Nonlinear Structures & Systems, Volume 1

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.

While experimental modal analysis for linear systems is well established and widely used, no universal procedure for identification of nonlinear oscillatory systems is currently available. Thus, the authors develop an automated tool to robustly identify nonlinear oscillators from data. Frequency response curves measured with periodic shaker excitation are arguably the best source to obtain a robust model for nonlinear oscillators. In this setting, an analysis in the frequency domain beneficially reduces noise and helps compress information. Then, data-driven identification techniques, curve fitting, as well as a judicious parameter elimination are combined to yield a low-order and sparse nonlinear oscillator, which can be used to accurately capture the considered system’s forced response curve.

Modern engineering solutions often aim to improve the energy efficiency of their structures by including lightweight, flexible, and slender designs. In practice, it is not always possible to maximise these characteristics and hence the efficiency of the structures, as they exhibit complex, nonlinear structural dynamics. Unless this behaviour is accurately predicted or controlled, the system may encounter extremely destructive behaviour, which can lead to catastrophic mechanical failure. Non-intrusive reduced-order models (NIROMs)—which project the system dynamics onto a reduced set of modes and approximate the nonlinear components of the behaviour—have, therefore, been of great interest and have the potential to greatly increase the industrial uptake of high-efficiency, nonlinear structures. Existing methodologies for NIROM generation apply linear regression to static force and displacement cases, but this approach has previously been demonstrated to be overly dependent on the scale of these characteristics, a point that has prevented wider application. In this work, initial steps are taken to utilise physics-informed recurrent neural networks (RNNs) in place of the static step, allowing the dynamic behaviour to be more accurately captured in the NIROM. First, the use of random and periodic data series is applied in the training stage, with low-pass filtered white noise shown to provide the more reliable model. Following this, long short-term memory RNNs are developed both with and without a physics-informed loss function, with the former demonstrating faster convergence and more accurate predictions. This study represents the first steps taken in a wider project that aims to improve the accuracy and reliability of nonlinear NIROMs, so that they may be more readily applied in a real-world setting.

In this chapter, nonlinear vibration analysis of L-shaped beams is performed for different structural parameters, and the effects of these parameters are observed. The L-shaped beam is composed of two beams joined end to end and perpendicular to each other; therefore, the system is considered as two separate beams with the same boundary conditions at their mutual ends. In addition to this, concentrated masses are attached to each beam on the L-shaped beam. The dynamic model is obtained by using Euler-Bernoulli Beam Theory and Hamilton’s principle. These equations are further simplified by disregarding the axial motions of the beams and only the transverse motions are considered in calculations. Galerkin’s method is utilized to discretize the obtained nonlinear partial differential equations into a set of nonlinear ordinary differential equations. These nonlinear ordinary differential equations are converted into a set of nonlinear algebraic equations by using Harmonic Balance Method (HBM), which are then solved numerically by using Newton’s method with arc-length continuation. In order to observe the effect of the nonlinearity, a linear solution is also obtained and compared with the nonlinear solution. Several case studies are performed in order to observe the effect of system parameters on the nonlinear steady-state response of the L-shaped beam.

Creating mathematical models of mechanical structures with bolted joints is still a challenging topic, which is the base of ongoing research efforts to develop methods to characterize and quantify nonlinearities in mechanical structures. An approach to efficiently handle the identification of large nonlinear structural systems using measured data is to formulate the identification problem in a low-dimensional space, that is, nonlinear modal reduced order model (NMROM). This chapter discusses the effects of truncating the order of reduced order model in their prediction accuracy and its impact on the nonlinear model identification process that uses the reduced order model. This chapter also presents results of a virtual sensing strategy that is used alongside to the model reduction to lift the necessity of direct measurement of the degrees of freedom where the active nonlinear element is located. An experimental setup of a bolted joint structure is used to demonstrate the application of reduced order modeling for identification purpose. The results quantify the accuracy of the selected model in predicting global responses of the structure such as amplitude-dependent frequencies and damping ratios.

In this work, we evaluate the usefulness of nonsmooth basis functions for representing the periodic response of a nonlinear system subject to contact/impact behavior. As with sine and cosine basis functions for classical Fourier series, which have C∞ smoothness, nonsmooth counterparts with C0 smoothness are defined to develop a nonsmooth functional representation of the solution. Some properties of these basis functions are outlined, such as periodicity, derivatives, and orthogonality, which are useful for functional series applied via the Galerkin method. Least-squares fits of the classical Fourier series and nonsmooth basis functions are presented and compared using goodness-of-fit metrics for time histories from vibro-impact systems with varying contact stiffnesses. This formulation has the potential to significantly reduce the computational cost of harmonic balance solvers for nonsmooth dynamical systems. Rather than requiring many harmonics to capture a system response using classical, smooth Fourier terms, the frequency domain discretization could be captured by a combination of a finite Fourier series supplemented with nonsmooth basis functions to improve convergence of the solution for contact-impact problems.

This chapter presents a completely novel framework for the prediction and understanding of nonlinear system behavior. The idea is simply that all nonlinear systems can be represented as a combination of linear systems between which information is exchanged. Under harmonic excitation, the periodic responses of such hybrid systems may be easily calculated when the switching between linear systems is specified in terms of time. These time-switching hybrid systems provide a useful stepping stone to more realistic piecewise linear systems where the switching criteria are specified in terms of displacement and/or velocity. This chapter details the framework and illustrates its ability to efficiently predict the periodic responses from piecewise linear systems. The framework is also shown to be capable of predicting both stable and unstable periodic responses for conditions where jump behavior is possible. The extension of the work to continuous nonlinear systems is also briefly discussed.

In the past two decades, it is fair to say that there has been an explosion in the use of machine learning technology or ‘data-driven’ methods, across the whole subject of engineering; this is no less true of the subdiscipline of structural dynamics. A modern dynamicist needs, at the least, some familiarity with these technologies. This paper attempts to give an overview of some of the main ideas in ‘data-based’ engineering, by focussing on the (comparatively) smaller area of nonlinear dynamics—indeed on nonlinear system identification. A particular viewpoint is adopted, based on modern Bayesian methods of regression. Considerable attention is paid here to the desirability of combining measured data with physical insight when modelling dynamic systems and structures. Although this view naturally begins with the idea of ‘grey-box’ models, this generalises into the emerging subject of physics-informed machine learning. Although this tutorial necessarily focusses down on a narrow application context, the many references allow the curious reader to explore further afield.

The constant drive to improve the performance of aeronautic structures is leading to new designs where nonlinearity is ubiquitous. Accurately predicting the dynamic behavior of nonlinear systems is very challenging because they can exhibit a wide range of behaviors that have no linear equivalent and are very sensitive to parameter changes. In this work, we consider a physics-based model to capture the underlying linear behavior of the system. This linear model is then augmented with a data-driven, machine-learnt model that captures the nonlinearities present in the system. Standard ML models have, however, several important shortcomings from an engineering point of view. They often require large training datasets, do not generalize well to unseen conditions, and can even be physically inconsistent. To overcome these limitations, we investigate the use of Lagrangian Neural Networks (LNNs) where a neural network is used to directly model the Lagrangian function of the system. To enforce physical consistency, the Euler-Lagrange equations of motion of the system are obtained by differentiating this neural network using automatic differentiation techniques. The potential of this modeling approach is numerically and experimentally shown on a range of systems with stiffness and damping nonlinearities.

Mechanical joints, which are indispensable for almost all mechanical systems, are often an important source of nonlinearity due to frictional, backlash, and/or preload effects. Recent studies have shown that the contact pressure distribution at bolted joint interfaces is the key parameter that governs joint friction and, therefore, the nonlinear damping mechanism in these systems. The problem is that this pressure distribution is susceptible to several different factors: bolt preload, bolt tightening order, surface roughness, surface flatness, and misalignments during the assembly process. These issues lead to considerable variability and repeatability problems in the nonlinear dynamics of jointed structures. Consequently, the accurate identification of nonlinear damping in jointed structures is still a challenging task. The combined use of the response-controlled stepped-sine testing (RCT) and the harmonic force surface concept (HFS) constitutes a framework that determines frequency response curves from the same measurement in two different ways: either by directly measuring them or by synthesizing them from the identified nonlinear modal parameters. Since any possible discrepancy of the frequency response curves obtained from the same measurement cannot be attributed to the repeatability issue, the RCT-HFS framework validates the accuracy of the identified nonlinear modal parameters in a sense bypassing the repeatability problem. In this study, this novel feature of the RCT-HFS framework is used in identifying and validating the accuracy of the modal model of a benchmark beam with a bolted lap joint.

The nonlinear extension of linear modal analysis is a problem that has received a great deal of attention in the literature. A recently proposed framework casts the problem of nonlinear modal analysis within a machine-learning framework that generates modes that are statistically independent from each other. Thus far, several techniques from machine learning have been applied to learn the nonlinear modal transformation, including a multinomial expansion and a neural-network model. Recent effort has been given to understand the nonlinear modal dynamics within the statistically independent framework from a theoretical perspective; of particular interest are ways that the nonlinear transformations might be specified exactly from the equations of motion. This paper uses the Fokker-Planck-Kolmogorov (FPK) equation to construct a transformation in the reduced single degree-of-freedom nonlinear case; in this way, the dynamics can be made to respect an arbitrary target distribution. It is furthermore shown that setting the target distribution to the Gaussian distribution corresponding to the underlying linear dynamics (whereby all of the nonlinear elements are removed) is not sufficient to produce global amplitude invariance in the modal transformation.

Self-excitation and beating phenomena are the result of nonlinear constitutive behavior of vibrating structures with nonlinear components. These behaviors require externally supplied excitation and can be induced by the combination of near equilibrium damping and nonlinear damping operating far from equilibrium. In this study, dry friction, a mechanism known for producing self-excitation in structures, is explored as a mechanism for producing beat phenomena in structures. The system consists of two masses connected via a linear spring with one of the masses damped via a grounded dashpot that is modeled using a five-parameter friction contact model. The system is modeled and solved using the RK-4 time-integration scheme. We perform system parameter identification of experimental data using the STFT (short-time Fourier transform) and wavelet-bounded empirical mode decomposition (WBEMD) to determine system model variables that may simulate the self-excitation and beat phenomena observed in the structural dynamics. Beat phenomena may also be a result of the existence of two or more closely separated damped natural frequencies. We also investigate the degree to which self-excitation in the structure is driving the nonlinear beat phenomenon as opposed to it being caused by choosing closely separated damped natural frequencies. We address this question using nonlinear normal modes (NNM) analysis which provides frequency-energy dependence of the modes as system parameters change. The approach developed here is useful for the design of energy harvesting and vibration isolation systems that are subjected to sliding friction.

While recent research has greatly improved our ability to test and model nonlinear dynamic systems, it is rare that these studies quantify the effect that the nonlinearity would have on failure of the structure of interest. While several very notable exceptions certainly exist, such as the work of Hollkamp et al. on the failure of geometrically nonlinear skin panels for high speed vehicles (see, e.g., Gordon and Hollkamp, Reduced-order models for acoustic response prediction. Technical Report AFRL-RB-WP-TR-2011-3040, Air Force Research Laboratory, AFRL-RB-WP-TR-2011-3040, Dayton, 2011. Issue: AFRL-RB-WP-TR-2011-3040AFRL-RB-WP-TR-2011-3040), other studies have given little consideration to failure. This work studies the effect of common nonlinearities on the failure (and failure margins) of components that undergo durability testing in dynamic environments. This context differs from many engineering applications because one usually assumes that any nonlinearities have been fully exercised during the test.

Determining the similarity of an existing structure with a reference structure is an important problem in structural dynamics. For this purpose, many metrics have been developed to quantify the similarity of frequency spectra, such as two transfer functions. However, these approaches yield an aggregate or single numerical score for the similarity over an entire frequency range. This paper, instead, applies these common similarity metrics across a range of frequencies and plots the results to illustrate instances where counterintuitive results can occur. For example, the highest degree of similarity often occurs at a frequency where the two frequency spectra appear to diverge.The result is that counterintuitive cases can be corrected by applying a log frequency shift to the response, enabling better comparisons. Additionally, a similarity metric that compares the phase of the frequency spectra can be applied to make further comparisons. This paper seeks to verify the new methods presented in Manring et al. (J Sound Vib 539:117255, 2022) using a modified experiment and proposes a windowing method as another tool for comparing similar transfer functions. The authors investigate these approaches while applying historical measures of similarity, to provide a more intuitive result for a similarity score. While the shifted frequency spectra can now provide more intuitive comparisons of the degree of similarity, the degree of shifting the frequency segments provides an additional opportunity to quantify the differences in the frequency spectra. The developed approaches were applied to both theoretical and experimental systems.

The study of nonlinear normal modes has become a very popular subdiscipline in the structural dynamics community. This is principally due to the fact that they allow for a practical generalization of the concept of spectral invariant manifolds in linear dynamics. There have been a lot of analytical successes achieved in this area through the application of the method of multiple scales which decomposes the response of a nonlinear oscillator into fast-changing dynamics occurring on a slowly varying manifold. Numerical calculations have also been carried out using a periodic ansatz (with resonance-based constraints). Most of such investigations in the past have focused on cases where a single nonlinear mode is studied in isolation. Although this alone provides very interesting results, such as invariant manifold characterization and internal resonance detection, these are usually not sufficient to study the coupling between multiple nonlinear modes. Recent experimental studies have shown that such coupling can lead to very nontrivial trends in the resonant characteristics, making it difficult to correlate computational realizations with experimental measurements. The present paper takes a computational approach frequency-domain numerical simulations conducted using quasi-periodic harmonic balance to obtain numerical insights into the steady-state multi-resonant behavior. These results are compared with signal processing conducted on transient ringdown responses to determine the ramifications of commonly employed signal processing techniques like frequency-domain filtering for mode isolation. Analytical results are also computed using multiple scales for comparisons. Results are presented for a simplified system with two kinds of nonlinearities: geometric and dry friction.

This study focuses on the continuation process that is inherent to control-based continuation. Existing continuation procedures can be separated in two families. Similarly to numerical continuation, derivative-based methods find the solution of an objective function, the derivatives of which are estimated using finite differences. In mapping-based methods, the input parameter space is exhaustively or partially explored during the experiment. The features of interest can then be extracted during a post-processing phase or in parallel to the experiment. A novel arclength continuation procedure is developed in this paper. It requires neither the estimation of derivatives nor the identification of responses outside the features of interest, thus simplifying and accelerating the continuation process. The method is demonstrated numerically using a Duffing oscillator.

In the last decade, various promising nonlinear modal identification techniques have been developed based on the nonlinear normal mode (NNM) concept. Most of these techniques rely on the phase resonance testing approach where the identification of nonlinear modal damping is still an unresolved issue. The response-controlled stepped-sine testing (RCT) framework provides a convenient way of accurately quantifying nonlinear modal damping by applying standard linear modal analysis techniques to frequency response functions (FRFs) measured at constant displacement amplitude levels with standard modal test equipment. Various studies by the authors have shown that these constant-response FRFs come out in quasi-linear form even in the case of a high degree of nonlinearities. The RCT approach has been validated so far on several systems including a real missile structure with moderate damping nonlinearity mostly due to bolted connections and a micro-electromechanical device with a stack-type piezo-actuator. This study makes a step further by validating the method on a real control fin actuation mechanism that exhibits very high and nonlinear modal damping; the maximum value of viscous modal damping ratio goes up to 15% and the percentage change of the damping with respect to vibration amplitude is about 70%.

In recent years, the prediction of the behavior of structures with high-level nonlinearities has been a challenging area of research. In 2021, the Tribomechadynamics Research Challenge was proposed to evaluate the current state of the art in modeling in the community of jointed structures: the task was a blind prediction of the nonlinear dynamic response of a system including a frictional and a geometric nonlinearity. Participants of the challenge were given only the technical drawings, including material and surface specifications required to manufacture and assemble the system and were asked to predict the frequency and damping ratio of the lowest-frequency elastic mode as function of the amplitude. The behavior of the real system was experimentally characterized during the Tribomechadynamics Research Camp 2022. This contribution presents the experimental work performed during the research camp. As the nature of the structure requires a base excitation, two recently developed nonlinear testing techniques have been explored to extract the modal parameters: the response-controlled testing method and the phase-resonant testing method. The results obtained with the different methods are compared and the blind predictions are confronted with the experimental results in order to assess their accuracy.

Automotive and aerospace structures are increasingly making use of thin panels to reduce weight while seeking to maintain durability and minimize noise transmission. These panels can exhibit geometrically nonlinear behavior due to bending-stretching coupling. Additionally, the use of mechanical fasteners results in nonlinear hysteretic behavior due to friction between the contact surfaces. The Tribomechadynamics benchmark structure, consisting of a thin panel clamped at the ends using bolted joints, was developed as part of a research challenge to test the ability of the nonlinear dynamics community to predict the dynamic behavior of a structure with both friction and geometric nonlinearity. Simulating the dynamic response of a high-fidelity nonlinear FE model is highly computationally expensive, even for such a small-scale structure. Therefore, quasi-static methods have been gaining popularity. This paper builds on our previous efforts to predict the amplitude-dependent frequency and damping of the first bending mode of this structure using quasi-static modal analysis (QSMA). A 3D FE model of the TMD structure was analyzed. The paper shows how Python, an open-source programming language, can be integrated with a commercial finite element package to perform QSMA. This minimizes file input/output compared to our previous approach and speeds up the process. We also investigate using the pseudo-inverse of the mode shape matrix, rather than the mass matrix times the mode shape matrix, to further accelerate the computations. The QSMA results are used to fit a reduced-order model to the structure, which comprises a single DOF implicit condensation and expansion (or SICE) ROM for geometric nonlinearity and an Iwan model to characterize friction nonlinearity. This model is able to reproduce the nonlinear modal behavior with high fidelity while significantly reducing the computational cost.

Structural modification methods provide powerful tools to calculate the dynamic behavior of a modified structure from that of the original one. In general, either the modal properties or responses of the original structure are used to calculate the responses of the modified system. These methods reduce the computational efforts drastically compared to a complete system re-analysis, especially when the modification is local. However, such methods do not apply to nonlinear systems due to response-dependent nature of the frequency response functions (FRFs). One of these methods for linear systems, called the “matrix inversion method,” uses the FRFs of the original structure and the spatial properties of the modification to estimate the FRFs of the modified system. Recently, a new method utilizing the response-controlled step-sine testing (RCT) approach was proposed for obtaining the quasi-linear FRFs and response-dependent modal properties of nonlinear structural systems. Full-duality between the quasi-linear constant amplitude FRFs and the nonlinear constant force FRFs was shown around the nonlinear structure’s resonance frequencies. In this chapter, a novel structural modification approach is proposed, which utilizes the matrix inversion method (so far used to modify only linear systems) and the RCT-based quasi-linear FRFs of the nonlinear structure. This approach enables obtaining the modified structure’s quasi-linear FRFs, similar to the linear system structural modification problem. Combining the matrix inversion method for linear systems with the RCT approach enables the efficient calculation of receptances of structures with local nonlinearities placed around response-controlled degree of freedom, or when the modifications are such that the mode shapes do not change significantly, even if the nonlinearity is distributed.

Traditional electronics assemblies are typically packaged using physically or chemically blown potted foams to reduce the effects of shock and vibration. These potting materials have several drawbacks including manufacturing reliability, lack of internal preload control, and poor serviceability. A modular foam encapsulation approach combined with additively manufactured (AM) silicone lattice compression structures can address these issues for packaged electronics. These preloaded silicone lattice structures, known as foam replacement structures (FRSs), are an integral part of the encapsulation approach and must be properly characterized to model the assembly stresses and dynamics. In this study, dynamic test data is used to validate finite element models of an electronics assembly with modular encapsulation and a direct ink write (DIW) AM silicone FRS. A variety of DIW compression architectures are characterized, and their nominal stress-strain behavior is represented with hyperfoam constitutive model parameterizations. Modeling is conducted with Sierra finite element software, specifically with a handoff from assembly preloading and uniaxial compression in Sierra/Solid Mechanics to linear modal and vibration analysis in Sierra/Structural Dynamics. This work demonstrates the application of this advanced modeling workflow, and results show good agreement with test data for both static and dynamic quantities of interest, including preload, modal, and vibration response.

Closed-cell polymer foams are commonly employed as support structures to absorb shock and vibration in mechanical systems. Engineering analysts responsible for system designs that incorporate these foams must understand the effects that intrinsic and extrinsic conditions have on their dynamic responses. Parameters intrinsic at the system level, such as preloading and material properties, along with extrinsic environmental parameters, such as forcing energy and frequency, have the potential to drive the system into nonlinear or chaotic regimes. A suite of simulation-based studies is performed utilizing finite element (FE) analysis to investigate both intrinsic and extrinsic model parameters to understand such effects on the nonlinear system dynamics. A high-fidelity FE model of the mass-foam testbed needs to be developed to perform two tasks – implicitly determine stress states from precompression in the foam and explicitly solve for the system’s response when subject to various dynamic inputs. Using prior knowledge of stochastic quantities in the model, input parameter distributions will be generated to quantify the influences on the system’s dynamic response behavior. An existing testbed consisting of a mass suspended by two pieces of closed-cell polymer foam will be used to perform various experiments for model validation. This project aims to explore the parameter spaces to predict, validate, and confidently bound the nonlinear dynamic behaviors of a system.

The coupling between the frictional energy dissipation and the dynamic response at the joint interface leads to mechanisms that are challenging to identify relying mainly on numerical data. The need for experimental analyses of such mechanisms is vital for better understanding of the inherent features in the contact interface dynamic response and for creating models that can evolve with changes in the system and environment. This work looks to gain insights about the different frictional mechanisms in the contact interface such as micro-vibro-impacts and micro-slip and macro-slip under different resonance modes and different environmental conditions. A bolted flange system was designed and manufactured at the Laboratory of Verification and Validation (LVV). Two main sensing systems are used for the testing using accelerometers and displacement sensors. All tests were performed in an environmental control chamber that contains a large multi-axial shaker. The interest in this work focuses on the hysteresis information near resonance which provides insight into the interaction between two substructures along with the contact interface.

To design a vehicle suspension the knowledge of wheel loads is required. These loads are due to road unevenness and can be identified thanks the acquisition of measurements during vehicle rolling on roads or tracks. Some offline methods are used to identify them; most of these methods are based on transfer functions between points of measurements and consider the hypothesis of linear dynamic behavior of the vehicle. This hypothesis leads to misestimation of the exceptional load. We propose an approach based on a nonlinear multi-body model of the half-vehicle and an extended Kalman filter augmented and constrained for the data fusion with measurements from accelerometers, gyrometer, tachometer, and GPS. This half vehicle model lies in a 2D plane. The Kalman state vector is composed of positions and velocities of each solid, the road/track loads are unknown but estimated by the filter, and the state prediction is constrained by kinematic links between bodies.

In this work, the frequency response of a simplified shaft-bearing assembly is studied using numerical continuation. Roller-bearing clearances give rise to contact behavior in the system, and past research has focused on the nonlinear normal modes of the system and its response to shock-type loads. A harmonic balance method (HBM) solver is applied instead of a time integration solver, and numerical continuation is used to map out the system’s solution branches in response to a harmonic excitation. Stability analysis is used to understand the bifurcation behavior and possibly identify numerical or system-inherent anomalies seen in past research. Continuation is also performed with respect to the forcing magnitude, resulting in what are known as S-curves, in an effort to detect isolated solution branches in the system response.

Stiffness of spindles, which are high-speed rotors, has to be determined accurately for precision metal cutting. Dependence of rotor natural frequencies and damping on both amplitude and rotation speed is studied experimentally. The spindle shaft is excited using a noncontact electromagnetic exciter: the excitation suddenly stops for free vibration after the response reaches steady state. The peak finding and picking (PFF) method is used to determine the amplitude-dependent natural frequency and damping. It is found that the nonlinearity trends obtained from different excitation methods are similar.

Limit cycle oscillations (LCOs) in nonlinear aeroelastic systems can be problematic for aircraft and structures, but can also be exploited for energy-harvesting applications. In the presence of constant flow conditions and structural characteristics, once initiated, aeroelastic LCO can persist indefinitely. However, the introduction of external forces from impinging vortices injected upstream of the wing has been shown to modulate the self-sustaining LCO. Under the right conditions, a static bluff body placed upstream of an aeroelastic wing has been experimentally shown to annihilate preexisting LCO and produce a cessation of oscillation. However, the exact conditions and characteristics of both the bluff body and the aeroelastic system must be tailored to produce this behavior. A rotation-oscillating cylinder with an attached splitter plate actuated by a servomotor is capable of producing a well-defined von Kármán vortex street at a range of oscillation frequencies. This system has been shown to both excite and annihilate LCO in a downstream aeroelastic wing when oscillated just below the inherent LCO frequency of the wing. Using real-time sensing on the aeroelastic wing to measure pitch angle and heave displacement, a controller can be designed to trigger the appropriate motion of the disturbance generator to inject prescribed disturbances if wing LCOs are detected. The variable frequency disturbance generator can then be used to either enhance or reduce the wing LCO amplitude or cancel out the LCO entirely. The controller will be designed using state machine control theory. The state machine determines whether the wing is not in LCO, entering LCO, in LCO, or exiting LCO. In this chapter, we present wind tunnel experiments that demonstrate and characterize the ability of an automatically controlled upstream flow disturbance generator to produce or suppress LCO in an aeroelastic wing. The experiments provide insight into how controlled interactions between the aeroelastic wing and prescribed flow disturbances can be used to produce desired LCO behavior and may spur follow-on development of energy-harvesting enhancement devices and on-wing disturbance generator devices that could be implemented in flight vehicles.

Meta-heuristic optimisation algorithms are high-level procedures designed to discover near-optimal solutions to optimisation problems. These strategies can efficiently explore the design space of the problems; therefore, they perform well even when incomplete and scarce information is available. Such characteristics make them the ideal approach for solving nonlinear parameter identification problems from experimental data. Nonetheless, selecting the meta-heuristic optimisation algorithm remains a challenging task that can dramatically affect the required time, accuracy, and computational burden to solve such identification problems. To this end, we propose investigating how different meta-heuristic optimisation algorithms can influence the identification process of nonlinear parameters in mechanical systems. Two mature meta-heuristic optimisation methods, i.e. particle swarm optimisation (PSO) method and genetic algorithm (GA), are used to identify the nonlinear parameters of an experimental two-degrees-of-freedom system with cubic stiffness. These naturally inspired algorithms are based on the definition of an initial population: this advantageously increases the chances of identifying the global minimum of the optimisation problem as the design space is searched simultaneously in multiple locations. The results show that the PSO method drastically increases the accuracy and robustness of the solution, but it requires a quite expensive computational burden. On the contrary, the GA requires similar computational effort but does not provide accurate solutions.

In the present study, experimental continuation is applied to investigate the dynamic behavior of a slender steel beam with an attached particle damper. The slender beam is excited to vibration amplitudes at which it behaves geometrically nonlinear by which multiple coexisting vibration states are possible. The forced response curves are measured while the excitation force is held harmonic. To this end, the higher harmonic components of the excitation voltage are controlled using a Newton algorithm. In addition the backbone curve is measured and a nonlinear modal damping is identified.

Additive manufacturing has become increasingly popular in the last decades and has shown great potential for designing and manufacturing innovative design solutions. Recently it has been demonstrated that additive manufacturing can be used to create monolithic compliant mechanisms that can avoid assembly and relative movement between components, showing considerable advantages in their use in harsh environments (i.e. space applications). In this paper, we explore the possibility of adopting 3D-printed compliant mechanisms as tuned-mass vibration absorbers: the challenge is to identify the characteristics of an equivalent nonlinear oscillator that can be used to assess the performance of the absorber. The experimental and numerical results show that the proposed compliant mechanism offers a complex nonlinear dynamic behaviour and it can effectively act as a vibration absorber for a simple cantilever beam.

Methods to reduce the vibration amplitude of blisks are essential to ensure the safe operation of turbomachinery. Resonant vibration absorbers (RVAs) can be used to mitigate vibrations of structures through energy transfer and absorption. Specifically, an RVA is a friction damper with possible impacts that is tuned to the natural frequency of the blisk, allowing energy transfer from the beam to the damper. The state of the art for blisk nonlinear dynamics includes contact models that capture the energy dissipation from friction dampers. Several recent studies have focused on modeling Coulomb friction and repetitive impacts to analyze their effects on reducing vibration amplitudes in blisks. This chapter explores the effectiveness of RVAs in dissipating energy in blisks through impacts. To do so, methods combining finite element modeling, reduced order modeling, and nonlinear structural dynamics are used to analyze the forced response of a beam that resembles a blade in a blisk equipped with an RVA. Specifically, an RVA is attached near the root of the beam where the beam experiences only small displacements similar to locations under the rim of blisks. The RVA device provides impacts at two surfaces connected to the host structure, namely, the beam. The dynamics of the system is analyzed using the harmonic balance method to account for nonlinear impact forces. Dynamic effects of impacts are simulated with a novel microscopic model that captures local material elasticity and viscosity. Results show that increasing the local stiffness of the impact areas shifts the first resonance of the beam tip response to higher frequencies. In addition, increasing the damping of the material at the impact areas decreases the beam response amplitude at resonance. Furthermore, a reduction in the impact time per cycle is observed with increasing material damping. Finally, the best RVA performance is observed when the beam and damper are moving out of phase.

The dynamic response of plate systems in which similar size plates are joined with bolts is modeled. Specifically, the modal response of analytical, numerical, and experimental models is compared and is shown to vary with respect to bolt pattern, number of joint interfaces, plate thickness, and bolt preload. The contrast between the numerical (validated with experimental data through model updating) and the analytical models quantifies the complexity of a joined plate system as opposed to a homogenous plate. Comparing and accurately modeling these configurations further the understanding of the dynamic response of layered joined plates.

Electrostatic speakers and ultrasound transmitters, including capacitive micromachined ultrasound transducers (CMUTs), may be thought of as systems with applied voltage as the input variable and diaphragm displacement and sound pressure as the output variables. These electrostatically actuated systems are inherently nonlinear. Electrostatic pressure is proportional to the square of the applied voltage and is inversely proportional to the cube of the time-varying diaphragm-backplate gap spacing. The squeeze film damping coefficient is also dependent on the time-varying gap spacing. For large diaphragm displacement, strain stiffening of the diaphragm can also affect the transducer dynamics. These nonlinear systems are most commonly driven in a linear regime, whereby the moving diaphragm is biased to an operating point with a large static voltage and a relatively small dynamic signal voltage is superimposed to provide actuation of the diaphragm. In recent work, we have demonstrated advantages of exploiting the full operating range of electrostatic transducers, not restricted to a linear operating regime. Our recent focus has been on using commonly available MEMS microphone structures as airborne ultrasonic projectors at frequencies of up to 100 kHz. Although designed as receivers to work in the audible range, these commercial devices are effective as ultrasound transmitters. A large dynamic input voltage is applied to force the diaphragm to traverse the full diaphragm-backplate gap, typically 2 μ $$\upmu $$ m. Electrostatic pull-in,whereby the diaphragm contacts the backplate,can be exploited as a braking feature to instantaneously eliminate diaphragm ring-down and increase the bandwidth of generated waveforms. Having an accurate model for these nonlinear devices is advantageous. A state-space model based on a segmented diaphragm approach has recently been summarized in prior work. The model is suitable for time-domain simulations of the movement of the diaphragm in response to input drive signals and addresses the nonlinear system aspects listed above. In this work, diaphragm waveforms in response to various input signals are measured using a laser doppler vibrometer (LDV) and compared against simulation results to verify the model’s accuracy.