Rotating Machinery, Structural Health Monitoring, Shock and Vibration, Volume 5

River Rapids Conference Proceedings of the Society for Experimental Mechanics Series Rotating Machinery, Structural Health Monitoring, Shock and Vibration, Volume 5 Tom Proulx Proceedings of the 29th IMAC, A Conference on Structural Dynamics, 2011 River Publishers

Conference Proceedings of the Society for Experimental Mechanics Series

River Publishers Tom Proulx Editor Volume 5 Proceedings of the 29th IMAC, A Conference on Structural Dynamics, 2011 Monitoring, Shock and Vibration, Rotating Machinery, Structural Health

Published, sold and distributed by: River Publishers Broagervej 10 9260 Gistrup Denmark www.riverpublishers.com ISBN 978-87-7004-851-4 (eBook) Conference Proceedings of the Society for Experimental Mechanics An imprint of River Publishers © The Society for Experimental Mechanics, Inc. 2011 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, or reproduction in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Preface Rotating Machinery, Structural Health Monitoring, and Shock and Vibration Topics represents one of six clusters of technical papers presented at the 29th IMAC, A Conference and Exposition on Structural Dynamics, 2011 organized by the Society for Experimental Mechanics, and held in Jacksonville, Florida, January 31 - February 3, 2011. The full proceedings also include volumes on Advanced Aerospace Applications; Modal Analysis, Linking Models and Experiments, Civil Structures; and Sensors, Instrumentation, and Special Topics. Each collection presents early findings from experimental and computational investigations on an important area within Structural Dynamics. The current volume on Rotating Machinery, Structural Health Monitoring, and Shock and Vibration includes studies on Random Vibrations, Wind Turbine Blade Sensing and Health Monitoring, Rotating Machinery, Machinery Condition Monitoring and Diagnostics, Structural Health Monitoring, Shock and Vibration, Acoustics, and Damage Detection. Condition monitoring, diagnostics and damage detection are a series of related topics of increasing importance. Structural Health Monitoring (SHM) has been an evolving technology for a number of years. Machinery monitoring and non-destructive testing topics have been active areas in IMAC and other forums. With the development of more powerful computational capability and miniaturized sensors, SHM schemes are being explored that can be designed into the structure, providing a comprehensive health evaluation. The organizers would like to thank the authors, presenters, session organizers and session chairs for their participation in this track. Bethel, Connecticut Dr. Thomas Proulx Society for Experimental Mechanics, Inc

Contents 1 1 G.C. Khoury, D.C. Zimmerman, University of Houston 2 Spectral Analysis Methodology for Acoustical and Mechanical Measurements Relative to Hydraulic Turbine’s Generator 15 3 Transverse Vibrations of Tapered Materially Inhomogeneous Axially Loaded Shafts 25 A.J. Mazzei, Jr., Kettering University; R.A. Scott, University of Michigan 4 Converting a Driven Base Vibration Test to a Fixed Base Modal Analysis R.L. Mayes, Sandia National Laboratories; M.S. Allen, University of Wisconsin-Madison 5 Model Form Error of Alternate Modeling Strategies: Shell Type Wind Turbine Blades 53 6 Operational Modal Analysis of Operating Wind Turbines: Application to Measured Data 65 S. Chauhan, D. Tcherniak, Bruel and Kjaer Sound and Vibration Measurement A/S; J. Basurko, 7 Amplitude Dependent Crack Characterization of Growing Fatigue Cracks 83 A.U. Rehman, K. Worden, J.A. Rongong, University of Sheffield 8 Utilization of Localized Panel Resonant Behavior in Wind Turbine Blades 93 D.T. Griffith, Sandia National Laboratories 9 105 T.L. Paez, Thomas Paez Consulting 10 Reduction of Vibration Transmission in String Trimmers 129 P. Rajbhandary, J. Leifer, B.J. Weems, Trinity University 37 F. Lafleur, S. Bélanger, E. Coutu, A. Merkouf, Institut de Recherche d’Hydro-Québec, IREQ K.L. Van Buren, S. Atamturktur, Clemson University O. Salgado, I. Urresti, Ikerlan-IK4; C.E. Carcangiu, M. Rossetti, Alstom Wind Development of Pretest Planning Methodologies for Load Dependent Ritz Vectors Random Vibration – History and Overview

viii 11 Insight Into Strong Motion Behavior of Large Concrete Structures Based on Low Level Response Monitoring in the Field 139 N. von Gersdorff, Southern California Edison Company; Z. Duron, Harvey Mudd College; V. Chiarito, ERDC/WES 12 Stability Monitoring of Burning Structures Based on Fire-induced Vibration Monitoring 149 13 Application of Energy Methods in Mechanical Shock to Study Base Excited Nonlinear System Response 157 J.E. Alexander, BAE Systems, US Combat Systems 14 Experimental Assessment of Gear Meshing Excitation Propagation Throughout Multi Megawatt Gearboxes 177 J. Helsen, Katholieke Universiteit Leuven; F. Vanhollebeke, B. Marrant, Katholieke Universiteit Leuven 15 An Advanced Numerical Model of Gear Tooth Loading From Backlash and Profile Errors 191 A. Sommer, J. Meagher, X. Wu, California Polytechnic State University 16 A Differential Planetary Gear Model With Backlash and Teeth Damage 203 X. Wu, J. Meagher, A. Sommer, California Polytechnic State University 17 Determination of Wind Turbine Operating Deflection Shapes Using Full-field 3D Point-tracking 217 C. Warren, C. Niezrecki, P. Avitabile, University of Massachusetts Lowell 18 Dynamic Performance and Integrity Assessment of an Electricity Transmission Tower 227 19 Operational Modal Analysis of Resiliently Mounted Marine Diesel Generator/Alternator 237 20 Eigenvalues and Nonlinear Behaviour of Levitron® 245 E. Bonisoli, C. Delprete, M. Silvestri, Politecnico di Torino 21 Optical Measurements and Operational Modal Analysis on a Large Wind Turbine: Lessons Learned 257 M. Ozbek, D.J. Rixen, Delft University of Technology 22 Operational Damage Detection of Turbine Rotors Using Integrated Blade Sensors 277 S.R. Dana, D.E. Adams, Purdue University Z.H. Duron, Harvey Mudd College H. Clarke, J. Stainsby, E.P. Carden, Lloyd’s Register EMEA E.P. Carden, J.R. Maguire, Lloyd’s Register EMEA Hansen Transmissions International nv; D. Berckmans, D. Vandepitte, W. Desmet,

ix 23 Uncertainty Assessment in Structural Damage Diagnosis 287 S. Sankararaman, S. Mahadevan, Vanderbilt University 24 A Framework for Embedded Load Estimation From Structural Response of Wind Turbines 295 A.V. Hernandez, R.A. Swartz, Michigan Technological University; A.T. Zimmerman, University of Michigan 25 A Review of Gearbox Condition Monitoring Based on Vibration Analysis Techniques Diagnostics and Prognostics 307 A.S. Sait, Y.I. Sharaf-Eldeen, Florida Institute of Technology 26 Experimental Results of Wind Turbine Operational Monitoring With Structural and Aerodynamic Measurements 325 J.R. White, Sandia National Laboratories; D.E. Adams, Purdue University 27 Torsional Response of Structure with Mass Eccentricity 333 28 Examination of a Feeling of Pulse Control Method for Cruiser-type Motorcycle 341 29 Seeding Cracks Using a Fatigue Tester for Accelerated Gear Tooth Breaking 349 30 Operational Modal Analysis of a Rectangular Plate Using Noncontact Acoustic Excitation 359 Y.F. Xu, W.D. Zhu, University of Maryland, Baltimore County 31 Algorithm Hybridization for Automated Modal Identification and Structural Health Monitoring 375 C. Rainieri, G. Fabbrocino, University of Molise 32 Fiber Optic Sensor Installation for Monitoring of 4 Span Model Bridge in UCF 383 I.B. Kwon, M. Malekzadeh, Q. Ma, H. Gokce, T.K. Terrell, A. Fedotov, F.N. Catbas, University of Central Florida 33 Transmissibility Measurements for System Identification 389 Z. Mao, M. Todd, University of California, San Diego 34 A Modified Whiteness Test for Damage Detection Using Kalman Filter Innovations 399 D. Bernal, Y. Bulut, Northeastern University 35 Vibration Reduction of an Atomic Force Microscope in the Point of the Mechanical Design 405 C. Kim, J. Jung, J. Jeong, K. Park, Gwangju Institute of Science and Technology H. Wu, Edith Cowan University D.G. Lewicki, NASA Glenn Research Center A Model of Uncertainty Quantification in the Estimation of Noise-contaminated N. Tsujiuchi, T. Koizumi, T. Tonomura, Doshisha University N.G. Nenadic, J.A. Wodenscheck, M.G. Thurston, Rochester Institute of Technology;

x 36 Modal Analysis and SHM Investigation of CX‐100 Wind Turbine Blade 413 K. Deines, New Mexico State University; T. Marinone, University of Massachusetts Los Alamos National Laboratory 37 Model-based Diagnostics and Fault Assessment of Induction Motors With Incipient Faults M. Nakhaeinejad, J. Choi, M.D. Bryant, The University of Texas at Austin 38 Use of the Cepstrum to Remove Selected Discrete Frequency Components From a Time Signal R.B. Randall, N. Sawalhi, University of New South Wales 39 A Review of Signal Processing and Analysis Tools for Comprehensive Rotating Machinery Diagnostics T. Reilly, Data Physics Corporation 40 Simple Tools for Simulating Structural Response to Underwater Explosions F.A. Costanzo, Naval Surface Warfare Center Carderock Division, UERD 41 Using Vibration Signatures Analysis to Detect Cavitation in Centrifugal Pumps 42 Modes Indicate Cracks in Wind Tubine Blades S.N. Ganeriwala, V. Kanakasabai, SpectraQuest, Inc.; M. Richardson, Vibrant Technology, Inc. 43 Estimating Shock Severity H.A. Gaberson, Consultant 44 H.C. Pusey, SAVIAC Manager of Technical Services 45 Rationale for Navy Shipboard Shock & Vibration (S&V) Requirements J.E. Howell, III, Naval Surface Warfare Center, Carderock Division 46 Deformation and Vibration Measurement and Data Evaluation on Large Structures Employing Optical Measurement Techniques H. Berger, M. Klein, GOM Gesellschaft für Optische Messtechnik mbH 439 451 463 481 499 509 515 533 545 547 Lowell; R. Schultz, Michigan Technological University; K. Farinholt, G. Park, S.N. Ganeriwala, V. Kanakasabai, SpectraQuest, Inc. Progress in Shock and Vibration Technology Over 80 Symposia

Development of Pretest Planning Methodologies for Load Dependent Ritz Vectors George C. Khoury Graduate Research Assistant Department of Mechanical Engineering University of Houston Houston, TX 77204-4792, USA David C. Zimmerman Professor Department of Mechanical Engineering University of Houston Houston, TX 77204-4792, USA ABSTRACT Load dependent Ritz vectors have been found to be suitable alternatives to mode shape vectors for structural vibration analysis. While much research has focused on how to best set-up and perform modal property identification, there has been no investigation on such pretest planning for Ritz vectors. Pre-test planning is a vital part of successful vibration tests, especially when dealing with large complex structures. In such cases, it is common that a limited number of sensors and actuators must be placed in a configuration to obtain the most important dynamic information. In this study, previously developed modal sensor placement techniques were utilized to determine Ritz vector sensor sets for the NASA eight-bay truss structure. The techniques used were the eigenvector product, kinetic energy product, and effective independence. The Modal Assurance Criteria was employed to determine how well each technique performed. Since Ritz vectors are subject to change due to the excitation locations, the paper also investigated the effect of various loading configurations on the Ritz vector sensor placement results. Finally, Ritz vectors and mode shapes were combined in an effort to verify that a single sensor set could be used to identify both sets of basis vectors. Introduction Pre-test planning is an integral part of successful structural dynamic testing. When dealing with large complex structures, typically a limited number of sensors and actuators must be placed in a configuration to obtain the most important dynamic information. Since sensor and actuator locations strongly influence structural testing, it is important that the selected locations be optimized. Optimal sensor placement methods tend to utilize information obtained from the baseline finite element model (FEM). Although much research has been performed in this area for mode shape vectors [1-3], there has been no investigation on such pre-test planning for Ritz vectors. The motivation for this paper is to apply sensor placement techniques to optimize the information obtained during experimental Ritz vector testing. Load dependent Ritz vectors have been found to be suitable alternatives to mode shape vectors for structural vibration analysis. The Ritz vector analysis is a computational method that allows accurate dynamic analyses to be obtained at a reduced computational cost [4]. Ritz vectors have several advantages over traditional mode shapes. (1) They automatically include the static correction term. (2) Once a system realization is obtained, Ritz vectors are less computationally expensive to extract. (3) All Ritz vectors generated by a load will be excited by that load. (4) Each Ritz vector describes behavior across a range of frequencies, so fewer Ritz vectors than mode shapes are typically needed to achieve the same level of accuracy in transient response prediction and model reduction. (5) Because the Ritz vector extraction process depends on the solution of linear algebraic equations instead of an eigensolution, Ritz vectors may be less sensitive to noise than measured mode shapes. An important step in the advancement of Ritz vector applications was the development of a method to extract Ritz vectors from dynamic testing data by Cao and Zimmerman [5], which was further refined by Boxoen and Zimmerman [6]. The sensor placement techniques presented in this paper were previously developed to perform the same function for mode shape vectors, but can be extended for use with any basis vector. The Eigenvector product uses the basis vector product from the reduced FEM to identify possible sensor or actuator locations [2]. The kinetic energy product uses the modal kinetic energy calculated utilizing the FEM mass matrix and target modes to place sensors or actuators at points of maximum kinetic energy [1, 2]. Finally, the Effective Independence is a more elaborate sensor placement method that attempts to maintain the linear independence of the basis vectors, by maximizing the determinant of the Fisher information matrix, as candidate sensor locations are eliminated [1-3]. 1 T. Proulx (ed.), Rotating Machinery, Structural Health Monitoring, Shock and Vibration, Volume 5, Conference Proceedings of the Society for Experimental Mechanics Series 8, DOI 10.1007/978-1-4419-9428-8_1, © The Society for Experimental Mechanics, Inc. 2011

The techniques above were used to find sensor placement locations for mode shapes, Ritz vectors, as well as a combination of both basis vectors. The NASA eight-bay truss model was used as the analytical test-bed. The Modal Assurance Criteria (MAC) was utilized to compare the sensor placement results and obtain a representation of how well each technique performed. Section 2 presents a detailed look into the Ritz vector extraction method. Section 3 details the sensor placement techniques used. The numerical studies and results are presented in section 4. Finally, the concluding remarks are given in section 5. 1 Ritz Vector Analysis The dynamic equilibrium equation for a discrete n-degree-of-freedom (DOF) structure can be expressed as ( ) ( ) ( ) ( ) xt+ xt+ xt =fut M C K & && (1) where M, C, and K are the ( ) n n× mass, damping and stiffness matrices, respectively, ( ) x t is the ( )1×n position vector, f is the ( )1×n force influence vector, and ( ) u t is the scalar force input signal. The over-dots represent differentiation with respect to time. The first Ritz vector, representing the deflection of the structure under a unit static load [4], is found from the solution of v = f * 1 K , (2) where * 1v is the non-normalized Ritz vector. Although Wilson [4] mass-normalized the Ritz vectors, it was found that in experimental identifications it is more appropriate that the Ritz vectors be unit-normalized as the mass matrix is unknown [5, 6]. The first Ritz vector is unit-normalized as ( ) * 1 * 1 * 1 1 v v v v T = . (3) The subsequent Ritz vectors are found from the solution of i-1 * i v= v K M . (4) Each Ritz vector is orthogonalized using a Gram-Schmidt orthogonalization process ∑ = i-1 j 1 j j * i ivˆ = v - c v (5) where * j T j j c = v v . (6) After each Ritz vector is orthogonalized, it is unit-normalized i Ti i i vˆ vˆ vˆ v = . (7) 2

2 Sensor Placement Techniques 2.1 Vector Product (VP) This technique is generally referred to as the Eigenvector Product technique [2]. In this study the technique is referred to as the Vector Product as it is used with basis vectors other than mode shapes. The basis vectors found from the finite element model (FEM) can be used to identify possible sensor or actuator locations. The shapes of interest are chosen as shown below ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ Φ= nm n2 n1 1m 12 11 φ φ φ φ φ φ M L M M , (8) where Φis the set of mode shapes, Ritz vectors or a combination of both. The VP is calculated by multiplying the Φcomponents, such that i1 i2 im i φ φ φ VP L = , (9) where subscript i corresponds to the degree-of-freedom (DOF) of interest. The total set of vector products, VP, is sorted so that the maximum values are found. A high value of VPi corresponds to nodes with high displacement and thus a candidate location. Vector product will exclude node point DOFs of any vector as VPi will give a value of zero. 2.2 Kinetic Energy Product (KE) The KE product can be used for sensor placement identification [1, 2]. The technique uses the assumption that maximum observability will occur by placing sensors at locations with maximum kinetic energy. Using the FEM basis vectors, the kinetic energy is found as follows ∑ = = n j 1 ij jm im im φ φ ke M , (10) where i corresponds to the DOF of interest, m is the target basis vector, and n corresponds to the total number of DOFs in the system. The total kinetic energy matrix is as follows ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = nm n2 n1 1m 12 11 ke ke ke ke ke ke M L M M ke . (11) Finally, the KE product is found by multiplying the ke components im i1 i2 i KE ke ke ke L = , (12) so that the total KE product, KE, can be sorted from highest to lowest. As with VP, a high value of KEi corresponds to a candidate location. The KE product will also negate node points associated with a single shape since the product will be zero at that DOF. The mass weighting inherent to the KE product causes the sensor and actuator placement to be dependent on the finite element discretization, such that there is a bias against areas with fine mesh sizes (and thus small masses). 3

2.3 Effective Independence (EI) Effective independence was developed as a technique to select sensor locations for large space structures [1]. The candidate sensor sets are ranked according to their contribution to the linear independence of the target basis vectors. The first step is to find the Fisher information matrix ∑ = = n i 1 i T i o φ φ A , (13) where iφ is the ith row of the target basis vector and n is the total number candidate sensor locations in the system. Equation 13 demonstrates that information can be added to or subtracted from the Fisher information matrix with the inclusion or exclusion of DOFs. The number of DOFs in the sensor set can be reduced by eliminating locations that do not contribute to the independence of the target basis vectors. The analysis starts by solving the eigenvalue equation for oA . [ ] 0 λ Ψj j o = − I A , m j 1,2, ,K = (14) where m is the total number of basis vectors in the system, and j Ψ are orthogonal vectors resulting in the relations j j o T j Ψ Ψ λ= A and 1 Ψ Ψj T j = . (15) The EI coefficients of the candidate sensors are computed as [ ] [ ] Ψ Ψ G= Φ ⊗Φ , (16) where the symbol ⊗ represents term-by-term matrix multiplication, and [ ]mΨ , , Ψ, Ψ1 2 K =Ψ . Next, Eq. (16) is postmultiplied by the inverse of the matrix of eigenvalues, -1 λ F G E = , (17) where FE represents the fractional eigenvalue distribution, and [ ] ( ) mλ , , λ, λ diag 1 2 K = λ . Finally, the terms within each row of FE are added to obtain ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = ∑ ∑ ∑ n 2 1 EI E E E F F F M . (18) Alternatively, EI can be found using the following formulation [ ] ( ) T 1 T EI diag = ΦΦ Φ Φ− . (19) The values of EI will range from 0 to 1, with a value of 0 indicating that the DOF does not contribute to the observability of the system. The smallest value of the EI vector is removed and the above process is repeated iteratively until the desired number of sensors is found. 4

3 Numerical Study 3.1 Test Bed Description A sensor placement analysis was performed to compare the various placement techniques utilizing mode shapes, Ritz vectors, and a combination of both basis vectors. The analysis was based on the eight-bay hybrid-scaled truss structure designed for research in dynamic scale model ground testing of large space structures at the NASA Langley Research Center [7]. The truss was cantilevered at one end and each truss member was modeled as a rod element. Concentrated masses were attached at each node to represent joint and instrumentation mass properties. The structure was modeled as an undamped 96-DOF structure. The model is shown in figure 1. Figure 1: NASA Eight-bay Truss From the above figure it can be seen that the first 32 nodes are unconstrained in the x-, y-, and z-directions. The last four nodes are constrained in all three directions. Using the supplied mass and stiffness matrices, the mode shapes and Ritz vectors were calculated. It was assumed that the first five mode shapes and Ritz vectors were the target shapes to be identified. 3.2 Mode Shape Sensor Placement Using the first five full mode shapes from the FEM model, 15 sensor locations were found using the three placement techniques. The results are shown in figure 2. Figure 2: Sensor Locations using Mode Shapes The reduced mode shapes were found by selecting the modal data from the sensor location DOFs. The modal assurance criteria (MAC) of the first five reduced mode shapes were compared, as shown in figure 3 below. 5

Figure 3: MAC of Identified Mode Shapes As can be seen in the MAC plots, the VP and EI gave similar results. KE gave slightly worse results than the other two placement techniques. The MAC norm values above each plot show the norm of the off-diagonal values of the MAC matrix. 3.3 Ritz Vector Sensor Placement For all Ritz vector sensor placement cases the force was placed in the x- and z-directions located at node 1 with an angle, θ, measured from the x-axis, as shown in figure 4. Figure 4: Eight-Bay Truss Force Application For the first case, a force was placed at node 1 with a 45 degree angle. Using the first five Ritz vectors from the FEM model, 15 sensor locations were found using the three placement techniques. The sensor placement results are shown in figure 5. Figure 5: Sensor Locations using Ritz Vectors (Force at node 1, 45 degrees) 6

Figure 6 shows the MAC plots of the above sensor placement results. Figure 6: MAC of Identified Ritz Vectors (Force at node 1, 45 degrees) The VP and KE gave nearly identical results since the sensor placements were very similar. EI gave the best results of the three. Since Ritz vectors are subject to the force locations and direction, an optimization process can be performed to improve the MAC results. In this case, a sub-optimal solution was found by changing the x- and z-direction force magnitudes at node 1 by varying the force angle on the x-axis from 0 to 180 degrees. The results are shown in figure 7. Figure 7: MAC Norm Range for Ritz Vectors In Figure 7, o, *, and x identify the minimum MAC norm for VP, KE, and EI, respectively. The figure shows clearly that EI gave the best MAC results in most cases, while VP tended to give the worst. When the forces were completely in the x- or zdirections, the VP and KE MAC results were at their worst. Although one would expect symmetry about 90 degrees, the complex lacing pattern creates motion in multiple axes for a force purely aligned in one axis. Note that the MAC norm plots are symmetric about 45 and 135 degrees from 0 to 90 and 90 to 180 degrees, respectively. This plot repeats every 180 7

degrees. It’s apparent that the force selection for case 1 above does not give good MAC results compared to other force directions. From the MAC norm results from figure 7, a sensor set was found in an attempt to obtain better MAC results. The force was selected at node 1 with an angle of 75 degrees from the x-axis. This selection was made as it nears the minimum for all three placement techniques and would allow for slight alignment error in an experimental test. The first five Ritz vectors from the FEM model were used to find 15 sensor locations. The sensor locations and MAC results are shown in figures 8 and 9 respectively. Figure 8: Sensor Locations using Ritz Vectors (Force at node 1, 75 degrees) Figure 9: MAC of Identified Ritz Vectors (Force at node 1, 75 degrees) Comparing the sensor placements for cases 1 and 2 in figures 5 and 8 shows that the force placed at 75 degrees located more sensors in the z-direction, due to the increased force in that axis. Figure 9 shows greatly improved MAC results for the VP and KE methods, while EI was improved slightly. 3.4 Combination Sensor Placement Sensor placement locations using a mode shape and Ritz vector combination was examined. The aim of combining the two shapes was to determine whether a single sensor set could be used to accurately find both basis vectors experimentally. In this study, the first five mode shapes and first five Ritz vectors were concatenated to a single matrix. The three sensor placements algorithms were then performed using this ten column shape matrix. Because some Ritz vectors could resemble mode shapes, a second procedure was to perform a singular value decomposition (SVD) of the ten column matrix and retain the first number of columns containing 95% of the singular value information. As with the Ritz vectors alone, an optimization process was applied to improve the MAC results. The sub-optimal solution was found by varying the force angle on the x-axis from 0 to 180 degrees as before. The MAC for the reduced mode shapes and Ritz vectors were found separately and then normalized together. Figure 10 shows the MAC norm plots for both cases. 8

Figure 10: MAC Norm Range for Mode Shapes and Ritz Vectors The x and o symbols in figure 10 represent the minimum MAC norm values for the SVD and non-SVD cases, respectively. There are a few items to note from figure 10. First, there is no clear-cut choice between the SVD and non-SVD approach. Therefore, it is recommended to calculate both the SVD and non-SVD and make a selection on a case-by-case basis. Secondly, as with the Ritz vectors alone, the VP tends to give the worst results, while EI gives the best. Using a force at node 1 with a 75 degree angle, sensor sets were found for the mode shape and Ritz vector combination. For comparison purposes, both the SVD and non-SVD cases were considered. Figures 11 and 12 show the sensor location results for the non-SVD and SVD cases, respectively. Figure 11: Non-SVD Sensor Locations using Mode Shapes and Ritz Vectors (Force at node 1, 75 degrees) 9

Figure 12: SVD Sensor Locations using Mode Shapes and Ritz Vectors (Force at node 1, 75 degrees) Comparing the non-SVD and SVD placement results from figures 11 and 12 above shows that for the selected force the SVD spatially spread out the sensors for VP and KE, but grouped the sensors closer together for EI. The MAC was found for mode shapes and Ritz vectors separately for all three sensor placement techniques. The mode shape MAC results are shown in figures 13 to 15. Figure 13: MAC of Identified Mode Shapes using Vector Product (Force at node 1, 75 degrees) Figure 14: MAC of Identified Mode Shapes using Kinetic Energy (Force at node 1, 75 degrees) 10

Figure 15: MAC of Identified Mode Shapes using Effective Independence (Force at node 1, 75 degrees) The mode shape MAC figures above show that mode shapes improved significantly with SVD for VP and KE, while EI gave slightly worse results. It is important to note that the results for EI only changed slightly and still gave better results than the other two methods. The MAC results were generally not as good when compared to the mode shapes calculated alone in figure 3, although the difference is not great. This indicates that combining the mode shapes with Ritz vectors still gave acceptable mode shape results. The only outlier was the KE case with SVD, where the results for the combined basis vectors improved over the mode shapes alone. The improvement may be due to greater spatial distribution of the sensor placement for KE that the combination basis vectors offer. It should also be noted that the results in section 4.2 for KE were not very good. The MAC was found for the Ritz vectors as shown in figures 16 through 18 below. Figure 16: MAC of Identified Ritz Vectors using Vector Product (Force at node 1, 75 degrees) 11

Figure 17: MAC of Identified Ritz Vectors using Kinetic Energy (Force at node 1, 75 degrees) Figure 18: MAC of Identified Ritz Vectors using Effective Independence (Force at node 1, 75 degrees) The Ritz vector MAC plots above show that the VP and EI methods improved using the SVD. KE did worsen in terms of the MAC norm, but it is important to note that the MAC went from having coupling between two sets of Ritz vectors (1 and 4, 2 and 5) to one set (2 and 5) using the SVD procedure. Again, EI gave the best results of the three techniques. When comparing the MAC results for Ritz vectors alone with the 75 degree force angle, as shown in figure 9, it can be easily deduced that the results deteriorate when the combination basis vectors were used. In the case of EI, the difference between the combination basis vectors and the Ritz vectors alone was negligible. However, as with the mode shape results, the combination basis vectors still gave acceptable Ritz vector results 4 Conclusion Pre-test planning is an important aspect of structural dynamic analyses. This study aimed at investigating sensor placement utilizing Ritz vectors for the NASA eight-bay truss. Three techniques developed for mode shape analysis in previous studies were utilized: the Vector Product (VP), the Kinetic Energy Product (KE), and Effective Independence (EI). The three techniques were used to find 15 sensor locations for mode shapes, Ritz vectors, and a combination of both. The results showed that VP, KE, and EI successfully identified sensor locations and gave good MAC results for Ritz vectors. Varying the force direction and location was found to change the Ritz vectors, affecting the sensor sets and in turn changing the MAC results. The off-diagonal norms of the MAC plots were used in an optimization process to select a force that would optimize 12

the results. Generally, the sensor placement for VP and KE gave similar Modal Assurance Criteria (MAC) results, although KE was slightly better in some cases. As was expected, EI gave the best MAC results and therefore the best sensor location results. In an effort to determine if a single sensor set could be used to find both mode shapes and Ritz vectors, the basis vectors were combined and used with the three sensor placement techniques. The results showed that although the basis vectors deteriorated slightly when combined, a single sensor set could identify both mode shapes and Ritz vectors with reasonable accuracy. 13 5 References [1] Journal of Guidance, Control, and Dynamics, Volume 15 Issue 2, pp.251-259, 1991. [2] and Sensor Placement Using the NASA 8-Bay Truss,” Proceedings of the 12th International Modal Analysis Conference, pp. 205-211, 1994. [3] 18th International Modal Analysis Conference, Volume 4062 Issue 2, pp. 607-612, 2000. [4] Engineering and Structural Dynamics, Volume 10 Issue 6, pp. 813-821, 1982. [5] Structural Engineering, Volume 125 Issue 12, pp. 1393-1400, 1999. [6] Structural Engineering, Volume 129 Issue 8, pp. 1131-1140, 2003. [7] Location Techniques,” NASA-LaRC Technical Memorandum 107626, 1992. Kammer, D.C., “Sensor Placement for On-Orbit Modal Identification and Correlation of Large Space Structures,” Larson, C.B., Zimmerman, D.C., and Marek, E.L, “A Comparison of Modal Test Planning Techniques: Excitation Yap, K.C. and Zimmerman, D.C., “Optimal Sensor Placement for Dynamic Model Correlation,” Proceedings of the Wilson, E.L., Yuan, M.W. and Dicken, J.M., ‘‘Dynamic Analysis by Direct Superposition of Ritz Vectors,’’ Earthquake Cao, T. and Zimmerman, D.C., “Procedure to Extract Ritz Vectors from Dynamic Testing Data,” ASCE Journal of Boxoen, T. and Zimmerman, D.C., “Advances in Experimental Ritz Vector Identification,” ASCE Journal of Kashangaki, T.A.L., “Ground Vibration Tests of a High Fidelity Truss for Verification of On Orbit Damage

SPECTRAL ANALYSIS METHODOLOGY FOR ACOUSTICAL AND MECHANICAL MEASUREMENTS RELATIVE TO HYDRAULIC TURBINE’S GENERATOR Lafleur F., Bélanger S., Coutu E., and Merkouf A. Institut de Recherche d’Hydro-Québec, IREQ 1800 Boul-Lionel-Boulet Varennes, Québec Canada, J3X 1S1 ABSTRACT This paper presents a spectral analysis methodology for acoustical and mechanical measurements performed on hydraulic turbine’s generator in the context of power increase diagnostic. The purpose of this analysis is to link the different excitation frequencies to the electromagnetic or mechanical sources. A multidisciplinary team was necessary to provide the necessary inputs and the specific workload for this analysis. This methodology included a number of activities such as: o Choice of measurements points and sensors (microphones, accelerometers, strain gages); o Input of the generator's electromagnetic frequency analysis; o Input of the rotor and stator's Finite Element Analysis (FEA) modes; o Knowledge of the hydraulic turbine’s generator operating frequencies; o Measurement and recording of the different time based signals and computing their respective FFTs; o Identifying peaks in the frequency domain from different mechanical signals; o And several analyses that leads to identifying excitation sources and frequencies. The methodology will be presented along with an example of an analysis performed on specific mechanical signals. INTRODUCTION In the context of a study of existing generators' power increase, full instrumentation was installed on high-power hydro-generators (> 60 MW). The main goal of this project is to assess the thermal limits of the stator in relation to the increase in power [1,2]. The mechanical limits are also accessed using vibration measurements performed on the generator's rotor and stator. The vibration measurements, combined with other mechanical measurements such as thermal dilatation of the stator, stress measurements on the rotor, and acoustic measurements, allow us to identify the forces acting on both structures. This paper will present the spectral methodology used in this project and will show preliminary results. T. Proulx (ed.), Rotating Machinery, Structural Health Monitoring, Shock and Vibration, Volume 5, Conference Proceedings of the Society for Experimental Mechanics Series 8, DOI 10.1007/978-1-4419-9428-8_2, © The Society for Experimental Mechanics, Inc. 2011 15

ALTERNATOR CHARACTERISTICS A newly refurbished hydraulic power generator was instrumented and monitored. Figure 1 shows the generator's rotor and stator during refurbishing. Table 1 shows the generator’s characteristics. Figure 1: Rotor and stator of the hydraulic generator Generator’s characteristics Total power (Winter/Sumer) (75/65) MVA Power factor 0,85 Real Power (Winter/Sumer) (64/55) MW Rotation speed 94,7 rpm, 1,58 Hz Number of poles on the rotor 76 Number of slots on the stator 396 Rotor External diameter 9,08 m Stator Internal diameter 9,11 m Gap 12,7 mm Rotor height 1,6 m Table 1: Generator characteristics DESCRIPTION OF MEASUREMENTS Performance measurements Generator performance measurements were taken for several operating conditions, namely: Speed No Load (SNL), 0%, 70%, 85% and 100% of the nominal summer power specifications (55 MW). 16

Acoustical and Mechanical measurements The mechanical measurements performed on the generator's rotor and stator include sound pressure level, vibration and stress for each of the abovementioned operating condition. Acoustic measurements were performed at 8 different locations on the alternator floor. These measurements were used to evaluate the overall noise level for monitoring the working environment and machinery by non-contact sensing. Frequency analyses (FFT 0-2 kHz and 010kHz) were also performed to correlate the acoustic signals to the operating conditions. Mechanical (vibration and strain gages) instrumentation was installed on the rotor and stator. The rotor instrumentation includes 3 accelerometers installed in the axial, radial, and tangential directions and 16 strain gages (one per cross arm, with their position optimized by finite-element analysis and previous measurements) (Figure 2). The signals coming from the rotor instrumentation were transferred to the acquisition system by RF transmission. The stator's instrumentation allows for acceleration measurements (radial and tangential) of the stator core and stator frame (Figure 3). These mechanical measurements, combined with a thorough frequency analysis, were used to investigate the results obtained during different operating conditions. Figure 2: Rotor instrumentation stator core and stator frame relative displacement laser sensor Stator core and stator frame accelerometers Figure 3: Stator instrumentation 17

SPECTRAL ANALYSIS METHODOLOGY A spectral analysis methodology was developed for the various mechanical signals. Figure 4 presents the various steps of this methodology in the form of a logical diagram. The analysis allows the identification of the hydro generator excitation sources using the different sensor inputs (accelerometers, strain gages and microphones). This methodology consists of several steps and has two key goals that can be found at the bottom line of the logical diagram of figure 4. The first goal is to determine the source of the various frequencies detected by comparing those with the spectral contents coming from the stator and rotor. The second goal is to ensure the follow-up of the evolution of a spectral component on a particular signal for the various operating conditions. There are two types of inputs for this analysis which can be found at the top line of the logical diagram (figure 4). Firstly, the output generated by the various sensors is recorded as a time based signal. These signals, representing acceleration and stress endured by the generator rotor and stator, are analyzed in the frequency domain using a FFT transformation over a frequency range of 0-2kHz. We also use other types of data sets which allow us to identify the excitation frequencies and their causes These data sets include the knowledge of the operational and mechanical characteristics of the hydraulic generator, the FEA and analytical analysis of the rotor/stator and electromagnetic simulations of the system. These data set allows the identification of the synchronous frequency, the vibration modes and the electromagnetic excitation. These are then compared to the peaks found in the frequency domain of the different mechanical signals. There are three types of comparison which can be made: o For a specific sensor and a specific operating condition to find the equivalent peaks from the mechanical signal and excitation frequencies. o Between the peaks of different sensors for the same operating condition to find the common excitation frequencies. o For a specific sensor and a specific and different operating condition to follow the evolution of one or more excitation peak. The logical diagram of figure 4 shows the overall process performed during this analysis. Each box of the logic diagram is an action on the input data or on the intermediate results that helps achieve the two key goals described above. 18

Figure 4: Spectral analysis methodology 19

TYPICAL SPECTRAL ANALYSIS RESULTS The typical results of the analysis show that the signals coming off the rotor and the stator have different spectral contents. Figure 5 and 6 respectively show the spectra of the generator’s rotor and the stator radial acceleration with an aim of comparing the frequencies content in each signal (the amplitude of the graphs is adjusted to the maximum of each signal). The vibration content of the stator consists principally of harmonics of the even multiple of the electric fundamental frequency (120 Hz, 240 Hz…). Table 2 presents a list of the rotor's main frequencies of vibration as well as their associated causes. Figure 7 presents the stress spectrum measured on a cross arm of the generator’s rotor. We notice that the maximum stress is reached at a frequency of 94.8 Hz, which is similar to the rotor radial acceleration. Another stress peak is reached at a frequency of 1.58 Hz, which corresponds to the actual rotor rotational speed. The stress level at the rotation speed of the rotor (1.58 Hz) is also comparable with the one at the principal frequency, which corresponds to the actual rotor rotational speed. 0 0.5 1 1.5 2 2.5 0 200 400 600 800 1000 Frequency (Hz) Acceleration (m/s2) Figure 5: Radial acceleration spectrum of the stator 20

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 100 200 300 400 500 600 700 800 900 1000 Frequency (Hz) Acceleration (m/s2) Figure 6: Radial acceleration spectrum of the rotor 0 0.5 1 1.5 2 2.5 3 0 20 40 60 80 100 Frequency (Hz) Stress (MPa) Figure 7: Strain gage spectrum of the rotor cross arm 21

Frequency (Hz) Associated causes 94.8 Magneto motive force harmonic Rotation frequency harmonic 113.7 Magneto motive force harmonic Blade passing frequency harmonic 189.5 Magneto motive force harmonic Blade passing frequency harmonic 208.4 Blade passing frequency harmonic 265.2 Magneto motive force harmonic Blade passing frequency harmonic 303.2 Magneto motive force harmonic Blade passing frequency harmonic 227.4 Magneto motive force harmonic Blade passing frequency harmonic 1250.3 Magneto motive force harmonic 284.1 Magneto motive force harmonic 206.8 Rotation frequency harmonic 720.0 harmonics of the even multiple of the electric fundamental frequency 378.9 Rotation frequency harmonic 1.58 Rotation frequency harmonic 191.0 Coincidence with a mode of the rotor (analytical calculation) 625.2 Slot passing frequency Table 2: Main rotor vibration frequencies and their associated causes The end result obtained from the study of the frequency versus the operating condition evolution is used in the diagnosis of power increase. The spectral analysis methodology allows us to compare the amplitude of the major spectral components or principal excitations variation against the operating condition for the stator (figure 8) and rotor (figure 9). We notice that the various excitation components do not respond in the same manner when subjected to a power increase. For this type of analysis, we look at the tendency (rising, stable or falling) of a particular excitation according to the increase in power. The tendency informs us on the types of extrapolation of the levels of excitation according to the increase of power beyond the nominal output. The tendency cues us that a specific excitation might rise or fall when the generator's power is increased beyond its nominal output, which helps us determine the main mechanical limiting factors. 22

0 0.5 1 1.5 2 2.5 3 120 240 360 480 600 720 Frequency (Hz) Acceleration (m/s2) 70% 80% 100% Figure 8 : Acceleration level at the stator vs. operating condition 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 94.8 113.7 189.5 625.3 Frequency (Hz) Acceleration (m/s2) 70% 80% 100% Figure 9 : Acceleration level at the rotor vs. operating condition 23

CONCLUSION A spectral methodology analysis of the mechanical signals (accelerometers, strain gauges, microphones) was applied to the diagnostic of power increase of hydroelectric generators. This methodology allows us to better understand the excitation forces on the various parts of the generator. The methodology also allows the understanding of the evolution of a particular excitation in relation to the operating conditions. The complete analysis of the data obtained will enable us to refine the analysis methodology and to obtain a more complete mechanical diagnostic in the power increase of hydro generator. ACKNOWLEDGMENTS The authors acknowledge the technical staff of the Hydro-Québec research institute (IREQ) for the quality of their work. This team includes Luc Martell and Mathieu Soares. REFERENCES [1] C. Hudon, M. Chaaban, J. Leduc and D.N. Nguyen, Use of distributed Temperature Measurements to Explain a Generator Winding Faillure, CIGRE 2005, Lausanne, 6 pages. [2] F. Lafleur, S. Bélanger, L. Marcouiller and A. Merkouf, Acoustic and mechanical measurements of an hydraulic turbine’s generator in relation to power levels and excitation forces, IMAC 2009, Jacksonville, Fl, 7 pages. 24

Transverse vibrations of tapered materially inhomogeneous axially loaded shafts Arnaldo J. Mazzei, Jr. Department of Mechanical Engineering C. S. Mott Engineering and Science Center Kettering University 1700 University Avenue Flint MI, 48504, USA Richard A. Scott Department of Mechanical Engineering University of Michigan G044 W. E. Lay Automotive Laboratory 1231 Beal Avenue Ann Arbor MI, 48109, USA ABSTRACT Shafts loaded by axial compressive constant forces constitute an area of considerable technical importance. The transverse vibration of such shafts is the subject of the current work. Occasionally the shafts are tapered and of interest is the effect of employing functionally graded materials (FGM), with properties varying in the axial direction, on the buckling load and lowest natural frequency. The shaft cross section is circular and two types of taper are treated, namely, linear and sinusoidal. All shafts have the same volume and length and are subjected to a constant axial force below the static buckling load. Euler-Bernoulli theory is used with the axial force handled by a buckling type model. The problems that arise are computationally challenging but an efficient strategy employing MAPLE®’s two-point boundary value solver has been developed. Typical results for a linear tapered pin-pin shaft where one end radius is twice the other, and the FGM model varies in a power law fashion with material properties increasing in the direction of increasing area, include doubling of the buckling load and first bending frequency increase of approximately 43%, when compared to a homogeneous tapered shaft. For the same material and boundary conditions, a sinusoidal shaft, with mid-radius twice the value of the end ones, increases the buckling load by about 118% and the first frequency by 26%, when compared to a homogeneous sinusoidal shaft. NOMENCLATURE A , area of the shaft cross section ( 0A initial value) , , , , a m n λ real arbitrary constants E, Young’s modulus ( 0, , t b E E E , Young’s modulus values for different non-homogenous materials) F , axial compressive force acting on the shaft f , external transverse force per unit length acting on the shaft 1 2 3 4 , , , f f f f , non-dimensional functions for material / geometrical properties 1 2 , , F F non-dimensional parameters ( 1 2 2 0 0 0 F F AL ρ = Ω , 2 2 0 0 0 f F AL ρ = Ω ) I , area moment of inertia of the shaft cross section ( 0I initial value) 1K , non-dimensional stiffness ( 0 0 1 4 2 0 0 0 E I K AL ρ = Ω ) T. Proulx (ed.), Rotating Machinery, Structural Health Monitoring, Shock and Vibration, Volume 5, Conference Proceedings of the Society for Experimental Mechanics Series 8, DOI 10.1007/978-1-4419-9428-8_3, © The Society for Experimental Mechanics, Inc. 2011 25

L, length of shaft R, cylindrical shaft radius 1 2 , R R , conical shaft radii midspan R , sinusoidal shaft radius at mid-length 0R , sinusoidal shaft end radius ( ) S ξ , non-dimensional spatial function ,s numerical parameter ( ( ) 2 3 1 α α + + ) 1, s numerical parameter ( 2 2 ( 2 8 ) π β π π β + + ) t , time w, shaft displacement in the y direction xyz , inertial reference system (coordinates , , x y z ) ,Y non-dimensional shaft displacement in the y direction ,α numerical parameter ( 1 2 R R ) ,β numerical parameter ( 0 midspan R R ) 1 2 ,γ γ , numerical parameters ( 1 2 , b t b t E E γ γ ρ ρ = = ) ν , non-dimensional frequency υ , Poisson’s ratio ρ, mass density ( 0, , t b ρ ρ ρ, density values for different non-homogenous materials) ξ , non-dimensional spatial coordinate τ , non-dimensional time Ω , frequency of the shaft 0Ω , reference frequency – first bending frequency of a pin-pin homogeneous shaft INTRODUCTION This work is concerned with the buckling and transverse vibration of tapered shafts, both homogeneous and nonhomogeneous. The non-homogeneity is modeled by the use of functionally graded materials (FGM) with material properties varying in the axial direction. A brief, by no means exhaustive, survey of previous work will now be given. There is a considerable body of research on the buckling of homogeneous tapered shafts, motivated, in part, by the search for the “strongest column” (i.e., for a given volume and length, what is the maximum buckling load?). See Lee et al. [1] for numerous references on this. Buckling of variable cross-section columns was also treated by, for example, Darbandi et al. [2], Totry et al. [3] and Rahai and Kazemi [4]. The latter pointed out that tapered shapes are widely used in buildings, bridges, bearings etc. Work on buckling of structures made of functionally graded materials has also been done. See, for example, Naei et al. [5]. 26

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