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About this book

Dynamic Substructures, Volume 4: Proceedings of the 39th IMAC, A Conference and Exposition on Structural Dynamics, 2021, the fourth volume of nine from the Conference brings together contributions to this important area of research and engineering. The collection presents early findings and case studies on fundamental and applied aspects of the Dynamics of Coupled Structures, including papers on:

Methods for Dynamic Substructures Applications for Dynamic SubstructuresInterfaces & SubstructuringFrequency Based Substructuring Transfer Path Analysis

Table of Contents

Frontmatter

Chapter 1. Sensitivity-Based Substructure Error Propagation for Efficient Assembly Model Reduction

Abstract
Due to ever-increasing complexity of structural dynamic systems in various fields of engineering, model reduction techniques using a substructuring approach, a.k.a. component mode synthesis techniques, still form an active field of research. This paper proposes an efficient, novel method for error approximation for model reduction of coupled substructures in structural dynamics. When coupling multiple reduced substructure models, the influence of individual substructure modes on the dynamic behavior of the total reduced mechanical assembly is generally unknown. Rather than selecting substructure eigenmodes, which are used to constitute the reduction bases, solely based on their eigenfrequencies, this paper proposes a different selection method. This method inspects the influence of individual substructure modes on an assembly receptance using so-called modal receptance error contributions. These modal receptance error contributions are defined as the assembly receptance reduction error induced by truncating individual substructure modes. By determining the sensitivity of the receptance of the assembly with respect to the uncoupled substructure receptances, a substructure reduction error is propagated through the assembly model, resulting in a first-order approximation of the assembly error. To calculate this sensitivity, the receptance of the assembly is expressed in terms of the individual receptances of the uncoupled substructures and Boolean mapping matrices, used to couple substructures. Comparing different modal receptance error contributions, associated with the reduction of individual substructures, provides insight in the selection of substructure modes which results in an efficient reduction of the assembly. As such, a mode-selection criterion is defined by using the obtained information on the sensitivity of the quality of the assembly reduction to truncating individual substructure modes. This criterion helps to determine a more efficient reduction basis. To illustrate the proposed method, a cantilever Euler beam consisting of two substructures is used.
B. M. Kessels, M. L. J. Verhees, A. M. Steenhoek, R. H. B. Fey, N. van de Wouw

Chapter 2. Assessing the Quality of Real-Time Hybrid Simulation Tests with Deep Learning Models

Abstract
Hybrid simulation (HS) is an advanced dynamic testing method that combines experimental testing and analytical modeling simultaneously to provide a better understanding of the structural systems as well as the structural elements while maintaining cost-effective solutions. A complex analytical substructure in the HS can be challenging, especially to conduct real-time HS (RTHS) tests due to the nature of numerical solution algorithms. Therefore, alternative methods, such as machine learning models are being explored to represent the analytical substructures of the RTHS tests. This study investigates the quality of the RHTS tests when a deep learning algorithm is used as a metamodel of the analytical substructure. A one-bay one-story concentrically braced frame (CBF) is selected to be used in RTHS tests where the frame is the analytical substructure, and the brace is tested experimentally. The compact HS laboratory at the University of Nevada, Reno, was used to run the RTHS experiments. Deep long short-term memory (LSTM) networks were selected to be trained as a metamodel using the Python environment to represent the dynamic behavior of the analytical substructure CBF. The pure analytical solution of the CBF under earthquake excitation is used as a training dataset of the metamodels. Several RTHS tests were performed. The quality of the test results was evaluated against the pure analytical solutions obtained from both the finite element model (FEM) and machine learning (ML) model.
Elif Ecem Bas, Mohamed A. Moustafa

Chapter 3. Dynamic Substructuring Using a Combination of Softening and Hardening Connecting Elements

Abstract
The dynamic analysis of complex engineering structures represents a challenging task, since the reliability of the results significantly depends on the accuracy of the model. In general, linearized models represent a valid approximation, but in some cases, it is necessary to include also the most significant nonlinearities to obtain reliable results. In the present work, the case in which two beams are jointed together through softening and hardening connecting elements is analyzed. It is possible to account for their presence by modeling them as nonlinear substructures, and the connected subsystems are instead modeled as linear substructures. A Nonlinear Coupling Procedure (NLCP) is defined in the modal domain to analyze the dynamics of these systems. The iterative procedure has been modified with respect to the one used in previous works by selecting a different initial guess and by maintaining the energy of the system constant at each iteration. The theory of Nonlinear Normal Modes (NNMs) is used to account for the presence of the nonlinear connections in the coupled assembly. The NLCP is employed to analyze the effects of modal truncation on the mode shapes and on the resonance frequency.
Jacopo Brunetti, Walter D’Ambrogio, Annalisa Fregolent, Francesco Latini

Chapter 4. On the Coupling of Reduced Order Modeling with Substructuring of Structural Systems with Component Nonlinearities

Abstract
The emergence of digital virtualization has brought Reduced Order Models (ROM) into the spotlight. A successful reduced order representation should allow for modeling of complex effects, such as nonlinearities, and ensure validity over a domain of inputs. Parametric Reduced Order Models (pROMs) for nonlinear systems attempt to accommodate both previous requirements (Benner et al., SIAM Rev 57(4), 1–14, 2015). Our work addresses a physics-based reduced representation of structural systems with localized nonlinear features. Via implementation of the approach described in Quinn (J Sound Vib 331(1), 81–93, 2012), we achieve a substructuring formulation similar to Component Mode Synthesis. However, instead of individual component modes, reduction modes of a global nature are obtained from the corresponding linear monolithic system. In turn, this technique allows for a divide and conquer strategy that naturally couples the response between linear and nonlinear subdomains. A pROM able to exploit this modular formulation is developed as a next step. The framework treats each component independently, and individual projection bases are assembled by applying a Proper Orthogonal Decomposition (POD) on snapshots of each subdomain’s response. Parametric dependencies on the boundary conditions and the structural and excitation traits are injected into the pROM utilizing clustering, following the methodology in Vlachas et al. (Vlachas, K., Tatsis, K., Agathos, K., Brink, A.R., Chatzi, E.: A local basis approximation approach for nonlinear parametric model order reduction. Journal of Sound and Vibration 502, 116055, 2021). To this end, a modified strategy is employed, relying on the Modal Assurance Criterion as a measure to indicate optimal training samples while defining regions with similar underlying dynamics on the domain of parametric inputs. A numerical case study of a 3D wind turbine tower featuring material nonlinearities exemplifies our approach. The derived pROM offers an accelerated approximation of the underlying high fidelity response and can be utilized for numerous tasks, including vibration control, residual life estimation, and condition assessment of hotspot locations.
Konstantinos Vlachas, Konstantinos Tatsis, Konstantinos Agathos, Adam R. Brink, Dane Quinn, Eleni Chatzi

Chapter 5. On Predicting Uncertainties in the Dynamic Response of a Welded Structure

Abstract
Present-day engineering projects are highly dependent on numerical models; thanks to the improvements in computing capabilities that have contributed significantly in this area. Although models these days are far better representations of engineering structures than before, every model is limited by its mathematical representation and the knowledge about the underlying physics. Validating numerical models involves obtaining test or performance data, which may not be practical in the case of many engineering structures. In such cases where data for the full structural model are not available, the subsystems or components can be tested and the associated models calibrated and validated separately. Inferring response at system level from these subsystem validation results is not straightforward and needs a proper uncertainty propagation strategy. Furthermore, the response of a structure depends not only on the components that it is made of, but also equally on the joints. Joints determine how the components interact within a structure, making validating the models for joints as important as validating the subcomponents. A joint does not physically exist on its own but co-exists with the components it connects, which makes it difficult to define a joint model. Isolated models, where the same type of joints is employed to connect ‘simple’ components with well-established numerical models can serve such a joint model for the purpose of validation. This paper describes a probabilistic approach to dealing with joints via such isolated models, where the uncertainties related to a welded joint are quantified and propagated into a target model of a welded structure to predict the dynamic response.
A. Muraleedharan, R. J. Barthorpe, K. Worden

Chapter 6. A Covariance-Based Approach for Recovering Phase-Consistent Loads from Random Vibration Analysis

Abstract
In the field of aerospace analysis, it is common practice to apply the root-mean-square (RMS) interface loads from a system-level random vibration analysis as static inputs to high-fidelity stress models. This process serves as an efficient means for evaluating several designs or design iterations; however, because the interface loads typically exhibit different magnitude and phase relationships at different times and frequencies, it is unrealistic to assume that all loads should be applied simultaneously at their maximum, in-phase magnitudes (often assumed to be the three-sigma or three times RMS loads). A more realistic approach for enveloping the problem space is to define a load vector (load case) for each interface degree of freedom (DOF), with a selected DOF applied at its maximum magnitude and all other DOFs applied at their phase-consistent magnitudes. These phase-consistent loads would seem easy to calculate through transient analysis by identifying the peak or valley with the desired magnitude for each DOF (e.g., three-sigma) and extracting the phase-consistent loads for the other DOFs; however, each peak or valley will produce a different load vector, since each peak or valley is merely a single data point in a statistical distribution. Thus, the preferred approach is to obtain phase-consistent loads in the frequency domain, such that the load vectors represent the expected magnitudes based on the statistical distribution of loads. This paper presents a frequency-domain, covariance-based approach for obtaining phase-consistent static loads from random vibration analysis. The load vectors obtained using this approach reasonably and realistically envelope the problem space and provide an efficient means for conducting high-fidelity random vibration stress analysis.
T. Van Fossen

Chapter 7. On the Estimation of Structural Admittances from Near-Field Acoustic Holography Using a Dynamic Substructuring Approach

Abstract
Near-field acoustic holography (NAH) provides a reconstruction of the normal particle velocity radiated by the source of interest on a nearby surface. Reconstructing structural admittance based on normal velocity establishes NAH as alternative to the conventional methods such as accelerometers or laser vibrometers. Measuring vibrating object using microphone array offers high-resolution measurements and with this the possibility to obtain spatially dense displacement field of the structure. Although useful, the method’s ill posed nature needs to be considered which introduces numerical noise to the reconstructed result. In this paper an alternative approach for improved estimation of FRFs from NAH is introduced by using dynamic substructuring methodology. Recently developed System Equivalent Model Mixing (SEMM) method is proposed that enables mixing of response models of the same structure into one hybrid model. In the hybrid formulation, each model contributes its own advantages to yield best combination of the two. NAH introduces a high spatial resolution to the hybrid model whereas discrete laser vibrometer measurements improve the accuracy of the reconstructed displacement field. Experimental validation of methodology is performed on a freely supported structure that is excited using electrodynamic shaker. Noisy acoustical environment introduces spurious peaks in FRFs acquired using NAH. The laser vibrometer measurement is not affected by a noisy environment, as only the structural response is measured. It is shown that hybrid model provides a more consistent FRFs of the structural response, especially with regard to the amplitude in the resonance regions.
Gregor Čepon, Domen Ocepek, Miha Kodrič, Miha Boltežar

Chapter 8. Experimental Framework for Identifying Inconsistent Measurements in Frequency-Based Substructuring

Abstract
The dynamic properties of modern products are analyzed using an experimental approach through the measurement of frequency-response functions (FRFs). For an individual measurement, the coherence offers an online check during the system acquisition. More general tools for determining the consistency of the complete measurement set are based on a comparison of the FRFs or the modal shapes with a numerical model. They are useful tools, but they rely on a comparison with a numerical model that might not reflect the behavior of the actual system. This paper aims to develop a comprehensive experimental method to check the consistency of individual measurements based on comparisons with the complete experimental response model. The numerical model is introduced only to enable the experimental model to be expanded using the System Equivalent Model Mixing method. The entire formulation is developed in the frequency domain, so that the transition to the modal domain, which might remove the physically relevant information from the system, is not required. In the frequency domain, it is possible to assess the consistency of the FRF across the entire frequency range of interest and not only in the region of the natural frequencies. This is of great importance in the area of frequency-based substructuring, where even small inaccuracies in the substructure’s FRFs (e.g., the position of the anti-resonance) can lead to erroneous coupling results due to the inversion process. The experimental case study demonstrates the efficiency of the proposed approach. By removing the identified inconsistent measurements, it was possible to significantly increase the accuracy of the final coupling process.
Miha Kodrič, Gregor Čepon, Miha Boltežar

Chapter 9. Weakening of the Interface Constraints in Modal Substructuring Using Singular Value Decomposition

Abstract
An experimental approach to modal substructuring often proves to be problematic. One of the issues to be addressed is to determine a suitable set of constrain equations to be applied when considering an interface with multiple connection points. An approach to solving this problem was presented in the transmission simulator method (TSM), where a truncated set of mode shapes of a flexible fixture is used to project the constraints on the fixture’s vector subspace. In this paper, the possibility of using the singular value decomposition (SVD) to obtain an appropriate subspace basis for weakening the constrain equations is explored. The most dominant response modes, acquired by performing SVD on the frequency response functions at the interface, are selected to weaken the constrain equations. The proposed approach is validated in a numerical case study.
Miha Pogačar, Tomaž Bregar, Gregor Čepon, Miha Boltežar

Chapter 10. Introducing pyFBS: An Open-Source Python Package for Frequency Based Substructuring and Transfer Path Analysis

Abstract
pyFBS is an open-source Python package for Frequency Based Substructuring. The package implements an object-oriented approach for dynamic substructuring. State-of-the-art methodologies in frequency based substructuring, such as virtual point transformation and system equivalent model mixing, are available within the pyFBS. Each method can be used as a standalone or interchangeably with others. Tools are provided to easily visualize components and configure the measurement setup. Also operational deflection shapes and mode shapes can be animated directly within the 3D display. Furthermore, basic and application examples are available, together with numerical and experimental datasets, to enable the user to get familiar with the work flow of the package. This paper showcases the use of the pyFBS on two example structures. Firstly, a simple beam-like structure is used to depict the use of the 3D display, FRF synthetization, virtual point transformation and system equivalent model mixing. Secondly, an automotive test-structure is used to show the use of the pyFBS on real-life complex structure, where the in-situ transfer path analysis is used to characterize blocked forces. The development of the pyFBS is an ongoing effort, as it is actively being used as a research tool. Additional features and new methods will be integrated within the pyFBS in the near future.
Tomaž Bregar, Ahmed El Mahmoudi, Miha Kodrič, Gregor Čepon, Miha Boltežar, Daniel J. Rixen

Chapter 11. Analysis of Friction Induced Mode Coupling Instabilities Using Dynamic Substructuring

Abstract
In complex vibrating systems, contact and friction forces can produce a dynamic response of the system (friction induced vibrations). They can arise when different parts of the system move one with respect to the other generating friction force at the contact interface. Component mode synthesis and more in general substructuring techniques represent a useful and widespread tool to investigate the dynamic behavior of complex systems, but classical techniques require that the component subsystems and the coupling conditions (compatibility of displacements and equilibrium of forces) are time invariant. In previous papers, it was shown that contact problems can be cast in the framework of dynamic substructuring by considering the models of the component substructures as time invariant, while the coupling conditions must be time dependent. In this paper a substructuring method is proposed that, depending on the contact assumption, is able either to account only for the macroscopic sliding between substructures, or to consider also the local vibrations of the contact points or to consider also the geometric nonlinearity due to the elastic deformation. This allows to adapt the contact algorithm to the contact problem that must be tackled, i.e. position dependent dynamics or friction induced vibrations.
Jacopo Brunetti, Walter D’Ambrogio, Annalisa Fregolent

Chapter 12. Road Noise: Embedding Suspension Test Benches in Sound & Vibration Design using virtual points and the Transfer Path Analysis Framework

Abstract
Many Car Manufacturers (OEMs) use dedicated test bench setups for testing and developing components separately from their vehicle platforms. Relating these results to full-vehicle assemblies is not immediately possible because of the difference in boundary conditions and incompatibility between test bench and vehicle measurement points.
This study focuses on the use of tire noise test benches in the engineering of full-vehicle NVH. We show how virtual points, easily obtained using DIRAC, ensure compatibility between various test assemblies. A total of 10 contact points between the suspension parts and the bodywork is described using 3, 5 or 6 degrees of freedom per point. The TPA framework is extensively used to create and compare source descriptions: direct blocked forces from a rigid test bench, in situ blocked forces from a compliant test bench, and several component TPA approaches on the full vehicle. Blocked forces are estimated both before and after the bushings. Results are evaluated for simulation of road noise at the driver’s ear in the vehicle.
Maarten V. van der Seijs, Julie M. Harvie, David P. Song
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