Experimental Modal Analysis

Transcription

1 Experimental Modal Analysis Modal Analysis 1 Shanghai Jiaotong University 2006

2 m f(t) x(t) SDOF and MDOF Models c k Different Modal Analysis Techniques Exciting a Structure = Measuring Data Correctly Modal Analysis Post Processing Modal Analysis 2 Shanghai Jiaotong University 2006

3 Simplest Form of Vibrating System Displacement D d = D sin n t Displacement Time T 1 T Frequency m Period, T n in [sec] k Frequency, f n = n = 2 f n = 1 T n k m in [Hz = 1 /sec] Modal Analysis 3 Shanghai Jiaotong University 2006

4 Mass and Spring time n 2f n k m m 1 m 1 m Increasing mass reduces frequency Modal Analysis 4 Shanghai Jiaotong University 2006

5 Mass, Spring and Damper time Increasing damping reduces the amplitude m k c 1 + c 2 Modal Analysis 5 Shanghai Jiaotong University 2006

6 Basic SDOF Model m f(t) x(t) c k M = mass (force/acc.) C = damping (force/vel.) K = stiffness (force/disp.) M x( t) Cx ( t) Kx( t) f ( t) x(t) x(t) x(t) f(t) Acceleration Vector Velocity Vector Displacement Vector Applied force Vector Modal Analysis 6 Shanghai Jiaotong University 2006

7 SDOF Models Time and Frequency Domain F() H() X() m f(t) x(t) H() 1 k 1 2 m 1 c c k H() 0º 90º 180º 0 = k/m f ( t) mx ( t) cx ( t) kx( t) X ( ) H( ) 2 F( ) m 1 jc k Modal Analysis 7 Shanghai Jiaotong University 2006

8 Modal Analysis 8 Shanghai Jiaotong University 2006 Modal Matrix Modal Model (Freq. Domain) F H H H H H H H X X X X nn n n n X 1 X 2 X 3 X 4 F 3 H 21 H 22

9 MDOF Model Magnitude 1+2 d 1 + d m d 1 Frequency df Phase Frequency Modal Analysis 9 Shanghai Jiaotong University 2006

10 Why Bother with Modal Models? Physical Coordinates = C H A OS Modal Space = Simplicity Rotor 1 1 q 1 Bearing Bearing q 2 Foundation q Modal Analysis 10 Shanghai Jiaotong University 2006

11 Definition of Frequency Response Function F(f) H(f) X(f) F H X f f f H H(f) X(f) F(f) H(f) is the system Frequency Response Function F(f) is the Fourier Transform of the Input f(t) X(f) is the Fourier Transform of the Output x(t) Modal Analysis 11 Shanghai Jiaotong University 2006

12 Benefits of Frequency Response Function F(f) H(f) X(f) Frequency Response Functions are properties of linear dynamic systems They are independent of the Excitation Function Excitation can be a Periodic, Random or Transient function of time 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 12 Shanghai Jiaotong University 2006

13 Different Forms of an FRF Compliance (displacement / force) Mobility (velocity / force) Inertance or Receptance (acceleration / force) Dynamic stiffness (force / displacement) Impedance (force / velocity) Dynamic mass (force /acceleration) Modal Analysis 13 Shanghai Jiaotong University 2006

14 Alternative Estimators F(f) H(f) X(f) H( f ) X ( f ) F( f ) H ( f ) G G ( f ) ( ) FX 1 FF f H ( f ) G G ( f ) ( ) XX 2 XF f H GXX GFX ( f H1 H G G 3 ) FF FX 2 2 G 2 FX G G* FX FX ( f ) G FF G XX G FF G XX H H 1 2 Modal Analysis 14 Shanghai Jiaotong University 2006

15 Modal Analysis 15 Shanghai Jiaotong University 2006 Alternative Estimators ) ( ) ( ) ( 1 f G f G f H FF FX X(f) F(f) H(f) + N(f) SISO: MIMO: s n N s p F p n H s n X H F N H F F H H F X p n nf G p p ff G p n H p n nf G 1 1 p n ff G p n xf G p n H

16 Alternative Estimators F(f) + H(f) X(f) M(f) SISO: H ( f ) G G ( f ) ( ) XX 2 XF f MIMO: X X ns G xx H 2 X H H nn nn n p H H G xx F ps M n s H H F X M X n p nn G fx pn G fx 1 nn G mx pn Modal Analysis 16 Shanghai Jiaotong University 2006

17 Which FRF Estimator Should You Use? Accuracy Definitions: H ( f ) G G ( f ) ( ) FX 1 FF f H ( f ) G G ( f ) ( ) XX 2 XF f H3( f ) G G XX FF G G FX FX Accuracy for systems with: H 1 H 2 H 3 Input noise - Best - Output noise Best - - Input + output noise - - Best Peaks (leakage) - Best - Valleys (leakage) Best - - User can choose H 1, H 2 or H 3 after measurement Modal Analysis 17 Shanghai Jiaotong University 2006

18 m f(t) x(t) SDOF and MDOF Models c k Different Modal Analysis Techniques Exciting a Structure = Measuring Data Correctly Modal Analysis Post Processing Modal Analysis 18 Shanghai Jiaotong University 2006

19 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 19 Shanghai Jiaotong University 2006

20 Different Types of Modal Analysis (Pros) Hammer Testing Quick and easy Typically Inexpensive Can perform poor man modal as well as full modal Shaker Testing More repeatable than hammer testing Many types of input available Can be used for MIMO analysis 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 20 Shanghai Jiaotong University 2006

21 Different Types of Modal Analysis (Cons) 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 Shaker Testing More difficult test setup (stingers, exciter, etc.) Usually more equipment and channels needed Skilled operator(s) needed Operational Modal Analysis Unscaled modal model Excitation assumed to cover frequency range of interest Long time histories sometimes required Computationally intensive Modal Analysis 21 Shanghai Jiaotong University 2006

22 Frequency Response Function [m/s 瞉 Time(Response) - Input Working : Input : Input : FFT A nalyzer m 80m 120m 160m 200m 240m [s] Output FFT H Output Input Motion Force Response Excitation Input [m/s 瞉 A utospectrum (Respo nse) - Input Working : Input : Input : FFT A nalyzer m 10m 1m k 1,2k 1,4k 1,6k [Hz] [N] A utospectrum (Excit atio n) - Input Working : Input : Input : FFT A nalyzer 1 100m [(m/ Frequency s?/n] Respo nse H1(Response,Excitation) - Input (M agnitud Working : Input : Input : FFT A nalyzer m k 1,2k 1,4k 1,6k [Hz] Inverse FFT [(m/ s?/n/s] Impulse Response h1(response,excitation) - Input (Real P art) Working : Input : Input : FFT A nalyzer 2k 1k 0-1k -2k 0 40m 80m 120m 160m 200m 240m [s] 10m 1m 100u k 1,2k 1,4k 1,6k [Hz] [N] 200 Time(Excitatio n) - Input Working : Input : Input : FFT A nalyzer FFT Frequency Domain Time Domain m 80m 120m 160m 200m 240m [s] Modal Analysis 22 Shanghai Jiaotong University 2006

23 Hammer Test on Free-free Beam Roving hammer method: Response measured at one point Excitation of the structure at a number of points by hammer with force transducer FRF s between excitation points and measurement point calculated Modes of structure identified Amplitude First Mode Second Mode Third Mode Beam Acceleration Force Force Force Force Force Force Force Force Force Force Force Force Press anywhere Modal Analysis 23 to advance animation Shanghai Jiaotong University 2006

24 Measurement of FRF Matrix (SISO) One row One Roving Excitation One Fixed Response (reference) SISO X 1 H 11 H 12 H 13...H 1n F 1 X 2 H 21 H 22 H 23...H 2n F 2 X 3 = H 31 H 32 H 33...H 3n F 3 : : : X n H n1 H n2 H n3...h nn F n Modal Analysis 24 Shanghai Jiaotong University 2006

25 Measurement of FRF Matrix (SIMO) More rows One Roving Excitation Multiple Fixed Responses (references) SIMO X 1 H 11 H 12 H 13...H 1n F 1 X 2 H 21 H 22 H 23...H 2n F 2 X 3 = H 31 H 32 H 33...H 3n F 3 : : : X n H n1 H n2 H n3...h nn F n Modal Analysis 25 Shanghai Jiaotong University 2006

26 Shaker Test on Free-free Beam Shaker method: Excitation of the structure at one point by shaker with force transducer Response measured at a number of points FRF s between excitation point and measurement points calculated Modes of structure identified Amplitude First Mode Second Mode Third Mode Beam Acceleration White noise excitation Force Modal Analysis 26 Press anywhere Shanghai Jiaotong University 2006 to advance animation

27 Measurement of FRF Matrix (Shaker SIMO) One column Single Fixed Excitation (reference) Single Roving Response SISO or Multiple (Roving) Responses SIMO Multiple-Output: Optimize data consistency X 1 H 11 H 12 H 13...H 1n F 1 X 2 H 21 H 22 H 23...H 2n F 2 X 3 = H 31 H 32 H 33...H 3n F 3 : : : X n H n1 H n2 H n3...h nn F n Modal Analysis 27 Shanghai Jiaotong University 2006

28 Why Multiple-Input and Multiple-Output? Multiple-Input: For large and/or complex structures more shakers are required in order to: get the excitation energy sufficiently distributed and avoid non-linear behaviour Multiple-Output: Measure outputs at the same time in order to optimize data consistency i.e. MIMO Modal Analysis 28 Shanghai Jiaotong University 2006

29 Situations needing MIMO 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 DOF s One shaker moved to different reference DOF s MIMO Modal Analysis 29 Shanghai Jiaotong University 2006

30 Measurement of FRF Matrix (MIMO) More columns Multiple Fixed Excitations (references) Single Roving Response MISO or Multiple (Roving) Responses MIMO X 1 H 11 H 12 H 13...H 1n F 1 X 2 H 21 H 22 H 23...H 2n F 2 X 3 = H 31 H 32 H 33...H 3n F 3 : : : X n H n1 H n2 H n3...h nn F n Modal Analysis 30 Shanghai Jiaotong University 2006

31 Operational Modal Analysis Modal (classic): Analysis FRF (OMA): = Response/Excitation only! [m/s²] 80 Time(Response) - Input Working : Input : Input : FFT Analyzer m 80m 120m 160m 200m 240m [s] FFT Frequency Domain Time Domain Input Output Natural Excitation [m/s²] Autospectrum(Response) - Input Working : Input : Input : FFT Analyzer m 10m 1m k 1,2k 1,4k 1,6k [Hz] [N] Autospectrum(Excitation) - Input Working : Input : Input : FFT Analyzer 1 100m 10m 1m 100u k 1,2k 1,4k 1,6k [Hz] [(m/s²)/n] Frequency Response H1(Response,Excitation) - Input (Magnitude) m Working : Input : Input : FFT Analyzer k 1,2k 1,4k 1,6k [Hz] Frequency Response Function Inverse FFT [(m/s²)/n/s] Impulse Response h1(response,excitation) - Input (Real Part) 2k 1k 0-1k -2k Working : Input : Input : FFT Analyzer 0 40m 80m 120m 160m 200m 240m [s] Impulse Response Function [N] Time(Excitation) - Input Working : Input : Input : FFT Analyzer m 80m 120m 160m 200m 240m [s] FFT Output Vibration H() = = = Input Force Response Excitation Modal Analysis 31 Shanghai Jiaotong University 2006

32 m f(t) x(t) SDOF and MDOF Models c k Different Modal Analysis Techniques Exciting a Structure = Measuring Data Correctly Modal Analysis Post Processing Modal Analysis 32 Shanghai Jiaotong University 2006

33 The Eternal Question in Modal a F 1 F 2 Modal Analysis 33 Shanghai Jiaotong University 2006

34 Impact Excitation Measuring one row of the FRF matrix by moving impact position Accelerometer # 1 # 2 # 3 # 4 # 5 H 11 () H 12 () H 15 () LAN Force Transducer Impact Hammer Modal Analysis 34 Shanghai Jiaotong University 2006

35 Impact Excitation a(t) t Magnitude and pulse duration depends on: Weight of hammer Hammer tip (steel, plastic or rubber) Dynamic characteristics of surface Velocity at impact Frequency bandwidth inversely proportional to the pulse duration a(t) 1 G AA (f) t f Modal Analysis 35 Shanghai Jiaotong University 2006

36 Weighting Functions for Impact Excitation How to select shift and length for transient and exponential windows: Criteria Transient weighting of the input signal Exponential weighting of the output signal Leakage due to exponential time weighting on response signal is well defined and therefore correction of the measured damping value is often possible Modal Analysis 36 Shanghai Jiaotong University 2006

37 Compensation for Exponential Weighting b(t) With exponential weighting of the 1 output signal, the measured time constant will be too short and the calculated decay constant and damping ratio therefore too large shift Length = Window function W Record length, T Original signal Weighted signal Time Correction of decay constant and damping ratio z: Correct value m W Measured value W 1 W z 0 m 0 W 0 z m z W Modal Analysis 37 Shanghai Jiaotong University 2006

38 Range of hammers Description Application 12 lb Sledge 3 lb Hand Sledge 1 lb hammer General Purpose, 0.3 lb Mini Hammer Building and bridges Large shafts and larger machine tools Car framed and machine tools Components Hard-drives, circuit boards, turbine blades Modal Analysis 38 Shanghai Jiaotong University 2006

39 Impact hammer excitation Advantages: Speed No fixturing No variable mass loading Portable and highly suitable for field work relatively inexpensive Conclusion Best suited for field work Useful for determining shaker and support locations 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 Modal Analysis 39 Shanghai Jiaotong University 2006

40 Shaker Excitation Measuring one column of the FRF matrix by moving response transducer Force Transducer Accelerometer # 1 # 2 # 3 # 4 # 5 H 11 () H 21 () H 51 () LAN Vibration Exciter Power Amplifier Modal Analysis 40 Shanghai Jiaotong University 2006

41 Attachment of Transducers and Shaker Accelerometer mounting: Stud Cement Wax (Magnet) Force Transducer and Shaker: Stud Stinger (Connection Rod) a Accelerometer F Advantages of Stinger: Force Transducer No Moment Excitation Properties of Stinger No Rotational Inertia Loading Protection of Shaker Protection of Transducer Helps positioning of Shaker Shaker Axial Stiffness: High Bending Stiffness: Low Modal Analysis 41 Shanghai Jiaotong University 2006

42 Connection of Exciter and Structure Force Transducer Measured structure Accelerometer Slender stinger Exciter Force and acceleration measurements unaffected by stinger compliance, but... Minor mass correction required to determine actual excitation F m Structure F s F s (m M) X Tip mass, m F m M X Piezoelectric material Shaker/Hammer mass, M F s F m m M M Modal Analysis 42 Shanghai Jiaotong University 2006

43 Shaker Reaction Force Reaction by external support Structure Suspension Reaction by exciter inertia Example of an improper arrangement Structure Suspension Exciter Support Exciter Suspension Modal Analysis 43 Shanghai Jiaotong University 2006

44 Sine Excitation A RMS a(t) Time Crest factor A 2 RMS For study of non-linearities, e.g. harmonic distortion B(f 1 ) For broadband excitation: Sine wave swept slowly through the frequency range of interest Quasi-stationary condition A(f 1 ) Modal Analysis 44 Shanghai Jiaotong University 2006

45 Swept Sine Excitation Advantages Low Crest Factor High Signal/Noise ratio Input force well controlled Study of non-linearities possible Disadvantages Very slow No linear approximation of non-linear system Modal Analysis 45 Shanghai Jiaotong University 2006

46 Random Excitation a(t) Time Random variation of amplitude and phase Averaging will give optimum linear estimate in case of non-linearities G AA (f), N = 1 G AA (f), N = 10 System Output B(f 1 ) Freq. Freq. A(f 1 ) System Input Modal Analysis 46 Shanghai Jiaotong University 2006

47 Random Excitation Random signal: Characterized by power spectral density (G AA ) and amplitude probability density (p(a)) a(t) p(a) Time Can be band limited according to frequency range of interest G AA (f) Baseband G AA (f) Zoom Frequency range Freq. Frequency range Freq. Signal not periodic in analysis time Leakage in spectral estimates Modal Analysis 47 Shanghai Jiaotong University 2006

48 Random Excitation Advantages Best linear approximation of system Zoom Fair Crest Factor Fair Signal/Noise ratio Disadvantages Leakage Averaging needed (slower) Modal Analysis 48 Shanghai Jiaotong University 2006

49 Burst Random Characteristics of Burst Random signal : Gives best linear approximation of nonlinear system Works with zoom a(t) Advantages Best linear approximation of system Disadvantages Time No leakage (if rectangular time weighting can be used) Relatively fast Signal/noise and crest factor not optimum Special time weighting might be required Modal Analysis 49 Shanghai Jiaotong University 2006

50 Pseudo Random Excitation Pseudo random signal: Block of a random signal repeated every T a(t) T T T T Time Time period equal to record length T Line spectrum coinciding with analyzer lines No averaging of non-linearities G AA (f), N = 1 G AA (f), N = 10 System Output B(f 1 ) Freq. Freq. A(f 1 ) System Input Modal Analysis 50 Shanghai Jiaotong University 2006

51 Pseudo Random Excitation Pseudo random signal: Characterized by power/rms (G AA ) and amplitude probability density (p(a)) a(t) p(a) T T T T Time Can be band limited according to frequency range of interest G AA (f) Baseband G AA (f) Zoom Freq. range Freq. Freq. range Freq. Time period equal to T No leakage if Rectangular weighting is used Modal Analysis 51 Shanghai Jiaotong University 2006

52 Pseudo Random Excitation Advantages Disadvantages No leakage Fast Zoom Fair crest factor Fair Signal/Noise ratio No linear approximation of non-linear system Modal Analysis 52 Shanghai Jiaotong University 2006

53 Multisine (Chirp) For sine sweep repeated every time record, T r T r Time A 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 smooth Applications: Measurement of structures with non-linear behaviour Modal Analysis 53 Shanghai Jiaotong University 2006

54 Periodic Random A combined random and pseudo-random signal giving an excitation signal featuring: No leakage in analysis Best linear approximation of system Pseudo-random signal changing with time: A A A B B B C C C transient response T T T steady-state response Analysed time data: (steady-state response) A B C Disadvantage: The test time is longer than the test time using pseudo-random or random signal Modal Analysis 54 Shanghai Jiaotong University 2006

55 Periodic Pulse Special case of pseudo random signal Rectangular, Hanning, or Gaussian pulse with user definable D t repeated with a user definable interval, D T D t D T Time The line spectrum for a Rectangular pulse has a shaped envelope curve sin x x 1/D t Frequency Leakage can be avoided using rectangular time weighting Transient and exponential time weighting can be used to increase Signal/Noise ratio Gating of reflections with transient time weighting Effects of non-linearities are not averaged out The signal has a high crest factor Modal Analysis 55 Shanghai Jiaotong University 2006

56 Periodic Pulse Advantages Fast No leakage (Only with rectangular weighting) Gating of reflections (Transient time weighting) Excitation spectrum follows frequency span in baseband Easy to implement Disadvantages No linear approximation of non-linear system High Crest Factor High peak level might excite non-linearities No Zoom Special time weighting might be required to increase Signal/Noise Ratio. This can also introduce leakage Modal Analysis 56 Shanghai Jiaotong University 2006

57 Guidelines for Choice of Excitation Technique For study of non-linearities: Swept sine excitation For slightly non-linear system: Random excitation For perfectly linear system: Pseudo random excitation For field measurements: Impact excitation For high resolution field measurements: Random impact excitation Modal Analysis 57 Shanghai Jiaotong University 2006

58 m f(t) x(t) SDOF and MDOF Models c k Different Modal Analysis Techniques Exciting a Structure = Measuring Data Correctly Modal Analysis Post Processing Modal Analysis 58 Shanghai Jiaotong University 2006

59 Garbage In = Garbage Out! A state-of-the Art Assessment of Mobility Measurement Techniques Result for the Mid Range Structure ( Hz) D.J. Ewins and J. Griffin Feb Transfer Mobility Central Decade Transfer Mobility Expanded Frequency Frequency Modal Analysis 59 Shanghai Jiaotong University 2006

60 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 60 Shanghai Jiaotong University 2006

61 Where Should Excitation Be Applied? H ij X F j i Response"i" Excitation " j" Driving Point Measurement i = j 2 1 j i X X 1 2 X H H H F H H F F 1 2 Transfer Measurement j X 1 H11 F1 H12 F2 i j i X H F H F2 Modal Analysis 61 Shanghai Jiaotong University 2006

62 Check of Driving Point Measurement All peaks in Im H ij X(f) X(f) Im, Re F(f) F(f) X(f) andim F(f) Mag H ij An anti-resonance in Mag H ij must be found between every pair of resonances Phase H ij Phase fluctuations must be within 180 Modal Analysis 62 Shanghai Jiaotong University 2006

63 Driving Point (DP) Measurement The 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 A ij Log MagA ij Measured R e A ij = Without rigid body modes Modal Analysis 63 Shanghai Jiaotong University 2006

64 Tests for Validity of Data: Coherence Coherence 2 (f) G FF G FX (f) (f) G 2 XX (f) 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 64 Shanghai Jiaotong University 2006

65 Reasons for Low Coherence Difficult measurements: Noise in measured output signal Noise in measured input signal Other inputs not correlated with measured input signal Bad measurements: Leakage Time varying systems Non-linearities of system DOF-jitter Propagation time not compensated for Modal Analysis 65 Shanghai Jiaotong University 2006

66 Tests for Validity of Data: Linearity Linearity X 1 = H F 1 X 1 +X 2 = H (F 1 + F 2 ) X 2 = H F 2 a X 1 = H (a F 1 ) H() F() X() More force going in to the system will equate to more response coming out Since FRF is a ratio the magnitude should be the same regardless of input force Modal Analysis 66 Shanghai Jiaotong University 2006

67 Tips and Tricks for Best Results Verify measurement chain integrity prior to test: Transducer calibration Mass Ratio calibration 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 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 67 Shanghai Jiaotong University 2006

68 m f(t) x(t) SDOF and MDOF Models c k Different Modal Analysis Techniques Exciting a Structure = Measuring Data Correctly Modal Analysis Post Processing Modal Analysis 68 Shanghai Jiaotong University 2006

69 From Testing to Analysis Measured FRF H(f) Curve Fitting (Pattern Recognition) Frequency H(f) Modal Analysis Frequency Modal Analysis 69 Shanghai Jiaotong University 2006

70 From Testing to Analysis Modal Analysis H(f) Frequency SDOF Models c m k f(t) x(t) c m k f(t) x(t) c m k f(t) x(t) Modal Analysis 70 Shanghai Jiaotong University 2006

71 Mode Characterizations All Modes Can Be Characterized By: 1. Resonant Frequency 2. Modal Damping 3. Mode Shape Modal Analysis 71 Shanghai Jiaotong University 2006

72 Modal Analysis Step by Step Process 1. Visually Inspect Data Look for obvious modes in FRF Inspect ALL FRFs sometimes 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 Shanghai Jiaotong University 2006

73 Modal Analysis Inspect Data 1. Visually Inspect Data Look for obvious modes in FRF Inspect ALL FRFs sometimes 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 Shanghai Jiaotong University 2006

74 Modal Analysis Curve Fitting 1. Visually Inspect Data Look for obvious modes in FRF Inspect ALL FRFs sometimes 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 74 Shanghai Jiaotong University 2006

75 How Does Curve Fitting Work? Curve Fitting is the process of estimating the Modal Parameters from the measurements H 2 R/ Find the resonant frequency Frequency where small excitation causes a large response d Find the damping What is the Q of the peak? Phase Frequency Find the residue Essentially the area under the curve Modal Analysis 75 Shanghai Jiaotong University 2006

76 Residues are Directly Related to Mode Shapes! H ( ) ij r H ijr r R ijr j p r R * ijr j p * r Residues express the strength of a mode for each measured FRF Amplitude First Mode Second Mode Therefore they are related to mode shape at each measured point! Third Mode Beam Acceleration Force Force Force Force Force Force Force Force Force Force Force Force Modal Analysis 76 Shanghai Jiaotong University 2006

77 SDOF vs. MDOF Curve Fitters Use SDOF methods on LIGHTLY COUPLED modes Use MDOF methods on HEAVILY COUPLED modes You can combine SDOF and MDOF techniques! H SDOF (light coupling) MDOF (heavy coupling) Modal Analysis 77 Shanghai Jiaotong University 2006

78 Local vs. Global Curve Fitting Local means that resonances, damping, and residues are calculated for each FRF first then combined for curve fitting Global means that resonances, damping, and residues are calculated across all FRFs H Modal Analysis 78 Shanghai Jiaotong University 2006

79 Modal Analysis Analyse Results 1. Visually Inspect Data Look for obvious modes in FRF Inspect ALL FRFs sometimes 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 79 Shanghai Jiaotong University 2006

80 Which Curve Fitter Should Be Used? Frequency Response Function Hij () Real Imaginary Nyquist Log Magnitude Modal Analysis 80 Shanghai Jiaotong University 2006

81 Which Curve Fitter Should Be Used? Frequency Response Function Hij () Real Imaginary Nyquist Log Magnitude Modal Analysis 81 Shanghai Jiaotong University 2006

82 Modal Analysis and Beyond Experimental Modal Analysis Dynamic Model based on Modal Parameters Structural Modification Response Simulation F Hardware Modification Resonance Specification Simulate Real World Response Modal Analysis 82 Shanghai Jiaotong University 2006

83 Conclusion All Physical Structures can be characterized by the simple SDOF model Frequency Response Functions are the best way to measure resonances There are three modal techniques available today: Hammer Testing, Shaker Testing, and Operational Modal Planning and proper setup before you test can save time and effort and ensure accuracy while minimizing erroneous results There are many curve fitting techniques available, try to use the one that best fits your application Modal Analysis 83 Shanghai Jiaotong University 2006

84 Literature for Further Reading Structural Testing Part 1: Mechanical Mobility Measurements Brüel & Kjæ r Primer Structural Testing Part 2: Modal Analysis and Simulation Brüel & Kjæ r Primer Modal Testing: Theory, Practice, and Application, 2 nd Edition by D.J. Ewin Research Studies Press Ltd. Dual Channel FFT Analysis (Part 1) Brüel & Kjæ r Technical Review # Dual Channel FFT Analysis (Part 1) Brüel & Kjæ r Technical Review # Modal Analysis 84 Shanghai Jiaotong University 2006

85 Appendix: Damping Parameters 3 db bandwidth D f 3dB 2 2, D 2 3dB Loss factor 1 Q Df f 3db 0 D 3dB 0 Damping ratio z Df3dB D 2 2f 2 0 3dB 0 Decay constant z 0 Df 3dB D3 2 db Quality factor Q f Df 0 3dB D 0 3dB Modal Analysis 85 Shanghai Jiaotong University 2006

86 Appendix: Damping Parameters h(t) 2 R e t sin d t, where the Decay constant is given by e -t The Envelope is given by magnitude of analytic h(t): h(t) h 2 ~ (t) h 2 (t) Decay constant Time constant : Damping ratio Loss factor Quality z 0 2f z f 0 1 Q 0 f h(t) The time constant,, is determined by the time it takes for the amplitude to decay a factor of e = 2,72 or e t 10 log (e 2 ) = 8.7 db Time Modal Analysis 86 Shanghai Jiaotong University 2006