River Rapids Conference Proceedings of the Society for Experimental Mechanics Series Rotating Machinery, Vibro-Acoustics & Laser Vibrometry, Volume 7 Dario Di Maio Proceedings of the 36th IMAC, A Conference and Exposition on Structural Dynamics 2018 River Publishers
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River Publishers Rotating Machinery, Vibro-Acoustics & Laser Vibrometry, Volume 7 Proceedings of the 36th IMAC, A Conference and Exposition on Structural Dynamics 2018 Dario Di Maio Editor
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Preface Rotating Machinery, Vibro-Acoustics and Laser Vibrometry represents one of nine volumes of technical papers presented at the 36th IMAC, A Conference and Exposition on Structural Dynamics, organized by the Society for Experimental Mechanics, and held in Orlando, Florida, February 12–15, 2018. The full proceedings also include volumes on Nonlinear Dynamics; Dynamics of Civil Structures; Model Validation and Uncertainty Quantification; Dynamics of Coupled Structures; Special Topics in Structural Dynamics; Structural Health Monitoring, Photogrammetry and DIC; Sensors and Instrumentation, Aircraft/Aerospace and Energy Harvesting; and Topics in Modal Analysis and Testing. Each collection presents early findings from experimental and computational investigations on an important area within structural dynamics. Topics represent papers on enabling technologies, rotating machinery, vibro-acoustics and laser vibrometry, and advances in wind energy. The organizers would like to thank the authors, presenters, session organizers, and session chairs for their participation in this track. Bristol, UK D. DiMaio v
Contents 1 Summarizing Results for Scaling OMA Mode Shapes by the OMAH Technique................................ 1 Anders Brandt, Marta Berardengo, Stefano Manzoni, Marcello Vanali, and Alfredo Cigada 2 Delamination Identification of Laminated Composite Plates Using a Continuously Scanning Laser Doppler Vibrometer System............................................................................................ 9 Da-Ming Chen, Y. F. Xu, and W. D. Zhu 3 Rapid and Dense 3D Vibration Measurement by Three Continuously Scanning Laser Doppler Vibrometers .............................................................................................................. 19 Da-Ming Chen and W. D. Zhu 4 Modal Control of Magnetic Suspended Rotors ...................................................................... 31 Marcus Vinicius Fernandes de Oliveira, Felipe Carmo Carvalho, Adriano Borges Silva, Aldemir Ap Cavalini Jr., and Valder Steffen Jr. 5 On the Implementation of Metastructures in Rotordynamics...................................................... 43 Carlo Rosso, Elvio Bonisoli, and Fabio Bruzzone 6 Analysis of the Dynamic Response of Coupled Coaxial Rotors .................................................... 53 Alexander H. Haslam, Christoph W. Schwingshackl, and Andrew I. J. Rix 7 Operational Modal Analysis of Rotating Machinery................................................................ 67 S. Gres, P. Andersen, and L. Damkilde 8 Characterization of Torsional Vibrations: Torsional-Order Based Modal Analysis............................. 77 Emilio Di Lorenzo, C. Colantoni, F. Bianciardi, S. Manzato, K. Janssens, and B. Peeters 9 Long-Term Automatic Tracking of the Modal Parameters of an Offshore Wind Turbine Drivetrain System in Standstill Condition......................................................................................... 91 Mahmoud El-Kafafy, Nicoletta Gioia, Patrick Guillaume, and Jan Helsen 10 Dynamic Modelling and Vibration Control of a Turbomolecular Pump with Magnetic Bearings in the Presence of Blade Flexibility.................................................................................... 101 Alysson B. Barbosa Moreira and Fabrice Thouverez 11 Pushing 3D Scanning Laser Doppler Vibrometry to Capture Time Varying Dynamic Characteristics ....... 111 Bryan Witt and Brandon Zwink 12 Dynamic Measurements on Miniature Springs for Flaw and Damage Detection ................................ 123 Daniel P. Rohe 13 Using High-Resolution Measurements to Update Finite Element Substructure Models ........................ 137 Daniel P. Rohe 14 Determination of Representative Offshore Wind Turbine Locations for Fatigue Load Monitoring by Means of Hierarchical Clustering.................................................................................. 149 Andreas Ehrmann, Cristian Guillermo Gebhardt, and Raimund Rolfes vii
viii Contents 15 Effect of Friction-Induced Nonlinearity on OMA-Identified Dynamic Characteristics of Offshore Platform Models ......................................................................................................... 153 Tobias Friis, Antonios Orfanos, Evangelos Katsanos, Sandro Amador, and Rune Brincker 16 Remote Damage Detection of Rotating Machinery.................................................................. 163 Peter H. Fickenwirth, Charles H. Liang, Tyrel C. Rupp, Eric B. Flynn, and Adam J. Wachtor 17 Experimental Demonstration of a Tunable Acoustoelastic System................................................ 179 Deborah Fowler, Garrett Lopp, Dhiraj Bansal, Ryan Schultz, Matthew Brake, and Micah Shepherd 18 Numerical Modeling of an Enclosed Cylinder ....................................................................... 191 Ryan Schultz and Micah Shepherd 19 Exploiting Laser Doppler Vibrometry in Large Displacement Tests .............................................. 199 E. Copertaro, P. Chiariotti, M. Martarelli, and P. Castellini 20 A Rational Basis for Determining Vibration Signature of Shaft/Coupling Misalignment in Rotating Machinery ................................................................................................................ 207 Changrui Bai, Surendra (Suri) Ganeriwala, and Nader Sawalhi 21 Parametric Experimental Modal Analysis of a Modern Violin Based on a Guarneri del Gesù Model ........ 219 Elvio Bonisoli, Marco Casazza, Domenico Lisitano, and Luca Dimauro 22 Influence of the Harmonics on the Modal Behavior of Wind Turbine Drivetrains............................... 231 N. Gioia, P. J. Daems, C. Peeters, M. El-Kafafy, P. Guillaume, and J. Helsen 23 The Influence of Geometrical Correlation in Modal Validation Using Automated 3D Metrology ............. 239 Tarun Teja Mallareddy, Daniel J. Alarcón, Sarah Schneider, and Peter G. Blaschke
Chapter 1 Summarizing Results for Scaling OMA Mode Shapes by the OMAH Technique Anders Brandt, Marta Berardengo, Stefano Manzoni, Marcello Vanali, and Alfredo Cigada Abstract Methods for scaling mode shapes determined by operational modal analysis (OMA) have been extensively investigated in the last years. A recent addition to the range of methods for scaling OMA mode shapes is the so-called OMAH technique, which is based on exciting the structure by harmonic forces applied by an actuator. By applying harmonic forces in at least one degree-of-freedom (DOF), and measuring the response in at least one response DOF, while using at least as many frequencies as the number of mode shapes to be scaled, the mode shape scaling (modal mass) of all modes of interest may be determined. In previous publications on the method the authors have proven that the technique is easy and robust to apply to both small scale and large scale structures. Also, it has been shown that the technique is capable of scaling highly coupled modes by using an extended multiple reference formulation. The present paper summarizes the theory of the OMAH method and gives recommendations of how to implement the method for best results. It is pointed out, as has been shown in previous papers, that the accuracy of the mode scaling is increased by using more than one response DOF, and by selecting DOFs with high mode shape coefficients. To determine the harmonic force and responses, it is recommended to use the three-parameter sine fit method. It is shown that by using this method, the measurement time can be kept short by using high sampling frequency and bandpass filtering whereas spectrum based methods require long measurement times. This means that even for structures with low natural frequencies, the extra measurement time for scaling the mode shapes can be kept relatively short. Keywords Operational modal analysis · OMA · Mode shape scaling · OMAH · Sine excitation 1.1 Introduction Operational modal analysis (OMA) naturally leads to unscaled mode shapes, since the forces acting on the structure are not measured. It is not uncommon that scaled mode shapes are desired, however. In such cases, several methods exist by which the mode shapes obtained by the OMA parameter extraction may be scaled. Most of the methods developed to scale OMA mode shapes can be divided into the following categories: 1. methods based on several OMA tests, with different mass or stiffness configurations, see for example [1–4]; 2. methods based on knowing the mass matrix of the structure, expand the OMA mode shapes to the size of the mass matrix, and scale the mode shapes using the weighted mode vector orthogonality property, see [5]; 3. methods based on exciting the structure by a known force, and use this force for scaling, usually referred to as OMAX, see for example [6, 7]. Of the methods above, the last method has the advantage that it uses an actual measurement of the force, and is thus, in some sense, scaling the modal model to some calibrated force value. On the other hand, it is generally difficult to excite large structures with broadband force. The authors recently suggested to use harmonic forces for the excitation, since this requires less performance of the actuator used [8]. The method, called OMAH, was extended with a global formulation in A. Brandt ( ) Department of Technology and Innovation, University of Southern Denmark, Odense M, Denmark e-mail: abra@iti.sdu.dk M. Berardengo · M. Vanali Department of Engineering and Architecture, Università degli Studi di Parma, Parma, Italy S. Manzoni · A. Cigada Department of Mechanical Engineering, Politecnico di Milano, Milan, Italy © The Society for Experimental Mechanics, Inc. 2019 D. Di Maio (ed.), Rotating Machinery, Vibro-Acoustics & Laser Vibrometry, Volume 7, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-319-74693-7_1 1
2 A. Brandt et al. [9], allowing to use multiple degrees of freedom (DOFs) for force as well as response locations. The global OMAH method is therefore capable of scaling mode shapes also in cases where there is no single DOF to be chosen for excitation of all modes. Furthermore, using several response points for the scaling reduces the variance in the estimates of the modal mass of the structure. Using harmonic force to scale OMA mode shapes has the advantage that it puts little demand on the actuator, as the actuator only needs to produce a narrowband excitation. Relatively inexpensive actuators can readily be designed for harmonic excitation – even for exciting large structures at low frequencies with relatively high force levels. Furthermore, the estimation of the harmonic signal, hidden in random noise from wind, traffic, and other possible sources, can be achieved under poor signal-to-noise ratios (SNRs), with well-known signal processing methods (mainly the so-called three-parameter sine fit method), see Sect. 1.2.2. 1.2 Theory The theory of the global formulation of the OMAH method is presented in this section. First, in Sect. 1.2.1 by laying out the method for scaling, based on estimates of the frequency response of the structure at a number of frequencies. Secondly, in Sect. 1.2.2 the method to accurately determine the harmonic force and responses at a particular frequency, is discussed. 1.2.1 OMAH Mode Shape Scaling Scaling mode shapes is identical to determining the modal mass of each mode. We start by assuming a frequency response function (FRF) in receptance format (displacement over force) between excitation in DOFq and response in DOFp, which can be written as a function of angular frequency, !, as Hp;q.j!/ D N X rD1 q r p r mr.j! sr/.j! s r / (1.1) where mr denotes the modal mass of mode r, and denotes complex conjugate. Moreover, p r and q r are the eigenvector coefficients (from the OMA) for mode r at DOFs p and q, respectively. The poles, sr, are defined by the undamped natural frequencies (in rad/s), !r, and the relative damping ratios, r, as sr D r!r Cj!rq1 2 r : (1.2) Finally, j is the imaginary unit. After OMA parameter extraction all factors on the right-hand side are known, except the modal mass, and scaling the modal model thus requires to determine the modal mass of each mode. The OMAH method relies on first making an OMA test, whereafter a number of frequency responses, Hp;q.j!/, are estimated at a number of response DOFs p Dp1; p2; : : : ; pm and one or more excitation DOFs q Dq1; q2; : : : ; qv. Then, an equation system is set up to estimate the modal masses and, potentially, residual terms accounting for out-of-band modes. In the simplest of cases, however, Eq. (1.1) can be used directly employing a single FRF estimate, assuming a single-DOF approximation and no effects of surrounding modes. For the general case, we define a global scaling method by first assuming we wish to scale a number, g, modes, from mode number h to hCg 1, using the set of measured FRFs. We also define constant residual terms Cpq (for modes below the modes of interest) and Dpq (for modes above the modes of interest) by approximating the FRF by Hp;q.j!/ hCg 1 X rDh p r q r mr.j! sr/.j! s r / C Cpq !2 C Dpq: (1.3) Next, we define the FRF column vector fHgl, containing measured FRFs, by fHgl D Hp1;q1.j!ex;1/ Hp1;q1.j!ex;2/ : : : Hp2;q1.j!ex;1/ Hp2;q1.j!ex;2/ : : : Hpm;q1.j!ex;1/ Hpm;q1.j!ex;2/ : : : Hp1;q2.j!ex;1/ Hp1;q2.j!ex;2/ : : : Hpm;qv.j!ex;1/ Hpm;qv.j!ex;2/ : : : T (1.4)
1 Summarizing Results for Scaling OMA Mode Shapes by the OMAH Technique 3 where the superscript Œ T denotes vector transpose. Furthermore, we define the column vector fxgl with the unknown modal masses and residual terms, by fxgl D 1 mh 1 mhCg 1 : : : Cp1q1 Dp1q1 Cp2q1 Dp2q1 : : : Cpmq1 Dpmq1 Cp1q2 Dp1q2 : : : Cpmqv Dpmqv T : (1.5) We now introduce the function .p; q; r;!ex/, defined by: .p; q; r;!ex/ D p r q r .j!ex sr/.j!ex s r / (1.6) at one of the experimental frequencies !ex where the FRF is measured. Finally we build a matrixŒA l, defined by ŒA l D 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 .1;1;h;!ex;1/ .1;1;hC1;!ex;1/ : : : .1;1;hCg 1;!ex;1/ 1=!2 ex;1 1 0 0 : : : .1;1;h;!ex;2/ .1;1;hC1;!ex;2/ : : : .1;1;hCg 1;!ex;2/ 1=!2 ex;2 1 0 0 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : .2;1;h;!ex;1/ .2;1;hC1;!ex;1/ : : : .2;1;hCg 1;!ex;1/ 0 0 1=!2 ex;1 1 : : : .2;1;h;!ex;2/ .2;1;hC1;!ex;2/ : : : .2;1;hCg 1;!ex;2/ 0 0 1=!2 ex;2 1 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : .pm;1;h;!ex;1/ .pm;1;hC1;!ex;1/ : : : .pm;1;hCg 1;!ex;1/ : : : .pm;1;h;!ex;2/ .pm;1;hC1;!ex;2/ : : : .pm;1;hCg 1;!ex;2/ : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : .1;2;h;!ex;1/ .1;2;hC1;!ex;1/ : : : .1;2;hCg 1;!ex;1/ : : : .1;2;h;!ex;2/ .1;2;hC1;!ex;2/ : : : .1;2;hCg 1;!ex;2/ : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : .pm;qv;h;!ex;1/ .pm;qv;hC1;!ex;1/ : : : .pm;qv;hCg 1;!ex;1/ : : : .pm;qv;h;!ex;2/ .pm;qv;hC1;!ex;2/ : : : .pm;qv;hCg 1;!ex;2/ : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 (1.7) at various frequencies !ex;k for some integers k. Using the vectors and matrices thus defined, we can form an equation system ŒH l DŒA lfxgl (1.8) which can be solved for the vector fxgl by a least squares solution or a pseudo inverse. It is important to note that the frequencies in each line of the matrix ŒA l, although for simplicity denoted !ex;1;!ex;2; : : :, may actually be arbitrary frequencies, albeit in most cases it will be practical to calculate many of the FRFs from simultaneous measurements of several responses. Next, it is important to consider which requirements apply in order to make the matrix ŒA l well conditioned. First, we need to ensure that we have more rows in the equation system than the number of unknowns (i.e. modal masses plus residual terms). If we have measured Munique FRF locations (p; q), then it is sufficient that the number of lines in the equation system, L >gC2M. In most cases this will be fulfilled without any extra thought, if the recommendations in Sect. 1.2.3 are followed. Next, we note that each function .p; q; r;!ex;k/ in the matrix ŒA l, belongs to the FRF value Hp;q.j!ex;k/ due tomode r, at frequency!ex;k. The natural way to ensure that ŒA l is well conditioned, is to excite the structure at frequencies close to the eigenfrequencies and including all the modes to be scaled. Also it should be ensured that the excitation DOF in each case, is at a point of the structure where the mode in question is well excited (i.e. has a large mode shape coefficient). This will ensure that each line in ŒA l has at least one large function .p; q; r;!ex;k/.
4 A. Brandt et al. 1.2.2 Estimating Harmonics in Noise One of the strengths with the OMAH method, is that harmonics hidden in noise can be accurately and efficiently estimated even in cases with very low SNRs. The method therefore does not rely on large excitation forces. Furthermore, as will be shown in the present section, the measurement time does not have to be very long even in cases with poor SNRs. The method we recommend for estimating the complex amplitudes of the force and response signals Fq.j!ex/ andUp.j!ex/ is the so-calledthree-parameter sine fit method [10]. The name implies that the DC value as well as two Fourier coefficients aandbare unknown, whereas in our case the DC does not apply since we remove the mean of the signals prior to estimating the harmonics. Regardless, the method is usually referred to as the three-parameter sine fit. For simplicity, in the following we assume that we are exciting the structure at a frequency ! (corresponding to one of the frequencies previously denoted !ex;k). We furthermore assume we have measured N samples of the signal y.t/ at y.n/ Dy.n t/ (y.t/ being a force or a response signal) using a sampling frequency fs D1= t. The measured signal will then conform to the model y.t/ Dacos.!t/ Cbsin.!t/ Ce.t/ (1.9) where the Fourier coefficients a and b are the desired unknowns, and e.t/ contains the random part of the signal, due to wind, traffic, and other random contributions, and possibly higher (and undesired) harmonics of the excitation frequency. To calculate the complex FRF values, we calculate the amplitude and phase of y.n/ as pa2 Cb2 exp.jarctan.b=a//. Based on the measurements of y.n/, we can define the matrixB ŒB D 2 6 6 6 6 6 6 6 6 4 cos.! 0 t/ sin.! 0 t/ cos.! 1 t/ sin.! 1 t/ cos.! 2 t/ sin.! 2 t/ : : : cos.! .N 1/ t/ sin.! .N 1/ t/ 3 7 7 7 7 7 7 7 7 5 (1.10) and the unknown coefficient vector fxgDŒa b T, and the measurement vector fygD 2 6 6 6 6 6 4 y.0/ y.1/ y.2/ : : : y.N 1/ 3 7 7 7 7 7 5 : (1.11) The model in Eq. (1.9), can now be written as: ŒB fxgDfygCfeg (1.12) which we can solve for the estimate Ox of x in Eq. (1.12), by a least squares solution. Furthermore, having solved for the estimate Ox, the remaining signal Oe may be calculated by f OegDfyg fBgf Oxg: (1.13) after which we can estimate the variance of the remaining signal Oe by 2 Oe D f Oeg T f Oeg N : (1.14) Since the power of the unknown harmonic is .Oa2 C Ob2/=2the SNR can be calculated as SNRD O a2 C Ob2 2 2 Oe D O xTOx 2 2 Oe : (1.15)
1 Summarizing Results for Scaling OMA Mode Shapes by the OMAH Technique 5 When the coefficients infOxg are computed, the absolute value of OFq that we need for the FRF estimates, may be computed by ˇ ˇ ˇ OFq.j!/ˇ ˇ ˇ D p OxTOx (1.16) In [10] it is shown that the mean square error (MSE) of ˇ ˇ ˇ OFq.j!/ˇ ˇ ˇ in Eq. (1.16) is approximately MSEŒ OFq 2 2 Oe N (1.17) which is a conservative error. The result in Eq. (1.17) is a very important result. It shows, that at any SNR (i.e. any level of variance of the random part of the response, independent of the harmonic amplitude), the relative error in the amplitude can be made arbitrarily small, by increasing the number of samples, N. Furthermore, it is important to realize that this number of samples can be recorded in a short time, by increasing the sampling frequency without increasing the bandwidth of the measurement system, since this will not affect the SNR, but will result in more samples in a given time. In most modern measurement systems, this will not be allowed because the cutoff frequency is chosen as a factor of the selected sampling frequency. But a higher performance can be achieved by sampling at a high frequency, and then bandpass filtering the data after the data acquisition is finished (which will improve the SNR as well as create more samples). If we want to have a particular maximum uncertainty on the estimate of the FRF value, we can allow half that uncertainty on each of the force and response values. Furthermore, since the MSE includes both random and bias errors, a conservative measure may be to use three times the square root of the MSE as the maximum deviation from the measured values. Say, for example, that we may allow 5% inaccuracy on the FRF estimate. For the force (and similar for the response) measurements, we then need to assure that, 100 3 qMSEŒ OFq ˇ ˇ ˇ OFqˇ ˇ ˇ 2:5 (1.18) 1.2.3 Guidelines for Applying OMAH Using the results in Sects. 1.2.1 and 1.2.2, we will now present some guide lines for the design of a test for scaling mode shapes from an OMA test. The following procedure can thus be used. 1. Complete the OMA measurements. This is usually done by measuring batches of channels, keeping some references. Keep all the sensors in place after the last batch measurement. If all DOFs can be measured simultaneously, the better. 2. Complete the OMA parameter estimation, resulting in poles and unscaled mode shapes. 3. Study the mode shapes, and choose a DOF with large mode shape coefficients for some (if possible most) modes. Attach the actuator in this DOF. (In many cases where the mode shapes are, at least approximately, known beforehand, this actuator position can be determined prior to performing the OMA test.) 4. Add an accelerometer on the mass of the actuator, for measuring the force. If there is no available channel, replace one of the response channels with the accelerometer for measuring the force. 5. Investigate a proper force amplitude, by measuring an arbitrary amount of time, and estimate the signals Fq.t/ and Up.t/, where pdenotes all the DOFs in the last measured batch. Furthermore, Uq must be included in the set of Up. This requires estimating the MSE using Eq. (1.17), and to calculate an appropriate relative error. If necessary, increase the sampling frequency and follow up by lowpass filtering the signals prior to estimating the amplitudes and phases, to ensure the measurement time is kept appropriately short. This step may also involve changing the moving mass of the actuator, to produce an appropriate force. 6. For each frequency, !ex;k, compute the amplitudes and phases of the force and all responses, and compute the FRF values Hp;q.j!ex;k/ and store these FRF values. 7. Make consecutive measurements of frequencies near all modes which are well excited by the DOFqwhere the actuator is located. This means that data for creating a relatively large number of rows for the matrixŒA l in Eq. (1.7) will be acquired.
6 A. Brandt et al. 8. For the mode or modes not well excited by the first chosen excitation DOF, move the actuator to a DOF where one or more of these modes will be well excited. Excite the mode or modes near their natural frequency (frequencies), measure the force and all response channels, produce the new FRF values, and store these. In most cases, each of these modes needs to be excited only at a single frequency near the natural frequency of the mode in question in order to be able to solve Eq. (1.8). However, exciting more frequencies, as long as they are close to a mode that is well excited by the DOF q, can result in higher accuracy of the estimated modal masses. Repeat this step until all modes have been well excited at least at one frequency. 9. Now build the measurement vector fHgl and the matrixŒA l by Eqs. (1.4) and (1.7), respectively, and solve for the unknown modal masses and residual terms by solving Eq. (1.8). 1.3 Discussion To see how an OMAH test may be conducted, we assume we would scale the first 8 vertical modes of the Little Belt Bridge, see [11]. The information about the modes is found in Table 1.1. We also assume that we have an eight-channel measurement system, and that we use seven accelerometers for the response measurements for scaling, including the reference (in the same DOF as the force). The final channel is used to measure the acceleration of the moving mass of the actuator, so we can compute the excitation force. Using the information in Table 1.1 and information about the mode shapes (that are essentially the shapes of a pinned narrow plate), we define three excitation points, in order to be able to excite all modes well: 1. vertically on one side, at 1/4 from one end, to excite modes 1, 2, 5, 7, 8 2. vertically on one side, at 1/3 from one end, to excite modes 4, and 6 3. vertically on one side, at 1/2 from one end, to excite mode 3 Note that these may not be optimal points, but are given as an example. We thus start the test by positioning the actuator in position 1, and set the frequency close to 0.155 Hz. Since mode 1 has a damping value of 1.5%, we need to wait approximately 500 s for the structure to reach its steady-state condition. After this time, we acquire a number of samples, for example corresponding to five periods of the harmonic, i.e. 32.3 s. After calculating the complex sines of the force and the 7 responses, we check that Eq. (1.18) is fulfilled, and if not, we may set the sampling frequency higher, acquire data again, and BP filter data to a narrow bandwidth around the frequency of interest, and then recompute the complex amplitudes and using them to produce seven estimates of the FRFs. After this, we tune the frequencies of modes 2, 5, 7, and 8, in turn, and for each of them wait for steady state, acquire data, and check the accuracy. When the accuracy is adequate, we compute the complex amplitudes, and then the FRF values. Once this is accomplished, the shaker is moved to position 2, the frequency tuned to the frequencies of modes 4 and 6 and the procedure is repeated. Finally, the same is done for position 3 for mode 3. When all this is done, we have acquired 7 FRFs times 5 frequencies from position 1, 7 FRFs times 2 frequencies from position 2 and finally 7 FRFs times 1 frequency for position 3. This thus produces 56 rows in the matrix ŒA l in Eq. (1.7). The number of unknowns we have are eight modal masses, plus 2 residuals times 7 response DOFs times 3 excitation DOFs, which equals 50 unknowns. We can thus solve the equation system. It would be advantageous, however, if time allows, to measure a few more frequencies to obtain a more overdetermined system of equations. This could easily be achieved by adding those frequencies, in each position of the actuator, for which there is a reasonably large mode shape coefficient in the forcing DOF. Table 1.1 First 8 modes of the Little Belt Bridge, used as an example for a mode shape scaling case Mode Frequency Damping Description # [Hz] [%] 1 0:155 1:49 First vertical bending 2 0:171 11:41 Second vertical bending 3 0:258 0:69 Third vertical bending 4 0:402 0:75 Fourth vertical bending 5 0:524 0:65 First torsion 6 0:572 0:45 Fifth vertical bending 7 0:769 0:71 Sixth bending 8 0:807 0:72 Second torsion
1 Summarizing Results for Scaling OMA Mode Shapes by the OMAH Technique 7 Table 1.2 Values of settling times (time until the response is steady-state, defined by less than 1% change in RMS level from 5 periods to next 5 periods), and measurement times (5 periods) for the entire test. See text for details. The total time in columns 5 and 6 adds up to 2595 s, or approx. 45 min Meas. Exc. Mode Frequency Settling Measurement # pos. # [Hz] time [s] time [s] 1 1 1 0:155 500 32:3 2 1 2 0:171 75 29:2 3 1 5 0:524 300 9:5 4 1 7 0:769 220 6:5 5 1 8 0:807 200 6:2 6 2 4 0:402 300 12:4 7 2 6 0:572 400 8:7 8 3 3 0:258 600 19:4 So, how long would this whole test take? In Table 1.2 we present all measurements with the settling time and the time taken for measuring five periods of each frequency of excitation. We have defined steady-state conditions as the time when the RMS of the response from one block of data containing five periods to the next five periods, does not change more than 1%. As can be seen in the table, the total time of data acquisition is approximately 45 min, of which the major time is spent waiting for the system to reach steady-state conditions. The time to move the actuator is not taken into account in this example. Also note, that the measurement time is independent of the SNR used. So the actuator can, for example, be set to generate 10% of the RMS of the random response. This example shows that even on a relatively low-frequency structure like this, the OMAH method does not require very long measurement times. A further advantage with the OMAH technique is that, once the modes are scaled, the accuracy of the scaling can be investigated. This was demonstrated in the previous papers presenting the method, see [8, 9]. It is done by comparing the measured harmonic response amplitudes in the response DOFs, with the computed responses using the synthesized FRF (using the scaled modal model) times the excitation force. Finally, it should also be mentioned that an advantage of the OMAH method is that the response of the structure to changes in the amplitude of the force can easily be included in the measurements. Thus, the linearity of the structure may be investigated. This is not easily done with most other methods for scaling OMA mode shapes. 1.4 Conclusions In this paper we have described the theory of global OMAH scaling of mode shapes, using harmonic excitation. The method depends on exciting the structure in one or more DOFs, and,although not strictly necessary, we recommend exciting at frequencies close to the eigenfrequencies of the structure. The least squares global solution method described in the paper can handle structures with closely coupled modes. Furthermore, it gives modal masses for all the modes taken into account as well as residual terms for all the pairs of measuring and forcing points. It has been shown that the technique offers several attractive properties: • It puts low demands on the actuator, as the force level can be low relative to ambient response. • The accuracy of the method may be increased by using more response measurements. • The method can handle closely coupled modes. • The sine hidden in noise can be accurately determined without needing long measurement times. • The method allows to investigate the accuracy of the mode shape scaling, by comparing the measured responses with those from synthesized frequency response multiplied by the harmonic force applied. • The method allows to easily investigate the linearity, by observing the response for several different force levels. An example of how to scale the first eight modes of a bridge with eigenfrequencies from approx. 0.15 to 0.8 Hz was shown to require approximately 45 min of total measurement time, exciting the structure in three DOFs, at a total of eight frequencies.
8 A. Brandt et al. References 1. Parloo, E., Verboven, P., Guillaume, P., Van Overmeire, M.: Sensitivity-based operational mode shape normalisation. Mech. Syst. Signal Process. 16(5), 757–767 (2002) 2. Bernal, D.: Modal scaling from known mass perturbations. J. Eng. Mech. 130(9), 1083–1088 (2004) 3. Coppotelli, G.: On the estimate of the FRFs from operational data. Mech. Syst. Signal Process. 23, 288–299 (2009) 4. Aenlle, M.L., Fernandez, P., Brincker, R., Fernandez-Canteli, A.: Scaling-factor estimation using an optimized mass-change strategy. Mech. Syst. Signal Process. 24(5), 1260–1273 (2010) 5. Aenlle, M.L., Brincker, R.: Modal scaling in operational modal analysis using a finite element model. Int. J. Mech. Sci. 76, 86–101 (2013) 6. Reynders, E., Degrauwe, D., De Roeck, G., Magalhães, F., Caetano, E.: Combined experimental-operational modal testing of footbridges. J. Eng. Mech. 136(6), 687–696 (2010) 7. Cara, J.: Computing the modal mass from the state space model in combined experimental operational modal analysis. J. Sound Vib. 370, 94–110 May (2016) 8. Brandt, A., Berardengo, M., Manzoni, S., Cigada, A.: Scaling of mode shapes from operational modal analysis using harmonic forces. J. Sound Vib. 407, 128–143 Oct (2017) 9. Brandt, A., Berardengo, M., Manzoni, S., Vanali, M., Cigada, A.: Global scaling of OMA modes shapes with the OMAH Method. In: Proceedings of International Conference on Structural Engineering Dynamics (ICEDyn), Ericeira, Portugal (2017) 10. Händel, P.: Amplitude estimation using IEEE-STD-1057 three-parameter sine wave fit: statistical distribution, bias and variance. Measurement 43, 766–770 (2010) 11. Christensen, S.S., Andersen, M.S., Brandt, A.: Dynamic characterization of the little belt suspension bridge by operational modal analysis. In: Proceedings of the 36th international modal analysis conference (IMAC), Orlando (2018)
Chapter 2 Delamination Identification of Laminated Composite Plates Using a Continuously Scanning Laser Doppler Vibrometer System Da-Ming Chen, Y. F. Xu, and W. D. Zhu Abstract Delamination frequently occurs in a laminated composite structure and can cause prominent local anomalies in curvature vibration shapes associated with vibration shapes of the composite structure. Spatially dense vibration shapes of a structure can be rapidly obtained by use of a continuously scanning laser Doppler vibrometer (CSLDV) system, which sweeps its laser spot over a vibrating surface of the structure. This paper extends two damage identification methods for beams to identify delamination in laminated composite plates using a CSLDV system. One method is based on the technique that a curvature vibration shape from a polynomial that fits a vibration shape of a damaged beam can well approximate an associated curvature vibration shape of an undamaged beam and local anomalies caused by structural damage can be identified by comparing the two curvature vibration shapes, and the other is based on the technique that a continuous wavelet transform can directly identify local anomalies in a curvature vibration shape caused by structural damage. In an experimental investigation, delamination identification results from the two methods were compared with that from a C-scan image of a composite plate with delamination. Keywords Delamination identification · Composite plate · Continuously scanning laser Doppler vibrometer system · Polynomial fit · Wavelet transform 2.1 Introduction Among various types of structural damage, delamination is one that frequently occurs in laminated composite structures. Since delamination is usually hidden from external view, it can be difficult to identify. When delamination occurs, local stiffness of a laminated composite structure in neighborhoods of the delamination can change and modal parameters of the structure, including natural frequencies, modal damping ratios and mode shapes, can subsequently change [1–3]. Effects of delamination on modal parameters of composite structures have been studied with their analytical, semi-analytical and finite element models [4–6]. Specifically, vibration shapes, such as mode shapes and operating deflection shapes, have been widely used for structural damage identification since spatial derivatives of vibration shapes, such as curvature vibration shapes [7] and slope vibration shapes [8], better manifest local anomalies caused by the damage, and identification methods that use curvature vibration shapes of laminated composite structures have been developed [3, 9, 10]. A polynomial-based method was proposed in Ref. [7], where curvature mode shapes of damaged and undamaged beams are compared and damage can be identified in neighborhoods with large differences between them. In the polynomial-based method, a curvature vibration shape of an undamaged beam is not needed as it is obtained from a polynomial that fits a vibration shape of a damaged beam; the method was extended to identify damage in plates using vibration shapes measured in a point-by-point manner by a scanning laser Doppler vibrometer [11]. A wavelet-based method was proposed in Ref. [8], where slope vibration shapes of damaged beams are wavelet-transformed and damage can be identified in neighborhoods with large wavelet-transform coefficients. The polynomial- and wavelet-based methods inspect smoothness of vibration shapes of beams by isolating prominent local anomalies in curvature and slope vibration shapes caused by damage, and they can be extended to identify delamination in laminated composite plates, since delamination can also cause local anomalies in curvature vibration shapes of plates. D.-M. Chen · W. D. Zhu ( ) Department of Mechanical Engineering, University of Maryland, Baltimore County, Baltimore, MD, USA e-mail: damingc1@umbc.edu; wzhu@umbc.edu Y. F. Xu Department of Mechanical and Materials Engineering, University of Cincinnati, Cincinnati, OH, USA e-mail: xu2yf@uc.edu © The Society for Experimental Mechanics, Inc. 2019 D. Di Maio (ed.), Rotating Machinery, Vibro-Acoustics & Laser Vibrometry, Volume 7, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-319-74693-7_2 9
10 D.-M. Chen et al. A laser Doppler vibrometer is capable of accurate, non-contact surface vibration measurement thanks to Doppler shifts between the incident light from and scattered light back to the vibrometer [12]. A laser Doppler vibrometer becomes a scanning laser Doppler vibrometer when equipped with a scanner that consists of a pair of orthogonal scan mirrors and its laser beam can be directed to any positions visible to the scanner. However, point-by-point acquisition of a vibration shape of a large-sized structure with a dense measurement grid can be time-consuming even though it can be achieved in an automatic and well-controlled manner. The idea of continuous scanning was first proposed in Refs. [13, 14], where a scanning laser Doppler vibrometer continuously sweeps its laser over a surface of a structure under sinusoidal excitation to obtain its vibration shapes that are approximated by Chebysev series. Two vibration shape measurement methods for continuous scanning, including demodulation and polynomial methods, were later proposed in Refs. [15–17]. The demodulation method was extended for structures under impact and multi-sine excitation in Refs. [18] and [19], respectively. A lifting method was proposed to obtain mode shapes of a structure under impact excitation by treating its free response measured by a CSLDV system as that of a linear time-periodic system [20]. The demodulation and polynomial methods were synthesized to identify damage in beams, where damage behind an intact surface of a beam could be identified by use of a CSLDV system that scanned the intact surface [21]. A type of vibration shapes called free-response shapes was developed and a corresponding damage identification method was proposed to identify damage in beams undergoing free vibration [22]. Damage indices associated with multiple elastic modes of beams can be obtained in one measurement by a CSLDV system and damage can be identified near neighborhoods with consistently high damage index values. In this work, the polynomial- and wavelet-based identification methods are extended from beam damage identification to laminated composite plate delamination identification using a CSLDV system. A vibration shape of a plate is formed by vibration shapes along multiple straight scan paths spanning a scan area assigned on the plate. Damage indices on each scan path from the two methods constitute corresponding damage indices of a whole plate and edges of delamination areas can be identified near neighborhoods with high damage index values. A composite plate with delamination was manufactured to experimentally investigate effectiveness of the two methods and experimental delamination identification results were compared with that of a C-scan image of the plate. 2.2 Methodology 2.2.1 Damage Identification Using Curvature Vibration Shapes A curvature vibration shape of a beam is the second-order spatial derivative of a vibration shape z along the direction of its length x0, and it can be calculated using a central finite difference scheme: zx0x0 .x0/ D z.x0 h/ 2z.x0/ Cz.x0 Ch/ h2 (2.1) wherehis the distance between two neighboring measurement points of z alongx0. Structural damage can cause local changes of bending stiffness of a beamEI. Since the relationship between zx0x0 and EI can be expressed by [8] zx0x0 x0 D M.x0/ EI .x0/ (2.2) with Mdenoting the bending moment of the beam, prominent local anomalies can be observed in zx0x0 in neighborhoods of the damage, and the damage can be identified by inspecting zx0x0 for anomalies. A similar relationship between bending stiffness of a plate Eh3 12. 2 1/ and its curvatures along an arbitrary scan path of a CSLDV system, withx and y denoting its tangential and normal directions, respectively, can be expressed by [23] Mxx Myy D Eh3 12. 2 1/ 1 1 zxx zyy (2.3) where zxx and zyy denote curvatures of the plate along x and y, respectively; Mxx, Myy and denote bending moments along x and y and Poison’s ratio of the plate, respectively. As structural damage can cause changes to Eh3 12. 2 1/ , prominent local anomalies can be observed in neighborhoods of the damage, and the damage can be identified by inspecting zxx and/or zyy. Straight scan paths are assigned on a plate and its vibration shapes are formed by those along assigned straight scan paths on the plate. The central finite difference scheme in Eq. (2.1) can be applied to calculate zxx, and it can be expressed by zxx .x/ D z.x Ch/ 2z.x/ Cz.x h/ h2 (2.4)
2 Delamination Identification of Laminated Composite Plates Using a Continuously Scanning Laser Doppler Vibrometer System 11 By assuming that measurement points on a straight scan path are equally spaced, the smaller hin Eq. (2.4) the more accurate the resulting zxx, which is true when z is free of measurement noise. When z is contaminated by measurement noise, it can be expressed by z.x/ Dz NF .x/ C .x/ (2.5) wherezNF and denote a noise-free vibration shape and measurement noise, respectively. Substituting Eq. (2.5) into Eq. (2.7) yields zxx .x/ D zNF.xCh/C .xCh/ 2z NF.x/ 2 .x/Cz NF.x h/C .x h/ h2 D z NF.xCh/ 2z NF.x/CzNF.x h/ h2 C .xCh/ 2 .x/C .x h/ h2 (2.6) When h is small, adverse effects of can be drastically amplified in the resulting zxx due to the term .xCh/ 2 .x/C .x h/ h2 in Eq. (2.6). One can modify Eq. (2.4) by introducing a resolution parameter n: zxx .x/ D z.x Cnh/ 2z.x/ Cz.x nh/ .nh/2 (2.7) Substituting Eq. (2.5) into Eq. (2.7) yields zxx .x/ D zNF.xCnh/C .xCnh/ 2z NF.x/ 2 .x/Cz NF.x nh/C .x nh/ .nh/2 D z NF.xCnh/ 2z NF.x/CzNF.x nh/ .nh/2 C .xCnh/ 2 .x/C .x nh/ .nh/2 (2.8) By comparing the term .xCh/ 2 .x/C .x h/ h2 in Eq. (2.6) and the term .xCnh/ 2 .x/C .x nh/ .nh/2 in Eq. (2.8), one can see that adverse effects of measurement noise on zxx can be alleviated in the latter by n2 times due to the introduction of n. A suitable value of n is the least one with which zxx seems free of averse effects of measurement noise and one can obtain such a value by increasingnfrom a small value and observing resultingzxx.Whenx in Eq. (2.4) is close to a boundary of z, either z.x Ch/ or z.x h/ may not exist and the central finite difference scheme can fail. A mode shape extension technique was proposed in Ref. [7] to append virtual mode shape extensions to boundaries of a mode shape so that both z.x Ch/ and z.x h/ become available even whenxin Eq. (2.4) is close to a boundary of z. The mode shape extension technique is applicable to a vibration shape. 2.2.2 Demodulation Method Application of the demodulation method for obtaining a vibration shape along a straight scan path using a CSLDV system is summarized below. Steady-state response of a structure under sinusoidal excitation with a frequency f measured by a CSLDV system can be expressed by r.t/ Dz.x.t//cos.2 ft ˛ / (2.9) where x.t/ denotes the spatial position of the laser spot from the CSLDV system on the structure, ˛is the difference between a phase determined by initial conditions of the structure and that determined by a mirror feedback signal, and is a phase variable that controls amplitudes of in-phase and quadrature components of z, which can be expressed by zI .x/ Dz.x/cos.˛C / (2.10) and zQ.x/ Dz.x/sin.˛C / (2.11)
12 D.-M. Chen et al. respectively [21]. Each obtained vibration shape from the demodulation method corresponds to a half-scan period. A halfscan period starts when the laser spot of the CSLDV system arrives at one end of a scan path and ends when the laser spot arrives at the other end. The steady-state response r.t/ in Eq. (2.9) can be further expressed by r.t/ DzI .x.t//cos.2 ft/ CzQ.x.t//sin.2 ft/ (2.12) Multiplying Eq. (2.12) by cos.2 ft/ and sin.2 ft/ yields r.t/cos.2 ft/ DzI .x.t//cos 2 .2 ft/ Cz Q.x.t//sin.2 ft/cos.2 ft/ D 1 2 zI .x.t// C 1 2 zI .x.t//cos.4 ft/ C 1 2 Q .x.t//sin.4 ft/ (2.13) and r.t/sin.2 ft/ DzQ.x.t//cos.2 ft/sin.2 ft/ CzQ.x.t//sin2 .2 ft/ D 1 2 zQ.x.t// 1 2 zQ.x.t//cos.4 ft/ C 1 2 zQ.x.t//sin.4 ft/ (2.14) respectively. A low-pass filter is applied to r.t/cos.2 ft/ and r.t/sin.2 ft/ to obtain 1 2 zI and 1 2 zQ, respectively, and terms corresponding to the frequency2f in Eqs. (2.13) and (2.14) can be removed. Further, zI andzQ can be obtained by multiplying the corresponding filtered response by two. The value of in Eq. (2.11) can be optimized so that zI and zQ attain their maximum and minimum amplitudes, respectively. In what follows, all vibration shapes from the demodulation method are presented as their in-phase components with maximum amplitudes, which are denoted by z for convenience. 2.2.3 Polynomial-Based Damage Identification As mentioned in Sect. 2.2.1, damage can cause prominent local anomalies in curvature vibration shapes and be identified by inspecting curvature vibration shapes. Such anomalies can be identified by comparing curvature vibration shapes of undamaged and damaged structures. However, vibration shapes and curvature vibration shapes of undamaged structures are usually unavailable in practice. It was shown in Ref. [7] that curvature vibration shapes of an undamaged beam can be well approximated by those from polynomials that fit vibration shapes of a damaged beam outside their boundary regions. Though vibration shapes from polynomial fits may not satisfy boundary conditions of damaged and undamaged beams, curvature vibration shapes associated with the vibration shapes can be used for damage identification purposes outside boundary regions. The mode shape extension technique in Ref. [7] can yield curvature mode shapes from polynomial fits that well approximate those of whole undamaged beams. However, this technique is inapplicable to vibration shapes of beams that are obtained using the demodulation method, as their curvature vibration shapes have inherent local distortions near boundaries of scan paths [21]. Due to the same reason, the mode shape extension technique mentioned in Ref. [7] is not applicable to vibration shapes of plates. A polynomial that fits z of a damaged plate along a straight scan path with order r can be expressed by zp .x/ D r X qD0 aqx q (2.15) where aq are coefficients of the polynomial that can be obtained by solving a linear equation Ua Dz (2.16) inwhich Uis an M .r C1/ Vandermonde matrix withMbeing the number of measurement points of z: UD 2 6 6 6 4 1 x1 x2 1 x r 1 1 x2 x2 2 x r 2 : : : : : : : : : : : : : : : 1 xM x2 M x r M 3 7 7 7 5 (2.17)
2 Delamination Identification of Laminated Composite Plates Using a Continuously Scanning Laser Doppler Vibrometer System 13 a D a0 a1 : : : ar T is an .r C1/-dimensional coefficient vector, and z is the vibration shape vector to be fit. To avoid ill-conditioning of U, it is proposed that x in Eq. (2.15) be normalized using the “center and scale” technique [24] before formulating the linear equation in Eq. (2.15). The normalized coordinate Ox can be expressed by Ox D 2x 2Nx l (2.18) where Nx is the x-coordinate of the center point of a straight scan path and l is its length. In order to determine the proper value of r, a fitting index fit and a convergence index con can be used. Fitting index fit at r can be expressed by fit .r/ D RMS.z/ RMS.z/ CRMS.e/ 100% (2.19) whereRMS. / denotes the root-mean-square value of a vector ande DUa z is an error vector. Convergence index con can be expressed by con.r/ Dfit .r/ fit .r 2/ (2.20) where r 3. When fit is close to 100%, zp completely fits z; the lower fit, the lower the level of fitting of zp to z.When con is close to 0, increasing r cannot further improve how well zp fits z; the lower con, the higher the level of convergence of zp. It is proposed in this work that the proper value of r be the minimum value of r with which con.r/ is below0:05%.Whenzp is obtained from a polynomial with a proper order, a curvature damage index can be defined along a straight scan path at x [7]: .x/ Dˇ ˇ zxx .x/ z p rxx .x/ˇ ˇ 2 (2.21) and the damage can be identified in neighborhoods with high . When associated with n different vibration shapes are available, an auxiliary curvature damage index a is proposed: a .x; y/ DXN k .x; y/ (2.22) where N k is the normalized damage index associated with the k-th available vibration shape, whose maximum index value is one. Due to inherent local distortions in curvature vibration shapes near boundaries of scan paths, only outside boundary regions are considered for delamination identification purposes here. 2.2.4 Wavelet-Based Damage Identification A wavelet transformation can be defined by W z.u; s/ DZ C1 1 z.x/ u;s .x/dx (2.23) where W denotes a wavelet transformation operator with a wavelet function u;s .x/ D 1 ps x u s (2.24) inwhichuands are spatial and scale parameters of u;s, respectively, and the superscript denotes complex conjugation. In this work, a wavelet function is defined in the real domain and the superscript in Eq. (2.23) can be dropped. The wavelet transformation in Eq. (2.23) can be expressed in the form of a convolution: W z.u; s/ D 1 ps Z C1 1 z.x/ u x s dx D 1 ps z ˝ (2.25) where an overbar denotes function reflection over the y-axis, i.e., .x/ D . x/, and ˝denotes convolution.
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