Modal Analysis ProjectL7005A – Senior Sound Design Project Andre Lundkvist [email protected] Lule˚ January 28, 2010 a, Abstract This report is about a modal analysis project done in the course L7005A – Senior Sound Design Project. It is based upon modal measurements from a project in the course L7001A – Experimental Acoustics and Dynamics, which were previously analysed in I-DEAS Test. The measurements were imported into Matlab. Functions to find poles (frequency, damping and amplitude) were constructed using the Complex Mode Indicator Function (CMIF) as a base, and mode shapes were extracted from the measurements. From these parameters, a multiple degrees of freedom (MDOF) modal-parameter model was constructed by summation of many single degree of freedom (SDOF) models. The modal-parameter model was analysed by different methods, including mathematical versus measured FRF comparisons, MAC matrix and visual interpretation of the mathematical mode shapes. Contents 1 Theory 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 1.1.2 1.2 Analytical mathematical models . . . . . . . . . . . . . . . . . . . . . . . . Experimental mathematical models . . . . . . . . . . . . . . . . . . . . . . . 2 2 3 3 4 4 4 4 5 5 5 5 8 8 9 10 10 11 11 12 13 14 15 15 17 Modal parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 1.2.2 1.2.3 Mode shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modal frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modal damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Mathematical theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 1.3.2 1.3.3 Degrees of Freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mobility matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mathematical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Parameter estimation methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 1.4.2 Complex Mode Indicator Function (CMIF) . . . . . . . . . . . . . . . . . . . Least Squares Frequency Domain method (LSFD) . . . . . . . . . . . . . . . 1.5 Modal parameter model validation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Modal Assurance Critereon (MAC) matrix . . . . . . . . . . . . . . . . . . . 1.6 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.1 1.6.2 1.6.3 1.6.4 Measurement estimation and validation . . . . . . . . . . . . . . . . . . . . Measurement errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Excitation and response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Response transducers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Measurements 2.1 2.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Measured FRFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 3 Method 3.1 3.2 3.3 Complex Mode Indicator Function (CMIF) . . . . . . . . . . . . . . . . . . . . . . . Mode shape estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modal-parameter model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 20 21 21 22 22 23 24 28 31 33 34 35 35 36 37 37 39 42 43 44 45 46 47 48 4 Results 4.1 4.2 4.3 4.4 4.5 4.6 Comparison between I-Deas Test and Matlab poles . . . . . . . . . . . . . . . . . . Complex Mode Indicator Function (CMIF) . . . . . . . . . . . . . . . . . . . . . . . Mathematical vs. measured FRFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mathematical vs. measured mode shapes . . . . . . . . . . . . . . . . . . . . . . . Modal Assurance Criterion Matrix (MAC) . . . . . . . . . . . . . . . . . . . . . . . . Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A I-Deas Test stability diagram B Lists B.1 Target modal parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.2 List of detected frequencies and dampings . . . . . . . . . . . . . . . . . . . . . . . C Matlab code C.1 Import FRFs (impfiles.m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.2 Process FRFs (process frfs.m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.3 Complex Mode Indicator Function (cmif.m) . . . . . . . . . . . . . . . . . . . . . . C.4 Find frequencies (find freqs.m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.5 Find dampings (find damps.m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.6 SDOF modal model (sdof.m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.7 MAC matrix (mac.m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.8 Animate mode (animate mode.m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . References 2 Chapter 1 Theory 1.1 Introduction Modal analysis is a method to determine modal parameters of a system. The system can be defined as a simple structure or a more complex system involving several subsystems [1]. For the purpose of this project, the system will be a simple structure. Mathematical dynamic models may be used for a number of reasons. • Gain understanding how structures behave under dynamic loads • Curve fitting, to smoothen and reduce data • Simulation or prediction when external forces are existent • Physical modifications to the structure, simulate the change of dynamic characteristics Mathematical models describe generally the dynamic behavior, not the structure itself. They are often constrained by assumptions and boundary conditions. The modal parameters that completely describe the dynamics of a system are: • Modal frequency • Modal damping • Mode shape Any forced dynamic deflection of a structure can be represented as a weighted sum of its mode shapes. Each mode can be described by an Single Degree of Freedom (SDOF) model. There are mainly two approaches for the study of system vibrations, analytical and experimental. The analytical starts out with knowledge about the structure geometry, boundary conditions and material characteristics (mass, stiffness and damping). The experimental utilize measurements of dynamic input forces and output responses on the structure or a prototype of the structure. 3 Which is the ratio between output and input as a function of frequency. 1. Reciprocity: Derives from the Linearity property. Usually measurements are observed in the frequency domain as Frequency Response Functions (FRFs). Time Invariant: The properties of the system should not change over time. See figure 1. data acquisition. validation and using the obtained information for improving the system in a systematic way [2]. H(ω). Causal: The structure should not be influenced by vibration before excitation. system identification. and implies that the force and response can switch place and still produce the same result.1: Frequency Response Function (FRF) example [3] 1 Finite Element Method 4 . the excitation level should not affect the behavior. Also.1 Analytical mathematical models Analytical models are often created by using FEM1 based on calculated mass and stiffness under specific boundary conditions. which are coupled [2].2 Experimental mathematical models Experimental modal analysis consists of five different phases. and the modal parameters can therefore be extracted from the FRFs. The model will contain a number of differential equations. These are: Linearity: The system is linear. Figure 1.1.Modal analysis requires some assumptions of the system to work. Stable: The vibrations should die out.1. The FRFs can be described by the modal parameters. which means the response is proportional to the force.1. and not continue to oscillate and converge to infinity when the force is removed. 1. These span from setup. The amount of samples is directly linked to the degrees of freedom (DOF) used. and expresses a deflection pattern in the structure [1]. Unreliable measurements can wrongly indicate complex modes instead of normal modes. This can be a problem on a lightly dampened structure. and these describe modes that can be considered as propagating waves.1.2. From measurements. It is connected to the modal frequency and therefore the pole location of the mode.1 Modal parameters Mode shape The mode shape Ψ is as the name suggests the shape of the mode. meaning Ψ is a real number vector. Standing waves is considered as a normal mode.2 1. 5 . 1. Often these vectors contain complex numbers that describe both magnitude and phase. These can be determined by observing the maximum magnitudes of the measured or calculated FRF. as the resonance peaks are very narrow. a mode shape Ψ is a vector containing sampled values. 1. and can be determined from an FRF by finding the −3 dB bandwidths of the modes.2 Modal frequency The resonance frequency of the specific mode is called modal frequency.2.2.3 Modal damping Modal damping is the damping to the specific mode. 3) is the residue.3 1. figures 1.1.2 describes an SDOF system. Equation 1.1 H1.2) (1. the mobility matrix would look like equation 1.1) H = H2. For a three-DOF system.2 and 1. There are not many rotational transducers available.5) jω − λ1 6 . The diagonal contains the driving point FRF [1]. The residue describes the mode strength. where λ (equation 1. The number of possible input/output (force/response) combinations will be n2 . see equation 1. both rotation and translation on axes x. H1. A1 H(jω) ≈ (1.2 H3.3) λ1 = σ + jω1 λ∗ = σ − jω1 1 (1. geometry and the frequency range. and each column have a common excitation DOF. 1.1 H3.3 1.2 Mobility matrix The mobility matrix contains all different combinations of the FRFs. but three transitional is usually enough to describe the the displacement and motion [1].2. σ is the damping factor and ω1 is the damped natural resonance frequency. where n is the selected DOF. but can be expanded to an MDOF system [2]. y & z.3.3. Because of the complex conjugate part is almost negligible around resonance.5.4) In the pole equation.3 Mathematical models SDOF modal-parameter model Below is described a Modal-Parameter Model FRF. A free point in space has 6 DOFs. H(jω) = with A1 = −j 1 2M ω1 1 A∗ = j 1 2M ω1 A1 A∗ 1 + jω − λ1 jω − λ∗ 1 (1.4) is the pole location and A (equation 1.2 H1.3. and is related to the mode shape. the model can be approximated to equation 1.2 H2.3 H3.1. Each row contains FRFs with a common response DOF.1 Mathematical theory Degrees of Freedom Specifying the Degrees of Freedom (DOF) is based on the purpose of the test.1 H2.3 (1.3. 6) Figure 1.3: Example of a phase plot of the FRF for a SDOF system [2] The impulse response is described in equation 1. h(t) = A1 eλ1 t + A∗ eλ1 t = eσ1 t A1 ejω1 t + A∗ e−jω1 t 1 1 ∗ (1. and it can look similar to figure 1.Figure 1.2: Example of a magnitude plot of the FRF for a SDOF system [2] Figure 1. The impulse response is the Fourier transform of the FRF [2].4: Example of an impulse response for a SDOF system [2] 7 .6.4. 8. σN + jωN σ1 − jω1 0 .8) If the poles are considered as eigenvalues of a system.7 [2].9. λ∗ {Ψ}∗ N N {Ψ}∗ N (1.....9) The residues..10) By using the notation above.11) Figure 1. . Ar = Qr {Ψ}r {Ψ}t r (1.. . . σN − jωN 0 (1. See equation 1.. λN {Ψ}N {Ψ}N λ∗ {Ψ}1 1 {Ψ}∗ 1 . and {Ψ} is the modal shape matrix for the specific mode.5: Example of a FRF magnitude plot constructed from two SDOF FRFs [2] 8 .10. Φ= λ1 {Ψ}1 {Ψ}1 ..MDOF modal-parameter model The MDOF model can be seen in equation 1..11.7) where the poles can be expressed in matrix form as in equation 1. N H(jω) = r=1 A∗ Ar r + jω − λr jω − λ∗ r (1. the FRF can be expanded to the form described in equation 1. λr = σ1 + jω1 . N H(jω) = r=1 Q∗ {Ψ}∗ {Ψ}∗t Qr {Ψ}r {Ψ}t r r r + r (jω − λr ) (jω − λ∗ ) r (1. where Qr is a scaling factor. Ar is described by equation 1. where each eigenvector corresponds with a specific eigenvalue. . its eigenvectors can be interpreted as the modal shapes.. It is also capable of yielding the corresponding mode shape and/or participation vector [2]. The diagonal Singular Matrix ([ Σ ]) is used as the CMIF.13) (1.1 Parameter estimation methods Complex Mode Indicator Function (CMIF) The CMIF is based upon the singular value decomposition of the FRF matrix. The Singular Value Decomposition (SVD) of FRF matrix [ H ] is expressed as.4.12) Complex Mode Indicator Function (CMIF) of FRF Matrix [H] 120 100 Amplitude (dB) 80 60 40 20 500 1000 1500 Frequency (Hz) 2000 2500 3000 Figure 1. containing all possible input – output FRF combinations.1. [ H ]=[ U ]·[ Σ ]·[ V ] where. It indicates the existence of real (normal) or complex modes and also gives the relative magnitude of each mode. [H [U [Σ [V ] ] ] ] is is is is the the the the Frequency Response Function Matrix Left Singular Matrix (unitary) Singular Matrix (diagonal) Right Singular Matrix (unitary) (1.4 1.6: Example of a CMIF for a FRF matrix 9 . 4. Ψir .3 on page 7.19) (1.20) (1. The LSFD method is mostly used for model updating. Nf (1. the derivatives of each parameter from the total error E should converge to 0. If equation 1. Lrj . No Ni E= i=1 j=1 Eij (1. The difference between the measured FRF and the model can be expressed as.2 Least Squares Frequency Domain method (LSFD) The non-linear Least Squares Frequency Domain (LSFD) method estimates poles.15) Eij = f =0 eij (jωf ) · e∗ (jωf ) ij (1.1. because of the complexity and amount of parameters that needs to be determined [2]. The amount of parameters is Nu . U Rij and LRij by minimizing the error between the measured FRF and the model.14) + Hij (jω) = jω − λr jω − λ∗ ω2 r r=1 where i denotes a response position and j denotes an input station.e for parameter rk . U Rij and LRij is the upper and lower residual terms. U Rij .16) The total error between all inputs and outputs can be expressed as.7 is expanded. The modal participation factors can only be estimated if there are multiple inputs.21) 10 . These terms approximates the modes below and above the frequency band of interest. j). The method is about iteratively determine the unknown parameters λr . dE dr1 dE dr2 dE drNu = = .Nm and the total squared error over the frequency range of interest (0 – Nf ) becomes. . λr . = 0 0 0 (1. mode shapes and modal participation factors of a system. This method is based on the frequency domain modal-parameter model described in section 1. the form will be described as N Ψ∗ L∗ LRij Ψir Lrj ir rj + U Rij − (1.17) To minimize the global error.18) (1. including the modal participation factors.3. Ψir . eij (jω) = Hij (jω) − Gij (jω. i. Hij (jω) is the measured FRF at point (i. Lrj . LRij )|r=1. . It can be used to investigate the validity of estimated modes [2].5. the MAC value should be low.22) The MAC will approach the value 1 if {Ψ}r and {Ψ}s are the same mode shape. It {Ψ}r and {Ψ}s are different mode shapes. {Ψ}s ) = (1.5 1.7: Example of a MAC matrix on 5 randomly generated vectors (Gaussian) 11 . {Ψ}∗T {Ψ}s r ({Ψ}∗T {Ψ}r ) ({Ψ}∗T {Ψ}s ) r s 2 MAC({Ψ}r . due to the orthogonality condition of the mode shapes.1. The MAC between two mode shape vectors {Ψ}r and {Ψ}s is defined as. Figure 1.1 Modal parameter model validation Modal Assurance Critereon (MAC) matrix The Modal Assurance Criterion (MAC) matrix is a mathematical tool to compare two vectors to each other. The magnitude of the cross spectrum describes the coherent product of power in the spectrums. The complex conjugate is the same spectrum but with opposite sign for the imaginary part.23. the magnitude of the squared cross spectrum is smaller than the product of the autospectrum. The function H1 is derived by using least squares method. H1 (ω) ≡ GF X (ω) GF F (ω) (1. and for the value 0. seen in equation 1. H2 (ω) ≡ GXX (ω) GXF (ω) (1. When the coherence is 1. the spectrum is multiplied with its complex conjugate.1 Measurements Measurement estimation and validation There are a couple of estimators used to calculate the FRF. For the value 1. |QXF (ω)|2 ≤ GXX (ω) · GF F (ω) |GF X (ω)|2 GXX (ω) · GF F (ω) (1. Cross spectrum is a complex entity which describes the phase shift between the different spectrums. and is defined as the autospectrum of the response divided by the cross spectrum.23) For noise at the input H2 is a useful estimator for the FRF. estimator H1 and H2 will yield the same result.26) 0 ≤ γ(ω)2 ≤ 1 (1. there is pure noise in the measurement.25) γ(ω)2 ≡ (1. the measurement contains no noise. The coherence function have the boundaries described in equation 1.[1] 12 . If averaging the FRFs measured with H1 . equation 1. The cross spectrum is calculated by multiplying a spectrum with the complex conjugate of a different spectrum. The coherence function also indicates the linearity between the input and output signal.27.27) To calculate the autospectrum.25 which states that if the autospectrum contains non-coherent noise. see equation 1. and is the cross spectrum divided by the autospectrum of the force F. The autospectrum is always real. The noise at the input is removed more and more from the cross spectrum with increased averages.24) A way to validate a measurement is to observe the coherence function.6 1. It is derived from the cross spectrum inequality. For instance the spectrum of the force and the response. therefore the estimators are overcompensated.6. the random noise will suppress and H1 will converge towards the true H [1]. For noise at the output. and the true FRF will be somewhere in between. It is derived from the same principle as the H1 estimator.1.26. a useful estimator is H1 . depending on the nature of the cause. The different errors can be divided into different classes.1.2 Measurement errors There are some error that can occur when performing mobility measurements.1: Possible errors for mobility measurements Type of Error H1 H2 Noise at the output R B Noise at the input B R Random excitation / non-linearity B/R B/R Deterministic excitation B B Scatter of impact (point / direction) R R Random excitation B (B) Deterministic impact B B γ2 + + + 0 + + 0 Abbreviation B R + 0 Table 1. because the noise is uncorrelated. Random noise can be minimized by averaging over many measurements. The only way to minimize bias errors is to select another estimator.2: Error table abbreviations Description Systematic Bias error Random error Coherence Function can indicate error Coherence Function can not indicate error 13 . These are listed in table 1.2.1. and can occur both at the input and the output of the signal chain. Random errors are caused by noise. Table 1. The classes are Random & Bias errors [1].6. and its abbreviations in table 1. Systematic bias error are the same for every measurement (both magnitude and phase). The other factor is that nonlinearity might be introduced due to the high peak forces. the structure etc [1]. the attached exciters give a more controllable excitation than a non-attached. Noise will be introduced because of the sharp peaks. see table 1. although there are ways of making linear approximations of a non-linear system.1. The choice of exciter depends on the excitation conditional parameters. The crest factor is the ratio between the peak and the standard deviation RMS of the signal. whereas the non-attached does not. However. which decreases the signal to noise ratio SNR. Because modal analysis assumes that the system is linear and time invariant.6. For modal analysis. attached and non-attached. such as shakers. There are advantages and disadvantages for both categories. There is a list of certain parameters to consider.4: Excitation waveform comparison Factor Analysis Speed Leakage Error Approximation Crest Factor Spectrum Control Zoom Analysis Detect Linearity Detect Leakage Sinusoid Very Slow Yes No Good High Yes No No Random Slow Yes Yes Fair High Yes Yes Yes Waveform Pseudo Random Fast No No Fair High Yes No No Impact Fastest No No Poor Limited No No No Multiple Impact Slow Yes Some Poor-Fair Limited No (Yes) Yes Excitation sources There are many different exciters to choose from.3 Excitation and response Waveform The choice of excitation waveform depends on which type of measurement is to be performed. These are categorized into two fastening categories. 14 . this can be a problem. The attached exciters affects the structure quite a lot. as it contributes to a couple of factors.3: Excitation conditional parameters Description What is the targeted system. hammers.3 [1]. pendulums etc. a high crest factor in the excitation waveform is undesired. which type of measurement Capability to control the frequency range Ratio between the peak and RMS in the signal Linear and time invariant system How much time is needed for a measurement Parameter Application Spectrum Control Crest Factor Linearity Speed of Test Table 1. Table 1. The impedance head is a transducer with integrated force and response accelerometers. Accelerometers can also be very light.6. • Steel stud • Beeswax • Cement stud • Thin tape • Thick tape • Magnet For measurements in the driving point. about 1 g. see figure 1. The piezoelectric accelerometer has good linearity. and are easy to mount by glue. broad dynamic range. Another typical setup is by placing a standard accelerometer transducer on the opposite side of the structure to where the force is applied to. The resonance frequency of the accelerometer is dependent on its weight though. wide frequency range and a strong construction. so this have to be considered. screw etc.1.2 Hz to at least 10 kHz. The most commonly used transducer is an accelerometer. Figure 1. the need for a transducer is imminent.8. Beeswax is a commonly used method. The standard measurement for an accelerometer is acceleration. an impedance head may be used if possible by the setup. See a list of selected mounting techniques below. but velocity or displacement can be calculated by integration [1]. wax. with a linearity of 5 %.4 Response transducers Accelerometers To be able to measure the structural response. The dynamic range is typically around 160 dB and ranges in frequency between 0.8: Example of a few accelerometers mounted with beeswax 15 . and the mounting also affects the applicable frequency range. Chapter 2 Measurements 2. column from left). and it is also the excitation point used. Right: The measurement grid used for measuring the metal plate 16 .1) would be the lower left hand corner. The measurement grid is showed to the right in figure 2. The reference to the measurement grid is counting from bottom left (row from bottom. Figure 2. It is hung by springs to reduce external excitation and to avoid influence on the structure. where the point (1.1 Introduction This chapter discusses the measured data of the metal plate showed to the left in figure 2.1.1: Left: The metal plate used for measurements. mounted with springs.1. throughout the report. An impact hammer was used for excitation.The settings for the measurements were as listed below in table 2.1: Measurement settings Parameter Value Resolution 6400 lines Break Frequency 3200 Hz Measure Time 2 s Means 3. and it had a transient window applied to avoid multiple excitations.3 ms Trigger Signal Accelerometer 2 Level 5 % Hold-Off 2 s Setting Base 17 . Accelerometer 2 was used as a trigger to start the measurement. linearly weighted Overload Reject Accelerometer Window Exponential —τ 0. Table 2.1.3 s Excitation Window Transient — Start 0.9 ms — Stop 1. The hold-off was set to the same length as the measurement time to avoid the measurement to restart before completion. 1) The measurement in point (3. and the apparent ones has lower amplitude which might suggest higher damping.1. except at about 680 Hz.2: Example of a phase and magnitude plot of the FRF associated with the point (1. In comparison to the driving point FRF. 18 . The driving point FRF is showed in figure 2.1). Imaginary Response Values (1. This FRF has clear separated resonances. where there could be a double pole.3.4 shows the FRF at point (5. is showed in figure 2.2 Measured FRFs This section presents some chosen FRFs measured on the plate described in figure 2.1) 50 Magnitude [dB (m/s2/N)] 0 −50 500 1000 1500 Frequency [Hz] 2000 2500 3000 Figure 2. according to the grid in figure 2.1) 3 2 Radians 1 0 −1 −2 −3 500 1000 1500 Frequency [Hz] 2000 2500 3000 Real Response Values (1. just above the middle of the plate.4). The phase response has a weird behavior in the low frequency range. and overall they are quite separated.2 below.2. this response has fewer clear resonance peaks. probably due to measurement problems.1) which is located at the left edge. There are clear resonance peaks. Figure 2. 1) 19 .4: Example of a phase and magnitude plot of the FRF associated with the point (5.Imaginary Response Values (3.4) Magnitude [dB (m/s2/N)] Imaginary Response Values (5.4) 10 5 0 −5 −10 −15 −20 −25 500 1000 1500 Frequency [Hz] 2000 2500 3000 Figure 2.4) 3 2 Radians 1 0 −1 −2 −3 500 1000 1500 Frequency [Hz] 2000 2500 3000 Real Response Values (3.1) 40 Magnitude [dB (m/s2/N)] 20 0 −20 −40 500 1000 1500 Frequency [Hz] 2000 2500 3000 Figure 2.3: Example of a phase and magnitude plot of the FRF associated with the point (3.1) 3 2 Radians 1 0 −1 −2 −3 500 1000 1500 Frequency [Hz] 2000 2500 3000 Real Response Values (5. 3) Magnitude [dB (m/s2/N)] 20 .Imaginary Response Values (7.3) 30 20 10 0 −10 −20 −30 500 1000 1500 Frequency [Hz] 2000 2500 3000 Figure 2.5: Example of a phase and magnitude plot of the FRF associated with the point (7.3) 3 2 Radians 1 0 −1 −2 −3 500 1000 1500 Frequency [Hz] 2000 2500 3000 Real Response Values (9.6: Example of a phase and magnitude plot of the FRF associated with the point (9.2) Imaginary Response Values (9.2) 3 2 Radians 1 0 −1 −2 −3 500 1000 1500 Frequency [Hz] 2000 2500 3000 Real Response Values (7.2) 20 Magnitude [dB (m/s2/N)] 10 0 −10 −20 500 1000 1500 Frequency [Hz] 2000 2500 3000 Figure 2. 1 Complex Mode Indicator Function (CMIF) The complex mode indicator function (CMIF) (see section 1. and when the curve has reached a maximum. The Matlab code used to calculate the CMIF can be found in appendix C. page 8) is used to evaluate the resonate frequencies and get an approximation for relative amplitude and damping.5 on page 44.4 on page 43.1. Complex Mode Indicator Function (CMIF) of FRF Matrix [H] 120 100 80 Amplitude (dB) 60 40 20 0 500 1000 1500 Frequency (Hz) 2000 2500 3000 Figure 3. see figure 3.3 on page 42.1.1: CMIF for all FRFs The algorithm for finding damping ratios is similar. The code for it can be found in appendix C.1. but it will start out from the detected frequencies and iteratively search for the -3 dB decay limits. The curve is similar to the average FRF used in I-DEAS Test to calculate the stability diagram (see appendix A at page 34). 21 . it will indicate a new pole by a red line in the CMIF. see figure 3.Chapter 3 Method 3. The -3 dB limits will be indicated on the CMIF as green lines.4. The algorithm to search for frequencies is an iterative method. It searches for peaks by stepping through the curve. The code can be found in appendix C. The amplitude of the mode was used as scaling.2 Mode shape estimation The mode shapes are estimated from the imaginary part of the measured FRF response matrix {H} at each system pole detected by the CMIF [4].j (jω) ≈ r=1 Q∗ {Ψi. indicating resonance frequency and damping for each pole. Figure 3. The Matlab code can be found in appendix C.j }∗T Qr {Ψi. Complex Mode Indicator Function (CMIF) of FRF Matrix [H] 70 60 50 Amplitude (dB) 40 30 20 10 250 260 270 280 290 300 310 Frequency (Hz) 320 330 340 350 Figure 3. where FP is the pole location and d2 is the right limit.abs(d2). This produces a mode shape matrix for each pole frequency.It tries to approximate the damping by taking the closest limit for both the left and right limit. but the right will not. Meaning.6 on page 45.3 Modal-parameter model The modal model was created by using equation 1. it will approximate the left limit to FP . 22 . that if for example the left limit will diverge. both in samples. To use the equation in Matlab. these are summed together to create the MDOF model.1).j }r is the modal shape amplitude and direction information for the specific mode and node position. σr is the damping factor and ωr is the damped natural resonance frequency for the specific mode.1) where Qr is a scaling factor and {Ψi.7 (page 7).j }T r r r + r jω − σr − jωr jω − σr + jωr (3.2: Detailed CMIF for all FRFs 3. 3. After creating all SDOF models.j }r {Ψi.2 shows a close up display of four detected poles. with the dimensions of the measurement grid (figure 2. it was rewritten like N Hi.j }∗ {Ψi. 017 23 .452 Damping Difference 1.9430 -3.093 0.175 0.071 0. with values both higher and lower to I-DEAS Test.074 —— 0.145 0.149 0.Chapter 4 Results 4.744 2089.53 3094.714 1999.9000 -1.6210 -2.403 641.181 0.113 0.1420 2.286 1294.0240 -2.348 620.222 0.280 0.80 1664.906 —— 1221.042 0.110 0.145 0.214 0. Poles number 7.273 0.458 331.377 965.17 2373.31 1296.214 0.Matlab).4040 0.158 0.21 1998.435 Damping Matlab 0.451 0.232 —— 0.1 Comparison between I-Deas Test and Matlab poles A comparison between the detected poles in I-DEAS Test and Matlab can be found in table 4.698 1853.63 2901.665 0.756 962.355 —— —— 0.1: Comparison between detected frequencies and damping ratios Pole Nr 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 Frequency I-DEAS Test 272. 14 and 15 was not detected by I-DEAS Test.090 0.229 0. Note on pole number 18 and 19.9730 Damping I-DEAS Test 1.182 0. Table 4.1 below. this might be a coincidence.2520 —— —— 1.67 2110.187 0.607 3093.092 0.708 3023.448 621.1190 -4.219 0.048 0.032 0.55 3023.212 0.5040 1.268 0.174 0.843 783.047 Frequency Matlab 270.9430 —— -6.719 0.056 0.0420 1.287 0. The frequencies detected in Matlab is quite similar to I-DEAS Test.135 0.849 973.1580 0.069 0.0770 -0.74 1852.238 0. The difference indicates the difference (I-DEAS Test . but can differ a bit on some.19 2090.034 2901. However.0860 3.437 0.199 -0.127 0.900 787.336 0. The damping is relatively the same for some poles.5540 -1. both the frequency and damping is quite close to each other.187 0.418 —— —— 2401.261 644.382 0.054 0.848 1227.02 Frequency Difference 2.061 -0.544 335.681 1660.136 —— —— 0.13 2399.202 0. and higher frequency poles has lower amplitude than measured. The overall trend seems to be that lower frequency poles has higher amplitude than the measured.1 below. first the CMIF function was used.2 Complex Mode Indicator Function (CMIF) To evaluate the generated mathematical FRFs. The first pole. This error could probably have something to do with the difference described in the previous section 4.1: CMIF for all mathematical FRFs 24 . Overall.4. the mathematical poles seem quite close to the measurements. which has almost double amplitude. located at about 270 Hz. A comparison between the measurement CMIF and the mathematical FRF CMIF can be seen in figure 4.1. Reconstructed FRF CMIF from SDOF Data 120 Reconstructed CMIF 100 80 Amplitude 60 40 20 500 1000 1500 Frequency (Hz) 2000 2500 3000 Figure 4. measured FRFs Another evaluation of the generated mathematical FRFs is to compare them to the measured FRFs.4.3. but still not good.4) can be seen in figure 4. More comparisons can be seen in figures 4.1)) seen in figure 4.1) The comparison for position (3.2: Comparison of the mathematical FRF against the measured FRF in point (1. 25 .1) 50 Magnitude [dB (m/s2/N)] 0 Math Real −50 500 1000 1500 Frequency (Hz) 2000 2500 3000 Figure 4.4 to 4. Also the phase is better than in the driving point comparison.7. which indicates that the model is not accurate at this position of the plate.2 below. The first comparison done is in the driving point (position (1.1) 3 2 Phase [rad] 1 0 −1 −2 −3 500 1000 1500 Frequency (Hz) 2000 2500 3000 Math Real Comparison Reconstructed FRF and Real FRF (1. Why this is the case has not been thoroughly investigated. The mathematical phase is not coherent with the measurement phase either. The mathematical does not correspond well to the measurement. Comparison Reconstructed FRF and Real FRF (1.3 Mathematical vs. An overall conclusion is that the mathematical FRFs is mostly accurate for the center positions of the plate. especially between 800 – 2300 Hz. It correspond much more with the measured response at this position. 4) 20 Magnitude [dB (m/s2/N)] 10 0 −10 Math −20 500 1000 1500 Frequency (Hz) 2000 2500 Real 3000 Figure 4.4) Comparison Reconstructed FRF and Real FRF (5.4: Comparison of the mathematical FRF against the measured FRF in point (5.1) 26 .4) 3 2 Phase [rad] 1 0 −1 −2 −3 500 1000 1500 Frequency (Hz) 2000 2500 3000 Math Real Comparison Reconstructed FRF and Real FRF (3.Comparison Reconstructed FRF and Real FRF (3.1) 3 2 Phase [rad] 1 0 −1 −2 −3 500 1000 1500 Frequency (Hz) 2000 2500 3000 Math Real Comparison Reconstructed FRF and Real FRF (5.1) 40 Magnitude [dB (m/s2/N)] 20 0 −20 Math −40 500 1000 1500 Frequency (Hz) 2000 2500 Real 3000 Figure 4.3: Comparison of the mathematical FRF against the measured FRF in point (3. 5: Comparison of the mathematical FRF against the measured FRF in point (7.2) 3 2 Phase [rad] 1 0 −1 −2 −3 500 1000 1500 Frequency (Hz) 2000 2500 3000 Math Real Comparison Reconstructed FRF and Real FRF (7.6: Comparison of the mathematical FRF against the measured FRF in point (8.2) 3 2 Phase [rad] 1 0 −1 −2 −3 500 1000 Math Real 1500 Frequency (Hz) 2000 2500 3000 Comparison Reconstructed FRF and Real FRF (8.Comparison Reconstructed FRF and Real FRF (7.2) 15 Magnitude [dB (m/s2/N)] 10 5 0 −5 −10 −15 Math Real 500 1000 1500 Frequency (Hz) 2000 2500 3000 Figure 4.2) Comparison Reconstructed FRF and Real FRF (8.2) Magnitude [dB (m/s2/N)] 27 .2) 15 10 5 0 −5 −10 −15 −20 500 1000 1500 Frequency (Hz) 2000 2500 Math Real 3000 Figure 4. 7: Comparison of the mathematical FRF against the measured FRF in point (9.3) 3 2 Phase [rad] 1 0 −1 −2 −3 500 1000 1500 Frequency (Hz) 2000 2500 3000 Math Real Comparison Reconstructed FRF and Real FRF (9.3) 28 .3) 30 Magnitude [dB (m/s2/N)] 20 10 0 −10 −20 500 1000 1500 Frequency (Hz) 2000 2500 3000 Math Real Figure 4.Comparison Reconstructed FRF and Real FRF (9. 8: Measured (left) and Mathematical (right) mode shape for 270 Hz Mode Animation: 331.4.5 −1 9 8 7 6 5 4 3 3 Nodes Y 2 1 1 2 Nodes X 4 5 Figure 4. The modal-parameter model used to create the mathematical FRFs uses an estimate of the mode shape from the measurements (section 3.458 Hz Mode Animation: 270. these rows appear to be 180 degrees out of phase. Mode Animation: 270.2).5 0 −0. the mode shape from the measurement data is presented. To the left.5 −1 9 8 7 6 5 4 3 3 Nodes Y 2 1 1 2 Nodes X 4 5 1 0.5 0 −0. The reason for this has not been investigated.448 Hz 1 0.5 −1 9 8 7 6 5 4 3 3 Nodes Y 2 1 1 2 Nodes X 4 5 Figure 4.9: Measured (left) and Mathematical (right) mode shape for 331 Hz 29 . and to the right is the mathematical mode shape.4 Mathematical vs. Overall the mode shapes for the lower frequency band looks reasonable.5 0 −0.5 0 −0. measured mode shapes This section presents the shapes of some of the detected modes.458 Hz 1 0. Intuitively. The mode shapes indicates that there are torsional and bending waves present in the modes.448 Hz Mode Animation: 331. but the “top” and “bottom” of the plate indicates possible errors.5 −1 9 8 7 6 5 4 3 3 Nodes Y 2 1 1 2 Nodes X 4 5 1 0. 848 Hz 1 0.5 0 −0.5 0 −0.5 −1 9 8 7 6 5 4 3 3 Nodes Y 2 1 1 2 Nodes X 4 5 Figure 4.5 0 −0.5 −1 9 8 7 6 5 4 3 3 Nodes Y 2 1 1 2 Nodes X 4 5 1 0.10: Measured (left) and Mathematical (right) mode shape for 621 Hz Mode Animation: 641.403 Hz 1 0.5 −1 9 8 7 6 5 4 3 3 Nodes Y 2 1 1 2 Nodes X 4 5 Figure 4.5 −1 9 8 7 6 5 4 3 3 Nodes Y 2 1 1 2 Nodes X 4 5 1 0.9 Hz 1 0.9 Hz Mode Animation: 641.5 −1 9 8 7 6 5 4 3 3 Nodes Y 2 1 1 2 Nodes X 4 5 Figure 4.403 Hz Mode Animation: 621.Mode Animation: 621.5 0 −0.5 0 −0.848 Hz Mode Animation: 973.11: Measured (left) and Mathematical (right) mode shape for 642 Hz Mode Animation: 973.5 −1 9 8 7 6 5 4 3 3 Nodes Y 2 1 1 2 Nodes X 4 5 1 0.12: Measured (left) and Mathematical (right) mode shape for 974 Hz 30 .5 0 −0. 74 Hz 1 0.5 −1 9 8 7 6 5 4 3 3 Nodes Y 2 1 1 2 Nodes X 4 5 Figure 4.5 0 −0.53 Hz 1 0.5 0 −0.5 0 −0.53 Hz Mode Animation: 3023.14: Measured (left) and Mathematical (right) mode shape for 1665 Hz Mode Animation: 3023.5 0 −0.5 −1 9 8 7 6 5 4 3 3 Nodes Y 2 1 1 2 Nodes X 4 5 Figure 4.5 −1 9 8 7 6 5 4 3 3 Nodes Y 2 1 1 2 Nodes X 4 5 1 0.74 Hz Mode Animation: 1664.31 Hz Mode Animation: 1227.5 −1 9 8 7 6 5 4 3 3 Nodes Y 2 1 1 2 Nodes X 4 5 1 0.13: Measured (left) and Mathematical (right) mode shape for 1227 Hz Mode Animation: 1664.Mode Animation: 1227.31 Hz 1 0.5 0 −0.5 0 −0.5 −1 9 8 7 6 5 4 3 3 Nodes Y 2 1 1 2 Nodes X 4 5 Figure 4.15: Measured (left) and Mathematical (right) mode shape for 3024 Hz 31 .5 −1 9 8 7 6 5 4 3 3 Nodes Y 2 1 1 2 Nodes X 4 5 1 0. If mode 7 is a faulty mode.4. Compared to the I-DEAS Test MAC matrix. 14 and 15 are not detected by I-DEAS Test. this could explain the similarity between mode 6 and 7. For the Matlab created mathematical modes. the model created in Matlab is not optimal. Only modes 7.2: Most similar modes Modes Frequency Similarity Mode 1 Mode 2 Mode 1 Mode 2 MAC Value 4 13 642 2090 0. This is the wanted result.2886 5 12 787 1998 0. Many modes are too similar to be trusted.7 on page 46.2226 32 . It indicates that the modes are not similar to other modes except for themselves.7367 6 7 966 974 0.16.3144 9 17 1297 2902 0. the MAC matrix can be found in figure 4.16: MAC for all mathematical FRFs from I-DEAS Test Table 4. The code to create this MAC matrix can be found in appendix C. The four most similar modes are listed in table 4. Figure 4.17.2.5 Modal Assurance Criterion Matrix (MAC) The MAC matrix of the mathematical model created with I-DEAS Test can be seen below in figure 4. 17: MAC for all mathematical FRFs from Matlab 33 .Figure 4. The method is referred to as “hard” to perform successfully.6 Discussion By comparing the detected modes (frequency and damping). all modes are handled as standing waves. In the model used. that the model is not very good. described in section 1. the Matlab model is fairly close to the model created with I-DEAS Test. and might cause problems if this is the case. to iteratively improve the parameters. There are other approaches available to update the model [5].4. The mathematical FRFs. 34 .2. This process has not been done. A reason for this could be that the mode shape estimation process is not very good. and requires a lot of experience to make the iteration to converge.4. A way to enhance the model is to use the Least Squares Frequency Domain method. Another reason could be that some of the modes detected are propagating waves through the surface. or explored further. MAC matrix and the mode shapes all indicate however. 1: Stability diagram from I-DEAS Test 35 .Appendix A I-Deas Test stability diagram Figure A. 214 0.571 1.5141E+01 4.999 1.261 644.8062E+01 6.571 1.28Z+ 1Z-.28Z+ 1Z-.7722E+01 5.571 1.000 1.28Z+ 1Z-.348 620.28Z+ 1Z-.28Z+ 1Z-.607 3093.708 3023.000 1.28Z+ 36 .000 1.451 0.000 1.1940E+01 8.28Z+ 1Z-.437 0.5194E+01 5.681 1660.1 PARM LABEL 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Target modal parameters SHAPE REC 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 FREQUENCY (HERTZ) 272.2707E+00 5.1796E+01 2.090 0.843 783.0770E+00 1.571 -1.202 0.0356E+02 4.034 2901.28Z+ 1Z-.RES 1.287 0.5229E+02 1.714 1999.28Z+ 1Z-.28Z+ 1Z-.000 1.906 1221.28Z+ 1Z-.571 -1.8477E+02 5.571 MCF REF.418 2401.28Z+ 1Z-.047 DAMPING (%) 1.8301E-01 5.28Z+ 1Z-.214 0.571 -1.571 1.756 962.268 0.000 1.744 2089.336 0.571 -1.286 1294.571 1.000 1Z-.665 0.232 0.000 1.000 1.000 1.355 0.1247E+02 6.000 0.28Z+ 1Z-.571 1.698 1853.544 335.28Z+ 1Z-.571 1.28Z+ 1Z-.000 1.5767E+02 1.571 1.571 1.212 0.000 1.435 AMPLITUDE PHASE (RAD) 1.000 1.571 1.4988E+02 1.571 -1.Appendix B Lists B.000 1.127 0.9071E+01 6.280 0. 3822 0.0687 0.992 57.765 77.287 54.489 131.1867 0.4519 Amplitude (Rel.048 74.264 81.67 2110.74 1852.0419 0.923 119.814 22.1584 0.1814 0.526 37 .787 103.848 1227.7185 0.533 69.2188 0.2 Pole (Nr) 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 List of detected frequencies and dampings Frequency (Hz) 270.17 2373.849 973.403 641.0932 0.2222 0.584 97.31 1296.111 75.0561 0.1098 0.017 89.982 107.186 124.024 68.377 965.1351 0.0321 0.458 331.02 Damping 0.1134 0.13 2399.900 787.19 2090.320 89.55 3023.53 3094. dB) 37.21 1998.B.2380 0.80 1664.2727 0.63 2901.448 621.634 68.1872 0. 1) ) + downsample ( FRF_C1_A1 . importfile ( ’ conv / FRF_C1_A2 .1) ) + downsample ( FRF_C5_A3 . FRF_C2_A2 = 1 i *( downsample ( FRF_C2_A2 .2) .1) ) + downsample ( FRF_C3_A4 .2 .2 . FRF_C3_A4 = 1 i *( downsample ( FRF_C3_A4 . txt ’) . txt ’) . txt ’) . importfile ( ’ conv / FRF_C2_A4 .2) . importfile ( ’ conv / FRF_C8_A2 .m) % Import all frequency files to workspace importfile ( ’ conv / FRF_C1_A1 . txt ’) .2) . txt ’) .2) . importfile ( ’ conv / FRF_C6_A5 . txt ’) .2 . FRF_C1_A4 = 1 i *( downsample ( FRF_C1_A4 . importfile ( ’ conv / FRF_C7_A3 . FRF_C2_A5 = 1 i *( downsample ( FRF_C2_A5 . importfile ( ’ conv / FRF_C4_A4 .2 .2) .2 . txt ’) . importfile ( ’ conv / FRF_C7_A2 . FRF_C4_A2 = 1 i *( downsample ( FRF_C4_A2 . FRF_C1_A5 = 1 i *( downsample ( FRF_C1_A5 .2) . importfile ( ’ conv / FRF_C3_A4 .2) .1) ) + downsample ( FRF_C3_A1 .1) ) + downsample ( FRF_C6_A1 . txt ’) .2) . .1) ) + downsample ( FRF_C1_A4 .2 . 4 . 38 .1) ) + downsample ( FRF_C2_A1 .2) . txt ’) . FRF_C5_A3 = 1 i *( downsample ( FRF_C5_A3 . importfile ( ’ conv / FRF_C4_A2 .Appendix C Matlab code C.1) ) + downsample ( FRF_C4_A2 . importfile ( ’ conv / FRF_C2_A1 . importfile ( ’ conv / FRF_C7_A5 . FRF_C4_A5 = 1 i *( downsample ( FRF_C4_A5 . txt ’) . FRF_C5_A5 = 1 i *( downsample ( FRF_C5_A5 . FRF_C4_A4 = 1 i *( downsample ( FRF_C4_A4 . FRF_C4_A1 = 1 i *( downsample ( FRF_C4_A1 . txt ’) . FRF_C3_A1 = 1 i *( downsample ( FRF_C3_A1 . (2 . txt ’) . txt ’) . importfile ( ’ conv / FRF_C3_A2 . FRF_C1_A3 = 1 i *( downsample ( FRF_C1_A3 .2) . importfile ( ’ conv / FRF_C5_A3 . importfile ( ’ conv / FRF_C2_A5 .2 .2 . txt ’) . txt ’) .2 . importfile ( ’ conv / FRF_C8_A3 . importfile ( ’ conv / FRF_C4_A3 . importfile ( ’ conv / FRF_C6_A2 . importfile ( ’ conv / FRF_C3_A1 . importfile ( ’ conv / FRF_C3_A3 . txt ’) . . txt ’) . txt ’) . txt ’) .2) .1) ) + downsample ( FRF_C5_A4 .2 . txt ’) . FRF_C5_A1 = 1 i *( downsample ( FRF_C5_A1 . txt ’) .2) . txt ’) .1) ) + downsample ( FRF_C4_A5 .2) .1) ) + downsample ( FRF_C1_A5 .2 . txt ’) . importfile ( ’ conv / FRF_C4_A5 .2 . importfile ( ’ conv / FRF_C2_A3 .2 . FRF_C2_A3 = 1 i *( downsample ( FRF_C2_A3 .2) . txt ’) . txt ’) .1) ) + downsample ( FRF_C3_A5 .1) ) + downsample ( FRF_C5_A5 . FRF_C3_A5 = 1 i *( downsample ( FRF_C3_A5 . importfile ( ’ conv / FRF_C1_A4 . txt ’) . txt ’) . FRF_C5_A2 = 1 i *( downsample ( FRF_C5_A2 . txt ’) . importfile ( ’ conv / FRF_C4_A1 . txt ’) .1) ) + downsample ( FRF_C2_A2 .2) . importfile ( ’ conv / FRF_C6_A4 .2) .1) ) + downsample ( FRF_C2_A4 .2) . FRF_C5_A4 = 1 i *( downsample ( FRF_C5_A4 . txt ’) . importfile ( ’ conv / FRF_C5_A5 . importfile ( ’ conv / FRF_C2_A2 .2 . txt ’) .2 .. importfile ( ’ conv / FRF_C7_A1 ..2 .1) ) + downsample ( FRF_C3_A2 . txt ’) .2) . FRF_C2_A4 = 1 i *( downsample ( FRF_C2_A4 . txt ’) .1) ) + downsample ( FRF_C5_A1 . FRF_C6_A1 = 1 i *( downsample ( FRF_C6_A1 . n = IMAG ) FRF_C1_A1 = 1 i *( downsample ( FRF_C1_A1 . txt ’) .2) . 3 . FRF_C1_A2 = 1 i *( downsample ( FRF_C1_A2 . importfile ( ’ conv / FRF_C8_A5 .2 . importfile ( ’ conv / FRF_C9_A2 .2) . importfile ( ’ conv / FRF_C5_A1 .2) . txt ’) . . importfile ( ’ conv / FRF_C7_A4 . txt ’) .2) ..2) .1) ) + downsample ( FRF_C4_A1 . importfile ( ’ conv / FRF_C1_A3 .1) ) + downsample ( FRF_C2_A3 .1) ) + downsample ( FRF_C1_A2 . importfile ( ’ conv / FRF_C6_A3 . importfile ( ’ conv / FRF_C9_A3 . FRF_C4_A3 = 1 i *( downsample ( FRF_C4_A3 .2 . importfile ( ’ conv / FRF_C1_A5 . FRF_C3_A2 = 1 i *( downsample ( FRF_C3_A2 .1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 Import FRFs (impfiles. importfile ( ’ conv / FRF_C9_A1 .1) ) + downsample ( FRF_C4_A4 . txt ’) .2 . FRF_C3_A3 = 1 i *( downsample ( FRF_C3_A3 .2 . importfile ( ’ conv / FRF_C9_A5 . FRF_C2_A1 = 1 i *( downsample ( FRF_C2_A1 .2 . txt ’) . importfile ( ’ conv / FRF_C8_A1 . . txt ’) .2 .2 . importfile ( ’ conv / FRF_C5_A2 . txt ’) . importfile ( ’ conv / FRF_C5_A4 . importfile ( ’ conv / FRF_C3_A5 . txt ’) . % Reconstruct the FRFs by (1 .1) ) + downsample ( FRF_C2_A5 . importfile ( ’ conv / FRF_C6_A1 . txt ’) .1) ) + downsample ( FRF_C5_A2 . n -1 = REAL ) .2 . txt ’) . txt ’) .1) ) + downsample ( FRF_C3_A3 .2) .2 . importfile ( ’ conv / FRF_C8_A4 .2) .1) ) + downsample ( FRF_C4_A3 .2 . txt ’) .1) ) + downsample ( FRF_C1_A3 .2) .. importfile ( ’ conv / FRF_C9_A4 . 2 ..1) ) + downsample ( FRF_C8_A3 .2 . 1 i *( downsample ( FRF_C6_A4 .. % Use only up to approx 1 kHz % Set up the frequency scale t = 1: length( FRF_C1_A1 ) . 1 i *( downsample ( FRF_C9_A4 ... FRF_C6_A5 ’ .1) ) + downsample ( FRF_C9_A3 .1) ) + downsample ( FRF_C7_A5 .2 .1) ) + downsample ( FRF_C9_A1 .2) . % t = downsample (t .2) . 1 i *( downsample ( FRF_C9_A5 .. FRF_C1_A4 ’ FRF_C2_A4 ’ FRF_C3_A4 ’ FRF_C4_A4 ’ FRF_C5_A4 ’ FRF_C6_A4 ’ FRF_C7_A4 ’ FRF_C8_A4 ’ FRF_C9_A4 ’ FRF_C1_A5 ’ . 1 i *( downsample ( FRF_C8_A1 .2) .1) ) + downsample ( FRF_C7_A4 . 1 i *( downsample ( FRF_C6_A3 .2 ..2) ... 1 i *( downsample ( FRF_C8_A5 .2 .2) ...1) ) + downsample ( FRF_C9_A5 . FRF_C2_A5 ’ ..1) ) + downsample ( FRF_C7_A3 .1) ) + downsample ( FRF_C6_A5 .2) .2 .:) . % t = t (1: round ( end /3) ) .2 ..2 .2 . 1 i *( downsample ( FRF_C9_A2 . 1 i *( downsample ( FRF_C7_A2 .2) . % H = H (1: round ( end /3) .2) . FRF_C1_A1 ’ FRF_C1_A2 ’ FRF_C1_A3 ’ FRF_C2_A1 ’ FRF_C2_A2 ’ FRF_C2_A3 ’ FRF_C3_A1 ’ FRF_C3_A2 ’ FRF_C3_A3 ’ FRF_C4_A1 ’ FRF_C4_A2 ’ FRF_C4_A3 ’ FRF_C5_A1 ’ FRF_C5_A2 ’ FRF_C5_A3 ’ FRF_C6_A1 ’ FRF_C6_A2 ’ FRF_C6_A3 ’ FRF_C7_A1 ’ FRF_C7_A2 ’ FRF_C7_A3 ’ FRF_C8_A1 ’ FRF_C8_A2 ’ FRF_C8_A3 ’ FRF_C9_A1 ’ FRF_C9_A2 ’ FRF_C9_A3 ’ ].2) .2) ..1) ) + downsample ( FRF_C8_A1 .1) ) + downsample ( FRF_C6_A3 .1) ) + downsample ( FRF_C8_A4 .1) ) + downsample ( FRF_C8_A2 .2) .2 .1) ) + downsample ( FRF_C8_A5 .2 .1) ) + downsample ( FRF_C7_A2 . FRF_C3_A5 ’ . 1 i *( downsample ( FRF_C7_A1 . 1 i *( downsample ( FRF_C9_A1 .. 1 i *( downsample ( FRF_C8_A2 . 1 i *( downsample ( FRF_C7_A4 .2) .2 . FRF_C7_A5 ’ . 1 i *( downsample ( FRF_C8_A3 . % Construct FRF Matrix H H = [ ..1) ) + downsample ( FRF_C7_A1 .2 ..2 .. 1 i *( downsample ( FRF_C8_A4 ..2) .2) .1) ) + downsample ( FRF_C9_A4 .2 .2) . t = t / t (end) *3200.2) . FRF_C4_A5 ’ . 1 i *( downsample ( FRF_C9_A3 .2) . FRF_C5_A5 ’ .2) .2 .. 1 i *( downsample ( FRF_C6_A5 .1) ) + downsample ( FRF_C9_A2 .53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 FRF_C6_A2 FRF_C6_A3 FRF_C6_A4 FRF_C6_A5 FRF_C7_A1 FRF_C7_A2 FRF_C7_A3 FRF_C7_A4 FRF_C7_A5 FRF_C8_A1 FRF_C8_A2 FRF_C8_A3 FRF_C8_A4 FRF_C8_A5 FRF_C9_A1 FRF_C9_A2 FRF_C9_A3 FRF_C9_A4 FRF_C9_A5 = = = = = = = = = = = = = = = = = = = 1 i *( downsample ( FRF_C6_A2 .1) ) + downsample ( FRF_C6_A4 .2) . FRF_C9_A5 ’ .2 .2) . 1 i *( downsample ( FRF_C7_A3 . FRF_C8_A5 ’ .1) ) + downsample ( FRF_C6_A2 .2 . 1 i *( downsample ( FRF_C7_A5 .2 .. % Use only up to approx 1 kHz 39 . ’g . ’) . ’) .C.25) . % Find Frequencies iteratively [ freqz . animation = 0.’) . d_err = 5. % Reset loop counter while k <= lf i f d ( k ) > d_err % If wrong pole ( damping factor unrealistic ) % Remove this pole from list d ( k ) = []. xlabel ( ’ Frequency ( Hz ) ’) . outfile ) 40 .depsc ’ . d_i2 ] = find_damps ( hs . % Find Damping ratios iteratively [d .2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 Process FRFs (process frfs. end. freqz ( k ) . f_i .-. f_l ) . f_i . f_i .d ( k ) . outfile = ’ . ylabel ( ’ Amplitude ( dB ) ’) . plot (t . clear y2 . d_i2 . % Smoothen the CMIF by a 25 sample long Running Average filter hs = smooth (h . freqz ( k ) = []. % Set to ’1 ’ to animate mode shapes % Create Complex Mode Indicator Function from all FRFs in H h = cmif ( H ) . % Plot Right Damping Limit Indicator hold off . k = k +1. close a l l . f_l ( k ) = []. f_a ( k ) = []. f_a ( k ) ) . f_a ] = find_freqs ( hs . clear y . t ) . % Plot Pole Indicator Lines stem(t . d_i1 . print ( ’ . lf = length( freqz ) . ’r -. ’g . t i t l e ( ’ Complex Mode Indicator Function ( CMIF ) of FRF Matrix [ H ] ’ ) . stem(t .t . ’b . hs .m) % =================================================== % MODAL FREQUENCY AND DAMPING RATIO % =================================================== format long % Start clock tic . % Plot CMIF hold on . % Plot figure . % Damping ’ error ’ threshold % Print out the findings f p r i n t f ( ’ Nr Frequency Damping Amplitude \ n ’) . ’) ./ report / figures / cmif ’.. lf = length( freqz ) . f_l .-. % Update length else f p r i n t f ( ’ %02 g % 7g % 7g % 7 g \ n ’ . end.k . d_i1 . % Will be updated in loop k = 1. axis tight . % Plot Left Damping Limit Indicator stem(t . :) = real ( ys (k . end 41 . % Create the SDOF modal model for this position and mode y (k . l ) = H ( f_l ( k ) . % SDOF summed % Somehow the real part of the generated frf is the imaginary .:) = sdof (t .4 .: .3. end. end.. for k = 1: length( freqz ) Resid = squeeze ( Psi (k .1 i .:) ) . end end end ys = squeeze (sum(y .1) ) . end. freqz ( k ) ) . % In long vector form u = u + 1. print ( ’ .:) . print ( ’ . u = u +1..:) ..4 . ys (k . . n ) .:) ) . l ) = ys (u . Psi (k .k ) ) .:) ) .78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 % =================================================== % MODE SHAPE VECTORS % =================================================== % Step through all detected poles for k = 1: length( f_l ) u = 1.20* d ( k ) .5 . i f ( animation == 1 ) % Animate the mode ( export to . % Cycle through all measurement positions for j = 1:9 for l = 1:5 % Save the mode shape information to Psi Psi (k . freqz ( k ) .: .:) = ys (k . outfile ) end. freqz ( k ) . outfile = [ ’ .j . % Let ’ s flip them .:) ) .: . for j = 1:9 for l = 1:5 Psi_Math (k . outfile = [ ’ .k ) ) .m .:) .depsc ’ .u .:) ) .:) ) . % =================================================== % MATHEMATICAL MODE SHAPE VECTORS % =================================================== for k = 1: length( f_l ) u = 1. animate_mode (imag( Psi_Math (k ../ report / figures / modeshape_math_ ’ num2str(round( freqz ( k ) ) ) ]. u = u + 1. end.u ) . for k = 1:45 ys_temp (k ./ report / figures / modeshape_ ’ num2str(round( freqz ( k ) ) ) ]. for m = 1:9 for n = 1:5 % Calculate the mode shape scalar for the specific position PsiL = Resid (: . avi ’ . end.. PsiG (k .j . n ) * Resid (m .45 . animate_mode (imag( Psi (k . u = 1.depsc ’ .1/ f_a ( k ) ) . sprintf ( ’ anims / math / test_math % g .:) = imag( ys (k . i f ( animation == 1 ) animate_mode (imag( Psi_Math (k . f_l ( k ) ) .* ys_temp (k . end % =================================================== % MATHEMATICAL FREQUENCY RESPONSE FUNCTIONS % =================================================== y = zeros (length( freqz ) . freqz ( k ) ) .: . u ) = Psi (k .: .5 . sprintf ( ’ anims / real / test % g ..3. outfile ) end. freqz ( k ) .:) ) . ys (k .file and save image ) animate_mode (imag( Psi (k . l ) . avi . length( t ) ) . avi ’ .j . for m = 1:9 for n = 1:5 clear y y = squeeze ( ys (l . xlabel ( ’ Frequency ( Hz ) ’) . axis tight . l ) ) . t i t l e ( ’ Reconstructed FRF CMIF from SDOF Data ’) ./ report / figures / cmif_generated ’ ]. ’ Location ’ . % Plot the CMIF plot (t . axis tight .t . % print ( ’ . % How much time is elapsed ? toc./max(abs( real ( y ) ) ) . hs ) ./ report / figures / frf_generated_ ’ num2str( l ) ]. ’ Best ’) .. axis tight .angle( H (: . 42 . xlabel ( ’ Frequency ( Hz ) ’) . ycmif . ’ num2str( n ) ’) ’ ]) . legend( ’ Reconstructed ’ .’ . ’ Real ’ .158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 % =================================================== % PLOTTING % =================================================== % Create the CMIF for the generated FRF matrix ’ys ’ ( the summe d sdofs ) figure . ’b . end end % Generate MAC Matrix MAC = mac ( PsiG .depsc ’ . ’r . outfile = [ ’ . ’ Location ’ . ylabel ( ’ Phase [ rad ] ’) . t i t l e ([ ’ Comparison Reconstructed FRF and Real FRF ( ’ num2str( m ) ’ .*max(abs( hs ) ) . legend( ’ Math ’ .1 . outfile = [ ’ .1 . ylabel ( ’ Amplitude ’) .angle( y ) . l ) ) . % . ’ Best ’) . legend( ’ Math ’ .’) . ylabel ( ’ Magnitude [ dB ( m / s ^2/ N ) ] ’) . % Smoothen the CMIF by a 25 sample long Running Average filter ycmif = smooth ( yc . real ( y ) . l ) ) ) ) . ’ num2str( n ) ’) ’ ]) . includi ng phase response l = 1..*2+ pi t i t l e ([ ’ Comparison Reconstructed FRF and Real FRF ( ’ num2str( m ) ’ . figure (5) .*max(abs( real ( H (: . subplot (2 .’) ./max(abs( ycmif ) ) . outfile ) % Plot all generated FRFs against the measured FRFs . ’r .’ . ’ Real ’ . outfile ) l = l +1.2) . freqz ) . yc = cmif ( ys ’) .t .25) .t . ’ Best ’) .1) . plot (t . real ( H (: . ’b . xlabel ( ’ Frequency ( Hz ) ’) . ’ Location ’ . % print ( ’ .:) ) . ’ CMIF ’ . subplot (2 .depsc ’ . plot (t . i ) = svd(abs( squeeze ( H (i .3 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 Complex Mode Indicator Function (cmif.2) .3) .: .:) ) ) ) . n u m b e r _ o f _ r e s p o n s e s = s i z e (H .1) . k dim rem .m) function h = cmif ( H ) % Function to compute the Complex Mode Indicator Function ( CMIF ) % % h = cmif ( H ) . end 43 . with dimensions % number_of_linex x n u m b e r _ o f _ r e s p o n s e s x n u m b e r _ o f _ r e f e r e n c e s % % Extract the resolution of the given FRF matrix number_of_lines = s i z e (H . n u m b e r _ o f _ r e f e r e n c e s = s i z e (H . % % It will compute for the FRF matrix ’H ’ .C. number_of_lines ) .) h (: . % Allocate memory for the result h = zeros ( number_of_references . % Step through all values for i = 1: number_of_lines % Calculate the Singular Value Decomposition of the squeeze d matrix H % Squeeze removes singleton dimensions ( if only m x n matrix . 1) . % Pole indicator lines for plot f_i = zeros (length( H ) . f_i ( i ) = max( H ) . F ) % where : % f : Output vector containing the found freqency poles % f_i : Pole indicator lines output % f_l : Pole indicator lines sample position output % f_a : Amplitude of the pole % % H : Input FRF data % F : Input Frequency line data % k = 1.4 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 Find frequencies (find freqs. f_a ] = find_freqs (H . for i = 1+ u : length( H ) .C. u = 1.u i f H(i+u) < H(i) i f H (i . end end end 44 .u ) < H ( i ) f(k) = F(i). % % Function to iteratively find poles of a sampled frf . % Syntax : % [ f ] = find_freqs (H . f_l ( k ) = i .m) function [f . f_a ( k ) = H ( i ) . f_i . F ) . f_l . k = k + 1. % Frequency for damping limit left d2 = 0. zeros (length( H ) .FP ( i ) ) ) .abs( FP ( i ) . k1 = FP ( i ) .thr ) && (( H ( k ) ) > 0) && ( k > 1) ) % Find limits k = k .k1 ) . i f ( abs( FP ( i ) . F ) % where : % d : Output vector containing the found dampings % f_i : Pole indicator lines output % d_i1 : Left freq line for pole % d_i2 : Right freq line for pole % % H : Input FRF data % F : Input Frequency line data % FP : Input Pole Location data p = 1. zeros (length( H ) .d1 ) / H ( FP ( i ) ) .k1 ) ) .C. end i f ( abs( FP ( i ) . % Try to correct if there are many poles close to each other by % approximating the correct damping . k2 = FP ( i ) + abs( FP ( i ) .k1 ) ) d2 = F ( FP ( i ) + abs( FP ( i ) . d_i1 .k2 ) ) d1 = F ( FP ( i ) . d2 = F ( k ) .1) . k2 = k . d_i2 ( FP ( i ) + abs( FP ( i ) . end % Calculate damping ratio for the specified pole d ( p ) = ( d2 .1. d_i2 ] = find_damps (H . k1 = k . end.m) function [d .k2 ) . d_i1 ( FP ( i ) .thr ) && (( H ( k ) ) > 0) && ( k < length( H ) -1) ) k = k + 1. f_i .1) . This is done by using the % pole as center location and adjusting the limit that is far away % to be the same length from the pole as the ’ correct ’ limit . % Start sample position for this frequency d1 = 0. % Frequency for damping limit right f_i ( k ) = max( H ) .k1 ) > abs( FP ( i ) . % Save left limit sample position d1 = F ( k ) .abs( FP ( i ) . while (( H ( k ) > H ( FP ( i ) ) . % Pole f_i = d_i1 = d_i2 = indicator lines for plot zeros (length( H ) .F . % Syntax : % [d .k2 ) > abs( FP ( i ) .1) . % Go to next pole p = p +1.5 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 Find dampings (find damps. end 45 .k1 ) ) = max( H ) .k2 ) ) = max( H ) . % Save frequency of this position k = FP ( i ) . d_i2 ] = find_damps (H . FP ) % % Function to iteratively find damping and poles of a sampled frf . else d_i1 ( k1 ) = max( H ) . end. f_i .abs( k2 . thr = 10. else d_i2 ( k2 ) = max( H ) . while (( H ( k ) > H ( FP ( i ) ) . d_i1 . % Step through all detected frequencies for i = 1 : length( FP ) % Reset parameters k = FP ( i ) . wd .C. Psi . Psi .i * wd . s . end. for k = 1: length( w ) ret ( k ) = ( A /( i * w ( k ) .6 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 SDOF modal model (sdof. Q ) % % Function to create a SDOF modal model . Lc = s . wd . s . 46 . L = s + i * wd . Q ) % where : % w : Frequency vector % wd : Damped natural resonance frequency % s : Damping factor % Psi : Mode shape value % Q : Scaling factor % Calculate residuals A = Q * Psi .Lc ) ) . % Syntax : % ret = sdof (w .m) function ret = sdof (w .L ) ) + ( Ac /( i * w ( k ) . Ac = conj( Q ) *conj( Psi ) . 20]) lighting gouraud t i t l e ( ’ Modal Assurance Critereon ( MAC ) Matrix ’) . l ) = ( PsiRed1 * PsiRed2 ’) ^2 / (( PsiRed1 * PsiRed1 ’) *( PsiRed2 * PsiRed2 ’) ) . axis tight .. % Allocate memory MAC = zeros ( Freqs ) . Mode Shape Vector ] % Optional parameter freqz .round( RES /2) : RES : Freqs * RES +round( RES /2) ) set (gca. l +1) .[8 . xlabel ( ’ Frequency ( Hz ) ’) . RES ) .. freqz . ’ YTickLabel ’ . figure . % Create temorary vectors PsiRed2 = squeeze ( Psi (l . end end MAC = abs( MAC ) . freqz ) colormap(hsv) view(57 . 47 . % Calculate MAC value MAC (u . mesh( MAC_PLOT ) . % If the optional freqz argument is not existant i f nargin < 2 freqz = 1: Freqs . freqz . outfile = [ ’ . end. axis square .round( RES /2) : RES : Freqs * RES +round( RES /2) ) set (gca. MAC_PLOT = zeros ( Freqs * RES ) .C. for plotting resolution ( if <= 0 . zlabel ( ’ Value ’) .43) . freqz ) set (gca. % Optional parameter RES . RES ) % % Modal Assurance Critereon Matrix Function % Takes a modal shape matrix in form : % Psi = [ Frequency . containing the frequency infor mation for axis . -8 . end.7 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 MAC matrix (mac./ report / figures / mac_generated ’ ]. for u = 0: Freqs -1 for l = 0: Freqs -1 MAC_PLOT ( u * RES +1:( u +1) * RES -( RES /2) .m) function MAC = mac ( Psi .1) . light ( ’ Position ’ . % Standard resolution for plot end. outfile ) end. ’ XTick ’ . set (gca. no plot is generated ) % % Syntax : % MAC = mac ( Psi . ’ YTick ’ . i f nargin < 3 RES = 100. ’ XTickLabel ’ .:) ) . % Compare each frequency for u = 1: Freqs for l = 1: Freqs PsiRed1 = squeeze ( Psi (u . ylabel ( ’ Frequency ( Hz ) ’) .l * RES +1:( l +1) * RES -( RES /2) ) = MAC ( u +1 . i f ( RES > 0 ) % For a nicer stem 3 d plot .:) ) . % % Find out how many frequencies Freqs = s i z e ( Psi .. end.depsc ’ . print ( ’ . given it ’ s matrix form % Syntax : % y = animate_mode ( Psi . e = 1.1: Maxtime * pi figure (1) surf ( Psi . xlabel ( ’ Nodes X ’) . for x = .file % i f nargin < 3 Freq = NaN. i f e == 1 aviobj = close ( aviobj ) . % Normalize Psi = Psi . i f nargin < 4 e = 0.1: Maxtime * pi ) . t i t l e ( sprintf ( ’ Mode Animation : %6 g Hz ’ . end f p r i n t f ( ’\ n \ n ’) .8 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 Animate mode (animate mode. axis equal . % Don ’ t return anything useful y = 0. colormap hot. k / la ) == 0) f p r i n t f ( ’. % where : % Psi : The mode shape matrix in the size of the measurement grid % Maxtime : Animation time % Freq : Frequency to animate ( just for printing the title ) % Name : For exporting to . Freq . Freq ) ) .30) .pi :. k = 0. end. Maxtime .C. Maxtime . f p r i n t f ([ sprintf ( ’ Exporting Mode Animation %6 g Hz to ’ . Freq ) Name ’ . else aviobj = avifile ( Name . i f e == 1 frame = getframe(gca) . Name ) ./max(max(abs(abs( Psi ) ) ) ) . k = k +1. Name = ’ NaN ’. end. la = length( . end. aviobj = addframe ( aviobj .pi :. end. avi . ’ fps ’ . Freq . end % Remove singleton dimensions Psi = squeeze ( Psi ) .m) function y = animate_mode ( Psi . frame ) . i f ( mod ( la . ’ ]) . 48 .* sin ( x ) ) . ’) . zlim ([ -1 1]) . ylabel ( ’ Nodes Y ’) . Name ) % % Function to animate a mode shape . Allemang and D.” August 1999. Journal of Sound and Vibration 211(3). 49 . Sas and W. March 1998. [2] P. & Kjaer. Heylen. eds. [3] “Vibration. [5] R. “Structural testing.” April 1988. International Seminar on Modal Analysis. Brown. [4] P.” http://en..org/wiki/Vibration. “Modal space . Avitabile. October 2009. (Heverlee).in our little world (sem experimental techniques).wikipedia. L. Dossing and B. A Unified Matrix Polynomial Approach to Modal Identification. September 1993. Theory and Practice. 301-322: Academic Press Limited.Bibliography [1] O. J.
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