River Rapids Conference Proceedings of the Society for Experimental Mechanics Series Model Validation and Uncertainty Quantification, Volume 3 Robert Barthorpe Roland Platz Israel Lopez Babak Moaveni Costas Papadimitriou Proceedings of the 35th IMAC, A Conference and Exposition on Structural Dynamics 2017 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 Robert Barthorpe • Roland Platz • Israel Lopez • Babak Moaveni Costas Papadimitriou Editors Model Validation and Uncertainty Quantification, Volume 3 Proceedings of the 35th IMAC, A Conference and Exposition on Structural Dynamics 2017
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Preface Model Validation and Uncertainty Quantification represents one of ten volumes of technical papers presented at the 35th IMAC, A Conference and Exposition on Structural Dynamics, organized by the Society for Experimental Mechanics and held in Garden Grove, California, on January 30–February 2, 2017. The full proceedings also include the following volumes: Nonlinear Dynamics; Dynamics of Civil Structures; Dynamics of Coupled Structures; Sensors and Instrumentation; Special Topics in Structural Dynamics; Structural Health Monitoring & Damage Detection; Rotating Machinery, Hybrid Test Methods, Vibro-Acoustics and Laser Vibrometry; Shock & Vibration, Aircraft/Aerospace, and Energy Harvesting; and Topics in Modal Analysis & Testing. Each collection presents early findings from experimental and computational investigations on an important area within structural dynamics. Model Validation and Uncertainty Quantification (MVUQ) is one of these areas. Modeling and simulation are routinely implemented to predict the behavior of complex dynamical systems. These tools powerfully unite theoretical foundations, numerical models, and experimental data which include associated uncertainties and errors. The field of MVUQ research entails the development of methods and metrics to test model prediction accuracy and robustness while considering all relevant sources of uncertainties and errors through systematic comparisons against experimental observations. The organizers would like to thank the authors, presenters, session organizers, and session chairs for their participation in this track. Sheffield, UK Robert Barthorpe Darmstadt, Germany Roland Platz Livermore, CA, USA Israel Lopez Medford, MA, USA Babak Moaveni Thessaly, Greece Costas Papadimitriou v
Contents 1 Lateral Vibration Attenuation of a Beam with Piezo-Elastic Supports Subject to Varying Axial Tensile and Compressive Loads........................................................................................ 1 Benedict Götz, Roland Platz, and Tobias Melz 2 Correlation of Non-contact Full-Field Dynamic Strain Measurements with Finite Element Predictions ..... 9 Ibrahim A. Sever, Martyn Maguire, and Jose V. Garcia 3 Nonlinear Prediction Surfaces for Estimating the Structural Response of Naval Vessels ....................... 21 Alysson Mondoro, Mohamed Soliman, and Dan M. Frangopol 4 A Case Study in Predictive Modeling Beyond the Calibration Domain........................................... 29 Philip Graybill, Eyob Tarekegn, Ian Tomkinson, Kendra Van Buren, François Hemez, and Scott Cogan 5 A Brief Overview of Code and Solution Verification in Numerical Simulation................................... 39 François Hemez 6 Robust Optimization of Shunted Piezoelectric Transducers for Vibration Attenuation Considering Different Values of Electromechanical Coupling..................................................................... 51 Anja Kuttich, Benedict Götz, and Stefan Ulbrich 7 Parameter Estimation and Uncertainty Quantification of a Subframe with Mass Loaded Bushings.......... 61 Mladen Gibanica and Thomas J.S. Abrahamsson 8 Vibroacoutsic Modelling of Piano Soundboards through Analytical Approaches in Frequency and Time Domains....................................................................................................... 77 B. Trévisan, K. Ege, and B. Laulagnet 9 Combined Experimental and Numerical Investigation of Vibro-Mechanical Properties of Varnished Wood for Stringed Instruments ........................................................................................ 81 Sarah Louise Lämmlein, David Mannes, Francis Willis Mathew Schwarze, Ingo Burgert, and Marjan Sedighi Gilani 10 Towards Robust Sustainable System Design: An Engineering Inspired Approach.............................. 85 Mario Holl and Peter F. Pelz 11 Linear Parameter-Varying (LPV) Buckling Control of an Imperfect Beam-Column Subject to Time-Varying Axial Loads.............................................................................................. 103 Maximilian Schaeffner and Roland Platz 12 Quantification and Evaluation of Uncertainty in the Mathematical Modelling of a Suspension Strut Using Bayesian Model Validation Approach ......................................................................... 113 Shashidhar Mallapur and Roland Platz 13 Unsupervised Novelty Detection Techniques for Structural Damage Localization: A Comparative Study... 125 Zilong Wang and Young-Jin Cha vii
viii Contents 14 Global Load Path Adaption in a Simple Kinematic Load-Bearing Structure to Compensate Uncertainty of Misalignment Due to Changing Stiffness Conditions of the Structure’s Supports ............. 133 Christopher M. Gehb, Roland Platz, and Tobias Melz 15 Assessment of Uncertainty Quantification of Bolted Joint Performance.......................................... 145 Nedzad Imamovic and Mohammed Hanafi 16 Sensitivity Analysis and Bayesian Calibration for 2014 Sandia Verification and Validation Challenge Problem................................................................................................................... 159 Ming Zhan, Qin-tao Guo, Lin Yue, and Bao-qiang Zhang 17 Non-probabilistic Uncertainty Evaluation in the Concept Phase for Airplane Landing Gear Design......... 161 Roland Platz and Benedict Götz 18 Modular Analysis of Complex Systems with Numerically Described Multidimensional Probability Distributions ............................................................................................................. 171 J. Stefan Bald 19 Methods for Component Mode Synthesis Model Generation for Uncertainty Quantification.................. 177 A.R. Brink, D.G. Tipton, J.E. Freymiller, and B.L. Stevens 20 Parameterization of Large Variability Using the Hyper-Dual Meta-model....................................... 189 Matthew S. Bonney and Daniel C. Kammer 21 Similitude Analysis of the Frequency Response Function for Scaled Structures................................. 209 Mohamad Eydani Asl, Christopher Niezrecki, James Sherwood, and Peter Avitabile 22 MPUQ-b: Bootstrapping Based Modal Parameter Uncertainty Quantification—Fundamental Principles .. 219 S. Chauhan and S.I. Ahmed 23 MPUQ-b: Bootstrapping Based Modal Parameter Uncertainty Quantification—Methodology and Application.......................................................................................................... 239 S. Chauhan 24 Evaluation of Truck-Induced Vibrations for a Multi-Beam Highway Bridge .................................... 255 Kirk A. Grimmelsman and John B. Prader 25 Innovations and Info-Gaps: An Overview............................................................................ 263 Yakov Ben-Haim and Scott Cogan 26 Bayesian Optimal Experimental Design Using Asymptotic Approximations..................................... 273 Costas Argyris and Costas Papadimitriou 27 Surrogate-Based Approach to Calculate the Bayes Factor ......................................................... 277 Ramin Madarshahian and Juan M. Caicedo 28 Vibrational Model Updating of Electric Motor Stator for Vibration and Noise Prediction..................... 283 M. Aguirre, I. Urresti, F. Martinez, G. Fernandez, and S. Cogan 29 A Comparison of Computer-Vision-Based Structural Dynamics Characterizations ............................ 295 Aral Sarrafi, Peyman Poozesh, and Zhu Mao 30 Sequential Gauss-Newton MCMC Algorithm for High-Dimensional Bayesian Model Updating.............. 303 Majid K. Vakilzadeh, Anders Sjögren, Anders T. Johansson, and Thomas J.S. Abrahamsson 31 Model Calibration with Big Data...................................................................................... 315 Guowei Cai and Sankaran Mahadevan 32 Towards Reducing Prediction Uncertainties in Human Spine Finite Element Response: In-Vivo Characterization of Growth and Spine Morphology ................................................................ 323 E.S. Doughty and N. Sarigul-Klijn 33 Structural Damage Detection Using Convolutional Neural Networks............................................. 331 Nur Sila Gulgec, Martin Takácˇ, and Shamim N. Pakzad
Contents ix 34 Experimental Model Validation of an Aero-Engine Casing Assembly ............................................ 339 D. Di Maio, G. Ramakrishnan, and Y. Rajasagaran 35 Damage Detection in Railway Bridges Under Moving Train Load ................................................ 349 Riya C. George, Johanna Posey, Aakash Gupta, Suparno Mukhopadhyay, and Sudib K. Mishra 36 Multi-Fidelity Calibration of Input-Dependent Model Parameters ............................................... 355 G.N. Absi and S. Mahadevan 37 Empirically Improving Model Adequacy in Scientific Computing ................................................ 363 Sez Atamturktur, Garrison N. Stevens, and D. Andrew Brown 38 Mixed Geometrical-Material Sensitivity Analysis for the Study of Complex Phenomena in Musical Acoustics.................................................................................................................. 371 R. Viala, V. Placet, and S. Cogan 39 Experimental Examples for Identification of Structural Systems Using Degree of Freedom-Based Reduction Method....................................................................................................... 375 Heejun Sung, Seongmin Chang, and Maenghyo Cho
Chapter 1 Lateral Vibration Attenuation of a Beam with Piezo-Elastic Supports Subject to Varying Axial Tensile and Compressive Loads Benedict Götz, Roland Platz, and Tobias Melz Abstract In this paper, vibration attenuation of a beam with circular cross-section by resonantly shunted piezo-elastic supports is experimentally investigated for varying axial tensile and compressive beam loads. Varying axial beam loads manipulate the effective lateral bending stiffness and, thus, lead to a detuning of the beams resonance frequencies. Furthermore, varying axial loads affect the general electromechanical coupling coefficient of transducer and beam, an important modal quantity for shunt-damping. The beam’s first mode resonance frequency and coupling coefficient are analyzed for varying axial loads. The values of the resonance frequency and the coupling coefficient are obtained from a transducer impedance measurement. Finally, frequency transfer functions of the beam with one piezo-elastic support either shunted to a RL-shunt or to a RL-shunt with negative capacitance, the RLC-shunt, are compared for varying axial loads. It is shown that the beam vibration attenuation with the RLC-shunt is less influenced by varying axial beam loads. Keywords Piezo-elastic support • Resonant shunt • Vibration attenuation • Beam • Axial load 1.1 Introduction Structural vibration may occur in mechanical systems leading to fatigue, reduced durability or undesirable noise. In this context, resonant shunting of piezoelectric transducers can be an appropriate measure for attenuating vibrations. Shunt-damping in general has been subject to research for several decades [1] and resulted in many diverse shunt concepts such as mono- or multi-modal resonant shunts [2], shunts with negative capacitances [3] or switched shunts [4]. Shunting a piezoelectric transducer with resistor and inductance, the RL-shunt, a tuned electrical oscillation circuit with the inherent capacitance of the transducer is created. This electromechanical system acts similar to a mechanical vibration absorber. RL-shunts are easy to implement and no stability limits or switching laws have to be taken into account. However, the achieved vibration attenuation significantly depends on the tuning of the shunt parameters and the amount of the general electromechanical coupling coefficient of transducer and structure [5]. By adding a negative capacitance, the RLC-shunt achieves higher vibration attenuation but stability issues have to be considered. In mechanical and civil engineering, truss structures bear and withstand constant and variable loads that may lead to vibrations. Truss structures comprise truss members such as beams that are connected to each other via the relatively stiff truss supports. On the one hand, truss structures show global vibration modes with lateral moving or rotating truss supports. On the other hand, local modes exist that are dominated by the lateral vibration behavior of each beam. Additional varying quasi static loading may result in axial tensile and compressive loads of the beams. This affects the resulting lateral bending stiffness of the beams leading to a permanent change in the resonance frequencies. For vibration attenuation with resonant shunt-damping, the detuning of the resonance frequencies affects the achievable vibration attenuation capability. In truss structures, piezoelectric shunt-damping has been investigated in [6–8]. Axial piezoelectric stack transducers are integrated within one strut of the truss, resulting in compression and elongation of the transducer in normal axial direction of the strut [6, 8]. In [7], a beam support with integrated piezoelectric washers alongside the beam that are strained in shear under dynamic loading has been investigated and vibration attenuation of bending modes in a truss substructure was achieved. Due to a planar washer design, only one bending direction of the beams could be influenced. Nevertheless, research B. Götz ( ) • T.Melz System Reliability, Adaptive Structures, and Machine Acoustics SAM, Technische Universität Darmstadt, Magdalenenstraße 4, 64289 Darmstadt, Germany e-mail: goetz@sam.tu-darmstadt.de R. Platz Fraunhofer Institute for Structural Durability and System Reliability LBF, Bartningstraße 47, 64289 Darmstadt, Germany © The Society for Experimental Mechanics, Inc. 2017 R. Barthorpe et al. (eds.), Model Validation and Uncertainty Quantification, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series, DOI 10.1007/978-3-319-54858-6_1 1
2 B. Götz et al. investigating the effect of varying quasi static axial beam loads on the vibration attenuation of beams with resonant shuntdamping are not known to the authors. In this paper, a new concept of a piezo-elastic support [9] for vibration attenuation of truss structures comprising beams with circular cross-section by shunted transducers is investigated for varying axial tensile and compressive beam loads. Within the piezo-elastic support, deflections in both lateral directions of a beam are transferred into an axial deformation of the transducers that are arranged perpendicular to one free end of the beam. By that design, local and global truss modes can attenuated without manipulating the beam’s surface. In the following only one beam with two piezo-elastic supports is investigated for varying axial tensile and compressive loads. One piezo-elastic support mechanically excites the beam laterally using white noise excitation while the second support is either shunted to a RL- or to a RLC-shunt. First, the experimental beam’s first mode resonance frequency and its general electromechanical coupling coefficient are calculated from impedance measurements for different axial tensile and compressive loads. Changes in both, the resonance frequency and the coupling coefficient may influence the vibration attenuation. Considering the electrical transducer impedance in the frequency domain is a known method for identifying the transducer capacitance, structural resonance frequencies or the coupling coefficient and has several advantages. E.g., all important quantities are obtained from one measurement at one time, no structural transfer function has to be measured and the estimation of the resonance frequency as well as the coupling coefficient is less influenced by the used frequency resolution [10]. Second, the experimental frequency transfer functions in lateral direction of the excited beam in case one transducer is shunted to a RL-shunt and in case one transducer is shunted to a RLC-shunt are compared for uncertain axial tensile and compressive beam loads. 1.2 System Description The investigated system is a beam under axial load made of aluminum alloy EN AW-7075 with length lb D400mm and circular solid cross-section of radius rb D 5mm, Fig. 1.1. The circumferential lateral stiffness is homogeneous and has no preferred direction of lateral deflection, so the beam may vibrate in any plane lateral to the longitudinal x-axis. The beam is supported by two piezo-elastic supports A and B at location x D 0 and location x D lb. Elastic membrane-like spring elements made of spring steel 1.1248 in both supports A and B at location x D0 and x Dlb bear axial and lateral forces at the beam’s ends in x-, y- and z-direction, and allow rotation 'y and 'z in any plane perpendicular to the x-axis, see Fig. 1.2. In Fig. 1.1, the membrane-like spring elements for both supports A and B are represented by axial stiffness kx;A Dkx;B D1:53 10 4 N=mm, not shown in the figure, lateral stiffness k y;A Dkz;A Dky;B Dkz;B Dkl D9:98 10 4 N=mm iny- andz-direction and rotational stiffness k'y;A Dk'z;A Dk'y;B Dk'z;B Dkr D2:69 10 5 Nmm=rad around the y- andzaxes. All spring element stiffness values are obtained from a finite element simulation and they are not experimentally verified yet. In each piezo-elastic support A and B at x D lext andx DlbClext, two piezoelectric stack transducers P1 andP2 aswell as P3 and P4 are arranged in the support housing at an angle of 90ı to each other orthogonal to the beam’s x-axis, Fig. 1.1b. All transducers are mechanically prestressed by a stack of disc springs with stiffness kpre D2:6 10 3 N=mm. The transducers are connected to the beam via a relatively stiff axial extension made of hardened steel 1.2312 with lengthlext D6:75mmand edge lengthtext D12mm. With that, lateral beam deflections iny- andz-direction due to vibration excitation are transformed into the stack transducer’s axial deformation. Each piezoelectric transducer P1 to P4 is a PI P-885.51 stack transducers with the capacitance Cp D1:65 F at constant mechanical stress, internal series resistance Rp D7 and the mechanical stiffness kp D50 10 3 N=mm with short circuited electrodes, defined as the ratio of the transducer’s block force and the maximum free stroke. The input current to the transducer is I.t/ and the potential difference at the transducer electrodes is the voltage U.t/. For vibration attenuation, a RL- and a RL-shunt with negative capacitance C, the RLC-shunt, are taken into account, Fig. 1.1c. In Fig. 1.1c, the RL-shunt is obtained by neglecting the negative capacitance. By adjusting the inductance L and the damping resistance R, the transducer shunted to a RL-shunt attenuates vibrations similar to a mechanical vibration absorber [1]. For vibration attenuation with shunt-damping, the generalized electromechanical coupling coefficient K33 is an important modal quantity indicating the vibration attenuation capability. The higher the value of K33 is, the higher the achievable vibration attenuation with a RL-shunt becomes. By adding a negative capacitance C 1:02 Cp in series to R andL, the coupling coefficient K33 increases effectively and, hence, the vibration attenuation capability is significantly higher compared to the simple RL-shunt [5]. Apart from that, RLC-shunts may destabilize the beam vibration, therefore, stability limits for the value of Chave to be considered. Basically, the achievable vibration attenuation capability with a RLC-shunt is almost independent of the coupling coefficient K33. However, the smaller the coupling coefficient is, the close the value of the negative capacitance has to be chosen to the stability limit [5]. In the experiment, the inductance Lis implemented by the use of a gyrator circuit and the negative capacitance by the use of negative admittance converter [2]. Both circuit designs are not discussed in detail in this paper.
1 Lateral Vibration Attenuation of a Beam with Piezo-Elastic Supports Subject to Varying Axial Tensile and Compressive Loads 3 (a) (c) (b) x y ϕy z ϕz kϕz,A kϕy,A kz,A ky,A kpre rb text kϕz,B kϕy,B ky,B kz,B kpre kpre P1 P2 P3 P4 lb lext lext xs a Fx y z kpre kpre P1 P2 I1 U1 I U R L C Fig. 1.1 Beam system, (a) beam with piezo-elastic supports A and B, (b) arrangement of piezoelectric transducers, (c) shunt circuit disc springs beam transducer P3 axial extension membrane-like spring element Fig. 1.2 Sectional view of piezo-elastic support [11]
4 B. Götz et al. For all experimental vibration attenuation investigations, the piezoelectric transducer P1 excites the beam in y-direction via a controlled voltage signal U1.t/. The transducer P3, either shunted to the RL-shunt or the RLC-shunt, attenuates the vibration acceleration a.t/ at the sensor location xs while the transducers P2 and P4 are operated with short circuited (sc) electrodes. Varying axial tensile and compressive loads 1000N Fx 1500N, with compressive loads in positive xdirection, are applied to the beam at x D0via a spindle-type lifting gear and measured by a force sensor. 1.3 The Beam’s First Mode Eigenfrequency and Coupling Coefficient for Varying Axial Loads The vibration attenuation capability bears on the piezoelectric transducers P3 that is either shunted to a tuned RL- or a tuned RLC-shunt. Varying axial beam loads may change the beam’s first mode resonance angular frequency !sc with short circuited (sc) transducer electrodes and the beam’s first mode general coupling coefficient K33 of transducer P3 iny-direction. Changes in both, !sc and K33 may influence the vibration attenuation with shunted transducers. To investigate the influence of varying loads Fx on!sc andK33, the mathematical receptance model of the transducer P3 is derived in frequency domain. The values of !sc and K33 for axial loads Fx DŒ 1000; 500; 0; 500; 1000; 1500 N are extracted from a least squares fit of the receptance model to the experimental data in frequency domain. As already shown by Kozlowski et al. [10], obtaining !sc and K33 from a curve fitting of the transducer receptance model results in a smaller error since the calculation of both parameters is less influenced by the used frequency resolution in the measurement. 1.3.1 Transducer Receptance Model Figure 1.3 shows the electrical network representation of the piezoelectric transducer P3 connected to the beam. The transducer P3 is described by a gyrator-like two-port transducer network with its electrical capacitance Cp, a internal series resistance Rp and its transducer constant Y [12]. The vibration behavior of the beam’s first mode with short circuited transducer electrodes is modelled by the modal mass m, the modal stiffness k and the assumed hysteretic damping with loss factor resulting in the complex stiffness k0 Dk.1 i /. The complex network receptance seen from the terminals 1and 2 in Fig. 1.3 is obtained by ˛.!/ D 1 i !Z.!/ (1.1) with excitation frequency !. The impedance Z.!/ D U3.!/ I3.!/ D Zp.!/Z1.!/ Zp.!/ CZ1.!/ (1.2) results from the parallel connection of the structural impedance Z1.!/ of the first mode seen from the terminals 3 and 4 the transducer impedance Zp.!/ DRp i !Cp : (1.3) Fig. 1.3 Electrical network model of the piezoelectric transducers and the beam’s first mode k m Cp Rp U3 I3 Y 2 1 3 4
1 Lateral Vibration Attenuation of a Beam with Piezo-Elastic Supports Subject to Varying Axial Tensile and Compressive Loads 5 The structural impedance of the first mode iny-direction Z1.!/ DY2 1 Zm.!/ C 1 Zk0 .!/ D Y2 i !m C k .1Ci / i ! (1.4) seen from the terminals 3 and 4 of the gyrator is the result of the parallel impedance of the modal mass Zm.!/ D 1 i !m and the complex stiffness Zk0 .!/ D i ! k0 . From (1.1) and (1.2), the receptance becomes ˛.!/ D Cp 1C!Cp Rp i C 1 Y2 m !2 sc !2 Ci !2 sc (1.5) with the angular eigenfrequency !sc Dr k m . Furthermore, the term 1 Y2 m in (1.5) is replaced by K33 Cp !sc, as suggested in [10], leading to the final expression of the transducer receptance ˛.!/ D Cp 1C!Cp Rp i CK33 Cp !sc !2 sc !2 Ci !2 sc : (1.6) 1.3.2 Transducer Receptance Model Fit In the model fit process, the parameters Cp; Rp; K33; !sc and in (1.6) are varied to solve the least squares curve fitting problem min Cp; Rp; K33;!sc; jj ˛.Cp; Rp; K33; !sc; ; !/ ˛exp.!/jj 2 2 (1.7) where ˛exp.!/ is the experimental data of the transducer receptance. Therefore, the lsqnonlin algorithm in MATLAB is used. Figure 1.4 shows the amplitude and phase response j˛.!/j and arg˛.!/ of the experimental data and the calculated receptance after the curve fitting for the axial load Fx D0N. Both, the model and the experimental data show a very good agreement. Fig. 1.4 Calculated transducer receptance (red solid line)with fitted parameters and experimental data (black solid line) for Fx D0N 150 200 250 0 −30 ω/2π in Hz argα in ◦ 150 200 250 1.2 1.6 2.1 |α| in F
6 B. Götz et al. 1.3.3 Experimental Results of the First Eigenfrequency and Coupling Coefficient for Varying Axial Loads As introduced before, a change of the resonance frequency !sc due to axial loads Fx will lead to a detuned vibration behavior of the RL- and RLC-shunt and, hence, the vibration attenuation capability will decrease. Apart from that, tensile and compressive axial loads may affect the electromechanical coupling coefficient K33 of transducer P3. An increased coupling coefficient K33 will also increase the vibration attenuation capability with a RL-shunt, while a decreased K33 will also decrease the vibration attenuation potential. Furthermore, vibration stability issues with a RLC-shunt due to a changing coupling coefficient K33 are not investigated in this paper. Figure 1.5a,b show the beam’s first mode resonance frequency !sc and the beam’s first mode coupling coefficient K33 of transducer P3 extracted from the fitted receptance model (1.6) for axial tensile and compressive loads Fx DŒ 1000; 500; 0; 500; 1000; 1500 N. In Fig. 1.5a, the resonance frequency !sc increases for an axial tensile load and decreases for an axial compressive load significantly. Compared to the axially unloaded beam with Fx D 0N, the resonance frequency !sc increases by 6% and decreases by 10% for the extremes of the applied axial loads at Fx D 1kN and Fx D1:5kN. In contrast to the behavior of the frequency!sc, in Fig. 1.5b, the coupling coefficient K33 decreases for an axial tensile load and increases for an axial compressive load, as also shown by Lesieutre and Davis [13]. Compared to the axially unloaded beamwithFx D0N, the coupling coefficient K33 decreases by6% and increases by7% for the extremes of the applied axial loads at Fx D 1kNand Fx D1:5kN. Considering the absolute changes of the resonance frequency !sc and the coupling coefficient K33 in Fig. 1.5a,b axial tensile and compressive loads will decrease the vibration attenuation capability with the RL- and the RLC-shunt due to a detuning. For the RL-shunt with axial compressive loads, the increase in coupling coefficient and the associated theoretically increase of vibration attenuation capability will not be able to compensate the effects of a detuned !sc on the vibration attenuation capability, as it will be shown in the next section. −1 −0.5 0 0.5 1 1.5 170 180 190 200 210 220 Fx in kN ωsc /2π in Hz (a) −1 −0.5 0 0.5 1 1.5 1 6 7 8 10 Fx in kN K33 in% (b) Fig. 1.5 Beam’s first mode (a) resonance frequency !sc and (b) electromechanical coupling coefficient K33 of transducer P3, for varying tensile and compressive axial loads Fx (cross symbol) and the axially unloaded beamFx D0N(open circle)
1 Lateral Vibration Attenuation of a Beam with Piezo-Elastic Supports Subject to Varying Axial Tensile and Compressive Loads 7 1.4 Experimental Vibration Attenuation with RL- and RLC-Shunt for Varying Axial Loads The vibration attenuation of the beam in case of the transducer P3 is shunted to a RL-shunt and in case of P3 is shunted to a RLC-shunt is compared for axial tensile and compressive loads Fx DŒ 1000; 500; 0; 500; 1000 N, Fig. 1.6. As a measure for the vibration attenuation of the RL- and RLC-shunt, the frequency transfer function H.!/ D a.!/ U1.!/ (1.8) of the voltage excitationU1.!/ of transducer P1 to the beam vibration accelerationa.!/ is considered, Fig. 1.1. Additionally for the axially unloaded beam withFx D0N, the vibration attenuation potential of the RL- and RLC-shunt is obtained from the comparison of H.!/ when the P3 electrodes are short circuited and when shunted. Figure 1.6a shows the amplitude and phase response jH.!/j and argH.!/ when transducer P3 is shunted to a RL-shunt with R D 48:4 and L D 402mH. The values for R and L were tuned experimentally to give the highest reduction in amplitude in the considered frequency range. For the axially unloaded beam at Fx D 0N, the vibration attenuation with RL-shunt is 15dB. For varying axial tensile and compressive loads Fx, a significant decline in the vibration attenuation potential can be observed in the amplitude response. The decline in vibration attenuation is slightly higher for tensile loads. Furthermore, the detuning of !sc due to axial tensile and compressive loads is observed in the phase response by a shift of the 90ı crossing frequency. Figure 1.6a shows the amplitude and phase response jH.!/j and argH.!/ when transducer P3 is shunted to a RLC-shunt with RD4:8 , L D29:6mHand CD 1:69 F. The values for R, Land Cwere tuned experimentally to give the highest and stable reduction in amplitude in the considered frequency range. For the unloaded beam, the vibration attenuation with RCL-shunt is26dB and, as expected, is significantly higher than with the RL-shunt. For varying axial tensile and compressive loads Fx, the observed decline in the vibration attenuation capability is smaller compared to the RL-shunt. The decline in vibration attenuation, again, is slightly higher for tensile loads. 100 200 300 0 −180 ω/2π in Hz argHin ◦ 100 200 300 10−1 102 |H| inm/s2/V (a) 100 200 300 0 −180 ω/2π in Hz argHin ◦ 100 200 300 10−1 102 |H| inm/s2/V (b) Fig. 1.6 Amplitude and phase response of H.!/, short circuited (black solid line), with (a) RL-shunt and (b) RLC-shunt, for the unloaded beam (blue solid line) with Fx D0N and varying axial loads Fx W C500N (gray solid line), C1000N (gray dashed line), 500N (red solid line), 1000N(red dashed line)
8 B. Götz et al. To conclude, vibration attenuation with a RLC-shunt is less sensitive to varying axial tensile and compressive loads. However, using a negative capacitance may lead to stability issues, but they were not observed in the performed experiments and are not part of this paper. 1.5 Conclusion Vibration attenuation of a beam with a circular cross-section by piezo-elastic supports with one transducer shunted to RL- or RLC-shunt subject to varying axial tensile and compressive loads is experimentally investigated. When no shunt is connected to the transducer, the first beam’s mode resonance frequency significantly increases and decrease in consequence of axial tensile and compressive loads. Compared to the unloaded beam, the relative changes of the electromechanical coupling coefficient are in the same order of magnitude as for the resonance frequency, but, the absolute change of the coupling coefficient has no significant effect an the vibration attenuation with RL- and RLC-shunt. The resonance frequency detuning due to axial tensile and compressive loads results in a declined vibration attenuation when shunting the transducer to a RLor a RLC-shunt. As observed in the measured frequency transfer functions, the RLC-shunt is less sensitive to uncertain axial loads then the RL-shunt. Acknowledgements The authors like to thank the German Research Foundation DFG for funding this research within the SFB 805. References 1. Hagood, N.W., von Flotow, A.H.: Damping of structural vibrations with piezoelectric materials and passive electrical networks. J. Sound Vib. 146(2), 243–268 (1991) 2. Moheimani, S.O.R., Fleming, A.J.: Piezoelectric Transducers for Vibration Control and Damping. Springer, London (2006) 3. Beck, B.S.: Negative capacitance shunting of piezoelectric patches for vibration control of continuous systems. PhD thesis, Georgia Institute of Technology (2012) 4. Niederberger, D.: Smart damping materials using shunt control. PhD thesis, Swiss Federal Institute of Technology Zürich (2005) 5. Neubauer, M., Oleskiewicz, R., Popp, K., Krzyzynski, T.: Optimization of damping and absorbing performance of shunted piezo elements utilizing negative capacitance. J. Sound Vib. 298, 84–107 (2006) 6. Hagood, N.W., Crawley, E.F.: Experimental investigation of passive enhancement of damping for space structures. J. Guid. Control. Dyn. 14(6), 1100–1109 (1991) 7. Hagood, N.W., Aldrich, J.B., von Flotow, A.H.: Design of passive piezoelectric damping for space structures. NASA Contractor Report 4625 (1994) 8. Preumont, A., de Marneffe, B., Deraemaeker, A., Bossens, F.: The damping of a truss structure with a piezoelectric transducer. Comput. Struct. 86, 227–239 (2008) 9. Götz, B., Schaeffner, M., Platz, R., Melz, T.: Lateral vibration attenuation of a beam with circular cross-section by a support with integrated piezoelectric transducers shunted to negative capacitances. Smart Mater. Struct. 25(095045), 10 (2016) 10. Kozlowski, M.A., Cole, D.G., Clark, R.L.: A comprehensive study of the RL series resonant shunted piezoelectric: a feedback controls perspective. J. Vib. Acoust. 133, 011012-1–011012-10 (2011) 11. Enss, G.C., Gehb, C.M., Götz, B., Melz, T., Ondoua, S., Platz, R., Schäffner, M.: Device for bearing design elements in lightweight structures (Festkörperlager) (2016) 12. Lenk, A., Ballas, R.G., Werthschützky, R., Pfeifer, G.: Electromechanical Systems in Microtechnology and Mechatronics. Number 978-3540-89320-2. Springer, Berlin (2011) 13. Lesieutre, G.A., Davis, C.L.: Can a coupling coefficient of a piezoelectric device be higher than those of its active material? J. Intell. Mater. Syst. Struct. 8, 859–867 (1997)
Chapter 2 Correlation of Non-contact Full-Field Dynamic Strain Measurements with Finite Element Predictions Ibrahim A. Sever, Martyn Maguire, and Jose V. Garcia Abstract It is highly desirable to have the capability to measure strain maps on components directly and in a full-field fashion that addresses shortcomings of conventional approaches. In this paper, use of a 3D laser measurement system is explored for direct and full-field dynamic strain measurements on compressor and turbine rotor blades. More importantly, the results obtained are numerically correlated to corresponding FE predictions in a systematic manner. The ability to measure strain maps on real engine hardware is demonstrated not only for low frequency fundamental modes, but also for challenging high frequency modes. Correlation results show a high degree of agreement between measured and predicted strains, demonstrating the maturity of the technology and the validity of the method of integration used here. The measurements are repeated for a number of different loading amplitudes to assess the variations in strain fields. Although the application of 3D laser systems to measurements of full-field strain were explored in previous studies, to the best knowledge of authors, full-field numerical correlation of full-field strain on a wide range of real, complex components to this extent is presented here for the first time. Keywords Model validation • Full-field strain • 3D SLDV • Correlation • Non-contact 2.1 Introduction The ability to measure dynamic strain on components subjected to high vibratory stresses is very important as these measurements then directly feed into all important endurance/life calculations. Historically this requirement has been fulfilled in two main ways. The first and most widely used approach is the application of strain gauges. Although a direct measurement and still a very popular practice; there are a number of shortcomings. Firstly, they are intrusive as they have to be bonded to the component. Typically only a few of these can be used which do not provide a representative spatial coverage nor are they enough to evaluate changing strain patterns due to load variations. Their nontrivial footprint means that they can only provide average strain under the area they cover. The second approach, albeit less common, is to validate a finite element model of a given component through direct measurements of displacement, velocity or acceleration and then to use that improved model for predictions of strain and stresses. Although effectively used, particularly in case of full-field measurement systems such as Scanning Laser Doppler Vibrometers (SLDV), good level of correlation with these measurements does not always translate to a good correlation in strain. Moreover, when it comes to components showing complex phenomena or those made from novel materials, the confidence in original FE models is often low or such FE models may not even exist; rendering strain predictions obtained this way even less reliable. Non-contact and full-field measurements of 3D vibration responses have been explored via a number of different technologies over the years. Earlier systems using double-pulse Electronic Speckle Pattern Interferometry (ESPI) exploited different combinations of viewing and observation directions [1, 2] vibration measurements. Systems with 3-observation and 1-illumination directions as well as 3-illumination and 1-observation directions were explored however recovery of full-field dynamic strain for industrial applications and high frequency complex mode shapes were not reported. A creative way in I.A. Sever ( ) Rolls-Royce Plc, SinA-33, PO Box 31, Derby, DE24 8BJ, UK e-mail: ibrahim.sever@rolls-royce.com M.Maguire Rolls-Royce MTOC, Kiefernstra“e 1, D-15827 Blankenfelde-Mahlow, Germany J.V. Garcia GP4-4, PO Box 3, Bristol, BS34 7QE © The Society for Experimental Mechanics, Inc. 2017 R. Barthorpe et al. (eds.), Model Validation and Uncertainty Quantification, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series, DOI 10.1007/978-3-319-54858-6_2 9
10 I.A. Sever et al. which a 1-D SLDV system is used in combination with a short-focus lens to recover 3D vibration information was given in [3] however obtaining full-field coverage this way is simply not practical. A detailed review of Digital Image Correlation (DIC) techniques applied to vibration measurements explored suitability of DIC technology compared with more conventional methods and SLDV based systems [4], however, similar to double pulse ESPI, the use of DIC based systems outside quasi static regime and for high frequency complex modes has been limited. Recently 3D SLDV based measurement systems have gained popularity due to their practicality for complete dynamic deformation field (i.e. 3D) measurements. A through study of strain measurements with a 3D SLDV system is given in [5]. Although comparison with FE and conventional strain gauge results are presented, these are done at the locations of strain gauges only, rather than in full-field sense, such as in the form of strain MAC and strain CoMAC. The pursuit of more direct, high density and high accuracy measurements in this study is motivated in particular by their potential to provide better model validation opportunities. Valid models (i.e. models that are demonstrated to be adequate representation of real life behaviour) provide unique opportunities as they can enable simulation of behaviour for a wide range of parameter ranges and constraints that may not be practicable or cost effective to do through testing. Given the criticality of the use of these models, such as in estimating the stress and strain fields and ultimately the structural integrity of aero-engine components, ensuring that they are valid to an acceptable degree is essential. This is something that has been mainly done via measurements of displacement mode shapes as these are the easiest to measure. The inferred conclusion from such measurements is that when the displacement shapes are shown to match with a sufficient degree of correlation, the resulting stress and strain distributions will follow the same trend. However the more direct and the more detailed the measurements of parameters of interest are, such as strain and stress distributions, the higher will be the confidence one can have in simulation models these data are used to asses and, if necessary, to correct. 2.2 Measurement Campaign Measurements of full-field strain on aero-engine blades using a 3D SLDV system were reported in an earlier publication [6]. Although in this paper the focus is on the correlation of results, particularly those of full-field strain, with the FE model predictions, it is worthwhile reviewing the basic principles of measurement system as well as hardware tested and the setup used. 2.2.1 Measurement System A 3D Scanning LDV system is used in acquisition of displacement and strain measurements. A picture of the measurement system in use is given in Fig. 2.1a. The principle behind the operation of an LDV transducer can be explained simply as follows [7]: light produced by a laser source is split into two beams of the same amplitude by a beam splitter, one directed to a fixed reference and the other to the vibrating target. Following the same path back, the beams are combined by the same splitter and send to a photodetector. Since the light from the target is optically mixed with an equally coherent reference beam and heterodyned on the photodetector surface, the resolution of the sign of the vibration velocity is achieved by preshifting the reference beam’s frequency by a known amount. The signal received by the photodetector is then frequency demodulated by a suitable Doppler processor and the vibration velocity of the target is worked out. The 3D SLDV system used during this measurement campaign was a PSV 500 3D. All simple out-of-plane 1D measurements were made using a Polytec PSV 400 HS. The 3D system consists of three independent SLDV heads as shown in Fig. 2.1a. Fundamental mode of operation for 1D and the 3D systems are identical in that each laser transducer captures the vibration response on the structure along its own line of sight. The fact that there are three such observation directions in the 3D case is being exploited to recover the complete vibration response in three orthogonal directions. This requires that all three SLDVs are coordinated and that the measurement surface is precisely aligned to a degree where laser beams from all SLDVs are coincident to within an acceptable tolerance. Alignment requirements are stricter for strain estimations than they are for the displacement measurements.
2 Correlation of Non-contact Full-Field Dynamic Strain Measurements with Finite Element Predictions 11 Fig. 2.1 (a) 3D SLDV measurement system in use, and (b) Intermediate pressure compressor blade (left) and intermediate pressure turbine blade (right)
12 I.A. Sever et al. 2.2.2 Test Hardware The measurement campaign is carried out on a number of aero-engine components including intermediate pressure compressor and turbine blades and a full-size fan blade (not shown here). Some of these components are shown in Fig. 2.1b. In case of compressor and turbine blades, tests are repeated for a number of different excitation levels. These components feature a number of different challenges in terms of clamping conditions, frequency range they cover and the complexity of modes of vibration they poses. As such they should provide appropriate coverage for demonstration of the capability being presented. Strain as well as displacement measurements are carried out for all components however in this paper the correlation of strain measurements is carried out for the compressor blade alone. 2.2.3 Test Environment and Setup In order to eliminate the adverse effects that the environment might have on the measurements, the testing was carried out in a state-of-the-art vibration test facility. Vibration isolation is achieved through the use of large air-sprung bed plates, and the temperature is maintained at a suitable level. Thick, well insulated test cell walls ensure that there is no interference from external sources. The measurement process was largely automated which meant that once alignment was achieved no user intervention was required. As alignment was based on the engine coordinate system it was repeatable. Components tested here were fixed at their roots with appropriate clamping mechanisms, mimicking similar boundary conditions to those present in engine. Various excitation techniques were used depending on the size of the component. Turbine blades were excited via an acoustic horn pressure unit and a bespoke piezoelectric resonator. Most tests on these components were performed using the pressure unit as it proved more effective. For larger components (i.e. compressor and fan blades) acoustic speakers were found to be more appropriate where suitable speakers were selected proportional to size of the components being excited. Measurement grids on blade surfaces to be scanned were carefully optimised in a separate test planning process to maximise observability of the modes on interest (e.g. maximise ability to distinguish them without any ambiguity) using nominal FE models present. This ensured that the measurement grid was defined in the engine coordinate system. This is a major advantage as this grid is then transferred to the measurement system and measurement volume is calibrated in a way that corresponds to the FE environment, making the alignment and correlation of FE and test points much easier. Much denser grids were used in strain measurements, compared with the ones used in displacement mode shapes. Also a much more accurate laser alignment process had to be used in strain measurement case which in return made the strain measurements a longer campaign. 2.3 FE Model and Test Planning Test planning for the measurement campaign was carried out using the nominal FE models for the blades tested. Provided that there are no fundamentally significant deviations, this is acceptable as the character of modes and the frequency ranges derived from the nominal models provide appropriate guidelines for defining the overall boundaries of the test campaign. However, it is well known that due to manufacturing tolerances, the physical parts show variations from their design intent. As these tolerances are often defined by manufacturability and performance constraints, their impact on structural dynamics may be non-trivial. The impact of these variations on the overall correlation study will be explored in future publications. For the sake of introducing the correlation methodology, all FE models used in this study will be those derived from the nominal geometry but with appropriate boundary conditions to reflect the test configuration. Planning of the test campaign consists of defining the measurement grid and assessing the suitability of this grid in capturing vibration modes of interest. FE model of the compressor blade, measurement grids for displacement and strain mode shapes, and, auto correlation matrix of the displacement modes captured by identified measurement grid are all given in Fig. 2.2a, b and c, respectively. In the case of displacement mode shapes, the effective independence method [8] is used for down selection of measurement points. Given that an optical measurement system is used, the candidate nodes to choose from are the ones that lie on the surface that can be measured (see Fig. 2.2a), rather the whole FE model. Suitability of this grid is confirmed by Auto Modal Assurance Criterion (autoMAC) plot given in Fig. 2.2c. Here the predicted modes of the nominal FE model
2 Correlation of Non-contact Full-Field Dynamic Strain Measurements with Finite Element Predictions 13 (a) (b) (c) 5 10 15 20 2 4 6 8 10 12 14 16 18 20 FE Modes Test Modes Modal Assurance Criterion (%) 10 20 30 40 50 60 70 80 90 Fig. 2.2 (a) Compressor blade FE model, (b) Measurement grid used displacement mode shapes, (c) resulting auto-MAC plot sampled at measurement nodes are compared to themselves. Cross-mode correlation amplitudes as evident from trivial offdiagonal terms (all below 10%) suggest that all measured modes should be uniquely identifiable. Measurement grid for the strain mode shapes is significantly denser than the one for displacement mode shapes. This is required to capture the local strain variations faithfully. Note that by this stage the modes are already identified through correlation of displacement mode shapes. As such autoMAC check performed for displacements is not necessary to repeat for strain. 2.4 Correlation of Mode Shapes Measurement grid given above was identified for the displacement mode shapes using the full deformation field. In other words all X, Y and Z displacement DOFs were used at each measurement grid as the same DOFs would be captured from the tests. Measurement of all DOFs at each point is a requirement for strain measurements but it is not essential for displacement mode shapes. In fact 1-D SLDVs, measuring a projection of total deformation field in the line of sight, have been used for decades. Nevertheless, availability of all DOFs brings significant advantages in the form of increased independent information which even in the case of displacement mode shapes can make a big difference. Figure 2.3 shows a particular mode measured on the intermediate pressure turbine blade by 1-D and 3-D SLDV systems, together with the predicted FE mode shape where FE and the 3-D measured mode shape are almost identical. Although the 1-D measurement appears to be very different, a direct comparison is inappropriate. 1-D SLDV measures a projection of overall response in the viewing direction whereas distributions shown for the FE and the 3-D SLDV are for the resultant displacements from all DOFs computed and measured. A correct correlation in the case of 1-D SLDV measurements would be with FE predictions projected in a similar way to reflect the operation of 1-D SLDV system. Having said that, the fact remains that the 3-D SLDV provides a lot more information (three times as much) about the deformation field, which in return allows better identification of measured mode shapes. This is demonstrated in Fig. 2.4. Here there are two correlation scenarios shown where mode shapes captured by 1-D and 3-D SLDV systems are correlated in the form of Modal Assurance Criterion (MAC) with their corresponding FE predictions in Fig. 2.4a and b, respectively. Significant off-diagonal values in 1-D SLDV case which lead to difficulties in identifying mode shapes unambiguously are greatly reduced in the 3-D case where the identification of the modes is now straight forward. Displacement mode shape measurement campaign performed on the compressor blade is summarised in Fig. 2.5. As evident from the sub-set of measured and predicted mode shapes given in Fig. 2.5a, not only the global behaviour but also the local variations are extremely closely matched. MAC matrix given in Fig. 2.5b shows a remarkable degree of correlation between measurements and the predictions with MAC values at 95% and above, and, with all off-diagonal values below 10%. It is worth noting the extraordinary similarity between autoMAC plot generated in Fig. 2.2c and the MAC plot given in Fig. 2.5b. As such the FE model is demonstrated to be a good representation of the measurement hardware from mode shapes point of view. This level of accuracy in the resultant correlation also demonstrates that tests are carried out as planned and that the alignment of FE model and the test model is performed adequately. The latter is a critical factor, particularly for mode shapes
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