Nonlinear Structures & Systems, Volume 1

A proper understanding of the complex physics associated with nonlinear dynamics can improve the accuracy of predictive engineering models and provide a foundation for understanding nonlinear response during environmental testing. Several researchers and studies have previously shown how localized nonlinearities can influence the global vibration modes of a system. This current work builds upon the study of a demonstration aluminum aircraft with a mock pylon with an intentionally designed, localized nonlinearity. In an effort to simplify the identification of the localized nonlinearity, previous work has developed a simplified experimental setup to collect experimental data for the isolated pylon mounted to a stiff fixture. This study builds on these test results by correlating a multi-degree-of-freedom model of the pylon to identify the appropriate model form and parameters of the nonlinear element. The experimentally measured backbone curves are correlated with a nonlinear Hurty/Craig-Bampton (HCB) reduced order model (ROM) using the calculated nonlinear normal modes (NNMs). Following the calibration, the nonlinear HCB ROM of the pylon is attached to a linear HCB ROM of the wing to predict the NNMs of the next-level wing-pylon assembly as a pre-test analysis to better understand the significance of the localized nonlinearity on the global modes of the wing structure.

This paper presents a set of tests on a bolted benchmark structure called the S4 beam with a focus on evaluating coupling between the first two modes due to nonlinearity. Bolted joints are of interest in dynamically loaded structures because frictional slipping at the contact interface can introduce amplitude-dependent nonlinearities into the system, where the frequency of the structure decreases, and the damping increases. The challenge to model this phenomenon is even more difficult if the modes of the structure become coupled, violating a common assumption of mode orthogonality. This work presents a detailed set of measurements in which the nonlinearities of a bolted structure are highly coupled for the first two modes. Two nominally identical bolted structures are excited using an impact hammer test. The nonlinear damping curves for each beam are calculated using the Hilbert transform. Although the two structures have different frequency and damping characteristics, the mode coupling relationship between the first two modes of the structures is shown to be consistent and significant. The data is intended as a challenge problem for interested researchers; all data from these tests are available upon request.

Morphing wings have great potential to dramatically improve the efficiency of future generations of aircraft and to reduce noise and emissions. Among many camber morphing wing concepts, shape changing fingerlike mechanisms consist of components, such as torsion bars, bushings, bearings, and joints, all of which exhibit damping and stiffness nonlinearities that are dependent on excitation amplitude. These nonlinearities make the dynamic response difficult to model accurately with traditional simulation approaches. As a result, at high excitation levels, linear finite element models may be inaccurate, and a nonlinear modeling approach is required to capture the necessary physics. This work seeks to better understand the influence of nonlinearity on the effective damping and natural frequency of the morphing wing through the use of quasi-static modal analysis and model reduction techniques that employ multipoint constraints (i.e., spider elements). With over 500,000 elements and 39 frictional contact surfaces, this represents one of the most complicated models to which these methods have been applied to date. The results to date are summarized and lessons learned are highlighted.

Calibrating a finite element model to test data is often required to accurately characterize a joint, predict its dynamic behavior, and determine fastener fatigue life. In this work, modal testing, model calibration, and fatigue analysis are performed for a bolted structure, and various joint modeling techniques are compared. The structure is designed to test a single bolt to fatigue failure by utilizing an electrodynamic modal shaker to axially force the bolted joint at resonance. Modal testing is done to obtain the dynamic properties, evaluate finite element joint modeling techniques, and assess the effectiveness of a vibration approach to fatigue testing of bolts. Results show that common joint models can be inaccurate in predicting bolt loads, and even when updated using modal test data, linear structural models alone may be insufficient in evaluating fastener fatigue.

The development of predictive computational models of joints is an ongoing challenge within the community. Unlike monolithic structures, the addition of friction in joints introduces nonlinearities in the vibration response of the structure. Frictional contact models can be applied to reproduce the nonlinear behavior, but the best predictive modeling framework is not clear. Elastic dry friction is a popular choice for predictive modeling, but recent work has highlighted its inability to recreate experimental behavior. As an alternative, several microslip rough contact models have been derived from distributions of asperity heights. Unlike elastic dry friction, these models have a smooth transition from sticking to slipping allowing them to capture smoother experimental trends. However, these models have often used the Masing assumptions and constant (over the interface and a cycle) normal pressures. The assumption of constant normal pressures neglects the kinematics of jointed interfaces, while the Masing assumptions do not generally hold for normal pressures that vary throughout a cycle. The present work seeks to further develop a microslip rough contact modeling framework without the simplifying assumptions to realize more physical simulations. Experiments on a benchmark structure, along with interfacial scans, are used to assess the validity of the proposed modeling framework.

There have been several studies developing calculus on graph domains, defining and generalizing the concepts of differential and integral operators on discrete domains. This chapter considers potential applications for such ideas in the field of modal analysis and identification. The thesis of the chapter lies in generalizing the “weak form” integral equations of the wave equation on a weighted graph domain using said developments (graph operations, Lebesgue integrals, etc.), leading to the definition of a parametric finite element model with sufficient flexibility to allow for model identification. There exist several results in the mathematical discipline of graph theory with regard to the physical interpretations of graphs and its subsets based on the relative weight distributions of the graph members (nodes and edges). The chapter will consider if there is merit for considering such ideas in the context of structural dynamics, specifically modal testing.

Friction interfaces are commonly used in large-scale engineering systems for mechanical joints. They are known to significantly shift the resonance frequencies of the assembled structures due to softening effects and to reduce the vibration amplitude due to frictional energy dissipation between substructural components. It is also widely recognized that the geometrical characteristics of interface geometry have a significant impact on the nonlinear dynamical response of assembled systems. However, the full FE modeling approaches including these geometrical characteristics are extremely expensive. In this work, the influence of geometry of friction interfaces is investigated by using a multi-scale approach. It consists in integrating a semi-analytical contact solver into a high-fidelity nonlinear vibration solver. A highly efficient semi-analytical solver based on the boundary element method is used to obtain the pressure and gap distribution from the contact interface with different geometrical characteristics. The static pressure and gap distribution are then used as input for a nonlinear vibration solver to evaluate nonlinear vibrations of the whole assembled structure. The effectiveness of the methodology is shown on a realistic “Dogbone” test rig, which was designed to assess the effects of blade root geometries in a fan blade disk system. The friction joints with different interface profiles are then investigated. The obtained results show that the effects of the surface geometrical characteristics can have a significant impact on the damping and resonant frequency behavior of the whole assembly.

Physical systems that are subject to intermittent contact/impact are often studied using piecewise-smooth models. Freeplay is a common type of piecewise-smooth system and has been studied extensively for gear systems (backlash) and aeroelastic systems (control surfaces like ailerons and rudders). These systems can experience complex nonlinear behavior including isolated resonance, chaos, and discontinuity-induced bifurcations. This behavior can lead to undesired damaging responses in the system. In this work, bifurcation analysis is performed for a forced Duffing oscillator with freeplay. The freeplay nonlinearity in this system is dependent on the contact stiffness, the size of the freeplay region, and the symmetry/asymmetry of the freeplay region with respect to the system’s equilibrium. Past work on this system has shown that a rich variety of nonlinear behaviors is present. Modern methods of nonlinear dynamics are used to characterize the transitions in system response including phase portraits, frequency spectra, and Poincaré maps. Different freeplay contact stiffnesses are studied including soft, medium, and hard in order to determine how the system response changes as the freeplay transitions from soft contact to near-impact. Particular focus is given to the effects of different initial conditions on the activation of secondary- and isolated-resonance responses. Preliminary results show isolated resonances to occur only for softer-contact cases, regions of superharmonic resonances are more prevalent for harder-contact cases, and more nonlinear behavior occurs for higher initial conditions.

Dynamical systems containing contact/impact between parts can be modeled as piecewise-smooth reduced-order models. The most common example is freeplay, which can manifest as a loose support, worn hinges, or backlash. Freeplay causes very complex, nonlinear responses in a system that range from isolated resonances to grazing bifurcations to chaos. This can be an issue because classical solution methods, such as direct time integration (e.g., Runge-Kutta) or harmonic balance methods, can fail to accurately detect some of the nonlinear behavior or fail to run altogether. To deal with this limitation, researchers often approximate piecewise freeplay terms in the equations of motion using continuous, fully smooth functions. While this strategy can be convenient, it may not always be appropriate for use. For example, past investigation on freeplay in an aeroelastic control surface showed that, compared to the exact piecewise representation, some approximations are not as effective at capturing freeplay behavior as other ones. Another potential issue is the effectiveness of continuous representations at capturing grazing contacts and grazing-type bifurcations. These can cause the system to transition to high-amplitude responses with frequent contact/impact and be particularly damaging. In this work, a bifurcation study is performed on a model of a forced Duffing oscillator with freeplay nonlinearity. Various representations are used to approximate the freeplay including polynomial, absolute value, and hyperbolic tangent representations. Bifurcation analysis results for each type are compared to results using the exact piecewise-smooth representation computed using MATLAB^® Event Location. The effectiveness of each representation is compared and ranked in terms of numerical accuracy, ability to capture multiple response types, ability to predict chaos, and computation time.

Thin structures comprising the skin panels of advanced aircraft will experience extreme thermal stresses as well as dynamic loads at hypersonic speeds, leading to highly nonlinear behaviors such as buckling. In order to determine whether a model correctly captures changes in the dynamics due to heating, the linearized natural frequencies can be compared between test and the FE model at a certain thermal state. This is considerably more difficult if the panel is subjected to localized heating. This work presents a case study in model updating for a non-uniformly heated, geometrically nonlinear panel and evaluates the effect of uncertainty. A curved panel was subjected to localized heating, and measurements of the temperature distributions and of the initial shape were mapped to the FE model and parameterized to use in model updating. The model was then updated for the baseline thermal state, after which the updated model was used to compute the linear natural frequencies and mode shapes with respect to varying temperature fields and those were compared with the experimental data, revealing that the modal properties are highly sensitive to the model’s design parameters. It proved difficult to find an exact correlation by deterministic model updating. The uncertainties in some of the design parameters were then evaluated using a Monte Carlo simulation. The results suggest that even modest uncertainties in the model parameters cause large changes in the natural frequencies, so that the uncertain model bounds the range of the measured natural frequencies.

Symbolic regression has emerged from the more general method of Genetic Programming (GP) as a means of solving functional problems in physics and engineering, where a functional problem is interpreted here as a search problem in a function space. A good example of a functional problem in structural dynamics would be to find an exact solution of a nonlinear equation of motion. Symbolic regression is usually implemented in terms of a tree representation of the functions of interest; however, this is known to produce search spaces of high dimension and complexity. The aim of this chapter is to introduce a new representation—the affine symbolic regression tree. The search space size for the new representation is derived, and the results are compared to those for a standard regression tree. The results are illustrated by the search for an exact solution to several benchmark problems.

A popular technique to control dynamical systems is the implementation of tuned-mass dampers. Most tuned-mass dampers only transfer the mechanical energy of the primary system to a secondary system, but it is desirable to convert the primary systems’ mechanical energy into usable electric energy. A piezoelectric energy harvester is used in this study. Furthermore, amplitude stoppers are included to possibly generate a broadband region by causing a nonlinear interaction. Mechanical stoppers have been investigated to sufficiently widen the response of piezoelectric energy harvesters. The effectiveness of the stoppers type is also investigated by comparing magnetic stoppers to mechanical stoppers. A nonlinear reduced-order model using Galerkin discretization and Euler-Lagrange equations is developed. The goal of this study is to maximize the energy harvested from the absorber without negatively affecting the control of the primary structure.

While many algorithms have been proposed to identify nonlinear dynamic systems, nearly all methods require that the form of equation of motion is known a priori. Examples of very effective methods of this kind are NIFO, CRP, and NARMAX. Several works have sought to extend NARX or NARMAX to a black-box modeling technique. They have proven to be successful in finding accurate mathematical models for certain types of nonlinear systems, yet no method has proved universally successful. This work presents and evaluates a new black-box identification approach based on a new NIFO/CRP type algorithm called Nonlinear Identification through eXtended Outputs (NIXO). The proposed algorithm expresses the nonlinear part of equation of motion as a polynomial of high order and then removes the terms that are classified (with high probability) as irrelevant in the mechanical system’s response. This division into dominant and irrelevant nonlinear terms relies on the values of two novel indicators that are particular to NIXO. This technique is demonstrated on a numerical case study employing a curved beam. Then, the method will be used to estimate the NLEOM of flat and curved beams that were manufactured using a 3D printer. The experimental results will be validated against those obtained using phase resonance testing, which identifies a nonlinear normal mode (NNM) of the system using a vastly different approach.

In this work, the contact/impact problem in a mechanical system with various locations of stoppers is investigated. Finite element simulations and analytical verification in the form of Euler-Bernoulli beam theory are utilized to initially understand the ideal linear system and identify an expected frequency range of interest. The complex nonlinear behavior of varying stopper’s location on the characteristics of a cantilever beam system with a tip mass is experimentally highlighted through the analysis of time histories and frequency response functions. Experimental analyses using free and forced vibration methods, such as impulse, harmonic, and random vibration tests, are performed to extract linear and nonlinear dynamic characteristics of the system. This work seeks to increase awareness of the effects of nonlinearities in the design of dynamical systems and expand the understanding of how these effects can be positively exploited. Results can aid in prevention of premature wear and extend the overall lifetime of systems by avoiding ranges of frequencies that exhibit chaotic responses.

The everlasting competition in the industry to achieve higher performance in aircraft, satellites, and wind turbines encourages lightweight design more than ever, which eventually gives birth to more flexible engineering structures exhibiting large deformations in operational conditions. Accordingly, continuously distributed geometrical nonlinearity resulting from large deformations is currently an important design consideration. Being guided with this motivation, this paper investigates the performance of a recently developed promising nonlinear experimental modal analysis method on a clamped-clamped beam structure which exhibits geometrical nonlinearity continuously distributed throughout the entire structure. The method is based on response-controlled stepped-sine testing (RCT) where the displacement amplitude of the excitation point is kept constant during the frequency sweep. In this study, the nonlinear beam structure is instrumented with multiple accelerometers at several different locations along its length and is excited at a single point. Tests are conducted at energy levels where no internal resonance occurs, yet the beam structure exhibits strong stiffening nonlinearity which results in jump phenomenon in the case of classical constant-force sine testing. Nonlinear modal parameters are experimentally identified as functions of modal amplitude by applying standard linear modal identification methods to quasi-linear frequency response functions (FRFs) measured with RCT. Validation of the identified modal parameters is accomplished by comparing the constant-force FRFs synthesized using the identified modal parameters with the ones obtained from constant force testing and also with the ones extracted from the harmonic force surface (HFS).

Analytical solutions to nonlinear differential equations—where they exist at all—can often be very difficult to find. For example, Duffing’s equation for a system with cubic stiffness requires the use of elliptic functions in the exact solution. A system with general polynomial stiffness would be even more difficult to solve analytically, if such a solution was even to exist. Perturbation and series solutions are possible but become increasingly demanding as the order of solution increases. This chapter aims to revisit, present and discuss a geometric/algebraic method of determining system response which lends itself to automation. The method, originally due to Fliess and co-workers, makes use of the generating series and shuffle product, mathematical ideas founded in differential geometry and abstract algebra. A family of nonlinear differential equations with polynomial stiffness is considered; the process of manipulating a series expansion into the generating series follows and is shown to provide a recursive schematic, which is amenable to computer algebra. The inverse Laplace–Borel transform is then applied to derive a time-domain response. New solutions are presented for systems with general polynomial stiffness, for both deterministic and Gaussian white noise excitations.

Finite element methods (FEM) are commonly used to analyze the behavior of various dynamic systems. Several model updating techniques are available that can be used for linear or nonlinear dynamic systems and help reduce discrepancies between characteristics obtained from the FE models and their experimental counterparts. Linear model updating techniques employ eigen solutions or the frequency response functions (FRF) to reduce errors between the discretized mass [M], damping [C], and stiffness [K] terms or FRF terms directly. However, model updating of structures exhibiting nonlinear behavior is seldom straightforward and requires special consideration of various terms contributing to the nonlinear responses when modeling the system. This paper investigates a hybrid static and dynamic model updating technique for structures exhibiting geometric nonlinearity. The static model updating minimizes errors in stiffness modeling and the dynamic model updating minimizes errors in modeling non-stiffness terms of the FE model.

Structures and parts are increasingly being fabricated using Additive Manufacturing (AM) due to low material waste, ability to quickly generate prototypes, and potential to create complex geometries that would otherwise be difficult or expensive to manufacture. While there are many benefits to using AM processes, the dynamic behavior of the structure is dependent upon the orientation of the construction. This requires investigation of orthotropic materials which have varying material properties in each of the primary Cartesian axes. Some materials, such as the common 3D printing filament Polylactic Acid (PLA), behave orthotropically. Orthotropic materials have often been represented as isotropic materials to simplify numerical models. While such an assumption may be valid in some cases, this study seeks to characterize PLA beams with a tip mass. Experimental PLA beams are printed in the same orientation and have the same external geometry, but the angle of the filament measured from the major axis of the beam varies. The dynamical characteristics are acquired using free and random vibration experiments. The results show that the angle of filament has a significant effect on the damping ratio and natural frequencies of the system.

Using phase-locked loop (PLL) controllers for obtaining frequency response curves (FRC) and backbone curves for nonlinear systems is gaining prominence. Such controllers deliberate a phase lag between the response obtained and the excitation provided. The use of such feedback controllers provides many advantages against traditional sine-sweep methods and helps better characterize nonlinear behavior of the test structure. This paper focuses on obtaining the nonlinear frequency response curves (FRC) and backbone curves of an isolated mode of a structural system exhibiting geometric nonlinearity. The capabilities of testing with such PLL controllers are highlighted and other important characteristics such as stability of the system are discussed. A qualitative comparison is provided between nonlinear modal testing using such feedback controllers as against other traditional methods such as sine-sweep methods.

Phase-locked loop (PLL) controllers are increasingly employed for obtaining nonlinear frequency response curves (FRC) and backbone curves. Such controllers provide a specific phase lag between the response obtained and the excitation signal and isolate the nonlinear mode under consideration. The use of such feedback controllers provides many advantages against traditional sine-sweep methods and helps better characterize nonlinear behavior of dynamic systems. This paper briefly discusses about obtaining the nonlinear frequency response curves (FRC) and backbone curves of both symmetric and asymmetric modes of a double-clamped thin beam exhibiting geometric nonlinearity including a qualitative analysis such as stability and other prominent issues that arise during nonlinear modal testing particularly when the technique is applied to a thin and light structure. A comparison of nonlinear modal testing using PLL controllers as against other traditional methods such as sine-sweep methods is also demonstrated.

The frequency response of multi-degree-of-freedom mechanical systems with weak forcing and damping is highly influenced by the nonlinear normal modes of their conservative limits. Indeed, conservative backbone curves have often been observed to shape the resonance regions in forced frequency responses of the damped system. However, a relation between conservative backbone curves and forced-damped responses has only been established via formal perturbation studies for small-amplitude oscillations or for small nonlinear terms appearing in the equations of motion. In this paper, we prove that the classic energy-balance principle is sufficient to determine exactly which members of conservative backbone curves persist in the damped-forced frequency response. In particular, we derive mathematical criteria for the existence and stability of forced-damped oscillations emerging from the conservative limits of strongly nonlinear multi-degree-of-freedom systems. Unlike previous approaches, our analysis is valid without any assumption on the integrability of the conservative limit, and without any restriction on the response amplitude or on the form of forcing and damping. As we show on specific examples, the location of maximal amplitude response, the existence of isolated response curves, and the presence of subharmonic resonances can all be analytically predicted solely from the knowledge of the conservative nonlinear normal modes. Moreover, our analytic results justify mathematically the experimental use of the phase-lag quadrature criterion (e.g., in force appropriation and control-based continuation) for a broader class of mechanical systems and motions than previous studies have suggested.

Bolted joints are vital constituents of almost every small- to large-scale built-up structure. An especially important aspect of bolted joint modeling is the prediction of the contact area inside the interface of the joint based on the torque applied to the joint and surface characteristics. This is because the joint interface contact area effectively determines the stiffness of the joint as well as the dissipative qualities resulting from slip-stick friction interactions. This research introduces a novel method of estimating actual contact areas inside the interfaces of bolted joints by measuring the strains on the external surfaces induced by the tightening of a bolt. This method can determine contact areas without directly interfering with or altering the interface like existing methods. The experimental specimen used to demonstrate the technique consists of two flat plates with a carriage bolt on the top plate. A carriage bolt instead of a conventional bolt has been used to facilitate surface strain measurements, measured using the digital image correlation measurement technique. Contact areas inside the interface are measured using pressure-sensitive films, and the torque is measured using a digital torque wrench.

Great progress has been made in the last two decades on the construction of non-intrusive reduced order models (ROMs) for the prediction of the response of structures with nonlinear geometric effects subjected to mechanical loading. Nevertheless, some challenges remain when the technique is extended to coupled structural-thermal problems. One such challenge is the construction of basis functions to account for the thermal effects on the structural deformations, especially when the temperature field is local and varies with time. The basis construction considered here starts with the basis relevant to the structure without temperature effects and then adds “enrichment modes” that capture the specificities of the thermal response. A systematic analysis of such possible enrichments and their potential benefits was recently performed (Wang and Mignolet, Proceedings of the 38th IMAC, conference and exposition on structural dynamics, Houston, TX, 2020). In the present study, the curved panel studied in that investigation is considered again but the optimal enrichment strategy established there is extended to a two-temperature-field local heating scenario, heating near the quarter of the panel in one case and near its middle in the second. The established enrichment strategy is firstly used to construct the enriched structural basis that captures the response of the panel under any linear combination of the two temperature fields which serve as two thermal modes. Two approaches are then followed to reduce the number of nonlinear enrichment modes and construct compact ROM bases. The first approach invokes the recently developed “progressive POD” method which was originally used for the reduction of the CFD data stored in multidimensional arrays (Wang et al., J. Aircr., 56:2248–2259, 2019). A notable reduction in the size of the basis is observed with this method. The second approach is using the static condensation to incorporate the in-plane components of the enrichment modes. The three ROMs constructed were found to lead to predictions of the structural responses that closely matched their counterparts determined from the underlying full finite element model.

Systems experiencing high-rate dynamic events, termed high-rate systems, typically undergo accelerations of amplitudes higher than 100 g in less than 10 ms. Examples include adaptive airbag deployment systems, hypersonic vehicles, and active blast mitigation systems. Given the critical functions of such systems, accurate and fast modeling tools are necessary for ensuring the target performance. However, the unique characteristics of these systems, which consist of (1) large uncertainties in the external loads, (2) high levels of nonstationarities and heavy disturbances, and (3) unmodeled dynamics generated from changes in system configurations, in combination with the fast-changing environment, limit the applicability of physical modeling tools. In this chapter, a neural network-based approach is proposed to model and predict high-rate systems. It consists of an ensemble of recurrent neural networks (RNNs) with short-sequence long short-term memory (LSTM) cells which are concurrently trained. To empower multi-step-ahead predictions, the input space for each RNN is selected individually using principal component analysis that extracts different resolutions on the dynamics. The proposed algorithm is validated on experimental data obtained from a high-rate system. Results showed that this algorithm significantly improves the quality of step-ahead predictions over a heuristic approach in constructing the input spaces.

The joints in an assembled structure represent a significant source of energy dissipation and may lead to overall stiffness variation, which may affect high cycle fatigue failure. Many approaches have been developed to model and simulate the dynamics of bolted joint structures. However, the inherent dynamics of the contact interfaces still need further investigation in order to be able to generate accurate models to predict the behaviour in the contact interface. In this paper, the modelling of the contact interface of a bolted lap-joint and the prediction of its pressure distribution are considered using 2D and 3D FE models. A 3D finite element model with solid elements is developed to simulate the behaviour of the contact interface. The model is a modified thin-layer element where the material properties of a thin layer are distributed over the contact interface. Due to the high computational cost of the 3D model, a reduced-order model is proposed for the lap-joint in which beam elements are used. The material properties are introduced in these models to account for the variability in the contact parameters. Finally, experimental modal properties were used to identify the joint parameters. A good agreement is obtained between the detailed model and the reduced-order model in the prediction of the pressure distribution in the contact interface.