River Rapids Conference Proceedings of the Society for Experimental Mechanics Series Dynamic Substructures, Volume 4 Andreas Linderholt Matthew S. Allen Randall L. Mayes Daniel Rixen Proceedings of the 37th IMAC, A Conference and Exposition on Structural Dynamics 2019 River Publishers
Conference Proceedings of the Society for Experimental Mechanics Series Series Editor Kristin B. Zimmerman, Ph.D. Society for Experimental Mechanics, Inc., Bethel, CT, USA
River Publishers Dynamic Substructures, Volume 4 Proceedings of the 37th IMAC, A Conference and Exposition on Structural Dynamics 2019 Andreas Linderholt • Matthew S. Allen • Randall L. Mayes • Daniel Rixen Editors
Published, sold and distributed by: River Publishers Broagervej 10 9260 Gistrup Denmark www.riverpublishers.com ISBN 978-87-7004-985-6 (eBook) Conference Proceedings of the Society for Experimental Mechanics An imprint of River Publishers © The Society for Experimental Mechanics, Inc. 2020 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 Dynamic Substructures represents one of eight volumes of technical papers presented at the 37th IMAC, A Conference and Exposition on Structural Dynamics, organized by the Society for Experimental Mechanics and held in Orlando, Florida, on January 28–31, 2019. The full proceedings also include volumes on Nonlinear Structures & Systems; Dynamics of Civil Structures; Model Validation and Uncertainty Quantification; Special Topics in Structural Dynamics & Experimental Techniques; Rotating Machinery, Optical Methods & Scanning LDV Methods; Sensors and Instrumentation, Aircraft/Aerospace, Energy Harvesting & Dynamic Environments Testing; and Topics in Modal Analysis &Testing. 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. Växjö, Sweden A. Linderholt Madison, WI, USA M. Allen Albuquerque, NM, USA R.Mayes Garching, Germany D. Rixen v
Contents 1 Using Laser Vibrometry for Precise FRF Measurements in Experimental Substructuring .................... 1 Francesco Trainotti, Tobias F. C. Berninger, and Daniel J. Rixen 2 A Priori Interface Reduction for Substructuring of Multistage Bladed Disks.................................... 13 Lukas Schwerdt, Lars Panning-von Scheidt, and Jörg Wallaschek 3 Using Hybrid Modal Substructuring with a Complex Transmission Simulator to Model an Electrodynamic Shaker.............................................................................................. 23 Benjamin Moldenhauer, Matt Allen, Washington J. DeLima, and Eric Dodgen 4 Hybrid Substructure Assembly Techniques for Efficient and Robust Optimization of Additional Structures in Late Phase NVH Design: A Comparison ............................................................. 35 Benjamin Kammermeier, Johannes Mayet, and Daniel J. Rixen 5 Workpiece Coupling in Machine Tools Using Experimental-Analytical Dynamic Substructuring ............ 47 Prateek Chavan, Christian Brecher, Marcel Fey, and Matthäus Loba 6 Mechanical Characterization and Numerical Modeling of High Density Polyethylene Pipes .................. 57 Mehrzad Taherzadehboroujeni and Scott W. Case 7 Study on Dynamic Stiffness Characteristic of Primary Suspension for High-Speed EMU..................... 67 Xiugang Wang, Xiaoning Cao, Ai qin Tian, Jian Su, Wei Xue, and Shen Zhan 8 Test-Based Modeling, Source Characterization and Dynamic Substructuring Techniques Applied on a Modular Industrial Demonstrator............................................................................... 73 A. M. Steenhoek, M. W. van der Kooij, M. L. J. Verhees, D. D. van den Bosch, and J. M. Harvie 9 Development of a Low Cost Automatic Modal Hammer for Applications in Substructuring .................. 77 Johannes Maierhofer, Ahmed El Mahmoudi, and Daniel J. Rixen 10 Using SEMM to Identify the Joint Dynamics in Multiple Degrees of Freedom Without Measuring Interfaces ................................................................................................................. 87 S. W. B. Klaassen and D. J. Rixen 11 Overview of Free Interface Substructuring Approaches for Systems with Arbitrary Viscous Damping in Dynamic Substructuring............................................................................................. 101 Fabian M. Gruber, Dennis Berninger, and Daniel J. Rixen 12 Model Updating of Fluid-Structure Interaction Effects on Piping System....................................... 133 Srijan Rajbamshi, Qintao Guo, and Ming Zhan 13 Vehicle Driveline Benchmarking to Support Predictive CAE Modeling Development .......................... 141 J. Furlich, J. Blough, and D. Robinette 14 A Proposal of Dynamic Behaviour Design Based on Mode Shape Tracing: Numerical Application to a Motorbike Frame................................................................................................... 149 Elvio Bonisoli, Domenico Lisitano, Luca Dimauro, and Lorenzo Peroni vii
viii Contents 15 Rapid Seismic Risk Assessment of Structures with Gaussian Process Regression............................... 159 Mohamadreza Sheibani, Ge Ou, and Shandian Zhe 16 Modeling Rail-Vehicle Coupled Dynamics by a Time-Varying Substructuring Scheme ........................ 167 Luigi Carassale, Paolo Silvestri, Roald Lengu, and Paolo Mazzaron 17 Planning of a Black-Box Benchmark Structure for Dynamic Substructuring.................................... 173 D. Roettgen and A. Linderholt 18 Study on the Technology of Reliable Life Prediction of Plate Heat Exchanger for Ship ........................ 177 Longbo Liu, Na Han, Lingli Fu, and Jun Yao 19 Real-Time Hybrid Substructuring Results of the Mars Pathfinder Parachute Deployment.................... 183 Michael J. Harris and Richard E. Christenson
Chapter 1 Using Laser Vibrometry for Precise FRF Measurements in Experimental Substructuring Francesco Trainotti, Tobias F. C. Berninger, and Daniel J. Rixen Abstract The acquisition of high quality FRF measurements is a key factor for a successful implementation of coupling/decoupling techniques in Experimental Dynamic Substructuring. Although the use of piezo accelerometers as response transducers is very popular for impact testing due to its easy and fast implementation, the level of accuracy could not be adequate in certain applications. The laser technology provides a non-invasive alternative to standard piezo devices. The choice of a non-contact measurement technique allows to minimize the impact of external dynamic systems on the test component during the measurement process. In this paper, a validation of Lagrange Multiplier—Frequency Based Substructuring coupling by means of a Virtual Point Reduction is performed on a benchmark structure with a non-stiff interface. The necessary FRF data is acquired twice, using accelerometers and a laser Doppler vibrometer respectively. Both coupling results are compared to each other and are shown to match very well simulation data up to a high frequency range. The results underline the potential of high quality, non-intrusive measurements for Frequency Based Substructuring. Keywords Experimental dynamics · Dynamic Substructuring · Frequency Based Substructuring · Virtual Point Transformation · FRF measurements · Laser vibrometry 1.1 Introduction In Dynamic Substructuring (DS) the concept of modular design can be reinterpreted from a structural dynamics point of view, making possible the modelling, analysis and optimization of a complex system on a substructure level. Although various methodologies and technical solutions within DS are well documented [1], it still remains challenging to validate theoretical concepts in the framework of an industrial application. Significant difficulties in Experimental Dynamic Substructuring (EDS) concern the coupling/decoupling of the measured components. In this context, a frequency-based formulation of the problem is recommended as it directly includes the measured Frequency Response Functions (FRFs) in the implemented methods. Furthermore, a proper coupling of substructures strongly depends on a complete and accurate modelling of interface dynamics. It is common practice to ‘weaken’ the interface problem by projecting the dynamics into a subspace composed by Interface Deformation Modes (IDMs), which aren’t global vibration modes but rather kinematic assumptions of the local deformation behaviour at the interface. This approach uses so-called virtual points to connect the components as in finite element models, overcoming some of the issues arising from experimental practice [2]. Inaccuracies and sources of error related to the application of experimental Frequency Based Substructuring (FBS) techniques by means of Virtual Point Transformation (VPT) arise from the complexities associated with both reliable and accurate data acquisition and high quality interface modeling. Regarding the errors exclusively generated by FRF measurements, different sources of disturbances can be further distinguished: • Modification of the signal arising from data acquisition and signal processing [3, 4]. • Influence of external dynamic systems on the measured component (e.g. support mechanisms, attached measurement devices) [5]. • Random and systematic errors (e.g. environmental noise, sensor noise and positioning). Moreover, the uncertainty on the measurements can be highly amplified by the DS algorithm due to numerical instabilities and induces spurious peaks and inaccuracies in the coupling results [6, 7]. F. Trainotti ( ) · T. F. C. Berninger · D. J. Rixen Faculty of Mechanical Engineering, Technical University of Munich, Garching, Germany e-mail: francesco.trainotti@studenti.unitn.it; t.berninger@tum.de; rixen@tum.de © Society for Experimental Mechanics, Inc. 2020 A. Linderholt et al., Dynamic Substructures, Volume 4, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-12184-6_1 1
2 F. Trainotti et al. In particular, the use of standard measurement devices like piezo accelerometers in the FRFs acquisition process may not be suitable for applications requiring a high level of accuracy. Therefore, a valid alternative for motion detection can be identified in the use of laser interferometry, thanks to its non-invasive and high quality measurement technique. In this contribution, the potential of laser technology in the acquisition of FRFs within FBS is investigated. The reliability of laser measurements compared to those taken with standard accelerometers is evaluated through the analysis of the final coupling results. In Sect. 1.2, the theoretical background on Experimental Dynamic Substructuring is recalled, with particular focus on frequency-based approaches and interface modelling techniques. The laser vibrometry is shortly discussed in Sect. 1.3 along with its advantages and drawbacks over traditional measurement techniques. In Sect. 1.4, two measurement campaigns are carried out, one with accelerometers and the other with a laser Doppler vibrometer. A DS coupling is performed and the results of both case studies are compared. A brief summary of findings and conclusions is given in Sect. 1.5. 1.2 Experimental Dynamic Substructuring This section briefly reviews the theoretical concepts underlying the Experimental Dynamic Substructuring. In particular, a general overview of Frequency Based Substructuring and Virtual Point Transformation is provided in Sects. 1.2.1 and 1.2.2 respectively. 1.2.1 Frequency Based Substructuring The assembly procedure in the frequency domain according to a dual approach is named Lagrange Multiplier—Frequency Based Substructuring (LM-FBS) [1, 8, 9]. This method, which operates with admittance notation, evaluates locally a set of interface DoFs for each single substructure in the system and considers the interface forces as unknown variables. The aim of LM-FBS is to derive the admittance of the assembled systemYAB from the separate admittances of the two subsystems YA and YB. Consider the system depicted in Fig. 1.1. The subsystems’ admittances are known and the substructures’ DoFs are grouped in internal DoFs ((∗)A 1 and (∗) B 3 ) and interface DoFs ((∗) A 2 and (∗) B 2 ). The vectors of displacements, applied forces and reaction forces are denoted by u, f and g respectively. The governing equation of motion for the uncoupled system in the frequency domain is written in a compact form: u=Y(f +g) ⇒ ⎡ ⎢⎢ ⎣ uA 1 uA 2 uB 2 uB 3 ⎤ ⎥⎥ ⎦ = ⎡ ⎢⎢ ⎣ YA 11 Y A 12 0 0 YA 21 Y A 22 0 0 0 0 YB 22 Y B 23 0 0 YB 32 Y B 33 ⎤ ⎥⎥ ⎦ ⎛ ⎜⎜ ⎝ ⎡ ⎢⎢ ⎣ fA 1 fA 2 fB 2 fB 3 ⎤ ⎥⎥ ⎦ + ⎡ ⎢⎢ ⎣ 0 gA 2 gB 2 0 ⎤ ⎥⎥ ⎦ ⎞ ⎟⎟ ⎠ (1.1) Thematrix Y represents the admittance of the uncoupled system, built in a block diagonal form by the admittances of the subsystems YA and YB. Fig. 1.1 Assembly of subsystems A and B at the interface DoFs u2
1 Using Laser Vibrometry for Precise FRF Measurements in Experimental Substructuring 3 The knowledge of the uncoupled system at substructure level and the proper application of boundary conditions on the common interface allow to obtain the dynamical properties of the coupled system. The key step of the assembly procedure is the definition of interface conditions [9]: • Compatibility condition. • Equilibrium condition. The first condition defines the compatibility of displacements at the common boundary: Bu=0 ⇒ uB 2 −uA 2 =0, B= 0 −I I 0 (1.2) The Bmatrix or ‘signed Boolean matrix’ matches the interface DoFs, enforcing the continuity at the boundary. The second condition describes the force equilibrium between matching interface DoFs according to Newton’s action and reaction principle: g =−BTλ ⇒ ⎧ ⎪⎪⎪⎨ ⎪⎪⎪⎩ gA 1 =0 gA 2 =λ gB 2 =−λ gB 3 =0 (1.3) Here λis a set of Lagrange multipliers denoting the intensities of interface forces (reaction forces at the common boundary). The application of the two interface conditions Eqs. (1.2), (1.3) to the equation of motion in Eq. (1.1) leads to the following set of equations: u=Y(f −BTλ) Bu=0 (1.4) The continuity of displacements is directly enforced with the compatibility condition and the reaction forces defined via Lagrange multipliers automatically fulfill the equilibrium condition. Solving the system of Eq. (1.4) for λ: λ= BYBT −1 BYf (1.5) This result can be interpreted as follows: as a product of the excitationf, a gap BYf is formed between the still uncoupled subsystems’ interface. The interface force λ is applied in order to close this gap and keep the subsystems together. The stiffness operator between the applied force and the gap is called Interface Dynamic Stiffness, obtained by the inversion of the Interface Flexibility MatrixBYBT. The coupled response is then obtained substituting back Eq. (1.5) in Eq. (1.4): u=YABf, YAB = I −YBT BYBT −1 B Y (1.6) Note that, according to the dual formulation of the problem, the assembled admittanceYAB contains twice the interface DoFs and has the same size of the original uncoupled admittanceY. Hence, the redundant rows and columns may be removed when deemed necessary. 1.2.2 Virtual Point Transformation The application of the FBS coupling technique presented in Sect. 1.2.1 requires both compatibility and equilibrium conditions to be satisfied at the interface of the subsystems. In finite element models this two boundary conditions are easily imposed to every coinciding node. In experimental practice, this geometric coincidence can rarely be assured and therefore it is common practice to reduce the interface problem to one or more connecting points [9].
4 F. Trainotti et al. Fig. 1.2 Illustration of a virtual point interface connection. Red: Impacts; Blue: Triaxial sensors; Green: Virtual DoFs A simple 3 DoFs translational Single-Point Connection (SPC) completely neglects the rotational DoFs. The Equivalent Multi-Point Connection (EMPC) method was introduced to overcome this issue [10]: by coupling translational directions of multiple points in proximity of the interface, which is assumed to be fully rigid locally, rotations are implicitly accounted for. This approach results however in an overdetermination of the coupling problem, leading to unwanted ‘stiffening’ effects and numerical instabilities. Hence the Virtual Point Transformation method was proposed [2]: translational DoFs are projected into a subspace composed by six rigid Interface Displacement Modes (IDMs), retaining only the dynamics that load the interface in a purely rigid manner. The residual flexibility is left uncoupled and the interface problem is ‘weakened’. The interface dynamics is condensed into a single virtual point characterized by a 6 DoFs nodal description. In a typical experimental measurement setup, measured displacements and forces around the interface are non-collocated, thus making it infeasible to apply the coupling technique presented in Eq. (1.6). A reduction of the measurements in the proximity of the interface, assumed to be fully rigid, into the virtual point displacements q and virtual point forces mis necessary. The transformation of the uncoupled admittance Y from measured to virtual DoFs is performed: Yqm =TuYT T f (1.7) The matrices Tu and Tf apply the transformation and the Virtual Point Admittance Yqm represents the transformed FRF matrix in the matching generalized DoFs q and m. The coupled equation (Eq. 1.6) can so be rewritten in terms of virtual DoFs: q = I −YqmBT BYqmBT −1 B Yqmm (1.8) A back transformation from virtual to measured DoFs is then also possible. A brief derivation of the transformation of displacements and forces for the subsystem interface of Fig. 1.2 is described in Sects. 1.2.2.1 and 1.2.2.2. More details can be found in [11]. 1.2.2.1 Virtual Point Displacements The measured interface displacements u2 are projected onto the virtual point and rewritten as a function of the generalized coordinates q. Let us assume that only rigid modes, sorted in translational qv t = [q v x,q v y,q v z ] and rotational q v θ = [qv θx ,qv θy ,qv θz] components, compose the IDM subspace. The relation between u2 and q is geometrically obtained providing the orientation of the sensor axis Iek and the sensors distance from the virtual point Irk. The subscript (∗)v denotes the virtual point v, while (∗)k describes the triaxial sensor k, projected on the absolute frame I(∗).
1 Using Laser Vibrometry for Precise FRF Measurements in Experimental Substructuring 5 uk =( Ie k)T(qv t +q v θ ×I r k) =( Ie k)T ⎡ ⎢⎣ 1 0 0 0 Ir k z −Ir k y 0 1 0 −Ir k z 0 Ir k x 0 0 1 Ir k y −Ir k x 0 ⎤ ⎥⎦ ⎡ ⎢⎢ ⎢⎢ ⎢⎢ ⎢⎣ qv x qv y qv z qv θx qv θy qv θz ⎤ ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎦ =Rukq v (1.9) The reduction operator Ruk is a 3 ×6 matrix relating the sensor displacement uk to the generalized coordinates qv. The relation can be extended to all sensor channels u2 and all virtual points q: u2 =Ruq (1.10) This spatial reduction onto the IDM is based on the assumption of an almost rigid behaviour of the interface around the virtual point. Since in reality this condition is not always fully satisfied, a residual termμ, which represents the unprojected flexible motion, must be added: u2 =Ruq +μ (1.11) As the number of IDMs is typically lower than the number of interface DoFs, the problem is overdetermined and is handled with a least square procedure. To find the q that best approximates the measured response u2, a residual cost function, namely the squared error μTμ, has to be minimized. The least square projection is performed by applying the Moore-Penrose pseudo-inverse of Ru: q = RT uRu −1 RT uu2 =Tuu2 (1.12) In general, a weighting matrix can be used to gain more control over the error minimization by adjusting the importance of certain DoFs in the transformation [2, 11]. 1.2.2.2 Virtual Point Forces The derivation of the force reduction matrix is similar. The full set of input forces f2 has to be related to the generalized forces mv t =[mv x,mv y,mv z] and moments mv θ =[mv θx ,mv θy ,mv θz] . For one single impact f h with orientation Ie h at a distance Irh from the virtual point, the relation can be written as follows: mv = ⎡ ⎢⎢ ⎢⎢ ⎢⎢ ⎢⎣ mv x mv y mv z mv θx mv θy mv θz ⎤ ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎦ = ⎡ ⎢⎢ ⎢⎢ ⎢⎢ ⎢⎣ 1 0 0 0 1 0 0 0 1 0 −Ir h z Ir h y Ir h z 0 −Ir h x −Ir h y Ir h x 0 ⎤ ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎦ Ie hf h =RT fh f h (1.13) The operator RT fh is the6×1 matrix describing the reduction of input forces f h to the generalized forces mv. The extended formulation for all virtual points and the full set of forces can be written: m=RT ff2 (1.14) Note that Rf assumes the same exact formulation as Ru if a ‘collocated’ setup (sensor channels and forces in the same position and direction) is chosen. Unlike the displacements transformation, the problem is underdetermined and therefore a standard least square is not applicable. The goal is to find the forces ˜f2 that realize the generalized forces mwith a minimal ‘effort’, or in mathematical
6 F. Trainotti et al. language the solution to a minimization problem with Eq. (1.14) as constraint and the quadratic set of forces ˜f T 2 ˜f2 as cost function: ˜f2 =Rf(RT fRf)−1 m=TT fm (1.15) Similar to virtual displacements, it is also possible to introduce a weighting matrix, such that preferences in measurement data may be given [2, 11]. 1.3 Laser Technology The acquisition of high quality measurement data is a key step for a successful implementation of the coupling techniques described in Sect. 1.2. It is necessary to ensure maximum reliability and accuracy of the measured data and the results obtained by its subsequent processing. The choice of the measurement equipment is the first step in the delicate process of FRFs acquisition. For motion detection, a distinction can be made between attached and non-attached transducers [12]. The former are widely used in impact testing because of the large displacements in softly supported components induced by the impact. A typical response transducer of this type is the piezo accelerometer, whose mechanism exploits the piezoelectric effect: an internal vibrating mass applies a force, proportional to the acceleration of the mass itself, to a crystal, which generates a voltage. A large variety of sensor types based on different sensitivities, sizes and weights is available on the market. In addition, a broad dynamic range (up to 160 dB), a wide frequency range (from 0.5 Hz to 10 kHz) in common structural dynamics applications, high environmental resistance, easy operation and simple mounting methods are some of the advantages that make piezo sensors so popular in impact testing [5]. An alternative to the standard piezo devices is represented by the laser technology. The laser Doppler vibrometer (LDV) is a scientific instrument that uses light to perform non-invasive displacement and velocity measurements. The physical principle underlying this technology is the Doppler effect. The velocity measurements are based on the Doppler frequency shift of laser light that is reflected by the moving surface of the test object. A vibrometer generally uses a laser interferometer to measure the frequency difference between an internal reference beam and the measurement beam. A comprehensive review of the operating principles and applications of the LDV can be found in [13]. The choice of a non-intrusive measurement technique is mainly driven by the desire to reduce the influence of external dynamic systems on the measured component throughout the data acquisition process. Indeed, the presence of a physical device attached to the structure slightly modifies the overall system dynamics. Loading the structure can affect its mass, stiffness and damping. The dominant effect is mass loading: the spatial distribution and the magnitude of the masses of the sensors can actually lower the eigenfrequencies of the global system. A sensor positioned on an anti-node of a particular mode will highly contribute to the inertial energy of that mode. The effect of additional mass depends on the quantity of inertial energy already present in the mode. For this reason, the relative added mass effect is more significant at high frequencies [5]. In addition, the cables of the sensors as well as a loose sensor attachment and friction phenomena at the mounting interface can introduce damping into the system, altering the dynamic behaviour of the structure [5]. Other benefits of using laser technology compared to traditional triaxial accelerometers are: • The high level of optical sensitivity allows to obtain a high signal-to-noise ratio even at very low frequencies (0–30 Hz). • The non-contact technology does not experience the problem of cross-talk between sensors. • The ‘long-distance’ operating principle, combined with an extremely small laser target, makes it possible to measure areas that would otherwise be difficult to access. • In the VPT, for displacements a high degree of overdetermination of the problem can be guaranteed even for very small interfaces. Most of the disadvantages in using a LDV are associated with the large size and the limited flexibility of the equipment in the measurement setup. In addition, the laser instrumentation is significantly more expensive than standard accelerometers. With regard to measurements, large displacements of the measuring point may be a problem due to the non-contact nature of the transduction system. In Table 1.1, the main features of laser vibrometry and piezoelectric sensors are compared.
1 Using Laser Vibrometry for Precise FRF Measurements in Experimental Substructuring 7 Table 1.1 Properties of measurement devices in FBS Type Laser Doppler vibrometer Piezo accelerometer Accuracy ++ + Structure interaction ++ −− Ease of use − ++ Size − + Time-consuming − + Cost −− + Fig. 1.3 Benchmark structure for DS. Blue: Substructure A; Green: Substructure B 1.4 Measurements The analysis is performed on the benchmark structure depicted in Fig. 1.3, consisting of two subcomponents A and B. The former is an ‘L-shaped’ construction welded on a rectangular supporting plate, causing an overall very low damping. The latter is a simple free beam. The structure, made entirely of aluminium, is characterized by a small-sized and highly flexible connection interface, manufactured by means of a CNC milling process. A hex head screw and a locking nut guarantee a proper assembly of the substructures, which are in contact with each other in correspondence of the vertical and upper horizontal flange. A tightening torque of 20 Nm is applied. The admittance models are developed for each subsystem from both an experimental and a numerical point of view. For the construction of the experimental FRF model, two case studies are carried out on the basis of the adopted response transduction system: one involves the use of triaxial accelerometers, the other exploits the laser technology. A numerical simulation is performed in parallel with the purpose of synthesizing the necessary FRFs starting from the modal properties of the system. Additionally, validation measurements are taken on the assembled structure. The experimental modelling is briefly described in Sect. 1.4.1. The application of the VPT is addressed in Sect. 1.4.2 and the LM-FBS coupling results are shown and discussed in Sect. 1.4.3. 1.4.1 Measurement Setup The substructure Ais fixed to a vibration-free table, component B is freely suspended in the air by elastic ropes. Particular focus is on the support setup of the free beam for the laser application: on the one hand, it is intended to ensure adequate flexibility and low friction of the support system; on the other hand, the actual movement of the structure has to be minimized in order to guarantee the stability of the laser focal point during the acquisition process. Note that a continuous change in the position of the measurement point during the acquisition may generate distortion around FRF antiresonances. The experimental FRF model is constructed by impact measurements. The source of excitations is an automatic modal hammer designed by the Chair of Applied Mechanics at the TU Munich equipped with a steel tip [14]. The response transducers are the 10 mV/g triaxial piezo accelerometers by Kistler in one case and a laser Doppler vibrometer, namely the RSV-150 Remote Sensing Vibrometer by Polytec, in the other case. In laser measurements, an adequate focal distance, a good isolation and the use of reflective tapes are some of the measures taken to maximize the signal-to-noise ratio.
8 F. Trainotti et al. Fig. 1.4 Layout of impacts and responses at the interface for VPT. Red: Impacts; Blue: Responses; Green: Virtual Point. (a) Component A—Piezo. (b) Component A—Laser. (c) Component B—Piezo. (d) Component B—Laser 1.4.2 Virtual Point Reduction The dynamics of the interface is condensed into a virtual point that is located in the center of the hole on the connecting surface of the two substructures. The placement and number of impacts and responses around the virtual point is chosen to ensure the observability of the entire set of selected virtual DoFs and a high quality transformation for both case studies [9, 11]. In the measurement campaign with the triaxial sensors the interface dynamics is reduced into a subspace composed by six rigid IDMs (three translations and three rotations). For this purpose, three sensors and eight excitation points are located on the interface of both substructures (Fig. 1.4a, c). The mono-dimensional nature of the available laser vibrometer, on the other hand, leads to the choice of a single-plane measurement campaign. Indeed, the FRFs are acquired exclusively along the vertical axis and consequently only the dynamics related to the vertical translational DoF and the two out-of-plane rotations is retained in the transformation. A total of six output signals and six impacts are used to describe the virtual point DoFs for each subcomponent (Fig. 1.4b, d). 1.4.3 Coupling Results The transformed FRF admittances YA qm and Y B qm are coupled according to LM-FBS (Eq. 1.8). An effective comparison between the two measurement campaigns is achieved by coupling only the dynamics associated with the translational DoF along the vertical axis. The experimental coupled FRF is plotted together with the correspondent assembly validation FRF for both case studies. To assess the accuracy of the experimentally coupled FRF, a cross-validation is performed by comparing the data with simulated results, in which only the vertical translation over the interface area is coupled. The outcome in magnitude and phase is shown in Fig. 1.5a,b. A focus on lower frequencies is depicted in Fig. 1.6a,b.
1 Using Laser Vibrometry for Precise FRF Measurements in Experimental Substructuring 9 Fig. 1.5 Experimental and numerical coupled FRF with validation. Frequency range [0−3000 Hz]. Only the dynamics associated with the vertical translational DoF is coupled for comparative purposes. (a) First case—piezo accelerometer. (b) Second case—laser Doppler vibrometer
10 F. Trainotti et al. (a) (b) Fig. 1.6 Experimental and numerical coupled FRF with validation. Frequency range [0−200 Hz]. Only the dynamics associated with the vertical translational DoF is coupled for comparative purposes. (a) First case—piezo accelerometer. (b) Second case—laser Doppler vibrometer
1 Using Laser Vibrometry for Precise FRF Measurements in Experimental Substructuring 11 The following observations are made: • Both experimental coupling results match very well numerical data in magnitude and phase up to a high frequency range (Fig. 1.5a,b). Note that the laser seems to be more accurate at very low (0−−30 Hz) and very high (above 2000 Hz) frequencies. In particular, the capability of the laser to measure static or quasi-static responses is highlighted in Fig. 1.6a,b. • The validation of the coupled FRF with the assembled FRF is highly affected by the rigid IDMs left uncoupled (two translations and three rotations) and the residual flexibility. This result is expected as the uncoupled horizontal DoF is highly correlated with the vertical one and plays an important role in most modes of the assembled structure. • The discrepancies in the shape of the coupled FRF (in the position of antiresonances) between piezo and laser measurements depend mainly on the different impact location in the observed FRF (Fig. 1.5a,b). In addition, the different dynamics acquired at the interface for the VPT in the two cases may slightly affect the coupling results. The added mass effect in piezo measurements is minimal due to the small number of lightweight sensors used. 1.5 Conclusions In this paper, the use of laser vibrometry in the context of experimental LM-FBS coupling via VPT is investigated. The noncontact measurement technique allows to minimize the disturbances in the overall system dynamics related to the presence of a physical device mounted on the measured component. In this sense, the use of a non-intrusive approach in sensing motion is strongly suggested when dealing with small-sized, lightweight structures. A further advantage of laser technology over standard accelerometers is the capability to reach inaccessible areas and measure a large number of points. A comparison between the coupling results of the two different measurement approaches is provided. Although the experimental data acquired with both accelerometers and LDV fit very well with the simulated data, the laser reveals great accuracy over a broader frequency range. Additional analysis can be conducted on more complex applications (e.g. 3D interface coupling) to explore the potential of laser vibrometry in FBS. References 1. De Klerk, D., Rixen, D.J., Voormeeren, S.: General framework for dynamic substructuring: history, review and classification of techniques. AIAAJ. 46.5, 1169–1181 (2008) 2. Van der Sejis, M., Van der Bosch, D., Rixen, D., De Klerk, D.: An improved methodology for the virtual point transformation of measured frequency response functions in dynamic substructuring. In: COMPDYN (2013) 3. Lyons, R.G.: Understanding Digital Signal Processing, 1st edn. Addison-Wesley Longman, Boston (1997) 4. Brown, D.L., Allemang, R.J., Phillips, A.W.: Forty years of use and abuse of impact testing: a practical guide to making good FRF measurements. In: Experimental Techniques, Rotating Machinery, and Acoustics. Proceeding of the Society for Experimental Mechanics, vol. 8 (2015) 5. Døssing, O., (Firm), B.K.: Structural Testing: Mechanical mobility measurements, pt. 1. Bruël & Kjær, Nærum (1988) 6. Rixen, D.: How measurement inaccuracies induce spurious peaks in frequency based substructuring. In: Proceedings of the Twenty Sixth International Modal Analysis Conference, Orlando, FL. Society for Experimental Mechanics, Bethel (2008) 7. Voormeeren, S.N., De Klerk, D., Rixen, D.J.: Uncertainty quantification in experimental frequency based substructuring. Mech. Syst. Signal Process. 24, 106–118, (2010) 8. De Klerk, D., Rixen, D.J., De Jong, J.: The frequency based substructuring (FBS) method reformulated according to the dual domain decomposition method. In: 24th International Modal Analysis Conference, St. Louis (2006) 9. Van der Sejis, M.: Experimental Dynamic Substructuring. PhD thesis, Delft University of Technology (2016) 10. De Klerk, D.: Solving the RDoF problem in experimental dynamic substructuring. In: Proceedings of the 26th International Modal Analysis Conference (IMAC) (2008) 11. Häußler, M., Rixen, D.: Optimal transformation of frequency response functions on interface deformation modes. In: Dynamics of Coupled Structures, vol. 4, pp. 25–237. Springer, Cham (2017) 12. Company, H.P.: The Fundamentals of Modal Testing: Application Note 243-3. Hewlett Packard Company, Palo Alto (1986) 13. Rothberg, S., Allen, M., Castellini, P., DiMaio, D., Dirckx, J., Ewins, D., Halkon, B., Muyshondt, P., Paone, N., Ryan, T., Steger, H., Tomasini, E., Vanlanduit, S., Vignola, J.: An international review of laser Doppler vibrometry: making light work of vibration measurement. Opt. Lasers Eng. 99, 11–22 (2017). https://doi.org/10.1016/j.optlaseng.2016.10.023 14. Maierhofer, J., El Mahmoudi, A., Rixen, D.J.: Development of a lowcost automatic modal hammer for applicaions in substructuring. In: Proceedings of the 37th International Modal Analysis Conference (IMAC) (2019)
Chapter 2 A Priori Interface Reduction for Substructuring of Multistage Bladed Disks Lukas Schwerdt, Lars Panning-von Scheidt, and Jörg Wallaschek Abstract When analyzing the dynamics of bladed disks in turbomachinery, most methods focus on a single stage at a time because of the challenges associated with multistage structures. Whereas the cyclic symmetry of individual bladed disks is commonly exploited to yield great savings of computational effort, multistage rotors lack this symmetry due to the differing number of blades in each stage. Substructuring methods can be used to overcome this problem but they still face challenges with non-conforming finite element meshes at the interface between stages. Some state of the art methods expect the nodes at the interface to be arranged in concentric rings and use a truncated Fourier series as basis for the displacement along each ring of nodes. In this paper, a reduction basis for the interface degrees of freedom between adjacent stages is proposed which uses polynomial basis functions in the radial direction in addition to a truncated Fourier series in the circumferential direction. This enables coupling the substructures of multiple stages with arbitrary meshes. Additionally, the resulting reduced order model (ROM) can be smaller while preserving accuracy. The proposed interface reduction is demonstrated in conjunction with a cyclic Craig-Bampton (CB) reduction of each stage. Different ROMs are compared to show the impact of the CB reduction as well as the interface reduction. Keywords Model order reduction · Component mode synthesis · Multistage · Interface reduction · Mistuning Nomenclature n Number of degrees of freedom x Physical degrees of freedom y Degrees of freedom in travelling wave coordinates F DFTmatrix H Fourier harmonic basis function I Identity matrix K Stiffness matrix M Mass matrix N Number of blades P Polynomial basis function T Transformation/reduction matrix V Interface reduction basis function W Matrix of interface reduction basis η Reduced/modal degrees of freedom ϕ Mode/eigenvector Matrix of fixed interface modes Matrix of constraint modes (r,α) (Polar) coordinates on the interface h Substructure harmonic index j Interface harmonic index l Mode index k Stage index L. Schwerdt ( ) · L. Panning-von Scheidt · J. Wallaschek Institute of Dynamics and Vibration Research, Faculty of Mechanical Engineering, Leibniz University Hannover, Hannover, Germany e-mail: schwerdt@ids.uni-hannover.de © Society for Experimental Mechanics, Inc. 2020 A. Linderholt et al., Dynamic Substructures, Volume 4, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-12184-6_2 13
14 L. Schwerdt et al. m Master/interface degrees of freedom s Slave/inner degrees of freedom ˜ Reduced model 2.1 Introduction Finite element models are a vital part of the product development process in turbomachinery to predict the vibratory response, but detailed models of multistage rotors are prohibitively large. Therefore models of single stages are used predominantly although the importance of multistage effects is known [1]. Increasingly more methods are developed to generate reduced order models for multistage assemblies [2–6]. Most use a variant of Component Mode Synthesis (CMS) to split the rotor into substructures, which are reduced separately. This can reduce the amount of memory required for an analysis and allows to perform multiple analyses efficiently using the ROM. In this paper, a model order reduction method for multistage bladed disks is presented, that uses CMS with the CraigBampton reduction (CB-CMS) with a priori interface reduction. As basis functions for the interface motion, a product of Fourier harmonics in circumferential direction and polynomials in radial direction is used. First, the CB-CMS method is presented by itself and then with interface reduction. afterwards, an overview of the existing model reduction methods for multistage bladed disks is given before the proposed method is presented. Finally, the new method is demonstrated on an academic two stage rotor. 2.2 Craig-Bampton Method and Interface Reduction One method to generate reduced order models is the Component Mode Synthesis. It is widely used in turbomachinery and other applications, by itself and as a basis for more advanced model order reduction methods. In the CMS method, the structure is split up in to multiple components, also known as substructures. Each of the components is then reduced individually and the reduced components are assembled to yield the reduced model of the complete structure. Different methods for the reduction and assembly are available, cf. [7]. 2.2.1 Craig-Bampton Method The most popular of these methods is the Craig-Bampton reduction[8]. Here the degrees of freedom (DOF) of each substructure are split intomaster (xm) and slave (xs) DOF and the stiffness and mass matrices are partitioned accordingly: K= Kmm Kms Ksm Kss M= Mmm Mms Msm Mss (2.1) The master DOF are kept in the reduced model while a truncated set of modal DOF (ηs) represent the slave DOF in the reduced system: xm xs = I 0 Ψ Φ xm ηs =T xm ηs (2.2) The constraint modes Ψ and fixed-interface normal modes Φ= ϕ1, ϕ2, . . . are obtained by: Ψ =−K−1 ss Ksm Kss −ω 2 i Mss ϕi =0 (2.3) The reduced stiffness and mass matrices are ˜K=T H KT ˜M=T H MT (2.4)
2 A Priori Interface Reduction for Substructuring of Multistage Bladed Disks 15 To facilitate the assembly of the reduced substructures, the DOF on the interface between adjacent substructures are selected as master DOF and kept in the reduced model. For this reason the modes in Φare called the fixed-interface normal modes although additional DOF can be selected as master DOF if desired. The biggest disadvantage of the regular CB-method is, that the interfaces between substructures are not reduced. This leads to two problems: firstly, the number of DOF of the reduced system is limited by the size of the interface. Secondly the reduced matrices and the matrix of constraint modes can contain more non-zero entries than the (sparse) matrices of the unreduced system (cf. [9]). This represents a major problem, making the regular CB-method unattractive for very large models. 2.2.2 Craig-Bampton Method with Interface Reduction To alleviate this problem, various interface reduction methods were developed [10–18]. Interface reduction methods can be classified into methods that reduce the interface prior to the reduction of the slave DOF or after the application of the regular CB-method. Methods that reduce the interface first avoid the problems associated with the partially reduced system altogether, whereas methods that use the reduced model of the regular CB-method generate vastly more computational effort for large-scale systems during the reduction process. Therefore, a prior interface reduction is preferred. One of these methods uses arbitrary assumed displacements of the interface as basis functions for the reduction basis. Carassale and Maurici [12] call this the Prior Basis Function Method. Using the Matrix W, where each column is one basis function evaluated at all interface DOF, the complete reduction is xm xs = I 0 Ψ Φ W 0 0 I ηm ηs = W 0 ΨW Φ ηm ηs (2.5) with ΨW=−K−1 ss (KsmW) (2.6) In Eq. (2.6) it is most efficient to perform the interface reduction first as indicated by the parenthesis. Some important properties that should be considered when selecting the basis functions: • To yield a reduced model of small size, a small number of basis functions should be able to represent the actual interface displacements in the frequency of interest. Optimally the interface portion of the modeshapes of the full system would be used if they were available. • The basis functions should be easy to evaluate. • The basis should be orthogonal to ensure the reduced matrices are not singular. Carassale and Maurici [12] use the GramSchmidt procedure to orthogonalize a general basis. It should be noted that perfect orthogonality is not necessary as long as the basis functions are linearly independent with enough margin to avoid numerical problems. The method allows arbitrary basis functions, but due to their simplicity polynomial bases were used previously[12, 16]. Because of the freedom afforded by the method to choose arbitrary functions, problem specific functions can be superior to multidimensional polynomials. As shown in the next section, Fourier basis functions are useful for systems with cyclic symmetry of the substructures [2]. The newly proposed method uses a combination of polynomials and Fourier basis functions. 2.3 Reduced Models of Multistage Bladed Disks Rotors consisting of multiple bladed disks, common in turbomachinery applications, represent a special case of mechanical system due to the cyclic symmetry of each bladed disk. This symmetry is commonly exploited to reduce the computational effort when analyzing such systems.
16 L. Schwerdt et al. 2.3.1 Single Stage Bladed Disks Single bladed disks are cyclic symmetric structures. This results in system matrices that are block circulant. By introducing travelling wave coordinates the matrices become block diagonal. This is only possible for identical disk sectors and blades. When deviations among the blades (mistuning) are considered, the matrices in travelling wave coordinates are fully populated in general. Nonetheless, some reduction methods for mistuned bladed disks make use of this representation as well (cf. [19]). The popular SNM method for example directly uses the modes of the tuned (i.e. cyclic symmetric) structure as a reduction basis[20]. The transformation from the physical coordinates x into travelling wave coordinates y is performed according to x =(F ⊗I)y (2.7) where F is the complex DFT matrix with a size of the number of sectors n and I is the identity matrix with a size of the number of DOF per sector. In travelling wave coordinates all analyses can be performed on the sector level i.e. one can don calculations where the matrices have the size of the number of DOF of a single sector. 2.3.2 Multistage Bladed Disks Due to the variable number of blades among the different stages, a multistage rotor does not in general exhibit cyclic symmetry. Therefore, other methods have to be applied to generate a reduced order model of the rotor. These methods rely on CMS and some exploit the cyclic symmetry of individual stages on the substructure level. Sternchüss uses individual sectors as components and couples them through intermediate rings which are only one layer of elements thick [21]. Cyclic symmetry is not exploited. Laxalde presented the multi-stage cyclic symmetry approach [3]. Here the coupling between adjacent stages is reduced to enable exploiting the computational advantages of cyclic symmetry. When representing each stage in travelling wave coordinates, in general all harmonic indices of adjacent stages are coupled together [4]. There can be exceptions depending on the number of blades. The multi-stage cyclic symmetry approach discards most of these connections, but retains the one belonging to the same number of nodal diameters. Thereby the complete rotor model is decomposed into systems with a size of one sector from every stage each. The resulting modes are similar to the actual system modes, but not identical. Nonetheless, they represent a good reduction basis to project the full system into. Disadvantages are the difficulty of connecting adjacent stages with non-matching meshes at the interface and the fact that the substructures are relatively big, as they have a size of multiple sectors, one from each stage. Song uses a CB-CMS approach where each harmonic index of each stage is a single substructure [2]. A Fourier basis is used to reduce the interfaces. To facilitate this, the interface is assumed to be ring shaped and meshed with concentric rings of nodes. For each of these rings, truncated Fourier series are used to represent the displacement along each direction. Thereby the number of interface DOF is reduced significantly, while retaining the substructure size of a single sector from single stage analyses. Using this prior basis function method adjacent sectors do not need matching meshes at the interface in circumferential direction. This is advantageous because the number of elements in circumferential direction for each stage needed to achieve a matching mesh is the least common multiple of the number of blades of the stage and its neighbors, e.g. 10,100 for a two stage design with 100 and 101 blades. Currently, Song’s method for the interface reduction is used with more advanced model order reduction methods for multistage assemblies [6, 22]. The interface reduction was later called Fourier Constraint Modes (FCM). 2.4 Proposed Model Order Reduction Method The proposed reduction method for multistage bladed disks uses CB-CMS with an a priori interface reduction based on the FCM method. The rotor is split into individual stages. The ring shaped interfaces between the stages are reduced using basis functions defined a priori. These functions consist of Fourier harmonics in the circumferential direction and polynomials in the radial direction. A cyclic Craig-Bampton reduction is then applied to each stage.
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