Modal Analysis

Dr. Peter Avitabile

Presentation Topics

Modal Analysis & Controls Controls Laboratory Intent Modal Overview

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 Analysis Dr. Peter Avitabile Avitabile

[email protected] [email protected]

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 deta iled 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 novice has some appreciation of the detailed material presented in these papers. This presentation is intended to identify ide ntify the basic methodology and techniques currently employed in this field and to expose one to the typical modal jargon used used in the field.

Intent of Young Engineer Program

1

Dr. Peter Avitabile Modal Analysis & Controls Laboratory

Experimental Modal Analysis A Simple Non-Mathematical 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


1



Analytical Modal Analysis Equation of motion

[M n ]{&x& n } + [C n ]{x& n } + [K n ]{x n } = {Fn ( t )}

Eigensolution


[[K n ] − λ[M n ]]{x n } = {0}

2


Could you explain modal analysis for me ? Simple time-frequency response relationship RESPONSE

increasing rate of oscillation

FORCE

time

frequency


3


Could you explain modal analysis for me ? Sine Dwell to Obtain Mode Shape Characteristics

MODE3

MODE 1

MODE 2


MODE 4

4


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

-1.0000

Phase Experimental Modal Analysis

Imaginary 5


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


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


7


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


8


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 ! ?


?

9


More measurements better defines the shape MODE # 1 MODE # 2 MODE # 3

DOF # 1

DOF #2

DOF # 3


10


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

Actual time signals

ANTIALIASING FILTERS

Analog anti-alias filter

AUTORANGE ANALYZER ADC DIGITIZES SIGNALS

OUTPUT

INPUT

Digitized time signals

APPLY WINDOWS

INPUT OUTPUT

Windowed time signals

COMPUTE FFT LINEAR SPECTRA LINEAR OUTPUT SPECTRUM

LINEAR INPUT SPECTRUM

Compute FFT of signal

AVERAGING OF SAMPLES

COMPUTATION OF AVERAGED INPUT/OUTPUT/CROSS POWER SPECTRA

INPUT POWER SPECTRUM

OUTPUT POWER SPECTRUM

CROSS POWER SPECTRUM

Average auto/cross spectra

COMPUTATION OF FRF AND COHERENCE

Compute FRF and Coherence FREQUENCY RESPONSE FUNCTION


COHERENCEFUNCTION

12


What’s most important in impact testing ? Hammers and Tips 40

COHERENCE

dB Mag

FRF INPUT POWER SPECTRUM -60 0Hz

800Hz

40

COHERENCE FRF

dB Mag


-60 0Hz


200Hz

13


What’s most important in impact testing ? Leakage and Windows ACTUAL TIME SIGNAL

SAMPLED SIGNAL

WINDOW WEIGHTING

WINDOWED TIME SIGNAL


14


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


15


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


16


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

frequency,

ζ

a ij2 a ij3

2

ζ

3

ω

1


ω

2

ω

3

17

damping, residue



MDOF

SDOF

The task for the modal test engineer is to determine the parameters that make up the pieces of the frequency response function


18



HOW MANY POINTS ???

RESIDUAL EFFECTS

Mathematical routines help to determine the basic parameters that make up the FRF

RESIDUAL EFFECTS

HOW MANY MODES ???


19


What is operating data ? Why and How Do Structures Vibrate?

INPUT TIME FORCE

f(t)

y(t)

FFT

IFT

INPUT SPECTRUM

f(j ω)


OUTPUT SPECTRUM

h(j ω)

20

y(j ω)


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 ???


21


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


MODE 2 CONTRIBUTION

22


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


23


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


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


25


So what good is modal analysis ?

EXPERIMENTAL MODAL TESTING

FINITE ELEMENT MODELING

MODAL PARAMETER ESTIMATION

PERFORM EIGEN SOLUTION

Repeat until desired characteristics are obtained

RIB STIFFNER

MASS

DEVELOP MODAL MODEL

SPRING STRUCTURAL CHANGES REQUIRED Yes

USE SDM TO EVALUATE STRUCTURAL CHANGES

No

DASHPOT DONE

STRUCTURAL

The dynamic model can be used for studies to determine the effect of structural changes of the mass, damping and stiffness

DYNAMIC MODIFICATIONS


26


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

STRUCTURAL DYNAMICS MODIFICATION

FORCED RESPONSE SIMULATION

MODIFIED MODAL DATA


REAL WORLD FORCES

STRUCTURAL RESPONSE

27


Single Degree of Freedom Overview

x(t)

f(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/ωn

ζ=10%

t1

ζ=5%

t2

ζ=2% ζ=1% ζ=0.1%

h (s) =

-180

ω/ω n

SDOF Overview

1

1 ms 2 + cs + k 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


SDOF Equations Equation of Motion d2x

dx m 2 +c dt dt

+ k x = f ( t )

or

m &x& + cx&

+ k x = f ( t )

Characteristic Equation ms 2

+ cs + k = 0

Roots or poles of the characteristic equation s1, 2

SDOF Overview

=−

c 2m

2

 c  + k    2m  m

±

3


SDOF Definitions Poles expressed as s1, 2 = − ζω n ±

j ω

(ζωn )2 −ωn 2 = − σ ± jωd POLE

Damping Factor

σ = ζωn

Natural Frequency

ωn =

% Critical Damping

ζ= c

k m ζωn

cc

Critical Damping

c c = 2mωn

Damped Natural Frequency

ωd = ωn 1−ζ 2

SDOF Overview

ωd

4

CONJUGATE


σ

SDOF - Harmonic Excitation 100

ζ=0.1%

0

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

x

δst

=

ω/ω n

1

φ=tan

(1 − β ) +(2ζβ )

SDOF Overview

2 2

2

5

 2ζβ   2   1 − β 

−1


SDOF - Damping Approximations MAG T = 2 π/ω n X1 X2

0.707 MAG

t1

ω1

ωn

ωn = Q= 2ζ ω2 − ω1 1

SDOF Overview

t2

ω2

δ = ln

6

x1 x2

≈ 2πζ


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 h (s) =

SDOF Overview

1 ms 2 + cs + k 7


SDOF - Transfer Function

h (s ) =

Polynomial Form

h (s) =

Pole-Zero Form Partial Fraction Form

h (s) =

Exponential Form

h(t) =

SDOF Overview

8

1 ms 2 + cs + k

1/ m (s − p1 )(s − p1* ) a1

(s − p1 ) 1 mωd

+

a1* (s − p1* )

e −ζωt sin ωd t


SDOF - Transfer Function & Residues

Residue a1

=

h (s)(s − p1 )

=

1 2 jmωd

related to mode shapes

Source: Vibrant Technology

SDOF Overview

s→ p1

9


SDOF - Frequency Response Function

h ( jω) = h (s)

SDOF Overview

s= jω

=

10

a1 ( jω − p1 )

+

a1* ( jω − p1* )


SDOF - Frequency Response Function Coincident-Quadrature Plot

Bode Plot

0.707 MAG

Nyquist Plot

SDOF Overview

11


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


Multiple Degree of Freedom Overview [B(s )]−1 = [H(s )] =

Adj[B(s )] det[B(s )]

=

[A(s )] det[B(s )]

f 1

p1

k1

f 2

p2

c1

f 3

p3

m2

m1

m3

k2

c2

k3

c3

R1 D1

MODE 1

MODE 2

MODE 3

R2 D2

\   

M

 \ { p  &&} +   \  

MDOF Overview

C

 \ { p& } +    \  

 T K { p} = [U ] {F}  \ 

1

R3 D3

F1

F2F3


MDOF Definitions Assumptions

f 2

• lumped mass

m2

• stiffness proportional to displacement

k2

• damping proportional to velocity • linear time invariant

MDOF Overview

f 1

2

c2 x1

m1 k1

• 2nd order differential equations

x2

c1


MDOF Equations Equation of Motion - Force Balance m1&x&1+(c1 + c 2 )x& 1−c 2 x& 2 +(k 1 + k 2 )x1−k 2 x 2 =f 1 (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 m1   &x&1      m 2  &x& 2   (c1 + c 2 ) − c 2   x& 1  +  x&  − c c  2 2  2 

Matrices and Linear Algebra are important !!!

(k 1 + k 2 ) − k 2   x1   f 1 ( t )  +  =   − k k x f ( t )  2  2 2  2  MDOF Overview

3


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

 =  \ 

ω12  2 ω 2  

   and [U ] = [{u1} {u 2 } \  4

L]


Modal Space Transformation Modal transformation

 p1    {x} = [U ]{ p} = [{u1} {u 2} L]p 2  M  

Projection operation

[U ]T [M ][U ]{ p&&} + [U ]T [C][U ]{ p& } + [U ]T [K ][U ]{ p} = [U ]T {F} Modal equations (uncoupled)  m1   

 p& 1  k 1

&&1   c1  p

m2

   p && 2 +      \  M  

MDOF Overview

c2

 p& +  2    \   M  

5

k 2

T  p1  {u1} {F}  p ={u }T{F}   2   2    \  M  M 


Modal Space Transformation Diagonal Matrices Modal Mass

\   

M

Modal Damping

 \ { p  &&} +   \  

C

Modal Stiffness

 \ { p& } +    \  

 T K { p} = [U ] {F}  \ 

Highly coupled system f 1

p1

transformed into

k1

simple system MDOF Overview

6

k2

m3 c2

MODE 2

f 3

p3

m2 c1

MODE 1

f 2

p2

m1

k3

c3 MODE 3


Modal Space Transformation .. . [M]{x} + [C]{x} + [K]{x} = {F(t)} = f 1

p1 m1

{x} = [U]{p} = [{u } {u } {u } 1

2

3

k1

]{p} +

MODAL

SPACE

c1 MODE 1 f 2

p2 m2 k2

.. . T [ M ]{p} + [ C ]{p} + [ K ]{p} = [U] {F(t)}

+

c2 MODE 2 f 3

p3 m3

{u }p + {u }p + {u }p 1 1 2 2 3 3

k3

c3 MODE 3

MDOF Overview

7


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 2 +[C]s+[K ] = 0 ⇒ p k = − σ k ± jωdk

Damping

MDOF Overview

8

Frequency


MDOF - Transfer Function System Equation

{x (s )} [B(s )]{x (s )} = {F(s )} ⇒ [H(s )] = [B(s )] = {F(s )} −1

System Transfer Function

[A(s )] [B(s )] = [H(s )] = = det[B(s )] det[B(s )] −1

Adj[B(s )]

[A(s )]

Residue Matrix

det[B(s )]

Characteristic Equation

MDOF Overview

9

Mode Shapes Poles


MDOF - Residue Matrix and Mode Shapes Transfer Function evaluated at one pole

[H(s )]s =s = {u k } k

q k s− p k

{u k }T

can be expanded for all modes m

[H(s )] = ∑ k =1

MDOF Overview

T

q k {u k }{u k }

(s− p k )

10

+

q k {u

}{u } (s− p )

* T k

* k

* k


MDOF - Residue Matrix and Mode Shapes

Residues are related to mode shapes as

[A(s )]k = q k {u k }{u k }T  a11k a12 k a13k a a a  21k 22 k 23k a 31k a 32 k a 33k  M M M 

MDOF Overview

L

 u1k u1k u1k u 2 k u1k u 3k  u u L u 2 k u 2 k u 2 k u 3k =q k  2 k 1k L  u 3k u1k u 3k u 2 k u 3k u 3k   M O M M 

11

L

  L  O L


MDOF - Drive Point FRF

h ij ( j )

a *ij1

a ij1 ( j

p1 )

p*1 )

( j

*

a ij 2 ( j

a ij 2

p2 )

p*2 )

( j

a *ij 3

a ij 3 ( j

h ij ( j )

MDOF Overview

p3 )

*

q1u i1u j1 ( j

p*3 )

( j

q1u i1u j1

p1 )

p*1 )

( j

*

q 2 u i 2u j 2

q 2u i 2u j 2

( j

( j

p2 )

p*2 )

q 3u i 3u j 3

q 3u i 3u j 3

( j

( j

12

p3 )

*

p*3 )


MDOF - FRF using Residues or Mode Shapes h ij ( j )

a*ij1

a ij1 ( j

p1 )

R1

p*1 )

( j

a*ij 2

a ij 2

D1

( j

p2 )

( j

R2 D2 R3 D3

F1

F2F3 a ij1

h ij ( j )

* 1

q1u i1u j1 ( j

p1 )

MDOF Overview

* j1

qu u

p2 )

2 3

p*1 )

( j

q 2u i 2 u j 2 ( j

* i1

1

a ij2 a ij3

q*2u*i 2u*j 2 ( j

p*2 )

1

2

3

L

13


p*2 )

L

Time / Frequency / Modal Representation PHYSICAL

TIME

FREQUENCY ANALYTICAL

MODAL f 1

p1 m1 k1

MODE 1

+

c1 MODE 1

+ p2

+

f 2

m2 k2

MODE 2

+

+

c2 MODE 2

p3

+

f 3

m3 k3 MODE 3

MODE 3

MDOF Overview

c3

14


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]

[ A(s) ] det [B(s)] [U]

FINITE ELEMENT MODEL

ANALYTICAL MODEL REDUCTION T

[K -

MODAL PARAMETER ESTIMATION

A

M]{X} = 0

N

H(j )

LARGE DOF MISMATCH

N

X j(t) H(j ) =

X j (j ) Fi (j )

MDOF Overview

U

A

FFT Fi (t)

15

MODAL TEST

EXPERIMENTAL MODAL MODEL EXPANSION


Measurement Definitions INPUT

SYSTEM

OUTPUT

u(t)

v(t) H

n(t)

x(t)

ACTUAL

m(t)

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


Measurement Definitions ANALOG SIGNALS

Actual time signals

OUTPUT

INPUT

ANTIALIASING FILTERS

Analog anti-alias filter

AUTORANGE ANALYZER ADC DIGITIZES SIGNALS

Digitized time signals

OUTPUT

INPUT

APPLY WINDOWS

Windowed time signals

INPUT 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


Average auto/cross spectra

OUTPUT POWER SPECTRUM

CROSS POWER SPECTRUM

COMPUTATION OF FRF AND COHERENCE

Compute FRF and Coherence FREQUENCY RESPONSE FUNCTION


COHERENCEFUNCTION

2


Leakage ACTUAL

F R E Q

DATA

CAPTURED DATA

T

T I M E

RECONTRUCTED DATA

Periodic Signal

U E N C Y

ACTUAL DATA

Non-Periodic Signal

CAPTURED DATA

T

RECONTRUCTED DATA

T


3

Leakage due to signal distortion


Windows 0

Time weighting functions are applied to minimize the effects of leakage

-10

-20

-30

-40

-50

Rectangular

-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


Windows Special windows are used for impact testing Force window

Exponential Window


5


Measurements - Linear Spectra x(t) INPUT

Sx(f)

h(t)

y(t)

SYSTEM

OUTPUT

FFT & IFT

H(f)

Sy(f)

FREQUENCY

TIME

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


6


Measurements - Linear Spectra +∞

+∞

∫

∫

x ( t )= Sx (f )e j2 πft df

Sx (f )= x ( t )e − j2 πft dt

−∞ +∞

∫

−∞ j2 πft

y( t )= S y (f )e

+∞

∫

S y (f )= y( t )e − j2 πft dt

df

−∞

−∞ +∞

+∞

∫

H(f )= h ( t )e − j2 πft dt

∫

h ( t )= H(f )e j2 πft df

−∞

−∞

Note: Sx and S are complex valued functions


7


Measurements - Power Spectra R xx(t) INPUT

Gxx(f)

Ryx(t)

Ryy(t)

SYSTEM

OUTPUT

FFT & IFT

Gxy(f)

Gyy(f)

FREQUENCY

TIME

R xx(t) - autocorrelation of the input signal x(t) R yy(t) - autocorrelation of the output signal y(t) R yx(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 )


8


Measurements - Linear Spectra R xx ( τ)=E[ x ( t ), x ( t + τ)]=

lim 1

x ( t )x ( t + τ)dt ∫ T → ∞T T

+∞

∫

G xx (f )= R xx ( τ)e − j2 πft dτ=Sx (f )•S*x (f ) −∞

R yy (τ)=E[ y( t ), y( t + τ)]=

lim 1

y( t )y( t + τ)dt ∫ T → ∞T T

+∞

∫

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 → ∞T T

+∞

∫

G yx (f )= R yx ( τ)e − j2 πft dτ=S y (f )•S*x (f ) −∞ Measurement Definitions

9


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

S y •S*x =HSx •S*x

H= = * Sx •Sx G xx

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

S y •S*y =HSx •S*y

= H= * Sx •S y G xy

COHERENCE

γ

2 xy

=


(S y •S*x )(Sx •S*y ) (Sx •S*x )(Sy •S*y ) 10

=

G yx / G xx G yy / G xy

=

H1 H2


Measurements - Noise H=G uv /G uu INPUT

   1  H1 =H  G 1+ nn   G uu   G mm  H 2 =H1+   G vv 


SYSTEM

OUTPUT

u(t)

v(t) H

n(t)

x(t)

m(t)

NOISE

y(t)

11

ACTUAL

MEASURED


Measurements - Auto Power Spectrum

x(t) OUTPUT RESPONSE

INPUT FORCE

G xx (f)

yy

AVERAGED INPUT

AVERAGED OUTPUT

POWER SPECTRUM

POWER SPECTRUM


12


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


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


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


15


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


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

7

1.8

Real

Real

-3

-200 -976.5625us

123.9624ms

-976.5625us

CH1 Pwr Spec

123.9624ms

CH1 Pwr Spec

-20

-10

dB Mag

dB Mag

-70

-110 0Hz

6.4kHz

0Hz

CH1 Time

6.4kHz

CH1 Time

3.5

8

Real

Real

-1.5

-2 -976.5625us

123.9624ms

-976.5625us

CH1 Pwr Spec

123.9624ms

CH1 Pwr Spec

-20

-10

dB Mag

dB Mag

-120

-110 0Hz


6.4kHz

0Hz

2

6.4kHz


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


-60 0Hz


200Hz 3


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


4


Excitation Considerations - Shaker Excitation AUTORANGING

AVERAGING WITH WINDOW

Leakage is a serious concern

Random with Hanning

1

2

AUTORANGING

4

3

Accurate FRFs are necessary

AVERAGING

Burst Random 1

AUTORANGING

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


Excitation Considerations - MIMO Multiple referenced FRFs are obtained from MIMO test

Energy is distributed better throughout the structure making better measurements possible Ref#1


6

Ref#2

Ref#3


Excitation Considerations - MIMO

Large or complicated structures require special attention


7


Excitation Considerations - MIMO

[G XF ]=[H ][G FF ] H12  H11 H H 22 21  [H ]= M  M H  No ,1 H No , 2

L L

L

H1, Ni 

H 2, Ni 

 M   H No, Ni 

Measurements are developed in a similar fashion to the single input single output case but using a matrix formulation

where

[H ]=[G XF ][G FF ]−1


No - number of outputs Ni - number of in uts

8


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


9


Modal Parameter Estimation Concepts j [A k ] A*k [A k ] A*k upper [A k ] A*k [H(s )]= ∑ + +∑ + + ∑ + * * s s ( ) s s ( ) s s (s−s*k ) − − − ( ) ( ) ( ) s s s s − − terms k = i terms k k k k k lower

SYSTEM EXCITATION/RESPONSE

SDOF POLYNOMIAL

PEAK PICK MULTIPLE REFERENCE FRF MATRIX DEVELOPMENT

INPUT FORCE

RESIDUAL COMPENSATION

LOCAL CURVEFITTING

IFT

INPUT FORCE

GLOBAL CURVEFITTING INPUT FORCE

POLYREFERENCE CUVREFITTING

COMPLEX EXPONENTIAL

Modal Parameter Estimation Concepts

1

MDOF POLYNOMIAL


Parameter Estimation Concepts NO COMPENSATION Y

y=mx X

COMPENSATION

Y

Y

y=mx+b X

WHICH DATA ??? Y

X X


2


Parameter Extraction Considerations HOW MANY POINTS ???

ORDER OF THE MODEL

RESIDUAL EFFECTS

AMOUNT OF DATA TO BE USED RESIDUAL EFFECTS

COMPENSATION FOR RESIDUALS

HOW MANY MODES ???

The test engineer identifies these items NOT THE SOFTWARE !!! Modal Parameter Estimation Concepts

3


Parameter Extraction Considerations HOW MANY POINTS ???

lower

H s terms

s

sk

j

k i

A*k

A k

A*k

A k s

sk

s

upper

terms

s*k

s

RESIDUAL EFFECTS

* k

s

A*k

A k s

sk

RESIDUAL EFFECTS

s

s*k HOW MANY MODES ???

[A k ] A*k [H(s )] = lower residuals + ∑ + + upper * (s−s k ) k = i (s−s k ) j


4


Parameter Extraction Considerations

The basic equations can be cast in either the time or frequency domain

h (s)=

a1

+

a1*

h ( t )=

(s − p1 ) (s − p1* )


5

1 mωd

e −σt sin ωd t


Parameter Extraction Considerations MODAL PARAMETER ESTIMATION MODELS Time representation

h ij ( n ) ( t ) + a 1h ij ( n −1) ( t ) + L + a 2 n h ij ( n − 2 N ) ( t ) = 0 Frequency representation

[( jω)

2 N

+ a 1 ( jω)

[( jω)


2M

2 N −1

+ b1 ( jω)

6

]

+ L + a 2 N h ij ( jω) = 2 M −1

+ L + b 2 M

]


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


7


Selection of Bands Select bands for possible SDOF or MDOF extraction for frequency domain technique. Residuals ???


Complex ???

8


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, a nd 3 (37,49,241) 4

10

3

10

2

10

1

10 CMIF

0

10

-1

10

-2

10 0

50

100

150

200

250 Frequency (Hz)

300

350

400

450

500

CMIF Modal Parameter Estimation Concepts

Stability Diagram 9


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


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

S1

S2

S3

S4

S5

S6

MAC


11


Linear Algebra Concepts s1  s 2 [A ] = [{u1} {u 2} {u 3} L] s3   

x 0  [U]= . .   0

Linear Algebra Concepts

x

.

.

.

x

.

.

.

0

x

.

.

.

0

x

.

.

0

[Adjo int[A ]] [A]−1= Det[A ]

 {v1}T   {v }T   2 T   {v3}    O  M 



  x  .

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 g

{X}m =[[U ]mm[S]gnm[V ]Tnn ]{B}n

1


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


2


Linear Algebra Common analytical and experimental equations needing linear algebra techniques G yf

[H ] = [G yf ][G ff ]−1

= [H ] [G ff ]

[[K ]−λ[M ]]{x}=0

[M ]{&x&}+[C]{x& }+[K ]{x}={F( t )}

[B(s )]−1 = [H(s )] =

[B(s )]{x (s )} = {F(s )}

O or [H(s )] = [U ]  S   Linear Algebra Concepts

3

Adj[B(s )] det[B(s )]

  [L ]T  O


Matrix Notation A matrix [A] can be described using row,column as

 a11 a12 a a  21 22 [A ] = a 31 a 32 a a  41 42 a 51 a 52 ( row ,

a 13

a 14

a 23

a 24

a 33

a 34

a 43

a 44

a 53

a 54

      

column )

[A] T -Transpose - interchange rows & columns [A] H - Hermitian - conjugate transpose


4


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 13

a 14

a 22

a 23

a 24

a 32

a 33

a 34

a 42

a 43

a 44

a 52

a 53

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 a15

a 11   [A ] =    

Triangular

a 12

a 13

a 14

a 22

a 23

a 24

a 23

a 33

a 34

a 24

a 34

a 44

a 25

a 35

a 45

a 15 

a 25 

 a 35   a 45  a 55 


a 5 a  4 [A ] = a 3 a  2  a1

a9 

a7

a8

a5

a6

a7

a8 

a4

a5

a6

a7 

a3

a4

a5

a6

a2

a3

a4



  a 5 

a 15 

a 12

a 13

a 14

a 22

a 23

a 24

a 25 

0

a 33

a 34

a 35 

0

0

a 44

a 45

0

0

0

Vandermonde

a6

5

a11 0  [A ] =  0 0   0

1 1 [A ] =  1  1

a1

a 12 

a2

a 22 



a3

a 32 

a4

a 24 






  a 55 

Matrix Manipulation A matrix [C] can be computed from [A] & [B] as

 a 11 a12 a13 a 14 a a  21 22 a 23 a 24 a 31 a 32 a 33 a 34 c 21

 b11 a 15   b 21   a 25  b 31  a 35  b 41   b 51

b12 

b 22 

c c   11 12  b 32  = c 21 c 22    b 42  c 31 c32  b 52 

= a 21 b11 + a 22 b 21 + a 23 b31 + a 24 b 41 + a 25 b 51 c ij

= ∑ a ik b kj k


6


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)


7


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

− x + 2 y − 1z = 2 4x − 2 y = 2


 2 − 1 0  x  1  − 1 2 − 1  y  = 2       4 − 2 0   z  2

8


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 x  [L] =  x x   x

0

0

0

x

0

0

0

x 0  [U ] =  0 0   0

0

 x x 0 0  x x x 0  x x x x 


9

x

x

x

x

x

x

x x

 0 x x x  0 0 x x  0 0 0 x 


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


10


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


Householder Lanczos

11


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


12


Singular Valued Decomposition SVD allows this equation to be written as

s1  s 2 [A ] = [{u1} {u 2 } {u 3 } L] s3   

 {v1}T   {v }T   2 T   {v3 }    O  M 

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 }s 3 {v 3 }T +


13

L


Singular Valued Decomposition Assume a vector and singular value to be

1   u1= 2 and s1 = 1 3    Then the matrix [A ] 1 can be formed to be 1 1 2 3   [A1 ] = {u1} s1 {u1}T = 2 [1]{1 2 3} = 2 4 6   3  3 6 9   The size of matrix [A ] 1 is (3x3) but its rank is 1 There is only one linearly independent piece of information in the matrix Linear Algebra Concepts

14


Singular Valued Decomposition Consider another vector and singular value to be

1   u 2=  1  and s2 = 1 − 1   Then the matrix [A ] 2 can be formed to be 1  1 1 − 1   [A 2 ] = {u 2 } s 2 {u 2 }T =  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


Singular Valued Decomposition Now consider a general matrix [A ] 3 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


16


Singular Valued Decomposition The SVD of matrix [A ] 3 is

1    [A ] = 2 3

1   1 − 1  

0 1   {1 2 3}        0  1  {1 1 − 1}  0  0  {0 0 0}    

or

1 1 0       T T [A ] = 21{1 2 3} +  1 1{1 1 − 1} + 00{0 0 0}T 3  − 1 0       These are the independent quantities that make up the matrix which has a rank of 2 Linear Algebra Concepts

17


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  12 6L − 12 6L   4L2 − 6L 2L2  EI 6L  [k ] = 3  L − 12 − 6L 12 − 6L   6L 2L2 − 6L 4L    i

j

i

13L   156 − 22L 54 2  − 13L − 3L2  ρAL − 22L 4L  [m] = 420  54 − 13L 156 22L   13L − 3L2 22L 4L2   

j

E, I F

i

 σ x   C11  σ  C  y   21  σ  C {σ} = [C]{ε}⇒ z  = 31 τxy  C41 τ xz  C51    τ yz  C61 Linear Algebra Concepts

F

L

j

C12

C13

C14

C15

C 22

C 23

C 24

C 25

C32

C33

C34

C35

C 42

C 43

C 44

C 45

C52

C53

C54

C55

C 62

C 63

C 64

C 65

18

C16  ε x

    C 26 ε y   C36  ε z    C 46 γ xy   C56  γ xz    C 66  γ yz 


Linear Algebra Applications Finite element model development uses individual elements that are assembled into system matrices


19


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&&} +  C { p&} +  K { p} = [U]T{F}       \  \  \    


20


Linear Algebra Applications Multiple Input Multiple Output Data Reduction G yx

[H ] = [G yx ][G xx ]−1

= [H ] [G 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


21


Linear Algebra Applications Principal Component Analysis using SVD

s1  s 2 [G xx ] = [{u1} {u 2 } {0} L] 0    [Gxx]

 {v1}T   {v }T   2 T    {0}    O  M 

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


Linear Algebra Applications SVD of Multiple Reference FRF Data

 {v1}T   {v }T   2 T   {v3 }    O  M 

s1  s 2 [H ] = [{u1} {u 2 } {u 3 } L] s3    [H]

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


Linear Algebra Applications Least Squares or Generalized Inverse for Modal Parameter Estimation Techniques

[A k ] A*k + [H(s )] = ∑ (s−s*k ) k = i (s−s k ) j

Least squares error minimization of measured data to an analytical function Linear Algebra Concepts

24


Linear Algebra Applications Extended analysis and evaluation of systems

[K ][][U ] = [M I ][U ] `ω2

[U ][K ][U] = [U] [M ][U ][`ω ] T

T

2

I

I

[K I ] = [K S ] + [V ]T [`ω2 + K S ][V ]

− [[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


25


Modal Analysis

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