River Rapids Conference Proceedings of the Society for Experimental Mechanics Series Dynamics of Coupled Structures, Volume 1 Matt Allen Randy Mayes Daniel Rixen Proceedings of the 32nd IMAC, A Conference and Exposition on Structural Dynamics, 2014 River Publishers
Conference Proceedings of the Society for Experimental Mechanics Series Series Editor TomProulx Society for Experimental Mechanics, Inc., Bethel, CT, USA
River Publishers Matt Allen • Randy Mayes • Daniel Rixen Editors Dynamics of Coupled Structures, Volume 1 Proceedings of the 32nd IMAC, A Conference and Exposition on Structural Dynamics, 2014
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Preface Dynamics of Coupled Structures, Volume 1 represents one of the eight volumes of technical papers presented at the 32nd IMAC, A Conference and Exposition on Structural Dynamics, 2014, organized by the Society for Experimental Mechanics, and held in Orlando, Florida, February 3–6, 2014. The full proceedings also include volumes on Nonlinear Dynamics; Model Validation and Uncertainty Quantification; Dynamics of Civil Structures; Structural Health Monitoring; Special Topics in Structural Dynamics; Topics in Modal Analysis I; and Topics in Modal Analysis II. Each collection presents early findings from experimental and computational investigations on an important area within structural dynamics. Coupled structures, or substructuring, is one of these areas. Substructuring is a general paradigm in engineering dynamics where a complicated system is analyzed by considering the dynamic interactions between subcomponents. In numerical simulations, substructuring allows one to reduce the complexity of parts of the system in order to construct a computationally efficient model of the assembled system. A subcomponent model can also be derived experimentally, allowing one to predict the dynamic behavior of an assembly by combining experimentally and/or analytically derived models. This can be advantageous for subcomponents that are expensive or difficult to model analytically. Substructuring can also be used to couple numerical simulation with real-time testing of components. Such approaches are known as hardware-in-the-loop or hybrid testing. Whether experimental or numerical, all substructuring approaches have a common basis, namely the equilibrium of the substructures under the action of the applied and interface forces and the compatibility of displacements at the interfaces of the subcomponents. Experimental substructuring requires special care in the way the measurements are obtained and processed in order to assure that measurement inaccuracies and noise do not invalidate the results. In numerical approaches, the fundamental quest is the efficient computation of reduced order models describing the substructure’s dynamic motion. For hardware-in-the-loop applications difficulties include the fast computation of the numerical components and the proper sensing and actuation of the hardware component. Recent advances in experimental techniques, sensor/actuator technologies, novel numerical methods, and parallel computing have rekindled interest in substructuring in recent years leading to new insights and improved experimental and analytical techniques. The organizers would like to thank the authors, presenters, session organizers, and session chairs for their participation in this track. Madison, WI, USA M. Allen Albuquerque, NM, USA R.Mayes Garching, Germany D. Rixen v
Contents 1 Experimental-Analytical Dynamic Substructuring of Ampair Testbed: A State-Space Approach ............ 1 Mladen Gibanica, Anders T. Johansson, Anders Liljerehn, Per Sjövall, and Thomas Abrahamsson 2 Experimental Dynamic Substructuring of the Ampair Wind Turbine Test Bed ................................. 15 Jacopo Brunetti, Antonio Culla, Walter D’Ambrogio, and Annalisa Fregolent 3 Are Rotational DoFs Essential in Substructure Decoupling?....................................................... 27 Walter D’Ambrogio and Annalisa Fregolent 4 Validation of Blocked-Force Transfer Path Analysis with Compensation for Test Bench Dynamics .......... 37 D.D. van den Bosch, M.V. van der Seijs, and D. de Klerk 5 Prediction of Forced Response on Ancillary Subsystem Components Attached to Reduced Linear Systems............................................................................................. 51 Sergio E. Obando and Peter Avitabile 6 Towards Dynamic Substructuring Using Measured Impulse Response Functions .............................. 73 M.V. van der Seijs, P.L.C. van der Valk, T. van der Horst, and D.J. Rixen 7 Hybrid Modeling of Floating Raft System by FRF-Based Substructuring Method with Elastic Coupling ... 83 Huang Xiuchang, Zhang Zhenguo, Hua Hongxing, and Xu Shiyin 8 Experimental Based Substructuring Using a Craig-Bampton Transmission Simulator Model ................ 91 Mathew S. Allen, Daniel C. Kammer, and Randy L. Mayes 9 Consideration of Interface Damping in Shrouded Mistuned Turbine Blades .................................... 105 F. Schreyer, J. Gross, P. Reuss, M. Junge, and H. Schoenenborn 10 Coupling Elements for Substructure Modelling of Lightweight Multi-storey Buildings ........................ 113 Ola Flodén, Kent Persson, and Göran Sandberg 11 Deformation Mode Selection and Orthonormalization for an Efficient Simulation of the Rolling Contact Problem......................................................................................................... 125 Karim Sherif and Wolfgang Witteveen 12 Towards a Parallel Time Integration Method for Nonlinear Systems ............................................. 135 Paul L.C. van der Valk and Daniel J. Rixen 13 Efficient Model Order Reduction for the Nonlinear Dynamics of Jointed Structures by the Use of Trial Vector Derivatives.............................................................................................. 147 Wolfgang Witteveen and Florian Pichler 14 A Substructuring Method for Geometrically Nonlinear Structures ............................................... 157 Frits Wenneker and Paolo Tiso 15 Craig-Bampton Substructuring for Geometrically Nonlinear Subcomponents .................................. 167 Robert J. Kuether and Matthew S. Allen vii
viii Contents 16 Parameterized Reduced Order Models Constructed Using Hyper Dual Numbers .............................. 179 M.R. Brake, J.A. Fike, S.D. Topping, R. Schultz, R.V. Field, N.M. McPeek-Bechtold, and R. Dingreville 17 Efficient Stochastic Finite Element Modeling Using Parameterized Reduced Order Models .................. 193 R. Schultz, M.R. Brake, S.D. Topping, N.M. McPeek-Bechtold, J.A. Fike, R.V. Field, and R. Dingreville 18 Application of a Novel Method to Identify Multi-axis Joint Properties ........................................... 203 Scott Noll, Jason Dreyer, and Rajendra Singh 19 Experimental Identification and Simulation of Rotor Damping ................................................... 209 Lothar Gaul and André Schmidt 20 An Approach to Identification and Simulation of the Nonlinear Dynamics of Anti-Vibration Mounts ....... 219 A. Carrella, S. Manzato, and L. Gielen 21 Test Method Development for Nonlinear Damping Extraction of Dovetail Joints ............................... 229 C.W. Schwingshackl, C. Joannin, L. Pesaresi, J.S. Green, and N. Hoffmann 22 Microslip Joint Damping Prediction Using Thin-Layer Elements ................................................. 239 Christian Ehrlich, André Schmidt, and Lothar Gaul 23 Variability and Repeatability of Jointed Structures with Frictional Interfaces .................................. 245 Matthew R. Brake, Pascal Reuss, Daniel J. Segalman, and Lothar Gaul 24 Evaluation of North American Vibration Standards for Mass-Timber Floors ................................... 253 Joshua A. Schultz and Benton Johnson 25 Improving Model Predictions Through Partitioned Analysis: A Combined Experimental and Numerical Analysis................................................................................................. 261 Garrison Stevens, Sez Atamturktur, and Joshua Hegenderfer 26 Model Reduction and Lumped Models for Jointed Structures..................................................... 273 G. Chevallier, H. Festjens, J.-L. Dion, and N. Peyret 27 A Complex Power Approach to Characterise Joints in Experimental Dynamic Substructuring............... 281 E. Barten, M.V. van der Seijs, and D. de Klerk 28 Prediction of Dynamics of Modified Machine Tool by Experimental Substructuring ........................... 297 Christian Brecher, Stephan Bäumler, and Matthias Daniels 29 Static Torsional Stiffness from Dynamic Measurements Using Impedance Modeling Technique.............. 307 Hasan G. Pasha, Randall J. Allemang, David L. Brown, and Allyn W. Phillips 30 Full Field Dynamic Strain on Wind Turbine Blade Using Digital Image Correlation Techniques and Limited Sets of Measured Data from Photogrammetric Targets.............................................. 317 Jennifer Carr, Christopher Niezrecki, and Peter Avitabile 31 Comparison of Multiple Mass Property Estimation Techniques on SWiFT Vestas V27 Wind Turbine Nacelles and Hubs ............................................................................................. 329 Timothy Marinone, David Cloutier, Kevin Napolitano, and Bruce LeBlanc 32 Overview of the Dynamic Characterization at the DOE/SNL SWiFT Wind Facility............................ 337 Bruce LeBlanc, David Cloutier, and Timothy Marinone 33 Artificial and Natural Excitation Testing of SWiFT Vestas V27 Wind Turbines ................................. 343 Timothy Marinone, David Cloutier, Bruce LeBlanc, Thomas Carne, and Palle Andersen 34 Effects of Boundary Conditions on the Structural Dynamics of Wind Turbine Blades—Part 1: Flapwise Modes .......................................................................................................... 355 Javad Baqersad, Christopher Niezrecki, and Peter Avitabile
Contents ix 35 Effects of Boundary Conditions on the Structural Dynamics of Wind Turbine Blades. Part 2: Edgewise Modes ......................................................................................................... 369 Javad Baqersad, Christopher Niezrecki, and Peter Avitabile 36 Modal Testing and Model Validation Issues of SWiFT Turbine Tests ............................................. 381 Timothy Marinone, David Cloutier, and Bruce LeBlanc 37 Development of Simplified Models for Wind Turbine Blades with Application to NREL 5 MW Offshore Research Wind Turbine...................................................................................... 389 Majid Khorsand Vakilzadeh, Anders T. Johansson, Carl-Johan Lindholm, Johan Hedlund, and Thomas J.S. Abrahamsson 38 A Wiki for Sharing Substructuring Methods, Measurements and Information.................................. 403 Matthew S. Allen, Jill Blecke, and Daniel Roettgen 39 Novel Parametric Reduced Order Model for Aeroengine Blade Dynamics....................................... 413 Jie Yuan, Giuliano Allegri, Fabrizio Scarpa, and Ramesh Rajasekaran 40 Practical Seismic FSSI Analysis of Multiply-Supported Secondary Tanks System.............................. 427 Nam-Gyu Kim, Choon-Gyo Seo, and Jong-Jae Lee 41 DEIM for the Efficient Computation of Contact Interface Stresses ............................................... 435 M. Breitfuss, H. Irschik, H.J. Holl and W. Witteveen 42 Amplitude Dependency on Dynamic Properties of a Rubber Mount .............................................. 447 C.B. Nel and J. van Wyngaardt 43 Model Order Reduction for Geometric Nonlinear Structures with Variable State-Dependent Basis.......... 455 Johannes B. Rutzmoser and Daniel J. Rixen 44 Stochastic Iwan-Type Model of a Bolted Joint: Formulation and Identification................................. 463 X.Q. Wang and Marc P. Mignolet
Chapter1 Experimental-Analytical Dynamic Substructuring of Ampair Testbed: A State-Space Approach Mladen Gibanica, Anders T. Johansson, Anders Liljerehn, Per Sjövall, and Thomas Abrahamsson Abstract The Society for Experimental Mechanics (SEM) Substructuring Focus Group has initiated a research project in experimental dynamic substructuring using the Ampair 600W wind turbine as a testbed. In this paper, experimental as well as analytical models of the blades of said wind turbine are coupled to analytical models of its brackets. The focus is on a statespace based substructuring method designed specifically for experimental-analytical dynamic substructuring. It is shown (a) theoretically that the state-space method gives equivalent results to the second order methods under certain conditions, (b) that the state-space method numerically produces results equivalent to those of a well-known frequency-based substructuring technique when the same experimental models are used for the two methods and (c) that the state-space synthesis procedure can be translated to the general framework given by De Klerk et al. Keywords Substructuring • Experimental-analytical dynamic • Modal analysis • Vibration testing • State-space • Component mode synthesis • Frequency based substructuring • Automatic system identification • Wind turbines 1.1 Introduction The subject of substructuring has been an open research area ever since its advent in the 1960s [1]. Recently, often attributed to advances in computing capacity and experimental equipment, there has been a renewed interest in the subject. Substructuring builds on the idea that a complex structure can be decomposed into a number of simpler components, or substructures. The modeling approach of these substructures may vary, which allows creating mixed experimental and analytical models thereby increasing the modeling flexibility for complex mechanical structures such as cars, air planes, rocket launchers and wind turbines. Although the main concept of substructuring is simple—a matter of enforcing compatibility and equilibrium at the interfaces between substructures—there are a number of different methods presented in literature. These are typically categorized, by the type of substructure models used, as Frequency Response Function (FRF) based and modal based (CMS—component mode synthesis) methods. Early works falling in the former category are e.g. [2, 3] while the CraigBampton method [4] is a well-known early representative of the latter. The paper by de Klerk et al. [1] is frequently cited as a well written overview of the field. This paper is based on the MSc thesis by Mr. Mladen Gibanica [5]. The structure under investigation here is the SEM Substructuring Focus Group testbed, the Ampair 600 wind turbine, further detailed by Mayes [6]. Since its introduction at IMAC XXX, several attempts at substructuring have been reported for this structure. A group from Sandia National Laboratories have worked with experimental substructuring using the Transmission Simulator Technique [7, 8], a method also employed at the Atomic Weapons Establishment in the UK [9], M. Gibanica • A.T. Johansson ( ) • T. Abrahamsson Department of Applied Mechanics, Chalmers University of Technology, SE-412 96 Göteborg, Sweden e-mail: anders.t.johansson@chalmers.se A. Liljerehn AB Sandvik Coromant, R & D Metal Cutting Research, SE-811 81 Sandviken, Sweden P. Sjövall FS Dynamics, SE-412 63 Göteborg, Sweden M. Allen et al. (eds.), Dynamics of Coupled Structures, Volume 1: Proceedings of the 32nd IMAC, A Conference and Exposition on Structural Dynamics, 2014, Conference Proceedings of the Society for Experimental Mechanics Series, DOI 10.1007/978-3-319-04501-6__1, © The Society for Experimental Mechanics, Inc. 2014 1
2 M. Gibanica et al. while a frequency-based method was used by a group from TU Delft [10]. Also, a group of Italian scientists have studied the role of interface degrees of freedom for the coupling of the structure [11]. The purpose of the test bed is to serve as a reference for researchers around the world. Data is shared online through a wiki [12]. This paper considers the blades and the brackets of the turbine only—the hub, tower and base are thus left for later studies. The dynamics of the joints has not been considered (a presentation on nonlinear effects from varying contact geometry at different load levels was given by Dr. Pascal Reuss at IMAC XXXI [13]). Components are coupled using the state-space synthesis method of [14]. (Similar methods have been proposed also by e.g. Su and Juang [15].) The results are compared to experimental measurements of the assembly as well as experimental-analytical FRF-based coupling techniques and analytical models coupled using CMS. The Ampair 600 wind turbine blades, consisting of a composite hull around a solid core, have been thoroughly tested in dynamic and static measurements as well as destructive tests to quantify the material parameters [16–21]. A calibrated FE model of the blades has been developed by Johansson et al. [20] which is used for comparison here. A new FE model of the bracket is created from geometry measurements made available by the TU Delft and using standard material properties. The paper is structured as follows: Sect. 1.2 briefly introduces the theory behind the state space coupling technique and puts it into the framework of de Klerk et al. [1]. Section 1.3 describes the Finite Element models used. Section 1.4 elaborates on performed experiments and describes the experimentally derived models used. Section 1.5 relates results before the paper is rounded up by some conclusions in Sect. 1.6. 1.2 Theory In linear system theory, systems are sometimes separated into external and internal descriptions, where the former offer information on the input-output relation only whereas the latter includes information on the state of the process [22, 23]. The typical external model in structural dynamics is the time domain convolution relation which gives rise to the Frequency Response Function in the frequency domain: y.t/ Dy0 CZ 1 0 h.t /u. /d (1.1) Y.!/ DH.!/U.!/ (1.2) While internal models are given in the usual first- or second-order forms [24]: Px.t/ DAx.t/ CBu.t/ y.t/ DCx.t/ CDu.t/ (1.3) Kq.t/ CVPq.t/ CMRq.t/ Df.t/ (1.4) where the matrices have the usual meanings and the forces in Eq. (1.4) relate to the input in Eq. (1.3) through a selection matrix such that f.t/ DPuu.t/ [25]. A shaker or impact hammer testing campaign will thus result in an external model of the system, from which the analyst frequently attempts to derive an internal model. This process is often referred to as modal parameter extraction [26] in the structural dynamics community andsystem identificationelsewhere [27]. 1.2.1 System Identification System identification is performed on experimentally obtained outer models in the frequency domain, Eq. (1.2), using the state-space subspace system identification algorithm implemented as n4sidin MATLAB’s System Identification Toolbox [27, 28] to estimate a state-space model quadruple set fA; B; C; Dg. In order to automate the system identification process a method for automatic model order estimation by Yaghoubi and Abrahamsson [29] is used along with the N4SID method. All systems studied in this paper are considered to be linear time invariant (LTI) systems. By assumption they are also stable and passive. The systems are also assumed to be reciprocal (since the systems are non-gyroscopic and non-circulatory). As system identification is a general tool, these assumptions must be enforced explicitly (except for stability, which can be included as a condition on the system identification algorithm). Reciprocity is enforced through measuring using a limited
1 Experimental-Analytical Dynamic Substructuring of Ampair Testbed: A State-Space Approach 3 number of input nodes and applying Maxwell-Betti’s reciprocity principle (see [5]). The passivity criterion states that the power supplied to the system is non-negative and can be zero only for components without damping. Models derived from first principles satisfy the passivity criterion implicitly, but this is not necessarily the case for experimentally identified models. The passivity problem for state space models is commonly known, for details see for example [14,30]. The passivity of a state space model is linked to the positive real (PR) lemma. In this paper, an engineering solution to the passivity optimization problem defined in [14] based on averaging of the BandCmatrices has been used. 1.2.2 Substructuring Substructuring is the process of coupling two structures together by enforcing two conditions at their common interface; compatibility and force equilibrium. This paper will treat only what de Klerk et al. [1] refers to as the primal formulation, which implies that the displacements are defined and interface forces are eliminated. For brevity, the explicit time dependency is dropped from the equations in the following. For (analytically derived) internal models as in Eq. (1.4), enforcing compatibility and force equilibrium amounts to an assembly procedure parallel to that used for Finite Element Models, see further de Klerk et al. [1], such that synthesis of two systems 1 and 2 can be described by: 8 ˆˆ< ˆˆ: K1 0 0 K2 q1 q2 C V1 0 0 V2 Pq1 Pq2 C M1 0 0 M2 Rq1 Rq2 Df Cg Eq D0 LTg D0 (1.5) where Ecan be thought of as a signed boolean matrix enforcing compatibility,1 g are the interface forces, and the matrix L is the nullspace of E. For details, refer to [1]. When coupling two structures, which are described by m1 and m2 degrees of freedom (DOFs), the assembled structure will consist of m1 Cm2 mc coordinates, where mc is the number of coupling constraints. The de Klerk et al. paper [1] generalizes the formulation using the two matrices EandLto include also models using generalized coordinates and outer models (Eq. (1.2)), i.e. FRF coupling. In a following subsection, the state-space method [14] will be put into the same format. 1.2.3 State-Space Coupling The state-space coupling used here is described in the paper by Sjövall and Abrahamsson [14]. At its core is the coupling formof Eq. (1.6), where the i:th first-order system to be coupled is rewritten to second-order form at the interface in order for the compatibility and force equilibrium conditions to be applicable 8 ˆˆˆˆ< ˆˆˆˆ: 2 4 Ryc Pyc Pxb 3 5D 2 4 Ai vv Ai vd Ai vb I 0 0 0 Ai bd Ai bb 3 5 2 4 Pyc yc xb 3 5C 2 4 Bi vv Bi vb 0 0 0 Bi bb 3 5 uc ud yc yd D 0 I 0 Ci bv Ci bd Ci bb (1.6) Here, the subscript c refers to coupling degrees of freedom, bother degrees of freedom, vvelocity outputs andd displacement outputs. Note that for state-space coupling there exist twice as many states as DOFs (a product of rewriting a second-order differential equation on first-order form) and thus if n D 2m, for two structures with n1 and n2 states and mc coupling coordinates the coupled system will containn1 Cn2 2mc states. 1This matrix is denotedBin de Klerk et al. The notation cannot be adopted here as theBmatrix is strongly associated to the state-space formulation in Eq. (1.3).
4 M. Gibanica et al. 1.2.3.1 General Framework Here, the coupling procedure derived in [14] is recast into the general framework established by de Klerk et al. [1], thus using the EandLmatrices defined above. Starting from a general state-space formulation (Eq. (1.3)), a transformation matrixTis introduced which takes the system to the coupling form of Eq. (1.6) [14]. Let the system on coupling form be: ( PQx D QAQxC QBu y D QCQx (1.7) where QADT 1AT, QBDTB, QCDCT 1 and Qx DTx. Bvv of Eq. (1.6) is the inertia at the interface DOFs [31]. By forming of a new matrix denotedMB, the system can be rewritten such that the first block row is pre-multiplied by B 1 vv . The matrix MB will then be: MB Ddiag.diag.B1 1 vv ; I; I/;:::;diag.Bh 1 vv ; I; I// (1.8) Such that the system can now be written as: ( MBPQx DMB QAQxCMB QBu y D QCQx (1.9) Introducing the Eand Lmatrices such that compatibility is described by EQx D0 and equilibrium by LTu c;g D0, where subscript g is used to denote interface forces, the matrix Lis formed as L Dnull.E/ such that the following is satisfied: EQx DELQz D0 and Qx DLQz. The new state vector Qz represents the new states after coupling. The synthesised state-space model is then formed as follows: ( LTMBLPQz DLTMB QALQz CLTMB QBu y D QCLQz (1.10) Finally, .LTMBL/ 1, may be pre-multiplied to the first equation in Eq. (1.10). A new transformation matrix must however be introduced simply to reduce the excess inputs and outputs. The new matrix is denoted Lu and is formed from a matrix Eu as before, Lu Dnull.Eu/. In other words, for a system with two synthesised components the new reduction matrix is Eu D ŒI 0 0 0 . The implication of this transformation is that in the synthesised model, to which this new matrix is applied, the first component of the considered vector will be removed. This new matrix is introduced in the equations by the transformations u DLuNuand Ny DLT u y as shown in Eq. (1.12). The last step is only true for systems with an equal number of inputs and outputs but is easily generalised. The final system can be written as follows: 8 < : PQz D OAQzC OBNu Ny D OCQz (1.11) Where each part in the equation above is then identified as: 8 ˆˆ< ˆˆ: OAD.LTMBL/ 1.LTMB QAL/ OBD.LTMBL/ 1.LTMB QB/Lu OCDLT u . QCL/ (1.12) 1.2.3.2 Second Order Form Equivalent of Synthesised First Order Form An analytical model written on second order form can easily be rewritten on first order form by doubling the number of states [24]. The transformation from a first order form to a second order form with half the number of states is not as trivial, even when possible. Begin the second order system shown in Eq. (1.5). Methods for rewriting such a system to the first-order form of Eq. (1.3) are well-known [24, 25]. It is assumed that the output is the entire displacement vector, such that all DOFs are considered as
1 Experimental-Analytical Dynamic Substructuring of Ampair Testbed: A State-Space Approach 5 outputs. A transformation matrix is introduced in Eq. (1.13) which transforms a state-space system to coupling form under the assumption that the original system is analytically derived with state vector xT D q Pq T and displacement output such that q Dy DCx Ti D 2 66 4 Ci cAi Ci c Ci bAi Ci b 3 77 5 (1.13) This transformation will produce a new state vector, Qx D Pyi c yi c Pyi b yb T, where the superscript i represents the subcomponent system. Again, the subscript c denotes the coupling partition of the input and output matrices, i.e. the block row and column of the BandCmatrices, respectively. The difference between this transformation and the transformation in [14], is that the internal states are represented by physical coordinates here. In a theoretical representation this is valid, but in an experimental representation it would imply a significant constraint on the model to allow only twice as many states as there are sensors. The coupling form obtained through Eq. (1.13) is shown below 8 ˆˆˆˆˆˆˆˆˆˆˆˆ< ˆˆˆˆˆˆˆˆˆˆˆˆ: QAi DTi Ai Ti 1 D 2 66 4 Ai cvc Ai cdc Ai cvb Ai cdb I 0 0 0 Ai bvc Ai bdc Ai bvb Ai bdb 0 0 I 0 3 77 5 QBi DTi Bi D 2 66 4 Ci cAi Bi Ci cBi Ci bAi Bi Ci bBi 3 77 5 D 2 66 4 Ci cAi Bi 0 Ci bAi Bi 0 3 77 5 D 2 66 4 Bi c;c Bi c;b 0 0 Bi bc Bi bb 0 0 3 77 5 QCi DCi Ti 1 D I 0 0 0 0 0 I 0 (1.14) It can be shown that the transformation of the input matrixCAB, with the assumptions introduced above, is but the inverse of the mass matrix, thus simplifying the QBmatrix as follows (recall that the full system mass matrix is a block-diagonal matrix): 8 ˆˆˆˆ< ˆˆˆˆ: Bi cc DMi 1 cc Bi cb D0 Bi bc D0 Bi bb DMi 1 bb (1.15) Synthesising two systems by the procedure described above yields the following model, here reorganised such that it is recognisable as the usual state-space formulation of a second order system [25]: 8 ˆˆˆˆˆ< ˆˆˆˆˆ: PNyc Py1 b Py2 b RNyc Ry1 b Ry2 b 9 >>>>>= >>>>>; D 2 66 66 66 64 0 0 0 I 0 0 0 0 0 0 I 0 0 0 0 0 0 I NAcdc NA1 cdb NA2 cdb NAcvc NA1 cvb NA2 cvb A1 bdc A1 bdb 0 A1 bvc A1 bvb 0 A2 bdc 0 A2 bdb A2 bvc 0 A2 bvb 3 77 77 77 75 8 ˆˆˆˆˆ< ˆˆˆˆˆ: Nyc y1 b y2 b PNyc Py1 b Py2 b 9 >>>>>= >>>>>; C 2 66 66 66 64 0 0 0 0 0 0 0 0 0 NBcc 0 0 0 B1 bb 0 0 0 B2 bb 3 77 77 77 75 8 < : Nuc u1 b u2 b 9 = ; 8 < : Nyc y1 b y2 b 9 = ; D 2 4 I 0 0 0 0 0 0 I 0 0 0 0 0 0 I 0 0 0 3 5 8 ˆˆˆˆˆ< ˆˆˆˆˆ: Nyc y1 b y2 b PNyc Py1 b Py2 b 9 >>>>>= >>>>>; (1.16)
6 M. Gibanica et al. The lower part can be identified as a second order system which is directly comparable to the direct coupling procedure [1]. The two second order systems can be written with an interface part (c) and a body part (b) for two components, i D1;2. 8 ˆˆˆˆˆˆ< ˆˆˆˆˆˆ: Mi D Mi cc 0 0 Mi bb Ki D Ki cc Ki cb Ki bc Ki bb Vi D Vi cc Vi cb Vi bc Vi cc (1.17) The synthesised system then becomes 8 ˆˆˆˆˆˆˆˆˆˆˆ< ˆˆˆˆˆˆˆˆˆˆˆ: Mas D 2 4 M1 cc CM2 cc 0 0 0 M1 bb 0 0 0 M2 bb 3 5 Kas D 2 4 K1 cc CK2 cc K1 cb K2 cb K1 bc K1 bb 0 K2 bc 0 K2 bb 3 5 Vas D 2 4 V1 cc CV2 cc V1 cb V2 cb V1 bc V1 bb 0 V2 bc 0 V2 bb 3 5 (1.18) For comparison, the second order system from Eq. (1.16) is pre-multiplied by the inverse of its input matrix, which is diagonal. The resulting matrix is split in two, one part representing the stiffness and the other the damping. The following is then obtained 8 ˆˆˆˆˆˆˆˆˆˆˆ< ˆˆˆˆˆˆˆˆˆˆˆ: Mas Msss D 2 64 NB 1 cc 0 0 0 NB1 1 bb 0 0 0 NB2 1 bb 3 75D 2 4 M1 cc CM2 cc 0 0 0 M1 bb 0 0 0 M2 bb 3 5 Kas Ksss D 2 64 NB 1 cc NAcdc NB 1 cc NA1 cdb NB 1 cc NA2 cdb NB1 1 bb A1 bdc NB1 1 bb A1 bdb 0 NB2 1 bb A2 bdc 0 NB2 1 bb A2 bdb 3 75D 2 4 K1 cc CK2 cc K1 cb K2 cb K1 bc K1 bb 0 K2 bc 0 K2 bb 3 5 Vas Vsss D 2 64 NB 1 cc NAcvc NB 1 cc NA1 cvb NB 1 cc NA2 cvb NB1 1 bb A1 bvc NB1 1 bb A1 bvb 0 NB2 1 bb A2 bvc 0 NB2 1 bb A2 bvb 3 75D 2 4 V1 cc CV2 cc V1 cb V2 cb V1 bc V1 bb 0 V2 bc 0 V2 bb 3 5 (1.19) It is now obvious that both methods produce exactly the same systems. 1.3 Analytical Models The two analytical FE models that were used were created in FEMAP and solved with MD Nastran. NX Nastran was used for verification where non-reduced models were solved. 1.3.1 Blade A version of the blade FE model described in [20] (Nastran model can be found online at [12]) was used and is visualised in Fig. 1.1. The model consisted of 20,523 nodes and 96,416 elements. The model has a solid core, for which solid elements were used and a laminate skin model, for which laminate plate elements were used. For a thorough explanation of the composite material model and how it was calibrated, see Johansson et al. [20] (Fig. 1.2).
1 Experimental-Analytical Dynamic Substructuring of Ampair Testbed: A State-Space Approach 7 Fig. 1.1 FE model of the blade with numbering used in the FE model but also during the vibration tests Fig. 1.2 Close-up view of the three inputs of the blade The accelerometer positions in the vibrations test and for the FE blade model are shown in Fig. 1.1. These were chosen as close to the physical locations of the accelerometers as possible for the MAC correlations to yield correct results. Note that the numbering starts at the coupling positions. 1.3.2 Bracket The bracket geometry was received from TU Delft. It was modelled by 24,707 solid parabolic elements and an isotropic material model with stiffness E D2 1011 N/m2 and density D5;000kg/m3. The shaft which is inserted into the bracket was modelled as an integrated part of the bracket. Parabolic elements were used so that the shaft curvature would be described better. The bracket model is shown in Fig. 1.3. The numbers 1, 2 and 3 are the coupling positions coupled to the blade at positions 3, 4 and 5, thus matching DOFs were used at each of the components coupling locations. 1.3.3 FE Coupling The coupling between the blade and the bracket can be seen in Fig. 1.4a, b, where the area around the holes on the blade have been constrained with rigid links. All the DOFs on the bracket in contact with the blade when mounted were constrained with rigid links to a six DOFs node located at the centre of the three holes. The bracket coupling type was constrained at the holes, see Fig. 1.4b. The same configurations were used in the experimental-analytical coupling. All three configurations were used in coupling of the analytical models. It should be noted that the blade bracket constraints configuration are on the same side of the blade which creates asymmetric modeshapes. The assembled blade bracket structure along with the measurement points can be seen in Fig. 1.5.
8 M. Gibanica et al. Fig. 1.3 FE model of the blade with numbering used in the FE model and during the vibration tests Fig. 1.4 The coupling configurations for the blade and the bracket using rigid links attached to a six DOFs coupling node. (a) Blade coupling configuration. (b) Bracket coupling configuration Fig. 1.5 The FE model of the blade mounted to the FE model of the bracket with measurement locations 1.4 Experiments Three different blades from the same Ampair 600 wind turbine were considered. Each blade had a unique mounting brackets so that each blade was mounted to its own bracket. The three experimental blades will henceforth be denoted by their serial number; 841, 722 and 819 and the brackets will be numbered, 01, 02 and 03. Coupling between the blade and their corresponding bracket was blade 841 and bracket 01, blade 722 and bracket 02 and blade 819 and bracket 03. The bracket numbers will usually be omitted when the blade bracket configuration is discussed but occasionally they will be abbreviated as blade/bracket, e.g. 841/01 would represent a blade 841 which in turn is mounted to bracket 01. The experiments were performed at Chalmers Vibration and Smart Structures Lab. The measurements were performed with a stepped sine input with 0.25 Hz step size from 20 to 800 Hz for the blade and blade bracket measurements. For the fully assembled system the measurements were performed from 10 to 300 Hz with a step size of 0.1 and 0.01 Hz at some particular frequency intervals. A total of three measurements for the blade and blade bracket models were made. The configurations
1 Experimental-Analytical Dynamic Substructuring of Ampair Testbed: A State-Space Approach 9 Fig. 1.6 The input directions and test setup for the blades. (a) Input in the x direction. (b) Input in the y direction. (c) Input in the z direction Fig. 1.7 A detailed view of the blade bracket assembly is shown with the accelerometer placements considered were: the three blades alone and the three blades with brackets mounted. Note that measurements made in a specific direction (x, y or z) means that the input was aligned with that axis. See Fig. 1.1 for an orientation of an generic blade and Fig. 1.2 for the different input locations. 1.4.1 Test Setup The three different input directions are shown in Fig. 1.6. This coordinate system is consistent with the FE analysis coordinate system. The test setup shown in Fig. 1.4 is also general for all measurements. It was exactly the same, both for the measurement of blades alone and for the blade measurements with a bracket attached. It can be seen, from Fig. 1.6, that the measured blades (and bracket mounted blades) were hung at three locations when measured in the x and y directions, which made it easier to position the components relative to the load cell. Measurements in the z direction for the blade and blade with attached bracket were only hung at two positions. Further, it should be noted that the components were hung in fasteners which were glued to the components, the same fastener was used for the load cell attachment and can be seen in Fig. 1.7. All the fasteners were mounted on the blades during all measurements for consistency.
10 M. Gibanica et al. The interface locations of the blade are shown in Fig. 1.2 but with the interface accelerometer from position 6 located under the stinger and used as a direct accelerance for a comparison with the accelerometer at position 1. For a view of a bracket and blade assembled see Fig. 1.7. A detailed view of the interface accelerometers can also be seen. Different input directions to the bracket coupled with a blade had the same accelerometer locations and the same input locations were used as shown in Fig. 1.6. 1.5 Experimental Models To obtain experimental models, system identification was performed on the obtained frequency response data (FRD). The system identification of the substructures, see Sect. 1.4.1, was conducted using an automated method for order selection and identification developed by Yaghoubi and Abrahamsson, see [29, 32]. This method is based on obtaining a number of realisations of the same dataset, using MATLAB’s n4sidmethod, which is a state-space subspace method, see McKelvey et al. [28]. The realisations is then statistically evaluated, using the MOC, to discard non-physical modes. The only input required by the user is a high model order from start. In general, the method worked very well for the experimental data of the blade. Only occasionally did it give non-physical modes that could be removed manually. The synthesised models correlate well with measured data and the spread in response between the different blades can be seen in Fig. 1.8. 1.6 Results In this section correlation between the eigenvalues of the blade models and the blade bracket coupled models are presented. Then, analytical coupling between the components is presented together with the experimental-analytical coupling results. Comparisons between the state-space and the frequency-based methods compared to analytical coupling and finally results based on measurements compared to experimental-analytical coupling. Fig. 1.8 Spread between the synthesised FRFs of the blades with input in z and output from channel 7
1 Experimental-Analytical Dynamic Substructuring of Ampair Testbed: A State-Space Approach 11 Table 1.1 Eigenfrequencies (Hz) and mass (g) for all the blades along with mean ( , Hz and g), standard deviation ( , Hz and g) and coefficient of variation (COV, %) of all the experimental blades Blades Statistics Component 841 722 819 FEA COV Error Mode 1 45:37 45:00 43:72 45:82 44:70 0:86 1:93 2.51 Mode 2 128:76 124:08 129:24 130:67 127:36 2:85 2:24 2.60 Mode 3 190:79 203:49 186:66 197:65 193:65 8:77 4:53 2.06 Mode 4 248:64 240:32 248:44 254:21 245:80 4:75 1:93 3.42 Mode 5 322:12 333:15 315:29 339:93 323:52 9:01 2:79 5.07 Mode 6 395:36 388:16 398:94 412:58 394:16 5:49 1:39 4.67 Mode 7 404:56 410:33 423:24 453:22 412:71 9:56 2:32 9.82 Mode 8 468:74 489:54 463:86 495:47 474:05 13:64 2:88 4.52 Mode 9 572:88 557:26 573:00 598:12 567:71 9:05 1:59 5.36 Mode 10 635:25 654:52 621:72 660:11 637:16 16:49 2:59 3.60 Mode 11 746:52 742:21 747:93 758:33 745:55 2:98 0:40 1.71 Mass 830:50 829:70 830:10 797:11 830:10 0:40 4.89e 04 3:97 The FE model eigenfrequencies (Hz) and mass (g) are also shown along with the relative error (%) between the measured mean and the FE element blade Table 1.2 Eigenfrequencies (Hz) for all the bracket mounted blades along with the error (%) between the measured mean and the experimentalanalytical synthesised components with the state-space method Set Blades/brackets Error Component SS841/FEA SS 722/FEA SS819/FEA Error 1 Error 2 Error 3 Mode 1 40:19 43:01 41:92 41:08 5:39 4:24 3:89 Mode 2 115:73 119:52 114:17 120:55 2:04 2:04 2:03 Mode 3 193:79 187:35 201:35 184:91 1:11 1:3 1:59 Mode 4 222:7 229:24 220:29 229:33 1:54 0:64 2:63 Mode 5 321:03 308:69 322:83 304:88 3:21 2:92 2:15 Mode 6 365:02 391:92 355:8 391:27 6:51 0:34 5:72 Mode 7 383:58 410:04 405:19 408:84 9:37 4:46 5:38 Mode 8 486:16 437:3 466:13 442:78 7:93 8:24 6:89 Mode 9 536:26 548:52 518:58 537:05 0:51 1:9 0:49 Mode 10 653:62 584:9 606:98 582:37 7:75 13:3 7:09 Mode 11 726:88 634:01 645:84 617:42 14:68 11:58 12:68 Error 1 stand for the relative error between SS 841/FEA and measured mean, Error 2 between SS 722/FEA and measured mean and Error 3 between SS 819/FEA and measured mean 1.6.1 Analytical and Synthesised Blade Models The 11 first eigenfrequencies for the three experimental blade models and the FE blade are shown in Table 1.1 along with themean ( , Hz), the standard deviation ( , Hz) and the coefficient of variance (COV, %) for the three experimental blades. Also, the relative error (%) between the measured mean relative to the FE blade is given. It can be seen that the spread between the blades is considerable and that the errors for modes 8 and 10 are considerably larger compared to the other modes. The mass (gramme) is also given in the table for the three blades and the FE blade along with the mean (gramme) of the three blades, the standard deviation (gramme), the coefficient of variance and the relative error with respect to the FE model. It should also be noted that the spread of the mass is very small. 1.6.2 Blade Bracket From Table 1.2 it can be seen that a relatively large spread is obtained for mode 7, 8, 10 and 11. In Fig. 1.9 the FRFs for the difference between the coupled models are shown for models coupled using frequency based and state space coupling. It can be seen that the two coupling procedures produces identical results. In Fig. 1.10 the difference between the individual measured FRFs of the blade bracket assemblies as well as the state space coupled systems becomes eminent.
12 M. Gibanica et al. Fig. 1.9 Difference between frequency based and state-space based substructuring methods when CBD0, for experimental-analytical coupling of the blade bracket system Fig. 1.10 FRF comparison between the coupled state space blade bracket models and the measured models In Fig. 1.11, the three considered configurations of the analytical couplings are compared to the measured components. The analytical model correlated well with the four first modes. After that the analytical model did not correlate very well with any of the blades. 1.7 Conclusions and Further work In this paper, an experimental-analytical synthesis of components for the SEM Substructuring focus group test bed, the Ampair 600W wind turbine, has been presented using a state-space method for the coupling. Experimental models of the blades of the turbine were identified through system identification techniques and coupled to analytical models of the brackets
1 Experimental-Analytical Dynamic Substructuring of Ampair Testbed: A State-Space Approach 13 Fig. 1.11 MAC matrix for the 11 first modes between the experimental (measured) models and the three different coupling configurations of the analytical bracket mounted blade. Four colour codes are given where red marks a correlation between 0.95 and 1, cyan a correlation between 0.9 and 0.95, green a correlation between 0.85 and 0.9 and yellow a correlation between 0.8 and 0.85 (Color figure online) normally used to fix them to the hub. The results were compared to those of a frequency-based method and the results were equivalent. A comparison with results obtained from coupling a finite element blade model to the same bracket model as the experimental models has been performed. When using the same coupling configuration, the experimental-analytical coupling version outperforms the analytical-analytical one. Owing to the greater flexibility in adapting the interface between the two analytical models however, the analytical-analytical model can be tuned to show slightly better results than the experimental ones, see further the full MSc report by Gibanica [5]. This motivates investigation of interfacing techniques such as the transmission simulator method in the context of state-space substructuring. A theoretical comparison to a second order system coupling technique was performed for the state-space synthesis. It shows that, under certain conditions, the system produced by the method is equal to that obtained by component mode synthesis/direct coupling techniques. The state-space method is also translated into the general framework of de Klerk et al. [1]. An ad-hoc solution for enforcing passivity to systems identified through system identification has been utilized in this paper. In the future, further theoretical development of this technique is needed. Also, the next step will be to couple the blades together through the hub. References 1. de Klerk D, Rixen DJ, Voormeeren SN (2008) General framework for dynamic substructuring: history, review and classification of techniques. AIAA J 46(5):1169–1181 2. Bishop RED, Johnson DC (1960) The mechanics of vibration. Cambridge University Press, Cambridge 3. Jetmundsen B, Bielawa RL, Flannelly WG (1988) Generalized frequency domain substructure synthesis. J Am Helicopter Soc 33(1):55–54 4. Craig RR Jr, Bampton MCC (1968) Coupling of substructures for dynamic analyses. AIAA J 6:1313–1319 5. Gibanica M (2013) Experimental-analytical dynamic substructuring: a state-space approach. Master’s thesis, Chalmers University of Technology 6. Mayes RL (2012) An introduction to the sem substructures focus group test bed - the ampair 600 wind turbine. In: Conference proceedings of the Society for Experimental Mechanics series. 30th IMAC, a conference on structural dynamics, pp 61–71 7. Mayes RL (2012) Tutorial on experimental dynamic substructuring using the transmission simulator method. In: Conference proceedings of the Society for Experimental Mechanics series. 30th IMAC, a conference on structural dynamics, pp 1–9 8. Rohe DP, Mayes RL (2013) Coupling of a bladed hub to the tower of the ampair 600 wind turbine using the transmission simulator method. In: Conference proceedings of the Society for Experimental Mechanics series. 31th IMAC, a conference on structural dynamics
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