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About this book

Nonlinear Structures & Systems, Volume 1: Proceedings of the 38th IMAC, A Conference and Exposition on Structural Dynamics, 2020, the first volume of eight 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 Nonlinear Dynamics, including papers on:

Nonlinear Reduced-order Modeling

Jointed Structures: Identification, Mechanics, Dynamics

Experimental Nonlinear Dynamics

Nonlinear Model & Modal Interactions

Nonlinear Damping

Nonlinear Modeling & Simulation

Nonlinearity & System Identification

Table of Contents


Chapter 1. Convergence Study on the C-Beam Using Joint Modes Based on Trial Vector Derivatives

In the last years joint modes based on trial vector derivatives have been presented. These joint modes allow an accurate consideration of nonlinear contact and friction forces inside jointed structures. If these joint modes are used for model reduction, even high fidelity finite element models can be time-integrated with acceptable effort. In recent research on joint nonlinearity and reduced order modelling a generic structure called “C-beam” or “S4 beam” was used. This structure consists of two C-shaped beams assembled at their ends by two bolted joints. In this work, the previously mentioned model reduction approach is applied to the C-Beam. A convergence study with respect to the number of additional joint modes for different excitation levels is presented. In addition, it is investigated whether joint modes are able to reproduce well-known damping phenomena due to friction inside joints at different preload levels. For this purpose, a detailed observation of local sticking/slipping effects is given.

Florian Pichler, Wolfgang Witteveen

Chapter 2. Semi Hyper-Reduction for Nonlinear Surface Loads on Finite Element Structures by the Use of Stress Modes

The determination of nonlinear state-dependent surface loads, acting on finite element (FE) structures, represents a computationally challenging and costly task in dynamic simulations. While for time integration an enormous reduction of the FE models number of degrees of freedom (DOFs) can be achieved by subspace projection, the computation of nonlinear surface loads usually depends on the non-reduced physical DOFs. In order to overcome this issue, so-called Hyper-Reduction (HR) methods have been introduced. These methods try to compute the surface loads in a reduced subspace as well. In this publication, an intermediate approach is proposed, which is called “Semi Hyper-Reduction” (SHR). The equations for computing the surface loads are built up in the full space and then projected into a lower dimensional subspace via proper force trial vectors. The required force trial vectors, called “stress modes”, thereby can be determined a priori without any nonlinear computations using the full DOF model. As a numerical example, a 3D crank drive is used, where the piston and the cylinder are separated by a hydrodynamic lubrication film, which is considered by Reynolds equation.

Lukas Koller, Wolfgang Witteveen, Florian Pichler

Chapter 3. Parameter Identification of a Linear Substitution Model for Nonlinear Contact and Damping Inside Lap Joints Using Distributed Optimization

Study Program “Mechanical Engineering”

The numerical simulation of flexible system dynamics considering nonlinear contact and friction forces inside joints is a very time-consuming process. In this work, a simple linear substitution model for the joint contact is proposed and the parameters are identified. Subsequently, it is numerically examined whether this model is predictive and transferable. The identified parameters are the tangential stiffness, normal stiffness, tangential damping and the contact area. All parameters are assumed to be a function of the bolt preload. The reference is a high fidelity Finite Element (FE) model of two strips of metal bolted together with three bolts. The solution of the reference model is calculated with distributed, nonlinear contact and friction forces. During the optimization, the Fourier coefficient of the first mode was fitted. To speed up the large number of time integrations a distributed optimization method is investigated. This decentralized method accelerates the optimization process almost by the factor of the used computers. The determined parameters are used in a modified setting and compared with a corresponding reference solution. The aim is to use the obtained substitute model for predictive and tendential investigations of jointed structures.

Stefan Pöchacker, Thomas Lauß, Florian Pichler, Stefan Oberpeilsteiner, Wolfgang Witteveen

Chapter 4. Flexible Multibody Dynamics: Which Terms of the Equations of Motion Have to Be Considered for Which Problem?

Scientists often face the task to carry out a “proof of concept” of a research idea. If this is related to multibody simulation of flexible bodies, it is sometimes necessary to implement a simple and problem-oriented multibody simulation code in programs like Matlab or Scilab. The question then arises, which terms of the equation of motion have to be considered when the flexible body is modelled along the Floating Frame of Reference Formulation. The consideration of all effects leads to very complex expressions and also require costly FE preprocessing. A remarkably simplified version can be obtained, by a strict application of the small deformation assumption to the level of the kinetic energy. But what is the truth in the context of a certain problem? Clear rules are provided, so that a researcher can decide for a particular mechanical system, which terms are required and which ones can be neglected. This work reviews a paper that has been recently published by the authors with respect to this question.

Wolfgang Witteveen, Florian Pichler

Chapter 5. Application of the Bouc-Wen Model to Bolted Joint Dynamics

Various numerical models have been developed to capture the dynamic, hysteretic behavior of different mechanical systems. One such semi-physical model is the Bouc-Wen model, which relates the input displacement to the output restoring force in a hysteretic way. This formulation is intended for any form of hysteresis and was originally applied to force – deflection and flux – current diagrams of mechanical and ferromagnetic hysteresis. Built-up structures are also known to show hysteretic behavior due to the slipping that occurs between interfaces bolted together. This paper tests how effective the Bouc-Wen model is in capturing the power-law damping behavior observed in bolted joints by comparing it with another commonly used numerical model – the Iwan model. While the Iwan element has been proven to be robust and well-suited at capturing the power-law increase in energy dissipation and slow decrease in stiffness with vibration amplitude exhibited by bolted interfaces, numerical integration of the same is currently computationally expensive. Time integration of the Bouc-Wen model, on the other hand, is much more efficient, thus warranting the proposed study.

Drithi Shetty, Matthew S. Allen, Joseph D. Schoneman

Chapter 6. Some Aspects of Using the Random Decrement Technique for Nonlinear Systems

The random decrement (RD) technique can be used to analyze the response signal from a system that has amplitude dependent modal parameters. The implementation of RD for this purpose is usually done by simply applying the technique at multiple amplitudes in the measured response signal. Modal parameters are then estimated based on RD signatures using well known time domain modal parameter estimation methods. This analysis procedure originates from the invention of the RD technique, and is described by several studies in the literature. However, the RD technique is developed for linear systems, and caution must be exercised when applying it to nonlinear systems. In this study, several aspects of applying of the RD technique on signals exhibiting nonlinear behavior are addressed. The principle of superposition does not apply for a nonlinear system. This means the averaging process in RD can yield corrupted results. The benefit of a sufficiently high sampling rate is described.

Karsten K. Vesterholm, Rune Brincker, Anders Brandt

Chapter 7. Investigation of String Motions of Bowed String Instruments: A Finite Element Approach

The bowed string motion of violin strings is examined using finite element analysis. The string is modeled as a bar with a finite number of elements, whereas the bow is one single node. The cause of the sound is the stick-slip effect, which occurs at steady bowing of a string. The contact area between the bow node and one string node is modeled as nodal contact. This work aims at investigating the Schelleng Diagram as well as the Guettler diagram. The Schelleng diagram shows bow force over bowing position and identifies the area of normal sound, i.e. the Helmholtz motion (von Helmholtz, Von den Tonempfindungen als physiologische Grundlage für die Theorie der Musik. Vieweg, Braunschweig, 1863), overtone sound and raucous sound, the latter depicts bow force over bow acceleration in order to find a proper parameter configuration for “perfect bowing starts”. The influence of different friction curves of the rosin are modeled and simulated as well, which has been tested and investigated extensively in the literature mainly experimentally, see e.g (Smith, Woodhouse, J Mech Phys Solids 48(8):1633–1681, 2000).An overall research of the finite-element modeling of bowed strings is started with this new simulation and modeling set-up. The numerical results will be compared to experimental results in future work.

Özge Akar, Kai Willner

Chapter 8. FreeDyn: A Free, Flexible and State of the Art Multibody Simulation Package for Education, Research and Industrial Applications

FreeDyn ( ) is a free multibody simulation (MBS) package consisting of a graphical user interface (GUI) and a C++ solver in which a modified version of the HHT time integration method is implemented. In order to underline the solvers efficiency with respect to CPU time, comparative simulations of FreeDyn and a commercial software product are presented in this paper. The main modelling elements of FreeDyn are rigid and flexible bodies, constraints, forces and measures (angle, position, velocity …). The FreeDyn standard force elements can be specified by user defined expressions. However, if a more advanced force is required, the user can link a user written subroutine (DLL) to FreeDyn. FreeDyn can be used as standalone software, which is controlled via the mentioned GUI. Alternatively, FreeDyn can be linked via a C-Interface as a dynamic link library (DLL) to other software like MATLAB, Scilab or self-written codes. The C-interface provides control functions for time integration and access to model data, which are normally hidden (e.g. mass matrix, constraint forces, Jacobi matrices etc.). Beside industrial applications, FreeDyn can be used in education and research.In Education: Beside the classical teaching elements like the creation and simulation of MBS models, it is furthermore possible, to look deeper inside the structure of the equations. This can be easily done by the use of the access routines of the C-Interface.In Research: FreeDyn enables quick implementations of particular research ideas in the context of MBS. The integration of those ideas can be realized with a user written subroutine and/or alternatively, by the use of the C-Interface. A researcher can focus on the research topic, while FreeDyn handles the MBS model. Moreover, due to the availability of very specific information, the C-Interface of FreeDyn is suitable for extraordinary optimization problems, which is underlined by an example and several literature citations.The paper contains a brief review of the theoretical concepts on which FreeDyn is based. The previously mentioned possibilities are briefly discussed, and underlined by illustrative and meaningful examples.

Wolfgang Witteveen, Lauss Thomas, Oberpeilsteiner Stefan

Chapter 9. Force Probing to Access Potential Energy

Perhaps the two most common forms of potential energy are those associated with gravitational and elastic forces. In an experimental setting, if we can measure the force required to maintain equilibrium, then the extraction of potential energy is relatively straightforward, since the force is the negative vector gradient of the potential. In this paper we apply this approach to a small mass that is placed on various shapes under the action of gravity.

Yawen Xu, Lawrence N. Virgin

Chapter 10. Modeling and Experimental Validation of a Pylon Subassembly Mockup with Multiple Nonlinearities

The industrial approach to nonlinearities in structural dynamics is still very conservative, particularly from an experimental point of view. A demo aluminum aircraft has been equipped with discrete nonlinear elements designed to replicate real-world engine pylon subassemblies to increase awareness on the effects of nonlinearities in design, and understand how these effects can be positively exploited, if properly understood. After some preliminary experiments aimed at understanding the coupled behavior of the aircraft-pylon mockup, it became clear that more in-depth numerical and experimental analyses are required on the pylon subassembly alone. For this paper, experimental data is collected to analyze the nonlinear dynamic behavior of the pylon, leading to better understanding of the subassembly once it connects to the aircraft. The pylon element has three main sources of nonlinearities: (1) geometric nonlinearities of the connecting beam, (2) contact as the beam presses into the tapered block surface and (3) friction in the bolted connections. Backbone curves are generated, which map the evolution of natural frequency and damping ratio with excitation amplitude. Using the experimental data, a low-order nonlinear model is identified to replicate the backbone characteristics and response of the pylon.

Connor Ligeikis, Adam Bouma, Justin Shim, Simone Manzato, Robert J. Kuether, Daniel R. Roettgen

Chapter 11. Comparison Between Control-Based Continuation and Phase-Locked Loop Methods for the Identification of Backbone Curves and Nonlinear Frequency Responses

Control-based continuation (CBC) and phase-locked loops (PLL) are two experimental testing methods that have demonstrated great potential for the non-parametric identification of key nonlinear dynamic features such as nonlinear frequency responses and backbone curves. Both CBC and PLL exploit stabilizing feedback control to steer the dynamics of the tested system towards the responses of interest and overcome important difficulties experienced when applying conventional testing methods such as sine sweeps to nonlinear systems. For instance, if properly designed, the feedback controller can prevent the system from exhibiting untimely transitions between coexisting responses or even losing stability due to bifurcations. This contribution aims to highlight the similarities that exist between CBC and PLL and present the first thorough comparison of their capabilities. Comparisons are supported by numerical simulations as well as experimental data collected on a conceptually simple nonlinear structure primarily composed of a thin curved beam. The beam is doubly clamped and exhibits nonlinear geometric effects for moderate excitation amplitudes.

Florian Müller, Gaëtan Abeloos, Erhan Ferhatoglu, Maren Scheel, Matthew R. W. Brake, Paolo Tiso, Ludovic Renson, Malte Krack

Chapter 12. Implementing the Restoring Force Surface Method to Fit Experimentally Measured Modal Coupling Effects

In complex structures, particularly those with jointed interfaces, the dynamic response of individual modes can behave nonlinearly. To simplify analyzing and modeling this response, it is typically assumed that modes are uncoupled, in that each responds independently of the excitation level of other modes. This assumption is derived from the belief that, while modal coupling generally exists in physical structures, its effects are relatively small and negligible. This practice is reinforced by the fact that the actual causes of modal coupling are poorly understood and difficult to model. To that end, this work attempts to isolate and fit a model to the effects of modal coupling in experimental data from a nonlinear structure. After performing a low-level test to determine the linear natural frequencies and damping ratios of several modes, sine beat testing is used to individually excite each mode and record its nonlinear dynamic response. The Restoring Force Surface (RFS) method is then implemented to fit a nonlinear model to each isolated modal response. Sine beats are then done on multiple modes simultaneously, in which the response is assumed to be a combination of the nonlinear models of each isolated mode and some coupling term between them. As the terms modeling the individual modes are known, the only unknown is the coupling term. This procedure is performed on several mode pairs and excitation levels to evaluate the effectiveness of all proposed coupling models and gauge the significance of modal coupling in the structure.

Benjamin Moldenhauer, Daniel R. Roettgen, Benjamin Pacini

Chapter 13. Identification of Backbone Curves and Nonlinear Frequency Responses using Control-based Continuation and Local Gaussian Process Regression

Control-based continuation (CBC) is a general and systematic method to probe the dynamics of nonlinear experiments. In this paper, CBC is combined with a novel continuation algorithm that is robust to experimental noise and enables the tracking of important nonlinear dynamic features such as backbone and nonlinear frequency response curves. The method uses Gaussian process regression to create a local model of the response surface on which standard numerical continuation algorithms can be applied. The local model evolves as continuation explores the experimental parameter space, exploiting previously captured data to actively select the next data points to collect such that they maximise the potential information gain about the feature of interest. The method is demonstrated experimentally on a nonlinear structure featuring harmonically-coupled modes. The regression model is also exploited to estimate the uncertainty of the identified features.

Ludovic Renson

Chapter 14. Model Correlation to a Nonlinear Bolted Structure Using Quasi-Static Modal Analysis

Bolted joints are common in many engineering structures, yet they introduce complexity when the interest is predicting the dynamic response of a system. Under large load amplitudes, the joint contact interfaces will slip and cause the response to be nonlinear. A method proposed by Festjens et al., and later elaborated upon by the authors and dubbed Quasi-Static Modal Analysis (QSMA), has made it feasible to model the contact between structures in detail and thus predict the nonlinear dynamic behavior. Prior works have shown that this is feasible (Wall et al, Predicting S4 beam joint nonlinearity using quasi-static modal analysis. In: IMAC XXXVII; Jewell et al, J Sound Vib 479:115376, 2020, yet) no work has rigorously correlated such a finite element model with experimental measurements. This work takes a step in that direction by quantifying the effect of various features in the FEM to the amplitude dependent damping and natural frequency predicted by QSMA. Specifically, the effect of the friction coefficient and the curvature of the contact interface on the QSMA predictions is found and quantified. The results so far show that some of these effects could improve model agreement.

Mitchell Wall, Seyed Iman Zare Estakhraji, Matthew S. Allen

Chapter 15. Numerical Continuation of Periodic Orbits for Harmonically Forced Nonlinear Systems with Iwan Joints

A common numerical model for bolted or riveted joints is the Iwan model, which uses a number of discrete Jenkins elements to capture the hysteretic behaviour of the system. Previously, the Iwan model has been primarily implemented within time-domain simulations using the Newmark integrator. However, in many applications it is desirable to predict the nonlinear Frequency Response Functions (FRFs) of a structure that contains joints. In order to do so, the steady-state response must be estimated over a range of frequencies, and it is very time consuming to simply compute the time response until steady-state is reached. Furthermore, the implicit nature of the state variables (i.e. the appearance of “hidden” state variables in the form of slider displacements) makes it non-trivial to use continuation to compute the frequency response using already established techniques such as the shooting method. This paper presents a novel method to numerically compute the non-linear FRFs of a single degree-of-freedom (SDOF) system with an Iwan element. The maximum displacement over the response period is included as a state variable, along with the initial displacement and velocity, and we demonstrate that the position of all sliders can be calculated using these states. The shooting method is modified to account for the added state<?pag ?>variable. The method has been tested by computing the FRFs of a SDOF system containing of an Iwan element at multiple force amplitudes. The results show that the proposed method is able to compute the steady-state response even at large force amplitudes, when the system behaves quite non-linearly.

Seyed Iman Zare Estakhraji, Matthew S. Allen, Drithi Shetty

Chapter 16. A Digital Absorber for Nonlinear Vibration Mitigation

In this study, a digital impedance is used to realize both a linear and a nonlinear piezoelectric tuned vibration absorber in order to mitigate the vibrations of a nonlinear structure. The digital processing unit enables the synthesis of impedances with arbitrary functional forms, thereby easing the implementation of nonlinear absorbers. The superior performance of the nonlinear absorber over its linear counterpart is demonstrated experimentally. Various nonlinear functional forms are also tested in the absorber and illustrate the relevance of the principle of similarity (i.e. the same nonlinear functional form as that in the host structure should be used in the absorber).

Ghislain Raze, Sylvain Guichaux, Andy Jadoul, Valery Broun, Gaetan Kerschen

Chapter 17. Control-Based Continuation of Nonlinear Structures Using Adaptive Filtering

Control-Based Continuation uses feedback control to follow stable and unstable branches of periodic orbits of a nonlinear system without the need for advanced post-processing of experimental data. CBC relies on an iterative scheme to modify the harmonic content of the control reference and obtain a non-invasive control signal. This scheme currently requires to wait for the experiment to settle down to steady-state and hence runs offline (i.e. at a much lower frequency than the feedback controller). This paper proposes to replace this conventional iterative scheme by adaptive filters. Adaptive filters can directly synthesize either the excitation or the control reference adequately and can operate online (i.e. at the same frequency as the feedback controller). This novel approach is found to significantly accelerate convergence to non-invasive steady-state responses to the extend that the structure response can be characterized in a nearly-continuous amplitude sweep. Furthermore, the stability of the controller does not appear to be affected.

Gaëtan Abeloos, Ludovic Renson, Christophe Collette, Gaetan Kerschen

Chapter 18. Tuning of Finite Element Model Parameters to Match Nonlinear Reduced Order Models

There has been a growing demand for nonlinear model updating procedures in the structural dynamics community as advanced vehicles have started to incorporate nonlinearity into their designs. Finite element (FE) model updating is difficult for a nonlinear structure for several reasons: there may be many unknown parameters in the FE model so that multiple solutions may exist, and it is very expensive to compute the structure’s nonlinear normal modes from a full FE model that are an excellent metric to use for nonlinear updating. A recent work showed that some of these challenges can be overcome by updating a nonlinear reduced order model (ROM) rather than the full FE model. The updated ROM can be used to compute response statistics, stresses and ultimately life of the structure. However, one drawback of this approach is that one does not gain physical insight into which parameters in the FE model were in error, and so it is difficult to transfer the lessons learned to future FE models. This work explores the feasibility of updating a FE model to correlate with a ROM with a set of known parameters. The target ROM parameters consist of the linear stiffness and the nonlinear stiffness coefficients which have been identified previously by updating the ROM to correlate with experimental measurements. In this process an optimization routine is setup in which the free parameters are those of the FE model, such as boundary stiffness springs, imperfections, pre-stress, etc. The optimization procedure is wrapped around a ROM creation algorithm that iteratively tunes the FE parameters, creates a ROM, and evaluates the ROM parameters with respect to the target ROM in order to minimize the objective function. This work will demonstrate the procedure on a numerical example of a curved beam in order to verify its effectiveness on the nonlinear model updating.

Kyusic Park, Matthew S. Allen

Chapter 19. Towards an Understanding of the Transient Behavior of the Five-Parameter Iwan-Type Model

Mignolet (J Sound Vib 349:289–298, 2015) demonstrated mathematically that his proposed five-parameter Iwan-type model weakens the coupling between the change in effective stiffness and change in effective damping of the model as vibration amplitude changes. Several experimental studies on bolted-joint structures have experienced difficulty in fitting a traditional Iwan model to match measurements, so this advantage is sorely needed. However, Mignolet’s work only considered steady-state harmonic motion, and the stiction behavior of the internal sliders was not studied in great detail. In this work, the force-constitutive formulation of the five-parameter Iwan-type model is implemented computationally and then examined to understand its behavior in more general transient scenarios. The five-parameter model is found to have a complicated dependence on its displacement history. If the model reaches a maximum steady-state displacement after a ring-up response (such as occurs when the system is excited at resonance by a shaker), the stiffness and damping it exhibits is consistent with those formulated by Mignolet. When the vibration decays below the maximum-achieved displacement, however, the effective stiffness and damping revert to power-law behavior with amplitude. The power-law behavior is functionally similar to that of the four-parameter Iwan model (on which the five-parameter model is based), so the advantage of the weakened coupling between the stiffness and damping is lost in a ring-down response.

Robert M. Lacayo, Matthew S. Allen

Chapter 20. On the Solution of Nonlinear Algebraic Equations Following Periodic Forced Response Analysis of Nonlinear Structures Using Different Nonlinear Solvers

In periodic forced response analysis of nonlinear structures, most of the time analytical solutions cannot be obtained due to the complex behavior of the nonlinearity and/or due to the number of nonlinear equations to be solved. Therefore, numerical methods are widely used. For periodic forced response analysis of nonlinear systems, generally Harmonic Balance Method (HBM) or Describing Function Method (DFM), which transform the nonlinear differential equations into a set of nonlinear algebraic equations, are used. In the literature, there exist several nonlinear algebraic equation solvers based on Newton’s method which have different convergence properties and computational expense. In this paper, comparison of computational performance of different nonlinear algebraic equation solvers are studied where, solvers with different convergence order are selected based on the number of Jacobian matrix and vector function evaluations. In order to compare the performance of these selected nonlinear solvers, a lumped parameter model with cubic stiffness nonlinearity is considered. Several case studies are performed and nonlinear solvers are compared to each other in terms of solution time based on error tolerance used.

H. Sefa Kizilay, Ender Cigeroglu

Chapter 21. Vibration-Based Bolt Tension Estimation for Multi-bolt Joints

Critical bolted connections exist in many engineering structures, from pressurized pipelines to wind turbines. Often there are legal demands for maintaining necessary bolt tension in such joints to prevent failure. The available tools for tightening makes it challenging to obtain the correct tension in a bolt, as well as subsequently checking if a bolt is in fact tightened to the correct level. Recent work proposes to use vibrations for estimating tension in a bolt. Estimation is possible by measuring and analysing transverse natural frequencies and damping ratios induced by e.g. a transversal or longitudinal hammer impact on the bolt itself. The work so far has focused on a single bolt. Most bolted joints consist of many bolts, e.g. a flange connection often has a ring of almost identical bolts. Identical bolts, with almost the same tension, will also have very similar boundary conditions and thus almost the same natural frequencies. If there is only very light damping between two adjoining bolts, a frequency response measured on one bolt after an impact might include the vibrational response of both bolts (the coupling might even be so strong that the two bolts cannot be viewed as entities on their own) leading to the question: How to separate which frequencies belong to which bolt?

M. Brøns, A. Plaugmann, J. J. Thomsen, A. Fidlin

Chapter 22. Experimental Application of Control-Based-Continuation for Characterization of Isolated Modes on Single- and Multiple-Degree-of-Freedom Systems

In this study, two dynamic systems are experimentally investigated using control-based-continuation (CBC) near their resonant frequencies. One system is a single mass oscillator with a stiffness nonlinearity, the other system is a pair of slim beams with a friction element in between and thus significant nonlinear damping. The focus of this study is on the implementation of the CBC-method and the interpretation of the experimental results. The dynamic load-displacement characteristics of both systems (s-curves) at different frequencies and the backbone-curves are presented.

Gleb Kleyman, Martin Paehr, Sebastian Tatzko

Chapter 23. Nonlinear Function Selection and Parameter Estimation of Structures with Localised Nonlinearities, Part 1: Numerical Analysis

Although techniques for system identification of linear dynamical systems are well developed, for nonlinear systems, it is still an open research question. Particularly, finding the correct form of the nonlinear functions (e.g. polynomial, multi-linear) and their parameters for dynamic systems with unknown nonlinearities still represents a crucial aspect to be addressed. Many methods have been presented in the literature to characterize and estimate the parameters of nonlinear dynamical systems; yet, most of them require pre-assumption of the form of the nonlinear function. In order to overcome this issue, in this study a data driven nonlinear autoregressive with exogenous inputs (NARX) model is considered together with a polynomial basis of model terms. We present here the mathematical background and numerical simulations of the proposed methodology. We use Forward Regression Orthogonal Least Square (FROLS) algorithm to select the terms in the NARX model considered. FROLS is a recursive algorithm able to find the more significant model terms based on the Error Reduction Ratio (ERR) criterion. We also explore ways to improve the selection of nonlinear terms using an optimization procedure during which models corresponding to various candidate structures are estimated based on Prediction Error Method (PEM), and the one providing the best fitness to the simulation data is selected. The procedure is motivated by the desire to quantify the degree of nonlinearity of a system, with the ultimate goal of updating a finite-element or other mathematical model to capture the nonlinear effects accurately. Single and multi-degrees of freedom systems that include a wide variety of nonlinearities are considered in this study. The results show that selecting the nonlinear function based on ERR is not affected by noise while the coefficient estimation depends on signal-to-noise ratio (SNR) value and the combination of different nonlinear functions. Discussions are made on sufficiency and efficiency of various optimization tools dealing with the coefficient estimation for MDOF system with localised nonlinearities.

Sina Safari, Julian M. Londono Monsalve

Chapter 24. Tutorial on Nonlinear Reduced Order Modeling for Nominally Cyclic Symmetric Structures and Rotating Machinery

Many important engineering structures such as rotating machinery, including turbine bladed disks, gears, flywheels and satellites are comprised of repeated (nominally identical) substructures arranged circumferentially with cyclic symmetry. Due to this unique arrangement, the system matrices and consequently the dynamics of such structures exhibit specific characteristics (Mitra and Epureanu, ASME Appl Mech Rev. , 2019; Olson et al. ASME Appl Mech Rev, 66(4):040803, 2014). Extensive scientific study and analysis has been conducted on this topic in recent decades. Of particular interest is the change in dynamic behavior when there are deviations in substructures from their nominal, even to a small extent. Colloquially termed mistuning, such deviations are practically impossible to avoid. They manifest as material or geometric differences due to causes such as manufacturing tolerances, wear and differential operation conditions (Castanier and Pierre, J Propuls Power 22/2:384, 2006). Mistuning can lead to strain energy localization, higher system responses and reduction of the operational life cycle and should therefore be carefully considered in the design and analysis of structures. The current industrial practice is to use Monte Carlo simulations to characterize mistuning effects using randomly generated deviations in substructures of the nominal design (Mitra and Epureanu, ASME Appl Mech Rev. , 2019; Castanier and Pierre, J Propuls Power 22/2:384, 2006). Since thousands of dynamic simulations might be required to characterize a single design, full order high fidelity models remain prohibitively expensive. For such tasks, reduced order models (ROMs) are employed instead (Castanier and Pierre, J Propuls Power 22/2:384, 2006; Baek and Epureanu, ASME J Vib Acoust, 139(6):061011, 2017). However, obtaining fast and accurate ROMs for cyclic structures with nonlinearities (Mitra and Epureanu, ASME Appl Mech Rev. , 2019; Baek and Epureanu, ASME J Vib Acoust, 139(6):061011, 2017; Zucca and Firrone, J Sound Vib, 333:916–926, 2014) remains a challenging task. This tutorial aims at summarizing and highlighting some of the most relevant techniques that have been proposed to date, with a specific focus on nonlinear ROMs including contact nonlinearities.

Mainak Mitra, Andrea Lupini, Andrew Madden, Chiara Gastaldi, Bogdan Epureanu

Chapter 25. Excitation Techniques for Nonlinear Dynamic Systems: A Summary

Experimental analysis of nonlinear dynamic mechanical systems is a challenging task. Although well established and widely accepted techniques for Modal Parameter Estimation (MPE) of structures within the linearity assumption exist, several challenges arise once these assumptions cease to be valid i.e. nonlinear systems. Foremost of these challenges in analyzing nonlinear systems is the excitation technique used. While several traditional and non-traditional techniques are available for exciting these systems, each has its own merits and demerits during experimentation and post-processing. This paper summarizes some of these techniques applied to a physical structure and provides a comprehensive discussion on the results obtained. The excitation techniques include impact hammer, electrodynamic shakers, pneumatic system and step relaxation.

Mahesh Nagesh, Akhil Sharma, Randall J. Allemang, Allyn W. Phillips

Chapter 26. Flutter of Double-Bay Panels with Finite Midbay Stiffness

Partitioned or multibay panel designs are a common aerostructural configuration and a potential source of concern for dynamic response and flutter analysis. Fortunately, past work has shown that the flutter boundary of an equally partitioned multibay panel is substantially similar to that of a single, isolated panel—identical, in fact, for the double-bay case. A related question concerns the flutter of multibay panels when the intermediate supports are of only finite stiffness. Preliminary investigation on this front reveals that when the intermediate support is not sufficiently stiff, a very large drop in the flutter boundary can occur due to frequency coalescence between even and odd structural modes. This finding has possible design ramifications but may be of even more interest on the experimental front, providing experimental aeroelasticians an additional parameter with which to investigate flutter and postflutter behavior of panels.

J. D. Schoneman

Chapter 27. Nonlinear System Identification of a Jointed Structure Using Full-Field Data: Part II Analysis

Mechanical joints have a significant influence on the dynamic response of assembled structures. Due to friction, wear, and non-idealized boundary conditions, joints introduce significant nonlinearity into the dynamics of assembled structures. To better understand and, in the future, tailor the nonlinearities, accurate methods are needed to characterize the dynamic properties of jointed structures. In this research, the response analysis for a beam with a bolted lap joint is studied with the help of several available identification techniques. The experimental setup and data capture are described in Part I of this work, providing high spatial resolution data for a variety of excitation methods. The nonlinear identification of the data is the focus of this paper, aiming to perform nonlinear modal analysis and to localize the nonlinear characteristics of the structure with a series of different approaches.

Giancarlo Kosova, Mengshi Jin, Mattia Cenedese, Wei Chen, Aryan Singh, Debasish Jana, Matthew R. W. Brake, Christoph W. Schwingshackl, Satish Nagarajaiah, Keegan J. Moore, Jean-Philippe Noël

Chapter 28. Control Parameters in Non-linear Properties of Linear Guideway in Lateral Direction

To build reliable digital twins of machine tools (also known as “virtual machine tools”), the dynamic properties of nonlinear components in the model should be close to the actual counterpart. The nonlinear dynamics mean that the frequency response functions (FRFs) derived from tap tests are inaccurate because of difference in excitation levels. In this study, dynamic behavior of a linear guideway is sought experimentally using harmonic excitation. It is found that the dynamic properties are affected by (1) excitation level, (2) lubrication, (3) specified preload and (4) static lateral load, and the dynamic properties cannot be fully described using the Hertzian contact model.

Ting-Yen Wu, Yi-Chun Lo, Yum Ji Chan

Chapter 29. Inferring Unstable Equilibrium Configurations from Observed Dynamics

Unstable equilibria play an important organizing role in nonlinear dynamic systems in a global sense. However, it is difficult to measure them directly in a physical experiment. In this study, a digital image correlation (DIC) system is used to capture the transient behavior of a post-buckled beam in which trajectories are generated by repeated impacts. The dynamic data collected by the DIC system, with relatively high temporal and spatial resolution, is used estimate equilibrium configurations of the post-buckled beam, including a detection of the presence of unstable equilibria. The results show good agreement with the equilibrium configurations obtained from two-mode models for the beam.

Yawen Xu, Lawrence N. Virgin, Richard Wiebe

Chapter 30. Bolt Preload Loss Due to Modal Excitation of a C-Beam Structure

Bolted joints often risk failure due to the loss of fastener preload when subjected to dynamic, multiaxial loads. This process is a complex problem that depends on multiple attributes such as loading direction, rate, contact within the threads and the interface, material properties, and many others. Current literature suggests that oscillatory shearing loads appear to be most detrimental to the loss of preload in threaded fasteners. To investigate the effect of less idealized loading conditions, an experimental setup employing a bolted c-beam structure is used to study loss of preload from various initial preloads during harmonic excitation near specific resonant frequencies of the structure. The preload force is measured using bolts equipped with internal strain gauges and the structure is excited at specific modes via sine dwell excitation with an electrodynamic shaker. The experiments were designed to measure loss of preload as a function of excitation duration and strength. A finite element model incorporating a fully-threaded joint is developed in parallel to investigate the effectiveness of each at measuring and predicting bolt loosening.

Max Miller, Chris Johnson, Noah Sonne, John Mersch, Robert J. Kuether, Jeff Smith, Jonel Ortiz, Gustavo Castelluccio, Keegan J. Moore

Chapter 31. Nonlinear Dynamic Analysis of Bolted Joints: Detailed and Equivalent Modelling

A standard finite element analysis of individual components in aero engine and other systems shows a high accuracy compared to experimental measurements of the system response. However when it comes to assemblies, the conventional linear approaches fail to deliver good accuracy. This is due to the uncertain physical phenomena in the contact interface of the joints. A nonlinear contact problem is introduced by the joint and influences the overall dynamic behavior of the engine assembly. Therefore, the linear dynamic models must be coupled with nonlinear analysis of the assembly to investigate the accurate dynamics of the nonlinear system. Flanges are widely used joints that represent the main source of nonlinearities in assemblies. In this study, a finite element simulation of two bolted flanges is considered to identify the nonlinear behavior of the bolted flange joint caused by the presence of friction in contact interfaces. A detailed model of a bolted joint was built in ANSYS in order to evaluate the energy dissipated in a bolted joint and therefore provide accurate modeling of joint interfaces. Due to the high cost of the detailed model, an equivalent model was derived and predictions from this model are compared to the detailed model results in order to provide a robust model for designing bolted joints.

N. Jamia, H. Jalali, J. Taghipour, M. I. Friswell, H. H. Khodaparast

Chapter 32. The Relevance of Nonlinear Normal Modes for Randomly Excited Nonlinear Mechanical Systems

Nonlinear normal modes are of great importance to understand the dynamical behavior of nonlinear mechanical systems and serve as natural candidates for model order reduction. Therefore, their accurate computation as well as mathematical foundation has extensively been studied. While numerous results are available for unforced and periodically forced nonlinear mechanical systems, the case of random external forcing is commonly not considered in the literature. Here, we clarify the relevance of nonlinear normal modes in the case of small white noise excitation. We demonstrate our results on explicit mechanical systems.

Thomas Breunung, George Haller

Chapter 33. Analysis of an Actuated Frictional Interface for Improved Dynamic Performance

Friction in assembled structures is of great interest due to its ability to reduce the vibration amplitude of critical components. The nonlinear behaviour of a structure depends on a variety of physical parameters. Among these parameters, the contact pressure distribution and the contact area have shown to be critical for the behaviour of the joint and the responses of assembled structures. In most application cases the impact of the interface geometry is not considered as a design parameter, although some attempts have been reported to shape the interface geometry for a specific dynamic response.Taking this idea of designing an interface geometry for a better dynamic performance a step further, the concept presented here propose an actively controlled interface geometry and contact pressure distribution, to change the joint behaviour during a vibration cycle. The concept consists of a device capable of manipulating the shape and pressure of a flexible membrane in contact with a rigid punch, subjected to a normal load and a tangential excitation, via a row of piezoelectric actuators.

M. Lasen, Y. Sun, C. W. Schwingshackl, D. Dini

Chapter 34. Vibration Reduction of a Structure by Using Nonlinear Tuned Vibration Absorbers

Tuned Vibration Absorbers (TVA) are commonly used in reducing undesirable vibrations of mechanical structures. However, TVAs work in a very limited frequency range and if the excitation frequency is outside of this range, they become ineffective. In order to solve this problem, researchers started to consider nonlinear TVAs for vibration attenuation. In this study, dynamic behavior of a linear Euler-Bernoulli beam coupled with a nonlinear TVA is investigated. The system is subjected to sinusoidal base excitation. Parameters of the nonlinear TVA is optimized to minimize vibration amplitudes of the primary system. Assumed modes method is used to model the Euler-Bernoulli beam. Nonlinear differential equations of motion are converted to a set of nonlinear algebraic equations by using Harmonic Balance Method (HBM). The resulting set of nonlinear algebraic equations is solved by Newton’s Method with Arc-Length continuation. Nonlinearities used in the TVA are cubic stiffness, cubic damping and dry friction damping. Hill’s method is used to evaluate stability of the solutions obtained. Results of the system with optimum nonlinear TVAs are compared with that of optimum linear TVA. Although, NES show to exhibit good vibration reduction performance – which is in parallel with the results given in literature, due to instability of the frequency domain solutions, it is observed that, actually, it is not as effective as other nonlinear TVAs.

Muhammed Emin Dogan, Ender Cigeroglu

Chapter 35. Simulation-Free Reduction Basis Interpolation to Reduce Parametrized Dynamic Models of Geometrically Non-linear Structures

Virtual design studies for the dynamics of structures that undergo large deformations, such as wind turbine blades or Micro-Electro-Mechanical Systems (MEMS), can be a tedious task. Such studies are usually done with finite element simulations. The equations of motion that result from the finite element discretization typically are high-dimensional and nonlinear. This leads to high computation costs because the high-dimensional nonlinear stiffness term and its Jacobian must be evaluated at each Newton-Raphson iteration during time integration. Model reduction can overcome this burden by reducing the high-dimensional model to a smaller problem. This is done in two steps: First, a Galerkin projection on a reduction basis, and, second, hyperreduction of the geometric nonlinear restoring force term.The first step, namely finding a proper reduction basis, can be performed by either simulation-based or simulation-free methods. While simulation-based methods, such as the Proper Orthogonal Decomposition (POD), rely on costly preliminary simulations of full high-dimensional models, simulation-free methods are much cheaper in computation. For this reason, simulation-free methods are more desirable for design studies where the amount of the so called ‘offline costs’ for reduction of the high-dimensional model are of high interest. However, simulation-free reduction bases are dependent on the system’s properties, and thus depend on design parameters that typically change for each design iteration. This dependence must be taken into account if the parameter space of interest is large.This contribution shows how design iterations can be performed without the need for expensive simulations of the high-dimensional model. We propose to sample the parameter space, compute simulation-free reduction bases at the sample points and interpolate the bases at new parameter points. As hyperreduction technique, the Energy Conserving Sampling and Weighting method and the Polynomial expansion are used for hyperreduction of the nonlinear term. In this step, we also avoid simulations of the high-dimensional nonlinear model. The coefficients of the hyperreduction are updated in each design iteration for the new reduction bases.A simple case study of a shape parameterized beam shows the performance of the proposed method. The case study also accounts for a last challenge that occurs in models that are parametric in shape: The topology of the finite element mesh must be maintained during the design iterations. We face this challenge by using mesh morphing techniques.

Christian H. Meyer, Daniel J. Rixen

Chapter 36. Experimental Spectral Submanifold Reduced Order Models from Machine Learning

Nonlinear system identification is a challenging problem in experimental modal analysis. It is currently tackled using a toolbox approach, where different techniques are employed depending on the structural system under investigation, the identification goals and the type of excitation used. In this contribution, we exploit analytic reduction to spectral submanifolds combined with machine learning techniques in order to obtain the nonlinear coefficients up to cubic order of a single-degree-of-freedom reduced order model. The system measurements aimed at model fitting can be performed using any type of excitation techniques, ranging from free-decay to sine-sweeps or random shaker testing. We illustrate the accuracy of our method using both simulated and real experimental data.

Mattia Cenedese, George Haller

Chapter 37. Development of an Experimental Rig for Emulating Undulatory Locomotion

Some creatures, such as eels, snakes and slender fish use body undulations to move the fluid around them and create propulsion; similarly, travelling waves in structures can be used as an alternative propulsion system. Based on this bio-inspired mechanism, the application of travelling waves is widespread in engineering, e.g. motors, pump systems and transport devices. However, generating high amplitude travelling waves is not as easy as generating standing waves, since travelling waves are generally observed away from resonance. Also, the reflection of the waves at the boundaries can interact destructively further reducing their amplitude. This paper investigates the characteristics of the experimental rig built for emulating the propulsion mechanism used in undulatory locomotion, introducing the conceptual design, and investigating the features that promote travelling waves. Finally, an analysis of the influence of individual components in the design is carried out.

S. N. H. Syuhri, A. McCartney, A. Cammarano

Chapter 38. Identification and Modeling of a Variable Amplitude Fatigue Experiment Apparatus with Damaged Beam Specimen

The useful remaining life of engineering structures under variable amplitude (VA) fatigue loading remains a major unresolved engineering problem. The existing proposed life prediction models are usually based on empirical approximation from experimental results (Fatemi, Yang Int J Fatigue 20(1):9–34, 1998, Santecchia et al. Adv Mater Sci Eng 2016:1–26, 2016). The variable fatigue experiment apparatus in this extended abstract was designed for simulating structural fatigue with a high testing frequency, variable R-ratio as well as modifiable experimental layout (Falco et al. J Vib Acoust 136(4):041001, 2014). In previous studies, the inherent nonlinearity of the testing rig was detected, the obtained parameters allow one to properly use this testing rig within its linear region. As damage accumulates, however, the corresponding dynamic characteristics of the specimen alter accordingly. Therefore, proper modeling considering the interaction between the inherent nonlinearity and the damage induced nonlinearity for both (1) opening crack and (2) breathing crack is necessary for future fatigue life estimation under complex fatigue loading. Here, nonlinear system identification of the lately modified variable amplitude fatigue experiment apparatus is presented based on a combination of first-principles and data-driven modeling techniques. Eventually, structure-damage interaction dynamics will be described to model the underlying fatigue evolution and structural dynamics interactions.

Hewenxuan Li, David Chelidze

Chapter 39. Higher-Order Decompositions for Modal Identification and Model Order Reduction

Output only modal analysis is an essential tool for monitoring operations of complex large structures like offshore platforms or studying complex flow dynamics. Here we consider a higher-order singular value and non-Hermitian matrix decompositions and describe how they can be used in linear modal analysis to enhance the currently available output only modal analysis methods such as dynamic mode decomposition or eigenvalue realization algorithm. In addition, we show how these methodologies can be used for empirical nonlinear modal identification to obtain the slow flow dynamics of nonlinear dynamical systems. Finally, we show how this information can be used to obtain high-fidelity robust reduced-order models of nonlinear systems.

David Chelidze

Chapter 40. Utilizing Modal Testing for Monitoring the Structural Health of Wind Tunnel Facility Hardware

The 10- by 10-Foot Abe Silverstein Supersonic Wind Tunnel (10 × 10) is the largest and fastest wind tunnel facility at NASA’s Glenn Research Center(GRC) and is specifically designed to test supersonic propulsion components from inlets and nozzles to full-scale jet and rocket engines (10 × 10 Abe Silverstein Supersonic Wind Tunnel, ). Recently, a critical part of the wind tunnel failed and required a redesign before reintegrating into the facility. The design requirements of this new component required that clearances between large metallic components exist, which have the potential for undesirable nonlinear dynamics to occur, in particular rattling. Rattling is feared to occur when the wind tunnel is being operated in certain flow regimes that induce cyclic aero loads on the new component near its natural frequencies. This paper describes the approach taken to better understand and resolve this vibration problem using modal testing. A modal test was developed and executed by GRC’s Structural Dynamics Lab to quantify the modal parameters of the structure, namely which specific excitation frequencies caused the structure to rattle. These results were shared with facility operators as frequency ranges that should be avoided to ensure maximum lifespan of the new structure. Additional means of structural health monitoring (SHM) as well as Vortex shedding are briefly discussed in this paper.

Kenneth Pederson, Vicente Suarez

Chapter 41. An Efficient Coupled Modal Quasi-static Approach for Characterizing Non-linear Modal Properties of Prestressed Structures

Linear modal analysis is a powerful tool in studying linear dynamical systems with several Degrees-of-Freedom (DoFs). There has been an increasing interest in how this can be extended to large (in the sense of number of DoFs) non-linear dynamical systems. The current study proposes an extension to the stationarity of Rayleigh quotients, a classical technique for linear modal analysis, and demonstrates its applicability to conservative and non-conservative non-linear systems. Apart from offering a theoretical motivation of modal analysis in non-linear dynamics, the approach also circumvents several limitations in previous quasi-static non-linear modal analysis methods. The method is demonstrated on a simplified model of a bolted-joint which includes unilateral springs and elastic dry friction elements describing the non-linearities. The results are compared with the Extended Periodic Motion Concept (EPMC), a frequency domain approach based on periodic solutions.

Nidish Narayanaa Balaji, Matthew R. W. Brake

Chapter 42. An Assessment of the Applicability and Epistemic Uncertainties Inherent to Different Classes of Friction Models for Modeling Bolted Interfaces

Modeling the contact in bolted structures is a persisting challenge in the community. One of the greatest obstacles in developing predictive models is a lack of understanding of the relative epistemic uncertainties (model form errors) inherent to different choices of contact constitutive modeling approaches. The contact constitutive models affect the stiffness and hysteresis of the contact resulting in different frequency and damping properties. Multi-Objective Optimization (MOO) is applied to find solutions that minimize the deviation in both frequency and damping from experimentally measured properties. Then the concept of non-domination of design parameter sets is used from MOO approaches to develop a quantitative understanding of the epistemic uncertainties, i.e., the inexactness of each approach to represent experimentally measured properties. The current study investigates the uncertainty associated with conventionally popular simplified approaches such as the use of phenomenological models (e.g., Iwan or Bouc-Wen models) and constitutive models (e.g., Jenkins models) within a whole-jointed framework.

Justin H. Porter, Clayton R. Little, Nidish Narayanaa Balaji, Matthew R. W. Brake

Chapter 43. Hyper-Reduction Approaches for Contact Modeling with Small Tangential Displacements: Applications for a Bolted Joint

The current paper presents a study of two approaches for the hyper-reduction of contact interfaces for non-linear structural dynamics analyses. The first approach is a reformation of the “whole-joint approach” wherein different regions of the contact interface are grouped together using a “non-linear patch”; and the second approach is based on a selective remeshing of the interface. Both are conducted based on some field objectives such as contact traction, etc. A recently developed quasi-static modal analysis technique is applied to characterize the systems in terms of amplitude-dependent modal properties, which are compared as a qualification metric.

Nidish Narayanaa Balaji, Tobias Dreher, Malte Krack, Matthew R. W. Brake

Chapter 44. Fatigue Damage of a Single-Edge Notched Beam Specimen Under Variable Amplitude Loading with Similar Probabilistic and Cycle Counting Statistics

Fatigue life estimation under variable amplitude (VA) loading is still a major unresolved engineering problem. The linear cumulative damage rule (LDR) based methodology is inadequate to predict fatigue damage or useful remaining life if the fatigue loading is complex (not sinusoidal). In addition, the LDR based damage estimation methods rely on statistics of various cycle counting methodologies (e.g., rain-flow counting) where the load interaction effects are being ignored. It has been shown that the damage estimation fails if two load-time (load-cycle) histories with different temporal dynamics have the same or similar load spectra. A robust damage estimation should take the temporal dynamics, i.e., the overload/underload information, stress memory statistics, into consideration. In this extended abstract, the shortcomings of the LDR based methods will be presented through a comparison where two VA load spectra with similar stress history and cycle counting statistics were applied to a singled-edge notched beam specimen.

Hewenxuan Li, David Chelidze
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