Nonlinear Structures & Systems, Vol. 1

Aircraft wings can experience dangerous flutter instabilities due to fluid–structure interactions. These instabilities can occur before the typical flutter point, posing a significant risk as they involve multiple stable solutions with large amplitudes. Besides, they become more dominant with the increased nonlinearities that arise in novel flexible designs. Accurate prediction of such behavior is vital for aircraft safety. Traditional modeling methods for aeroelastic systems are computationally expensive and challenging when accurate models are lacking, especially for large and complex systems. On the other hand, in-flight flutter tests are risky and only provide approximate flutter speed estimates based on gradual vibration observation. As a result, safe and model-free methods for testing and evaluating experimental aeroelastic systems are desired. This chapter investigates application of a proposed data-driven forecasting method for predicting flutter and its bifurcation diagrams in a 2-degree-of-freedom experimental airfoil. The bifurcation forecasting approach, relying on measurements collected in safe pre-flutter regime, offers insights into post-bifurcation dynamics without relying solely on expensive and risky in-flight tests. The proposed stability analysis approach eliminates the need for complex models or unsafe wing oscillations by using pre-bifurcation transient measurements. Results show the strong potential of this approach to enhance aeroelastic systems safety and performance in a more efficient and cost-effective manner.

This chapter presents an example of how challenging can be the validation of system responses when a product must be certified before being commissioned for a launch or flight. Several divisions of industry dealing with dynamic models and simulations of forced vibrations are discovering how the nonlinear dynamics of joints can lead to inaccuracies in the simulated vibration responses, mainly due to the nonlinear relationships of frequency and damping with the vibration amplitude (which is high for qualification tests). Hence, the example of this chapter refers to analysis of test data gathered from a solar array in qualification tests. It is shown how dissimilar simulated and measured vibration data can be, and how the qualification levels enhance the nonlinear dynamics, often not predicted by dynamic models. Finally, it presents an effort to analyse a cut-out of the solar array in terms of frequency and damping nonlinearity.

The nonuniqueness of residual tractions is an epistemic uncertainty that can lead to significant variability in the dynamic behavior of friction-damped structures. While numerous studies have addressed this phenomenon with deterministic perspectives, probabilistic approaches have never been considered in its consideration. In this study, we treat the nonuniqueness of residual tractions as a source of stochastic uncertainty and use a probabilistic approach, namely polynomial chaos expansion, to investigate its influence on the variability of dynamics of friction-damped structures. A planar three-mass oscillator representing a simplified model of two adjacent blades with an underplatform damper is employed as a numerical benchmark for the study. The uncertainty is first quantified on the amplitude-dependent modal parameters, and it is then propagated to forced response for a given harmonic excitation level with the closed-form expression governing the amplitude-frequency curves. The outputs are also parameterized by uncertain contact parameters, i.e., coefficient of friction and contact stiffness, to compare which parameter exerts a greater impact on the observed variability in the system’s behavior. The results indicate that in fully stuck and gross slip regimes, the response variability is dominated by variances in contact stiffnesses, but in the partial slip regime, the influence of nonunique residual tractions exhibits a pronounced dominance.

The presence of contact and backlash often leads to complex and rich dynamic responses in mechanical structures. This typically occurs in systems such as jointed structures and geared mechanisms, where the inherent complexity leads to responses which are difficult to predict. In particular, identifying models capable of performing reliable predictions is extremely challenging, both for the definition of the equations of motion and for the characterisation of their parameters. In this chapter, we present a systematic approach based on the restoring force surface method for analysing and identifying the localised piecewise nonlinear characteristics in multi-degree-of-freedom systems. Our approach builds on the knowledge of the underlying linear system and outlines a procedure to identify equivalent nonlinear characteristics via a graphic representation of the piecewise restoring forces. The approach is validated against experimental measurements of a test rig representative of a two-degree-of-freedom system with a localised nonsmooth characteristic. The reconstruction of the nonsmooth restoring force, identification of the piecewise characteristics, and the validation of the model against experimental time histories are performed following our systematic procedure, proving the flexibility of this practical approach.

The global climate effort is increasingly dependent on lightweight, flexible designs to provide engineering solutions capable of meeting ambitious emissions targets. Examples of these designs include high-aspect-ratio wings, which are capable of achieving extended flight times using significantly less energy, but their complexity introduces geometric nonlinearity to the system, leading to a substantial increase in complexity. Although these nonlinear dynamics can be accurately modelled using finite element (FE) software, the required magnitude of such models is extremely computationally expensive, preventing their use in real-time applications or extensive modelling procedures. Non-intrusive reduced-order models (NIROMs) for nonlinear behaviour are of great interest to the mechanical engineering community, as they are capable of capturing the full system dynamics using a significantly reduced coordinate system (typically a subset of the vibration modes). However, the generation of reliable NIROMs remains an active challenge. This chapter combines the projection-based strategy adopted by the implicit condensation method with recent results from the field of machine learning to create a novel NIROM generation technique based on time series data. Specifically, a variational recurrent autoencoder is applied to the system dynamics on a reduced modal basis. To complement the ability of VRAEs to reproduce time series and create statistically consistent synthetic data, a second decoder is added to recreate the true parameterization of the nonlinear system of equations.

Large-scale structures are becoming increasingly common for space applications, driven by advances in in-orbit manufacturing and construction, lightweight materials, and robotics. While these structures have the potential to be key enablers for the next generation of space applications—see, for example, recent interest in space-based solar power—their ability to deliver such goals depends on the stability and controllability of their dynamics. Large-scale space structures, which often have architectures at the kilometre length scale, face several important dynamic challenges. In this work, we propose a novel control strategy for flexible satellites based on a dual unscented Kalman filter strategy. The methodology is outlined and then applied to a smaller satellite, which is assumed to be linear. A key aim of this research is the accurate identification of the moment of inertia of the system, which we demonstrate the methodology can achieve within 1% (or significantly closer), even when the initial assumptions are inaccurate. Finally, we discuss the implications for larger structures, and how the work will be expanded in the future.

In a number of applications, predicting the maximum displacement, velocity, or acceleration amplitude that can be undergone by a forced nonlinear system is of crucial importance. Existing resonance tracking methods rely on phase resonance or approximate the response via a single-harmonic Fourier series, which constitutes an approximation in both cases. This chapter addresses this problem and proposes a harmonic balance tracking procedure to follow the amplitude extrema of nonlinear frequency responses. Means to compute the amplitude of multi-harmonic Fourier series and their time derivatives are first outlined. A set of equations describing the local extrema of the amplitude of a nonlinear frequency response is then derived. The associated terms and their derivatives can be computed via an alternating frequency–time procedure without resorting to finite differences. The whole method can be embedded in an efficient predictor–corrector continuation framework to track the evolution of amplitude resonances with a changing parameter such as the external forcing amplitude. The proposed approach is illustrated on two examples: a Helmholtz–Duffing oscillator and a doubly clamped von Kàrmàn beam with a nonlinear tuned vibration absorber.

This tutorial introduces nonlinear methods for model order reduction of structures discretised with finite elements, with a particular emphasis on the case of geometric nonlinear structures. The aim of model order reduction (MOR) is to reduce the dimensionality of a large system of nonlinear ordinary differential equations by performing a change of coordinates from the original ones to new reduced ones. The two main ingredients of each (MOR) method are (i) the change of coordinates and (ii) the reduced dynamics in the new coordinate system. Specifically, nonlinear methods differ from linear-based techniques, as they rely on a nonlinear change of coordinates rather than the addition of new vectors to enlarge the linear projection basis.

A variety of reduced order modeling (ROM) methods for geometrically nonlinear structures have been developed over recent decades, each of which takes a distinct approach, and may have different advantages and disadvantages for a given application. This research challenge is motivated by the need for a consistent, reliable, and ongoing process for ROM comparison. In this chapter, seven state-of-the-art ROM methods are evaluated and compared in terms of accuracy and efficiency in capturing the nonlinear characteristics of a benchmark structure: a curved, perforated plate that is part of the exhaust system of a large diesel engine. Preliminary results comparing the full-order and ROM simulations are discussed. The predictions obtained by the various methods are compared to provide an understanding of the performance differences between the ROM methods participating in the challenge. Where possible, comments are provided on insight gained into how geometric nonlinearity contributes to the nonlinear behavior of the benchmark system.

Prof. Lothar Gaul, who passed away on December 18, 2018, had a tremendous impact on the joint dynamics community. His contributions to joint mechanics range from the well-known Gaul resonator for the experimental study of isolated joints over numerical methods bridging the micro-scale of surface roughness to the macro-scale of structures to semi-active joints for adaptive damping. However, his interests were much broader, spanning vibrations, wave transmission, and acoustics in structures and fluids including fluid–structure interaction with applications in soil–foundation interaction and structural health monitoring. A characteristic of his research was the elegant combination of experimental and numerical methods, where he made significant contributions to the development of the boundary element method and the numerical modeling of viscoelasticity using fractional derivatives.

The tangential friction force of a permanently sticking contact (referred to as residual traction) can have an arbitrary (nonunique) value within the limits defined by the Coulomb cone. However, the actual value of residual traction is unknown due to its dependence on the load history. This uncertainty can introduce significant variability, ranging from 20 to 300 percent, in the level of resonant vibration, as observed in experiments of structures with friction dampers. In this study, we systematically investigate, for the first time, the influence of nonunique residual traction on the vibration response variability of a system with bolted joints. The selected benchmark consists of an L-shaped beam and a cross beam (stiffener) connected by two joints. The joints are aligned under 90-degree angle, establishing a structural coupling between the tangential and normal directions of the contacts. Both experimental and numerical analyses are conducted to assess the effects in detail. In the experimental phase, phase-locked-loop testing is employed to measure the amplitude-dependent modal parameters. On the computational side, the structure is modelled using commercial finite element software, and quasi-static modal analysis is employed. The results show to what extent the nonuniqueness of residual traction contributes to the non-repeatability of the vibration response and the accuracy of the correlation between experiments and numerical analyses.

The instantaneous relationship between frequency and amplitude (backbone curve) of a decaying trajectory represents an important feature of a system, as it provides information about the forced response, if forcing and damping are small. The backbone curve of a system can be directly approximated by the reduced dynamics on spectral submanifolds (SSMs) constructed either from governing equations or from data, without the necessity for extensive numerical computations (numerical continuation or time integration). In this chapter, we study oscillating backbone curves observed in experiments and numerical simulations. Oscillations become more apparent, as the initial condition of decaying trajectory moves farther away from the primary SSM, which is indeed incapable of reproducing this phenomenon. Conversely, a new class of manifolds, fractional (or secondary) SSMs, offer a clear explanation for this observation.

Accurately analyzing systems with frictional contact interfaces poses computational challenges due to the large number of degrees of freedom involved. Although component mode synthesis can effectively reduce internal degrees of freedom, it does not apply to nonlinearly coupled degrees of freedom required for evaluating the nonlinear force law. Consequently, when using methods like the harmonic balance method, a considerable computational effort is needed to compute a large number of variables, specifically the Fourier coefficients. To tackle this issue, we present several interface reduction techniques suitable for implementation within the framework of the harmonic balance method. These reduction techniques can be categorized as either polynomial-based or mode-based approaches. By employing these methods, our objective is to decrease the computational effort associated with calculating numerous Fourier coefficients while retaining sufficient accuracy. To assess the effectiveness of the proposed reduction methods, we compare them in terms of computational efficiency and the accuracy of computed results. We utilize example systems to evaluate the performance of these reduction techniques. Our findings demonstrate that the interface reduction methods significantly enhance computational efficiency by reducing the number of variables. By comparing the solutions with reduced interfaces degrees of freedom to those with full interfaces, we provide insights into the trade-offs between accuracy and computational efficiency.

Modeling the response of nonlinear structures due to random excitation is crucial for the design of mechanical systems, including the estimation of loading on mechanical joints and the fatigue life of nonlinear components. This chapter presents a method for bounding the maximum variance of the output response of a nonlinear system under random excitation of known power spectral density. The proposed approach leverages integral quadratic constraints (IQCs) that enclose the relationship between inputs and outputs of the nonlinearity sufficiently for analysis. While IQCs have traditionally been employed in robust control to analyze stability and performance, recent advancements have extended its applications to analyzing optimization algorithm rate of convergence and stability of transitional flows. In this chapter, we explore an optimization-based algorithm that harnesses different IQCs to bound the nonlinearities in the system. To validate the efficacy of the proposed algorithm, we apply it to the analysis of the Duffing equation, a well-known nonlinear oscillator. Results demonstrate the effectiveness of the algorithm in bounding the maximum variance of the system’s response and its potential for application in the design and analysis of nonlinear structures subject to random vibration.

We compute the periodic responses of geometrically nonlinear beam structures modelled using the Special Euclidean \(SE(2)\) Lie group formulation using shooting and pseudo-arclength continuation. Nonlinear normal modes (NNMs) are calculated for a cantilever beam and are compared to those obtained using a reference displacement-based finite element (FE) model with von Kàrmàn strains. Some specificities of the shooting algorithm which are unique to the underlying Lie group framework are discussed.

Design changes often occur during engineering applications. Although there are numerous methods to implement these changes and efficiently calculate dynamic responses of modified linear systems, modification methods applicable to nonlinear systems are few in number. Considering the cost of re-analysis of modified large-scale nonlinear system models, the development of nonlinear modification methods has major importance. A numerically efficient approach has been developed for the re-analysis of a modified nonlinear system using frequency response function (FRFs) of the original system and the mass, stiffness, and damping properties of the modification. However, this method is applicable if the nonlinearity is confined to a single nonlinear element between a linear structure and ground. In recent research, the authors of this chapter proposed a structural modification method applicable to systems with multiple local nonlinearities distributed at different points on the structure and validated the method on lumped systems that exhibit conservative nonlinearities. This method utilizes the response-controlled stepped-sine testing (RCT) to determine quasi-linear FRFs of the original nonlinear system. These quasi-linear FRFs allow the implementation of an efficient re-analysis method previously developed for linear systems to accurately estimate frequency responses of nonlinear systems modified with linear elements. The current study demonstrates the applicability of the method to nonlinear systems exhibiting nonconservative nonlinearities and severe modification (causing a considerably high shift in the natural frequency) by employing a numerical example. The method is also verified by using a real experiment.

Vibrating slender structures often deform considerable, which trigger nonlinear behavior. For example, when the curvature is no longer small, the equations of motion contain nonlinear terms, which are often neglected assuming small vibration amplitudes. In this chapter, we experimentally observe nonlinear vibration behavior of a slender beam under harmonic excitation. We excite the bending modes of the beam and observe the forced response for different excitation levels. The frequency response shows a softening behavior with a jump phenomenon. The nonlinearity is more pronounced in higher bending modes but is already detected in the first clamped free bending motion. The experimental results are compared with an analytical approximation using the single nonlinear mode theory. The single-mode approach is appropriate for isolated modes and predicts a softening Duffing equation as a minimal nonlinear model of geometrically nonlinear beams. Finally, we obtain the governing equations directly from the measurements utilizing data-driven techniques. The underlying nonlinear differential equation is derived using test functions and sparse identification. The identified parameters are then compared to the analytical model.

Often, system identification is performed on a single data set at a time. When multiple data sets exist, an approach to considering them is to analyze the resulting identified parameters statistically (such as the average frequency, or 95% confidence interval of extracted parameters, etc.). An alternative could be a method to identify the parameters of a system based on data from multiple measurements; then this would potentially lead to an identified system model that is valid over a much larger operating range. In this chapter, new methods are investigated for the estimation of nonlinear characteristics when large amounts of data are available. The methods include direct nonlinear optimization–based identification techniques like the more commonly used sparse identification package, SINDy, and a more customizable sequential learning algorithm. Besides, parameter initialization (such that any optimized models are able to avoid bad local minima) is studied to accomplish a successful identification. Multiple data sets from an experimental setup of nonlinear structures with the expected source of nonlinearity, i.e., the bolted joints are used in this study to evaluate the performance of the identification methods. For assessing the quality of the resulting models, the responses from simulations are compared to the measured responses of the structures such as amplitude-dependent frequencies and damping ratios.

Periodic forced response analysis of nonlinear real systems is a computationally demanding task. In order to reduce the computational burden, different approaches are proposed in the literature. The reduced computational effort required for the response-dependent nonlinear normal modes (RDNMs) developed recently, making them suitable for the computation of the steady-state harmonic response of nonlinear systems by employing modal superposition method (MSM). RDNMs are obtained by representing the nonlinear internal forces as a nonlinearity matrix multiplied by the displacement vector using describing function method (DFM). The nonlinearity matrix is considered as a structural modification to the linear system, and RDNMs are calculated by solving the eigenvalue problem of this modified system. However, the solution of a large eigenvalue problem is computationally demanding. Therefore, a further reduction is made by applying the dual modal space method. A detailed study is conducted on the finite element model of a two-blade system having a shroud-to-shroud contact in order to investigate the performance of utilizing RDNMs in MSM. The finite element model of the system is obtained in commercial finite element software, and one-dimensional friction elements with normal load variation are used at the contact interface. Harmonic balance method (HBM) is used to obtain the nonlinear algebraic equations representing the steady-state response of the system which are solved by Newton’s method. Several case studies are performed, and the effect of using different number of RDNMs is studied.

This chapter explores the application of recent nonlinear identification methods to measurements from structures where traditional methods have failed. Specifically, curved beams that exhibit geometric nonlinearity, softening–hardening, and snap-through behavior and strong modal coupling. System identification is performed using steady-state input–output measurements, using two recently presented methods, the inverse SNRM-based algorithm by Kwarta and Allen, and the sparse identification method by Breunung et al. The latter produces excellent results and new insights regarding what terms are needed to model a structure such as this.

In this chapter, an electromagnetic energy transducer for vibration damping is investigated. The device consists of a magnetic circuit, a flux linking coil, and a variable air gap between a fixed horseshoe type magnet with a moving magnetic core. Due to the magnetic effect caused by the reluctance forces in the air gap, the horseshoe magnet and its counterpart attract each other. The corresponding force–displacement behavior is strongly nonlinear and can be characterized as negative stiffness. We now introduce passive shunts of the coil to create a phase shift in the reluctance force dynamics. This way, the resulting hysteresis between reluctance force and air gap causes damping of an oscillating motion of the moving magnetic core. The nonlinear state equation is solved by applying the harmonic balance method to obtain the magnetic flux for a given harmonic input signal. For harmonic air gap oscillation, resistive as well as resonant shunts are considered at different frequencies. In contrast to a simple energy dissipation, the resonant shunt leads to an amplification of the damping effect as the electric circuit is capable of vibration itself. Furthermore, additional resonance effects can occur if the oscillation frequency is close to, for example, half or one-third of the electrical resonance frequency. This is due to the strong nonlinear characteristic of the system activating the electric resonance with the oscillation’s higher harmonics.

Traditionally, solving the Volterra series for nonlinear differential equations subject to an impulse excitation warrants the use of contour integration or a method of exponential inputs. Both of these solutions may be problematic when implemented on a computer and involve error-prone calculations when performed by hand. This chapter aims to address these issues by presenting a computer-based approach for the impulse response of a nonlinear oscillator with quadratic and cubic stiffness terms.

The generating-series method offers a technique to transform nonlinear differential equations with polynomial nonlinearities into a domain where the calculus problem is converted into an algebraic and combinatorial problem. Within this transformed domain, an iterative scheme determines higher-order Volterra kernels, and the nonlinear terms are expanded using the shuffle product. The generating series is then converted back into the time domain. This chapter showcases the computation of the first six terms in the Volterra series expansion and provides an error analysis by comparing the results to numerical approximations. The findings presented in this chapter contribute to the broader field of nonlinear dynamics and modal analysis; namely, the proposed method not only enhances the understanding of Duffing’s equation but also presents a practical tool for analysing nonlinear differential equations.

In the field of aerospace and mechanical engineering, the identification of mathematical models capable of accurately representing the dynamics of mechanical structures is extremely important to improve the design and accelerate the certification of new systems and structures. Recent developments in the field of machine learning have demonstrated that Neural Networks (NNs) can accurately model the dynamics of linear and nonlinear systems in the time domain. Physics-Informed Neural Networks (PINNs) exploit this property and utilise physics-informed loss functions to identify the parameters of systems. Nonetheless, it is still not completely clear how the loss functions affect the learning process of the NNs and which type of function improves/deteriorates the identification process of the parameters associated with mechanical systems.

In this chapter, we investigate the effect of three different loss functions on the learning and identification capabilities of PINNs when mechanical problems are considered. To this end, classic Forward Neural Networks (FNNs) are embedded in a parameter identification scheme based on physics-informed loss functions, and the parameters (natural frequency and damping) of a single-degree-of-freedom (SDOF) mechanical oscillator are identified via numerical experiments. In order to minimise the required training data, the loss functions are built considering the governing equations of motion and a single dynamics response of the oscillator in the form of either acceleration, velocity, or displacement. Their effect is then evaluated in terms of the accuracy of the identified unknown parameters and the capacity of the NN to predict the unknown physical dynamic responses. We demonstrate that loss functions based on the acceleration time series allow the NN to correctly learn the unknown physical dynamic responses, i.e., the velocity and the displacement, with great accuracy; this results in faster and more efficient learning of the dynamic behaviour of the system which, in turn, allows to identify the correct mechanical parameters.

Resonant plate shock testing is used in the Mechanical Shock Lab at Sandia National Laboratories to simulate pyroshock and other severe shock environments on aerospace components. At this time, damping is required in the resonant plate system to better match specified shock requirements. Current damping systems added to resonant plates, known as damping bars, can disrupt the desired plate modes, resulting in responding modes of the test system far from the design frequency. Alternative damping treatments could be beneficial in maintaining the desired modal response of the resonant plate. A prior study investigated the modal response (natural frequency and damping ratio) of layered square plate assemblies under typical experimental modal test conditions with an impact hammer. This work evaluates the response of layered plate assemblies under shock loading in addition to modal test loading. Shock loading is achieved via a projectile impact, which delivers an impulsive force much greater than achievable with impact hammer loading. A simple empirical model is developed and used to estimate the damping response of a layered resonant plate systems exposed to aerospace component test shock levels.