Brüel and Kjaer

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Modal Analysis 1 Experimental Modal Analysis Copyright© 2003 Brüel & Kjær Sound & Vibration Measurement A/S All Rights Reserved
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  • Modal Analysis 1

    ExperimentalModal Analysis

    Copyright 2003Brel & Kjr Sound & Vibration Measurement A/SAll Rights Reserved

  • Modal Analysis 2

    SDOF and MDOF Models

    Different Modal Analysis Techniques

    Exciting a Structure

    Measuring Data Correctly

    Modal Analysis Post Processing

    f(t)

    x(t)

    kc

    m

    = ++ + +

  • Modal Analysis 3

    Simplest Form of Vibrating System

    D

    d = D sinnt

    m

    k

    T

    Time

    Displacement

    Frequency1T

    Period, Tn in [sec]

    Frequency, fn= in [Hz = 1/sec] 1Tn

    Displacement

    kmn= 2 fn =

  • Modal Analysis 4

    Mass and Spring

    time

    m1

    m

    Increasing mass reduces frequency

    1n mm

    k2 +== nf

  • Modal Analysis 5

    Mass, Spring and Damper

    Increasing damping reduces the amplitude

    time

    m

    k c1 + c2

  • Modal Analysis 6

    )()()()( tftKxtxCtxM =++ &&&M = mass (force/acc.)C = damping (force/vel.)K = stiffness (force/disp.)

    =)t(x&&=)t(x&=)t(x

    Acceleration VectorVelocity VectorDisplacement Vector

    =)t(f Applied force Vector

    Basic SDOF Model

    f(t)x(t)

    kc

    m

  • Modal Analysis 7

    SDOF Models Time and Frequency Domain

    kcjmFXH ++==

    2 1)()()(

    H()F() X()

    |H()|1k

    H()

    0 90

    180

    0 = k/m

    12m

    1c

    )()()()( tkxtxctxmtf ++= &&&

    f(t)x(t)

    kc

    m

  • Modal Analysis 8

    Modal Matrix

    Modal Model(Freq. Domain)

    ( )( )( )

    ( )

    ( ) ( ) ( )( )( )

    ( ) ( )

    ( )

    =

    3

    1

    23

    22

    12111

    3

    2

    1

    F

    HH

    HH

    HHH

    X

    XXX

    nnn

    n

    n

    X1

    X2X3

    X4

    F3H21

    H22

  • Modal Analysis 9

    MDOF Model

    0

    -90

    -180

    2 1

    1+2

    1 2

    1+2

    Frequency

    FrequencyPhase

    Magnitude

    m

    d1+ d2

    d1

    dF

  • Modal Analysis 10

    Why Bother with Modal Models?

    Physical Coordinates = CHAOS Modal Space = Simplicity

    q1

    11

    20112 2

    q21

    20222

    q31

    20332

    3

    Bearing

    Rotor

    Bearing

    Foundation

  • Modal Analysis 11

    Definition of Frequency Response Function

    F X

    f f f

    H(f) is the system Frequency Response FunctionF(f) is the Fourier Transform of the Input f(t)X(f) is the Fourier Transform of the Output x(t)

    F(f) H(f) X(f)

    )f(F)f(X)f(H =

    H

    H

  • Modal Analysis 12

    Benefits of Frequency Response Function

    F(f) H(f) X(f)

    z Frequency Response Functions are properties of linear dynamic systems

    z They are independent of the Excitation Function

    z Excitation can be a Periodic, Random or Transient function of time

    z The test result obtained with one type of excitation can be used for predicting the response of the system to any other type of excitation

  • Modal Analysis 13

    Compliance Dynamic stiffness(displacement / force) (force / displacement)

    Mobility Impedance(velocity / force) (force / velocity)

    Inertance or Receptance Dynamic mass(acceleration / force) (force /acceleration)

    Different Forms of an FRF

  • Modal Analysis 14

    Alternative Estimators

    )()()( fFfXfH =

    )()()(1 fGfGfH

    FF

    FX=

    )()()(2 fGfGfH

    XF

    XX=

    213 )( HHGG

    GGfH

    FX

    FX

    FF

    XX ==

    2

    1

    22 *)( H

    HGG

    GG

    GGG

    fXX

    FX

    FF

    FX

    XXFF

    FX ===

    X(f)F(f) H(f)

  • Modal Analysis 15

    Which FRF Estimator Should You Use?

    Definitions:

    User can choose H1, H2 or H3 after measurement

    Accuracy for systems with: H1 H2 H3Input noise - Best -

    Output noise Best - -

    Input + output noise - - Best

    Peaks (leakage) - Best -

    Valleys (leakage) Best - -

    Accuracy

    )()()(1 fGfGfH

    FF

    FX= )()()(2 fGfGfH

    XF

    XX=FX

    FX

    FF

    XX

    GG

    GGfH =)(3

  • Modal Analysis 16

    SDOF and MDOF Models

    Different Modal Analysis Techniques

    Exciting a Structure

    Measuring Data Correctly

    Modal Analysis Post Processing

    f(t)

    x(t)

    kc

    m

    = ++ + +

  • Modal Analysis 17

    Three Types of Modal Analysis

    1. Hammer Testing Impact Hammer taps...serial or parallel measurements Excites wide frequency range quickly Most commonly used technique

    2. Shaker Testing Modal Exciter shakes product...serial or parallel measurements Many types of excitation techniques Often used in more complex structures

    3. Operational Modal Analysis Uses natural excitation of structure...serial or parallel measurements Cutting edge technique

  • Modal Analysis 18

    Different Types of Modal Analysis (Pros)

    z Hammer Testing Quick and easy Typically Inexpensive Can perform poor man modal as well as full modal

    z Shaker Testing More repeatable than hammer testing Many types of input available Can be used for MIMO analysis

    z Operational Modal Analysis No need for special boundary conditions Measure in-situ Use natural excitation Can perform other tests while taking OMA data

  • Modal Analysis 19

    Different Types of Modal Analysis (Cons)

    z Hammer Testing Crest factors due impulsive measurement Input force can be different from measurement to measurement

    (different operators, difficult location, etc.) Calibrated elbow required (double hits, etc.) Tip performance often an overlooked issue

    z Shaker Testing More difficult test setup (stingers, exciter, etc.) Usually more equipment and channels needed Skilled operator(s) needed

    z Operational Modal Analysis Unscaled modal model Excitation assumed to cover frequency range of interest Long time histories sometimes required Computationally intensive

  • Modal Analysis 20

    Frequency Response Function

    Input

    Output

    Time(Response) - InputWorking : Input : Input : FFT Analyzer

    0 40m 80m 120m 160m 200m 240m

    -80

    -40

    0

    40

    80

    [s]

    [m/s] Time(Response) - InputWorking : Input : Input : FFT Analyzer

    0 40m 80m 120m 160m 200m 240m

    -80

    -40

    0

    40

    80

    [s]

    [m/s]

    Time(Excitation) - InputWorking : Input : Input : FFT Analyzer

    0 40m 80m 120m 160m 200m 240m

    -200

    -100

    0

    100

    200

    [s]

    [N] Time(Excitation) - InputWorking : Input : Input : FFT Analyzer

    0 40m 80m 120m 160m 200m 240m

    -200

    -100

    0

    100

    200

    [s]

    [N]

    FFT

    FFTFrequency Domain

    Autospectrum(Excitation) - InputWorking : Input : Input : FFT Analyzer

    0 200 400 600 800 1k 1,2k 1,4k 1,6k

    100u

    1m

    10m

    100m

    1

    [Hz]

    [N] Autospectrum(Excitation) - InputWorking : Input : Input : FFT Analyzer

    0 200 400 600 800 1k 1,2k 1,4k 1,6k

    100u

    1m

    10m

    100m

    1

    [Hz]

    [N]

    Autospectrum(Response) - InputWorking : Input : Input : FFT Analyzer

    0 200 400 600 800 1k 1,2k 1,4k 1,6k

    1m

    10m

    100m

    1

    10

    [Hz]

    [m/s] Autospectrum(Response) - InputWorking : Input : Input : FFT Analyzer

    0 200 400 600 800 1k 1,2k 1,4k 1,6k

    1m

    10m

    100m

    1

    10

    [Hz]

    [m/s]

    Frequency Response H1(Response,Excitation) - Input (Magnitude)Working : Input : Input : FFT Analyzer

    0 200 400 600 800 1k 1,2k 1,4k 1,6k

    100m

    10

    [Hz]

    [(m/s)/N]Frequency Response H1(Response,Excitation) - Input (Magnitude)Working : Input : Input : FFT Analyzer

    0 200 400 600 800 1k 1,2k 1,4k 1,6k

    100m

    10

    [Hz]

    [(m/s)/N]

    Time Domain

    Impulse Response h1(Response,Excitation) - Input (Real Part)Working : Input : Input : FFT Analyzer

    0 40m 80m 120m 160m 200m 240m

    -2k

    -1k

    0

    1k

    2k

    [s]

    [(m/s)/N/s]Impulse Response h1(Response,Excitation) - Input (Real Part)Working : Input : Input : FFT Analyzer

    0 40m 80m 120m 160m 200m 240m

    -2k

    -1k

    0

    1k

    2k

    [s]

    [(m/s)/N/s]InverseFFT

    ExcitationResponse

    ForceMotion

    InputOutputH ===

  • Modal Analysis 21

    Hammer Test on Free-free Beam

    Frequency

    Distan

    ceAmplitude First

    Mode

    Beam

    SecondMode Third

    Mode

    ForceForceForceForceForceForceForceForceForceForceForce

    Roving hammer method:z Response measured at one point z Excitation of the structure at a number of points

    by hammer with force transducer

    Frequency Domain View

    z Modes of structure identifiedz FRFs between excitation points and measurement point calculated

    Press anywhereto advance animation

    ForceMo

    dal

    Domain

    View

    Acceleration

  • Modal Analysis 22

    Measurement of FRF Matrix (SISO)

    One rowz One Roving Excitationz One Fixed Response (reference)

    SISO

    =

    X1 H11 H12 H13 ...H1n F1X2 H21 H22 H23 ...H2n F2X3 H31 H32 H33...H 3n F3: : :Xn Hn1 Hn2 Hn3...Hnn Fn

  • Modal Analysis 23

    Measurement of FRF Matrix (SIMO)

    More rowsz One Roving Excitationz Multiple Fixed Responses (references)

    SIMO

    =

    X1 H11 H12 H13 ...H1n F1X2 H21 H22 H23 ...H2n F2X3 H31 H32 H33...H 3n F3: : :Xn Hn1 Hn2 Hn3...Hnn Fn

  • Modal Analysis 24

    z FRFs between excitation point and measurement points calculated

    Whitenoiseexcitation

    Shaker Test on Free-free Beam

    Frequency

    Distan

    ceAmplitude First

    Mode SecondMode

    ThirdMode

    BeamAcceleration

    Press anywhereto advance animation

    Modal

    Domain

    View

    Frequency Domain View Force

    Shaker method:z Excitation of the structure at one point

    by shaker with force transducerz Response measured at a number of points

    z Modes of structure identified

  • Modal Analysis 25

    Measurement of FRF Matrix (Shaker SIMO)

    One columnz Single Fixed Excitation (reference)

    z Single Roving Response SISOor

    z Multiple (Roving) Responses SIMOMultiple-Output: Optimize data consistency

    X1 H11 H12 H13 ...H1n F1X2 H21 H22 H23 ...H2n F2X3 H31 H32 H33...H 3n F3: : :Xn Hn1 Hn2 Hn3...Hnn Fn

    =

  • Modal Analysis 26

    Why Multiple-Input and Multiple-Output ?z Multiple-Input: For large and/or complex structures more

    shakers are required in order to: get the excitation energy sufficiently distributedand avoid non-linear behaviour

    z Multiple-Output: Measure outputs at the same time in order to optimize data consistency

    i.e. MIMO

  • Modal Analysis 27

    Situations needing MIMO

    z One row or one column is not sufficient for determination of all modes in following situations:

    More modes at the same frequency (repeated roots), e.g. symmetrical structures

    Complex structures having local modes, i.e. reference DOF with modal deflection for all modes is not available

    In both cases more columns or more rows have to be measured - i.e. polyreference.

    Solutions: Impact Hammer excitation with more response DOFs One shaker moved to different reference DOFs MIMO

  • Modal Analysis 28

    Measurement of FRF Matrix (MIMO)

    =

    More columnsz Multiple Fixed Excitations (references)

    z Single Roving Response MISOor

    z Multiple (Roving) Responses MIMO

    X1 H11 H12 H13 ...H1n F1X2 H21 H22 H23 ...H2n F2X3 H31 H32 H33...H 3n F3: : :Xn Hn1 Hn2 Hn3...Hnn Fn

  • Modal Analysis 29

    Modal Analysis (classic): FRF = Response/Excitation

    Output

    Frequency Response H1(Response,Excitation) - Input (Magnitude)Frequency Response H1(Response,Excitation) - Input (Magnitude)

    Working : Input : Input : FFT Analyzer

    0 200 400 600 800 1k 1,2k 1,4k 1,6k

    100m

    10

    [Hz]

    [(m/s)/N]Working : Input : Input : FFT Analyzer

    0 200 400 600 800 1k 1,2k 1,4k 1,6k

    100m

    10

    [Hz]

    [(m/s)/N]

    Autospectrum(Excitation) - Input

    Working : Input : Input : FFT Analyzer

    0 200 400 600 800 1k 1,2k 1,4k 1,6k

    100u

    1m

    10m

    100m

    1

    [Hz]

    [N] Autospectrum(Excitation) - Input

    Working : Input : Input : FFT Analyzer

    0 200 400 600 800 1k 1,2k 1,4k 1,6k

    100u

    1m

    10m

    100m

    1

    [Hz]

    [N]

    Autospectrum(Response) - Input

    Working : Input : Input : FFT Analyzer

    0 200 400 600 800 1k 1,2k 1,4k 1,6k

    1m

    10m

    100m

    1

    10

    [Hz]

    [m/s] Autospectrum(Response) - Input

    Working : Input : Input : FFT Analyzer

    0 200 400 600 800 1k 1,2k 1,4k 1,6k

    1m

    10m

    100m

    1

    10

    [Hz]

    [m/s]

    Time(Excitation) - InputWorking : Input : Input : FFT Analyzer

    0 40m 80m 120m 160m 200m 240m

    -200

    -100

    0

    100

    200

    [s]

    [N] Time(Excitation) - InputWorking : Input : Input : FFT Analyzer

    0 40m 80m 120m 160m 200m 240m

    -200

    -100

    0

    100

    200

    [s]

    [N]

    Time(Response) - InputWorking : Input : Input : FFT Analyzer

    0 40m 80m 120m 160m 200m 240m

    -80

    -40

    0

    40

    80

    [s]

    [m/s] Time(Response) - InputWorking : Input : Input : FFT Analyzer

    0 40m 80m 120m 160m 200m 240m

    -80

    -40

    0

    40

    80

    [s]

    [m/s]

    FFT

    FFT

    Impulse Response h1(Response,Excitation) - Input (Real Part)Working : Input : Input : FFT Analyzer

    0 40m 80m 120m 160m 200m 240m

    -2k

    -1k

    0

    1k

    2k

    [s]

    [(m/s)/N/s]Impulse Response h1(Response,Excitation) - Input (Real Part)Working : Input : Input : FFT Analyzer

    0 40m 80m 120m 160m 200m 240m

    -2k

    -1k

    0

    1k

    2k

    [s]

    [(m/s)/N/s]

    InverseFFT

    Input

    ExcitationResponse

    ForceVibration

    InputOutput

    H() = = =

    FrequencyResponseFunction

    Frequency Domain

    ImpulseResponseFunction

    Time Domain

    Operational Modal Analysis (OMA): Response only!

    Natural Excitation

  • Modal Analysis 30

    SDOF and MDOF Models

    Different Modal Analysis Techniques

    Exciting a Structure

    Measuring Data Correctly

    Modal Analysis Post Processing

    f(t)

    x(t)

    kc

    m

    = ++ + +

  • Modal Analysis 31

    The Eternal Question in Modal

    F1a

    F2

  • Modal Analysis 32

    Impact Excitation

    H11() H12() H15()

    Measuring one row of the FRF matrix by moving impact position

    LAN

    # 1# 2

    # 3# 4

    # 5Accelerometer

    ForceTransducer Impact

    Hammer

  • Modal Analysis 33

    Impact Excitation

    t

    GAA(f)

    z Magnitude and pulse duration depends on: Weight of hammer Hammer tip (steel, plastic or rubber) Dynamic characteristics of surface Velocity at impact

    z Frequency bandwidth inversely proportional to the pulse durationa(t)

    a(t)

    ft

    12

    12

  • Modal Analysis 34

    Weighting Functions for Impact Excitation

    z How to select shift and length for transient and exponential windows:

    z Leakage due to exponential time weighting on response signal is well defined and therefore correction of the measured damping value is often possible

    Transient weighting of the input signal

    Criteria

    Exponential weighting of the output signal

  • Modal Analysis 35

    Compensation for Exponential Weighting

    With exponential weighting of the output signal, the measured time constant will be too short and the calculated decay constant and damping ratio therefore too large shift Length =

    Record length, T

    Original signal

    Weighted signal

    Window function

    Time

    1b(t)

    W

    Correction of decay constant and damping ratio :

    Wm =

    WW

    1=

    Wm0

    W

    0

    m

    0

    =

    ==

    Correctvalue

    Measuredvalue

  • Modal Analysis 36

    3 lb Hand Sledge

    Building and bridges

    12 lb Sledge

    Large shafts and larger machine toolsCar framed and machine tools

    1 lb hammer

    ComponentsGeneral Purpose, 0.3 lb

    Hard-drives, circuit boards, turbine blades

    Mini Hammer

    ApplicationDescription

    Range of hammers

  • Modal Analysis 37

    Impact hammer excitation

    z Advantages: Speed No fixturing No variable mass loading Portable and highly suitable for

    field work relatively inexpensive

    z Disadvantages High crest factor means

    possibility of driving structure into non-linear behavior

    High peak force needed for large structures means possibility of local damage!

    Highly deterministic signal means no linear approximation

    z Conclusion Best suited for field work Useful for determining shaker and support locations

  • Modal Analysis 38

    Shaker Excitation

    H11() H21() H51()

    Measuring one column of the FRF matrix by moving response transducer

    # 1# 2

    # 3# 4

    # 5Accelerometer

    ForceTransducer

    VibrationExciter PowerAmplifier

    LAN

  • Modal Analysis 39

    Attachment of Transducers and Shaker

    Accelerometer mounting:z Studz Cementz Waxz (Magnet)

    Force Transducer and Shaker:z Studz Stinger (Connection Rod)

    Properties of StingerAxial Stiffness: HighBending Stiffness: Low

    Accelerometer

    ShakerFa

    ForceTransducer

    Advantages of Stinger: z No Moment Excitationz No Rotational Inertia Loadingz Protection of Shakerz Protection of Transducerz Helps positioning of Shaker

  • Modal Analysis 40

    Connection of Exciter and Structure

    ExciterSlenderstinger

    Measured structure

    Accelerometer

    Force Transducer

    Tip mass, mShaker/Hammer

    mass, M

    FsFm

    Piezoelectricmaterial

    Structure

    MMmFF ms

    +=

    XMFm &&=X)Mm(Fs &&+=

    Force and acceleration measurements unaffected by stinger compliance, but ...Minor mass correction required to determine actual excitation

  • Modal Analysis 41

    Shaker Reaction Force

    Reaction byexternal support

    Reaction byexciter inertia

    Example of animproper

    arrangementStructure

    Suspension

    ExciterSupport

    StructureSuspension

    ExciterSuspension

  • Modal Analysis 42

    Sine Excitationa(t)

    Time

    A

    Crest factor 2RMSA ==

    z For study of non-linearities, e.g. harmonic distortion

    z For broadband excitation: Sine wave swept slowly through

    the frequency range of interest Quasi-stationary condition

    A(f1)

    B(f1)

    RMS

  • Modal Analysis 43

    Swept Sine Excitation

    Advantagesz Low Crest Factorz High Signal/Noise ratioz Input force well controlledz Study of non-linearities

    possible

    Disadvantagesz Very slowz No linear approximation of

    non-linear system

  • Modal Analysis 44

    Random Excitation

    A(f1)Freq.

    Time

    a(t)

    B(f1)GAA(f), N = 1 GAA(f), N = 10

    Random variation of amplitude and phase Averaging will give optimum linear

    estimate in case of non-linearities

    Freq.

    SystemOutput

    SystemInput

  • Modal Analysis 45

    Random Excitationz Random signal:

    Characterized by power spectral density (GAA) and amplitude probability density (p(a))

    a(t)

    p(a)

    z Can be band limited according to frequency range of interestGAA(f) GAA(f)Baseband Zoom

    Time

    z Signal not periodic in analysis time Leakage in spectral estimatesFrequency

    rangeFrequency

    range

    Freq. Freq.

  • Modal Analysis 46

    Random Excitation

    Advantagesz Best linear approximation of systemz Zoomz Fair Crest Factorz Fair Signal/Noise ratio

    Disadvantagesz Leakagez Averaging needed (slower)

  • Modal Analysis 47

    Burst Randomz Characteristics of Burst Random signal :

    Gives best linear approximation of nonlinear system Works with zoom

    a(t)

    Time

    Advantagesz Best linear approximation of systemz No leakage (if rectangular time weighting can be used)z Relatively fast

    Disadvantagesz Signal/noise and crest factor not optimumz Special time weighting might be required

  • Modal Analysis 48

    Pseudo Random Excitation

    a(t)

    z Pseudo random signal: Block of a random signal repeated every T

    Time

    A(f1)

    B(f1)

    T T T T

    z Time period equal to record length T Line spectrum coinciding with analyzer lines No averaging of non-linearities

    Freq.

    GAA(f), N = 1 GAA(f), N = 10

    Freq.

    SystemOutput

    System Input

  • Modal Analysis 49

    Pseudo Random Excitation

    T T T T

    a(t)

    Timep(a)

    Freq. range Freq. range

    GAA(f) GAA(f)Baseband Zoom

    z Can be band limited according to frequency range of interest

    Time period equal to T No leakage if Rectangular weighting is used

    z Pseudo random signal: Characterized by power/RMS (GAA) and amplitude probability density (p(a))

    Freq. Freq.

  • Modal Analysis 50

    Pseudo Random Excitation

    Advantagesz No leakagez Fastz Zoomz Fair crest factorz Fair Signal/Noise ratio

    Disadvantagesz No linear approximation of

    non-linear system

  • Modal Analysis 51

    Multisine (Chirp)For sine sweep repeated every time record, Tr

    TrA special type of pseudo random signal where the crest factor has been minimized (< 2)

    It has the advantages and disadvantages of the normal pseudo random signal but with a lower crest factor

    Additional Advantages: Ideal shape of spectrum: The spectrum is a flat

    magnitude spectrum, and the phase spectrum is smoothApplications:

    Measurement of structures with non-linear behaviour

    Time

  • Modal Analysis 52

    Pseudo-random signal changing with time:

    Analysed time data:(steady-state response)

    transient responsesteady-state response

    A B C

    A A A B B B C CC

    T T T

    Periodic RandomA combined random and pseudo-random signal giving an excitation signal featuring:

    No leakage in analysis Best linear approximation of system

    Disadvantage: z The test time is longer than the test time

    using pseudo-random or random signal

  • Modal Analysis 53

    Periodic Pulse

    z Leakage can be avoided using rectangular time weightingz Transient and exponential time weighting can be used to increase

    Signal/Noise ratioz Gating of reflections with transient time weightingz Effects of non-linearities are not averaged outz The signal has a high crest factor

    Rectangular, Hanning, or Gaussian pulse with user definable t repeated with a user definable interval, T

    The line spectrum for a Rectangular pulse has a shaped envelope curve

    sin xx

    t T Time

    1/ t Frequency

    Special case of pseudo random signal

  • Modal Analysis 54

    Periodic Pulse

    Advantages

    z Fastz No leakage

    (Only with rectangular weighting)

    z Gating of reflections(Transient time weighting)

    z Excitation spectrum follows frequency span in baseband

    z Easy to implement

    Disadvantages

    z No linear approximation of non-linear system

    z High Crest Factorz High peak level might excite

    non-linearitiesz No Zoomz Special time weighting might

    be required to increase Signal/Noise Ratio . This can also introduce leakage

  • Modal Analysis 55

    Guidelines for Choice of Excitation Technique

    Swept sine excitationz For study of non-linearities:

    Random excitationz For slightly non-linear system:

    Pseudo random excitationz For perfectly linear system:

    Impact excitationz For field measurements:

    Random impact excitationz For high resolution field measurements:

  • Modal Analysis 56

    SDOF and MDOF Models

    Different Modal Analysis Techniques

    Exciting a Structure

    Measuring Data Correctly

    Modal Analysis Post Processing

    f(t)

    x(t)

    kc

    m

    = ++ + +

  • Modal Analysis 57

    Garbage In = Garbage Out!A state-of-the Art Assessment of Mobility Measurement Techniques

    Result for the Mid Range Structure (30 - 3000 Hz) D.J. Ewins and J. Griffin

    Feb. 1981

    Transfer MobilityCentral Decade

    Transfer MobilityExpanded

    Frequency Frequency

  • Modal Analysis 58

    Plan Your Test Before Hand!

    1. Select Appropriate Excitation Hammer, Shaker, or OMA?

    2. Setup FFT Analyzer correctly Frequency Range, Resolution, Averaging, Windowing Remember: FFT Analyzer is a BLOCK ANALYZER!

    3. Good Distribution of Measurement Points Ensure enough points are measured to see all modes of interest Beware of spatial aliasing

    4. Physical Setup Accelerometer mounting is CRITICAL! Uni-axial vs. Triaxial Make sure DOF orientation is correct Mount device under test...mounting will affect measurement! Calibrate system

  • Modal Analysis 59

    Where Should Excitation Be Applied?

    Driving Point Measurement

    Transfer Measurement

    i = j

    i j

    i

    j

    j

    i

    "j"Excitation"i"sponseRe

    FXH

    j

    iij ==

    12

    2121111 FHFHX +=

    =

    2

    1

    2221

    1211

    2

    1

    FF

    HHHH

    XX

    }{ [ ]{ }FHX =

    2221212 FHFHX +=

  • Modal Analysis 60

    Check of Driving Point Measurement

    z All peaks in

    )f(F)f(XImand

    )f(F)f(XRe,

    )f(F)f(XIm

    &&&

    z An anti-resonance in Mag [Hij]must be found between every pair of resonances

    z Phase fluctuations must be within 180

    Im [Hij]

    Mag [Hij]

    Phase [Hij]

  • Modal Analysis 61

    Driving Point (DP) MeasurementThe quality of the DP-measurement is very important, as the DP-residues are used for the scaling of the Modal Model

    DP- Considerations: Residues for all modes

    must be estimated accurately from a single measurement

    DP- Problems: Highest modal coupling, as all

    modes are in phase Highest residual effect from

    rigid body modes

    m Aij

    Re Aij

    Log MagAij

    =

    Without rigidbody modes

    Measured

  • Modal Analysis 62

    Tests for Validity of Data: Coherence

    Coherence)f(G)f(G

    )f(G)f(

    XXFF

    2FX2 =

    Measures how much energy put in to the system caused the response

    The closer to 1 the more coherent Less than 0.75 is bordering on poor coherence

  • Modal Analysis 63

    Reasons for Low Coherence

    Difficult measurements:z Noise in measured output signalz Noise in measured input signalz Other inputs not correlated with measured input signal

    Bad measurements:z Leakagez Time varying systemsz Non-linearities of systemz DOF-jitterz Propagation time not compensated for

  • Modal Analysis 64

    Tests for Validity of Data: Linearity

    Linearity X1 = HF1X2 = HF2

    X1+X2 = H(F1 + F2)

    aX1 = H(a F1)

    More force going in to the system will equate to

    F() X()

    more response coming out Since FRF is a ratio the magnitude should be

    the same regardless of input force

    H()

  • Modal Analysis 65

    Tips and Tricks for Best Resultsz Verify measurement chain integrity prior to test:

    Transducer calibration Mass Ratio calibration

    z Verify suitability of input and output transducers: Operating ranges (frequency, dynamic range, phase response) Mass loading of accelerometers Accelerometer mounting Sensitivity to environmental effects Stability

    z Verify suitability of test set-up: Transducer positioning and alignment Pre-test: rattling, boundary conditions, rigid body modes, signal-to-noise ratio,

    linear approximation, excitation signal, repeated roots, Maxwell reciprocity, force measurement, exciter-input transducer-stinger-structure connection

    Quality FRF measurements are the foundation of experimental modal analysis!

  • Modal Analysis 66

    SDOF and MDOF Models

    Different Modal Analysis Techniques

    Exciting a Structure

    Measuring Data Correctly

    Modal Analysis Post Processing

    f(t)

    x(t)

    kc

    m

    = ++ + +

  • Modal Analysis 67

    From Testing to Analysis

    MeasuredFRF

    Curve Fitting(Pattern Recognition)

    Modal Analysis

    )f(H

    Frequency

    )f(H

    Frequency

  • Modal Analysis 68

    From Testing to Analysis

    Modal Analysis

    )f(H

    Frequency

    f(t)x(t)

    kc

    m

    f(t)x(t)

    kc

    m

    f(t)x(t)

    kc

    mSDOF Models

  • Modal Analysis 69

    Mode Characterizations

    All Modes Can Be Characterized By:1. Resonant Frequency

    2. Modal Damping

    3. Mode Shape

  • Modal Analysis 70

    Modal Analysis Step by Step Process1. Visually Inspect Data

    Look for obvious modes in FRF Inspect ALL FRFssometimes modes will show up in one

    FRF but not another (nodes) Use Imaginary part and coherence for verification Sum magnitudes of all measurements for clues

    2. Select Curve Fitter Lightly coupled modes: SDOF techniques Heavily coupled modes: MDOF techniques Stable measurements: Global technique Unstable measurements: Local technique MIMO measurement: Poly reference techniques

    3. Analysis Use more than 1 curve fitter to see if they agree Pay attention to Residue calculations Do mode shapes make sense?

  • Modal Analysis 71

    Modal Analysis Inspect Data1. Visually Inspect Data

    Look for obvious modes in FRF Inspect ALL FRFssometimes modes will show up in one

    FRF but not another (nodes) Use Imaginary part and coherence for verification Sum magnitudes of all measurements for clues

    2. Select Curve Fitter Lightly coupled modes: SDOF techniques Heavily coupled modes: MDOF techniques Stable measurements: Global technique Unstable measurements: Local technique MIMO measurement: Poly reference techniques

    3. Analysis Use more than 1 curve fitter to see if they agree Pay attention to Residue calculations Do mode shapes make sense?

  • Modal Analysis 72

    Modal Analysis Curve Fitting1. Visually Inspect Data

    Look for obvious modes in FRF Inspect ALL FRFssometimes modes will show up in one

    FRF but not another (nodes) Use Imaginary part and coherence for verification Sum magnitudes of all measurements for clues

    2. Select Curve Fitter Lightly coupled modes: SDOF techniques Heavily coupled modes: MDOF techniques Stable measurements: Global technique Unstable measurements: Local technique MIMO measurement: Poly reference techniques

    3. Analysis Use more than 1 curve fitter to see if they agree Pay attention to Residue calculations Do mode shapes make sense?

  • Modal Analysis 73

    How Does Curve Fitting Work?z Curve Fitting is the process of estimating the Modal

    Parameters from the measurements

    |H|

    2

    R/

    d0

    -180Phase

    Frequency

    z Find the resonant frequency

    Frequency where small excitation causes a large response

    z Find the damping What is the Q of the

    peak?

    z Find the residue Essentially the area

    under the curve

  • Modal Analysis 74

    *

    *

    r

    ijr

    r r

    ijr

    rijrij pj

    Rpj

    RH)(H +==

    Residues are Directly Related to Mode Shapes!

    z Residues express the strength of a mode for each measured FRF

    z Therefore they are related to mode shape at each measured point!

    Frequency

    Distan

    ceAmplitude First

    Mode

    Beam

    SecondMode

    ForceForceForceForceForceForceForceForceForceForceForceForce

    Acceleration

    ThirdMode

  • Modal Analysis 75

    SDOF vs. MDOF Curve Fitters

    z Use SDOF methods on LIGHTLY COUPLED modes

    z Use MDOF methods on HEAVILY COUPLED modes

    z You can combine SDOF and MDOF techniques!

    SDOF(light coupling)

    |H|

    MDOF(heavy coupling)

  • Modal Analysis 76

    Local vs. Global Curve Fitting

    z Local means that resonances, damping, and residues are calculated for each FRF firstthen combined for curve fitting

    z Global means that resonances, damping, and residues are calculated across all FRFs

    |H|

  • Modal Analysis 77

    Modal Analysis Analyse Results1. Visually Inspect Data

    Look for obvious modes in FRF Inspect ALL FRFssometimes modes will show up in one

    FRF but not another (nodes) Use Imaginary part and coherence for verification Sum magnitudes of all measurements for clues

    2. Select Curve Fitter Lightly coupled modes: SDOF techniques Heavily coupled modes: MDOF techniques Stable measurements: Global technique Unstable measurements: Local technique MIMO measurement: Poly reference techniques

    3. Analysis Use more than one curve fitter to see if they agree Pay attention to Residue calculations Do mode shapes make sense?

  • Modal Analysis 78

    Which Curve Fitter Should Be Used?Frequency Response Function

    Hij ()

    Real Imaginary

    Nyquist Log Magnitude

  • Modal Analysis 79

    Which Curve Fitter Should Be Used?

    Real Imaginary

    Nyquist Log Magnitude

    Frequency Response FunctionHij ()

  • Modal Analysis 80

    Modal Analysis and Beyond

    Hardware ModificationResonance Specification

    Simulate RealWorld Response

    Dynamic Modelbased on

    Modal Parameters

    ExperimentalModal Analysis

    StructuralModification

    ResponseSimulation

    F

  • Modal Analysis 81

    Conclusionz All Physical Structures can be characterized by the simple

    SDOF model

    z Frequency Response Functions are the best way to measure resonances

    z There are three modal techniques available today: Hammer Testing, Shaker Testing, and Operational Modal

    z Planning and proper setup before you test can save time and effortand ensure accuracy while minimizing erroneous results

    z There are many curve fitting techniques available, try to use the one that best fits your application

  • Modal Analysis 82

    Literature for Further Readingz Structural Testing Part 1: Mechanical Mobility Measurements

    Brel & Kjr Primer

    z Structural Testing Part 2: Modal Analysis and SimulationBrel & Kjr Primer

    z Modal Testing: Theory, Practice, and Application, 2nd Edition by D.J. EwinResearch Studies Press Ltd.

    z Dual Channel FFT Analysis (Part 1)Brel & Kjr Technical Review # 1 1984

    z Dual Channel FFT Analysis (Part 1)Brel & Kjr Technical Review # 2 1984

  • Modal Analysis 83

    Appendix: Damping Parameters

    3 dB bandwidth =

    22f dB3

    Loss factor0

    dB3

    0

    db3

    ff

    Q1

    ===

    Damping ratio0

    Bd3

    0

    dB3

    2f2f

    2 ===

    Decay constant

    Quality factordB3

    0

    dB3

    0

    ffQ

    ==

    2f dB3dB30

    ===

    = 2dB3,

  • Modal Analysis 84

    Appendix: Damping Parameters

    ( )tsineR2)t(h dt = , where the Decay constant is given by e-tThe Envelope is given by magnitude of analytic h(t):

    The time constant, , is determined by the time it takes for the amplitude to decay a factor of e = 2,72or 10 log (e2) = 8.7 dB

    )t(h~)t(h)t(h 22 +=

    Decay constant

    Time constant :

    Damping ratio

    Loss factor

    Quality

    =1

    =1

    ==

    00 f21

    == 0f12

    == 0f1Q

    h(t)

    Time

    te

    ExperimentalModal AnalysisSimplest Form of Vibrating SystemMass and SpringMass, Spring and DamperBasic SDOF Model SDOF Models Time and Frequency DomainModal MatrixMDOF ModelWhy Bother with Modal Models?Definition of Frequency Response FunctionBenefits of Frequency Response FunctionDifferent Forms of an FRFAlternative EstimatorsWhich FRF Estimator Should You Use?Three Types of Modal AnalysisDifferent Types of Modal Analysis (Pros)Different Types of Modal Analysis (Cons)Hammer Test on Free-free BeamMeasurement of FRF Matrix (SISO)Measurement of FRF Matrix (SIMO)Shaker Test on Free-free BeamMeasurement of FRF Matrix (Shaker SIMO)Why Multiple-Input and Multiple-Output ?Situations needing MIMOMeasurement of FRF Matrix (MIMO)The Eternal Question in ModalImpact ExcitationImpact ExcitationWeighting Functions for Impact ExcitationCompensation for Exponential WeightingRange of hammersImpact hammer excitationShaker ExcitationAttachment of Transducers and ShakerConnection of Exciter and StructureShaker Reaction ForceSine ExcitationSwept Sine ExcitationRandom ExcitationRandom ExcitationRandom ExcitationBurst RandomPseudo Random ExcitationPseudo Random ExcitationPseudo Random ExcitationMultisine (Chirp)Periodic RandomPeriodic PulsePeriodic PulseGuidelines for Choice of Excitation TechniqueGarbage In = Garbage Out!Plan Your Test Before Hand!Where Should Excitation Be Applied?Check of Driving Point MeasurementDriving Point (DP) MeasurementTests for Validity of Data: CoherenceReasons for Low CoherenceTests for Validity of Data: LinearityTips and Tricks for Best ResultsFrom Testing to AnalysisFrom Testing to AnalysisMode CharacterizationsModal Analysis Step by Step ProcessModal Analysis Inspect DataModal Analysis Curve FittingHow Does Curve Fitting Work?Residues are Directly Related to Mode Shapes!SDOF vs. MDOF Curve FittersLocal vs. Global Curve FittingModal Analysis Analyse ResultsWhich Curve Fitter Should Be Used?Which Curve Fitter Should Be Used?Modal Analysis and BeyondConclusionLiterature for Further ReadingAppendix: Damping ParametersAppendix: Damping Parameters