River Rapids Conference Proceedings of the Society for Experimental Mechanics Series Modal Analysis Topics, Volume 3 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 Proceedings of the 29th IMAC, A Conference on Structural Dynamics, 2011 Volume 3 Modal Analysis Topics,
Published, sold and distributed by: River Publishers Broagervej 10 9260 Gistrup Denmark www.riverpublishers.com ISBN 978-87-7004-849-1 (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 Modal Analysis 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; Linking Models and Experiments; Civil Engineering, Rotating Machinery, Structural Health Monitoring, and Shock and Vibration, and Sensors, Instrumentation, and Special Topics. IMAC covers the wide variety of subjects that are related to the broad field of Structural Dynamics. It is SEM’s mission to disseminate information on a broad selection of subjects. To this end, research and application papers in this volume relate to the broad field of Structural Dynamics. Modal Analysis is a major enabling technology in this area and consequently is a significant component of this volume. 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 Each collection presents early findings from experimental and computational investigations on an important area within Structural Dynamics. The current volume on Modal Analysis Techniques includes studies on Modal Analysis, Modal Parameter Identification, Modal Parameter Estimation, Modal Testing Vibration Damping Methods, Processing Modal Data, Experimental Techniques, Active Control, Nonlinear Systems, and
Contents 1 Selecting Appropriate Analytical Mode Basis for SEREP-expansion of Experimental Modes 1 A.T. Johansson, T.J.S. Abrahamsson, Chalmers University of Technology 2 Tutorial Guideline VDI 3830: Damping of Materials and Members 17 L. Gaul, University of Stuttgart 3 Optimal Second Order Reduction Basis Selection for Nonlinear Transient Analysis 27 P. Tiso, Delft University of Technology 4 Operating Vibration Measurements of Test Fuel Assembly in Reactor Thermo-hydraulic Test Condition 41 K.-H. Lee, C.-H. Shin, H.-S. Kang, D.-S. Oh, N.-K. Park, Korea Atomic Energy Research Institute 5 Output-only Modal Analysis Using Continuous-scan Laser Doppler Vibrometry and Application to a 20kW Wind Turbine 47 S. Yang, M.S. Allen, University of Wisconsin-Madison 6 Coupling a Compliant Structure With a Hand – arm System Using FBS 65 S. Perrier, Y. Champoux, J.-M. Drouet, Université de Sherbrooke 7 A Comparison of Accelerometer Selection Methods for Modal Pretest Analysis 79 D. Linehan, K. Napolitano, ATA Engineering, Inc. 8 Modal Impact Testing of Ground Vehicle Enabling Mechanical Condition Assessment 93 A. Meyer, B. Wang, S. Britt, R. Kazi, D.E. Adams, Purdue University 9 Identifying Parameters of Nonlinear Structural Dynamic Systems Using Linear Time-periodic Approximations 103 M.W. Sracic, M.S. Allen, University of Wisconsin-Madison 10 B-spline Laminate Shell Finite Element Updating by Means of FRF Measurements 127 A. Carminelli, G. Catania, The University of Bologna 11 Measurement of 2D Dynamic Stress Distributions With a 3D-Scanning Laser Doppler Vibrometer 141 M. Schüssler, M. Mitrofanova, Polytec GmbH; U. Retze, MTU Aero Engines GmbH
viii 12 153 University of Western Ontario 13 Tool Chatter in Turning With a Two-link Robotic Arm 161 14 An Identification Method for the Elastic Characterization of Materials 169 D. Yang, Day Software Systems, Inc. 15 An Alternating Least Squares (ALS) based Blind Source Separation Algorithm for Operational Modal Analysis 179 J. Antoni, University of Technology of Compiegne; S. Chauhan, Bruel and Kjaer Sound and Vibration Measurement A/S 16 189 E. Reynders, J. Houbrechts, G. De Roeck, Katholieke Universiteit Leuven 203 J.M. Liu, Tsinghua University/China Orient Institute of Noise & Vibration; S. Sheng, M. Ying, S.W. Dong, Tsinghua University 18 Vibrations in Floors 215 D.S. Nyawako, P. Reynolds, M. Hudson, University of Sheffield 19 Nonlinear Normal Modes of a Full-scale Aircraft 223 Office National d’Etudes et de Recherches Aérospatiales (ONERA) 20 243 C. Warren, C. Niezrecki, P. Avitabile, University of Massachusetts Lowell 21 Winer Filter 253 J. Lee, S. Kim, D. Kim, S. Wang, Gwangju Institute of Science and Technology 22 A New Broadband Modal Identification Technique With Applications 261 L. Zhang, Nanjing University of Aeronautics and Astronautics; Y. Tamura, Wind Engineering Research Center; T. Wang, X. Sun, Tokyo Polytechnic University 23 Advanced Dynamic Absorber Design Method for Practical Application 273 24 Use of Operational Modal Analysis in Solving Ship Vibration Issues 281 A. Boorsma, E.P. Carden, Lloyd’s Register EMEA 25 The Use of Layered Composites for Passive Vibration Damping 289 C.E. Lord, J.A. Rongong, A. Hodzic, University of Sheffield 26 Unbiased Estimation of Frequency Response in the Presence of Input and Output Noise 299 A. Brandt, University of Southern Denmark Automated Interpretation of Stabilization Diagrams 17 The Optimization and Autonomous Identification of Modal Parameters FRF Measurements and Mode Shapes Determined Using Image-based 3D Point-tracking Experimental Modal Analsis (EMA) Using Ibrahim Time Domain (ITD) Method and J.W. Lee, Ajou University A. Özer, Gyeongsang National University; S.E. Semercigil, Victoria University S.E. Semercigil, Ö.F. Turan, Victoria University; G.A. Kopp, A Particle Damper for Transient Oscillations Independent Modal Space Control Technique for Mitigation of Human-Induced M. Peeters, G. Kerschen, J.C. Golinval, University of Liège; C. Stéphan, P. Lubrina,
ix 27 Vibration Control Using the Non-model Based Algorithm 307 28 Active Suspension Systems for Passenger Cars: Operational Modal Analysis as a Tool for the Performance Assessment 313 J. Anthonis, H. Van der Auweraer, LMS International 29 Structural Dynamics With Coincident Eigenvalues: Modelling and Testing 325 E. Bonisoli, C. Delprete, M. Esposito, Politecnico di Torino; J.E. Mottershead, University of Liverpool 30 339 S. Tornincasa, E. Bonisoli, F. Di Monaco, S. Moos, M. Repetto, F. Freschi, Politecnico di Torino 31 Practical Trouble Shooting Test Methodologies 351 D.L. Brown, A.W. Phillips, M.C. Witter, University of Cincinnati 32 Autonomous Modal Parameter Estimation: Methodology 363 A.W. Phillips, R.J. Allemang, D.L. Brown, University of Cincinnati 33 Autonomous Modal Parameter Estimation: Statistical Considerations 385 R.J. Allemang, A.W. Phillips, D.L. Brown, University of Cincinnati 34 Autonomous Modal Parameter Estimation: Application Examples 403 D.L. Brown, R.J. Allemang, A.W. Phillips, University of Cincinnati 35 Combined State Order and Model Order Formulations in the Unified Matrix Polynomial Method (UMPA) 429 R.J. Allemang, A.W. Phillips, D.L. Brown, University of Cincinnati 36 Interaction Between Structures and Their Occupants 445 L. Pedersen, Aalborg University 37 Evaluation of Site Periods in the Metro Vancouver Region Using Microtremor Testing 451 J. Traber, K. Kutyn, C.E. Ventura, W.D. Liam Finn, University of British Columbia 38 Experimental Study of the Nonlinear Hybrid Energy Harvesting System 461 39 Using Transmissibility Measurements for Nonlinear Identification 479 A. Carrella, D.J. Ewins, University of Bristol; L. Harper, AgustaWestland 40 Steer Control of Motorcycle by Power Steering 491 41 Estimating Low-bias Frequency Response Using Random Decrement 497 R. Brincker, Aarhus University; A. Brandt, University of Southern Denmark H. Rhee, G. Hamm, G.H. Kim, S.J. Park, S.Y. Lee, Sunchon National University M.A. Karami, Virginia Polytechnic Institute and State University; P.S. Varoto, T. Koizumi, N. Tsujichi, T. Takemura, Doshisha University University of São Paulo; D.J. Inman, Virginia Polytechnic Institute and State University Nonlinear Dynamics of an Electro-mechanical Energy Scavenger L. Soria, Politecnico di Bari; A. delli Carri, Università degli Studi di Brescia; B. Peeters,
x 42 A Study of Mechanical Impedance in Mechanical Test Rigs Performing Endurance Testing Using Electromagnetic Shakers 503 43 A Global-local Approach to Nonlinear System Identification 513 Y.S. Lee, New Mexico State University; A. Vakakis, D.M. McFarland, L. Bergman, University of Illinois at Urbana-Champaign 44 Characterization of Rotating Structures in Coast-down by Means of Continuous Tracking Laser Doppler Vibrometer 525 45 Pyroshock Loaded MISO Response 533 J.C. Wolfson, J.R. Foley, Air Force Reserach Laboratory; A.L. Beliveau, Applied Research Associates, Inc.; G. Falbo, J. Van Karsen, LMS Americas 46 Development of a Long Term Viable Dynamic Data Archive Format 541 47 557 48 Understanding the Effect of Preload on the Measurement of Forces Transmitted Across a Bolted Interface 567 D. Di Maio, D.J. Ewins, University of Bristol Data Analysis Strategies for Characterizing Helmet-head Performance M. Martarelli, Università degli Studi e-Campus; C. Santolini, P. Castellini, A.W. Phillips, R.J. Allemang, University of Cincinnati Università Politecnica delle Marche C. Butner, D.E. Adams, Purdue University; J. Foley, Air Force Research Laboratory T. Robbins, D. Adams, Purdue University; S. Walsh, U.S. Army Research Laboratory
Selecting Appropriate Analytical Mode Basis for SEREP-Expansion of Experimental Modes Anders T. Johansson, Thomas J.S. Abrahamsson Abstract Since being introduced in 1986, the System Equivalent Reduction Expansion Process (SEREP) has been used to expand experimental eigenvector elements to the number of degrees-of-freedom of an associated FE-model. In fact, expansion for interpolation and extrapolation was its original purpose. Since then, studies of SEREP and other reduction/expansion methods have been abundant. A remarkable number of these have concentrated on the selection of master degrees of freedom for model reduction. Few have however considered the modal basis best used when SEREP is used for expansion. Expanded experimental modes are expected to correlate well with their analytical siblings. However, we argue that the degree of global correlation should only be in parity with the local correlation between the analytical and experimental modes at locations where measurements are made. Since SEREP is a method which basically approximates a measured mode by a linear combination of analytical modes, perfect agreement between the expanded experimental and analytical modes is easily achieved, e.g. by simply using only one single mode for expansion. Of course, in this way the expanded mode normally has very little in common with the measured mode. On the other hand, using too many modes may result in something similar to the well known problem of fitting a high-order polynomial to noisy data: Perfect agreement at measurement locations is achieved at the expense of unrealistic deviations and large curvatures between these. To make sure that the experimental mode has been expanded in a manner faithful to the actual measurements, it is therefore reasonable to use a correlation based criterion in the selection of the expansion basis. Such a criterion is presented in the present paper. Anders T. Johansson Chalmers University of Technology, Department of Applied Mechanics, SE-412 96 Go¨teborg, Sweden e-mail: anders.t.johansson@chalmers.se Thomas J.S. Abrahamsson Chalmers University of Technology, Department of Applied Mechanics, SE-412 96 Go¨teborg, Sweden e-mail: thomas.abrahamsson@chalmers.se T. Proulx (ed.), Modal Analysis Topics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 6, 1 DOI 10.1007/978-1-4419-9299-4_1, © The Society for Experimental Mechanics, Inc. 2011
1 Introduction The System Equivalent Reduction Expansion Process (SEREP) was introduced in the late eighties by O’Callahan et al. [16, 17] and, slightly altered, by Kammer [9]. Its use as an expansion method for experimental modes will be discussed here. Assume that the i:th experimental eigenvector φX L,i exists on some local level Ldefined as a subset of the degrees of freedom (DOFs) of an FE model, where the full set of FE DOFs is denoted the global level, G. Using a set of analytical modes in the modal matrix ΦA G, the SEREP expansion of φX L,i is φX G,i = ΦA G(ΦA L) +φX L,i (1) Where superscript (·)+ denotes the Moore-Penrose pseudoinverse. The method has been used and discussed vividly within the structural dynamics community. To mention a few examples of its use, Yun et al. [20] used SEREP reduction in a damage detection application, Mitra et al. [14] applied it to fit a piezoelectric beam into a control algorithm, and Das and Datt [7] presented a modified version of SEREP adapted to rotating machinery applications. SEREP, both by construction and name, is an expansion as well as a reduction method, and much work has been directed at reduction, with an emphasis on the choice of master nodes, i.e. the nodes to be retained in a reduction process, see for instance [3, 6]. The latter article, by Bonisoli et al., touches the subject of modal incompleteness, which by extension, or perhaps rather reversion, leads to the choice of basis for expansion. In fact, quite a few authors have noted the importance of the reduction/expansion basis. Noor [15] wrote: The effectiveness of reduction methods depends, to a great extent, on the appropriate choice of the global approximation vectors (or reduced-basis subspace). Which is to say that the choice of modal basis is highly important (also when using SEREP as a reduction method). Sastry et al. [19] considered this in devising a method for an efficient choice of modal basis when using a reduced model to try and reproduce time-domain system responses with a high frequency band-limited content. In studying expansion methods for experimental modes, Balme`s ([4, 5]) also pinpointed the main drawback of SEREP as being the sensitivity to the selection of FE modes that should form the expansion basis (the term used is ”modal methods” - Balme`s distinguishes between the work presented by O’Callahan, know as SEREP, [16, 17] and that by Kammer, known as Modal [9] - the difference being that Modal keeps the eigenvector elements at local DOF:s from measurements as is, while SEREP filters them by the FE-modes). Kammer [10, 11] also noted that the 2
accuracy of modal methods is dependent on the accurate correlation between analytical and experimental modes, that is; the goodness of the analytical modal basis used for expansion. Balme`s therefore discards modal methods in favor of a Least Squares framework minimizing the difference between the expanded model response and the measurements at local level, a methodology also advocated by Levine-West et al. [13]. Kammer went on to devise a Hybrid method which combines static expansion with Modal [10]. In contrast, we try to select an optimal basis of FE modes to achieve a good expansion. Pascual et al. [18] discussed such a selection of expansion basis for the specific purpose of damage detection. In their article, a metric defined by the maximum residual energy of a particular expansion is evaluated to verify that the expansion basis is large enough. When comparing experimental eigenvectors to analytical ditto, it is commonplace to use a correlation metric such as the Modal Assurance Criterion (MAC, [1]). In particular, and in constraining the discussion to the modal expansion methods, such a metric is most likely to be used in practice to select the set of analytical modes to be used for expansion. Realizing this, it is but a small step to include this information into a discrete optimization scheme aimed at selecting the best modal base for SEREP-expansion. 2 Selection through optimization This article aims to improve the performance of the SEREP modal expansion method by approaching the selection of FE modes forming the basis of the expansion as a discrete optimization problem. Let the analytical modes be sorted in ascending frequency order. If there are n analytical modes available such that the full set of analytical modes can be defined by the numbers: S:={k ∈N∗|k ≤n} (2) The problem is to find I ⊂S, a set of indices defining a subset of the analytical modes such that the i:th experimental mode φX L,i, is expanded as well as possible (using ΦA G,I in (1)). The problem of finding the optimal set of analytical modes I ? for SEREP expansion of φX L,i is: I? =argmin I⊂S J(I) (3) Where Jis the cost function defined below. Note that the optimization procedure hence needs to be rerun for each mode i to be expanded. The number of optimiza3
tion problems which needs to be solved is hence equal to the number of measured modes. Below, the steps for expanding a mode i are described; letting i run over all measured modes, the entire set is expanded. Note that all entities including i or j, which includes all correlations and candidate sets, hence both I:s and J, will vary for each experimental mode φX L,i to be expanded. While the optimization perspective comes with standard performance aids such as the use of constraints, the key to successful optimization is the cost function, the quantity to be minimized by the optimization routine. For the purpose of this article, the cost function is created from a correlation metric which quantifies the agreement between local and global levels. The cost function’s purpose is to assess the quality of the mode expansion, and so ideally, it should compare the actual measured mode at the full set of FE DOFs with the expanded mode. Since by its nature, the measured mode is defined at measured DOFs only, this metric is not available, and so the more indirect approach of including correlation comparisons using different sets of DOFs is needed. 2.1 Cost function Above, we state that an expanded experimental mode should correlate with its analytical counterpart only as well as to the degree of correlation between analytical and experimental modes at the DOFs where measurements were made. Upon defining the full FE model DOFs as the global level and the subset where measurements have been taken the local level, this can be concisely expressed in the following proposition: Proposition 1. The correlation between a measured mode and its analytical counterpart at the global level should be similar to their correlation at the local level. Several things combine to complicate this, however. Foremost is the fact that correlation between two modes may be defined in several ways. An interesting article by Allemang, [2], provides an overview of correlation metrics in use in modal analysis. We performed the comparisons of this article with several of the metrics described, achieving best results using the most well known of them, the Modal Assurance Criterion, MAC, introduced by Allemang and Brown 1982 [1] and deduced originally in a linear regression setting. It is calculated as MAC(φa, φb)= | φH a φb| 2 (φH a φa)(φH b φb) (4) Where φa and φb are two modal vectors and(·) H denotes Hermitian transpose. At face value, the construction of a cost function from a correlation metric is straightforward. Recall that we try to find the optimal basis for expanding mode 4
i, φX L,i. Let φA L, j be the analytical mode which correlates best with it at local level, and define the scalar correlations at local and global level as mL =MAC(φ X L,i, φ A L, j) (5a) mG =MAC(φ X G,i, φ A G, j) (5b) A cost function in keeping with proposition 1 is: J=kmG−mLk (6) Where k·k is a user-defined norm. But what if φX L,i does not correlate well with any one of the analytical modes? If the FE model is poor, this is a likely situation. But even if the analytical model is of high quality, this can occur when modes are present that correspond to closely spaced eigenvalues. To incorporate as much valuable information as possible into the cost function it may be beneficial to include other MAC values, i.e. degrees of correlation between the measured mode and analytical modes other than the one which correlates best with φX L,i. Recall that the experimental mode to be expanded, φX L,i, correlates best with analytical mode j. The index set of all modes except j becomes: S \j :=S\{j} (7) At this point, we can define the two correlation arrays: ML =MAC(φ X L,i, ΦA L,S \j ) (8a) MG =MAC(φ X G,i, ΦA G,S \j ) (8b) And a cost function similar to that of equation (6) can be written as: J=W1kmG−mLk+W2kMG−MLk (9) Most of the analytical modes will however have very low correlation with a given experimental mode. Furthermore, in increasing the number of degrees of freedom, we expect the correlation between two separate eigenvectors to drop because of the orthogonality properties of the eigenvectors. The second term in (9) may then force the correlation to be higher than desirable. To prevent this, it can be beneficial to include only analytical modes whose correlation with the experimental mode to be expanded, at the local level, is above a certain limit. Thus, the set S \j modifies to: ¯S \j :={k ∈S \j| MAC(φX L,i, φ A L,k) > σ} (10) 5
Where 0 < σ<1 is a user-defined limit value. The user must also set the relative weights W1 and W2, and define which norm to use. If taking the ` 2-norm, (9) can be generalized further to the quadratic formJ= εTWε popular in engineering sciences, where εT =[(mG−mL) (MT G−MT L )] andWis a positive definite weighting matrix. 2.2 Reduction of candidate sets The number of index sets I satisfying I ⊂S is vast. If a brute-force method is to be used for the solution of this problem, with n analytical modes under consideration, the number of combinations Nto be compared is N= n ∑k=1 n! (n−k)!k! (11) Even for a relatively moderate number of FE modes, this becomes prohibitive computational-wise. For instance to include up to 30 FE modes as basis would require considering just over one billion combinations. To examine the entire solution space must therefore be considered impractical, if not outright impossible. We propose to circumvent this through considering only combinations of analytical eigenvectors such that the corresponding eigenvalue indices make up an interval I :=[ jmin, jmax] (such that I ⊂S). This reduces the number of combinations needed, and yet it still requires the evaluation of a great many uninteresting sets, such as ones that do not contain the analytical eigenvector which correlates best with the mode to be expanded. Thus, the concept of an iterative method of candidate set construction arose.1 2.2.1 Constructing candidate sets The candidate set construction algorithm proposed here is based on closed frequency intervals; that is, to include all analytical modes in a frequency range. The optimization problem of (3) is then reduced to finding the limits of that frequency band. The algorithm consists of two steps; the first step expands the candidate set, while the second translates the frequency band in the frequency domain. 1 An idea inspired by Kammer’s method of Effective Independence [12] for sensor placement. A candidate set construction algorithm based on a methodology more closely resembling the EFI, in which the analytical eigenvector making the largest impact on the cost function was added without restricting the candidate sets to be intervals, was also tested. In this context, it did not perform as good as the algorithm in the subsequent section, however, and was left out. 6
ùj ùj-1 ùj+1 ùj+2 ùj+3 ùj-2 ùj-3 I I I I I I I 1 2 3 4 5 6 7 Fig. 1 Concept of sequential construction of candidate sets consisting of frequency intervals. Step 1: Expanding the candidate set In step 1, the candidate sets are created through sequential expansion, starting at a set including only the mode with the highest correlation with modei and iteratively increasing the interval through adding that of the two modes adjacent to the set which affects the cost function the most: Let the initial set consist only of j, the analytical eigenvector with the highest correlation to experimental mode i, so that I1 :={j}. This will result in a global diagonal MAC 1, see (5b). Next, add the mode adjacent to j which together with j gives the lowest cost function value to create set number two. Hence, I2 :={j −1, j} , or I2 :={j, j +1} (12) Assume I2 :={j −1, j}. To formI3, consider the modes adjacent to I2; denote the set of these two modes Γ2, where in this example Γ2 :={j −2, j +1} (13) Include the mode in Γ2 which together with I2 gives the lowest cost function. Continue the process until a=NDOF. See Figure 1. Step 2: Translating the candidate set When the number of analytical modes exceeds the number of measured DOFs, SEREP (1) corresponds to an over-determined least-squares problem. To avoid the problems associated, the size of each candidate set Ia must be less than or equal to the number of DOFs at local level, NDOF. If there are still analytical modes to be tried when a =NDOF, the mode adjacent to INDOF which gives the lowest cost function value can be added, compensated by through removing the last index at the other end of the set to render the next candidate set. In effect, this means sweeping the frequency window up or down the frequency axis. Concisely, the candidate set construction algorithm can be written as: 7
Step1 1.1) I1 :={j} 1.2) k? =argmin k Ji(Ia ∪{k}) , k ∈ Γa Γa :={s ∈{S(Ia) | ∃t ∈Ia s.t. s =arg min u∈{S(Ia) kt −uk} 1.3) Ia+1 :=Ia ∪{k?} 1.4) Repeat 1.2) and 1.3) until: a=NDOF Step2 2.1) k? =argmin k Ji(Ia ∪{k}) , k ∈ Γa \ a[ r=1 Ir 2.2) ¯Ia+1 :=Ia ∪{k?} Ia+1 :=¯Ia+1 \{s ∈Ia | s =extremum∗(¯Ia+1)} 2.3) Repeat 2.1) and 2.2) until: Γa \ a[ r=1 Ir =0/ ∗ Extremum is here defined as the largest and smallest value of the set, i.e. maximum and minimum. Each candidate set calculated is stored, as an array of indices, along with the corresponding cost function values. The set with the lowest cost function value is denoted I?; the expansion of φX L,i is then φX G,I?,i. Unfortunately, the algorithm described above does not yield monotonically decreasing cost function values, wherefore this cumbersome approach is necessary. Whether (3) is actually fulfilled, that is, whether I? is the optimal solution in the solution space of all sets I ⊂S, is not further investigated. 8
2.2.2 Constraints Constraints on the feasible domain is standard in optimization. In (3), the feasible domain is defined byI ⊂S. Constraining I further, the feasible domain becomes: I ⊂ ˆS⊂S (14) Where ˆS is a proper subset of S. In effect, this means excluding certain analytical eigenvectors from the full eigensolution. On what grounds can such an exclusion be justified? We have come up with two potential scenarios: First, if an analytical eigenvector is not observable at the local (measurement) level, its contribution to system motion cannot be identified from the measurements, and it should be excluded. Using the observability metric proposed by Hamdan and Nayfeh [8], the subset ˆS becomes: ˆS:={s ∈S | os > σ} (15a) where os =max k |ck φA G,s| kckkkφA G,sk =|{ φA G,s}k| kφA G,sk (15b) Where σis a user-defined limit value and ck is the k:th row of the observation matrix relating the local and global levels. Hence, ck is a row vector with unit value at the position of the DOF corresponding to measurement DOF k and zeros otherwise, which makes the observability metric simply the normalized local-level highest value of the analytical mode under scrutiny; the second identity in (15b). Second, if for some reason an analytical eigenvector correlates extremely little with the measured mode to be expanded at the local level, its use as one of the basis vectors is likely to be limited. Using MAC exclusion, the subset ˆS becomes: ˆS:={s ∈S | max k MAC(φA L,s, φ X L,k) > σ} (16) Where again σis user-defined. In using these excluding constraints, simply replace Swith ˆSin the candidate set construction algorithm above (making the Ia:s intervals only in transferred sense, i.e. such that Ia :=[ jmin, jmax]∩ˆS). Apart from excluding modes from the potential set, we also have the option of adding proper constraints, such that the optimization problem of equation (3) modifies to: I? i =argmin I⊂ˆS Ji(I) s.t. gi(I) ≤0 (17) Two constraints of this type have been considered. One verifies the numerical soundness of SEREP: When the expansion has been successful, the expanded mode should 9
resemble the original mode very closely at the local level (this is not applicable when using the Modal version of SEREP, see the discussion regarding differences between SEREP and Modal in the introduction): MAC(φX I,L,i, φ X L,i) > σ (18) Where σis a user-defined limit value somewhat smaller than unity. As for the other one, recall that in equation 10, we created a modified set ¯S \j which excluded modes with low local level correlation from the cost function, since we expected the correlation to drop as the number of DOFs increased. These can instead be kept below their initial value (with some slack): MAC(φA G,k, φ X I,i)−MAC(φ A L,k, φ X L,i) ≤ σ, ∀k ∈S\( ¯S \j ∪{ j}) (19) Again, σis a small user-defined limit value. Sets which do not comply with these constraints are deemed infeasible and discarded, provided that a feasible set can be found. If a feasible set cannot be found, the constraints are disregarded and the mode set with the lowest cost function value is used for expanding mode i. 3 Numerical validation The procedure described in the previous section has been evaluated on two simple numerical examples. In both cases, results are compared to ”SEREP classic”, interpreted as using the same set of analytical modes in the expansion of every experimental mode, letting the set of analytical modes included in the expansion basis start at the first mode and be a proper interval up to a given mode, such that the highest number of modes is accurately expanded. This is ensured in the following examples. The first example (see Figure 2) consists of a single flat 153x299.5mm titanium plate (E=108GPa, ν=0.22), which was modeled using 1352 plate elements with FEMAPr and MD NASTRANTM. The analytical model represents a 10mm thick plate, while a 12mm plate model emulates measurements. Emulated sensing is taken perpendicular to the plate at the 15 locations shown in figure 2, with added normal distribution eigenvector errors of a magnitude of 2% of the maximum magnitude vector element of each mode. In-plane modes are not included in the measured set. Using the method described above withW1 6=0 andW2 =0, exclusion based on observability with a limit value corresponding to 0.5% of the maximum observability value and the constraint ensuring a MAC-value when comparing the expanded mode with its original no lower than 0.99, all 36 observable eigenvectors below 19000Hz are accurately expanded, see Table 1. When increasing eigenvector element error levels to 5%, all experimental modes except for eigenvector number 25 are accu10
Fig. 2 Basic FE-model of planar plate used for validation, numerical example one. The bold dots indicate measurement locations. rately expanded. Eigenvector 25 is not accurately expanded due to the fact that, at the local level, it has MAC correlation of 0.99 with the second eigenvector of the analytical mode shape basis. When noise levels increase, a mode mismatch occurs. In comparison, the original SEREP algorithm accurately expands only at most 15 modes (several choices gave 15 accurate modes), see Table 2. The poor results of SEREP classic in this example are due to the fact that the analytical modal basis includes modes that are not observable at the local level - modes perpendicular to the measurement direction. Since these are zero vectors at the local level, bar numerical errors, SEREP is unable to calculate their contribution to the solution. Table 1 MAC values of SEREP-expanded modes and the modes used to emulate measurements. Results for example 1 with 2% white gaussian noise. First 38 modes used in SEREP classic. Mode optim classic 38 Mode optim classic 38 1 1.00 0.00 19 1.00 1.00 2 1.00 0.01 20 1.00 1.00 3 1.00 0.00 21 1.00 1.00 4 1.00 0.00 22 1.00 0.00 5 1.00 1.00 23 1.00 1.00 6 1.00 0.00 24 1.00 1.00 7 1.00 0.02 25 1.00 0.00 8 1.00 1.00 26 1.00 1.00 9 1.00 0.00 27 1.00 1.00 10 1.00 0.01 28 1.00 0.01 11 1.00 0.00 29 1.00 1.00 12 1.00 0.01 20 1.00 0.00 13 1.00 0.00 31 1.00 1.00 14 1.00 1.00 32 1.00 1.00 15 1.00 1.00 33 1.00 0.00 16 1.00 1.00 34 1.00 0.01 17 1.00 0.00 35 1.00 0.00 18 1.00 0.00 36 1.00 0.00 The second example is an assembly consisting of three titanium plates rigidly connected at their intersections. The resulting model, seen in figure 3, was made up of 11
Table 2 MAC values comparing expanded modes and the modes used to emulate measurements. Results for example 1 with 5% white gaussian noise. First 38 modes used in SEREP classic. Mode optim classic 38 Mode optim classic 38 1 1.00 0.01 19 1.00 0.99 2 0.99 0.01 20 1.00 1.00 3 1.00 0.01 21 1.00 0.99 4 1.00 0.00 22 1.00 0.00 5 1.00 0.99 23 1.00 0.99 6 1.00 0.00 24 1.00 0.98 7 0.99 0.01 25 0.01 0.00 8 1.00 1.00 26 1.00 1.00 9 0.99 0.00 27 1.00 0.99 10 1.00 0.01 28 1.00 0.01 11 0.99 0.00 29 1.00 0.99 12 0.99 0.01 20 1.00 0.00 13 1.00 0.00 31 1.00 0.99 14 1.00 1.00 32 0.96 0.98 15 0.99 0.99 33 1.00 0.00 16 1.00 1.00 34 1.00 0.01 17 1.00 0.00 35 1.00 0.00 18 1.00 0.00 36 1.00 0.00 3380 plate elements at a grand total of 21222 DOFs. The model emulating measurements is such that the left plate in Figure 3 has 12mm thickness, while the other two plates are 10mm thick. By contrast, all of the plates in the analytical model are 10mm thick. This means in effect that we are using a symmetric model as a basis for expanding non-symmetric modes. The experimental mode set consists of the first 34 elastic modes (the first 40 from FEM minus 6 rigid body modes). For the analytical solution, ten more modes were included, making the number of analytical candidate modes 44. In this case, all 44 elastic analytical candidate modes are observable, so no exclusion was used. The weighting parameters were set toW1 =W2 to account for the problem of finding a best fit symmetric or anti-symmetric counterpart to the non-symmetric mode to be expanded. No analytical modes were left out of the cost function. Furthermore, the constraint was again used with the local-level correlation after expansion no lower than 0.99, and the constraint with keeping off-diagonal MAC-values low was used for all modes with a local-level correlation below 0.7 and a slack variable of σ=0.05. The measurements are taken in a symmetric pattern at 33 nodes measuring in the direction normal to the plates; in total three measurements are perpendicular to the smaller plate. Again, 2% normal distribution eigenvector element error was added. At most, the classical SEREP routine managed to accurately expand 22 modes out of the first 34 flexible. Using the proposed method, 31 out of 34 were accurately expanded. The three modes that were not expanded properly, modes 24, 25 and 27, were such that the main deformation in the two larger plates was in-plane, i.e. largely unobservable. See Table 3. Figure 4 shows mode 23 as an example of a mode where the proposed method clearly outperforms SEREP classic. 12
Fig. 3 Basic FE-model used for validation, numerical example two. The bold dots indicate measurement locations. 153x111.5mm 153x299.5mm It should be noted that the results in this section are somewhat sensitive to the Table 3 MAC values comparing expanded modes and the modes used to emulate measurements. Results for example 2. First 24 modes used in SEREP classic. Mode optim classic 24 Mode optim classic 24 1 0.99 0.97 18 0.97 1.00 2 0.99 0.99 19 0.99 0.99 3 0.99 0.99 20 0.99 0.99 4 0.99 0.99 21 0.98 0.98 5 1.00 1.00 22 0.99 0.99 6 0.99 0.99 23 1.00 0.30 7 1.00 1.00 24 0.88 0.82 8 1.00 1.00 25 0.67 0.02 9 1.00 1.00 26 1.00 0.01 10 0.99 0.99 27 0.59 0.02 11 0.99 0.99 28 1.00 0.09 12 1.00 0.99 29 0.98 0.03 13 1.00 1.00 30 0.97 0.00 14 0.99 0.99 31 0.96 0.01 15 1.00 0.99 32 0.99 0.00 16 0.98 0.99 33 1.00 0.01 17 1.00 0.99 34 0.95 0.00 noise realization. For different realizations of the first example with 5% error, for instance, there were cases where both mode 2 and mode 25 were inaccurately expanded, and there were cases where they were both expanded correctly. Other times, other modes could not be properly expanded at this error level. 13
Actual SEREP optim SEREP classic 24 Fig. 4 Comparison of mode number 23, numerical example 2. First 24 modes used in SEREP classic. 4 Conclusions and further work The most important contribution of this work is the focus on the basis selection in modal expansion methods. A first example highlights the importance of excluding analytical modes that are not observable at measurement locations from the expansion set. A second example illustrates that even an FE model whose normal modes correlate quite poorly with the measured modes can be successfully used as an expansion basis. The original SEREP method also works quite well for this example in the lower end of the frequency range. The method proposed in this paper clearly outperformed the classical SEREP expansion method in the comparisons performed. Its drawbacks are however its computational inefficiency as an abundance of candidate expansions are to be tested for each mode and the extensive input needed to set the user-defined limit values of the various constraints. In the future, we hope to compare this methodology with other methods such as those proposed by Balme`s [4, 5] and Kammer [10, 11]. As the latter method also uses a modal basis, it would also be interesting to investigate the effects of basis selection on it as well. References 1. Allemang, R.J., Brown, D.L., A Correlation Coefficient for Modal Vector Analysis, in Proceedings of the first International Modal Analysis Conference, Orlando, Florida, November 1982. 14
2. Allemang, R.J., The Modal Assurance Criterion - Twenty Years of Use and Abuse, Journal of Sound and Vibration, 2003. 3. Avitabile, P., Pechinsky, F., O’Callahan, J., Study of Modal Vector Correlation Using Various Techniquies for Model Reduction, in Proceedings of the Ninth International Modal Analysis Conference, Florence, Italy, April 1991. 4. Balme`s, E., Review and Evaluation of Shape Expansion Methods, inProceedings of the 18:th International Modal Analysis Conference, San Antonio, Texas, February 2000. 5. Balme`s, E., Sensors, Degrees of Freedom, and Generalized Modeshape Expansion Methods, in Proceedings of the 17:th International Modal Analysis Conference, Kissimmee, Florida, February 1999. 6. Bonisoli, E., Delprete, C., Rosso, C., Proposal of a modal-geometrical-based master nodes selection criterion in modal analysis, Mechanical Systems and Signal Processing23, 606-620, 2009. 7. Das, A.S., Dutt, J.K., Reduced model of a rotor-shaft system using modified SEREP, Mechanics Research Communications 35, 398-407, 2008. 8. Hamdan, A. M. A. and Nayfeh, A. H., Measures of Modal Controllability and Observability for First- and Second-Order Linear Systems, Journal of guidance, control, and dynamics, Vol. 12, No. 3 (1989), pp. 421-428 9. Kammer, D.C., Test-analysis-model development using exact model reduction, The International Journal of Analytical and Experimental Modal Analysis 2 (4), 174-179, (1987). 10. Kammer, D.C., A Hybrid Approach to Test-Analysis-Model Development for Large Space Structures, Journal of Vibration and Acoustics 113, 325-332, 1991. 11. Kammer, D.C., Correlation considerations - Part 2, Model reduction using modal SEREP and Hybrid, in Proceedings of 16th International Modal Analysis Conference, Santa Barbara, California, February 1998. 12. Kammer, D., Sensor set expansion for modal vibration testing, Mechanical Systems and Signal Processing, 2005. 13. Levine-West, M., Kissil, A., Milman, M., Evaluation of Mode Shape Expansion Techniques on the Micro-Precision Interferometer Truss, inProceedings of the 12:th International Modal Analysis Conference, Honolulu, Hawaii, Jan-Feb 1994. 14. Mitra, M., Gopalakrishnan, S. and Seetharama Bhat. M, Vibration control in a composite box beam with piezoelectric actuators, Smart Materials and Structures 13, 676-690, 2004. 15. Noor, A.K., Recent advances and applications of reduction methods, Applied Mechanics Reviews 47 (5), 125-145, 1994. 16. O’Callahan, J.C., Avitabile, P., Madden, R., Lieu, I.W., An Efficient Method of Determining Rotational Degrees of Freedom From Analytical and Experimental Modal Data, in Proceedings of the Fourth International Modal Analysis Conference, Los Angeles, California, February 1986. 17. O’Callahan, J.C., Avitabile, P., Riemer, R., System Equivalent Reduction Expansion Process (SEREP), inProceedings of the Seventh International Modal Analysis Conference, Las Vegas, Nevada, February 1989. 18. Pascual, R., Scha¨lchli, Razeto, M., Improvement of damage-assessment results using highspatial density measurements, Mechanical Systems and Signal Processing19, 123-138, 2005. 19. Sastry, C.V.S., Roy Mahapatra, D., Gopalakrishnan, S., Ramamurthy, T.S., An iterative system equivalent reduction expansion process for extraction of high frequency response from reduced order finite element model, Computer Methods in Applied Mechanics and Engineering192, 1821-1840, 2003. 20. Yun, G.J., Ogorzalek, K.A., Dyke, S.J., Song, W., A parameter subset selection method using residual force vector for detecting multiple damage locations, Structural Control and Health Monitoring, 1987. 15
Tutorial Guideline VDI 3830: Damping of Materials and Members Lothar Gaul Committee Background It was Nov 10, 1982 when Prof. Federn, Prof. Gaul, Prof. Mahrenholtz, and Dr. Pieper VDI decided to work out a guideline on damping in the VDI/FANAK C13 Committee “Material Damping”. They were joined by Prof. Ottl, Prof. Kraemer, Prof. Pfeiffer, Prof. Markert, Prof. Wallaschek, and Mr. Hilpert VDI lateron in their names order. The idea was to comprise distributed theoretical and experimental knowledge and to homogenize the nomenclature of this subject. At the very beginning, important knowledge was provided by the books J.D. Ferry: Viscoelastic Properties of Polymers John Wiley & Sons, New York, 1960 B.J. Lazan: Damping of Matrials and Members in Structural Mechanics Pergamon Press, Oxford, 1968 Important contributions to the subject were made at conferences in the USA, such as – Damping Lynn Rogers – The Role of Damping in Vibration and Noise Control, ASME Boston Lynn Rogers, Lothar Gaul – Damping Sessions at IMAC Lothar Gaul et al and in Germany by the colloquium Lothar Gaul Institute of Applied and Experimental Mechanics, University of Stuttgart, Pfaffenwaldring T. Proulx (ed.), Modal Analysis Topics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 6, 17 DOI 10.1007/978-1-4419-9299-4_2, © The Society for Experimental Mechanics, Inc. 2011 9, 70569 Stuttgart, Germany e-mail: gaul@iam.uni-stuttgart.de
– Daempfungsverhalten von Werkstoffen und Bauteilen Kolloquium, TU Berlin, 1975 VDI-GKE H. Fuhrke, K. Federn, R. Gasch Results of five guidelines worked out by the named committee VDI-Richtlinie 3830, Blatt 1-5 have been presented at the conference Schwingungsdaempfung (Vibration Damping) October 16 and 17, 2007, Wiesloch near Heidelberg providing information about – modelling – numerical methods (Finite Elements, Boundary Elements, Modal Analysis) – experimental techniques for determining material damping properties from measured components or system characteristics alongwith – passive and adaptive practical applications. The guideline VDI 3830 “Damping of Materials and Members” The guideline VDI 3830 consists of the following parts: Part 1 Classification and survey Part 2 Damping of solids Preliminary note 1 Physical phenomena 2 Linear models 3 Nonlinear models Part 3 Damping of assemblies Preliminary note 1 From the material to the homogeneous member 2 Laminated members 3 Damping in joints 4 Damping due to fluids 5 Damping by squeezing 6 Assemblies Part 4 Models for damped structures Preliminary note 1 Basic model 2 Structures with finite number of degrees of freedom 3 Calculation of viscoelastic components using the boundary element method 18
Part 5 Experimental techniques for the determination of damping characteristics Preliminary note 1 Remarks on experimental techniques 2 Experimental techniques and possible instrumentation 3 Special experimental techniques for determining damping characteristics under aggravated conditions 4 Experimental Modal Analysis (EMA) 5 Experimental techniques for the damping measurement of subsoil Introduction All dynamic processes in mechanical systems are more or less damped. Consequently, damping is highly relevant in those fields of technology and applied physics which deal with dynamics and vibrations. These include • machine-, building-, and structural dynamics, • system dynamics, • control engineering, and • technical acoustics, because damping in these cases often has a considerable effect on the time history, intensity, or even the existence of vibrations. Important applications are: • transient vibrations (transient effects associated with the onset or decay of vibrations, shock-induced vibrations, reverberation effects) • resonance vibrations (unavoidable with random excitation) • wave propagation • dynamic-stability problems Accordingly, a multitude of scientific publications dealing with damping, or taking it into account at least, are found in technical literature. Due to different theory approaches, objects, and task definitions in the applications listed above, the designations, the characterisation of damping, the experimental techniques, and the analytical and numerical methods are not harmonised. The dynamic behaviour of damped structures can, in special cases, be calculated using generally valid material laws for inelastic materials based on continuum mechanics taking into account boundary effects (e.g. joints). In general, this approach is too elaborate or expensive, or not at all practicable. In most cases, therefore, phenomenological equivalent systems or mathematical models tailored to the task definition are used which are only valid assuming a special state of stresses and/or a special time history. Harmonic (sinusoidal) time histories are a preferred special case where complex quantities describe the elastic and damping properties. These depend on a number of parameters: material data, rate of deformation, frequency, temperature, number of load cycles, etc. In the case of nonlinear behaviour there is also a dependence on the amplitude. 19
For certain problems, it is sufficient to state, for one deformation cycle, the energy dissipated in a unit volume or within the system, or the energy released into the environment at the system boundaries, often related to a conveniently chosen elastic energy in a unit volume or in the system as a whole. In structural dynamics, the use of modal damping ratios has proven useful, which do no longer contain detailed information about the damping. This guideline is not a textbook; it cannot be a compilation of generally mandatory rules. It is intended • to contribute to a better understanding of the physical causes of damping, • to facilitate interdisciplinary cooperation by defining harmonised terms and pointing out the relations between different approaches to the modelling of damping, and • to allow an overview of the state of knowledge and experience gathered in various fields of application and research, in order to promote the application of existing knowledge. This guideline is structured in accordance with its objective. It starts off with the notion of damping and the causes of damping before dealing with different modelling approaches for the linear and nonlinear behaviour of solids, and establishing cross-references between these approaches. Linear viscoelastic materials being the best investigated. Their behaviour is discussed in great detail. They are followed by the damping of assemblies, relevant to the user, by its mathematical characterisation and its relation to material damping. Models for damped structures are discussed next, and the application of the boundary element method (BEM) is explained. Finally, as statements on damping rely on experiments, Part 5 describes established experimental techniques, possible instrumentation for the determination of damping characteristics, and analytical methods. The notion of damping Damping in mechanical systems is understood to be the irreversible transition of mechanical energy into other forms of energy as found in time-dependent processes. Damping is mostly associated with the change of mechanical energy into thermal energy. Damping can also be caused by releasing energy into a surrounding medium. Electromagnetic and piezoelectric energy conversion can also give rise to damping if the energy converted is not returned to the mechanical system. 20
Classification of damping phenomena The physical causes of damping are multifarious. In addition to friction, wave propagation or flow effects, other possible causes are phase transitions in materials or energy conversion by piezoelectric, magnetostrictive, or electromechanical processes. Forces associated with damping are non-conservative. They can be internal or external forces. If both action and reaction forces in a free body diagram the damping force, are effective within the system boundaries, the effect is said to be an internal damping effect. Where the reaction force is effective outside the system boundaries, the effect is an external damping effect. Examples of internal damping are: • material damping due to nonelastic material behaviour • friction between components, e.g. in slide ways, gears, etc. • conversion of mechanical vibration energy into electrical energy by means of the piezoelectric effect and dissipation due to dielectric losses Examples of external damping are: • friction against the surrounding medium • air-borne-sound radiation into the environment • structure-borne-sound radiation into the ground Phenomenologically, the damping in a mechanical system can be composed of the following contributions: • Material damping The energy dissipation within a material, due to deformation and/or displacement, is called material damping. Its physical causes are, in essence: – in solids • heat flows induced by deformation (thermomechanical coupling) • slip effects • microplastic deformations • diffusion processes – in fluids • viscous flow losses • Contact-surface damping Relative motion, friction Contact-surface damping is caused by relative motions in the contact surfaces of joined components such as screwed, riveted, and clamped joints. The physical causes are: – friction due to relative motions in the contact surface – pumping losses in the enclosed medium due to relative motion in a direction normal to the contact surface (e.g. gas pumping) The term “structural damping” includes: 21
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