2
D2
\



M

\
{&p&} + 


\ 

MDOF Overview
C

\
{p& } + 


\ 

K

{p} = [U ]T {F}

\ 
1
R3
D3
F1
F2F3
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
MDOF Definitions
Assumptions
f2
• lumped mass
m2
• stiffness proportional
to displacement
k2
• damping proportional to
velocity
• linear time invariant
MDOF Overview
f1
2
c2
x1
m1
k1
• 2nd order differential
equations
x2
c1
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
MDOF Equations
Equation of Motion - Force Balance
m1&x&1+(c1 + c 2 )x& 1−c 2 x& 2 +(k1 + k 2 )x1−k 2 x 2 =f1 (t )
m 2 &x& 2 −c 2 x& 1+c 2 x& 2 −k 2 x1 +k 2 x 2 =f 2 (t )
Matrix Formulation
  &x&1 
m1
 


m 2  &x& 2 

Matrices and
Linear Algebra
are important !!!
(c1 + c 2 ) − c 2   x& 1 
+
 x& 
−
c
c

2
2  2 
(k1 + k 2 ) − k 2   x1   f1 ( t ) 
+
 =


−
k
k
x
f
(
t
)

2 
2
2  2 
MDOF Overview
3
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
MDOF Equations
Equation of Motion
[M ]{&x&}+[C]{x& }+[K ]{x}={F( t )}
Eigensolution
[[K ]−λ[M ]]{x}=0
Frequencies (eigenvalues) and
Mode Shapes (eigenvectors)
\



Ω2
MDOF Overview
2

ω

1
=
 
\  
ω22


 and [U ] = [{u1}
\ 
4
{u 2 } L]
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Modal Space Transformation
Modal transformation
{x} = [U ]{p} = [{u1} {u 2 }
Projection operation
 p1 
 
L]p 2 
M
 
[U ]T [M ][U ]{&p&} + [U ]T [C][U ]{p& } + [U ]T [K ][U ]{p} = [U ]T {F}
Modal equations (uncoupled)
 m1



m2
 &p&1   c1
&p& +
 2  
\  M  
MDOF Overview
c2
 p& 1   k1
p& +
 2  
\  M  
5
k2
T
 p1  {u1} {F}
p ={u }T {F}

 2   2

\  M   M

Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Modal Space Transformation
Diagonal Matrices Modal Mass
\



M
Modal Damping

\
{&p&} + 


\ 

C
Modal Stiffness

\
{p& } + 


\ 

K

{p} = [U ]T {F}

\ 
Highly coupled system
transformed into
k1
6
k2
m3
c2
MODE 2
f3
p3
m2
c1
MODE 1
f2
p2
m1
simple system
MDOF Overview
f1
p1
k3
c3
MODE 3
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Modal Space Transformation
..
.
[M]{x} + [C]{x} + [K]{x} = {F(t)}
=
Σ
f1
p1
m1
{x} = [U]{p} = [{u } {u } {u }
1
MODAL
2
3
]{p}
k1
+
SPACE
c1
MODE 1
f2
p2
m2
k2
..
.
T
[ M ]{p} + [ C ]{p} + [ K ]{p} = [U] {F(t)}
{u }p + {u }p + {u }p
1 1
2 2
3 3
+
c2
MODE 2
f3
p3
m3
k3
c3
MODE 3
MDOF Overview
7
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
MDOF - Laplace Domain
Laplace Domain Equation of Motion
[[M]s
2
+ [C]s + [K ]]{x (s )} = 0 ⇒ [B(s )]{x (s )} = 0
System Characteristic (Homogeneous) Equation
[[M]s +[C]s+[K ]] = 0
⇒ p k = − σ k ± jωdk
2
Damping
MDOF Overview
8
Frequency
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
MDOF - Transfer Function
System Equation
{
x (s )}
[B(s )]{x (s )} = {F(s )} ⇒ [H(s )] = [B(s )] =
{F(s )}
−1
System Transfer Function
[B(s )]
−1
[
Adj[B(s )]
A(s )]
= [H(s )] =
=
det[B(s )] det[B(s )]
[A(s )]
Residue Matrix
det[B(s )]
Characteristic Equation
MDOF Overview
9
Mode Shapes
Poles
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
MDOF - Residue Matrix and Mode Shapes
Transfer Function evaluated at one pole
qk
T
[H(s )]s=s = {u k }
{u k }
s−p k
can be expanded for all modes
k
m
[H(s )] = ∑
k =1
MDOF Overview
q k {u k }{u k }
q k {u }{u }
+
(s−p*k )
(s−p k )
T
10
*
k
* T
k
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
MDOF - Residue Matrix and Mode Shapes
Residues are related to mode shapes as
[A(s )]k
 a11k
a
 21k
a 31k
 M

a12 k
a 22 k
a 32 k
M
MDOF Overview
a13k
a 23k
a 33k
M
= q k {u k }{u k }
T
L
 u1k u1k
u u
L
=q k  2 k 1k
L
 u 3k u1k
 M
O

11
u1k u 2 k
u 2k u 2k
u 3k u 2 k
M
u1k u 3k
u 2 k u 3k
u 3k u 3k
M
L
L

L
O
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
MDOF - Drive Point FRF
h ij ( jω ) =
a ij1
( jω − p 1 )
+
( jω − p 2 )
MDOF Overview
( jω − p*2 )
( jω − p 3 )
+
a *ij 3
( jω − p*3 )
*
( jω − p 1 )
+
q1 u i 1 u j 1
( jω − p*1 )
q 2u i 2u j 2
( jω − p 2 )
+
+
a *ij 2
a ij 3
q1 u i 1 u j 1
+
( jω − p*1 )
a ij 2
+
h ij ( jω ) =
+
a *ij1
*
+
q 2u i 2u j 2
( jω − p*2 )
q 3u i 3u j 3
( jω − p 3 )
12
*
+
q 3u i 3u j 3
( jω − p*3 )
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
MDOF - FRF using Residues or Mode Shapes
h ij ( jω ) =
R1
+
( jω − p1 ) ( jω − p*1 )
+
D1
a*ij1
a ij1
a*ij 2
a ij 2
+
+ L
( jω − p 2 ) ( jω − p*2 )
R2
D2
R3
D3
F1
F2F3
a ij1
h ij ( jω ) =
q1u i1u j1
* * *
1 i 1 j1
*
1
qu u
+
( jω − p1 ) ( jω − p )
+
q 2u i 2u j 2
+
* * *
2 i2 j2
*
2
qu u
( jω − p 2 ) ( jω − p )
MDOF Overview
ζ1
a ij2
a ij3
ζ2
ζ3
ω1
ω2
ω3
+ L
13
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Time / Frequency / Modal Representation
PHYSICAL
TIME
FREQUENCY
ANALYTICAL
MODAL
f1
p1
m1
k1
MODE 1
+
c1
MODE 1
+
p2
+
f2
m2
k2
MODE 2
+
+
c2
MODE 2
p3
+
f3
m3
k3
MODE 3
MODE 3
MDOF Overview
c3
14
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Overview Analytical and Experimental Modal Analysis
TRANSFER
FUNCTION
[B(s)] = [M]s2 + [C]s + [K]
LAPLACE
DOMAIN
[B(s)] -1 = [H(s)]
qk u j {u k}
[U]
ω
[ A(s) ]
det [B(s)]
ζ
ω
[U]
FINITE
ELEMENT
MODEL
T
A
[K - λ M]{X} = 0
MODAL
PARAMETER
ESTIMATION
ANALYTICAL
MODEL
REDUCTION
N
H(j ω)
LARGE DOF
MISMATCH
N
X j(t)
H(j ω) =
MDOF Overview
Xj (j ω )
Fi (j ω)
U
A
FFT
Fi (t)
15
MODAL
TEST
EXPERIMENTAL
MODAL MODEL
EXPANSION
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Measurement Definitions
INPUT
SYSTEM
u(t)
n(t)
Σ
x(t)
OUTPUT
v(t)
H
m(t)
ACTUAL
Σ
NOISE
MEASURED
y(t)
0
-10
-20
-30
-40
1.0000
-50
-60
-1.0000
dB
-70
- 80
- 90
-100
-16
Measurement Definitions
1
-14
-12
-10
-8
-6
-4
-2
0
2
4
6
8
10
12
14
15.9375
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Measurement Definitions
Actual time signals
ANALOG SIGNALS
OUTPUT
INPUT
ANTIALIASING FILTERS
Analog anti-alias filter
AUTORANGE ANALYZER
ADC DIGITIZES SIGNALS
Digitized time signals
OUTPUT
INPUT
APPLY WINDOWS
INPUT
Windowed time signals
OUTPUT
COMPUTE FFT
LINEAR SPECTRA
Compute FFT of signal
LINEAR
OUTPUT
SPECTRUM
LINEAR
INPUT
SPECTRUM
AVERAGING OF SAMPLES
COMPUTATION OF AVERAGED
INPUT/OUTPUT/CROSS POWER SPECTRA
INPUT
POWER
SPECTRUM
Average auto/cross spectra
OUTPUT
POWER
SPECTRUM
CROSS
POWER
SPECTRUM
COMPUTATION OF FRF AND COHERENCE
FREQUENCY RESPONSE FUNCTION
Measurement Definitions
Compute FRF and Coherence
COHERENCE FUNCTION
2
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Leakage
F
R
E
Q
ACTUAL
DATA
CAPTURED
DATA
T
I
M
E
T
RECONTRUCTED
DATA
ACTUAL
DATA
Periodic Signal
U
E
N
C
Y
Non-Periodic Signal
CAPTURED
DATA
T
RECONTRUCTED
DATA
T
Measurement Definitions
3
Leakage due to
signal distortion
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Windows
0
Time weighting functions
are applied to minimize
the effects of leakage
-10
-20
-30
-40
Rectangular
-50
-60
Hanning
dB
-70
Flat Top
- 80
- 90
-100
-16
-14
-12
-10
-8
-6
-4
-2
0
2
4
6
8
10
12
14
15.9375
and many others
Windows DO NOT eliminate leakage !!!
Measurement Definitions
4
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Windows
Special windows are
used for impact testing
Force window
Exponential Window
Measurement Definitions
5
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Measurements - Linear Spectra
x(t)
INPUT
Sx(f)
h(t)
y(t)
TIME
SYSTEM
OUTPUT
FFT & IFT
H(f)
Sy(f)
FREQUENCY
x(t)
- time domain input to the system
y(t)
- time domain output to the system
Sx(f)
- linear Fourier spectrum of x(t)
Sy(f)
- linear Fourier spectrum of y(t)
H(f)
- system transfer function
h(t)
- system impulse response
Measurement Definitions
6
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Measurements - Linear Spectra
+∞
+∞
x ( t )= ∫ Sx (f )e j2 πft df
Sx (f )= ∫ x ( t )e − j2 πft dt
−∞
+∞
y( t )= ∫ S y (f )e
j2 πft
−∞
+∞
S y (f )= ∫ y( t )e − j2 πft dt
df
−∞
+∞
h ( t )= ∫ H(f )e
−∞
+∞
j2 πft
H(f )= ∫ h ( t )e − j2 πft dt
df
−∞
−∞
Note: Sx and Sy are complex valued functions
Measurement Definitions
7
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Measurements - Power Spectra
Rxx(t)
INPUT
Gxx(f)
Ryx(t)
Ryy(t)
TIME
SYSTEM
OUTPUT
FFT & IFT
Gxy(f)
Gyy(f)
FREQUENCY
Rxx(t) - autocorrelation of the input signal x(t)
Ryy(t) - autocorrelation of the output signal y(t)
Ryx(t) - cross correlation of y(t) and x(t)
Gxx(f) - autopower spectrum of x(t)
G xx ( f ) = S x ( f ) • S*x ( f )
G yy(f) - autopower spectrum of y(t)
G yy ( f ) = S y ( f ) • S*y ( f )
G yx(f) - cross power spectrum of y(t) and x(t)
G yx ( f ) = S y ( f ) • S*x ( f )
Measurement Definitions
8
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Measurements - Linear Spectra
lim 1
x ( t )x ( t + τ)dt
R xx (τ)=E[ x ( t ), x ( t + τ)]=
∫
T → ∞TT
+∞
G xx (f )= ∫ R xx (τ)e − j2 πft dτ=Sx (f )•S*x (f )
−∞
lim 1
R yy (τ)=E[ y( t ), y( t + τ)]=
y( t )y( t + τ)dt
∫
T → ∞TT
+∞
G yy (f )= ∫ R yy (τ)e − j2 πft dτ=S y (f )•S*y (f )
−∞
R yx (τ)=E[ y( t ), x ( t + τ)]=
lim 1
y( t )x ( t + τ)dt
∫
T → ∞TT
+∞
G yx (f )= ∫ R yx (τ)e − j2 πft dτ=S y (f )•S*x (f )
−∞
Measurement Definitions
9
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Measurements - Derived Relationships
S y =HSx
H1 formulation
- susceptible to noise on the input
- underestimates the actual H of the system
S y •S*x G yx
H=
=
*
Sx •Sx G xx
S y •S*x =HSx •S*x
H2 formulation
- susceptible to noise on the output
- overestimates the actual H of the system
Other
formulations
for H exist
S y •S*y G yy
=
H=
*
Sx •S y G xy
S y •S*y =HSx •S*y
COHERENCE
γ
2
xy
=
Measurement Definitions
(S y •S*x )(Sx •S*y )
(Sx •S*x )(S y •S*y )
10
=
G yx / G xx
G yy / G xy
=
H1
H2
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Measurements - Noise
H=G uv /G uu


 1 
H1 =H

1+G nn 
 G uu 
 G mm 
H 2 =H1+

G

vv 
Measurement Definitions
INPUT
SYSTEM
u(t)
n(t)
OUTPUT
v(t)
H
m(t)
Σ
x(t)
ACTUAL
Σ
y(t)
11
NOISE
MEASURED
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Measurements - Auto Power Spectrum
x(t)
OUTPUT RESPONSE
INPUT FORCE
G xx (f)
yy
AVERAGED INPUT
AVERAGED OUTPUT
POWER SPECTRUM
POWER SPECTRUM
Measurement Definitions
12
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Measurements - Cross Power Spectrum
AVERAGED INPUT
AVERAGED OUTPUT
POWER SPECTRUM
POWER SPECTRUM
G yy (f)
G xx (f)
AVERAGED CROSS
POWER SPECTRUM
G yx (f)
Measurement Definitions
13
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Measurements - Frequency Response Function
AVERAGED INPUT
AVERAGED CROSS
AVERAGED OUTPUT
POWER SPECTRUM
POWER SPECTRUM
POWER SPECTRUM
G yx (f)
G yy (f)
G xx (f)
FREQUENCY RESPONSE FUNCTION
H(f)
Measurement Definitions
14
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Measurements - FRF & Coherence
Coherence
1
Real
0
0Hz
AVG: 5
200Hz
COHERENCE
Freq Resp
40
dB Mag
-60
0Hz
AVG: 5
200Hz
FREQUENCY RESPONSE FUNCTION
Measurement Definitions
15
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Excitation Considerations
1
2
3
h 13
1
1
2
3
2
1
3
2
h 23
3
h 33
h 31
h 33
h 32
Excitation Considerations
1
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Excitation Considerations - Impact
The force spectrum can be customized to some extent
through the use of hammer tips with various hardnesses.
CH1 Time
CH1 Time
1.8
7
Real
Real
-3
-200
-976.5625us
123.9624ms
-976.5625us
CH1 Pwr Spec
-20
123.9624ms
CH1 Pwr Spec
-10
dB Mag
dB Mag
-70
-110
0Hz
6.4kHz
0Hz
CH1 Time
3.5
6.4kHz
CH1 Time
8
Real
Real
-1.5
-2
-976.5625us
123.9624ms
-976.5625us
CH1 Pwr Spec
-20
123.9624ms
CH1 Pwr Spec
-10
dB Mag
dB Mag
-120
-110
0Hz
Excitation Considerations
6.4kHz
0Hz
2
6.4kHz
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Excitation Considerations - Impact/Exponential
The excitation must be sufficient to excite all the modes
of interest over the desired frequency range.
40
COHERENCE
dB Mag
FRF
INPUT POWER SPECTRUM
-60
0Hz
800Hz
40
COHERENCE
FRF
dB Mag
INPUT POWER SPECTRUM
-60
0Hz
Excitation Considerations
200Hz
3
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Excitation Considerations - Impact/Exponential
The response due to impact excitation may need an
exponential window if leakage is a concern.
ACTUAL TIME SIGNAL
SAMPLED SIGNAL
WINDOW WEIGHTING
WINDOWED TIME SIGNAL
Excitation Considerations
4
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Excitation Considerations - Shaker Excitation
AUTORANGING
Random
with
Hanning
Burst
Random
AVERAGING WITH WINDOW
1
2
AUTORANGING
Accurate FRFs are necessary
4
3
AVERAGING
1
AUTORANGING
Leakage is a serious concern
2
3
Special excitation
techniques can be
used which will result
in leakage free
measurements without
the use of a window
4
AVERAGING
Sine
Chirp
1
2
3
4
as well as other techniques
Excitation Considerations
5
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Excitation Considerations - MIMO
Multiple referenced FRFs are
obtained from MIMO test
Energy is distributed
better throughout the
structure making
better measurements
possible
Ref#1
Excitation Considerations
6
Ref#2
Ref#3
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Excitation Considerations - MIMO
Large or
complicated
structures
require
special
attention
Excitation Considerations
7
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Excitation Considerations - MIMO
[G XF ]=[H][G FF ]
 H11
H
[H]= 21
 M
H
 No,1
H12
H 22
M
H No , 2
[H ]=[G XF ][G FF ]−1
Excitation Considerations




L H No, Ni 
L
L
H1, Ni
H 2, Ni
M
Measurements are
developed in a
similar fashion to
the single input
single output case
but using a matrix
formulation
where
No - number of outputs
Ni - number of inputs
8
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Excitation Considerations - MIMO
Measurements on the same structure can show
tremendously different modal densities depending
on the location of the measurement
Source: Michigan Technological University Dynamic Systems Laboratory
Excitation Considerations
9
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Modal Parameter Estimation Concepts
j
[
[
A k ] [A*k ]
A k ] [A*k ] upper [A k ] [A*k ]
[H(s )]= ∑
+
+∑
+
+∑
+
*
*
*
terms (s−s k ) (s−s k )
k =i (s−s k ) (s−s k ) terms (s−s k ) (s−s k )
lower
SYSTEM EXCITATION/RESPONSE
SDOF POLYNOMIAL
PEAK PICK
MULTIPLE REFERENCE FRF MATRIX DEVELOPMENT
INPUT FORCE
RESIDUAL COMPENSATION
INPUT FORCE
INPUT FORCE
LOCAL CURVEFITTING
IFT
GLOBAL CURVEFITTING
POLYREFERENCE CUVREFITTING
COMPLEX EXPONENTIAL
Modal Parameter Estimation Concepts
1
MDOF POLYNOMIAL
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Parameter Estimation Concepts
NO COMPENSATION
y=mx
Y
X
COMPENSATION
Y
Y
y=mx+b
X
WHICH DATA ???
Y
X
X
Modal Parameter Estimation Concepts
2
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Parameter Extraction Considerations
HOW MANY POINTS ???
•
•
RESIDUAL
EFFECTS
RESIDUAL
EFFECTS
•
ORDER OF THE MODEL
AMOUNT OF DATA TO
BE USED
COMPENSATION FOR
RESIDUALS
HOW MANY MODES ???
The test engineer identifies these items
NOT THE SOFTWARE !!!
Modal Parameter Estimation Concepts
3
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Parameter Extraction Considerations
HOW MANY POINTS ???
lower
[Ak ]
[A ]
terms
k
*
k
*
k
[H( s) ] = ∑ ( s − s ) + s − s
( )
j
[Ak ]
[A ]
k
*
k
*
k
∑ ( s − s ) + (s − s )
k=i
RESIDUAL
EFFECTS
upper
[Ak ]
[A ]
terms
k
*
k
RESIDUAL
EFFECTS
*
k
∑ ( s − s ) + (s − s )
HOW MANY MODES ???
[
[
Ak ]
A*k ]
[H(s )] = lower residuals + ∑
+
+ upper
*
(s−s k )
k =i (s−s k )
j
Modal Parameter Estimation Concepts
4
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Parameter Extraction Considerations
The basic equations can be cast in either the
time or frequency domain
a1
a1*
h (s)=
+
(s − p1 ) (s − p1* )
Modal Parameter Estimation Concepts
1 −σt
h ( t )=
e sin ωd t
mωd
5
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Parameter Extraction Considerations
MODAL PARAMETER ESTIMATION MODELS
Time representation
h ij ( n ) ( t ) + a1h ij ( n −1) ( t ) + L + a 2 n h ij ( n − 2 N ) ( t ) = 0
Frequency representation
[( jω)
2N
+ a 1 ( jω) 2 N −1 + L + a 2 N ]h ij ( jω) =
[( jω)
Modal Parameter Estimation Concepts
2M
+ b1 ( jω) 2 M −1 + L + b 2 M ]
6
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Parameter Extraction Considerations
The FRF matrix contains
redundant information
regarding the system
frequency, damping and
mode shapes
Multiple referenced data
can be used to obtain
better estimates of
modal parameters
Modal Parameter Estimation Concepts
7
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Selection of Bands
Select bands for possible SDOF or MDOF
extraction for frequency domain technique.
Residuals ???
Modal Parameter Estimation Concepts
Complex ???
8
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Mode Determination Tools
Summation
MIF
A variety of tools assist in the determination
and selection of modes in the structure
1 Point Each From Panels 1,2, and 3 (37,49,241)
4
10
3
10
2
10
1
10
CMIF
0
10
-1
10
-2
10
0
50
100
150
CMIF
200
250
Frequency (Hz)
300
350
Modal Parameter Estimation Concepts
400
450
500
Stability Diagram
9
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Modal Extraction Methods
Peak Picking
Circle Fitting
SDOF Polynomial
A multitude of techniques exist
IFT
Complex Exponential
MDOF Polynomial Methods
Modal Parameter Estimation Concepts
10
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Model Validation
Synthesis
Validation tools exist
to assure that an
accurate model has
been extracted from
measured data
1.2
1
0.8
0.6
0.4
0.2
0
MAC
S1
S2
S3
S4
Modal Parameter Estimation Concepts
S5
S6
11
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Linear Algebra Concepts
[A] = [{u1} {u 2 } {u 3}
x . . . x 
0 x . . . 


[U]= . 0 x . . 
. . 0 x .


 0 . . 0 x 
s3
T
 {v1} 
T

 {v 2 } 
 {v 3 }T 


O  M 
[A]nm{X}m ={B}n
[A]nm =[V]nn [S]nm [U]Tmm
{X}m =[A ]gnm{B}n =[[V ]nn [S]nm [U ]Tmm ] {B}n
{X}m =[[U ]mm [S]gnm [V]Tnn ]{B}n
[A ]−1=[Adjo int[A ]]
Det[A ]
Linear Algebra Concepts
s1

s2
L]



g
1
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Linear Algebra
The analytical treatment of structural dynamic
systems naturally results in algebraic equations
that are best suited to be represented through
the use of matrices
Some common matrix representations and linear
algebra concepts are presented in this section
Linear Algebra Concepts
2
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Linear Algebra
Common analytical and experimental equations
needing linear algebra techniques
[G ] = [H][G ]
yf
[H ] = [G yf ][G ff ]
−1
ff
[[K ]−λ[M ]]{x}=0
[M ]{&x&}+[C]{x& }+[K ]{x}={F( t )}
[B(s )]−1 = [H(s )] = Adj[B(s )]
det[B(s )]
[B(s )]{x (s )} = {F(s )}
or
Linear Algebra Concepts
O

[H(s )] = [U]  S  [L]T
O

3
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Matrix Notation
A matrix [A] can be described using row,column as
 a 11
a
 21
[A ] = a 31
a
 41
a 51
a 12
a 22
a 32
a 42
a 52
a 13
a 23
a 33
a 43
a 53
a 14
a 24
a 34
a 44
a 54
( row , column )







[A]T -Transpose - interchange rows & columns
[A]H - Hermitian - conjugate transpose
Linear Algebra Concepts
4
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Matrix Notation
A matrix [A] can have some special forms
Diagonal
Square
 a 11
a
 21
[A ] = a 31
a
 41
a 51
a 12
a 22
a 32
a 13
a 23
a 33
a 14
a 24
a 34
a 42
a 52
a 43
a 53
a 44
a 54
a 15 
a 25 

a 35 
a 45 

a 55 
a 22
a 33
a 44






a 55 
Toeplitz
Symmetric
 a 11
a
 12
[A ] = a13
a
 14
a 15
a 11


[A ] = 



Triangular
a 12
a 22
a 23
a 13
a 23
a 33
a 14
a 24
a 34
a 24
a 25
a 34
a 35
a 44
a 45
a 15 
a 25 

a 35 
a 45 

a 55 
Linear Algebra Concepts
a 5
a
 4
[A ] = a 3
a
 2
 a 1
a 11
0

[A ] =  0
0

 0
a 12
a 22
0
a 13
a 23
a 33
a 14
a 24
a 34
0
0
0
0
a 44
0
Vandermonde
a6
a5
a4
a3
a2
5
a7
a6
a5
a4
a3
a8
a7
a6
a5
a4
a9 
a8 

a7 
a6 

a 5 
1

1
[A ] = 
1

1
a1
a2
a3
a4
a 12 

a 22 
a 32 

a 24 
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
a 15 
a 25 

a 35 
a 45 

a 55 
Matrix Manipulation
A matrix [C] can be computed from [A] & [B] as
 a 11
a
 21
a 31
a 12
a 22
a 32
a 13
a 23
a 33
a 14
a 24
a 34
 b11
a 15  b 21


a 25  b 31

a 35  b 41

 b 51
b12 
b 22   c11

b 32  = c 21


c
b 42
  31
b 52 
c12 
c 22 

c 32 
c 21 = a 21b11 + a 22 b 21 + a 23 b 31 + a 24 b 41 + a 25 b 51
c ij = ∑ a ik b kj
k
Linear Algebra Concepts
6
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Simple Set of Equations
A common form of a set of equations is
[A ] {x} = [b]
Underdetermined
# rows < # columns
more unknowns than equations
(optimization solution)
Determined
# rows = # columns
equal number of rows and columns
Overdetermined
# rows > # columns
more equations than unknowns
(least squares or generalized inverse solution)
Linear Algebra Concepts
7
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Simple Set of Equations
This set of equations has a unique solution
2x − y = 1
− x + 2 y − 1z = 2
−y+z =3
 2 − 1 0  x  1 
− 1 2 − 1  y  = 2
   

 0 − 1 1   z  3
whereas this set of equations does not
2x − y = 1
 2 − 1 0  x  1 
− 1 2 − 1  y  = 2
− x + 2 y − 1z = 2

   
4x − 2 y = 2
 4 − 2 0   z  2
Linear Algebra Concepts
8
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Static Decomposition
A matrix [A] can be decomposed and written as
[A ] = [L][U ]
Where [L] and [U] are the lower and upper
diagonal matrices that make up the matrix [A]
x
0

[U] =  0
0

 0
x 0 0 0 0 
x x 0 0 0 


[L] =  x x x 0 0 
x x x x 0 


 x x x x x 
Linear Algebra Concepts
9
x x
x x
0 x
0 0
0 0
x
x
x
x
0
x
x

x
x

x 
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Static Decomposition
Once the matrix [A] is written in this form then
the solution for {x} can easily be obtained as
[A ] = [L][U ]
[U ] {X} = [L]−1 [B]
Applications for static decomposition and inverse
of a matrix are plentiful. Common methods are
Gaussian elimination
Crout reduction
Gauss-Doolittle reduction
Cholesky reduction
Linear Algebra Concepts
10
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Eigenvalue Problems
Many problems require that two
matrices [A] & [B] need to be reduced
[[B] − λ[A ]] {x} = 0
[A ]{&x&} + [B]{x} = {Q( t )}
Applications for solution of eigenproblems are
plentiful. Common methods are
Jacobi
Givens
Subspace Iteration
Linear Algebra Concepts
Householder
Lanczos
11
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Singular Valued Decomposition
Any matrix can be decomposed using SVD
[A ] = [U ][S][V ]T
[U] - matrix containing left hand eigenvectors
[S] - diagonal matrix of singular values
[V] - matrix containing right hand eigenvectors
Linear Algebra Concepts
12
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Singular Valued Decomposition
SVD allows this equation to be written as
[A] = [{u1} {u 2 } {u 3}
s1

s2

L]



 {v1} 
T

 {v 2 } 
 {v 3 }T 



O  M 
T
s3
which implies that the matrix [A] can be written in
terms of linearly independent pieces which form
the matrix [A]
[A ] = {u1}s1{v1}T + {u 2 }s 2 {v 2 }T + {u 3}s3 {v3}T +
Linear Algebra Concepts
13
L
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Singular Valued Decomposition
Assume a vector and singular value to be
1 
 
u1= 2
and
s1 = 1
3 
 
Then the matrix [A1] can be formed to be
1 2 3
1 
 
T
[A1 ] = {u1} s1 {u1} = 2 [1] {1 2 3} = 2 4 6
3 
3 6 9
 
The size of matrix [A1] is (3x3) but its rank is 1
There is only one linearly independent
piece of information in the matrix
Linear Algebra Concepts
14
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Singular Valued Decomposition
Consider another vector and singular value to be
1
 
u 2=  1 
and
s2 = 1
− 1
 
Then the matrix [A2] can be formed to be
 1 1 − 1
1
 
T
[A 2 ] = {u 2 } s 2 {u 2 } =  1  [1] {1 1 − 1} =  1 1 − 1
− 1
− 1 − 1 1 
 
The size and rank are the same as previous case
Clearly the rows and columns
are linearly related
Linear Algebra Concepts
15
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Singular Valued Decomposition
Now consider a general matrix [A3] to be
2 3 2 
[A 3 ] = 3 5 5  = [A1 ] + [A 2 ]
2 5 10
The characteristics of this matrix are not
obvious at first glance.
Singular valued decomposition can be used to
determine the characteristics of this matrix
Linear Algebra Concepts
16
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Singular Valued Decomposition
The SVD of matrix [A3] is
1 
[A] = 2
3
or
1
 
1
− 1
 
  {1 2 3}
0 1
  
 {1 1 − 1}
0
1
  

0 
0  {0 0 0}
  





0
1
1 
[A] = 21{1 2 3}T +  1 1{1 1 − 1}T + 00{0 0 0}T
0
− 1
3 
 
 
 
These are the independent quantities that
make up the matrix which has a rank of 2
Linear Algebra Concepts
17
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Linear Algebra Applications
The basic solid mechanics formulations as well as
the individual elements used to generate a finite
element model are described by matrices
ν
i
θ
F
i
6L − 12 6L 
 12
 6L 4L2 − 6L 2L2 

[k ] = EI3 
L − 12 − 6L 12 − 6L 
 6L 2L2 − 6L 4L 


ν
j
i
θ
E, I
Linear Algebra Concepts
Fj
L
 σ x   C11
 σ  C
 y   21
 σ  C
{σ} = [C]{ε}⇒ z  = 31
τ xy  C 41
τ xz  C51
  
τ yz  C 61
54
13L 
 156 − 22L
− 22L 4L2 − 13L − 3L2 

[m] = ρAL 
− 13L 156
420  54
22L 
 13L − 3L2 22L
4L2 

j
C12
C13
C14
C15
C 22
C32
C 42
C 23
C33
C 43
C 24
C34
C 44
C 25
C35
C 45
C52
C 62
C53
C 63
C54
C 64
C55
C 65
18
C16  ε x 
C 26  ε y 
 
C36  ε z 
 
C 46 γ xy 

C56  γ xz 
 
C 66  γ yz 
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Linear Algebra Applications
Finite element model development uses individual
elements that are assembled into system matrices
Linear Algebra Concepts
19
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Linear Algebra Applications
Structural system equations - coupled
[M ]{&x&}+[C]{x& }+[K ]{x}={F( t )}
Eigensolution - eigenvalues & eigenvectors
[[K ]−λ[M ]]{x}=0
Modal space representation
of equations - uncoupled
\



M

\
{&p&} + 


\ 

Linear Algebra Concepts
C

\
{p& } + 


\ 

K
20

{p} = [U ]T {F}

\ 
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Linear Algebra Applications
Multiple Input Multiple Output Data Reduction
[G ] = [H][G ]
yx
[H ] = [G yx ][G xx ]−1
xx
[Gyx]
=
[H]
[Gxx]
=
RESPONSE
(MEASURED)
FREQUENCY RESPONSE FUNCTIONS
(UNKNOWN)
FORCE
(MEASURED)
Matrix inversion can only be performed if the
matrix [Gxx] has linearly independent inputs
Linear Algebra Concepts
21
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Linear Algebra Applications
Principal Component Analysis using SVD
[G xx ] = [{u1} {u 2 } {0}
[Gxx]
s1

s2

L]



 {v1} 
T

 {v 2 } 
0
  {0}T 



O  M 
T
SVD of the input excitation matrix identifies the
rank of the matrix - that is an indication of how
many linearly independent inputs exist
Linear Algebra Concepts
22
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Linear Algebra Applications
SVD of Multiple Reference FRF Data
[H] = [{u1} {u 2 } {u 3}
[H]
 {v1} 
T

 {v 2 } 
 {v 3 }T 



O  M 
T
s1

s2

L]



s3
FREQUENCY RESPONSE FUNCTIONS
0
50
100
150
200
250
Frequency (Hz)
300
350
400
450
500
SVD of the [H] matrix gives an indication
of how many modes exist in the data
Linear Algebra Concepts
23
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Linear Algebra Applications
Least Squares or Generalized Inverse for
Modal Parameter Estimation Techniques
[
[
Ak ]
A*k ]
[H(s )] = ∑
+
*
(
−
(
)
−
s
s
s
s
k =i
k
k)
j
Least squares error minimization of
measured data to an analytical function
Linear Algebra Concepts
24
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Linear Algebra Applications
Extended analysis and evaluation of systems
[K ][U ] = [M I ][U ][`ω2 ]
[U ][K ][U] = [U] [M ][U][`ω ]
[K ] = [K ] + [V] [`ω + K ][V]
T
T
2
I
I
T
I
[
2
S
S
] [
]
− [K S ][U ][U ] [M I ] − [K S ][U ][U ] [M I ]
T
T
T
[K I ] = [K S ] + [V ]T [`ω2 + K S ][V ] − [[K S ][U ][V ]] − [[K S ][U ][V ]]T
generally require matrix manipulation of some type
Linear Algebra Concepts
25
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory
Linear Algebra Applications
Many other applications exist
Correlation
Model Updating
Advanced Data Manipulation
Operating Data
Rotating Equipment
Nonlinearities
Modal Parameter Estmation
and the list goes on and on
Linear Algebra Concepts
26
Dr. Peter Avitabile
Modal Analysis & Controls Laboratory