Mechanical Vibrations Overview of Experimental Modal Analysis Peter Avitabile Mechanical Engineering Department University of Massachusetts Lowell

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1 Mechanical Vibrations Overview of Experimental Modal Analysis Peter Avitabile Mechanical Engineering Department University of Massachusetts Lowell.457 Mechanical Vibrations - Experimental Modal Analysis Dr. Peter Avitabile

2 Structural Dynamic Modeling Techniques Could you explain modal analysis and how is it used for solving dynamic problems? Illustration by Mike Avitabile Illustration by Mike Avitabile Illustration by Mike Avitabile.457 Mechanical Vibrations - Experimental Modal Analysis Dr. Peter Avitabile

3 Modal Analysis and Structural Dynamics DISK DRIVE INDUCED VIBRATIONS RESPONSE INPUT TIME FORCE OUTPUT TIME RESPONSE INPUT FORCE I F T BOARD CABINET RESPONSE F F T INPUT POWER SPECTRUM INPUT FORCE FAN INDUCED VIBRATIONS Modal Analysis is the study of the dynamic character of a system which is defined independently from the loads applied to the system and the response of the system. Structural dynamics is the study of how structures respond when subjected to applied loads. Many times, in one form or another, the modal characteristics of the structure is used to determine the response of the system..457 Mechanical Vibrations - Experimental Modal Analysis 3 Dr. Peter Avitabile

4 Analytical Modal Analysis Equation of motion [ M ]{&& x } + [ C ]{ x& } + [ K ]{ x } { F (t)} n n n n n n = Eigensolution [ K ] λ[ M ]]{ x } { 0} n n n = n.457 Mechanical Vibrations - Experimental Modal Analysis 4 Dr. Peter Avitabile

5 Finite Element Models Advantages Models used for design development No prototypes are necessary Disadvantages Modeling assumptions Joint design difficult to model Component interactions are difficult to predict Damping generally ignored.457 Mechanical Vibrations - Experimental Modal Analysis 5 Dr. Peter Avitabile

6 Finite Element Models Analytical models are developed to describe the system mass and stiffness characteristics of a component or system The model is decomposed to express the part in terms of its modal characteristics - its frequency, damping and shapes The dynamic characteristics help to better understand how the structure will behave and how to adjust or improve the component or system design.457 Mechanical Vibrations - Experimental Modal Analysis 6 Dr. Peter Avitabile

7 Experimental Modal Analysis [Y] MEASURED RESPONSE [F] APPLIED FORCE fref fref [H] FREQUENCY RESPONSE FUNCTIONS Advantages Modal characteristics are defined from actual measurements Damping can be evaluated Disadvantages Requires hardware Actual boundary conditions may be difficult to simulate Different hardware prototypes may vary.457 Mechanical Vibrations - Experimental Modal Analysis 7 Dr. Peter Avitabile

8 Experimental Modal Analysis Measured frequency response functions from a modal test can also be used to describe the structure s dynamic properties - its frequency, damping and shapes DOF # 3 DOF # DOF # MODE # MODE # MODE # 3 40 COHERENCE db Mag FRF INPUT POWER SPECTRUM -60 0Hz 800Hz AUTORANGING AVERAGING h h 3 3 h 3 h 33 h 3 h Mechanical Vibrations - Experimental Modal Analysis 8 Dr. Peter Avitabile

9 Experimental Data Reduction Measured frequency response functions from a modal test or operating data can be used to develop a model of the dynamic characteristics of the system.457 Mechanical Vibrations - Experimental Modal Analysis 9 Dr. Peter Avitabile

10 What Are Measurements Called FRFs? A simple inputoutput problem Magnitude Real 6 MODE # MODE # MODE # 3 DOF # DOF #.0000 DOF # 3 Phase Imaginary.457 Mechanical Vibrations - Experimental Modal Analysis 0 Dr. Peter Avitabile

11 What Are Measurements Called FRFs? Response at point 3 due to an input at point h h 33 h Mechanical Vibrations - Experimental Modal Analysis Dr. Peter Avitabile

12 Only One Row or Column of FRF Matrix Needed The third row of the FRF matrix - mode The peak amplitude of the imaginary part of the FRF is a simple method to determine the mode shape of the system.457 Mechanical Vibrations - Experimental Modal Analysis Dr. Peter Avitabile

13 Only One Row or Column of FRF Matrix 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.457 Mechanical Vibrations - Experimental Modal Analysis 3 Dr. Peter Avitabile

14 Only One Row or Column of FRF Matrix Needed Any row or column can be used to extract mode shapes - as long as it is not the node of a mode!??.457 Mechanical Vibrations - Experimental Modal Analysis 4 Dr. Peter Avitabile

15 More Measurements Define Shape Better MODE # MODE # MODE # 3 DOF # DOF # DOF # Mechanical Vibrations - Experimental Modal Analysis 5 Dr. Peter Avitabile

16 SDOF Equations Equation of Motion d x dt m + dx c dt + kx = f (t) or m && x + cx& + kx = f (t) Characteristic Equation ms + cs + k = 0 x(t) k m f(t) c Roots or poles of the characteristic equation s, = ± + c m c m k m.457 Mechanical Vibrations - Experimental Modal Analysis 6 Dr. Peter Avitabile

17 SDOF Definitions Poles expressed as j ω s ( ζωn ) ωn = σ ± ωd,= ζωn± j POLE ω d Damping Factor Natural Frequency % Critical Damping σ = ζω n ω n = ζ= c k m c c ζω n σ Critical Damping c c = mω n Damped Natural Frequency ω d = ω n ζ CONJUGATE.457 Mechanical Vibrations - Experimental Modal Analysis 7 Dr. Peter Avitabile

18 SDOF - Laplace Domain Equation of Motion in Laplace Domain (ms + cs+ k)x(s) = f (s) with b() s = (ms + cs+ k) System Characteristic Equation b (s) x(s) = f (s) and x(s) = b (s)f (s) = h(s)f (s) System Transfer Function h(s) = ms + cs + k Source: Vibrant Technology.457 Mechanical Vibrations - Experimental Modal Analysis 8 Dr. Peter Avitabile

19 SDOF - Transfer Function & Residues Polynomial Form (s) = ms + cs + k h Pole-Zero Form / m (s) = (s p )(s p ) h Residue related to mode shapes a = h(s)(s p = jmω d ) s p Partial Fraction Form a a h (s) = + (s p ) (s p ) Exponential Form ζωt h(t) = e sin ωdt mω d Source: Vibrant Technology.457 Mechanical Vibrations - Experimental Modal Analysis 9 Dr. Peter Avitabile

20 SDOF - Frequency Response Function a a h (jω) = h(s) = + s= jω (jω p ) (jω p ).457 Mechanical Vibrations - Experimental Modal Analysis 0 Dr. Peter Avitabile

21 SDOF - Frequency Response Function Bode Plot Coincident-Quadrature Plot MAG Nyquist Plot.457 Mechanical Vibrations - Experimental Modal Analysis Dr. Peter Avitabile

22 SDOF - Frequency Response Function DYNAMIC COMPLIANCE MOBILITY INERTANCE DISPLACEMENT / FORCE VELOCITY / FORCE ACCELERATION / FORCE DYNAMIC STIFFNESS MECHANICAL IMPEDANCE DYNAMIC MASS FORCE / DISPLACEMENT FORCE / VELOCITY FORCE / ACCELERATION.457 Mechanical Vibrations - Experimental Modal Analysis Dr. Peter Avitabile

23 MDOF Equations Equation of Motion [ M ]{}[ && x + C]{}[ x& + K]{}{ x = F(t) } Eigensolution [ K ] λ[ M] ]{}0 x = Frequencies (eigenvalues) and Mode Shapes (eigenvectors) \ Ω \ = ω ω \ and [ U] = [{ u } { u } L].457 Mechanical Vibrations - Experimental Modal Analysis 3 Dr. Peter Avitabile

24 MDOF - Laplace Domain Laplace Domain Equation of Motion [ ] () [ M] s + [ C] s + [ K] { x s } = 0 [ B( s) ]{ x() s } = 0 System Characteristic (Homogeneous) Equation [ M] s + [ C] s+ [ K] ] = 0 pk = σk ± jωdk Damping Frequency.457 Mechanical Vibrations - Experimental Modal Analysis 4 Dr. Peter Avitabile

25 MDOF - Transfer Function System Equation [ B() s ]{ x() s } = { F( s) } [ H( s) ] = [ B() s ] = { x() s } { F() s } System Transfer Function [ B() s ] = [ H() s ] = Adj det [ B() s ] [ B() s ] = [ A() s ] [ B() s ] det [ A() s ] det [ B() s ] Residue Matrix Mode Shapes Characteristic Equation Poles.457 Mechanical Vibrations - Experimental Modal Analysis 5 Dr. Peter Avitabile

26 MDOF - Residue Matrix and Mode Shapes Residues are related to mode shapes as [ A () s ] = q { u }{ u } T k k k k a a a M k k 3k a a a k k 3k M a a a 3k 3k 33k M L L = q L O k u u u k k 3k u M u u k k k u u u k k 3k u u u M k k k u u u k k 3k u u u M 3k 3k 3k L L L O.457 Mechanical Vibrations - Experimental Modal Analysis 6 Dr. Peter Avitabile

27 MDOF - Drive Point FRF h ij ( jω) = a ij a ij + ( jω p ) ( jω p ) aij a ij + + ( jω p ) ( jω p ) aij3 a ij3 + + ( jω p ) ( jω p ) 3 3 h ij ( jω) = qu iuj qu iuj + ( jω p ) ( jω p ) qu iuj qu iuj + + ( jω p ) ( jω p ) qu 3 i3uj3 qu 3 i3uj3 + + ( jω p ) ( jω p ) Mechanical Vibrations - Experimental Modal Analysis 7 Dr. Peter Avitabile

28 MDOF - FRF using Residues or Mode Shapes R R R3 D D D3 h ij ( jω) = aij aij + ( jω p ) ( jω p ) aij aij + + ( jω p ) ( jω p ) + L F FF3 h ij ( jω) = qu iuj qu iuj + ( jω p ) ( jω p ) qu iuj qu iuj + + ( jω p ) ( jω p ) + L a ij a ij a ij3 ζ ω ω ω ζ ζ Mechanical Vibrations - Experimental Modal Analysis 8 Dr. Peter Avitabile

29 Overview Analytical and Experimental Modal Analysis TRANSFER FUNCTION [B(s)] = [M]s + [C]s + [K] - [B(s)] = [H(s)] LAPLACE DOMAIN q u {u } k j k [U] [ A(s) ] ω ζ det [B(s)] ω [U] FINITE ELEMENT MODEL ANALYTICAL MODEL REDUCTION MODAL PARAMETER ESTIMATION H(j ) ω [K - λ M]{X} = 0 A T N LARGE DOF MISMATCH H(j ) = ω X (j ω) j F (j ω) i FFT X (t) j F (t) i MODAL TEST N U A EXPERIMENTAL MODAL MODEL EXPANSION.457 Mechanical Vibrations - Experimental Modal Analysis 9 Dr. Peter Avitabile

30 Measurement Definitions INPUT ANALOG SIGNALS OUTPUT Actual time signals INPUT ANTIALIASING FILTERS AUTORANGE ANALYZER ADC DIGITIZES SIGNALS OUTPUT Analog anti-alias filter Digitized time signals INPUT APPLY WINDOWS OUTPUT Windowed time signals LINEAR INPUT SPECTRUM COMPUTE FFT LINEAR SPECTRA LINEAR OUTPUT SPECTRUM Compute FFT of signal AVERAGING OF SAMPLES INPUT POWER SPECTRUM COMPUTATION OF AVERAGED INPUT/OUTPUT/CROSS POWER SPECTRA CROSS POWER SPECTRUM OUTPUT POWER SPECTRUM Average auto/cross spectra COMPUTATION OF FRF AND COHERENCE Compute FRF and Coherence FREQUENCY RESPONSE FUNCTION COHERENC E FUNCTION.457 Mechanical Vibrations - Experimental Modal Analysis 30 Dr. Peter Avitabile

31 Measurements - Linear Spectra x(t) h(t) y(t) TIME INPUT SYSTEM OUTPUT FFT & IFT Sx(f) H(f) Sy(f) FREQUENCY x(t) y(t) Sx(f) Sy(f) H(f) h(t) - time domain input to the system - time domain output to the system - linear Fourier spectrum of x(t) - linear Fourier spectrum of y(t) - system transfer function - system impulse response.457 Mechanical Vibrations - Experimental Modal Analysis 3 Dr. Peter Avitabile

32 Measurements - Linear Spectra x(t) y(t) h(t) + jπft = Sx (f )e df + jπft = Sy(f )e df + jπft = H(f )e df S S x y (f ) (f ) + jπft = x(t)e dt + jπft = y(t)e dt + jπft h(t)e H(f ) = dt Note: Sx and Sy are complex valued functions.457 Mechanical Vibrations - Experimental Modal Analysis 3 Dr. Peter Avitabile

33 Measurements - Power Spectra Rxx(t) Ryx(t) Ryy(t) INPUT SYSTEM OUTPUT Gxx(f) Gxy(f) Gyy(f) TIME FFT & IFT FREQUENCY Rxx(t) Ryy(t) Ryx(t) - autocorrelation of the input signal x(t) - autocorrelation of the output signal y(t) - cross correlation of y(t) and x(t) xx x x Gxx(f) - autopower spectrum of x(t) G ( f) = S ( f) S ( f) yy y y Gyy(f) - autopower spectrum of y(t) G ( f) = S ( f) S ( f) yx y x Gyx(f) - cross power spectrum of y(t) and x(t) G ( f) = S ( f) S ( f).457 Mechanical Vibrations - Experimental Modal Analysis 33 Dr. Peter Avitabile

34 Measurements - Linear Spectra lim R xx ( τ) = E[x(t), x(t + τ)] = T T G R G R G xx yy yy yx yx (f ) = lim ( τ) = E[y(t), y(t + τ)] = T T (f ) = lim ( τ) = E[y(t), x(t + τ)] = T T (f ) = R R R xx yy yx ( τ)e ( τ)e ( τ)e jπft jπft jπft dτ= S dτ= S dτ= S (f ) S (f ) (f ) S (f ) (f ) S (f ) x(t)x(t + τ)dt y(t)y(t + τ)dt y(t)x(t + τ)dt.457 Mechanical Vibrations - Experimental Modal Analysis 34 Dr. Peter Avitabile x y y T x T y T x

35 Measurements - Derived Relationships S y=hs x H formulation - susceptible to noise on the input - underestimates the actual H of the system S y S = HS x x S x Sy S G H = = S S G H formulation - susceptible to noise on the output - overestimates the actual H of the system S y S = HS y COHERENCE γ xy = (S (S x y x S y S )(S x x S )(S x y S ) = S ) y y G G x x x x Sy S G H = = S S G yx yy / G / G xx xy y y = H H yx xx yy xy Other formulations for H exist.457 Mechanical Vibrations - Experimental Modal Analysis 35 Dr. Peter Avitabile

36 Typical Measurements Measurements - Auto Power Spectrum Measurements - Cross Power Spectrum x(t) y(t) AVERAGED INPUT POWER SPECTRUM AVERAGED OUTPUT POWER SPECTRUM G (f) xx G (f) yy INPUT FORCE OUTPUT RESPONSE G (f) xx G (f) yy AVERAGED INPUT POWER SPECTRUM AVERAGED OUTPUT POWER SPECTRUM AVERAGED CROSS POWER SPECTRUM G (f) yx Measurement Definitions Dr. Peter Avitabile Measurement Definitions 3 Dr. Peter Avitabile Measurements - Frequency Response Function Measurements - FRF & Coherence Coherence Real AVERAGED INPUT AVERAGED CROSS AVERAGED OUTPUT POWER SPECTRUM POWER SPECTRUM POWER SPECTRUM 0 G (f) xx G (f) yx G (f) yy 0Hz AVG: 5 COHERENCE 00Hz Freq Resp 40 db Mag -60 0Hz AVG: 5 00Hz FREQUENCY RESPONSE FUNCTION FREQUENCY RESPONSE FUNCTION H(f) Measurement Definitions 4 Dr. Peter Avitabile Measurement Definitions 5 Dr. Peter Avitabile.457 Mechanical Vibrations - Experimental Modal Analysis 36 Dr. Peter Avitabile

37 Impact Excitation An impulsive excitation which is very short in the time window usually lasting less than 5% of the sample interval. ADVANTAGES -easy setup - fast measurement time - minimum of equipment -low cost DISADVANTAGES - poor rms to peak levels - poor for nonlinear structures - force/response windows needed - pretrigger delay needed - double impacts may occur - high potential for signal overload and underload of ADC.457 Mechanical Vibrations - Experimental Modal Analysis 37 Dr. Peter Avitabile

38 Impact Excitation - Hammer Tip Selection The force spectrum can be customized to some extent through the use of hammer tips with various hardnesses. A hard tip has a short pulse and excites a wide frequency range. A soft tip has a long pulse and excites a narrow frequency range. In addition, the local structure flexibility will also have an effect on the frequency range excited..457 Mechanical Vibrations - Experimental Modal Analysis 38 Dr. Peter Avitabile

39 Impact Excitation - Hammer Tip Selection METAL TIP HARD PLASTIC TIP Real Real us TIME PULSE 3.964ms us TIME PULSE 3.964ms db Mag db Mag 0Hz FREQUENCY SPECTRUM 6.4kHz 0Hz FREQUENCY SPECTRUM 6.4kHz SOFT PLASTIC TIP RUBBER TIP Real Real us TIME PULSE 3.964ms us TIME PULSE 3.964ms db Mag db Mag 0Hz FREQUENCY SPECTRUM 6.4kHz 0Hz FREQUENCY SPECTRUM 6.4kHz.457 Mechanical Vibrations - Experimental Modal Analysis 39 Dr. Peter Avitabile

40 Important Impact Testing Considerations Hammers and Tips 40 COHERENCE db Mag FRF INPUT POWER SPECTRUM -60 0Hz 800Hz COHERENCE 40 FRF db Mag INPUT POWER SPECTRUM -60 0Hz.457 Mechanical Vibrations - Experimental Modal Analysis 40 00Hz Dr. Peter Avitabile

41 Important Impact Testing Considerations Leakage and Windows ACTUAL TIME SIGNAL SAMPLED SIGNAL WINDOW WEIGHTING WINDOWED TIME SIGNAL.457 Mechanical Vibrations - Experimental Modal Analysis 4 Dr. Peter Avitabile

42 Impact Test Either a row or column of the FRF matrix is needed to estimate mode shapes Ref# Ref# Ref# Ref# Ref#3 Ref#3.457 Mechanical Vibrations - Experimental Modal Analysis 4 Dr. Peter Avitabile

43 Simple Peak Pick - SDOF System Substitute the pole into the SDOF FRF equation h(jω) ω ω n = (jω n a + σ jω d + ) (jω n a + σ + jω d ) a =σh(jω) ω ω n.457 Mechanical Vibrations - Experimental Modal Analysis 43 Dr. Peter Avitabile

44 Simple Peak Pick - Consider Additional Modes Substitute the first pole into the FRF equation h(jω) ω ω = (jω a + σ jω d + ) (jω a + σ + jω d + ) (jω n a + σ jω d + ) (jω + σ + ω n a j d ).457 Mechanical Vibrations - Experimental Modal Analysis 44 Dr. Peter Avitabile

45 Simple Peak Pick - Considerations and Use Peak pick is a quick and simple check Modes must be well spaced (ie: no significant modal overlap) Approximate since peak is determined from the frequency resolution Good quick check before attempting major modal parameter estimation Quickly identify erroneous measurement points (ie: phase reversal, incorrect direction, etc).457 Mechanical Vibrations - Experimental Modal Analysis 45 Dr. Peter Avitabile

46 SDOF Circle Fit - Kennedy-Pancu Simple equation of a circle is fit to the data in the Nyquist Noise and leakage have a pronounced effect on circle Use of windows tends to make the circle look egg-shaped Circle fit method not used as much today due to the availability of many MDOF methods Source: Heylen, Modal Analysis - Theory and Testing h(jω ) = σ φ = r U + jv + j( ω ω U + V ; tan( α ) σ r r + R + ) = U V ji.457 Mechanical Vibrations - Experimental Modal Analysis 46 Dr. Peter Avitabile

47 SDOF - Complex Exponential Response Time domain response can be used to extract parameters This time domain technique is generally used on multiple mode time response data Amplitude Period Damping Decay Generally not used for single mode response extraction h(t) = mω d e ζωt sin ω d t.457 Mechanical Vibrations - Experimental Modal Analysis 47 Dr. Peter Avitabile

48 Complex Exponential One of the first mdof estimators was the complex exponential which uses the Prony Algorithm to solve the set of equations. The Toeplitz equations are used to form the characteristic polynomial followed by the mode shape extraction using Vandemonde Equation formulation. ADVANTAGE h(t) = m k= numerically fast and stable handles many modes m ω k dk e σkt sin ω dk t I F T DISADVANTAGE time domain leakage is a concern must overspecify modes to handle residuals.457 Mechanical Vibrations - Experimental Modal Analysis 48 Dr. Peter Avitabile

49 SDOF Polynomial Form Simple equation of a polynomial for one mode is used to fit the function Additional terms are typically added to account for effects of adjacent or out-of-band modes Fast, simple, easy to use Inappropriate for use with very closely spaced modes (s) = ms h + cs + k compensation terms can be added to account for out of band effects.457 Mechanical Vibrations - Experimental Modal Analysis 49 Dr. Peter Avitabile

50 SDOF Rational Fraction Polynomial Simply the ratio of two polynomials While useful for estimating SDOF type characteristics, its real benefits are for multiple modes Orthogonal polynomials used to greatly simplify the numerical processing More discussion on this in the advanced curvefitting Source: Richardson, Rational fraction Polynomial.457 Mechanical Vibrations - Experimental Modal Analysis 50 Dr. Peter Avitabile

51 MDOF Polynomial This method uses a Rational Fraction polynomial form of the FRF in order to extract modal parameters. Both the numerator and denominator polynomials are used in a least squares fit to extract the polynomial coefficients. h ij (jω) a ij = (jω p aij3 + (jω p aij + ) (jω p 3 ) aij3 + ) (jω p 3 ) A key advantage of the frequency domain representation of the FRF is that the effects of out-of-band modes can be easily accounted for by adding extra terms to the numerator polynomial..457 Mechanical Vibrations - Experimental Modal Analysis 5 Dr. Peter Avitabile

52 Other Time Domain Techniques Other time domain techniques exist which extend the Complex Exponential technique described above. Techniques such as Ibrahim Time Domain and Polyreference LSCE utilize some variant of the equation below to formulate the problem O O Λt [ h(t) ] = [ V] e [ L] global parameters are extracted for poles and shapes uses MIMO time data for the estimation process.457 Mechanical Vibrations - Experimental Modal Analysis 5 Dr. Peter Avitabile

53 Other Frequency Domain Techniques Other frequency domain techniques exist which extend the polynomial technique described above. Techniques such as Least Squares Frequency Domain, Orthogonal Polynomial, Frequency Domain Parameter Identification utilize some variant of the rational fraction, partial fraction or reduced equation of motion to formulate the problem m [ ] ik kj hij(jω) = + + UR ij + k= u L (jω p global poles, MPF and shapes extracted LSFD nonlinear problem solved iteratively RFP - ill-conditioning possible for higher order polys use of orthogonal poly to minimize numerical problems k ) LR ω ij.457 Mechanical Vibrations - Experimental Modal Analysis 53 Dr. Peter Avitabile

54 Mode Shapes from the FRFs A simple peak pick of the imaginary part of the FRF will be used to estimate shapes MODE 4 MODE Advanced manipulation of FRF data is beyond the scope of this simple introduction to experimental modal analysis.457 Mechanical Vibrations - Experimental Modal Analysis 54 Dr. Peter Avitabile

55 .457 Mechanical Vibrations - Experimental Modal Analysis 55 Dr. Peter Avitabile