River Rapids Conference Proceedings of the Society for Experimental Mechanics Series Special Topics in Structural Dynamics & Experimental Techniques, Vol. 5 Matthew Allen Babak Moaveni David Epp 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
Matthew Allen· Babak Moaveni · DavidEpp Editors Special Topics in Structural Dynamics & Experimental Techniques, Vol. 5 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 97887-438-0150-4 (Hardback) ISBN 97887-438-0162-7 (eBook) https://doi.org/10.13052/97887-438-0150-4 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 Special Topics in Structural Dynamics & Experimental Techniques 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 Nonlinear Structures & Systems; 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. Special Topics in Structural Dynamics & Experimental Techniques represents papers highlighting new advances and enabling technologies for Experimental Techniques, Biomedical and Bioinspired Systems, Metamaterials, Metastructures and Additive Manufacturing and Rotating Machinery. The organizers would like to thank the authors, presenters, session organizers, and session chairs for their participation in this track. Editor: Matthew Allen – Brigham Young University, UT, USA; Babak Moaveni – Tufts University, MA, USA; David Epp – Sandia National Laboratories, NM, USA. v
Contents 1 Using Mode Shapes from Cell Phone Videos for Machinery Health Monitoring 1 Shawn Richardson and Mark Richardson 2 Rotordynamic Analysis of the Turbopump Using Superelement Approach 11 Yunus Ozcelik, Hakan Bayraktar, and Yunus Tufek 3 Metamaterial-Based Vibration Mitigation for Enhanced Reliability of an Automotive Inverter: A Numerical Study 21 Sara Tincani, Claus Claeys, Elke Deckers, Nimish Pandiya, and Christian Dindorf 4 Test-vs-Simulation Correlation of Noise Emission in a Bent-Axis Pump 33 Bhaskar Banerjee, Michael Kwarta, Zuher Abdel Mallak, Marcus Wiklund La˚ng, Per-Ola Vallebrant, Adarsh Chaurasia, and Omer Kayani 5 A Brief Review of Robotic Welding Technology for Structural Steel Applications 39 Ahmad Rababah, Feras Abla, Sahabeddin Rifai, Guilherme Pereira, Devin Huber, and Onur Avci 6 Bridging Minds and AI for Enhanced Autonomous Driving 49 Roveri Nicola, Silvia Milana, Antonio Culla, Gianluca Pepe, and Antonio Carcaterra 7 Rotordynamic Influence of Balance Pistons with Combined Axial and Radial Flow 53 Maximilian M.G. Kuhr, Zhengzhong Gao, and Peter F. Pelz 8 Investigating the Propulsive Efficiency of Bio-Inspired Fish-like Elastic Caudal Fin through Dynamic Analysis and Experimental Validation 65 E. Paifelman, S. Milana, A. Culla, G. Costantino Muniz, F. Passacantilli, and E. Ciappi 9 Bandgap Design of Metamaterial Structures by Varying Local Resonator Properties 71 Hannes Wo¨hler and Sebastian Tatzko 10 Experimental Investigation of the Damping Effect of Inherent Particle Damping Under Centrifugal Load 79 Mirco Jonkeren and Sebastian Tatzko 11 Greedy Source Placement Optimization for MIMO Control of Ground-Based Vibration Tests 85 Chandler Smith, Timothy Walsh, Wilkins Aquino, Drew Kouri, and Ryan Schultz 12 Design and Testing of Vibroacoustic Metamaterials for Active Damping of Traffic Noises 95 W. Kaal, M. M. Becker, L. S. Kollmannsperger, and S. C. L. Fischer 13 pyFBS: A Python Package for Frequency Based Substructuring and Transfer Path Analysis 103 Domen Ocepek, Miha Kodricˇ, Francesco Trainotti, Miha Pogacˇar, Tomazˇ Bregar, Daniel Jean Rixen, and Gregor Cˇ epon 14 Non-contact Video-based Heart Rate Monitoring using Hilbert Motion Magnification 109 Camilly S. Costa, Gustavo N. Castro, Yuri S. A. Alencar, Victor H. R. Cardoso, Joa˜o C. W. A. Costa and Moise´s F. Silva vii
viii Contents 15 Variable Atwood Number Rayleigh-Taylor InstabilitiesInduced by Volumetric Energy Deposition 117 Clara S. Bender, Anna C. Cardall, Fabian A. Rodriguez, Joseph A. Kerwin, Ricardo Mejia, and Adam J. Wachtor 16 Identification of Linear Time-variant Rotor Components in Resonance Crossing Experimental Validation 127 Luigi Carassale, Roberto Guida, and Davide Iaffaldano 17 Efficient Simulation of Soft Tissue for Human, Robotics, and Exoskeleton Applications 131 Stefan Holzinger, Andreas Zwo¨lfer, and Daniel Rixen 18 Piezo Elements to Improve Dynamic Response of Load Cells 135 M. Vanali, C. Rossi, S. Pavoni, M. Berardengo, and S. Manzoni 19 Monitoring the Interface Behaviour of Bolted Joints 145 T. Mace, S. Gilbert, T. Moghadam, and P. Pandey
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 Using Mode Shapes from Cell Phone Videos for Machinery Health Monitoring Shawn Richardson and Mark Richardson Abstract A cell phone video offers a low-cost non-contacting alternative to traditional accelerometer-based methods for monitoring the health of plant operating equipment. The frequency bandwidth and video enhancement technology in modernday cell phones have rendered them ideal for use as a non-contacting measurement device in a plantwide route-based monitoring program. During the past 30 years, trained neural networks have been used in a variety of applications to solve problems where the number of possible solutions is overwhelming. A neural network must be trained with lots of data, and it will diagnose mechanical faults in rotating machinery more accurately as it is trained with more vibration data In this paper, it is first shown how the Operational Modal Analysis (OMA) mode shapes of a rotating machine are obtained by using FRF-based curve fitting on vibration data extracted from a cell phone video recording of a rotating machine during operation. Then, a database search method calledFaCTsTMis used to identify various unbalance conditions of a rotating machine. FaCTsTMfunctions in the same manner as a trained neural network. FaCTsTMuses the current mode shapes of a machine together with a shape difference indicator (SDI) to find the closest match of the current mode shapes with mode shapes that were previously labeled and archived in a database. FaCTsTMdisplays a bar chart of the ten closest matches of the current mode shapes with the labeled mode shapes, thereby defining the current mechanical condition of a machine based on its mode shapes. Keywords Artificial Intelligence (AI) · Neural Network (NN) · Time Waveform (TWF) · Digital Fourier Transform (DFT) · Operational Modal Analysis (OMA) mode shape · Operating Deflection Shape (ODS) · Degree of Freedom (DOF) · Frames Per Second (fps) · Auto Power Spectrum (APS) · Cross Power Spectrum (XPS) · ODS-FRF(APSandPhase of anXPS) · Fault Correlation Tools (FaCTsTM) · Shape Difference Indicator (SDI) Rotating Machine In this paper, FaCTsTMis used to uniquely identifynine different unbalance cases of the rotating machine shown in Figure 1. The machine has a variable speed motor connected to the rotor with a rubber belt. The motor speed was adjusted so that the rotor speed was approximately 1000 RPM throughout all the video recordings. Introduction Most power plants, oil refineries, and manufacturing plants worldwide have implemented route-based machinery health monitoring programs for accessing the health of their rotating equipment. Digital vibration signals are the primary data used to detect and diagnose faults in operating equipment. Traditionally, machine health monitoring has been done by attaching accelerometers to the surfaces of the operating equipment and acquiring vibration signals from the accelerometers with a portable digital spectrum analyzer. This method Shawn Richardson· Mark Richardson Aerospace Centre of Excellence, University of Strathclyde, Montrose St, Glasgow, G1 1XJ, UK e-mail: shawn.richardson@vibetech.com © The Author(s), under exclusive license to River Publishers 2025 Matthew Allen et al. (eds.), Special Topics in Structural Dynamics & Experimental Techniques, Vol. 5, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.13052/97887-438-0150-4 1
2 S. Richardson and M. Richardson Fig. 1 Rotating Machine Showing Unbalance Screws Added to Its Rotors of data acquisition is time consuming and expensive compared with a cell phone video recording. Furthermore, because it is non-contacting, a cell phone can record vibration of machine parts that are hot or inaccessible where accelerometers cannot be used. In previous papers [3], [4] we applied various traditional signal processing methods to the TWFs extracted from the frames of a video to display its ODS’s in animation. Both TWFs and their associated DFTs can be used to display either time-based ODS’s or frequency-based ODS’s so that a machine’s deformation can be visualized at slower speeds with higher amplitudes. In a previous paper [4], a new database search method called FaCTsTMwas introduced, and was used to uniquely identify nine different unbalance cases of a rotating machine using only ODS data from the tops of the two bearing blocks on the machine. In this paper FaCTsTMis used to uniquely identify the same nine unbalance cases, but by using the OMA mode shapes of the machine. These results demonstrate the reliability and repeatability of usingOMAmode shapes extracted from a cell phone video for machine health monitoring. FaCTsTMalso functions like a neural network in that it becomes more accurate as more labeled and archived OMAmode shapes are made available for shape comparison. Artificial Intelligence (or Machine Learning) Artificial Intelligence, popularly known as AI, uses a “trained” neural network (NN) to interpret the meaning of a set of data. Vibration data is primarily used to diagnose the health of rotating machinery. Vibration data in the form of TWFs, DFTs, ODS-FRFs, ODS’s andMode Shapes can be used to train anNN. But to accurately diagnose a mechanical fault, an NNmust be trained with a lot of vibration data. Here are a couple definitions of AI from a Google search on the Internet. • A neural network is a machine learning model that uses a network of interconnected nodes, or artificial neurons, to process data in a way that mimics the human brain. • A neural network is a method in artificial intelligence (AI) that teaches computers to process data in a way that is inspired by the human brain. It is a type of machine learning (ML) process, called deep learning, that uses intercon-
Using Mode Shapes from Cell Phone Videos for Machinery Health Monitoring 3 nected nodes or neurons in a layered structure that resembles the human brain. Machine Learning (ML) using an NN mimics the learning of the human brain. AnNNis depicted in Figure 2 • Lots of labeled input datais required to train anNN • To diagnose mechanical faults, anNNmust be trained with data that is uniquely associated with a mechanical fault • When vibration data is input to an Inference Engine (a trained NN), it diagnoses a mechanical fault, as shown in Figure 3. Fig. 2 A Neural Network Fig. 3 NN Training & Inference Engine
4 S. Richardson and M. Richardson TWFs&DFTs When a video is processed in MEscope [8], a rectangular grid of points with rectangular surfaces between them is created. Frames of the video are attached to this surface during animated display of ODS’s extracted from the video. Using a rectangular point grid, millions of pixels in each frame of a cell phone video are processed to extract TWFs for the horizontal & vertical motion of thousands of points in the point grid. Grid points with little or no motion, (like background points), are hidden and their linked TWFs are removed from further analysis. A point grid with background points hidden is shown in Figure 4. ADFTis calculated for eachTWFthat is extracted from the video. Time-basedODS’s are displayed in animation from the TWFs using a sweeping Line cursor. Frequency-based ODS’s are displayed in animation from the DFTs using sine dwell modulation of the ODS at the cursor position. The magnitude & phase of the ODS at selected points can also be displayed, as illustrated in Figure 4. Fig. 4 First-Order ODS Animated from DFTs ODS-FRFs A unique frequency domain function, called an ODS-FRF, can be calculated from each response TWF extracted from a video. A set of ODS-FRFs calculated from the TWFs is typically more accurate because spectrum averaging can be used to reduce extraneous noise from the ODS-FRFs. The magnitude of an ODS-FRF is the APS of the response DOF at a grid point. The phase of the ODS-FRF is the phase of the XPSbetween the response DOFand the DOFof any reference grid point. ODS-FRFs carry the same displacement units as the response TWFs from which they are calculated. But because it is a frequency domain function, an ODS-FRF can be accurately differentiated to velocity units by multiplying it by the frequency variable. Vibration invelocity units is commonly used by vibration analysts to quantify vibration levels in rotating equipment.
Using Mode Shapes from Cell Phone Videos for Machinery Health Monitoring 5 Law of the FFT One of the laws of the FFTalgorithm is that ∆f =1/T, where ∆f is the frequency difference between samples of anODSFRF, and Tis the time length of TWF data from which the ODS-FRF was calculated. For example, if an ODS-FRF is calculated fromTWFdata over a 15 secondperiod, the frequency resolution (∆f ) of the ODS-FRFis 60/15 =4RPM. To increase the frequency resolution of anODS-FRF, TWFdataover alonger periodTis required. Therefore, the video from which the TWFs are extracted must be recorded over a longer period T. Fig. 5 ODS-FRF Showing First Three Order Peaks. Aliased Order Peaks A limitation of any video recording is that anti-alias filtering cannot be used to remove high-frequency signals from the video. Without anti-alias filtering, machine order peaks greater than one-half the sampling frequency, (called Fmax), are folded around Fmaxand appear at lower frequencies in the ODS-FRFs, as shown in Figure 5. All order-related resonance peaks between Fmax & 2 x Fmax are folded around (wrapped around) Fmax and appear at a lower frequency in the frequency band (0 to Fmax). Aliasing of higher frequencies occurs in both DFTs and ODSFRFs. The ODS-FRFshown in Figure 5 was calculated from a TWFthat was extracted from a video that was sampled at 60 fps, or 3600RPM. Therefore, theFmaxof theODS-FRFis 1800RPM. The first-order peak is at the machine running speed and is clearly visible at 1014 RPM. The second order resonance peak should be at 2028 RPMand third order resonance peak should be at 3042RPM, but they are both folded around 1800RPMand are clearly visible at lower frequencies in the ODS-FRF. The aliased frequency of an order higher thanFmaxcan be calculated fromFmaxand the expected order frequency. • Second order aliased frequency 1800 - (2028 - 1800) 1572RPM • Third order aliased frequency 1800 - (3042 - 1800) 558RPM
6 S. Richardson and M. Richardson The ODS-FRFin Figure 5 was calculated from a TWFwith a 10-second lengthT. Therefore ∆f = (1/10) Hz or 60/10 =6RPM. So, the aliased frequency of the third order peakis within one (∆f) of its calculated value. Curve Fitting the ODS-FRFs ODS-FRFs can be curve fit using an FRF-basedcurve fitter if they have been filtered with a special window. In MEscope [9] this filter is called a DeConvolution window. This filter reshapes anODS-FRFso that it closely resembles anFRFand therefore can be curve fit using FRF-basedcurve fitting. An FRF-basedcurve fit of the three order peaks in an ODS-FRF is shown in Figure 6. In the upper left-hand graph, the red curve fitting function is overlaid on a magnitude plot of the ODS-FRF. The aliased frequencies of the second and third orders are correctly estimated by curve fitting the ODS-FRF. More details on curve fittingODS-FRFs are given in a companion paper [1]. Fig. 6 Curve-fit of ODS-FRFs. Mode Shapes at Monitored Points Although curve fitting was applied to all the ODS-FRFs calculated from the videos, only the mode shape components are five grid points were labeled and archived in the machine database. These five points are shown in Figure 7. Damping Removal DeConvolution windowed is necessary before using an FRF-basedcurve fitter on a set of ODS-FRFs. But DeConvolution windowing adds a specific amount of damping to each OMAmode shape. Therefore, following FRF-based curve fitting, when theOMAmode shapes are stored into a Shape Table in MEscope [9], the damping added by DeConvolution windowing is removed from them. This is shown in Figure 8.
Using Mode Shapes from Cell Phone Videos for Machinery Health Monitoring 7 Fig. 7 Point Grid Showing Monitored Points. Fig. 8 Damping is Removed when OMA Mode Shapes are Saved. FaCTsTM FaCTsTM[4] is a database search algorithm used by MEscope [9]. FaCTs searches a database of labeled mode shapes, each mode shape associated with a particular machine fault.When a newmode shape is saved into the archival database, FaCTs searches the database of labeled mode shapes and displays a bar chart of the ten closest matching mode shapes together with the mechanical fault associated with the labeled mode shapes.
8 S. Richardson and M. Richardson FaCTsTMuses an algorithm called the Shape Difference Indicator (SDI) [8]. SDI calculates a correlation coefficient between two complex-valued shape vectors. FaCTs finds the ten closest matching mode shapes in the archival database based on the SDI value between each current mode shape and each labeled mode shape. • FaCTs has values between 0.0 and 1.0 • FaCTs =1.0 →two mode shapes are identical • FaCTs >=0.9 →two mode shapes are similar • FaCTs <0.9 →two mode shapes are different Baseline Case When no unbalance screws were added to either rotor of the rotating machine in Figure 1, its mode shapes were labeled as the Baseline case. When the Baseline case is archived into the database, the FaCTs bar chart in Figure 9 clearly identifies it by its unique OMAmode shapes compared to the mode shapes of other unbalance cases. Its FaCTs bars with all the other unbalance cases are much less than1.0. Fig. 9 Baseline OMA Mode Shapes Versus Other Eight Unbalance Cases Figures 10 through 18 show the FaCTs bar charts for the eight unbalance cases where screws were added to either the Inner or Outer rotor. Each unbalance case was uniquely identified byFaCTs because its correspondingOMAmode shapes were unique when compared to the OMAmode shapes of the other unbalance cases. But there is one exception. Unbalance cases with 2 Inner screws and 3 Inners screws have a FaCTs bar of 1.0, indicating that the OMAmode shapes for these two cases are essentially the same.
Using Mode Shapes from Cell Phone Videos for Machinery Health Monitoring 9 Fig. 10 One Outer Screw Fig. 15 One Inner Screw Fig. 11 Two Outer Screws Fig. 16 Two Inner Screws Fig. 13 Three Outer Screws Fig. 17 Three Inner Screws Fig. 14 Four Outer Screws Fig. 18 Four Inner Screws
10 S. Richardson and M. Richardson Conclusions Nine different unbalance cases were created on a rotating machine by adding screws to its Inner and Outer rotors. The first casewith no screws added was labeled as the Baseline case. In each case, a 16 secondcell phone video was recorded, with the machine running at approximately1000RPM. Using MEscope [9], TWFs were extracted from each cell phone video andODS-FRFs were calculated from the TWFs for each unbalance case. To reduce noise in the ODS-FRFs, five spectrum averages and overlap processing were used to calculate the ODS-FRFs. Then the ODS-FRFs were curve fit using FRF-based curve fitting and the OMAmode shapes for the first three machine orders were labeled and stored in an archival database. The first order OMAmode shape and the mode shapes of the aliased secondandthird orders were archived. Each set of three mode shapes was labeled with its corresponding unbalance case. Then, when the OMAmode shapes for each case were again stored into the archival database, FaCTs correctly identified each case by calculating its SDI value with each labeled set of OMAmode shapes in the database. This method of numerically comparing a current set of OMAmode shapes with sets of labeledOMAmode shapes uniquely identified each of the nine unbalance cases using onlyten mode shape components from five points on the video point grid of the machine. This low-cost approach using cell phone videos and FaCTs can be used by any plant maintenance department for monitoring the health of its rotating equipment and accurately identifying machine faults. References 1. S. Richardson, M. Richardson “Extracting 3D Mode Shapes from a Cell Phone Video” IMAC XLII, February 10-13, 2025, Orlando, FL. 2. B. Schwarz, S. Richardson, P. McHargue, M. Richardson “Using Modal Analysis and ODS Correlation to Identify Mechanical Faults in Rotating Machinery” IMAC XLII, January 29 - February 1, 2024, Orlando, TX. 3. D. Ambre, B. Schwarz, S. Richardson, M. Richardson, “Using Cell Phone Videos to Diagnose Machinery Faults” IMAC XLI, , Austin, TX, February 13-16, 2023. 4. B. Schwarz, S. Richardson, M. Richardson, “Using a Cell Phone Video and ODS Correlationto Diagnose Unbalance in Rotating Machinery” IMAC XLII, January 29 - February 1, 2024, Orlando, FL 5. M.H. Richardson, “Is It a Mode Shape or an Operating Deflection Shape?” Sound and Vibration magazine, March 1997. 6. B. Schwarz, M.H. Richardson, “Measurements Required for Displaying Operating Deflection Shapes” Proceedings of IMAC XXII, January 26, 2004 7. B. Schwarz, M.H. Richardson, “Introduction to Operating Deflection Shapes” CSI Reliability Week, Orlando, FL, October 1999 8. S. Richardson, J. Tyler, P. McHargue, M. Richardson “A New Measure of Shape Difference” IMAC XXXII February 3.6, 2014 9. MEscopeTMis a trademark of Vibrant Technology, Inc. www.vibetech.com
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 Rotordynamic Analysis of the Turbopump Using Superelement Approach Yunus Ozcelik, Hakan Bayraktar, and Yunus Tufek Abstract The use of turbopumps in aerospace propulsion systems and high-performance fluid transport applications necessitates thorough rotordynamic analysis to ensure efficiency and reliability. The complexity of turbopumps, with factors like high speeds, fluid-structure interactions, and flexible components, requires advanced modeling techniques for accurate predictions. Model reduction techniques like Component Mode Synthesis (CMS) are crucial for efficient dynamic analysis of complex systems. This study analyzed the rotordynamic behavior of the turbopump using a superelement approach. The rotor was modeled using 3D elements in ANSYS, with its dynamic characteristics evaluated under constant bearing stiffness. To enhance computational efficiency, the stator was represented as a superelement and coupled with the 3D rotor model. For validation, a full 3D stator model was also developed and coupled with the rotor. This comprehensive modeling strategy enabled a detailed assessment of the system’s dynamic behavior. Results validated against the full rotor–stator model demonstrated that stator flexibility has a notable effect on the rotor’s critical speeds. The study highlights the effectiveness of the superelement method in reducing computational costs while maintaining the accuracy required for high-fidelity rotordynamic analysis. Keywords Rotordynamic · Turbopump · Component Mode Synthesis · Superelement Introduction Turbopumps are critical subsystems in aerospace propulsion systems, commonly used in liquid rocket engines and other high-performance fluid transport applications. They consist of a pump driven by a turbine, which operates at extremely high rotational speeds to deliver the required power. Due to the high operating speeds and complex structural interactions, turbopumps are subject to various dynamic phenomena that can significantly affect their performance and reliability. One of the most crucial aspects of turbopump design and analysis is rotordynamic, which examines the dynamic behavior of the rotating components, including the rotors, bearings, seals, and the surrounding casings. Rotordynamic analysis is essential for predicting critical speeds, stability margins, and potential failure modes such as rotor whirl and resonance, which can lead to excessive vibrations and mechanical failure. A thorough understanding and optimization of turbopump rotordynamic behavior are key to ensuring efficient, stable, and reliable operation under a wide range of operating conditions. The high rotational speeds of turbopumps, coupled with the presence of fluid-structure interactions, bearings, seals, and flexible stator components, contribute to the complexity of their rotordynamic analysis. Factors such as rotor flexibility, bearing stiffness, and damping characteristics, as well as seal forces and casing flexibility, all play a role in determining the system’s dynamic response. Identifying and addressing rotordynamic issues early in the design phase is crucial to avoiding costly failures and ensuring smooth operation throughout the entire operating range. A rotor system must be designed to operate without excessive vibration throughout its entire speed range. The theoretical foundation of rotordynamic analysis dates to the late 18th century when Dunkerly [1] proposed that the critical speed of a rotor corresponds to the natural frequency of a simply supported beam. Early design approaches assumed rigid bearings. However, as operating speeds increased and computational methods advanced, rotor dynamics analysis evolved to account for bearing properties. It is well-established that the structure beyond the bearings, known as the foundation, tends to reduce Yunus Ozcelik· Hakan Bayraktar · Yunus Tufek Roketsan Inc. e-mail: yunuseozcelik@gmail.com; hakan.bayraktar@roketsan.com.tr; yunus.tufek@roketsan.com.tr © The Author(s), under exclusive license to River Publishers 2025 11 Matthew Allen et al. (eds.), Special Topics in Structural Dynamics & Experimental Techniques, Vol. 5, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.13052/97887-438-0150-4 2
12 Y. Ozcelik et al. the first critical speed frequency while increasing the amplification factor [2]. These studies demonstrate that considering both bearing and foundation effects in rotor dynamics analysis leads more accurate rotor dynamics predictions. Rotordynamic analysis using one-dimensional (1D) elements, where the shaft is represented by beam or bar elements and the disks are idealized as a concentrated mass, is widely employed in industry due to its simplicity and computational efficiency [3]. However, this approach has notable limitations, particularly in its inability to model complex rotor geometries or accurately capture the effects of large flexible disks and blades. For axisymmetric rotors, two-dimensional (2D) axisymmetric harmonic elements can be used, as they consider the three-dimensional (3D) nature of the problem while reducing the number of elements needed for modeling [4]. With advancements in modeling, analysis, optimization techniques, and material science, modern rotor configurations frequently incorporate non-uniform shaft diameters, hollow components, and large flexible structures. These features necessitate more sophisticated modeling techniques to accurately predict their dynamic behavior. 2D axisymmetric harmonic elements are insufficient for representing complex, non-uniform rotor blade geometries. This limitation has led to the increased use of 3D solid and shell elements, which offer a more accurate and geometrically complete representation of the rotor. These elements allow for analysis in both the fixed and rotating reference frames, reducing the need for geometric simplifications that remain prevalent in industrial practice. Modeling rotors with 3D shell and solid elements offers the advantage of accurately representing complex rotor geometries. However, 3D rotor models are more suitable for analysis in the rotating reference frame and tend to result in significantly larger problem sizes [5]. Despite advancements in computational resources, linear rotordynamic analyses can still take hours or even days to complete [6]. To integrate such models into the commercial engineering design of critical components, and to account for nonlinear effects from bearings and dampers, it is crucial to improve dynamic analysis performance by orders of magnitude. Crucially, this improvement must be achieved without sacrificing model complexity or solution accuracy. The application increase in model size for rotors modeled with 3D elements necessitates the use of model reduction techniques. Component Mode Synthesis (CMS) is a model reduction technique that allows for the efficient dynamic analysis of large and complex mechanical systems by breaking them down into smaller, manageable substructures. These substructures are individually analyzed, and their dynamic properties (e.g., mass, stiffness, and damping) are characterized through modal parameters such as natural frequencies and mode shapes. CMS enables the recombination of these modes to form a reduced-order model for the entire system, significantly decreasing the computational load while preserving the essential dynamic characteristics. The use of CMS to reduce the computational effort for analyzing large rotating machinery, particularly in turbochargers, has been demonstrated by Kim et al. [7], while maintaining accuracy in predicting critical speeds and response amplitudes. CMS methods have also been applied to analyze the dynamic behavior of rotor-bearing systems under unbalance forces, enabling the inclusion of nonlinear effects like bearing stiffness and damping [3]. CMS was employed to investigate the dynamics of a turbopump rotor system in a study conducted by Lund [8]. The research highlighted the method’s ability to predict natural frequencies and mode shapes accurately, facilitating the identification of critical speeds that could lead to resonance. Similarly, the superelement method has been used in the design optimization of rotor-bearing systems to predict how component modifications would affect overall system behavior, as shown in a study by Ehrich [9]. The combination of CMS with nonlinear effects has been gaining traction in recent years. CMS with nonlinear rotordynamic phenomena, such as oil whip and shaft bending, was integrated, demonstrating the method’s versatility in capturing complex real-world dynamics that are difficult to handle with traditional full-order models, by Gupta et al. [10]. In this study, the rotordynamic behavior of a turbopump, including both the rotor and stator, was investigated using a combination of 3D modeling and superelement techniques obtained through the CMS method. The rotor was initially modeled with 3D finite elements in ANSYS, and its dynamic characteristics were evaluated under the assumption of constant bearing stiffness. To further enhance the analysis, the rotor was coupled with a stator represented as a condensed superelement, simplifying the stator’s influence on the rotordynamic. Furthermore, a full 3D model of the stator was created in ANSYS and coupled with the rotor to validate the reduced model. This detailed approach provided an in-depth analysis of how the interaction between the rotor and stator influences the critical speeds of the turbopump. Background Substructuring is a technique that condenses a group of finite elements into a single equivalent element, represented by a reduced-order matrix. This condensed element is referred to as a superelement. In analytical procedures, a superelement is treated comparably to any other element type, with the exception that it necessitates an initial substructure generation analy-
Rotordynamic Analysis of the Turbopump Using Superelement Approach 13 sis for its creation. Substructuring significantly reduces computational effort and solution time of large-scale problems while reducing the requirement for computational resources. This method proves particularly beneficial in nonlinear analyses and in examinations involving structures with recurring geometric configurations, such as rotordynamic analysis. In nonlinear analysis, substructuring can be implemented in the linear sections of the model, thereby eliminating the necessity for repetitive recalculation of element matrices during each equilibrium iteration. For structures characterized by repetitive patterns, a singular superelement can be generated to represent the recurring geometry and replicated across various locations, resulting in substantial savings in computational time. In substructuring analysis, the nodal displacement vector {u} of a substructure is expressed in terms of reduced coordinates {ˆu}, by following transformation [11] {u} = [T] {ˆu}, (1) where [T] is the transformation matrix. In dynamic analysis, equation of motion can be expressed by, [M] {¨u}+[C] {˙u}+[K] {u} ={F}, (2) where [M] , [C] and[K] are mass, damping and stiffness matrix, respectively and{F} is the load vector. By substituting Eq. (1) into Eq. (2) and left multiplying by[T]T, the following reduced equation is obtained, h ˆMi{¨ˆu}+hˆCi{ˆ˙u}+h ˆKi{ˆu} ={ˆF} (3) where, h ˆMi = [T]T [M] [T] is the reduced mass matrix, hˆCi = [T]T [C] [T] is the reduced damping matrix, h ˆKi = [T]T [K] [T] is the reduced stiffness matrix and{ˆF} = [T]T{F} is the reduced load vector. Static condensation, often referred to as Guyan Reduction, and Component Mode Synthesis (CMS) are both widely used techniques for substructuring in structural dynamics. Guyan Reduction is typically insufficient to fully capture the dynamic behavior of a superelement. To address this, static condensation is enhanced with the inclusion of component modes and residual vectors. When applying CMS to reduce a complex engineering model, certain loading conditions may induce deformation states that cannot be accurately represented by static reduction and component modes alone. In such cases, the Residual Vector technique is used to incorporate high-frequency contributions, compensating for this deformation discrepancy and ensuring an accurate dynamic response across all loading conditions. In this study, CMS is utilized to model the stator component of the turbopump. By applying equilibrium and compatibility conditions at the component interfaces, the method defines the dynamic behavior of the entire system model. Dividing the matrix equation into degrees of freedom (DOFs) for the interface and interior, {u} = { um} {uc} , [M] = [Mmm] [Mmc] [Mcm] [Mcc] , [C] = [Cmm] [Cmc] [Ccm] [Ccc] , [K] = [Kmm] [Kmc] [Kcm] [Kcc] , {F} = { Fm} {Fc} (4) where, mis the master DOF that defined at interference, andc is all nodes except master nodes. Nodal displacement vector can be expressed by, {u} = { um} {uc} = [T] { um} {yδ} (5) where, {yδ} is a truncated set of generalized modal coordinates. Transformation matrix used in this study is obtained by [12], [T] = [I] [0] [Gcm] [Φc] (6) where, [Φc] is the eigenvector obtained with interference nodes fixed. In the reduced system, master DOFs serve to connect the CMS superelement with other elements or additional CMS superelements. For the fixed-interface method, when the fixed-interface normal modes are mass-normalized, the resulting reduced stiffness, mass, and damping matrices, as well as the reduced load vector, take the following final form [13]: [ ˆM] = [Mmm]+[Mmc] [Gcm]+[Gmc] [[Mcm]+[Mcc] [Gcm]] [[Mmc]+[Gmc] [Mcc] [[Φc] [Φc] T [[Mcm]+[Mcc] [Gcm]] [I] (7)
14 Y. Ozcelik et al. [ ˆC] = [Cmm]+[Cmc] [Gcm]+[Gmc] [[Ccm]+[Ccc] [Gcm]] [[Cmc]+[Gmc] [Ccc]] [Φc] [Φc] T [[Ccm]+[Ccc] [Gcm]] [Φc] T [Ccc] [Φc] (8) [ ˆK] = [Kmm]+[Kmc][Gcm] [0] [0] [Λ2] (9) {ˆF} = { Fm}+[Gmc] {Fc} [Φc] T {Fc} (10) where, [Gcm] = −[Kcc]−1 [Kcm] , [Gmc] = [Gcm]−T and , Λ2 is a diagonal matrix consisting of the eigenvalues of the retained fixed-interface normal modes. The displacements at the condensed or DOFs are obtained from Eq. (5). For the fixed-interface condition, they are computed using Craig-Bampton method as follows: {uc} = [Gcm] {um}+[ϕc] {yδ} (11) In rotordynamic analysis, the dynamic equation in a stationary reference frame can be expressed as: [M] {¨u}+([G]+[C]){˙u}+([B]+[K]){u} ={F}, (12) where, [G] is the gyroscopic matrix which plays a critical role in rotordynamic analysis as it depends on the rotational velocity and, [B] is the rotating damping matrix. When a structure rotates about an axis such as Z and a precession motion (a rotation about an axis perpendicular to Z Axis) is introduced, a reaction moment arises. This reaction, known as the gyroscopic moment, acts along an axis that is perpendicular to both the spin axis Z and the precession axis. The gyroscopic matrix [G] represents this effect by coupling degrees of freedom in planes that are perpendicular to the spin axis in finite element modeling. Notably, this matrix is skew-symmetric, reflecting the nature of gyroscopic forces. Whirling refers to the circular or elliptical motion exhibited by a rotating structure when it vibrates at its resonant frequency. This motion can occur in two distinct forms: forward whirl (FW), which occurs when the motion is in the same direction as the rotational velocity, and backward whirl (BW), which takes place when the motion opposes the direction of rotation. Both forms of whirling are critical in understanding the dynamic behavior of rotating systems, as they can influence stability and performance, particularly at resonance. The critical speed refers to the rotational speed at which a structure’s resonance frequency (or frequencies) coincides with the excitation frequency. This condition arises when the natural frequency of the system matches the frequency of the applied excitation, which can result from an unbalance that is synchronous with the rotational speed. To identify critical speeds, one effective approach is to conduct a Campbell diagram analysis. This involves calculating the intersection points between the frequency curves and the excitation line, which reveal the critical speeds of the system. This study focuses exclusively on the effect of unbalance forces synchronous with the rotational speed. Methodology In this study, the rotordynamic behavior of a turbopump, including the stator, was analyzed using a superelement approach. The rotor was modeled using 3D elements in ANSYS, and its dynamic characteristics were evaluated under the assumption of constant bearing stiffness. Subsequently, the rotor model was coupled with a superelement representing the entire stator. Additionally, the rotor was connected to the full stator model, which was also constructed using 3D elements in ANSYS, allowing for a comprehensive analysis of the system’s dynamic performance. Rotordynamic Model with Fixed Bearing Stiffness A simplified model of the turbopump rotor, excluding the casing, was created using constant stiffness properties. Figure 1 illustrates a rotor configuration for the analysis, showing several rotating components arranged sequentially from the left end of the shaft: nut, front inducer, front impeller, front bearing and rear inducer, rear impeller, rear bearing and turbine. Additionally, the lumped mass model of the rotating non-axisymmetric components, such as blades of impeller, inducer and turbine are also represented in Figure 1. Since the objective of this study is to evaluate the impact of using a superelement on solution time and accuracy, the stiffness and damping effects of the seals are excluded from the scope of the analysis.
Rotordynamic Analysis of the Turbopump Using Superelement Approach 15 Fig. 1 The Turbopump Rotor Configuration. The turbopump, powered by a turbine using hot, high-pressure gas, operates at a rotational speed of 23,500 RPM under steady-state conditions. It is assumed that both the front and rear bearings are isotropic with a stiffness of 108 N/mm. Inconel 718 superalloy, with a density of 8240 kg/m3, a Young’s modulus of 200 GPa, and a Poisson’s ratio of 0.3, was assigned to rotor components. As illustrated in Figure 2, the rotor model was discretized with 105,344 SOLID185 elements, which are first-order hexahedral elements in ANSYS. Bonded contacts were employed at all component interfaces to simulate interference. Additionally, COMBI214 elements were utilized to represent the bearings within the model. Fig. 2 Cross-sectional View of the 3D Finite Element Model of the Turbopump Rotor. A complex eigenvalue analysis was conducted to generate the Campbell diagram over a range of rotor speeds (5,00030,000 RPM), as shown in Figure 3. In this diagram, the 1X line denotes the rotor’s operating speed. Figure 4 illustrates the first backward (1st BW) and first forward (1st FW) mode shapes of the rotor when operating at a speed of 5,000 RPM. These mode shapes provide insight into the rotor’s dynamic behavior under these conditions. The overhang configuration of the turbine, along with its large diameter and mass, causes the first bending mode to predominantly involve shaft deflection on the turbine side, while the other rotor components remain largely unaffected and stationary.
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