Nonlinear Structures & Systems, Volume 1

In this contribution, we study the transient scattering of waves from a structural bi-stable element modelled as a Euler elastica. The non-linear elastic element has localized inertia at its end points, each of which is connected to a semi-infinite straight rod. The focus is on longitudinal waves reflected and transmitted into the semi-infinite rods by the non-linear structural interface. The derivation of the transient governing ordinary differential equations for the reflected and transmitted amplitudes is outlined. The key point is the assumption that the semi-infinite rods are linear elastic, with non-linearity embedded into special transient transmission conditions for the scattering amplitudes. The governing equations are solved using standard numerical integration techniques. From a physical point of view, we focus on the effect of the bi-stability on the transmission resonances, identified in the linearized regime. Special attention is given to the up- and downconversion of the frequency content of scattered amplitudes as a function of the amplitude of a harmonic impinging wave.

This paper presents an investigation of complex mode shape analysis caused by non-linear damping. Nowadays, most academics are accustomed to complex mode shapes, which are a characteristic of most axisymmetric structures. The topic was deeply investigated during the 1980s, sparking the sharpest debates about their physical existence or not. However, after nearly three decades, one question still stands, do we know all about complex mode shapes? This paper takes the dust off this topic again and explores how complex eigenvectors arise when the percentage frequency separation between two mode shapes is the same order of magnitude as the percentage damping. The difference between the past and present investigations relates to the non-linear damping that might arise from joint dynamics under various vibration amplitudes. Hence, the new research question is about the investigation of amplitude-dependent damping on the modal complexity. Why bother? There are several engineering applications in both space and aerospace where axisymmetric structures and joint dynamics can impair the numerical analysis that is currently performed. This paper does not offer any solutions but does expand the research on an unsolved challenge by identifying the questions posed.

When exposed to mechanical environments such as shock and vibration, electrical connections may experience increased levels of contact resistance associated with the physical characteristics of the electrical interface. A phenomenon known as electrical chatter occurs when these vibrations are large enough to interrupt the electric signals. It is critical to understand the root causes behind these events because electrical chatter may result in unexpected performance or failure of the system. The root causes span a variety of fields, such as structural dynamics, contact mechanics, and tribology. Therefore, a wide range of analyses are required to fully explore the physical phenomenon. This paper intends to provide a better understanding of the relationship between structural dynamics and electrical chatter events. Specifically, electrical contact assembly composed of a cylindrical pin and bifurcated structure were studied using high fidelity simulations. Structural dynamic simulations will be performed with both linear and nonlinear reduced-order models (ROM) to replicate the relevant structural dynamics. Subsequent multi-physics simulations will be discussed to relate the contact mechanics associated with the dynamic interactions between the pin and receptacle to the chatter. Each simulation method was parametrized by data from a variety of dynamic experiments. Both structural dynamics and electrical continuity were observed in both the simulation and experimental approaches, so that the relationship between the two can be established.

Data-driven machine learning models are useful for modeling complex structures based on empirical observations, bypassing the need to generate a physical model where the physics is not well known or readily otherwise model-able. One disadvantage of purely data-driven approaches is that they tend to perform poorly in regions outside the original training domain. To mitigate this limitation, physical knowledge about the structure can be embedded in the model architecture via the model topology or numerical constraints in the formulation. For large-scale systems, relevant physics, such as the system-state matrices, may be expensive to compute. One way around this problem is to use scalar functionals, such as energy, to constrain the network to operate within physical bounds. We propose a neural network framework based on Hamiltonian mechanics to enforce a physics-informed structure to the model. The Hamiltonian framework allows us to relate the energy of the system to the measured quantities (e.g., accelerations) through the Euler-Lagrange equations of motion. In this work, the potential, kinetic energy, and Rayleigh damping terms are each modeled with a multilayer perceptron. Auto-differentiation is used to compute partial derivatives and assemble all the relevant equations, including computing the generalized inertia matrix by forming the Hessian of the kinetic energy with respect to the generalized coordinates. Moreover, a Bayesian approach is used to estimate model-form error to predict domain shifts in the data and enable model correction. The network incorporates a numerics-informed loss function via the residual of a multistep integration term, allowing the ensemble of networks to be time-integrated with new initial conditions and an arbitrary external force after it has been trained. The approach is demonstrated on simple exemplars, such as a two degree-of-freedom (DOF) damped oscillator with cubic nonlinearities.

System identification of engineering structures is an established area in the structural dynamics research community. It is often used to characterise certain physical properties of a structure using the data measured from it. For structures exhibiting nonlinear behaviour, physics-based approaches are used where a form of nonlinearity is synthesised and parameters are estimated using the data, or probabilistic approaches are investigated to tackle the model uncertainty of structures. However, to build reliable models, the estimated parameters from the measurement data must reflect the true underlying physics of the structure. Therefore, Reduced-Order Models (ROMs) can be used as the surrogate models, where the nonlinear parameters of the ROMs are having a meaningful relation with the physical parameters of the system. In this work, we propose nonlinear system identification in the context of using some recently developed ROMs which account for the kinetic energy of unmodelled modes. It is shown how ROMs may be used to represent low-order, accurate models for system identification. Identification of a nonlinear system with strong modal coupling is demonstrated, using simulated data, while the estimated ROM response shows good convergence with that of full order system. Similarly, the estimated parameters match with those of directly computed ROM.

Engineering structures are often designed using finite element (FE) models. Performing non-linear dynamic analysis on high-fidelity FE models can be prohibitively computationally expensive, due to the very large number of degrees of freedom. Non-linear reduced-order modelling allows the salient dynamics of the FE model to be captured efficiently in a smaller, computationally cheap reduced-order model (ROM).Recent developments in indirect reduced-order modelling techniques enable ROMs to be developed efficiently, accurately, and robustly, for a wide range of structures. Nevertheless, these methods are applicable to conservative systems and are unable to capture the effects of, for example, damping and external forcing. In this work, we show how indirect reduced-order modelling methods can be extended to non-conservative non-linear structures, which offers invaluable insight into the behaviour of the system. We demonstrate the proposed method using a simple oscillator.

An established approach to reduce the effort for the numerical time integration of Finite Element (FE) structures is given by model order reduction (MOR) via subspace projection. These techniques lead to a significant decrease of the necessary degrees of freedom (DOF) by using proper deformation trial vectors. However, if nonlinear loads are applied on distributed regions of the FE structures surface, the computation of these forces is based on physical state-information of all involved nodes. To avoid this dependency, Hyper-Reduction (HR) methods provide a suitable framework to compute the nonlinearity with a reduced number of DOF too. In this contribution, the HR of the nonlinear surface load is based on stress trial vectors, which can be either determined in conjunction with the deformation trial vectors for the MOR (a priori) or as a result of given solution snapshots of the nonlinearity under consideration (a posteriori). In both cases, the stress trial vectors span a subspace, which is combined with a problem formulation via the calculus of variations and a procedure for a reduced selection of integration points (e.g., empirical cubature method). As a result, an HR approach is obtained that allows a more efficient evaluation of the acting nonlinear loads. A numerical comparison of an a priori and an a posteriori subspace is made by using a planar crank drive mechanism, where an elastohydrodynamic (EHD) contact is considered between the piston and the cylinder liner.

Engineers and scientists want mathematical models of the observed system for understanding, design, and control. Many mechanical and civil structures are nonlinear. This paper illustrates a combined nonparametric and parametric system identification framework for modelling a nonlinear vibrating structure. First step of the process is the analysis: measurements are (semiautomatically) preprocessed, and a nonparametric best linear approximation (BLA) method is applied. The outcome of the BLA analysis results in nonparametric frequency response function, noise and nonlinear distortion estimates. Second, based on the information obtained from the BLA process, a linear parametric (state-space) model is built. Third, the parametric model is used to initialize a complex polynomial nonlinear state-space (PNLSS) model. The nonlinear part of a PNLSS model is manifested as a combination of high-dimensional multivariate polynomials. The last step in the proposed approach is the decoupling: transforming multivariate polynomials into a simplified, alternative basis, thereby significantly reducing the number of parameters. In this work a novel filtered canonical polyadic decomposition (CPD) is used. The proposed methodology is illustrated on, but of course not limited to, a ground vibration testing measurement of an F16 aircraft.

Self-excited vibrations can be found in many engineering applications such as flutter of aerofoils, stick-slip vibrations in drill strings, and wheel shimmy. These self-excited vibrations are generally unwanted since they can cause serious damage to the system. To avoid such phenomena, an accurate mathematical model of the system is crucial. Self-excited systems are typically modelled as dynamical systems with Hopf bifurcations. The identification of such non-linear dynamical system from data is much more challenging compared to linear systems.In this research, we propose two different mathematical model identification methods for self-excited systems that use experimental bifurcation analysis data. The first method considers an empirical mathematical model whose coefficients are identified to fit the measured bifurcation diagram. The second approach considers a fundamental Hopf normal form model and learns a data-driven coordinate transformation mapping the normal form state-space to physical coordinates. The approaches developed are applied to bifurcation data collected on a two degree-of-freedom flutter rig and the two methods show promising results. The advantages and disadvantages of the methods are discussed.

Friction dampers are classically used in turbomachinery for bladed discs to control the levels of vibrations at resonance and limit the risk of fatigue failure. It consists of small metal components located under the platforms of the blades, which dissipate the vibratory energy through friction when a relative displacement between the blades and the damper appears. It is well known that the shape of such component has a strong influence on the damping properties and should be designed with a particular attention. With the arrival of additive manufacturing, new dedicated shapes for these dampers can be considered, determined with specific numerical methods as topological optimisation (TO). However, the presence of the contact nonlinearity challenges the use of traditional TO methods to minimise the vibration levels at resonance. In this work, the topology of the damper is parametrised with the moving morphable components (MMC) framework and optimised based on meta-modelling techniques: here kriging coupled with the efficient global optimisation (EGO) algorithm. The level of vibration at resonance is computed based on the harmonic balance method augmented with a constraint to aim directly for the resonant solution. It corresponds to the objective function to be minimised. Additionally, a mechanical constraint based on static stress analysis is also considered to propose reliable damper designs. Results demonstrate the efficiency of the method and show that damper geometries that meet the engineers’ requirements can be identified.

Flutter instabilities might lead to structural failure in aeroelastic systems. Recently, a flap control surface turned into a nonlinear energy absorber (flap-NES) has been introduced to suppress post-flutter oscillations. This nonlinear absorber not only benefits from aerodynamic damping but also acts in a broad range of frequencies. However, the parameters of the NES must be chosen carefully to avoid adverse effects. The optimal design of the flap-NES demands a comprehensive study on the effect of its parameters on the flutter speed and post-flutter dynamics of the aeroelastic system where traditional approaches require considerable analytical and computational efforts. In this work, we study the optimization of a flap-NES to control flutter instabilities of a typical pitch-plunge section. To facilitate efficient optimization and exploration of the design space, we employ genetic algorithm in conjunction with a bifurcation forecasting method, i.e., a data-driven nonlinear stability analysis algorithm developed to construct the bifurcation diagram using limited number of trajectories collected in the pre-flutter regime. Results show that an optimal design on the flap-NES system results in an increased critical flutter speed, reduced post-flutter limit cycle oscillation amplitudes, and improved nature of the bifurcation diagram from subcritical to supercritical.

This work presents the approach and results of the Dynamics Group at Imperial College in face of the Tribomechadynamics 2021 challenge. The challenge encourages to obtain the best blind prediction of a benchmark structure so that a transversal comparison, among the groups working in nonlinear studies, is done. The approach of the Dynamics Group consists in predicting the behaviour due to friction nonlinearities at the location where more energy dissipation is observed. The results show a slight softening in the contact with an overall shifting of the linear frequency of 2.6% and a damping increase of about 1.5% with respect to the linear damping. The effect of the contact is modest, given the lack of dissipated energy and the fact that geometric nonlinearities are not considered throughout this study.

Complex machinery holding several components in assembled structures requires a multitude of joints which inevitably develop frictional contacts under different dynamics conditions. Those conditions have been studied from a passive control point of view, predesigning the shape of the contact that will face later frictional circumstances. That passive approach can be extended into an active form of control.This paper proposes a novel concept of active manipulation of the contact shape and consequently the contact pressure distribution developed in frictional interfaces. This concept is initially tested with computational simulations that show sufficient confidence in the concept as to allow its continuation with the development of a physical prototype and experiments. The proposed system has been manufactured as a proof of concept, and here we present the setup along with the experimental measurements that will be carried out with it.

Additive manufacturing (AM) offers unique capabilities to incorporate the effects of different mechanics. Traditionally, AM parts are fabricated to reduce weight while maximizing strength. More recently, it has been observed that AM can also be used to fabricate components with internal particle dampers through the process of leaving a small pocket of unfused powder during the printing process. This small pocket of unfused powder can assist in eliminating unwanted vibrations in parts without the post-processing involved with traditional particle dampers. Previous works have reported on the damping capability of various AM designs. Those works focused on a narrow range of excitation amplitudes, over which approximately linear behavior was observed. This work reports on a study of damping capability over a much wider range of excitation amplitudes. Over this wider range, nonlinear behavior was observed. These results indicate certain pocket configurations and modes result in a system regime such that the response amplitude does not exceed a threshold value, up to a certain system excitation. This trend directly translates into an increasingly damped system as excitation increases to a certain level. Other configurations and modes result in a typical single DOF oscillator FRF, while the damping is dependent on the amplitude of the base excitation. Through further experiments, it was determined that there are certain ranges of base excitation that result in consistent nonlinear responses while other ranges result in a response similar to that of a linear system. These results suggest that a system intentionally designed to operate in scope of the nonlinear region may have significantly higher vibration reduction than was thought possible based on previous works.

Modern turbomachinery contains integrally bladed disks, or blisks, which are nominally cyclic symmetric structures manufactured as a single piece. Unlike traditional disks with inserted blades, blisks lack contact interfaces and thus have very low internal damping. Additionally, due to imperfections in material properties and geometry variations among sectors, called mistuning, energy localization can occur during operation creating amplified response amplitudes and stresses and greater risk of high cycle fatigue failure. To reduce vibration amplitudes, nonlinear friction damping via contact interfaces can be introduced. One such method is to add a friction ring damper to the underside of the disk portion within a groove, which is held in place by centrifugal forces. Due to blisk finite element models often containing millions of degrees of freedom, modeling the nonlinear dynamics of these systems necessitates the use of reduced-order models to be computationally feasible. Thus, physics-based nonlinear reduced-order models have been developed to predict the nonlinear dynamic behavior of blisks with friction interfaces. However, data-driven methods for predicting nonlinear blisk dynamics have remained largely unexplored. Here, we introduce a novel data-driven reduced-order model for predicted blisk dynamics based on two feed-forward neural networks. These networks are based on sector-level data, allowing for significantly fewer simulations and/or experiments needed for training data generation. Unlike previous physics-based methods, this approach does not use linear or nonlinear modal information. This approach is validated for a lumped-mass model representative of a blisk with a friction ring damper and small stiffness mistuning subject to a traveling-wave excitation of the type seen during turbomachinery operation.

Flutter is a phenomenon that occurs in wings or platelike structures as a result of aerodynamical forces when a certain flow speed, i.e., flutter speed, is reached. Flutter results in severe vibrations which eventually leads to fatigue failure of the wing. Many solutions are suggested against flutter phenomena. Wings or platelike structures under the effect of flowing air may contain nonlinearities due to connections or materials used. In this paper, effect of different structural nonlinear elements on the flutter speed is studied by using a 2D wing model. Aerodynamic lift and moment acting on the airfoil is obtained by utilizing Theodorsen’s unsteady aerodynamics which is only applicable to subsonic flow. In this paper, modified Theodorsen model for a 2D wing is used. To solve the flutter equation, several methods are suggested in the literature. Methods like k method and p-k method assume harmonic vibration in the generalized coordinates resulting in an eigenvalue problem. These methods are applied to linear systems. When nonlinearities are present in the system, numerical time marching solutions to differential equations are used; however, they are very costly in terms of computational time. In this study, state-space approach is utilized to obtain the flutter speed in frequency domain by using describing function method (DFM). The nonlinear system of differential equations is converted into a nonlinear eigenvalue problem utilizing state-space approach from which the flutter speed resulting in unstable solutions is obtained. Nonlinear eigenvalue problem obtained can be solved iteratively without time marching methods. This method of finding flutter speed is computationally much faster than solving nonlinear flutter problems in time domain. Free play nonlinearity is a frequently observed nonlinearity where there exists a gap at both sides of the wing after which it is restricted by stiffnesses. Piecewise linear stiffness is a symmetric nonlinearity similar to free play where finite stiffness exists instead of a zero stiffness in between the gap. Softening cubic stiffness is a nonlinearity where stiffness of the structure decreases as the amplitude of the vibration increases. In this study, free play (gap nonlinearity), piecewise linear stiffness, and cubic stiffness nonlinearities acting on the rotational degree of freedom are considered in the case studies. Results obtained for these nonlinearities are presented and compared with each other.

One of the major challenges encountered by civil structures throughout their life cycle pertains to exposure to dynamic loadings, with earthquakes representing an extreme case of such a load. The frequency content of a dynamic excitation is of primary importance not only because of the potential resonance it induces but also due to limitations that arise for the capabilities of vibration mitigation devices. A recently emerging technology for civil structure applications includes the development of metamaterial configurations. These are structures, which are formed by periodic arrangement of a fundamental component design, termed the unit cell. They can offer impressive filtering properties within specific frequency ranges, the so-called bandgaps. Considering the low-frequency content of earthquake excitation, a design that features a bandgap in the lower-frequency range is required. In this study, the potential of a geometrically nonlinear design for vibration mitigation purposes is investigated for lowering of the corresponding bandgap. The system consists in the periodic arrangement of nonlinear unit cells, each including a triangular arch configuration, which under large displacement considerations can produce not only geometrically nonlinear behavior but also negative stiffness effects. Analytical derivations result to the determination of the amplitude-dependent bandgap of the system. The proposed configuration is attached to a target structure subjected to protection. An assessment on the capabilities of the device toward this direction was performed via numerical analyses, revealing considerable effectiveness. Acceleration response and energy-related measures are considered in the evaluation of the system’s performance. An additional potential, which the proposed configuration can offer, refers to the applicability of the system for retrofitting purposes of existing structures. It is concluded that the system can offer significant vibration mitigation capabilities, while further study and development of the design, taking into consideration constructability limitations, can lead to an efficient passive vibration absorption device.

In this paper, nonlinear forced vibrations of uniform and functionally graded Euler-Bernoulli beams with large deformation are studied. Spectral and temporal boundary value problems of beam vibrations do not always have closed-form analytical solutions. As a result, many approximate methods are used to obtain the solution by discretizing the spatial problem. Spectral Chebyshev technique (SCT) utilizes the Chebyshev polynomials for spatial discretization and applies Galerkin’s method to obtain boundary conditions and spatially discretized equations of motions. Boundary conditions are imposed using basis recombination into the problem, and as a result of this, the solution can be obtained to any linear boundary condition without the need for re-derivation. System matrices are generated with the SCT, and natural frequencies and mode shapes are obtained by eigenvalue problem solution. Harmonic balance method (HBM) is used to solve nonlinear equation of motion in frequency domain, with large deformation nonlinearity. As a result, a generic method is constructed to solve nonlinear vibrations of uniform and functionally graded beams in frequency domain, subjected to different boundary conditions.

Experimental characterization of nonlinear structures usually focuses on fundamental resonances. However, there is useful information about the structure to be gained at frequencies far away from those resonances. For instance, non-fundamental harmonics in the system’s response can trigger secondary resonances, including superharmonic resonances. Using the recently introduced definition of phase resonance nonlinear modes, a phase-locked loop feedback control is used to identify the backbones of even and odd superharmonic resonances, as well as the nonlinear frequency response curve in the vicinity of such resonances. When the backbones of two resonances (either fundamental or superharmonic) cross, modal interactions make the phase-locked loop unable to stabilize some orbits. Control-based continuation can thus be used in conjunction with phase-locked loop testing to stabilize the orbits of interest. The proposed experimental method is demonstrated on a beam with artificial cubic stiffness exhibiting complex resonant behavior. For instance, the frequency response around the third superharmonic resonance of the third mode exhibits a loop; the fifth superharmonic resonance of the fourth mode interacts with the fundamental resonance of the second mode; and the second superharmonic resonance of the third mode exhibits a branch-point bifurcation and interacts with the fourth superharmonic resonance of the fourth mode.

Linear modal analysis has provided a robust and eminently useful framework for the analysis of structural dynamic systems. Considerable attention has been directed towards the development of a nonlinear variant of modal analysis that is effective in the presence of nonlinearities. Thus far, essentially two approaches to constructing nonlinear normal modes (NNM) have gained traction. The first was that of Rosenberg, whereby the modes are defined in terms of synchronous motions of the structure. The second is the geometrically more general approach of Shaw and Pierre, wherein modes are defined on invariant manifolds of the phase space of the system. A recent third approach from Worden and Green proposes a statistical definition of a nonlinear normal mode. Under this framework, the modal coordinates are defined by latent directions in the configuration space that result in statistically uncorrelated time series. This paper examines the properties of the NNMs generated by this framework, by applying techniques from nonlinear system identification (NLSI). Both linear and nonlinear time domain models are fitted to the physical and modal coordinates of a two degree-of-freedom simulated system with a cubic nonlinearity. It is demonstrated in this work, that the NNMs generated are able to decompose the system into independent functionals, and that the modal transformation generalises to lower excitation levels, while maintaining excellent reconstruction of the physical displacements.

Mechanical metamaterials are structures designed to obtain a predefined dynamical property. A typical example are band gaps, frequency bands in which only evanescent waves exist. At these frequencies, the metastructure can be used as an efficient vibration damper or isolator, without the use of viscoelastic materials. The advantage of such a design is its full tunability, beyond the range of bulk properties of classical linear construction materials.In the quest for vibration isolators that are at the same time highly attenuating and stiff enough to carry high loads, phononic crystals can take advantage of inertia amplification. In these designs, the main motion direction is kinematically coupled to small masses with separate degrees of freedom. This coupling results in a high dynamic mass due to the additional inertia. If the geometrical relation between the degrees of freedom is non-linear, the equation of motion and wave equation become non-linear as well. More precisely, it introduces a damping term proportional to the square of the velocity.In our work, we use analytical and numerical models to confirm the existence of amplitude-dependent vibration damping of mechanical resonators with inertia amplification. Two types of resonators are investigated: one with masses undergoing only translational degrees of freedom and one where translation and rotation are coupled. The damping of each structure can be tuned by altering the geometry of the resonator, in particular the angle of the connectors between the individual masses. Furthermore, we show how the amplitude dependency affects the wave propagation in phononic crystals that exploit inertia amplification to lower the start frequency of the band gap.

This work addresses the vibration reduction of nonlinear host structures by dynamic absorbers of Duffing type. For deriving the equations of motion, both host structure and absorber are modeled as nonlinear SDOF oscillators. The method of harmonic balance is applied to obtain an approximate analytical steady-state solution, which is used for parameter identification. Therefore, the frequency response function of either host structure or absorber is determined experimentally, and the dynamic system parameters are identified by a nonlinear least squares method. Since the focus of the work is on the absorber design, a generalization of Den Hartog’s equal-peak method renders the optimal parameters. These are strongly dependent on the operating point’s amplitude, and consequently, the optimization must be often repeated. To keep this effort minimal, DOE methods are applied in the test and design phase. After confirming a proper absorber design by numerical simulations, a physical model of the absorber is tested in a real-time hybrid simulation setup. Hence, the behavior of the nonlinear host structure is reproduced by a virtual simulation model coupled to the physical absorber using a transfer system. This method allows to focus on the physical absorber without the need to construct an expensive laboratory host structure model. Furthermore, all host structure parameters can be changed in the virtual model, without modifying the real-time hybrid simulation setup. Furthermore, the virtual host structure cannot be damaged or destroyed, and thus, all experiments can be repeated and optimized straight away. So far, all real-time hybrid simulation experiments are in good accordance with the theoretical predictions of this work and corresponding results already published in literature.

This paper considers the problem of simultaneous model selection and parameter estimation for dynamical systems with piecewise-linear (PWL) stiffnesses. PWL models are a series of locally linear models that specify or approximate nonlinear systems over some defined operating range. They can be used to model hybrid phenomena common in practical situations, such as, systems with different modes of operation, or systems whose dynamics change because of physical limits or thresholds. Identifying PWL models can be a challenging problem when the number of operating regions and the boundaries of the regions are unknown. This study focusses on the joint problem of identifying the regions (their number and boundaries) as well as their associated parameters. PWL-stiffness models with up to four regimes are considered, and the identification problem is treated as a combined model selection and parameter estimation problem, addressed in a Bayesian framework. Because of the varying number of parameters across the PWL models, traditional Bayesian model selection would typically require reversible-jump Markov chain Monte Carlo (RJ-MCMC) for switching between model spaces. Here instead, a likelihood-free Approximate Bayesian Computation (ABC) scheme with nested sampling is followed, which simplifies the jump between model spaces. To illustrate its performance, the algorithm has been used to select models and identify parameters from four PWL-stiffness systems—linear, bilinear, trilinear, and quadlinear stiffnesses. The results demonstrate the flexibility of using ABC for identifying the correct model and parameters of PWL-stiffness systems, in addition to furnishing uncertainty estimates of the identified parameters.

With the increasing trend toward lightweight assemblies, the designer has to face new NVH challenges. Particularly, various structural joints have a different impact on the vibration response of these assemblies, which should be numerically modelled since the early stage of the design. The goal of this work is to experimentally assess the accuracy of simplified finite element models for joints and to evaluate how model updating techniques can improve the model accuracy. Four different lap joint samples are considered, each of them constituted by two aluminum plates connected with a different technology: one epoxy adhesive, one toughened acrylic adhesive, one acrylic foam tape, and one bolted joint, which is assessed for multiple preload configurations. For each of these cases, an experimental modal analysis is performed, a finite element model with input parameters from the datasheet of the bonding method is set up, and finally these joint parameters are updated with respect to the experimental reference. We apply the same procedure on an industrial case: the adhesive connection between the tube and the bracket of an automotive shock absorber. It is shown that the initial finite element models generally show a good agreement with the experiments for stiff adhesives. However, for the more flexible adhesives, the model update offers a significant accuracy improvement which can be important to account for during the design.

The goal of this study is to find mathematical models for the dynamics of multiple local nonlinear attachments on a single primary structure. We focus on the application of the characteristic nonlinear system identification (CNSI) method to multiple attachments and attachments tuned higher than the primary structure’s first linear mode. The characteristic nonlinear system identification method is a data-driven method for building mathematical models of nonlinear attachments. To produce representative mathematical models, the method only requires the transient experimental response measurements, the general frequency content, and the mass of the attachments. A two-story tower with a nonlinear attachment installed on each floor in two configurations is used to demonstrate the applicability of the CNSI method for identifying multiple nonlinear attachments. The attachments are tuned to interact with the tower’s first mode in the first configuration. The linear stiffness of the attachment on the first floor is increased in the second configuration so that it interacts with the second mode rather than the first mode. We numerically integrate the models and compare the resulting displacements with experimental measurements to validate the CNSI method’s success and strength.

Event detection is often a predominant challenge in processing non-stationary signals. In engineering mechanics, events may result from non-smoothness in the form of loss of contact, impact, or the onset of sliding-friction. An interesting example of such a mechanical system is a wheel whose center of mass does not coincide with its geometric center. An eccentric wheel may evolve in three distinct phases: roll without slip, roll with slip, and hop. Therefore, this paper seeks to explore and compare supervised learning methods for phase identification (i.e., roll, slip, and hop) in simulated data from a driven eccentric wheel. The mechanics of a torque driven wheel on a flat surface are derived through an augmented Lagrangian formulation and Coulomb friction is adopted to model transverse contact forces. To accommodate for non-smoothness, the system is broken down in complementary sub-problems and the simulation is conducted using event-based methods. The simulated data is then used to train a Naive Bayes classifier, a Support Vector Machine (SVM), and an Extreme Gradient Boosting (XGBoost) classifier. Lastly, the methods as well as their performance, merits, and drawbacks are discussed in detail.

In jointed structures, both stiffness and damping properties are affected by the way that the contact interfaces behave. The overall behaviour of the contact interface is the result of different parameters; among them are the contact surface quality and interface pressure distribution. Small changes in these parameters result in a considerable change in the physics of the contact interface and hence introduce variability in structural dynamics properties. The variability in the contact interface can happen among nominally identical structures. These features raise the need for robust models of the joint contact interface to predict their performance in the structure. In this paper, experimental modal testing of a set of nominally identical Brake-Reuss beam structures is employed to investigate the effect of preload on the variability in the natural frequencies and damping ratios for different modes of the structure. The contact interface is modelled using a representative stochastic model to account for the variabilities in the contact interface parameters. Finally, the distribution of the joint model stiffness parameters is identified using the variability in dynamic properties.

Despite all the efforts made to investigate the dynamics of jointed structures, they still remain as one of the uncertain dynamical systems in structural dynamics. Modelling the dynamic behaviour of bolted joint contact interfaces is one of the most challenging tasks in structural dynamics. The multi-frequency response of nonlinear systems, particularly bolted joint structures, is an important characteristic for the model identification of such structures. The alternating frequency-time approach using a harmonic balance (AFTHB) is used in this study to identify the nonlinear model of the bolted joint of the Brake-Reuß beam. Vibration tests with harmonic excitation were performed over a range of frequencies to measure the dynamic response of the structure. The two beams of the structure are modelled using Timoshenko beam theory. A model selection procedure using the measured data is applied to a range of candidate models to model the dynamics of the joint contact interface. The AFTHB approach is then applied to estimate the unknown parameters of the assumed nonlinear model of the structure utilizing the experimentally measured data. The estimated parameters are used to reconstruct the measured dynamic response of the structure.

The assembly of components into a large-scale engineering system naturally leads to the presence of joints with frictional interfaces. The degree of agreement between numerical models and their experimental counterparts decreases when assemblies based in this kind of interfaces are studied due to the non-linear dynamic behaviour that joints introduce. This is, for example, the case in turbine blade root joints. The main cause for these deviations is the friction-related non-linear damping and stiffness effects influencing the dynamic behaviour of the assembly.The experimental measurement of these damping effects poses a challenge due to the presence of the excitation rig itself, which can introduce significant parasitic damping in the system. A free decay measurement is consequently the ideal way to extract the non-linear behaviour; however, the exciter must be initially in physical contact with the test fixture in order to reach the high excitation amplitudes that lead to macroslip friction in the fixture joints.The test setup proposed in this paper is developed for a beam on which two blade root designs have been machined at both ends (dog bone). This beam is fitted between two clamps equipped with dovetail roots and pulled into tension to simulate rotational centrifugal loading, thus creating a blade root contact joint at either end of the beam. The novel excitation method excites the beam harmonically with a rigidly connected shaker to macroslip deflection amplitudes before decoupling from the beam to release it into free decay. This test procedure allows the contactless measurement of the variation in vibrational decay in the beam and the subsequent extraction of the resulting non-linear frictional behaviour associated with the joints.

In this study, we investigate the performance of data-driven Koopman operator and nonlinear normal mode (NNM) on predictive modeling of nonlinear dynamical systems using a physics-constrained deep learning approach. Two physics-constrained deep autoencoders are proposed: one to identify eigenfunction of Koopman operator and the other to identify nonlinear modal transformation function of NNMs, respectively, from the response data only. Koopman operator aims to linearize nonlinear dynamics at the cost of infinite dimensions, while NNM aims to capture invariance properties of dynamics with the same dimension as original system. We conduct numerical study on nonlinear systems with various levels of nonlinearity and observe that NNM representation has higher accuracy than Koopman autoencoder with same dimension of feature coordinates.

We present a data-driven method based on deep learning for identifying nonlinear normal modes of unknown nonlinear dynamical systems using response data only. We leverage the modeling capacity of deep neural networks to identify the forward and inverse nonlinear modal transformations and the associated modal dynamics evolution. We test the method on Duffing systems with cubic nonlinearity and observe that the identified NNMs with invariant manifolds from response data agree with those analytical or numerical ones using closed-form equations.

In this contribution we present a method to directly compute asymptotic expansion of invariant manifolds of large finite element models from physical coordinates and their reduced-order dynamics on the manifold. We show the accuracy of the reduction on selected models, exhibiting large rotations and internal resonances. The results obtained with the reduction are compared to full-order harmonic balance simulations obtained by continuation of the forced response. We also illustrate the low computational cost of the present implementation for increasing order of the asymptotic expansion and increasing number of degrees of freedom in the structure. The results presented show that the proposed methodology can reproduce extremely accurately the dynamics of the systems with a very low computational cost.

Nonlinear identification methods seek to create a mathematical representation of a mechanical system, which can then be used to: (1) predict the structure’s motion or (2) design, redesign or optimize the structure. In a prior work the Nonlinear Identification through eXtended Outputs (NIXO) algorithm was found to work well if the model form is known a priori. Moreover, the black-box NIXO-based algorithm was successful for the data generated numerically. However, when it comes to actual experimental measurements, the black-box identification procedure has proven more challenging. This work builds on the previous efforts seeking to create a black-box NIXO and to demonstrate it on experimental measurements. The identification attempt is performed on a 3D-printed flat beam, and the results are validated against experimental measurements collected during sweep sine vibration testing.

Phase space warping (PSW) was proposed to resolve hidden slow-time deterministic processes from observable fast-time system responses in a hierarchical dynamical system (HDS). In an HDS, the two categories of dynamics with disparate time scales are assumed to be coupled through slowly varying system parameters. PSW was proposed as a practical algorithm to process sampled field measurement data, which has a lower dimensionality than the total number of system states, and resolve the hidden dynamics through nonlinear time series analysis. Since its introduction, PSW has been used successfully in many engineering applications. These applications range from estimating of slow-time multivariate parameter drifts in electromechanical systems, identifying physiological fatigue, to tracking and characterizing damage evolution in mechanical and biomechanical systems. PSW tries to resolve the underlying slow-time dynamics by investigating how the original high-dimensional fast-time nonlinear dynamical system’s manifold changes over a longer period of time over which the slow subsystem’s effects kick in. Despite its success in the applications mentioned above, there is no systematic discussion regarding selecting the parameters to ensure a reasonably good estimation of the underlying slow-time dynamics. Furthermore, there were six variables involved in the original PSW algorithm, randomly selecting these parameters takes one almost infinite amount of time to figure out a somewhat optimal set of parameters P $$\mathcal {P}$$ to make the tracking work. The PSW algorithm has a solid theoretical background but may seem like a gray-box to a black-box data-driven method if one uses it for the first time. Practitioners definitely should not anticipate a random set of parameters to work out, and they may get lost in searching for such a set of parameters. As a primer study, this extended abstract tries to provide a rough strategy to systematically select a set of parameters that yields reasonably good or even optimal tracking results. First, a brief graphical illustration of how the PSW algorithm works is discussed. Then, simulations with various types of fast-time and slow-time dynamics will be conducted to guide parameter selection. These systems are well-known nonlinear oscillators derived from mechanical and electrical systems to make the illustrations approachable. Simulations with known damage dynamics are conducted to further illustrate the power of PSW algorithm when an optimal parameter set is used.

A novel method for the numerical computation of the nonlinear normal modes (NNMs) of a highly flexible cantilever beam is presented. The flexible cantilever is modeled using a 2D finite element discretization of the geometrically exact beam model, wherein geometric nonlinearities relating to the rotation are kept entirely intact. The model is then solved using the proposed solution method, which is fully frequency domain-based and involves a novel combination of a harmonic balance (HBM) Fourier expansion with asymptotic numerical (ANM) continuation for periodic solutions. The NNMs are also calculated experimentally using a flexible cantilever specimen mounted to a shaker table. The experimental NNMs can be compared to their numerical counterparts in order to validate the frequency domain numerical technique.

In this work, we determine the quality of the harmonic balance method (HBM) using a single degree-of-freedom forced Duffing oscillator with free play. HBM results are compared to results obtained using direct time integration with an event location procedure to properly capture contact behavior and identify nonperiodic motions. The comparison facilitates an evaluation of the accuracy of nonlinear, periodic responses computed with HBM, specifically by comparing super- and subharmonic resonances, regions of periodic and nonperiodic (i.e., quasiperiodic or chaotic) responses, and discontinuity-induced bifurcations, such as grazing bifurcations. Convergence analysis of HBM determines the appropriate number of harmonics required to capture nonlinear contact behavior, while satisfying the governing equations. Hill’s method and Floquet theory are used to compute the stability of periodic solutions and identify the types of bifurcations in the system. Extensions to multi- degree-of-freedom oscillators will be discussed as well.

Tribomechadynamics is an emerging field seeking to advance the theoretical understanding of the dynamics of jointed structures by considering the influences of physical phenomena at different length scales (micro-level tribology, meso-level contact, and macro-level geometrical features). The current work presents a fully physics-based modeling approach attempting to make blind predictions of the near-resonant dynamics of a bolted joint through non-intrusive scans of the contacting interfaces aloe (from an interferometer). Drawing inspiration from earlier work, the study presents a novel rough contact modeling approach that considers micro-scale asperity distributions along with meso-scale interface geometry (surface topology) for the contact-constitutive modeling. The developed framework is validated against hammer impact tests conducted on a three-bolted lap joint benchmark structure known as the Brake-Reuß Beam (BRB). The results demonstrate that the current framework can predict the linearized (low amplitude) natural frequencies of the first three modes with an accuracy of less than 0.25%. The model, however, seems to yield relatively inferior predictions when it comes to the nonlinear trends, characterized here as amplitude-dependent resonant frequency and damping. The Rayleigh-Quotient based Nonlinear Modal Analysis (RQNMA) technique is employed for all the simulations. In order to explore the sensitivity of the model to errors/deviations in the parameters assumed (or measured), Polynomial Chaos (PC) based uncertainty propagation studies are undertaken. The factors studied in this stochastic manner are: the micro-level contact model parameters, bolt prestress levels, and deviations in the meso-scale topologies of the contact interfaces. Initial results indicate that the response is most sensitive to the micro-level contact model properties. Finally, the blind predictions of the model are also assessed for the case of a damaged beam, in which case the model performs relatively poorly in the quantitative sense, although capturing the qualitative trends satisfactorily well. It is inferred, from these observations, that further improvements in the micro-scale contact model are necessary in order to develop a modeling approach that is capable of predicting the dynamics of structures with considerably worn/damaged contact interfaces.

Model updating is an essential part of any dynamic system study. In an ideal situation, an experimental analysis and a corresponding modeling analysis must establish equivalent characteristics for the same dynamic system. This is a tall task, particularly for nonlinear systems where known parameters are surpassed by many unknown parameters crucial for establishing the right model. In contrast, many robust techniques and processes are available for model updating of linear vibration systems. However, the linear updated models typically provide erroneous characteristics when directly employed for nonlinear studies. In this paper, a test structure exhibiting geometric nonlinearity is experimentally studied, and the same is modeled using finite element method (FEM). A study of various parameters involved during the modeling processes such as any assumed parameters and boundary conditions is thoroughly reviewed. The deficiencies of linear model updating processes are highlighted, and suitable workarounds that provide more meaningful and well correlating models for nonlinear dynamic systems are discussed in this paper.

Traditional modal shakers are generally unsuitable for direct modal testing of extremely lightweight and thin structures. For small structures, a base excitation technique may be applied but the technique has many disadvantages. In this paper, a novel magnetic excitation system is explored for obtaining linear and nonlinear modal characteristics of an extremely thin double-clamped beam. The linear modal testing uses established experimental modal analysis techniques, and a software phase-locked loop (sPLL) system is employed for nonlinear modal testing of the double-clamped beam. Comprehensive results and challenges associated with the magnetic excitation system are discussed in this paper. A comparison is also provided to highlight the advantages of a magnetic excitation over traditional modal testing based on an earlier similar work.

Built-up structures exhibit nonlinear dynamic behavior due to friction between interfaces that are fastened together. On the other hand, aircraft, spacecraft, and even automotive structures consist of thin panels to reduce their weight, which can exhibit geometric nonlinearity for displacements on the order of the thickness. As part of the Tribomechadynamics Research Challenge (TRC), this work seeks to predict these effects a priori, whereas most prior works have focused on tuning a model to experimental measurements. While methods are beginning to mature that can predict micro-slip nonlinearity of structures, and methods are well established for reduced-order modeling of geometrically nonlinear structures, these have not been combined previously. This paper presents a simulation approach used to predict the nonlinear response of a benchmark structure proposed in the TRC, which exhibits both geometric nonlinearity and micro-slip due to friction in the bolted connections. A two-dimensional model of the structure is created to enable a wide range of simulations to be performed with minimal computational cost, including some dynamic simulations where both friction and geometric nonlinearity are considered. The nonlinear modal behavior is predicted using quasi-static modal analysis (QSMA) and a recent extension called single-degree-of-freedom implicit condensation and expansion (SICE). Static load-displacement data is also used to define a non-parametric Iwan element that reproduces the modal behavior with high fidelity and yet with minimal computational cost. Additionally, limited simulations are performed on three-dimensional models, which are much more expensive but should be predictive, at least so long as Coulomb Friction is appropriate to model the interactions at the interfaces.

Joints are a well-known source of nonlinearity in assembled structures, introducing amplitude-dependent stiffness and damping properties that are active areas of research in the community. The origins of these nonlinearities are understood to be frictional contact interactions between the members that are joined together. Due to the fact that frictional contact involves the gradual degradation and/or damage of the material in contact, the nonlinear influences of the joints on the system-level dynamics are constantly changing. At the level of the joint interfaces, this phenomenon is referred to as wear and there are numerous studies, mainly in the contact mechanics/tribology community, advancing the theoretical understanding of the various factors contributing to it. Tribomechadynamics is an emerging field seeking to syncretize research work in these fields with structural dynamics, with an aim to improve the state-of-the-art when it comes to structural dynamics models of assembled structure. The present study is conducted in this spirit, drawing from the understanding that the system-level dynamical properties of any jointed system are described by phenomena ranging from the material descriptions of the contacting interfaces including the grain structure and boundaries, plastic-flow, etc.; micro- to meso-level features of the interfaces such as asperities distributions, mean surface curvature, etc.; to macro-level features such as the geometry of the assembly, loading path, assembly misalignments, etc. While other studies focus on developing efficient computational approaches spanning the various factors above, the present work is aimed at making experimental observations of the joint-wear phenomenon affecting the global dynamics of a jointed structure. The paper presents the results of an experimental campaign consisting of 12 hours steady-state periodic excitation, with interferometry scans of the interface made at different intervals of time across the study. The tests were conducted on a three-bolted lap joint benchmark structure known as the Brake-Reuß Beam (BRB). The BRB employed here was freshly manufactured for the purpose of this study, and the interface showed signs of significant wear as well as degradation of the dynamical properties at the end of the experimental campaign.

Linear modal analysis offers a vital and mostly complete framework for the dynamic analysis of simplified engineering systems, with important insights offered from the extraction of natural frequencies and mode shapes. The extracted mode shapes further serve as an invariant basis upon which to build reduced-order models (ROMs) of linear systems. When moving to nonlinear systems, however, the principles upon which modal analysis are based, no longer hold. This issue motivates the development of a framework for nonlinear modal analysis, which can maintain some of the key features of modal analysis. This work is based upon the concept of nonlinear normal modes (NNMs); these consist of invariant manifolds upon which motion for a given NNM is constrained. NNMs can also provide insight into engineering systems as well as offer a basis for construction of nonlinear ROMs.