River Rapids Conference Proceedings of the Society for Experimental Mechanics Series Special Topics in Structural Dynamics, Volume 6 Randall Allemang Proceedings of the 33rd IMAC, A Conference and Exposition on Structural Dynamics, 2015 River Publishers
Conference Proceedings of the Society for Experimental Mechanics Series Series Editor TomProulx Society for Experimental Mechanics, Inc. Bethel, CT, USA
River Publishers Randall Allemang Editor Special Topics in Structural Dynamics, Volume 6 Proceedings of the 33rd IMAC, A Conference and Exposition on Structural Dynamics, 2015
Published, sold and distributed by: River Publishers Broagervej 10 9260 Gistrup Denmark www.riverpublishers.com ISBN 978-87-7004-910-8 (eBook) Conference Proceedings of the Society for Experimental Mechanics An imprint of River Publishers © The Society for Experimental Mechanics, Inc. 2015 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 Special Topics in Structural Dynamics represents one of ten volumes of technical papers presented at the 33rd IMAC, A Conference and Exposition on Structural Dynamics, 2015, organized by the Society for Experimental Mechanics, and held in Orlando, Florida, February 2–5, 2015. The full proceedings also include volumes on Nonlinear Dynamics; Dynamics of Civil Structures; Model Validation and Uncertainty Quantification; Dynamics of Coupled Structures; Sensors and Instrumentation; Structural Health Monitoring and Damage Detection; Experimental Techniques, Rotating Machinery and Acoustics; Shock and Vibration, Aircraft/Aerospace, and Energy Harvesting; and Topics in Modal Analysis. Each collection presents early findings from experimental and computational investigations on an important area within structural dynamics. Special Topics in Structural Dynamics represents papers on enabling technologies for modal analysis measurements and applications of modal analysis in specific application areas. Topics in this volume include: Analytical Methods Biological Systems Wind Turbine Applications The organizers would like to thank the authors, presenters, session organizers, and session chairs for their participation in this track. Lowell, MA, USA C. Niezrecki v
Contents 1 Development of Reduced Order Models to Non-modeled Regions ................................................ 1 Kevin Truong and Peter Avitabile 2 Prediction of Forced Response Using Expansion of Perturbed Reduced Order Models with Inexact Representation of System Modes ....................................................................... 13 Sergio E. Obando, Peter Avitabile, and Jason Foley 3 Estimation of Rotational Frequency Response Functions .......................................................... 35 T.A.N. Silva and N.M.M. Maia 4 Estimation of Spatial Distribution of Disturbances.................................................................. 49 Yalcin Bulut, Omer F. Usluogullari, and Ahmet Temugan 5 Body Wise Time Integration of Multi Body Dynamic Systems..................................................... 55 Wolfgang Witteveen 6 Structural Dynamic Modeling: Tales of Sin and Redemption...................................................... 63 Robert N. Coppolino 7 Muscle Property Identification During Joint Motion Using the NL-LTP Method..................................................................................................... 75 Michael W. Sracic 8 On the Detectability of Femoral Neck Fractures with Vibration Measurements................................. 85 Wolfgang Witteveen, Carina Wagner, Patrick Jachs, Stefan Froschauer, and Harald Schöffl 9 Static Calibration of Microelectromechanical Systems (MEMS) Accelerometers for In-Situ Wind Turbine Blade Condition Monitoring .......................................................................... 91 O.O. Esu, J.A. Flint, and S.J. Watson 10 Predicting Full-Field Strain on a Wind Turbine for Arbitrary Excitation Using Displacements of Optical Targets Measured with Photogrammetry ................................................................ 99 Javad Baqersad, Peyman Poozesh, Christopher Niezrecki, and Peter Avitabile 11 Predicting the Vibration Response in Subcomponent Testing of Wind Turbine Blades ......................... 115 Mohamad Eydani Asl, Christopher Niezrecki, James Sherwood, and Peter Avitabile 12 Linear Modal Analysis of a Horizontal-Axis Wind Turbine Blade ................................................ 125 Gizem Acar and Brian F. Feeny 13 Reduced-Order Modeling of Turbine Bladed Discs by 1D Elements .............................................. 133 Luigi Carassale, Mirko Maurici, and Laura Traversone 14 Damping Estimation for Turbine Blades Under Non-stationary Rotation Speed................................ 145 Luigi Carassale, Michela Marrè-Brunenghi, and Stefano Patrone 15 Finite Element Modeling of a 40 m Space Frame Wind Turbine Tower........................................... 153 S.A. Smith, W.D. Zhu, and Y.F. Xu vii
viii Contents 16 Experimental Validation of Modal Parameters in Rotating Machinery........................................... 171 Bram Vervisch, Kurt Stockman, and Mia Loccufier 17 Estimation of Modal Damping for Structures with Localized Dissipation........................................ 179 M. Krifa, N. Bouhaddi, and S. Cogan 18 Design of UAV for Surveillance Purposes ............................................................................. 193 F. Cheli, F. Ripamonti, and D. Vendramelli 19 An Innovative Solution for Carving Ski Based on Retractile Blades .............................................. 201 F. Cheli, L. Colombo, and F. Ripamonti
Chapter1 Development of Reduced Order Models to Non-modeled Regions Kevin Truong and Peter Avitabile Abstract Model reduction and model expansion techniques have been used in many structural dynamic modeling applications to map detailed FEM at the full set of DOF (NDOF) to an abbreviated set of active DOF (ADOF) while preserving the dynamic characteristics of interest. In order to perform model reduction and model expansion, the ADOF model must be a subset of the NDOF model. For complex structures with hollow spaces (i.e. wind turbine blades, airframes, hollow fuselage sections, etc.), model reduction to a “beam-like” neutral axis is not possible. An approach for model reduction to a “line element style model with a neutral axis” is developed in this work. Several analytical models using hollow cantilever beams as academic structures are presented to illustrate the technique. This technique can be extremely useful for development of simplistic beam models from full 3D FEM that possess hollow regions where a neutral axis would exist. These simplistic beam models can be desirable for analyses such as flutter where computation is intensive for structures such as fuselage, aircraft wings, wind turbine blades, etc. Keywords FEM • Reduced order modeling • Hollow beam • Correlation • Phantom nodes Nomenclature DOF Degrees of freedom NDOF Full space DOF ADOF Active DOF Xn Physical space coordinate system Xa Reduced physical space coordinate T Transformation matrix Mn Mass matrix Ma Reduced mass matrix Kn Stiffness matrix Ka Reduced stiffness matrix œ Eigenvalue matrix TU SEREP transformation matrix [Ua] g Generalized inverse Un Full space modal matrix (shape matrix) Ua Reduced space modal matrix U Modal matrix 1 ui ith vector from U V Modal matrix 2 vj jth vector from V K. Truong ( ) • P. Avitabile Structural Dynamics and Acoustic Systems Laboratory, University of Massachusetts Lowell, One University Avenue, Lowell, MA 01854, USA e-mail: kevin_truong@student.uml.edu © The Society for Experimental Mechanics, Inc. 2015 R. Allemang (ed.), Special Topics in Structural Dynamics, Volume 6, Conference Proceedings of the Society for Experimental Mechanics Series, DOI 10.1007/978-3-319-15048-2_1 1
2 K. Truong and P. Avitabile 1.1 Introduction Finite element models are commonly developed from a CAD model which describes the structure in its actual geometric configuration. Many packages available enable the finite element mesh to be made from this geometry definition. However, there are many cases where the CAD generated models developed are very detailed with plate or solid elements to model local details such as holes, slots, rivets, grooves, etc.; thus, the CAD generated finite element models tend to become very large for no real reason other than it is easy to generate from the CAD representation. While there are many computation resources available at this point in time, a more effective modeling methodology is proposed for efficient analysis for use in different applications. There are many instances where a simpler beam/line element model would enable much more efficient analyses to be performed to understand the overall system dynamics and perform perturbation studies to investigate a variety of effects. In fact, there are many applications where the analyst actually tries to develop a much reduced order line model from the complicated 3D model to allow for efficient dynamic response studies. One such case is shown in Fig. 1.1 for a generator where a complicated model is transformed to a simple beam type model with simple component connection elements. Another very important problem is related to the structural dynamic problem for flutter analysis where simplistic beam type models are used for the aerodynamic analysis to address the flutter problem. This is done for systems such as airplane wing structures and wind turbine blades to name just a few of the many applications. In most cases, the models of these structures tend to be a much more detailed rib-stiffened plate structure for airplane wing models and rib-stiffened, multi-ply composite plate structure for wind turbine blades. Figure 1.2 shows some typical cross sections for these types of structures to illustrate the inherent problem of defining the simpler configuration. In this case, the complicated wind turbine blade made from a complicated composite layup must be converted to a simple beam cross section; the ability to accurately develop such a reduced model from a complicated cross section is, of course, not a trivial task. In both cases, the models are very detailed and complicated and are not typically used for computationally intense analyses (such as flutter) because of the large size of the model and computational resources required to do so. These models are always transformed into a much simpler beam-type model, especially in the case of flutter analysis. However, the development of this simplistic model represented by line elements along a beam like neutral axis is not easily done in current finite element software packages. This work proposes an approach to address this problem of development of a very accurate reduced order model to an equivalent neutral axis where no existing geometry exists. The reduction will utilize the SEREP model reduction scheme to exactly preserve the dynamic characteristics of interest. The technique allows for the development of a very accurate reduced order model to convert very complicated 3D shell and solid type models to equivalent reduced order line elements for further dynamic response studies including computationally intense flutter type analyses. 1.2 Theoretical Background The development of this modeling technique is based on concepts related to model reduction and model expansion, which are summarized herein. 1.2.1 Model Reduction/Expansion Techniques Model reduction is typically performed to reduce the size of a large analytical model to develop a more efficient model for further analytical studies. These techniques have been presented in earlier work cited in the references; only summarizing equations are presented below. Several model reduction methods have commonly been employed for analysis but the SEREP (System Equivalent Reduction-Expansion Process) technique is used for this work. For any reduction/expansion process, the relationship between the full set of degrees of freedom and a reduced set of degrees of freedom can be written as fXngDŒT fXag (1.1)
1 Development of Reduced Order Models to Non-modeled Regions 3 Flywheel Housing Drive End Adapter Block Rear Face Pan Adapter Non Drive End Adapter Frame DETAILED MODEL SIMPLE BEAM REPRESENTATION 11 Z Y X Control Element & Short Circuit Torque Impulse KBrgF X,Y&Z KBrgR X&Y KMnt4 X,Y&Z, DampMnt4 X,Y&Z 12 16 15 27 24 26 23 25 14 21 22 20 13 34 36 33 35 38 30 32 31 4 3 5 2 1 TEng Z TEng Z 37 KMnt3 X,Y&Z, DampMnt3 X,Y&Z KMnt2 X,Y&Z, DampMnt2 X,Y&Z KMnt1 X,Y&Z, DampMnt1 X,Y&Z UY UY UY UZ UX UZ Connected Assembly Coupled DOF’s: U1 X = U11 X, U1 Y = U11 Y & U1 Z = U11 Z R1 X = R11 X, R1 Y = R11 Y Fig. 1.1 Detailed model of genset reduced to simplistic beam element model
4 K. Truong and P. Avitabile 2 Leading Edge Shear Web Trailing Edge Shear Web Spar Cap Skin DETAILED MODEL SIMPLE BEAM REPRESENTATION Fig. 1.2 Generation of a simplistic beam model from a detailed composite ply configuration The transformation matrix [T] is used to project the full mass and stiffness matrices to a smaller size. The reduced matrices can be formulated as ŒMa DŒT T ŒMn ŒT (1.2) ŒKa DŒT T ŒKn ŒT (1.3) For model reduction, it is important that the eigenvalues and eigenvectors of the original system are preserved as accurately as possible in the reduction process. If this is not maintained then the matrices are of questionable value. The eigensolution is then given by ŒŒKa œŒMa fXagDf0g (1.4) Because reduction schemes such as Guyan Condensation [1] and Improved Reduced System Technique [2] are based primarily on the stiffness of the system, the eigenvalues and eigenvectors will not be exactly reproduced in the reduced model. However, the System Equivalent Reduction Expansion Process (SEREP) [3] exactly preserves the eigenvalues and eigenvectors in the reduced model. 1.2.2 System Equivalent Reduction Expansion Process (SEREP) For the specific work in this paper, the SEREP has been used to make the reduced order models. SEREP produces reduced matrices for mass and stiffness that yield the exact frequencies and mode shapes as those obtained from the eigensolution of the full size matrix. The SEREP transformation is formed as ŒTU DŒUn ŒUa g (1.5) with ŒUa g Dh UT a Ua 1UT a i (1.6)
1 Development of Reduced Order Models to Non-modeled Regions 5 1.2.3 Modal Assurance Criterion (MAC) Modal Assurance Criterion is a correlation tool commonly used to compare mode shapes. MAC compares two vectors (ui andvj) and calculates a value from 0 to 1 that quantifies the degree of similarity between the vectors. The equation is MACij D hfuig T ˚vj i 2 hfuig T fuigih ˚vj T ˚vj i (1.7) A MAC value of 1 signifies perfect correlation and 0 signifies no correlation. 1.2.4 Pseudo Orthogonality Check (POC) Pseudo Orthogonality Check is a mass weighted orthogonality tool used to compare mode shapes and is given as POCDŒV T ŒM ŒU (1.8) The POC is mass weighted. If the shapes are scaled to unit modal mass, POC ranges from 0 to 1, similar to MAC. 1.3 Model Description Five finite element models of a cantilever beam were used to develop and demonstrate the technique. Models with nonphysical properties shall be referred to as the “phantom” model. The different models generated are: • Model 1: Line model using 3D beam elements (BEAM3D) • Model 2: Shell model using plate elements (QUAD4) • Model 3: Shell model with “phantom” beam model constrained at the neutral axis (QUAD4 and BEAM3D) • Model 4: Shell model with a “phantom” beam model located away from the neutral axis with same constraints as Model 3 (QUAD4 and BEAM3D) • Model 5: Shell model with “phantom” beam model constrained at the neutral axis using a subset of nodes from Model 3 (QUAD4 and BEAM3D) The properties used in the FEMs are shown in Table 1.1. These models were generated using the software, FEMtools [5]. Table 1.1 Cantilever beam dimensions and FEM characteristics Line Shell Nodes – – – 81 972 DOF 486 5832 Elastic modulus [psi] Density [lb/in**3] Poisson's ratio b [in.] h [in.] t (uniform) [in.] Length [in.] 4 2 0.5 80 Model Unit Property 10E6 0.098 0 b h Y Z Beam cross section t
6 K. Truong and P. Avitabile 1.3.1 Model 1 (Line) Model 1 uses simple beam elements to model the cantilever beam. Generally, the computation time for the solution is much quicker for models using line style elements than shell elements. Line models are common for beam structures with simple cross sections. However, for beam-like structures with complex geometries i.e. varying cross sections, rib stiffeners, etc., a FEM using beam elements would not be practical since the cross sectional properties for every element need to be specified. 1.3.2 Model 2 (Shell) Model 2 uses linear plate elements to model the cantilever beam. Typically, FEMs of the structure are created from existing CAD models readily available. From these CAD models, a solid or surface mesh can quickly be generated on the structure and an eigensolution can be performed to obtain the structures frequencies and mode shapes. Because the solid/surface models contain many nodes, the computation time tends to be lengthy. Thus, an accurate reduced order model that captures the dynamics of interest, while making computation more efficient is desirable. For beam-like structures with hollow cross sections, a reduced order model at the neutral axis is not possible. This provides the motivation for Model 3. 1.3.3 Model 3 (Hybrid) Model 3 is an extension of Model 2 with “phantom” beam elements constrained to the appropriate shell nodes. This modeling technique provides a set of DOFs to make a reduced order model at the neutral axis while obtaining the original frequencies and mode shapes of the shell model. A set of ADOF at the neutral axis were selected for the reduction process and expanded to the full space NDOF. Table 1.2 compares the reference mode shapes to the expanded mode shapes to illustrate the accuracy of the reduction/expansion process using SEREP. The “phantom” beam is constrained to the nodes on the surrounding four faces of the shell model. Note: To avoid numerical difficulties, small values for E and rho were used for the beam element properties. 1.3.4 Model 4 (Hybrid-Offset) Model 4 is similar to Model 3 with the only difference of the location of the “phantom” beam; this model had the “phantom” beam offset from the actual neutral axis of the beam. For beam-like structures with complex geometries where the neutral axis does not coincide with the center, this model was develop to study the differences of the technique when the “phantom” beam is not modeled at the neutral axis. 1.3.5 Model 5 (Hybrid-1 Node) The constraints in Model 5 differ that of Model 3. The “phantom” beam is constrained to only one face of the shell model. The frequencies obtained from the models are shown in Table 1.3. Only bending modes were considered for reduction. Frequencies highlighted in blue and purple correspond to bending about the weaker and stronger axes, respectively. The frequencies of Model 1 differ from Model 2–5 due to formulations of the elements used in the models. Model 1 uses beam elements while Model 2 uses plate elements; and Models 3–5 uses a combination of plate and beam elements. Note that only the plate elements in Model 3, 4 and 5 have physical properties. The frequencies from Model 2 are used to validate the modeling technique used in Model 3, 4 and 5.
1 Development of Reduced Order Models to Non-modeled Regions 7 Table 1.2 Comparison of reference mode shapes to expanded mode shapes from model 3 DOF Mode 1 Mode 2 Mode 3 . . . Mode 10 Mode 11 Mode 12 1 0.00 –0.01 –0.02 . . . –0.21 –0.26 0.34 2 0.00 0.00 –0.01 . . . –0.27 0.01 –0.01 3 0.00 0.00 0.00 . . . –0.03 0.21 –0.33 4 0.00 0.00 0.00 . . . 0.05 –0.03 0.05 5 0.00 0.00 0.00 . . . 0.01 –0.31 0.47 6 0.00 0.00 –0.03 . . . –0.41 0.00 0.00 7 0.00 –0.01 0.00 . . . 0.00 –0.21 0.26 8 0.00 0.00 –0.01 . . . –0.28 0.00 0.00 9 0.00 0.00 0.00 . . . 0.00 0.18 –0.28 10 0.00 0.00 0.00 . . . 0.01 0.00 0.00 11 0.00 0.00 0.00 . . . 0.00 –0.31 0.47 12 0.00 0.00 –0.02 . . . –0.27 0.00 0.00 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5821 0.00 0.00 0.00 . . . 0.00 0.00 0.00 5822 5.63 0.00 5.34 . . . 4.24 0.00 0.00 5823 0.00 5.62 0.00 . . . 0.00 4.34 4.03 5824 0.00 0.00 0.00 . . . 0.00 0.00 0.00 5825 0.00 –0.10 0.00 . . . 0.00 –0.87 –1.02 5826 0.10 0.00 0.34 . . . 1.04 0.00 0.00 5827 0.00 0.00 0.00 . . . 0.00 0.00 0.00 5828 5.72 0.00 5.67 . . . 5.31 0.00 0.00 5829 0.00 5.72 0.00 . . . 0.00 5.23 5.08 5830 0.00 0.00 0.00 . . . 0.00 0.00 0.00 5831 0.00 –0.10 0.00 . . . 0.00 –0.88 –1.03 5832 0.10 0.00 0.34 . . . 1.06 0.00 0.00 Tu*Ua - Expanded fullspace modeshapes DOF Mode 1 Mode 2 Mode 3 . . . Mode 10 Mode 11 Mode 12 1 0.00 –0.01 0.02 . . . 0.21 0.26 –0.34 2 0.00 0.00 0.01 . . . 0.27 –0.01 0.01 3 0.00 0.00 0.00 . . . 0.03 –0.21 0.33 4 0.00 0.00 0.00 . . . –0.05 0.03 –0.05 5 0.00 0.00 0.00 . . . –0.01 0.31 –0.47 6 0.00 0.00 0.03 . . . 0.41 0.00 0.00 7 0.00 –0.01 0.00 . . . 0.00 0.21 –0.26 8 0.00 0.00 0.01 . . . 0.28 0.00 0.00 9 0.00 0.00 0.00 . . . 0.00 –0.18 0.28 10 0.00 0.00 0.00 . . . –0.01 0.00 0.00 11 0.00 0.00 0.00 . . . 0.00 0.31 –0.47 12 0.00 0.00 0.02 . . . 0.27 0.00 0.00 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5821 0.00 0.00 0.00 . . . 0.00 0.00 0.00 5822 5.63 0.00 –5.34 . . . –4.24 0.00 0.00 5823 0.00 5.62 0.00 . . . 0.00 –4.34 –4.03 5824 0.00 0.00 0.00 . . . 0.00 0.00 0.00 5825 0.00 –0.10 0.00 . . . 0.00 0.87 1.02 5826 0.10 0.00 –0.34 . . . –1.04 0.00 0.00 5827 0.00 0.00 0.00 . . . 0.00 0.00 0.00 5828 5.72 0.00 –5.67 . . . –5.31 0.00 0.00 5829 0.00 5.72 0.00 . . . 0.00 –5.23 –5.08 5830 0.00 0.00 0.00 . . . 0.00 0.00 0.00 5831 0.00 –0.10 0.00 . . . 0.00 0.88 1.03 5832 0.10 0.00 –0.34 . . . –1.06 0.00 0.00 Unm - Reference modeshapes Modal matrix difference 1.4 Cases Studied Three cases were studied and presented in this section. Model reduction using SEREP is performed, and correlation tools such as the POC and MAC are applied to each of the cases. All reduction analyses were performed using Matlab [4] and correlation analyses were performed using FEMtools [5]. Note that only bending modes were considered for presenting the technique. The axial and torsion modes could have been preserved in the reduction process but provided no advantage in demonstrating the technique and therefore were not included or considered. The specified cases studied are: • Case X: Correlate Model 1 to Model 3 • Case Y: Correlate Model 3 to Model 4 • Case Z: Correlate Model 3 to Model 5 In each of the cases, the models are reduced down to a set of 6 nodes with two transverse DOF at each node corresponding to the bending directions. The DOF selected and the modes preserved for SEREP are chosen consistently in each of the cases. 1.4.1 Case X In Case X, the reduced line model (Model 1) is compared to the reduced hybrid model (Model 3) for similarities. First, the reduced mass matrices were studied and compared. Plots of the reduced mass matrices and the differences between the
8 K. Truong and P. Avitabile Table 1.3 FEM natural frequencies for model 1–5 [Hz] [Hz] [Hz] [Hz] [Hz] 1 15.65 15.41 15.41 15.41 15.41 2 26.16 25.83 25.83 25.84 25.83 3 98.06 95.03 95.03 95.04 95.03 4 163.93 158.72 158.72 158.73 158.72 5 274.51 259.61 259.61 259.62 259.61 6 438.79 335.71 335.71 335.71 335.71 7 458.93 431.32 431.32 431.32 431.32 8 537.85 491.42 491.42 491.43 491.42 9 620.54 620.54 620.56 620.56 620.56 10 888.96 778.34 778.34 778.35 778.34 11 899.17 811.90 811.91 811.92 811.91 12 1,316.19 992.20 992.20 992.21 992.21 13 1,327.73 1,106.18 1,106.18 1,106.20 1,106.20 14 1,486.15 1,280.54 1,280.55 1,280.57 1,280.55 15 1,854.13 1,461.16 1,461.16 1,461.19 1,461.19 16 1,861.38 1,595.51 1,595.52 1,595.53 1,595.53 17 2,193.09 1,816.79 1,816.80 1,816.84 1,816.80 18 2,219.69 1,830.62 1,830.63 1,830.66 1,830.69 19 2,468.11 1,861.38 1,861.45 1,861.45 1,861.45 20 3,069.14 2,089.71 2,089.71 2,089.74 2,089.75 Hybrid beam (1 Node) Model 1 Model 2 Model 3 Model 4 Model 5 Mode Line beam Shell beam Hybrid beam Hybrid beam (Offset) models are shown below in Fig. 1.3. There are some minor differences in the reduced mass matrices; this is to be expected because of the different formulations of the elements used in Model 1 and Model 3. As an initial tool, the MAC was used to analyze similarities of the mode shapes in the models. POC was then used for vector correlation on a mass weighted basis. Two POC calculations were performed; one calculation using Ma from Model 1 and one calculation using Ma from Model 3. The MAC and POC values and plots for all cases are shown below in Table 1.4 and Fig. 1.4, respectively. Both high MAC and high POC values were obtained from the calculations. Because the DOF preserved for reduction were located at modally active nodes, the MAC calculation was dominated by the high mode shape values at these locations which accounts for the high MAC values. The POC shows that the mode shapes from Model 1 and Model 3 differ by a small scale factor, which results from the use of different element properties. 1.4.2 Case Y In Case Y, the effects of modeling the “phantom” beam offset from the neutral axis was studied. The Ma from Model 3 and Model 4 have negligible differences as shown in Fig. 1.3. POC values of 1 indicate perfect model correlation shown in Table 1.4. Good model correlation is also indicated with the high MAC values. Identical results can be obtained independent of the “phantom” beam location. The nodes to which the “phantom” beam is constrained to, dictate the model behavior. The “phantom” beam does not have to be modeled at the neutral axis in order to use this technique. This is applicable for real-world structures with complicated geometries, where the neutral axis cannot be easily located.
1 Development of Reduced Order Models to Non-modeled Regions 9 1.4.3 Case Z In Case Z, Model 3 is correlated to Model 5. Each model had a different set of constraints; Model 5 constraints are a subset of the constraints applied in Model 3. The Ma from both models have negligible differences shown in Fig. 1.3. High POC and MAC values from Table 1.4 show good correlation. Since only a handful of the lower order modes were preserved for reduction, very similar results were obtained from the two models. The effects of the higher order breathing type modes when using the technique requires further study and is beyond the scope of this initial investigation. Fig. 1.3 Reduced mass matrix plots and their respective differences for Case X, Y, and Z
10 K. Truong and P. Avitabile Table 1.4 POC and MAC values for Case X, Y, and Z POC MAC CaseX CaseY Case Z Case X Case Y Case Z Mode (a)Ma from Model 1 (b)Ma from Model 3 (a)Ma from Model 3 (b)Ma from Model 4 (a)Ma from Model 3 (b)Ma from Model 5 (c) (c) (c) 1 0.9997 1.0003 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 2 0.9992 1.0008 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 1.0000 3 0.9979 1.0020 1.0000 1.0000 1.0003 0.9997 0.9999 0.9999 1.0000 4 0.9943 1.0057 1.0000 1.0000 0.9999 1.0001 0.9999 0.9996 1.0000 5 0.9959 1.0039 1.0000 1.0000 1.0019 0.9981 0.9995 0.9998 1.0000 7 0.9881 1.0120 1.0000 1.0000 0.9990 1.0010 0.9994 0.9992 1.0000 8 0.9952 1.0037 1.0000 1.0000 1.0066 0.9934 0.9980 0.9996 1.0000 10 1.0037 0.9967 1.0000 1.0000 1.0163 0.9839 0.9951 0.9994 0.9998 11 0.9830 1.0169 1.0000 1.0000 0.9966 1.0035 0.9979 0.9986 1.0000 13 0.9917 1.0097 1.0000 1.0000 1.0325 0.9685 0.9990 0.9992 0.9994 14 0.9891 1.0123 1.0000 1.0000 0.9915 1.0086 0.9951 0.9980 1.0000 18 0.9667 1.0361 1.0000 1.0000 0.9830 1.0173 0.9983 0.9973 1.0000 1.5 Conclusion This work demonstrates that an exact reduced order model can be developed and reduced to a region where there is no specific model defined such as the neutral axis of a hollow rectangular section. The technique relies on the development of a “phantom” set of elements that can be used to define the neutral axis. This work also shows that the neutral axis of the phantom beams need not be exactly defined on the neutral axis of the hollow structure i.e. the “phantom elements” need not be geometrically located at the neutral axis. Several cases were presented to demonstrate and validate the technique. The approach is useful for reduction of any large model where the specific neutral axis is unknown. The approach is also useful for reduction of large complicated models that need to be drastically reduced to very low order model for complicated analyses such as flutter analysis. This is useful for large airframe structures, airframe wing like structures and wind turbine blade structures to name a few examples. Acknowledgements Some of the work presented herein was partially funded by Air Force Research Laboratory Award No. FA8651-10-10009 “Development of Dynamic Response Modeling Techniques for Linear Modal Components”. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the particular funding agency. The authors are grateful for the support obtained.
1 Development of Reduced Order Models to Non-modeled Regions 11 Fig. 1.4 POC and MAC plots for Case X, Y, and Z References 1. Guyan RJ (1965) Reduction of stiffness and mass matrices. AIAA J 3(2):380 2. O’Callahan JC (1989) A procedure for an Improved Reduced System (IRS) model. In: Proceedings of the seventh international modal analysis conference, Las Vegas, publisher is Society for Experimental Mechanics, February 1989 3. O’Callahan JC, Avitabile P, Riemer R (1989) System equivalent reduction expansion process. In: Proceedings of the seventh international modal analysis conference, Las Vegas, publisher is Society for Experimental Mechanics, February 1989 4. MATLAB Matrix analysis software. The MathWorks, Inc., Natick 5. FEMtools FEMtools 3.0. Dynamic Design Solutions, Leuven
Chapter2 Prediction of Forced Response Using Expansion of Perturbed Reduced Order Models with Inexact Representation of System Modes Sergio E. Obando, Peter Avitabile, and Jason Foley Abstract Variability in measured data is a common problem in the engineering practice. Changes in the mass and stiffness of the same structural component can occur due to minor variability in the tolerances used during the production/manufacturing process. Differences can exist between the real physical structure and its mathematical model representation (FEM) as well as the predicted response and the actual dynamic behavior of the system. For models in which limited data exists or is collected, the quality of the equivalent reduced order model is dependent on the retained modal parameters as well as the level of correlation of the mode shapes. Prediction of system level forced response from the expansion of these reduced order models can be affected by the use of inexact representations of the system modes such as those from Guyan reduced models. Furthermore, the reduction methodology used, the degrees of freedom selected, as well as the number of retained modes can play an important role in the accuracy of the predicted dynamics of the system. In this work, a truth model (real answer) is created from the perfect analytical representation of a cantilevered beam. A perturbed variation of the analytical representation of the cantilever beam model is also created to correspond to the simulated imperfections of a FEM of the system. The analytical models will be created to investigate the prediction of the full field dynamic response obtained from the expansion of reduced model information (or data at limited number of DOF) and using the inexact mode shapes of the perturbed model (FEM). The perturbed system representation will have the same geometry and properties as the original unmodified beam (perfect analytical model) but imperfections will be introduced by the addition of mass. The models will be created first at full space as a reference and then reduction techniques will be used to determine the necessary information in order to accurately predict the response at all DOF. Aspects involved in model reduction/expansion, DOF selection, and number of retained modes for the analytical cantilever models are investigated for common reduction techniques such as Guyan condensation and SEREP. The use of a perturbed model (not perfectly correlated to the model) for the expansion of measured real time response data will be shown to produce very accurate full field response even though the model does not perfectly correlate to the real truth model. Keywords Forced linear response • Reduced order modeling • Perturbed models Nomenclature Symbols fXng Full Set Displacement Vector fXag Reduced Set Displacement Vector fXdg Deleted Set Displacement Vector [Ma] Reduced Mass Matrix [Mn] Expanded Mass Matrix [Ka] Reduced Stiffness Matrix [Kn] Expanded Stiffness Matrix S.E. Obando ( ) • P. Avitabile Structural Dynamics and Acoustic Systems Laboratory, University of Massachusetts Lowell, One University Avenue, Lowell, MA 01854, USA e-mail: sergio.e.obando@gmail.com J. Foley Air Force Research Laboratory, Munitions Directorate Fuzes Branch, Eglin Air Force Base, 306 W. Eglin Blvd., Bldg 432, Eglin AFB, FL 32542-5430, USA © The Society for Experimental Mechanics, Inc. 2015 R. Allemang (ed.), Special Topics in Structural Dynamics, Volume 6, Conference Proceedings of the Society for Experimental Mechanics Series, DOI 10.1007/978-3-319-15048-2_2 13
14 S.E. Obando et al. [Ua] Reduced Set Shape Matrix [Un] Full Set Shape Matrix [Ua] g Generalized Inverse [T] Transformation Matrix [TU] SEREP Transformation Matrix fpg Modal Displacement Vector [M] Physical Mass Matrix [C] Physical Damping Matrix [K] Physical Stiffness Matrix fFg Physical Force Vector fRxg Physical Acceleration Vector f PXg Physical Velocity Vector fxg Physical Displacement Vector ! Rx0 Initial Acceleration Vector ! Px0 Initial Velocity Vector ! x0 Initial Displacement Vector ! F0 Initial Force Vector ’ Parameter for Newmark Integration “ Parameter for Newmark Integration t Time Step Acronyms ADOF Reduced Degrees of Freedom NDOF Full Space Degrees of Freedom DOF Degrees of Freedom ERMT Equivalent Reduced Model Technique FEM Finite Element Model MAC Modal Assurance Criterion SEREP System Equivalent Reduction Expansion Process TRAC Time Response Assurance Criterion VIKING Variability Improvement of Key Inaccurate Node Groups 2.1 Introduction Structural components of typical systems are mass produced with allowable imperfections/tolerances in dimension and physical properties. Within the constraints of the manufacturing design, this variability should not affect the overall nominal values of performance of the component or part. However, often times during qualification testing or whenever components of similar specifications are being replaced/tested, the degree of variation and its effect on the operation of the system are highly important. In particular, if experimental data is being correlated with existing finite element models (FEM) these minor perturbations can result in inaccurate model characterization, poor correlation and poor predicted system response. Thus, there is significant motivation in understanding the effect of imperfections in the system’s representation (model perturbations) in the accuracy of reduced order model characteristics, expanded system mode shapes, and predicted full system response. Expansion of limited sets of data to the full NDOF of a large FEM is often performed to correlate test data with existing analytical models. The expanded mode shapes and the predicted response are affected by the level of correlation of the FEM and the tested structure as well as the location of the preserved ADOF in the reduced model. Recent work has investigated important aspects in the expansion of system modes used in the SEREP methodology for cases where poor correlation exists between test and FEM [1–5]. Mode shapes processed with the Variability Improvement of Key Inaccurate Node Groups (VIKING) method were shown to produce much more accurate results than the shapes expanded
2 Prediction of Forced Response Using Expansion of Perturbed Reduced Order Models with Inexact Representation of System Modes 15 with contaminated/perturbed expansion matrices [5]. The most significant aspect of these studies lies in the proper selection of sufficient modal information in the preserved projection vectors resulting in the smoothing of noise and variance of the inaccurate experimental data. In this paper, the predicted full field forced response of a cantilevered beam structure (from information at a limited number of points) is analyzed in the case where the expansion is performed with inexact system modes from a perturbed model representing an imperfect FEM of the truth model (real answer). The perfect analytical representation of the truth model is formed from the full space FEM of the cantilevered beam. Different reduction methodologies (Guyan condensation [6], SEREP [7], and KM_AMI [8]) are first studied with emphasis in DOF selection and accurate prediction of time response from reduced order models of this reference beam model. Then, a new system model is formed from the same geometry and physical properties of the original (unmodified reference) model but with simulated imperfections due to the addition of a point mass at the tip of the beam. Therefore, the modified/perturbed model will have different frequencies and mode shapes as well time response compare to the perfect analytical model. The forced response at selected DOF will be extracted from the full model response of the truth/unmodified model (much like having measurements of a system at a limited set of points) and the predicted response at full space obtained using both the perturbed and the accurate projection vectors (mode shapes) from a SEREP reduction process of the models. This study will show that when the preserved mode shapes contain sufficient information to span the solution space of the system, the perturbations do not distort the predicted expanded response at full space. Moreover, the selection of degrees of freedom during the reduction process will be explored in cases where the retained DOF are at less than ideal locations. This is of particular importance in real world measurements, as often times, transducers can only be placed at limited locations which may be highly susceptible to noise. 2.2 Theory The fundamental theory in the study of forced response estimation and expansion/reduction of the type of linear systems analyzed here require a variety of theoretical topics. The summary starts with a description of linear multiple degree of freedom systems and continues with an overview of structural dynamic modification, analytical model reduction and expansion, model updating and forced time response computations. Further information can be found in the respective references. 2.2.1 Equations of Motion for Multiple Degree of Freedom System The general equation of motion for a multiple degree of freedom system written in matrix form is ŒM1 fRxgCŒC1 fPxgCŒK1 fxgDfF.t/g (2.1) Assuming proportional damping, the eigensolution is obtained from Œ ŒK1 œ Œ M1 fxgDf0 g (2.2) The eigensolution yields the eigenvalues (natural frequencies) and eigenvectors (mode shapes) of the system. The eigenvectors are arranged in column fashion to form the modal matrix [U1]. Often times, only a subset of modes is included in the modal matrix to save on computation time and due to the fact that only certain modes actually contribute to the response. Exclusion of modes results in truncation error which can be serious if key modes are excluded; truncation error will be discussed in further detail in the structural dynamic modification section. The physical system can be transformed to modal space using the modal matrix as ŒU1 T ŒM1 ŒU1 fRp1gCŒU1 T ŒC1 ŒU1 fPp1gCŒU1 T ŒK1 ŒU1 fp1gDŒU1 T fF.t/g (2.3)
16 S.E. Obando et al. Scaling to unit modal mass yields 2 66 4 : : : I1 : : : 3 77 5 fRp1gC 2 66 4 : : : 2Ÿ¨n : : : 3 77 5 fPp1gC 2 66 4 : : : 2 1 : : : 3 77 5 fp1gD Un 1 T fF.t/g (2.4) where [I1] is the diagonal identity matrix, [ 1 2] is the diagonal natural frequency matrix and [2Ÿ¨n] is the diagonal damping matrix (assuming proportional damping). More detailed information on the equation development is contained in Ref. [9]. 2.2.2 Structural Dynamic Modification Structural Dynamic Modification (SDM) is a technique that uses the original mode shapes and natural frequencies of a system to estimate the dynamic characteristics due to changes in the mass and/or stiffness of the system; only mass and stiffness changes are considered in this work. First, the change of mass and stiffness are transformed to modal space as shown M12 DŒU1 T Œ M12 ŒU1 (2.5) K12 DŒU1 T Œ K12 ŒU1 (2.6) The modal mass and modal stiffness changes are added to the original modal space equations to obtain 2 66 4 2 66 4 : : : M1 : : : 3 77 5 C M12 3 77 5 fRp1g C 2 66 4 2 66 4 : : : K1 : : : 3 77 5 C K12 3 77 5 fp1gDŒ0 (2.7) The eigensolution of the modified modal space model is computed and the resulting eigenvalues are the new frequencies of the system. The modified vectors can be obtained from the original vectors with the use of [U12] from the eigensolution of (2.7) and is given as Œ U2 DŒU1 ŒU12 (2.8) The new mode shapes are [U2]. The new mode shapes are formed from linear combinations of the original mode shapes [U1]. The [U12] matrix shows how much each of the [U1] modes contributes to forming the new modes. Figure 2.1 shows the formation of the new mode shapes as seen in Eq. 2.8. See [9] for additional information on SDM. Fig. 2.1 Structural dynamic modification, mode contribution identified using U12 [10]
2 Prediction of Forced Response Using Expansion of Perturbed Reduced Order Models with Inexact Representation of System Modes 17 2.2.3 General Reduction/Expansion Methodology and Model Updating Model reduction is a tool used to reduce the number of degrees of freedom (DOF) in order to reduce the required computation time of an analytical model, while attempting to preserve the full DOF dynamic characteristics. The relationship between the full space and reduced space model can be written as fXngD Xa Xd DŒT fXag (2.9) where subscript ‘n’ signifies the full set of DOF (NDOF), ‘a’ signifies the reduced set of DOF (ADOF) and ‘d’ is the deleted or embedded DOF (those DOF not used during the reduced computation process). The transformation matrix [T] relates the full set of NDOF to the reduced set of ADOF. The transformation matrix is used to reduce the mass and stiffness matrices as ŒMa DŒT T ŒMn ŒT and ŒKa DŒT T ŒKn ŒT (2.10) The eigensolution of these ‘a’ set mass and stiffness matrices are the modes of the reduced model. These modes can be expanded back to full space using the transformation matrix ŒUn DŒT ŒUa (2.11) If an optimal ‘a’ set is not selected when using methods such as Guyan Condensation [6] or Improved Reduced System Technique [11], the reduced model may not perfectly preserve the dynamics of the full space model. If System Equivalent Reduction Expansion Process (SEREP) [7] is used, the dynamics of the selected modes will be perfectly preserved regardless of the ‘a’ set selected as long as the matrix is formed from a linearly independent set of vectors. Furthermore, a model improvement technique, KM_AMI, has been recently developed to update the mass and stiffness matrices obtained through the Guyan methodology and obtained exact modes and frequencies at the reduced ‘a’ space [8]. 2.2.3.1 System Equivalent Reduction Expansion Process (SEREP) The SEREP modal transformation relies on the partitioning of the modal equations representing the system using selected DOFs and modes to obtain a reduced model that perfectly preserves the eigenvalues and eigenvectors of interest [7]. The SEREP technique utilizes the mode shapes from a full finite element solution to map to the limited set of active DOF. SEREP is not performed to achieve efficiency in the solution but rather is intended to perform an accurate mapping matrix for the transformation. The SEREP transformation matrix is formed using a subset of modes at full space and reduced space as ŒTU DŒUn ŒUa g (2.12) where [Ua] g is the generalized inverse and [TU] is the SEREP transformation matrix. When the SEREP transformation matrix is used for model reduction/expansion as outlined in the previous section, the reduced model perfectly preserves the full space dynamics of the modes in [Un] [7]. 2.2.3.2 KM_AMI Reduction A more recent technique has been developed that utilizes Guyan Reduction (or any reduced matrix for that matter) along with direct updating of the reduced system matrices with the full space modal vectors as targets for the updating process [11]. This reduction technique also overcomes some of the rank problems associated with SEREP and provides a reduced set of ADOF that retain all the eigenvalues and eigenvectors of the full system matrices. The Guyan reduced mass and stiffness matrices are updated using ŒMI DŒMS CŒV T Œ I MS ŒV (2.13)
18 S.E. Obando et al. and Œ KI DŒKS CŒV T 2 REF C KS ŒV ŒŒ KS Œ UREF Œ V ŒŒ KS Œ UREF ŒV T (2.14) with ŒV D MS 1 ŒUREF T ŒMS (2.15) 2.2.4 System Forced Response Analysis The computation of the time response developed in this paper is based on the Equivalent Reduced Model Technique (ERMT), a technique developed by Avitabile and Thibault [10, 12]. This technique uses an exact reduced model representation for the calculation of the system response. Newmark integration technique [13] is used to perform the direct integration of the equations of motion for the ERMT solution process due to similarity with the HHT (Hilber-Hughes-Taylor [14]) method commonly used in FEA software. From the known initial conditions for displacement and velocity, the initial acceleration vector is computed using the equation of motion and the applied forces as ! Rx0 DŒM 1 ! F 0 ŒC ! Px0 ŒK ! x0 (2.16) Choosing an appropriate t, ’, and“, the displacement vector is ! xiC1 Dh 1 ’. t/ 2 ŒM C “ ’. t/ ŒC CŒK i 1 n! FiC1 CŒM 1 ’. t/ 2 ! xi C 1 ’. t/ ! Pxi C 1 2’ 1 ! Rxi CŒC “ ’. t/ ! xi C “ ’ 1 ! Pxi C “ ’ 2 t 2 ! Rxi o (2.17) The values chosen for ’ and “ were ¼ and ½, respectively. This assumes constant acceleration and the integration process is unconditionally stable, where a reasonable solution will always be reached regardless of the time step used. However, the time step should be chosen such that the highest frequency involved in the system response can be characterized properly to avoid numerical damping in the solution. Following the displacement vector calculation, the acceleration and velocity vectors are computed for the next time step using ! PxiC1 D ! Pxi C.1 “/ t ! Rxi C“ t ! RxiC1 (2.18) ! RxiC1 D 1 ’. t/2 ! xiC1 ! xi 1 ’ t ! Pxi 1 2’ 1 ! Rxi (2.19) This process is repeated at each time step for the duration of the time response solution desired. 2.2.5 Time Response Correlation Tools In order to quantitatively compare two different time solutions, two correlation tools were employed: The Modal Assurance Criterion (MAC) and the Time Response Assurance Criterion (TRAC). 2.2.5.1 Modal Assurance Criterion (MAC) The Modal Assurance Criterion (MAC) [15] is widely used as a vector correlation tool. In this work, the MAC was used to correlate all DOF at a single instance in time. The MAC is written as
2 Prediction of Forced Response Using Expansion of Perturbed Reduced Order Models with Inexact Representation of System Modes 19 MACij D hfX1ig T ˚X2j i 2 hfX1ig T fX1igi h ˚X2j T ˚X2j i (2.20) where X1 and X2 are displacement vectors. MAC values close to 1.0 indicate strong similarity between vectors, where values close to 0.0 indicate minimal or no similarity. 2.2.5.2 Time Response Assurance Criterion (TRAC) The Time Response Assurance Criterion (TRAC) [16] quantifies the similarity between a single DOF across all instances in time. The TRAC is written as TRACji D h ˚X1j .t/ T fX2i .t/gi 2 h ˚X1j .t/ T ˚X1j .t/ i hfX2i .t/g T fX2i .t/gi (2.21) where X1 and X2 are time response vectors for a particular DOF. TRAC values close to 1.0 indicate strong similarity between vectors, where values close to 0.0 indicate minimal or no similarity. In this work, the MAC is calculated between the shapes of the full space reference solution and estimated solution obtained from the reduced order model at each time step. Similarly the TRAC is used to compare the time response from the reduced order model to the time response from the full space finite element solution at each degree of freedom. A diagram detailing the two comparison techniques is shown in Fig. 2.2. 2.3 Model Description Analytical models of a cantilevered beam were created to investigate the prediction of the dynamic response using different reduction/expansion methodologies. Planar element beam models of the cantilevered beam were generated using MAT_SAP [17], which is a FEM program developed for MATLAB [18], and forced response calculations were performed in MATLAB using Newmark integration scripts. The beam models were set to have dimensions and characteristics as described in Fig. 2.3. For all models, 1 % of critical damping was used in the time response computation. The system was subjected to a double sided force pulse at the tip of the beam and this input force was designed as to only excite the modes in a frequency band of approximately 200 Hz as shown in Fig. 2.4. With all 20 elements of the system (i.e. 40 DOF) the full n-space reference solution to the system was calculated and served as a point of comparison for all subsequent reduced order model calculations. The frequencies of the beam are shown in Fig. 2.3. The modified/perturbed (imperfect FEM) cantilevered beam was created using a point mass of approximately 2 % of the total mass of the reference (unmodified perfect model representation) beam placed at the tip of the beam. All other properties and input force remained the same. Figure 2.5 depicts the two models (reference/unmodified and modified) and compares the frequencies for the first 6 modes of the system. 2.4 Cases Studied This work will be divided in two parts. Part A is a study of reduced order modeling in the context of prediction of forced time response. The forced time response of the full space reference model with 40 DOF was first calculated. Reduction techniques (Guyan, SEREP and KM_AMI) were used to reduce the active DOF of the system and preserve a selected number of modes. The time response was computed at the limited set of DOF and expansion used to predict the response at all NDOF. Two groups of ADOF were studied, illustrating instances of poor or limited DOF location and the effect on the predicted (through expansion) response. Part B focuses on the prediction of full NDOF time response from the expansion of data at limited points of the model for cases in which the projection matrices (i.e. mode shape vectors in SEREP methodology) differ from
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