River Rapids Conference Proceedings of the Society for Experimental Mechanics Series Rotating Machinery, Hybrid Test Methods, Vibro-Acoustics & Laser Vibrometry, Volume 8 James De Clerck David S. Epp Proceedings of the 34th IMAC, A Conference and Exposition on Structural Dynamics 2016 River Publishers
Conference Proceedings of the Society for Experimental Mechanics Series Series Editor Kristin B. Zimmerman, Ph.D. Society for Experimental Mechanics, Inc., Bethel, CT, USA
River Publishers James De Clerck • David S. Epp Editors Rotating Machinery, Hybrid Test Methods, Vibro-Acoustics & Laser Vibrometry, Volume 8 Proceedings of the 34th IMAC, A Conference and Exposition on Structural Dynamics 2016
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Preface Rotating Machinery, Hybrid Test Methods, Vibro-Acoustics, and Laser Vibrometry represent one of ten volumes of technical papers presented at the 34th IMAC, A Conference and Exposition on Structural Dynamics, organized by the Society for Experimental Mechanics and held in Orlando, Florida, on January 25–28, 2016. The full proceedings also include volumes on Nonlinear Dynamics; Dynamics of Civil Structures; Model Validation and Uncertainty Quantification; Dynamics of Coupled Structures; Sensors and Instrumentation; Special Topics in Structural Dynamics; Structural Health Monitoring, Damage Detection, and Mechatronics; and Shock and Vibration, Aircraft/Aerospace, Energy Harvesting, Acoustics and Optics, 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 for Modal Analysis measurements and applications of Modal Analysis in specific application areas. The organizers would like to thank the authors, presenters, session organizers, and session chairs for their participation in this track. Houghton, MI, USA James De Clerck Albuquerque, NM, USA David S. Epp v
Contents 1 Practical Techniques for Scaling of Optically Measured Operating Deflection Shapes ......................... 1 Peyman Poozesh, Danilo Damasceno Sabino, Javad Baqersad, Peter Avitabile, and Christopher Niezrecki 2 Prediction of the Coupled Impedance from Frequency Response Data........................................... 19 Ramona Fagiani, Elisabetta Manconi, and Marcello Vanali 3 Real-Time State Detection in Highly Dynamic Systems............................................................. 27 Ryan A. Kettle, Andrew J. Dick, Jacob C. Dodson, Jason R. Foley, and Steven R. Anton 4 Stereo-DIC Measurements of Thermal Gradient Effects on the Vibratory Response of Metals ............... 35 Ryan Berke, Ravinder Chona, Arthur Ding, John Lambros, Eann Patterson, and Christopher Sebastian 5 Modal Testing of a Nose Cone Using Three-Dimensional Scanning Laser Doppler Vibrometry............... 43 Daniel P. Rohe 6 A Mathematical Model for Determining the Pose of a SLDV ...................................................... 57 Da-Ming Chen and W.D. Zhu 7 Operational Modal Analysis with a 3D Laser Vibrometer Without External Reference........................ 75 Simon Marwitz and Volkmar Zabel 8 Scanning LDV Measurement Technology for Vibration Fatigue Testing ......................................... 87 Fabrizio Magi, Dario Di Maio, and Ibrahim Sever 9 Optically Detecting Wavefronts and Wave Speeds in Water Using Refracto-Vibrometry ...................... 95 Matthew T. Huber, Brent K. Hoffmeister, and Thomas M. Huber 10 Stochastic Wavenumber Estimation: Damage Detection Through Simulated Guided Lamb Waves .......... 105 Garrison N. Stevens, Kendra L. Van Buren, Eric B. Flynn, Sez Atamturktur, and Jung-Ryul Lee 11 Use of Continuous Scanning LDV for Diagnostics................................................................... 127 Dario Di Maio 12 A Cost Effective DIC System for Measuring Structural Vibrations ............................................... 139 Markus J. Hochrainer 13 Teaching DSP and Dynamic Measurements at the Graduate Level at Michigan Technological University... 147 Jason R. Blough 14 Flipping the Classroom for a Class on Experimental Vibration Analysis......................................... 155 Anders Brandt and Christopher Kjær 15 Lessons Learned from Operational Modal Analysis Courses at the University of Molise ...................... 161 Carlo Rainieri and Giovanni Fabbrocino 16 Authentic Engineering Assignments for an Undergraduate Vibration Laboratory Class....................... 169 James P. De Clerck and Jean S. De Clerck vii
viii Contents 17 Vibration and Acoustic Analysis of Acoustic Guitar in Consideration of Transient Sound..................... 177 Nobuyuki Okubo, Naoaki Iwanaga, and Takeshi Toi 18 Demarcation for the Coupling Strength in the MODENA Approach ............................................. 187 Peng Zhang, Shaoqing Wu, Yanbin Li, and Qingguo Fei 19 Vibro-Acoustic Modal Model of a Traction Motor for Railway Applications .................................... 197 Fredrik Botling, Hanna Amlinger, Ines Lopez Arteaga, and Siv Leth 20 Operational Deflection Shapes of a PWM-Fed Traction Motor.................................................... 209 Hanna Amlinger, Fredrik Botling, Ines Lopez Arteaga, and Siv Leth 21 Acoustic Fatigue and Dynamic Behavior of Composite Panels Under Acoustic Excitation..................... 219 Canan Uz and Tamer T. Ata 22 Evaluation of Microphone Density for Finite Element Source Inversion Simulation of a Laboratory Acoustic Test .......................................................................................... 231 Ryan Schultz and Tim Walsh 23 Experimental Mapping of the Acoustic Field Generated by Ultrasonic Transducers ........................... 243 Songmao Chen, Christopher Niezrecki, and Peter Avitabile 24 Enhanced Spin-Down Diagnostics for Nondestructive Evaluation of High-Value Systems ..................... 255 David Sehloff, Clark Shurtleff, Josh Pribe, Colin Haynes, and John Heit 25 Performing Direct-Field Acoustic Test Environments on a Sandia Flight System to Provide Data for Finite Element Simulation.......................................................................................... 267 Eric C. Stasiunas, Ryan A. Schultz, and Mike R. Ross 26 Smooth Complex Orthogonal Decomposition Applied to Traveling Waves in Elastic Media................... 281 Rickey A. Caldwell Jr. and Brain F. Feeny 27 Subspace Algorithms in Modal Parameter Estimation for Operational Modal Analysis: Perspectives and Practices.............................................................................................. 295 S. Chauhan 28 An Application of Multivariate Empirical Mode Decomposition Towards Structural Modal Identification..................................................................................................... 303 AyanSadhu 29 Dynamic Characterization of Milling Plant Columns ............................................................... 311 Filippo Cangioli, Steven Chatterton, Paolo Pennacchi, and Edoardo Sabbioni 30 Mixed Force and Displacement Control for Base-Isolation Bearings in RTHS .................................. 323 Richard Erb, Matthew Stehman, and Narutoshi Nakata 31 Leveraging Hybrid Simulation for Vibration-Based Damage Detection Studies................................. 333 Timothy P. Kernicky, Michael Tedeschi, and Matthew J. Whelan 32 Real Time Hybrid Simulation with Online Model Updating on Highly Nonlinear Device...................... 343 Ge Ou and Shirley J. Dyke 33 Discrete-Time Compensation Technique for Real-Time Hybrid Simulation ..................................... 351 Saeid Hayati and Wei Song 34 Evaluating the Effectiveness of a Lodengraf Damping Approach for String Trimmers......................... 359 Nicholas Swanson and Jack Leifer 35 Using Operating Data to Locate and Quantify Unbalance in Rotating Machinery.............................. 375 Shawn Richardson, Mark Richardson, Jason Tyler, and Patrick McHargue
Contents ix 36 Gear Dynamics Characterization by Using Order-Based Modal Analysis........................................................................................................... 387 Emilio Di Lorenzo, Antonio Palermo, Simone Manzato, Andrea Dabizzi, Bart Peeters, Wim Desmet, and Francesco Marulo 37 A Design Framework to Improve the Dynamic Characteristics of Double Planet Planetary Gearsets........ 405 Dylan C. Fyler, Murat Inalpolat, Sang Hwa Lee, and Hyun Ku Lee 38 Dynamics and Pareto Optimization of a Generic Synchronizer Mechanism..................................... 417 Muhammad Irfan, Viktor Berbyuk, and Håkan Johansson 39 Modeling and Characterization of a Flexible Rotor Supported by AMB......................................... 427 Marcus Vinicius Fernandes de Oliveira, Adriano Borges Silva, Adailton Borges Silva, Edson Hideki Koroish, and Valder Steffen Jr. 40 Nonlinear Reduced Order Modeling of a Curved Axi-Symmetric Perforated Plate: Comparison with Experiments ........................................................................................................ 437 David A. Ehrhardt and Matthew S. Allen 41 Reduced Order Models for Systems with Disparate Spatial and Temporal Scales............................... 447 Shahab Ilbeigi and David Chelidze 42 Using NNMs to Evaluate Reduced Order Models of Curved Beam............................................... 457 Christopher I. VanDamme and Matthew S. Allen 43 Simulation of Rotor Damping Assembled by Disc Shrink Fits ..................................................... 471 Lothar Gaul and André Schmidt 44 Developments in the Prediction of Full Field Dynamics in the Nonlinear Forced Response of Reduced Order System Models ..................................................................................... 481 Sergio E. Obando and Peter Avitabile 45 On the Behaviour of Structures with Many Nonlinear Elements .................................................. 509 T. Rogers, G. Manson, and K. Worden 46 Estimation of Instantaneous Speed for Rotating Systems: New Processing Techniques ........................ 521 Achyut Vemuri, Randall J. Allemang, and Allyn W. Phillips 47 Identification of Breathing Cracked Shaft Models from Measurements .......................................... 537 Michael I. Friswell, Ralston Fernandes, Nidhal Jamia, and Sami El-Borgi
Chapter 1 Practical Techniques for Scaling of Optically Measured Operating Deflection Shapes Peyman Poozesh, Danilo Damasceno Sabino, Javad Baqersad, Peter Avitabile, and Christopher Niezrecki Abstract Operational Modal Analysis (OMA) is used to identify vibration patterns of large structures under unknown operating conditions. However, operating data extracted from output-only measurements is not scaled and cannot be used for Structural Dynamic Modification (SDM), frequency response function (FRF) synthesis, force estimation and structural response simulation. Therefore, developing an algorithm that is able to extract scaled mode shapes using measured operating data is desirable. In the current paper, two different scaling techniques including drive point scaling as well as mass sensitivity scaling are employed to scale optically measured operating deflection shapes (ODS). To evaluate the capability of each scaling technique, the scaled optically measured operating shapes are compared to mode shapes extracted using input– output measurements (reference mode shapes). Additionally, the scaled operating shapes are used in structural dynamic modification to demonstrate the benefits and drawbacks associated with the mass sensitivity technique. The results reveal that both mass sensitivity and drive point scaling techniques are capable of effectively scaling optically measured operating deflection shapes of the structure. Keywords Operating deflection shapes • Scaling techniques • Operational modal analysis • Digital image correlation • Structural dynamics • Mass sensitivity Nomenclature f gk1 ODS of the unmodified structure [M1] Mass matrix of the unmodified structure f gk2 ODS of the modified structure [K1] Stiffness matrix of the unmodified structure [E1] Scaled ODS of the unmodified structure [M2] Mass matrix of the modified structure [E2] Scaled ODS of the modified structure [K2] Stiffness matrix of the modified structure [Eun 12] Unscaled mode contribution matrix [A(s)] Residue matrix [E˜2] Estimated mode shapes of the modified structure ˛k Initial scaling factor for unmodified structure [E12] Scaled mode contribution matrix ˇk Initial scaling factor for modified structure [Eref] Reference shapes extracted using input–output measurement !k1 Natural frequencies of unmodified structure pk Pole location !k2 Natural frequencies of modified structure [ M12] Mass modification matrix P. Poozesh ( ) • J. Baqersad • P. Avitabile • C. Niezrecki Structural Dynamics and Acoustic Systems Laboratory, University of Massachusetts Lowell, One University Avenue, Lowell, MA 01854, USA e-mail: Peyman_Poozeshstudent.uml.edu D.D. Sabino Unive Estadual Paulista, Ilha Solteira, SP, Brazil © The Society for Experimental Mechanics, Inc. 2016 J. De Clerck, D.S. Epp (eds.), Rotating Machinery, Hybrid Test Methods, Vibro-Acoustics & Laser Vibrometry, Volume 8, Conference Proceedings of the Society for Experimental Mechanics Series, DOI 10.1007/978-3-319-30084-9_1 1
2 P. Poozesh et al. 1.1 Introduction Experimental Modal Analysis (EMA), based on input–output system identification technique, is extensively employed to identify dynamic characteristics of structures. In this approach, modal parameters of a structure can be identified using known applied forces and the measured responses [1]. However, there are some cases that measuring input forces is an arduous task and requires expensive tools [2, 3]. Unlike EMA, Operational Modal Analysis (OMA) is a technique that estimates modal parameters of the structure using the vibration response measurements [4, 5]. Because the input forces are not measured in OMA, the extracted shapes that are called unscaled operating mode shape are not mass normalized [6, 7]. These unscaled operating deflections shapes cannot be used for further structural dynamic studies such as damage detection, model updating, structural modification and force estimation. Several approaches based on updating finite element model and structural dynamic modification have been proposed to scale operating deflection shapes [8, 9]. Researchers have also used the single drive point measurement and mass sensitivity techniques to scale operating deflection shapes measured using conventional sensors [10, 11]. Using a single drive point measurement, the scale of each mode shape at the drive point location can be computed. The value of mode shapes at the drive point is then used to scale the operating deflection shapes. With this method, it is possible to only scale a single mode without the knowledge of any other modes. It is worth mentioning that this technique requires at least one force measurement [10]. After extracting the scaled mode shapes and poles (frequency and damping values), a FRF synthesis can be performed using operating data. The newly synthesized set of FRFs can accurately describe the system’s dynamic behavior for the specified frequency range. Mass sensitivity technique uses the structure’s sensitivity to a change in mass to calculate the scaling factor. Adding a small mass to the structure will result in a shift in resonant frequencies. This frequency shift can be related to the derivative of the eigenvalue. Using this fact, the scaling factor can be extracted from operating data with no force measurement [12, 13]. The ideal mass modification in this technique depends on the mass and the flexibility of the structure. However, the mass modification must be small enough to prevent significant change in the mode shapes of the structure [14]. Aenlle et al. [11] proposed two mass sensitivity techniques based on structural dynamic modification and the accuracy of each method was evaluated by using simulated models. Furthermore, Parloo et al. [15] used mass sensitivity technique to scale the operating deflection shapes of a bridge. In another effort, Brincker et al. [14] utilized the mass change technique to estimate the scaling factors associated with operating deflections shapes of a four story building. In addition, the uncertainties on the estimated scaling factors using different mass sensitivity techniques were evaluated. In another work by Aenlle et al. [9], a technique was proposed to optimize the mass change strategy including the number, locations and the weight of the masses. Khatibi et al. [16] showed that mass-stiffness change can be effective in scaling operating deflection shapes. The numerical and experimental results revealed that the stiffness change method shows the same higher effect on the accuracy of the lower order modes than mass change method. Therefore, it was suggested that mass sensitivity technique in conjunction with the stiffness change technique is more effective than only the mass sensitivity technique in scaling operating deflection shapes extracted from output-only measurements. Three-dimensional (3D) digital image correlation (DIC) and three-dimensional point tracking (3DPT) are non-contacting measurement approaches that provide alternatives to traditional measurement sensors and LDVs [17–19]. Both 3D DIC and 3DPT are based on stereophotogrammetry principles and rely on a pair of digital cameras to capture images of the structure over a period of time. For the DIC approach, speckled patterns are applied to a structure and can be used to obtain full-field displacement and strain over the entire area of interest. Likewise, 3DPT can be used to measure displacement at discrete points by mounting optical targets to the structure. DIC and 3DPT have matured over the last two decades and have been primarily applied to the field of experimental solid mechanics [20, 21]. However, more recently researchers have begun to exploit optically based approaches for measuring vibration and transient phenomena in turbine blades and rotors. There have been several published papers that use photogrammetry in conjunction with Operational Modal Analysis to measure vibration in non-rotating and rotating turbine blades [20, 22]. The current paper is built on a foundation of research to understand how to use a non-contact stereophotogrammetry technique with output-only system identification methods to accurately estimate modal parameters in a structure during operation. Within this paper, the single drive point measurement techniques as well as the mass sensitivity technique are employed to evaluate the effectiveness of each technique in scaling operating deflections shapes. Furthermore, the traditional mass sensitivity technique is improved by introducing SDM process and using the mode contribution matrix. The modal assurance criteria (MAC) [23], orthogonally check, and structural dynamic modification (SDM) [25] are the tools that will be used to evaluate the effectiveness of each of the scaling techniques.
1 Practical Techniques for Scaling of Optically Measured Operating Deflection Shapes 3 1.2 Theoretical Background Within the current paper, two significant techniques are utilized, drive point scaling and mass sensitivity scaling. Both techniques are described in the following sections. 1.2.1 Drive Point Scaling Technique The operating deflections shapes tend to be highly correlated with the corresponding mode shape of the system. This correlation can be expressed as fugk D’kf‰gk (1.1) where ’0k is the scale factor between the operating deflection shapes (f‰gk) and the scaled mode shapes (fugk) at the kth mode of the structure. The scaled mode shapes and operating deflection shapes can be expressed as 8ˆ ˆ< ˆˆ: u1k u2k u3k : : : 9> >= >>; Dfugk; 8ˆ ˆ< ˆˆ: ods1k ods2k ods3k : : : 9> >= >>; Df‰gk (1.2) The drive point scaling technique relies on the use of a single drive point measurement to properly scale operating deflection shapes. The FRF obtained from the single drive point measurement can be shown as ŒH.s/ s Dj¨ DŒH.j¨/ D mX kD1 ŒAk .j¨ pk/ C A k j¨ p k (1.3) For a particular mode k, Eq. (1.3) can be expressed as ŒH.s/ s Dpk Df ukg qk s pkf ukg T (1.4) with the residue matrix calculated using ŒA.s/ k Dqk fukgfukg T (1.5) Equation (1.5) can be further simplified for a single reference point as 8ˆ ˆ< ˆˆ: a1k a2k a3k : : : 9> >= >>; Dqkurk 8ˆ ˆ< ˆˆ: u1k u2k u3k : : : 9> >= >>; !aijk Dqkuikujk ; qk D 1 2j¨k (1.6) where uik andujk are the kth mode of the structure at point i and j respectively. The residues are directly related to the system mode shapes and are scaled by qk, the scaling constant for the k th mode. From Eq. (1.6), it becomes evident that by taking a measurement at the drive point location, urk, a value for modal scaling can be determined. The drive point measurement provides a scaled mode shape for the drive point DOF as arrk Dqkurkurk (1.7) With this drive point scaling, the scaled mode shapes, for the corresponding drive point measurement location, can be used to scale the operating mode shapes
4 P. Poozesh et al. Scaled Operating Mode Shape ! 8ˆ ˆˆˆˆ< ˆˆˆˆˆ: ods1k ods2k : : : odsik : : : odsnk 9> >>>>= >>>>>; urk odsrk (1.8) It is important to note that the reference location selected for the operating deflection shape extraction does not need to be the same as the drive point selected for the scaling of the operating deflection shapes [10]. 1.2.2 Mass Sensitivity Scaling Technique Unlike the drive point scaling, the mass sensitivity scaling technique requires no FRF measurements. This technique relies on the sensitivity of the eigensolution of the system to an adequate change in the mass [15]. By adding masses to the structure, a shift in its natural frequency occurs. The frequency shift coupled with the value of the mass modification can be measured and used to determine the effective scaling constant. The eigensolution for a system without damping is ŒK1 ¨2 k1 ŒM1 ’kf‰gk1 D0 (1.9) where f gk1 is the unscaled mode shape of the original structure, !k1 the natural frequency, [M1] mass matrix and [K1] the stiffness matrix. After mass modification (ŒMC M ), the new eigenvalues and eigenvectors can be found using ŒK1 ¨2 k2 ŒM1 C M12 ’kf‰gk2 D0 (1.10) where f gk2 and !k2 are the operating deflection shape and natural frequencies of the modified structures. Subtracting Eq. (1.10) from Eq. (1.9) gives a relation between these two equations. This relation will be used further in determining the scaling factors ŒK1 .’kf‰gk1 ’kf‰gk2/ ŒM1 ’kf‰gk1¨2 k1 ’kf‰gk2¨2 k2 Œ M12 ’kf‰gk2¨2 k2 D0 (1.11) Equation (1.11) can be greatly simplified by assuming that change in the mode shape due to the small mass modification is unnoticeable. f‰gk1 D 8ˆ ˆˆˆˆ< ˆˆˆˆˆ: ods11k ods12k : : : ods1ik : : : ods1nk 9> >>>>= >>>>>; ; f‰gk2 D 8ˆ ˆˆˆˆ< ˆˆˆˆˆ: ods21k ods22k : : : ods2ik : : : ods2nk 9> >>>>= >>>>>; ! f‰gk1 Šf‰gk2 Šf‰gk (1.12) By assuming that mode shapes do not change after mass modification, Eq. (1.11) will be simplified as ’kf‰g T k ŒM1 ’kf‰gk ¨2 k1 ¨2 k2 ’kf‰g T k Œ M12 ’kf‰gk¨2 k2 D0 (1.13) Scaled mode shapes ([M1]) possesses the orthogonality properties given as ŒE1 T ŒM1 ŒE1 DŒI !’ 2 kf‰g T k ŒM1 f‰gk D1 (1.14)
1 Practical Techniques for Scaling of Optically Measured Operating Deflection Shapes 5 Taking into account the orthogonality of mode shapes with respect to mass, Eq. (1.13) can be simplified as ¨2 k1 ¨2 k2 ’ 2 kf‰g T k Œ M12 f‰gk¨2 k2 D0 (1.15) Equation (1.15) can now be solved for the scale factor, and the resultant equation is ’k Ds ¨2 k1 ¨2 k2 !2 k2f‰g T k Œ M12 f‰gk (1.16) Equation (1.16) is similar to the equation developed by Parloo et al. [24]. As can be seen from Eq. (1.16), only the operating deflection shapes and the resonant frequencies are needed to find the scaling factor associated to each mode. To determine scaling factors using Eq. (1.16), both modified and unmodified operating shapes can be used. 1.2.3 Structural Dynamic Modification Because of several assumptions made to develop the mass sensitivity scaling technique, some sources of errors are present [10]. The structural dynamic modification concepts in conjunction with the mass sensitivity technique are used to evaluate the merits associated with the mass sensitivity technique. The basis for the theory of structural dynamic modification is directly related to the equation of motion [11]. The equation of motion for the modified structure in modal space is represented as 2 664 2 664 : : : I : : : 3 775C M12 3 775fP2gC 2 664 : : : ¨2 2 : : : 3 775fP2gDŒ0 (1.17) Modal mass modification matrix ( M12), can be calculated using M12 DŒE1 T Œ M12 ŒE1 (1.18) It is important to note that while part of Eq. (1.17) contains a diagonal representation of the modal mass of the original system, the modal mass modification part ( M12 ) is not diagonal, and therefore Eq. (1.17) is coupled. The eigenvector ([E12]) is obtained from the eigensolution for Eq. (1.17) and can be used to transform from original modal space (unmodified system) to second modal space (modified structure). fP1gDŒE12 fP2g (1.19) Equation (1.19) uncouples the set of equations that were coupled from modification from the original unmodified state to the second modified state. The original modal transformation is given by fXgDŒE1 fPg (1.20) Substituting Eq. (1.19) into Eq. (1.20) will give fXgDŒE1 ŒE12 fP2g (1.21) Estimated scaled mode shapes of the modified structure are obtained from linear combination of the unmodified scaled modes shapes with the normalizing matrix given as QE2 DŒE1 ŒE12 (1.22) Equation (1.22), can be used to approximate the modified modal parameters due to structural changes using only the original modal parameters of the structure [25]. Figure 1.1 shows the mass modification process of a structure using SDM. The
6 P. Poozesh et al. Fig. 1.1 A schematic of the structural dynamic modification algorithm used to estimate the scaled mode shapes from operating deflection shapes of the modified and unmodified structure scaling factors for modified and unmodified structures can be found using ’k Ds ¨2 k1 ¨2 k2 !2 k2f‰g T k1 Œ M12 f‰gk1 (1.23) ˇk Ds ¨2 k1 ¨2 k2 !2 k2f‰g T k2 Œ M12 f‰gk2 (1.24) Because operating deflection shapes tend to be highly correlated with the corresponding scaled mode shapes of the system, the scaled mode shapes of the unmodified and modified structure can be expressed as ŒE1 DŒ‰1 2 664 : : : ˛ : : : 3 775 (1.25) ŒE2 DŒ‰2 2 664 : : : ˇ : : : 3 775 (1.26)
1 Practical Techniques for Scaling of Optically Measured Operating Deflection Shapes 7 Modal parameters - Unmodified 2 1 1 k k ω ψ Es mated Modified Mode Shape [ ] [ ] 2 1 12 ⎡ ⎤ ⎢ ⎥ ⎡⎣ ⎤⎦ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ E E ψ α ( ) [ ] 2 2 1 2 2 2 1 1 .{ } { } − = ⋅ Δ ⋅ k k k T k k k M ω ω α ω ψ ψ Ini al Scaling Factor Scaled Unmodified Mode Shape [ ] 1 1 ⎡ ⎤ ⎢ ⎥ ⎡⎣ ⎤⎦ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ E ψ α Structural Dynamic Modifica on - SDM [ ] [ ] ( ) 2 1 12 12 1 12 1 2 1 2 2 12 2 2 , = +Δ Δ = Δ = ⎡ ⎤ = ⎣ ⎦ T M M M M E M E K K E eigen K M ω Scaled Modified Mode Shape [ ] 2 2 ⎡ ⎤ ⎢ ⎥ ⎡⎣ ⎤⎦ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ E ψ β Compare Modal parameters - Modified 2 2 2 12 Δ k k M ω ψ System Iden fica on Op cal Measurement ( ) [ ] 2 2 1 2 2 2 2 2 .{ } { } − = ⋅ Δ ⋅ k k k T k k k M ω ω β ω ψ ψ Fig. 1.2 A flowchart showing the scaling of the modified and unmodified structure using mass sensitivity and comparison between estimated mode shapes and experimentally measured shapes Using Eqs. 1.25 and 1.26, the estimated mode shape of the modified structure can be shown as QE2 DŒ‰1 2 664 : : : ˛ : : : 3 775ŒE12 (1.27) The procedure shown in Fig. 1.2 is a general theory for the eigenvalue modification from SDM and scaling operating shapes using the mass sensitivity technique. 1.3 Experimental Test Setup A measurement to experimentally validate the proposed approach for scaling operating deflections shapes was performed on a 5-ft cantilevered aluminum beam (see Fig. 1.3). The cross section of the beam was a rectangular tube with dimensions of 1-in by 2-in and a thickness of 1/8-in. One stereovision system composed of two synchronized complementary metal-oxide
8 P. Poozesh et al. Fig. 1.3 A photo of the test setup showing the measurement system and test structure; (a) optical measurement system including two CMOS cameras and position of optical targets mounted on the cantilevered beam, (b) lumped masses distributed on the beam to quantify the sensitivity of the structure to the mass changes. (c) Locations of accelerometer mounted to the beam Opera ng Deflec on Shapes System Iden fica on Opera ng Response Op cal Measurement Scaling Techniques Scaling Factors Scaled Mode shapes Fig. 1.4 A flowchart showing the scaling of optically measured operating deflection shapes using output only system identification semiconductor (CMOS) high speed cameras were used to capture the operating response of the beam to the impulse excitation. Figure 1.4a shows a pair of four megapixel (2048 2048 pixels) PHOTRON high-speed cameras equipped with 14-mm lenses used for the measurement. The procedure to identify lens distortions was performed by calibrating individual cameras and capturing a series of images from a 1-m calibration cross in different positions/orientations for each camera. The final image was taken when the cameras were installed in their final positions relative to the beam allowing for the identification of the camera positions with respect to each other. The calibration deviation (average intersection error) for the test was calculated as 0.0225 pixels ( 0.007 mm for the measured field of view). According to the PONTOS™ user manual [26] for a good calibration, the calibration deviation needs to be less than or equal to 0.04 pixels. The stereo-vision system was placed approximately 1.3-m away from the beam such that the entire beam was fitted in the field of view of each camera (see Fig. 1.3). The system was configured to record at a rate of 1500 frame per second. Furthermore, proper illumination was provided using several lights. To conduct a good operational modal test, general testing rules similar to a traditional modal test must be followed. To capture the mode shapes of the system, 3DPT technique in conjunction with stochastic subspace system identification (SSI) [27, 28] were employed to estimate dynamic characteristics of the beam. A set of eleven optical targets were mounted along the length of the beam (see Fig. 1.3). This arrangement of optical targets results in a proper spatial resolution of the identified operating deflection shapes. The measurement locations and position of cameras are shown in Fig. 1.3. To perform the drive point measurement, a 100 mv/g accelerometer was mounted to the tip of the beam and an impulse signal generated by an impact hammer was used to excite the beam at the location of the accelerometer (tip of the beam). The acquired data was processed in LMS Test. Lab [29] to compute the FRF used to scale in the drive point scaling technique. The optical measurement system and 3DPT were used to capture the operating vibration response of the beam subjected to the single impact test. The measured response of the beam was processed using SSI to extract the operational deflection
1 Practical Techniques for Scaling of Optically Measured Operating Deflection Shapes 9 shapes and natural frequencies of the structure. Furthermore, eleven 100 mv/g PCB accelerometers were mounted to the opposite side of the beam in longitudinal locations similar to the optical targets to conduct a modal survey using input– output system identification as the reference data (Fig. 1.3c) 1.4 Test Cases Studied Two different scaling techniques were employed to scale deflection shapes extracted using the operating response of the cantilevered beam subjected to the single impact. The location of the reference point and drive point were chosen to be at the tip of the beam. The operating data was collected using the 3DPT technique and then modal parameters of the structure were extracted using SSI. Figure 1.4 represents the workflow that was used to scale the operating deflection shapes. The first operational modal analysis was carried out on the unmodified structure, and then operating shapes were scaled using the drive point scaling technique. Furthermore, the mass sensitivity technique was used to estimate appropriate scaling factors to scale operating deflection shapes. The effectiveness of each technique was evaluated using different correlation tools such as the modal assurance criteria (MAC) and orthogonality check. 1.4.1 Case A: Drive Point Measurement The first scaling approach uses a drive point measurement to scale the optically measured operating deflection shapes of the cantilevered beam. This technique is a linear transformation that scale the deflection shapes by a scaling factor. The scaling factor is calculated by dividing the value of mode shapes (extracted using experimental modal analysis) by the value of the ODS at the drive point degree of freedom, (see Eq. 1.8). These scaling factors can be represented by a diagonal matrix. To scale a deflection shape by a computed scaling vector, each point at different modes would need to be multiplied with this scaling vector. Orthogonality between the reference shapes (input–output measurement) and scaled operating shapes can be computed using 2 664Œ‰k 2 664 : : : ˛ : : : 3 775 3 775 T ŒM ŒEref )Orthogonality (1.28) Mass matrix in Eq. (1.28) is experimental mass given as ŒM DŒEref gT ŒEref g (1.29) Figure 1.5 compares the optically measured operating deflection shapes to reference mode shapes (EMA). The differences between unscaled operating shapes and scaled mode shapes (see Fig. 1.5) indicates the importance of finding an appropriate scaling factor. As can be seen in Fig. 1.6, the scaled ODS shows an excellent agreement with the reference mode shapes. The correlation of the scaled operating shapes and reference mode shapes are depicted in Fig. 1.7. All four scaled operating shapes show strong correlations to the reference mode shapes. As can be seen in Fig. 1.7, high orthogonality between impact test (input–output measurement) and scaled operating shaped indicates that operating shapes are well correlated with reference modes in both shapes and scale. 1.4.2 Case B: Mass Sensitivity Scaling Technique As the second approach in this paper, the operating data was scaled using the conventional mass sensitivity approach. The mass sensitivity technique relies on the amount of mass added to the structure as well as the size of the added masses. With small mass changes, the frequency shift will be minimal and it may be impossible to appropriately scale the data using mass sensitivity. On the other hand, large mass changes will result in mode distortion and consequently in-accurate scaling factors.
10 P. Poozesh et al. Fig. 1.5 Comparison of the operating deflection shapes (OMA test) and reference mode shapes (input–output measurement) for the first four bending modes Fig. 1.6 Comparison of the scaled operating deflection shapes (OMA test) using the drive point technique and reference mode shapes (input– output measurement) for the first four bending modes Therefore, selecting appropriate mass to evaluate the sensitivity of the structure to mass changes is critical. Within this work, five lumped mass modifications (6 % of the beam’s weight) are used to examine the sensitivity of the beam to the mass changes. The estimated natural frequencies of the unmodified and modified structure using SSI for the first four bending modes are shown in Fig. 1.8.
1 Practical Techniques for Scaling of Optically Measured Operating Deflection Shapes 11 MAC Scaled ODS UsingDrive Point 1 2 3 4 Freq(Hz) 14.19 88.13 244.63 467.50 EMA Impact Test 1 14.34 1.000 0.014 0.015 0.006 2 89.28 0.016 1.000 0.020 0.004 3 247.07 0.016 0.018 0.999 0.014 4 471.75 0.019 0.016 0.027 0.965 Scaled ODS UsingDrive Point 1 2 3 4 Freq (Hz) 14.19 88.13 244.63 467.50 EMA Impact Test 1 14.34 1.000 0.003 0.007 0.002 2 89.28 0.0074 0.989 0.006 0.003 3 247.07 0.0058 0.0185 0.967 0.007 4 471.75 0.045 -0.003 -0.025 1.043 MAC Orthogonality Orthogonality Check a b Fig. 1.7 Correlation between operating deflections shapes, scaled using drive point measurement, and reference mode shapes. (a) MAC values indicating degree of consistency between scaled operating shapes and reference mode shapes (input–output measurement). (b) Experimental mass weighted orthogonality check between the scaled operating shapes and reference modal vectors Fig. 1.8 Comparison of the natural frequencies of the modified (redbar) and unmodified (blue bar) cantilevered beam estimated using stochastic subspace identification (SSI) 1.4.3 Case B.1: Estimating Mode Shapes of the Modified Structure Using Unscaled Deflection Shapes Structural dynamic modification can be used to find the mode shapes of a modified structure using the modes of the original structure and the mode contribution matrix ([E12]). In order to demonstrate the necessity of scaling ODS, the operating shapes of the unmodified structure in conjunction with unscaled mode contribution matrix ([Eun 12]) were used in the SDM process to estimate the modified ODS (blue lines in Fig. 1.9). The mode contribution matrix extracted using ODS does not represent the real contribution of each original mode shapes in the response of modified structure. Figure 1.9 shows that the unscaled ODS used in the SDM process does not yield the appropriate modes shapes of the modified structure, and the estimated mode shapes do not correlate well to the final mode shapes of the modified structure. The inconsistency between the estimated and experimentally measured modes is more evident when third and fourth modes are compared (see Fig. 1.9). Orthogonality values, shown in Fig. 1.10, indicate the importance of scaling operating shapes before executing structural
12 P. Poozesh et al. Fig. 1.9 Comparison of the first operating deflection shapes (OMA test) and estimated mode shape of modified structure using SDM and unscaled mode shapes (ODS) of modified structure Es mated Modes Shapes of Modified Structure Using un-scaled Shapes 1 2 3 4 Freq(Hz) 13.51 83.11 224.92 438.23 Op cally Modified Str 1 13.850 1.000 0.016 0.017 0.008 2 85.591 0.014 1.000 0.023 0.008 3 234.530 0.016 0.015 0.997 0.041 4 452.90 0.007 0.009 0.026 0.976 Es mated Modes Shapes of Modified Structure Using un-scaled Shapes 1 2 3 4 Freq (Hz) 13.51 83.11 224.92 438.23 Op cally Modified Str 1 1 3.850 -0.964 0.000 0.000 0.000 2 85.591 0.006 0.996 0.008 0.000 3 234.530 -0.006 0.000 -1.803 0.001 4 452.90 0.012 0.043 -0.285 -0.830 MAC a b Orthogonality Check Orthogonality MAC (Ψk1Eun) 12 (Ψk1Eun) 12 Fig. 1.10 Correlation between experimentally measured operating deflection shapes and estimated shapes using SDM. (a) MAC values indicating degree of consistency between operating shapes and estimated mode shapes. (b) Experimental mass weighted orthogonality check between the measured operating shapes and estimated mode shapes dynamic modification. The unscaled mode contribution matrix can be calculated using 2 664 2 664 : : : I : : : 3 775Cf‰g T k Œ M f‰gk3 775fP2gC 2 664 : : : !2 2 : : : 3 775fP2gDŒ0 (1.30) Mode shape of the modified structure is estimated using QE2 DŒ‰1 E un 12 (1.31)
1 Practical Techniques for Scaling of Optically Measured Operating Deflection Shapes 13 Fig. 1.11 Comparison between the first four reference mode shapes (EMA) and the scaled operating deflection shapes (OMA test) using mass sensitivity 1.4.4 Case B.2: Mass Sensitivity Scaling Technique Figure 1.11 shows the shape comparison of scaled operating deflection shapes and reference mode shapes (impact test). The operating shapes scaled using mass sensitivity is well correlated with the reference mode shapes. The MAC values between the reference mode shapes and the scaled shapes are shown in Fig. 1.12a. The MAC is only a comparison of shape and has no regard for scale. The orthogonality, however, takes into account both scale as well as shape. While the diagonal values of MAC matrix are close to one, the orthogonality values are far from one, especially for mode three (see Fig. 1.12b). The orthogonality values for the second and third mode shapes indicate that the scale values for these modes are not very accurate. Therefore, some error is still present in the scaling factor. 1.4.5 Case B.3: SDM Using Scaled Operating Deflection Shapes To evaluate the accuracy of the scaling factors (Eq. 1.23) and the mode contribution matrix (Eq. 1.17), the estimated mode shapes of the modified structure are compared with to the scaled operating shapes and reference mode shapes (EMA) of the modified structure. Figure 1.13 shows the shape comparison between the estimated and measured mode shapes of the modified structure after being scaled. Figure 1.14 shows the correlation and orthogonality between the estimated mode shapes and optically measured shapes of the modified structure. Orthogonality and MAC values shown in Fig. 1.14 demonstrate that scaling obtained using mass sensitivity has appropriately scaled both modified and unmodified operating shapes. Figure 1.15 demonstrates the comparison between estimated mode shapes and reference mode shapes of the modified structure (EMA). The MAC results shown in Fig. 1.16a demonstrate that the estimated shapes of the modified structure are in strong correlation with the reference mode shapes. Furthermore, orthogonality values (see Fig. 1.16b) reveal that by using the mass sensitivity technique all the four modes have been scaled to reference mode shapes (EMA).
14 P. Poozesh et al. MAC a b Scaled ODS Using Mass Sensi vity 1 2 3 4 Freq(Hz) 14.19 88.13 244.63 467.50 Impact Test (EMA) 1 14.34 1.000 0.016 0.016 0.019 2 89.28 0.017 1.000 0.018 0.016 3 247.07 0.018 0.025 0.999 0.024 4 471.75 0.008 0.008 0.029 0.969 Scaled ODS UsingMass Sensi vity 1 2 3 4 Freq(Hz) 14.19 88.13 244.63 467.50 Impact Test (EMA) 1 14.34 0.964 0.002 0.001 0.001 2 89.28 0.008 1.172 0.006 0.004 3 247.1 0.006 0.022 1.338 0.010 4 471.75 0.046 0.034 0.022 1.088 MAC Orthogonality Orthogonality Check Fig. 1.12 Correlation between operating deflections shapes, scaled using mass sensitivity, and the reference mode shapes (EMA). (a) MAC values indicating the degree of consistency between scaled operating shapes and reference mode shapes (MAC). (b) Orthogonality check between the scaled operating shapes and reference modal vectors with the experimental mass matrix Fig. 1.13 Comparison between the first four scaled operating scaled deflection shapes of modified structure and estimated mode shapes of the modified structure using SDM 1.5 Conclusion The results of the study revealed that both drive point and mass sensitivity scaling techniques offer a great prospect to be used as a scaling technique. The choice for one of these techniques generally relies on the feasibility of measuring input force. Drive point scaling proves to be more accurate than mass sensitivity technique. However, mass sensitivity is more practical to be used for scaling operating shapes. Using a searching algorithm, scaling factors minimizing the error between
1 Practical Techniques for Scaling of Optically Measured Operating Deflection Shapes 15 MAC Es mated Modes Shapes of Modified Structure Using SDM 1 2 3 4 Freq(Hz) 13.51 83.11 224.92 438.23 Exp Measured) 1 13.850 1.000 0.014 0.015 0.006 2 85.591 0.014 1.000 0.017 0.003 3 234.530 0.016 0.016 0.998 0.021 4 452.36 0.007 0.009 0.029 0.971 Es mated Modes Shapes of Modified Structure Using SDM 1 2 3 4 Freq (Hz) 13.51 83.11 224.92 438.23 Exp Measured 1 13.850 1.025 0.003 0.003 0.002 2 85.591 0.001 1.038 0.011 0.026 3 234.530 0.000 0.003 1.028 -0.050 4 452.36 0.000 0.002 0.038 0.998 Orthogonality Check a b Orthogonality MAC (α0k.Ψk1.E12) (α0k.Ψk1.E12) (β0k·Ψk2) (β0k·Ψk2) Fig. 1.14 Correlation and orthogonality between scaled operating deflection shapes, and estimated mode shapes using SDM. (a) MAC values indicating the degree of consistency between scaled operating shapes and estimated mode shapes using SDM. (b) Experimental mass weighted orthogonality check between the scaled operating shapes and estimated mode shapes using SDM Fig. 1.15 Comparison between the first four reference mode shapes (EMA) and estimated mode shapes of the modified structure using SDM estimated and experimental mode shapes were determined. The selection of mass sensitivity scaling factor as a starting value is important to ensure that the algorithm converges to an optimal value for the scaling factor. The MAC and orthogonality values indicate that the proposed method was capable of obtaining optimal scaling factors and all resulting operating modes show strong correlations to their corresponding mode shapes. Acknowledgements The authors gratefully appreciate the financial support for this work provided by the Massachusetts Clean Energy Center (CEC), Task Order 13-2. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of Mass CEC.
16 P. Poozesh et al. Estimated Modes Shapes of Modified Structure Using SDM 1 2 3 4 Freq (Hz) 13.51 83.11 224.92 438.23 Impact Test (EMA) 1 14.01 1.000 0.013 0.015 0.007 2 87.32 0.016 1.000 0.016 0.008 3 242.49 0.017 0.018 0.998 0.028 4 463.49 0.020 0.016 0.038 0.990 Estimated Modes Shapes of Modified Structure Using SDM 1 2 3 4 Freq (Hz) 13.51 83.11 224.92 438.23 Impact Test (EMA) 1 14.01 0.924 0.012 -0.009 0.055 2 87.32 0.002 -1.152 -0.019 0.034 3 242.49 0.001 0.006 1.266 -0.018 4 463.49 0.000 -0.002 0.049 -1.126 MAC a b Orthogonality Orthogonality Check MAC (α0k.Ψk1.E12) (α0k.Ψk1.E12) Fig. 1.16 Correlation and orthogonality between estimated scaled operating deflections shapes, and the reference mode shapes (EMA). (a)MAC values indicating degree of consistency between estimated scaled operating deflections shapes, and the reference mode shapes. (b) Experimental mass weighted orthogonality check between estimated scaled operating deflections shapes, and the reference mode shapes References 1. Baqersad, J., Poozesh, P., Niezrecki, C., Avitabile, P.: Comparison of modal parameters extracted using MIMO, SIMO, and impact hammer tests on a three-bladed wind turbine. In: Proceeding of the 32nd IMAC, A Conference and Exposition on Structural Dynamics, Orlando, FL, pp. 185–197, February 3–6, 2014 2. Devriendt, C., Guillaume, P., Brussel, V.U., Reynders, E., Roeck, G.D.: Operational modal analysis of a bridge using transmissibility measurements. In: Proceeding of the 25th IMAC, Conference & Exposition on Structural Dynamics, Orlando, Florida, February 19–22, 2007 3. Rainieri, C., Fabbrocino, G.: Operational Modal Analysis of Civil Engineering Structures. Springer, New York (2014) 4. Peeters, B., Van der Auweraer, H., Vanhollebeke, F., Guillaume, P.: Operational modal analysis for estimating the dynamic properties of a stadium structure during a football game. Shock Vib. 14(4), 283–303 (2007) 5. Poozesh, P., Baqersad, J., Niezrecki, C., Avitabile, P.: A multi-camera stereo DIC system for extracting operating mode shapes of large scale structures. In: Proceeding of the SEM Annual Conference on Experimental and Applied Mechanics, Costa Mesa, CA, June 8–11, 2015 6. Tcherniak, D., Chauhan, S., Hansen, M.H.: Applicability limits of operational modal analysis to operational wind turbines. In: Proceeding of the 28th IMAC, A Conference and Exposition on Structural Dynamics, Springer New York, pp. 317–327, February 1–4, 2010 7. López-Aenlle, M., Brincker, R., Pelayo, F., Canteli, A.F.: On exact and approximated formulations for scaling-mode shapes in operational modal analysis by mass and stiffness change. J. Sound Vib. 331(3), 622–637 (2012) 8. Ventura, C.E., Lord, J.F., Turk, M., Brincker, R., Andersen, P., Dascotte, E.: FEM updating of tall buildings using ambient vibration data. In: Proceeding of the Sixth International Conference on Structural Dynamics (EURODYN), Paris, France, 2005 9. López-Aenlle, M., Fernández, P., Brincker, R., Fernández-Canteli, A.: Scaling-factor estimation using an optimized mass-change strategy. Mech. Syst. Signal Process. 24(5), 1260–1273 (2010) 10. Hout, B., Avitabile, P.: Application of operating data scaling techniques. In: Proceeding of the 22nd IMAC, A Conference on Structural Dynamics, Dearborn, MI, pp. 52–58, January 26–29, 2004 11. López-Aenlle, M., Brincker, R., Fernández-Canteli, A.: Some methods to determine scaled mode shapes in natural input modal analysis. In: Proceeding of the 23rd IMAC, Conference & Exposition on Structural Dynamics, Orlando, Florida, pp. 165–176, January 31–February 3 2005 12. Aenlle, M.L., Fernández, P.F., Brincker, R., Canteli, A.F.: Scaling factor estimation using an optimized mass change strategy, part 1: theory. In: Proceeding of the, International Operational Modal Analysis Conference (IOMAC), pp. 421–428, 2007 13. Fernández, P., Reynolds, P., López-Aenlle, M.: Scaling mode shapes in output-only systems by a consecutive mass change method. Exp. Mech. 51(6), 995–1005 (2011) 14. Brincker, R., Rodrigues, P., Andersen, P.: Scaling the mode shapes of a building model by mass changes. In: Proceeding of the 22nd IMAC, International Modal Analysis Conference, Dearborn, MI, pp. 119–126, January 26–29, 2004 15. Parloo, E., Verboven, P., Guillaume, P., Van Overmeire, M.: Sensitivity-based operational mode shape normalisation. Mech. Syst. Signal Process. 16(5), 757–767 (2002) 16. Khatibi, M.M., Ashory, M.R., Malekjafarian, A., Brincker, R.: Mass-stiffness change method for scaling of operational mode shapes. Mech. Syst. Signal Process. 26, 34–59 (2012)
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