River Rapids Conference Proceedings of the Society for Experimental Mechanics Series Nonlinear Structures & Systems, Vol. 1 Ludovic Renson Robert Kuether Paolo Tiso Drithi Shettya Proceedings of the 43rd IMAC, A Conference and Exposition on Structural Dynamics 2025 River Publishers
Conference Proceedings of the Society for Experimental Mechanics Series Series Editor Kristin B. Zimmerman Society for Experimental Mechanics, Inc., Bethel, USA i
The Conference Proceedings of the Society for Experimental Mechanics Series presents early findings and case studies from a wide range of fundamental and applied work across the broad range of fields that comprise Experimental Mechanics. Series volumes follow the principle tracks or focus topics featured in each of the Society’s two annual conferences: IMAC, A Conference and Exposition on Structural Dynamics, and the Society’s Annual Conference & Exposition and will address critical areas of interest to researchers and design engineers working in all areas of Structural Dynamics, Solid Mechanics and Materials Research. ii
Ludovic Renson • Robert Kuether • Paolo Tiso • Drithi Shetty Editors Nonlinear Structures & Systems, Vol. 1 Proceedings of the 43rd IMAC, A Conference and Exposition on Structural Dynamics 2025 River Publishers
Published, sold and distributed by: River Publishers Broagervej 10 9260 Gistrup Denmark www.riverpublishers.com ISBN 97887-438-0146-7 (Hardback) ISBN 97887-438-0158-0 (eBook) https://doi.org/10.13052/97887-438-0146-7 Conference Proceedings of the Society for Experimental Mechanics An imprint of River Publishers © The Society for Experimental Mechanics, Inc. 2025 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 Nonlinear Structures & Systems represents one of twelve volumes of technical papers presented at the 43rd IMAC, A Conference and Exposition on Structural Dynamics, organized by the Society for Experimental Mechanics, and held February 10-13, 2025. The full proceedings also include volumes on Dynamics of Civil Structures; Model Validation and Uncertainty Quantification; Dynamic Substructuring & Transfer Path Analysis; Special Topics in Structural Dynamics & Experimental Techniques; Computer Vision & Laser Vibrometry; Dynamic Environments Testing; Sensors & Instrumentation and Aircraft/Aerospace Testing Techniques; Topics in Modal Analysis & Parameter Identification Iⅈ Data Science in Engineering; and Structural Health Monitoring & Machine Learning. Each collection presents early findings from experimental and computational investigations on an important area within Structural Dynamics. Nonlinearity is one of these areas. The vast majority of real engineering structures behave nonlinearly. In many cases, nonlinearities can even be exploited to achieve novel functionalities. Therefore, it is necessary to consider nonlinear effects in all of the engineering design steps, both in the experimental analysis tools (so that the nonlinear parameters can be correctly identified) and in the mathematical and numerical models of the structure (to run accurate simulations). In so doing, it is possible to create models representative of the reality that can be used for better predictions once validated. Several papers address theoretical and numerical aspects of nonlinear dynamics (covering rigorous theoretical formulations and robust computational algorithms) as well as experimental techniques and analysis methods. There are also papers dedicated to nonlinearity in practice where real-life examples of nonlinear structures are discussed. The organizers would like to thank the authors, presenters, session organizers, and session chairs for their participation in this track. Editors: Ludovic Renson–Imperial College London, UK; Robert Kuether–Sandia National Laboratories, USA; Paolo Tiso–ETH Zu¨rich, Switzerland, Drithi Shetty – University of South Florida, USA. v
Contents 1 Linear and Nonlinear Response of a Continuous-Interface BARC: FEA Prediction and Test 1 Sean Magoffin, Brooklyn Andrus, Matthew S. Allen, and Raymond Joshua 2 www.FreeDyn.at A free, GUI based multi-body dynamics software for stand-alone use and with C API for linking it to Python, Matlab or custom code 15 Wolfgang Witteveen and Stefan Oberpeilsteiner 3 Nonlinear Frequency Response Surfaces, Part I: Theoretical Formulation 21 D. Di Maio, Jip van Tiggelen, and Marcel H.M. Ellenbroek 4 Fatigue Damage Potential in a Nonlinear Dynamical System with Isolated Resonances 27 Brian Evan Saunders 5 Robust Optimization of Surface Topology for Jointed Connections 31 Aalokeparno Dhar, Justin H. Porter, and Matthew R. W. Brake 6 Nonlinear Vibration Control on Complex Structures via Model Predictive Control Exploiting Koopman Model 41 Yichang Shen and Ludovic Renson 7 Stochastic Dynamics and Surrogate Modeling for Nonlinear Aerospace Nozzle Systems with Partial Observations 45 Evange´line Capiez-Lernout, Olivier Ezvan, and Christian Soize 8 Industrial Applications of Nonlinear Modal Modeling: Two Case Studies 49 Benjamin R Pacini 9 An Overview of ANSYS Harmonic Balance Method: Current Capabilities and Future Directions 71 Furkan K. C¸ elik, Taylan Karaag˘ac¸lı, M. Bu¨lent O¨ zer, and H. Nevzat O¨ zgu¨ven 10 Derivative-Free Arclength Control-Based Continuation for Secondary Resonances Identification 77 Michael Kwarta 11 Modelling a Nonlinear Single-Degree-of-freedom System from Experimental Data using Lagrangian Neural Networks 83 Alexandre Spits, Ghislain Raze, and Gae¨tan Kerschen 12 Computing Periodic Responses of Highly Flexible Beams Modelled under the SE(3) Lie Group Framework with Multiple Shooting 87 Alan Xavier and Ludovic Renson 13 Direct Parameterization of Invariant Manifolds 94 Amir K. Bagheri, Valentin Sonneville, and Ludovic Renson 14 Nonlinear Solvers for Computing Periodic Solutions 96 Tiago S. Martins, F. Trainotti, L. Kreuzer, and D. J. Rixen 15 Application and Discussion of Results 98 ˙Izzet Ulug˘ and Ender Cigeroglu vii
viii Contents 16 Flutter Analysis of Control Surface Free-play by Using a 3-DOF Airfoil Model 103 Cristiano Martinelli and Andrea Cammarano 17 The Nonlinear Restoring Force Method: An Overview with Numerical and Experimental Applications 111 Dersu Celiksoz and Ender Cigeroglu 18 Finding the Limits of Beam Elements: Modeling Bolted Joints as an Effective Joint Region 129 Nicholas Pomianek, Casey Whitworth, Enrique Gutierrez-Wing, Trevor Jerome, and J. Gregory McDaniel 19 Design of a Tuned Vibration Absorber with Friction Contact for High Location Ratio Application 137 Matthew W. Hancock, Nathan P. Lovell, Yusuf R. Shehata, Keegan J. Moore, Bogdan I. Epureanu, Thomas N. Thompson, and Sean T. Kelly 20 Topology Optimization of Isolated Response Curves in 3D Geometrically-nonlinear Beam 157 Enora Denimal Goy, Yichang Shen, Samuel Fruchard, Adrien Me´lot, and Ludovic Renson 21 Coupled Harmonic Balance based approach for the non-linear dynamics of spur gear pairs 161 Giacomo Saletti, Giuseppe Battiato, and Stefano Zucca 22 Multi-level Stochastic Model Identification of Complex Aeroelastic Systems Considering Aerodynamic Nonlinearities 171 Michael McGurk and Jie Yuan 23 Resonance Tracking of Nonlinear Structures by Amplitude Controlled Sine-Dwell Testing 175 Gleb Kleyman, Hans-Ju¨rgen Borutta, and Sebastian Schwarzendahl
Chapter 1 Chapter 1 On the Detection and Quantification of Nonlinearity via Statistics of the Gradients of a Black-Box Model Georgios Tsialiamanis and Charles R. Farrar Abstrac t Detection and identification of nonlinearity is a task of high importance for structural dynamics. On the one hand, identifying nonlinearity in a structure would allow one to build more accurate models of the structure. On the other hand, detecting nonlinearity in a structure, which has been designed to operate in its linear region, might indicate the existence of damage within the structure. Common damage cases which cause nonlinear behaviour are breathing cracks and points where some material may have reached its plastic region. Therefore, it is important, even for safety reasons, to detect when a structure exhibits nonlinear behaviour. In the current work, a method to detect nonlinearity is proposed, based on the distribution of the gradients of a data-driven model, which is fitted on data acquired from the structure of interest. The data-driven model selected for the current application is a neural network. The selection of such a type of model was done in order to not allow the user to decide how linear or nonlinear the model shall be, but to let the training algorithm of the neural network shape the level of nonlinearity according to the training data. The neural network is trained to predict the accelerations of the structure for a time-instant using as input accelerations of previous time-instants, i.e. one-step-ahead predictions. Afterwards, the gradients of the output of the neural network with respect to its inputs are calculated. Given that the structure is linear, the distribution of the aforementioned gradients should be unimodal and quite peaked, while in the case of a structure with nonlinearities, the distribution of the gradients shall be more spread and, potentially, multimodal. To test the above assumption, data from an experimental structure are considered. The structure is tested under different scenarios, some of which are linear and some of which are nonlinear. More specifically, the nonlinearity is introduced as a column-bumper nonlinearity, aimed at simulating the effects of a breathing crack and at different levels, i.e. different values of the initial gap between the bumper and the column. Following the proposed method, the statistics of the distributions of the gradients for the different scenarios can indeed be used to identify cases where nonlinearity is present. Moreover, via the proposed method one is able to quantify the nonlinearity by observing higher values of standard deviation of the distribution of the gradients for lower values of the initial column-bumper gap, i.e. for “more nonlinear” scenarios. Keyword s Structural health monitoring (SHM) · Structural dynamics · Nonlinear dynamics · Machine learning · Neural networks 1.1 Introduction In the pursuit of making everyday life safer, humans have extensively tried to model the environment around them. Structures are an important part of the environment, in which humans live. They are man-made and should be safe throughout their lifetime. Structures are exposed to numerous environmental factors, which may cause them to fail. Moreover, during operation, structures are subjected to dynamic loads, which, in time, may cause failure. Such failures will most probably result in economic damage to society and may even result in loss of human lives. Therefore, for the purpose of maintaining structures safe, the field of structural health monitoring (SHM) [1] has emerged. G. Tsialiamanis ( ) Dynamics Research Group, Department of Mechanical Engineering, University of Sheffield, Sheffield, UK e-mail: g.tsialiamanis@sheffield.ac.uk C. R. Farrar Engineering Institute, MS T-001, Los Alamos National Laboratory, Los Alamos, NM, USA e-mail: farrar@lanl.gov © The Society for Experimental Mechanics, Inc. 2024 M. R. W. Brake et al. (eds.), Nonlinear Structures & Systems, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-36999-5_1 1 Linear and Nonlinear Response of a Continuous-Interface BARC: FEA Prediction and Test Sean Magoffin, Brooklyn Andrus, Matthew S. Allen, and Raymond Joshua Abstract This paper presents the results of finite element modeling and linear and nonlinear testing of a BARC structure with a continuous interface (as opposed to the split-interface BARC). Before the actual hardware was constructed, blueprints were used to create a FEM in which the bolts were modeled in detail, including contact at the interfaces due the expected preload based on the bolt torque specs. This model was used to predict the linearized natural frequencies, to infer which modes might exhibit nonlinearity, and then quasi-static modal analysis was used to predict the amplitude dependent natural frequency and damping for the modes that were expected to exhibit the strongest nonlinearity. Once the FEM predictions were mostly complete, a BARC was constructed based on the blueprints and tests were performed to measure its linear modes and characterize those modes that exhibit nonlinear behavior. The linear modes measured in test are compared to those computed from the FEM about the preloaded state. Additionally, for those modes where significant nonlinearity was observed, quasi-static modal analysis (QSMA) was used to predict their amplitude dependent natural frequencies and damping ratios. Keywords BARC· Quasi-static modal analysis · Nonlinearities · FEA Introduction The Box Assembly with Removable Component (BARC) structure was created in 2017 to provide a common benchmark structure to evaluate dynamic environment testing methods [1]. State of the art environmental testing methods do not consider nonlinearity, although one could argue that they seek to capture it by testing the component of interest at high enough levels that any nonlinearities would be observed. In any event, while many studies have tested and modeled the BARC over the past decade, very few focused on its nonlinear behavior. On the other hand, because the removable component consists of three parts bolted together, it can behave nonlinearly and experimental campaigns have noted this since it was introduced. This work presents a first effort to characterize the nonlinear behavior of the continuous-interface BARC structure, both experimentally and using nonlinear finite element simulations. This study is a complement to the various studies that have used experimental modal analysis to extract the linear modes of the BARC and compared those with modes from finite element models. For example, Alvis and Schoenherr [2] used the BARC to perform a detailed study on the best practices for modeling bolted interfaces. Their study, as with others, Honeywell Federal Manufacturing & Technologies, LLC operates the Kansas City National Security Campus for the United States Department of Energy / National Nuclear Security Administration under Contract Number DE-NA0002839. UUR – NSC-614-6620 Sean Magoffin· Brooklyn Andrus Undergraduate Students, Brigham Young University, Department of Mechanical Engineering e-mail: sm996@byu.edu; brookand@byu.edu Matthew S. Allen Professor, Brigham Young University, Department of Mechanical Engineering e-mail: matt.allen@byu.edu Raymond Joshua Sr. Mechanical Engineer, Advanced Development Domain (ADD), Kansas City National Security Campus e-mail: rjoshua@kcnsc.doe.gov © The Author(s), under exclusive license to River Publishers 2025 Ludovic Renson et al. (eds.), Nonlinear Structures & Systems, Vol. 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.13052/97887-438-0146-7 1
2 Sean Magoffin et al. [3][4][5][6] focused on the low amplitude, linearized behavior. Several studies have had good success in creating finite element models that correlated very well with measurements of the natural frequencies and mode shapes of the BARC, although most have noted that care had to be taken in modeling the bolted connections appropriately. The authors found very few studies that quantified the nonlinear behavior of the BARC. Takeshita’s thesis [7] and some precursor papers did explore this somewhat, in particular the randomly excited response of the BARC to a base input. No other detailed studies were found before this paper had to be submitted. In this work the BARC is tested and simulated in free-free conditions. Note also that this study focuses on the continuous-interface BARC, while many prior studies have focused on the split-interface BARC. In undertaking this study, the authors began with blueprints of the BARC structure, and the finite element models were created from those before any hardware had been machined. The authors did have the benefit of comparing their results to other papers that had also modeled this structure, but other than that the predictions were blind in that the linear and nonlinear testing was not performed until after the finite element models had been created and debugged. The FEM included a simplified, three-dimensional model of the bolts, which were preloaded based on the torque specified in the assembly drawings. The FEM was created in Abaqus, which solved the nonlinear contact problem to obtain the preloaded state of the bolts and contact between the components. Linear modal analysis was performed about this preloaded state. This produced a prediction of the linearized natural frequencies of the BARC that didn’t depend on model updating. Similarly, the nonlinearity in the BARC was predicted using quasi-static modal analysis [8], which consists of applying static loads in the shape of a mode of interest in order to predict its amplitude dependent natural frequency and damping ratio. This was also performed before any nonlinear tests had been performed. Hence, the paper presents an idea of the predictions that might be obtained with these tools based only on the nominal design of the components. These predictions are then compared with the actual test measurements. The following section details the finite element model that was created, the validation steps that were performed, and the linear and nonlinear results obtained. Then Section 3 discusses the manufacturing of the actual BARC and the linear and nonlinear tests that were performed. The FEM and test results are then compared and conclusions are presented. Creating a Finite Element Model of the BARC A 3D finite element model (FEM) of the BARC was created using Abaqus from the provided drawings. The joints were modeled by creating simplified 3D bolts that were meshed with solid elements, as was found to work well in [2] as well as in the authors’ prior work [9, 10]. The structure was preloaded by applying a bolt load in Abaqus to the middle surface of each bolt (see Figure 1) and then fixing the bolt length in the subsequent step. The specified torque values were converted to preload values using the plot of preload force versus torque found by Wall et al. in [11]. Furthermore, in [10] the best agreement with measurements was found when using a friction coefficient of 0.2, so this was used in Abaqus even though this is a lower value than typically expected for steel-on-steel contact. (a) (b) Fig. 1 (a) The surface that the bolt load is applied to is highlighted in red. (b) Yellow arrows represent the applied preload to those surfaces of each bolt. The regions where contact was defined are shown in Figure 2. Notice that the interfaces between the bolts and the respective parts are tied in Abaqus instead of having contact defined because Jewell et al. [9] found these regions to contribute a negligible amount of damping and nonlinearity when similar joints were in a state of microslip. The model was meshed with 3D, 8 node linear brick elements with reduced integration (C3D8R in Abaqus). In order to reduce the number of nodes in the model a separate part in Abaqus was created for the contact region on the box and is shown in Figure 4. These parts were then bonded together using a tie in Abaqus.
Linear and Nonlinear Response of a Continuous-Interface BARC: FEA Prediction and Test 3 (a) (b) Fig. 2 (a) surfaces where contact was defined and (b) example of where a tie was used between the bolt heads and their mating surface. (a) (b) Fig. 3 (a) Separate parts created for the contact region, which had a fine mesh so that the contact could be resolved. (b) Remaining part of the box assembly, which had a coarser mesh. The parts shown in (a) were connected to the box using tie constraints. Splitting the box into three parts as shown allows one to use a coarser mesh on the box without losing the ability to mesh the contact region finely so that contact at the interface could be resolved. This technique brought the number of nodes from ≈218,000 in our initial model to ≈110,000, essentially cutting the size of the model in half. These models are denoted the 218k model and the 110k model respectively in the following. To verify this technique, modal analysis was performed on both models and the natural frequencies of the first ten elastic modes were compared. The maximum and average errors were 3.24% and 2.05% respectively between the two models. This confirmed that the two models were acceptably similar while also minimizing the number of nodes. The authors also created a third model, called the 190k model, which had a similar density of elements as the 110k model over the box, but a much finer mesh over the contact region. The 218k model was used to compute the linear natural frequencies and mode shapes in all results below, because that computation was fast with either model. In the following subsection the results of quasi-static modal analysis are shown for these models and the computational savings are discussed. Abaqus then solved for the linear mode shapes and natural frequencies of the preloaded structure with free-free boundary conditions. Figure 4 shows the first ten resulting mode shapes and natural frequencies. The results were compared to [5, 12] and found to agree reasonably well with their published natural frequencies and mode shapes. Nonlinear Predictions from the FEM Once the authors were comfortable that the model was giving accurate results, the mode shapes were examined to determine which might induce the most slip in the joints and hence the most nonlinearity. Mode 2 was chosen as the first promising candidate because vibration in that mode shears the RC. Quasi-static modal analysis (QSMA) [8] was applied to the FEM
4 Sean Magoffin et al. (a)Mode 1 391.77Hz (b)Mode 2 426.02Hz (c)Mode 3 523.77Hz (d)Mode 4 664.21Hz (e)Mode 5 977.38Hz (f)Mode 6 1072.9Hz (g)Mode 7 1359.7Hz (h)Mode 8 1516Hz (i)Mode 9 1655Hz (j)Mode 10 1699Hz Fig. 4 First ten elastic modes of the BARC and their natural frequencies in Hz to predict the nonlinearity in this mode. QSMA is performed by applying a force in the shape of the mode of interest, and solving for the nonlinear static response of the structure when subjected to that force. The static load displacement curve can then be used to compute the effective damping and natural frequency. Details on the QSMA process can be found in [8]. The predictions from QSMA were obtained before any test data was available. Section presents the test results, which reveal which modes actually exhibited the most nonlinearity. Figure 5 shows the damping versus amplitude and natural frequency versus amplitude curves obtained for Mode 2 using the 218k model. The process was repeated four times, with the force scaled to four different amplitudes, denoted by the 10-2 10-1 100 101 102 103 104 10-4 10-3 10-2 Damping Ratio -0.0254 mm -2.54 mm -5.08 mm -0.127 mm 10-2 10-1 100 101 102 103 104 Amplitude (g) 440 445 450 455 460 Natural Frequency (Hz) Fig. 5 Damping and Frequency vs. Amplitude for Mode 2. Each line represents a computation up to s different maximum force. These results are for the 218k model.
Linear and Nonlinear Response of a Continuous-Interface BARC: FEA Prediction and Test 5 maximum displacement that each force would produce for the linear structure. These values ranged from0.0254mm(0.001 in) to 5.08mm (0.2 in). The fact that the four curves agree with each other helps to verify that the behavior is physical and not an artefact of the numerical solution or function of the contact settings [9]. Figure 5 shows an increase in damping with amplitude as well as a decrease in frequency with amplitude. The damping increases with amplitude, up to a maximum value of about ζ = 0.005, while the natural frequency decreases by about 15 Hz (3.3%). One would expect to be able to detect this level of damping and stiffness nonlinearity in a test. The QSMA predictions below about 1.0g are noisy, presumably because the mesh isn’t fine enough to smoothly resolve the stick and slip transitions at this amplitude, so these should be ignored. At low amplitudes material damping would dominate so predictions are not needed for low amplitudes. At higher amplitudes the FEM shows microslip in the joints, which would likely be the source of the increase in damping observed here. Figure 7 shows the changing contact area and the slip in the joints as the analysis progresses. The mesh seems to be adequate to resolve a gradual change in the contact area and the sticking and slipping regions, providing additional confidence in the predictions. Figure 6 compares the results between the 218k and 110k models. Although there is a shift in the frequency the trends between the two match well which helps validate this modeling technique. The 218k model, which had a single part for the box, took approximately 5.5 hours for the entire QSMA process while the 110k model, which had separate parts for the contact regions, finished in just under 3 hours. The fact that the results match suggests that one can reduce the computation time while keeping accuracy when examining a contact interface. This is especially helpful when one needs to run the model many times, for example to examine nonlinearity in several modes, or to cover a wide range of amplitude for a single mode. The frequency and damping for the 190k model were also nearly identical to these, so they were not shown here for brevity. The contact status for the 190k model was shown in Figure 7, where one can see that the overall areas of slip and stick were similar between the two models. 100 101 102 103 104 10-4 10-3 10-2 Damping Ratio 110k Model 218k Model 100 101 102 103 104 Amplitude (g) 440 445 450 455 460 465 470 Natural Frequency (Hz) Fig. 6 Damping and Frequency vs. Amplitude for mode 2. The red line is from the 110k model and the black line is from the 218k model. The authors also suspected that Mode 6 might exhibit nonlinearity, as the deformation of the removable component tends to want to open and close the bolted joints when the structure vibrates in that mode. QSMA was also attempted on this mode. Force displacement curves were obtained for a force applied in the shape of Mode 6 in both the positive and negative directions (i.e. a force that drives the removable component upward and downward respectively). These curves were compared and found to be significantly different. As a result, the standard QSMA approach could not be used on this
6 Sean Magoffin et al. (a) (b) (c) (d) (e) (f) Fig. 7 The slip or stick status of the contact region on the box. (a), (c) and (e) show the 218k model, which has ≈5,600 nodes in the contact region while (b), (d) and (f) show the 190k model, which has ≈44,600 nodes on the contact region. (a) and (b) show the status at the beginning (≈1g), (c) and (d) the middle (≈500g), and (e) and (f) at the end (≈4000g) of the QSMA analysis. mode. One could instead use the approach developed by Shetty et al. [13], but time did now allow the authors to pursue this. instead it was simply noted that, based on the QSMA results, this mode might be expected to exhibit a bilinear-type nonlinearity. Physical Testing The BARC structure was manufactured per the drawings and instructions available on the Sandia website, which is in the process of being transitioned to another site hosted by SEM. A few of the washers had to be ground slightly so they wouldn’t interfere when everything was assembled, but otherwise the structure was built per the drawings and reasonable tolerances seemed to have been achieved.
Linear and Nonlinear Response of a Continuous-Interface BARC: FEA Prediction and Test 7 Linear Testing and Data Analysis The BARC structure was hung from a metal frame by looping rope through the box component, then connecting the rope to 0.125 inch bungee cord, which then attached to the frame. Accelerometers were attached to several points, denoted131Z+, 133Z+, 135Z+, 231X−, 231Y−, 231Z+, and237Y−and shown in Figure 8. Using a tiny PCB impact hammer, each point was hit 3 times and averaged to obtain the frequency response functions (FRF) between the hammer and all accelerometers. The CMIF for the set of FRFs that was obtained is shown in Figure 9, and confirms that each mode appears at a distinct frequency in the spectrum. (a) (b) Fig. 8 (a) Photo of test setup for the BARC structure and (b) Measurement Grid The set of FRFs was curve fit using the Algorithm of Mode Isolation and the modes below 2kHz were extracted. These are shown in Figure 10. Nonlinear Testing and Data Analysis A series of impacts was performed and the measurements processed to detect and characterize the nonlinearity in each of the first six modes. Based on the mode shapes shown in Figure 10, points for testing were chosen based on where the greatest displacement of the removable component was. A medium-sized hammer (PCB 086-C0x) was used for all modes except Mode 3. For that mode both a medium and large (PCB O86-D0x) hammers were used. Hence, the excitation location should have been that which best excites each mode. Data was processed in MATLAB using the Hilbert Transform as outlined in [14], which estimates the natural frequency and damping ratio as a function of vibration amplitude. The curves from various impacts were then combined into a single graph to check consistency. Mode 1 responds in both the X and Z directions, and so impacts were applied in both directions. Figure 11 shows natural frequency and damping that were obtained from each of various impacts. The legend gives the impact number, maximum force, and the location and direction of the impact (237X−,238X−or 231Z−). The response was measured at point 231X−for X-direction hits and 231Z−for Z-direction hits. The amplitude on the x-axis is that of the acceleration at that point, which should be close to the maximum acceleration for the mode in question. Both sets of measurements show minimal change in the damping ratio, as the impact with the greatest damping ratio range (impact 28) only has a net change in damping of 0.08%. The natural frequency also shows a small change with amplitude. The results are similar for the Z-direction in Figure 11(b), although these results seem to show a sudden decrease in the natural frequency at higher amplitudes. The Z-direction measurements were taken on two different days, and it is interesting to note that the linear
8 Sean Magoffin et al. 0 500 1000 1500 2000 2500 3000 3500 Frequency 100 101 102 103 Magnitude of Singular Values Complex Mode Indicator Function 1 2 3 4 5 6 7 Fig. 9 Complex Mode Indicator Function for FRFs obtained in the linear test of the BARC (a)Mode 1 388.7Hz (b)Mode 2 427.7Hz (c)Mode 3 525.8Hz (d)Mode 4 660.2Hz (e)Mode 5 980.9Hz (f)Mode 6 1166.8Hz (g)Mode 7 1350Hz (h)Mode 8 1508.2Hz (i)Mode 9 1631.6 (j)Mode 10 1721.1Hz Fig. 10 First ten elastic modes of the BARC and their natural frequencies in Hz natural frequency appears to have shifted by about 0.2 Hz from one day to the next. In summary, Mode 1 a slight softening (stiffness) nonlinearity, yet it is small relative to experimental uncertainty. Data for Mode 2 was taken on two separate days, 6/4/2024 and 6/7/2024, and as was the case for Mode 1, the linear natural frequency was slightly higher on the first day. The shift in the natural frequency is once again fairly small, only about 1.5Hz or 0.3%, all data does show the same trend. The damping increases by a factor of three or four. Both of these observations agree perfectly with the typical nonlinearity that is exhibited by bolted joints, as seen in [14, 15]. Data for Mode 3 was all collected on the same day, but using both the medium hammer (denotedmh, purple-yellow lines) and the big hammer (denoted bh, green and blue lines) in Figure 13. This mode was active along the top of the removable component near the center, so points 233-236 were excited in the negative Z direction. The big hammer didn’t seem to be
Linear and Nonlinear Response of a Continuous-Interface BARC: FEA Prediction and Test 9 (a) 10-3 10-2 10-1 100 101 2.2 2.4 2.6 2.8 3 3.2 Damping Ratio 10-3 Damping vs. Velocity Amplitude Im#28 (237-X), F max =34 Im#29 (237-X), F max =34 Im#30 (237-X), F max =114 Im#31 (237-X), F max =60 Im#32 (237-X), F max =209 Im#33 (237-X), F max =123 Im#34 (238-X), F max =116 Im#35 (238-X), F max =98 Im#36 (238-X), F max =204 Im#37 (238-X), F max =35 Im#38 (238-X), F max =37 10-3 10-2 10-1 100 101 Amplitude (g) 388.4 388.6 388.8 389 389.2 389.4 389.6 389.8 Natural Frequency (Hz) (b) 10-3 10-2 10-1 100 101 1.1 1.2 1.3 1.4 1.5 1.6 Damping Ratio 10-3 Damping vs. Velocity Amplitude 10-3 10-2 10-1 100 101 Amplitude (g) 388.5 388.6 388.7 388.8 388.9 389 389.1 Natural Frequency (Hz) Im#1 (231-Z), F max =33 Im#2 (231-Z), F max =37 Im#3 (231-Z), F max =69 Im#4 (231-Z), F max =93 Im#1 (231-Z), F max =30 Im#2 (231-Z), F max =6 Im#3 (231-Z), F max =61 Im#4 (231-Z), F max =110 Fig. 11 Mode 1 Natural Frequency and Damping vs. Amplitude. The legend gives the impact number, maximum force, and the location and direction of the impact (237X−,238X−or 231Z−). Amplitude is that measured at points 231X−and 231Z−respectively. 10-3 10-2 10-1 100 101 102 1 1.5 2 2.5 3 3.5 4 4.5 Damping Ratio 10-3 Im#1a, 6/7 F max =55 Im#1b, 6/7, F max =67 Im#3b, 6/7, F max =77 Im#2b, 6/7, F max =82 Im#4b, 6/7, F max =88 Im#3a, 6/7, F max =98 Im#5a, 6/7, F max =117 Im#9a, 6/7, F max =132 Im#11a, 6/7, F max =139 Im#6a, 6/7, F max =146 Im#7a, 6/7, F max =164 Im#13a, 6/7, F max =166 Im#4a, 6/7, F max =185 Im#8a, 6/7, F max =195 Im#10a, 6/7, F max =195 Im#12a, 6/7, F max =196 Im#16a, 6/7, F max =214 Im#5, 6/4, F max =10 Im#6, 6/4, F max =32 Im#7, 6/4, F max =53 Im#8, 6/4, F max =128 10-3 10-2 10-1 100 101 102 Amplitude (g) 423 424 425 426 427 428 Natural Frequency (Hz) Fig. 12 Mode 2 Natural Frequency and Damping vs. Amplitude. All impacts were applied at 232Y−and amplitude is that measured at points 231Y−. able to excite any nonlinearity in this mode, as the measurements showed little to no change in the damping and natural frequency. The tip on that hammer may have been too soft to excite this mode effectively. The medium hammer did appear to produce changes at high amplitudes, the overall changes in the damping ratio (5e-4) and natural frequency (0.6 Hz or 0.2%) were fairly small. Mode 4 was tested on 6/14/2024 (red-yellow lines) and 6/18/2024 (green-purple lines) at points 233 and 234 in the –Z direction. As with previous modes the linear natural frequencies were at slightly different values based on the day the data was taken, The damping shows no significant trend and while the frequency did decrease consistently on the second day of testing, the change was small (0.4 Hz or 0.06%). Modes 5 and 6, shown in Figure 14, were somewhat difficult to process using the Hilbert Transform because they are somewhat close together and decayed quickly relative to the other modes. Hence, the results were somewhat noisy. Even then, the variation in these modes is small suggesting that they are behaving linearly. To confirm that these modes were truly linear, the frequency response (FRF) was computed by dividing the FFT of the response by the FFT of the input force. The result, in Figure 15, shows that there is not a significant change in the stiffness
10 Sean Magoffin et al. (a) 10-3 10-2 10-1 100 101 1.2 1.4 1.6 1.8 2 2.2 Damping Ratio 10-3 Im#7, mh (233-Z), F max =77 Im#8, mh (233-Z), F max =53 Im#9, mh (234-Z), F max =77 Im#10, mh (234-Z), F max =59 Im#11, mh (235-Z), F max =45 Im#12, mh (235-Z), F max =68 Im#13, mh (236-Z), F max =71 Im#14, mh (236-Z), F max =60 Im#26, bh (233-Z), F max =139 Im#27, bh (233-Z), F max =108 Im#28, bh (234-Z), F max =131 Im#29, bh (234-Z), F max =251 Im#31, bh (235-Z), F max =165 Im#32, bh (236-Z), F max =153 Im#33, bh (236-Z), F max =91 10-3 10-2 10-1 100 101 Amplitude (g) 525.2 525.4 525.6 525.8 526 526.2 526.4 Natural Frequency (Hz) (b) 10-3 10-2 10-1 100 101 0.8 0.9 1 1.1 1.2 1.3 Damping Ratio 10-3 10-3 10-2 10-1 100 101 Amplitude (g) 658.8 659 659.2 659.4 659.6 659.8 660 Natural Frequency (Hz) Im#7, 6/14, F max =77 Im#8, 6/14, F max =53 Im#9, 6/14, F max =77 Im#10, 6/14, F max =59 Im#7, 6/18, F max =14 Im#8, 6/18, F max =51 Im#9, 6/18, F max =91 Im#10, 6/18, F max =80 Im#11, 6/18, F max =43 Im#12, 6/18, F max =98 Fig. 13 (a) Mode 3 and (b) Mode 4 Natural Frequency and Damping vs. Amplitude. For Mode 4, impacts 9 and 10 were at 234Z−and all others were at 233Z−. Amplitude is that measured at point 231Z−. (a) 10-3 10-2 10-1 100 1.5 2 2.5 3 3.5 4 4.5 Damping Ratio 10-3 Im#1, (231-Z), F max =30 Im#2, (231-Z), F max =6 Im#3, (231-Z), F max =61 Im#4, (231-Z), F max =110 Im#7, (233-Z), F max =14 Im#8, (233-Z), F max =51 Im#9, (233-Z), F max =91 Im#10, (233-Z), F max =80 Im#15, (235-Z), F max =108 Im#16, (235-Z), F max =80 Im#17, (235-Z), F max =57 Im#18, (235-Z), F max =19 Im#20, (237-Z), F max =62 Im#21, (237-Z), F max =52 Im#22, (237-Z), F max =42 Im#23, (237-Z), F max =28 10-3 10-2 10-1 100 Amplitude (g) 958 960 962 964 966 968 970 972 Natural Frequency (b) 10-3 10-2 10-1 100 101 102 10-2 Damping Ratio 10-3 10-2 10-1 100 101 102 Amplitude (g) 1120 1130 1140 1150 1160 1170 1180 1190 Natural Frequency (Hz) Fig. 14 (a) Mode 5 and (b) Mode 6 Natural Frequency and Damping vs. Amplitude. Amplitude is that measured at point 231Z−. or damping of these modes for the various impacts. Mode 5 does show an amplification of about 50% for Impact 17, as compared to the other tests, but the trend is not consistent for Impacts 18 and 19, which also had lower amplitudes. Comparing Test Results and Predictions Linear Mode Shapes and Natural Frequencies The mode shapes, shown previously in Figures 4 and 10 show a strong visual correspondence between the FEM and test. These are repeated in Figures 17 and 18 in the Appendix to allow an easier comparison. The natural frequencies are compared in Table 1. They differed by at most 1.7% except for Mode 6, which differed by 8.0%. This is an opening-closing mode; the other modes don’t involve this type of motion. Most of the natural frequencies agreed to within 1-2%.
Linear and Nonlinear Response of a Continuous-Interface BARC: FEA Prediction and Test 11 900 950 1000 1050 1100 1150 1200 1250 1300 Frequency (Hz) 101 102 103 Frequency Response @ 231:+Z Impact15, F max =107.9 Impact16, F max =79.67 Impact17, F max =57.14 Impact18, F max =19.29 Impact19, F max =62.02 Fig. 15 Frequency Response Functions (FRFs) computed from several high amplitude impacts, to quantify nonlinearity in Modes 5 and 6. Table 1 Comparison of Natural Frequencies between FEM and Test Mode FEM Test %Diff 1 391.77 388.7 0.79 2 426.02 427.7 -0.39 3 523.77 525.8 -0.39 4 664.21 660.2 0.61 5 977.38 980.9 -0.36 6 1072.9 1166.8 -8.0 7 1359.7 1350 0.72 8 1516 1508.2 0.52 9 1655 1629.3 1.6 10 1699 1721.1 -1.3 Nonlinearity As was expected based on the FE analysis described in Section , Mode 2 showed the strongest nonlinearity in the test data. Furthermore, that mode exhibited a classical bolted-joint type nonlinearity, so one would expect QSMA to be effective at predicting its behavior. Figure 5 and 12 both show an increase in damping with amplitude as well as a decrease in frequency with amplitude. The damping also increases by a similar amount in both cases (about half an order of magnitude for a twoorder of magnitude increase in vibration amplitude). These results are overlaid in Figure 16. To facilitate the comparison, material damping of ζ = 0.001 was added to the FEM predictions, as this was clearly manifest in the test data. Note that the horizontal axis for the FEM data is the peak modal amplitude, or the product of the modal acceleration and the mode shape at the point where the displacement is largest. For the test data the horizontal axis is the acceleration amplitude at point 231Y−, which appears to be the point of maximum displacement. The FEM and Test results agree qualitatively, and have similar orders of magnitude. The anomaly in the damping that was observed in the FEM predictions around 3-6g happens to be the same region in which the test data shows the strongest nonlinearity; hence it would be wise to refine the model to make sure that the contact is well resolved for this amplitude. In contrast, if one extrapolates the power law behavior that the FEM exhibits from 200-2000g, that seems to agree with the lower bound of the test data. The lower bound is typically of interest in measurements such as this, as it
12 Sean Magoffin et al. 10-2 10-1 100 101 102 103 Amplitude (g) - Peak Modal Amplitude (g) 10-3 10-2 Damping Ratio Damping vs. Velocity Amplitude Fig. 16 Comparison of damping vs vibration amplitude between FEM and test for Mode 2: (black dashed) FEM Prediction, (colored lines) Measurements would have the minimum influence from other modes (i.e. see [15]) and hence agrees best with the assumptions made in QSMA. It is surprising that the results agree this well. In [10] the authors found that one had to model the interface very carefully, including any lack of flatness, to accurately capture damping. The FEM analysis also noted that Mode 6 was likely to behave nonlinearly, with an opening-closing type nonlinearity. The measurements did not show significant nonlinearity in Mode 6. However, it should be noted that bilinear-type nonlinearities may not exhibit a significant change in the resonance frequency with vibration amplitude [16]. Further analysis would be required to see if the test data confirms any bilinear behavior of Mode 6. Conclusion The Box Assembly with Removable Component (BARC) was modeled using Abaqus FEA software in order to predict the linear mode shapes and natural frequencies. Then QSMA was performed on Mode 2, which was suspected to have nonlinear behavior, in order to predict the amplitude dependant damping ratio and natural frequency. These predictions from the FEM were complete before physical testing on the BARC structure was performed, thus giving blind predictions of the linear and nonlinear behavior based only on the nominal design of the components. Once the FEM predictions were well along, the BARC was manufactured and linear and nonlinear testing was performed on the physical BARC structure. Modal analysis was performed in MATLAB to extract the linear mode shapes and natural frequencies. The mode shapes were found to visually match those of the FEM and the linearized natural frequencies from the FEM matched the tests well with typical errors of less than 1% and a maximum of 1.7% error, except for mode 6, which had an error of 8.0%. Mode 6 involves opening and closing of the joint while the other modes do not have this behavior. The first six modes were examined for nonlinearity and Mode 2 was found to exhibit a typical joint-type nonlinear damping behavior while the other modes had minimal nonlinearity. The nonlinearity predicted by applying QSMA on Mode 2 agreed quite well with the measurements. This study demonstrates the feasibility of using finite element analysis and QSMA to predict the linear and nonlinear behavior of a component such as the BARC. The linear and nonlinear FEM predictions agreed very well with measurements in this study, but, as was cautioned earlier, research on other structures with joints [10] suggests that prediction may often be much more challenging. Still, these results are encouraging and motivate further use of these predictive tools.
Linear and Nonlinear Response of a Continuous-Interface BARC: FEA Prediction and Test 13 Appendix (a)Mode 1 391.77Hz (b)Mode 2 426.02Hz (c)Mode 3 523.77Hz (d)Mode 4 664.21Hz (e)Mode 5 977.38Hz (f)Mode 1 388.7Hz (g)Mode 2 427.7Hz (h)Mode 3 525.8Hz (i)Mode 4 660.2Hz (j)Mode 5 980.9Hz Fig. 17 Modes 1-5 with the FEM being on top and the mode shapes from tests on bottom. (a)Mode 6 1072.9Hz (b)Mode 7 1359.7Hz (c)Mode 8 1516Hz (d)Mode 9 1655Hz (e)Mode 10 1699Hz (f)Mode 6 1166.8Hz (g)Mode 7 1350Hz (h)Mode 8 1508.2Hz (i)Mode 9 1629.3Hz (j)Mode 10 1721.1Hz Fig. 18 Modes 6-10 with the FEM being on top and the mode shapes from tests on bottom.
14 Sean Magoffin et al. References 1. David E. Soine, Richard J. Jones, Julie M. Harvie, Troy J. Skousen, and Tyler F. Schoenherr. Designing hardware for the boundary condition round robin challenge. In Michael Mains and Brandon J. Dilworth, editors, Topics in Modal Analysis & Testing, Volume 9, pages 119–126, Cham, 2019. Springer International Publishing. 2. Alvis, Tyler and Schoenherr, Tyler. Best Practices for Modeling Bolted Joints: Calibrating the BARC System. Orlando, FL, United states, February 2024. 3. Ryan Schultz, Tyler Schoenherr, and Brian Owens. A proposed standard random vibration environment for barc and the boundary condition challenge. In Chad Walber, Matthew Stefanski, and Steve Seidlitz, editors, Sensors and Instrumentation, Aircraft/Aerospace, Energy Harvesting & Dynamic Environments Testing, Volume 7, pages 77–84, Cham, 2022. Springer International Publishing. 4. Sebastian Chirinos, Aneesh Pawar, Haley Tholen, Scott Ouellette, and Thomas Roberts. Parametric simulations of the BARC model in SDOF and MIMO configurations for estimating service environment severity. In Chad Walber, Matthew Stefanski, and Julie Harvie, editors, Sensors and Instrumentation, Aircraft/Aerospace and Dynamic Environments Testing, Volume 7, pages 19–40. Springer International Publishing. 5. Christopher L. Padilla, Jonah M. Madrid, Ezekiel C. Granillo, Antonio Flores, and Abdessattar Abdelkefi. Dynamics and nonlinear characterization of BARC systems with varying central cut widths. American Institute of Aeronautics and Astronautics, Inc, 2024. 6. Ezekiel C. Granillo, Jonah M. Madrid, Christopher L. Padilla, Jorge Perez, and Abdessattar Abdelkefi. Insights on the scalability of the BARC structure on its dynamical characteristics. American Institute of Aeronautics and Astronautics, Inc, 2024. 7. Adam Takeshita. Experimental and computational investigations on fixture interference and accelerometer placement for barc systems. Master’s thesis, New Mexico State University, Las Cruces, NM, USA, November 2022. 8. Robert M. Lacayo and Matthew S. Allen. Updating Structural Models Containing Nonlinear Iwan Joints Using Quasi-Static Modal Analysis. Mechanical Systems and Signal Processing, 118(1 March 2019):133–157, 2019. Number: 1 March 2019. 9. Emily Jewell, Matthew S. Allen, Iman Zare, and Mitchell Wall. Application of quasi-static modal analysis to a finite element model and experimental correlation. Journal of Sound and Vibration, 479(0022-460X):115376, 2020. 10. Seyed Iman Zare Estakhraji, Mitchell Wall, Jacob Capito, and Matthew S. Allen. A thorough comparison between measurements and predictions of the amplitude dependent natural frequencies and damping of a bolted structure. Journal of Sound and Vibration, 544:117397, February 2023. 11. Mitchell Wall, Matthew S. Allen, and Robert J. Kuether. Observations of modal coupling due to bolted joints in an experimental benchmark structure. Mechanical Systems and Signal Processing, 162(0888-3270):107968, 2022. 12. Daniel Peter Rohe. Modal data for the barc challenge problem test report. Technical Report SAND-2018-0640R; 660094, Sandia National Lab, 2018. 13. Drithi Shetty, Mathew S. Allen, and Kyusic Park. A New Approach to Model a System with Both Friction and Geometric Nonlinearity. Journal of Sound and Vibration, 522(26 May 2023):117631, 2023. 14. Daniel R. Roettgen and Matthew S. Allen. Nonlinear characterization of a bolted, industrial structure using a modal framework. Mechanical Systems and Signal Processing, 84:152–170, 2017. 15. Mitchell P. J. Wall, Allen Allen, Matthew S, and Robert J. Kuether. Observations of Modal Coupling due to Bolted Joints in an Experimental Benchmark Structure. Mechanical Systems and Signal Processing, 162:107968, 2022. 16. Benjamin R. Pacini, Wil A. Holzmann, and Randall L. Mayes. Performance of Nonlinear Modal Model in Predicting Complex Bilinear Stiffness. In Gaetan Kerschen, editor, Nonlinear Dynamics, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, pages 101–112, Cham, 2019. Springer International Publishing.
Chapter 2 Chapter 1 On the Detection and Quantification of Nonlinearity via Statistics of the Gradients of a Black-Box Model Georgios Tsialiamanis and Charles R. Farrar Abstrac t Detection and identification of nonlinearity is a task of high importance for structural dynamics. On the one hand, identifying nonlinearity in a structure would allow one to build more accurate models of the structure. On the other hand, detecting nonlinearity in a structure, which has been designed to operate in its linear region, might indicate the existence of damage within the structure. Common damage cases which cause nonlinear behaviour are breathing cracks and points where some material may have reached its plastic region. Therefore, it is important, even for safety reasons, to detect when a structure exhibits nonlinear behaviour. In the current work, a method to detect nonlinearity is proposed, based on the distribution of the gradients of a data-driven model, which is fitted on data acquired from the structure of interest. The data-driven model selected for the current application is a neural network. The selection of such a type of model was done in order to not allow the user to decide how linear or nonlinear the model shall be, but to let the training algorithm of the neural network shape the level of nonlinearity according to the training data. The neural network is trained to predict the accelerations of the structure for a time-instant using as input accelerations of previous time-instants, i.e. one-step-ahead predictions. Afterwards, the gradients of the output of the neural network with respect to its inputs are calculated. Given that the structure is linear, the distribution of the aforementioned gradients should be unimodal and quite peaked, while in the case of a structure with nonlinearities, the distribution of the gradients shall be more spread and, potentially, multimodal. To test the above assumption, data from an experimental structure are considered. The structure is tested under different scenarios, some of which are linear and some of which are nonlinear. More specifically, the nonlinearity is introduced as a column-bumper nonlinearity, aimed at simulating the effects of a breathing crack and at different levels, i.e. different values of the initial gap between the bumper and the column. Following the proposed method, the statistics of the distributions of the gradients for the different scenarios can indeed be used to identify cases where nonlinearity is present. Moreover, via the proposed method one is able to quantify the nonlinearity by observing higher values of standard deviation of the distribution of the gradients for lower values of the initial column-bumper gap, i.e. for “more nonlinear” scenarios. Keyword s Structural health monitoring (SHM) · Structural dynamics · Nonlinear dynamics · Machine learning · Neural networks 1.1 Introduction In the pursuit of making everyday life safer, humans have extensively tried to model the environment around them. Structures are an important part of the environment, in which humans live. They are man-made and should be safe throughout their lifetime. Structures are exposed to numerous environmental factors, which may cause them to fail. Moreover, during operation, structures are subjected to dynamic loads, which, in time, may cause failure. Such failures will most probably result in economic damage to society and may even result in loss of human lives. Therefore, for the purpose of maintaining structures safe, the field of structural health monitoring (SHM) [1] has emerged. G. Tsialiamanis ( ) Dynamics Research Group, Department of Mechanical Engineering, University of Sheffield, Sheffield, UK e-mail: g.tsialiamanis@sheffield.ac.uk C. R. Farrar Engineering Institute, MS T-001, Los Alamos National Laboratory, Los Alamos, NM, USA e-mail: farrar@lanl.gov © The Society for Experimental Mechanics, Inc. 2024 M. R. W. Brake et al. (eds.), Nonlinear Structures & Systems, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-36999-5_1 1 www.FreeDyn.at – A free, GUI based multi-body dynamics software for stand-alone use and with C API for linking it to Python, Matlab or custom code Wolfgang Witteveen and Stefan Oberpeilsteiner Abstract FreeDyn is a free multibody simulation (MBS) package developed by the Upper Austrian University of Applied Sciences. The precompiled Windows binaries can be downloaded from www.freedyn.at. It features a graphical user interface (GUI) and a C++ solver which is a modified version of the HHT time integration method. FreeDyn can be used as standalone software via its GUI or can be linked to other software such as Python, MATLAB, Scilab or custom codes through a CInterface. The new Python interface makes FreeDyn particularly attractive to universities and research groups. Researchers can outsource multi-body simulations to FreeDyn while addressing unique research questions using Python to call FreeDyn functions. FreeDyn supports rigid and flexible bodies, force elements and constraints. Users can define state and timedependent force and constraint elements using mathematical expressions. Additionally, advanced force elements may be linked to FreeDyn using a user written subroutine (DLL). The extended abstract contains a brief review of the underlaying theoretical concepts, a short summary of all modelling elements and of all callable FreeDyn functions. Furthermore, a simple example will be given in detail to demonstrate the binding of FreeDyn to Python. Keywords Multibody Dynamics · Free Software · Python Introduction and Theoretical Background FreeDyn [1] is a freely available Windows® software package for the pre- and post-processing and numerical time integration of multibody dynamic systems. A user interface (UI) supports the intuitive modelling of complex models via typical graphic oriented methods. The final model can be exported into a solver for the numerical time integration. The results can be read in again, animated and plotted. The automatically generated equations of motion take on the form M(q)q¨+Cq T(q)λ=Q(q, q˙, t) C(q, t)=0 (1) where the state vector qcontains the degrees of freedom (dof) of the system. It consists of n sub-vectors containing the dof of the n modeled bodies. The translation of a rigid body is characterized by the 3 coordinates of the center of mass. For the parameterization of the rotation, Euler parameters are chosen. The matrixMis the state dependent mass matrix withnblock diagonal entries according to the mass matrices of the individual bodies. The vector Qcontains the generalized forces and the quadratic velocity vector. The constraint equations are collected in the vector Cand depend on the state vector q and the time t. The constraint forces are computed via the constraint JacobianCq T and the vector of the Lagrange multipliers λ. Readers who are more interested can find more details in the following list: • On the numerical time integration via an HHT solver see [2]. Wolfgang Witteveen University of Applied Sciences Upper Austria, Degree Program Mechanical Engineering, 4600 Wels, Austria e-mail: wolfgang.witteveen@fh-wels.at Stefan Oberpeilsteiner principia MBS GmbH, 4113 St. Martin, Austria, www.principia-mbs.com e-mail: stefan@principia-mbs.com © The Author(s), under exclusive license to River Publishers 2025 15 Ludovic Renson et al. (eds.), Nonlinear Structures & Systems, Vol. 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.13052/97887-438-0146-7 2
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