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Many modern engineering structures exhibit nonlinear vibration. Characterizing such vibrations efficiently is critical to optimizing designs for reliability and performance. For linear systems, steady-state vibration occurs only at the forcing frequencies. However, nonlinearities (e.g., contact, friction, large deformation, etc.) can result in nonlinear vibration behavior including superharmonics — responses at integer multiples of the forcing frequency. When the forcing frequency is near an integer fraction of the natural frequency, superharmonic resonance occurs, and the magnitude of the superharmonics can exceed that of the fundamental harmonic that is externally forced. Characterizing such superharmonic resonances is critical to improving engineering designs. The present work extends the concept of phase resonance nonlinear modes (PRNM) to be applicable to general nonlinearities, and is demonstrated for eight different nonlinear forces. The considered forces include stiffening, softening, contact, damping, and frictional nonlinearities that have not been previously considered with PRNM. The proposed variable phase resonance nonlinear modes (VPRNM) method can accurately track superharmonic resonances for hysteretic nonlinearities that exhibit amplitude dependent phase resonance conditions that cannot be captured by PRNM. The proposed method allows for characterization of superharmonic resonances without constructing a full frequency response curve at every force level with the harmonic balance method. Thus, the present method allows for analysis of potential failures due to large amplitudes near the superharmonic resonance with reduced computational cost. The consideration of single degree of freedom systems in the present paper provides insights into superharmonic resonances and a basis for understanding internal resonances for multiple degree of freedom systems. • A new method for tracking superharmonic resonances across forcing levels is proposed. • Variable phase resonance nonlinear modes (VPRNM) is needed for the present tracking. • Key features are identified at significantly lower cost than using harmonic balance. • Superharmonic resonances are analyzed for eight nonlinear forces. • Hysteretic nonlinearities show superharmonic resonances over a limited force range. [ABSTRACT FROM AUTHOR]

Assembled structures are nonlinear. The sources of this nonlinearity could include the jointed interfaces, damage and wear, non-idealized boundary conditions, or other features inherent in real parts. To study these systems and to ascertain if they will be operating in a regime in which the nonlinearity is prominent, nonlinear system identification techniques are needed to assess and characterize the nature of the nonlinearity in the structure. Significant progress over the last few years has focused on using nonlinear system identification to identify damage and other deviations from idealized structures. This research reviews nine different methods for nonlinear system identification (restoring force surface, Hilbert transform, direct quadrature, zero-crossing, short-time Fourier transform, Gabor wavelet, Morlet wavelet, Morse wavelet, and a neural network-based algorithm) in order to assess their accuracy. The methods are compared by identifying characteristics of two systems: a single degree of freedom model of a Duffing oscillator and measured data from a jointed structure. As neural networks are not commonly used for system identification, multiple variations of the method are investigated to study its effectiveness. Perturbation analysis is conducted to see the efficacy of the different methods for identifying parameters across a large range of design spaces, and the advantages and disadvantages of each method are discussed. The primary contribution of this paper is a comparison on both analytical and experimental data of multiple widely used system identification methods, and an assessment of when each method is most and least applicable, specifically in the context of jointed structures. [ABSTRACT FROM AUTHOR]

Highlights • Physically perturbed variations of a bolted lap joint studied. • Modeling independent similarity metrics sought from parameter estimation studies. • The surrogate system hypothesis formulated and applied to the lap-jointed benchmarks. • Hypothesis tested in a multi-objective optimization framework. • Tuned whole-joint patch methods employed for the interfacial modeling. Abstract The main challenges in the design and analysis of jointed structures deal with the prediction of dissipation across bolted interfaces. Since friction is thought to be one of the main mechanisms for this, the current study investigates the utility of frictional parameter identification applied across different structures. Based on recent discoveries, a Surrogate System Hypothesis has been formulated to systematize this: the hypothesis states in brief, that the physical properties of a joint are independent of its structural context. The current work seeks to obtain a better understanding of the underlying systems by evaluating a confidence metric for the hypothesis for a relatively simple set of systems—physically perturbed configurations of the Brake-Reuß Beam. Interfacial friction is modeled using whole jointed patches with distributed hysteretic (Iwan) elements and simulations are conducted using the Quasi-Static Modal Analysis (QSMA) approach to estimate the modal characteristics of the system response (amplitude dependence of natural frequency and dissipation). Posing the estimation as a Multi-Objective Optimization Problem (MOOP) is shown to reveal important features of both the employed constitutive model as well as the structure that is modeled. Consequently, the approach is used to evaluate the epistemic uncertainty inherent in three different friction models. The studies reveal that a confidence metric for the hypothesis can be formulated in such a way that it is nearly independent of the friction constitutive model that is employed. [ABSTRACT FROM AUTHOR]

Bolted connections are ubiquitous in mechanical designs and pose a significant challenge to understanding and predicting the vibration response of assembled structures. The present paper develops a physics-based rough contact model of the frictional interactions within a joint. This model sums over the probable interactions of asperities – defined as locally maximum surface features – to determine the contact forces. Here, the tangential contact forces vary smoothly between sticking and slipping and allow the model to better capture the qualitative trends of experimental amplitude dependent frequency and damping than previous studies. Furthermore, the novel model is generalized to allow for arbitrarily varying normal pressure including potential separation to better represent the interfacial dynamics. This includes developing a new, computationally tractable approximation to the analytical Mindlin partial slip solution for tangential loading of contacting spheres. The results highlight the importance of accurately characterizing the as-built topology of the interface, the plastic behavior of the contacting asperities, the relevant length scale of asperities, and the eccentricity of asperities. A predictive friction coefficient based on plasticity provides a poor match to experiments, so fitting the friction coefficient is also considered. Numerical results are compared to experiments on the Brake-Reuß Beam to assess the predictive potential of the models. While blind predictions over-predict the slip limit, the current model presents a significant improvement in physics-based modeling and highlights areas for ongoing research. • A physics-based model of contact for jointed structures is derived. • A new model is derived to describe tangential contact of spheres under general motion • Sensitivities to plasticity, topology and asperity statistics are investigated. • The model is implemented in a tribomechadynamics framework to make blind predictions. • Sensitivity studies investigate the role of asperity density, orientation, and distribution. • Predictive friction coefficients are needed as current coefficients are inadequate. [ABSTRACT FROM AUTHOR]

• A multi-scale modeling framework bridging micro-surface features and macro-dynamics. • A coupled modal quasi-static solver employed for the dynamic characterization. • Rough contact models used for constitutive modeling with Zero-thickness elements. • Effectiveness of the framework demonstrated through validation with experimental data. A new framework for modeling the dynamics of bolted structures is proposed that considers the nonlinear interfacial modeling of bolted structures using multi-scale traction-based contact constitutive laws implemented through Zero-Thickness Elements (ZTE). Using rough contact theory, it is possible to establish fundamental constitutive relationships with parameters estimated from micro-scale surface scans. Such a model is employed in the framework for a three bolt lap-joint benchmark (the so-called "Brake-Reuß-Beam"). Since the characterization of the interface is conducted in a full-field manner on top of a finite element mesh, the framework is also demonstrated to be applicable for conducting full-field micro-scale interface evolution studies. Preliminary studies are conducted to establish correlations of local changes in relevant roughness parameters with predicted local tractions and dissipation fluxes. [ABSTRACT FROM AUTHOR]

The Hurty/Craig-Bampton method in structural dynamics represents the interior dynamics of each subcomponent in a substructured system with a truncated set of normal modes and retains all of the physical degrees of freedom at the substructure interfaces. This makes the assembly of substructures into a reduced-order system model relatively simple, but means that the reduced-order assembly will have as many interface degrees of freedom as the full model. When the full-model mesh is highly refined, and/or when the system is divided into many subcomponents, this can lead to an unacceptably large system of equations of motion. To overcome this, interface reduction methods aim to reduce the size of the Hurty/Craig-Bampton model by reducing the number of interface degrees of freedom. This research presents a survey of interface reduction methods for Hurty/Craig-Bampton models, and proposes improvements and generalizations to some of the methods. Some of these interface reductions operate on the assembled system-level matrices while others perform reduction locally by considering the uncoupled substructures. The advantages and disadvantages of these methods are highlighted and assessed through comparisons of results obtained from a variety of representative linear FE models. [ABSTRACT FROM AUTHOR]

The dynamic responses of assembled structures are greatly affected by the mechanical joints, which are often the cause of nonlinear behavior. To better understand and, in the future, tailor the nonlinearities, accurate methods are needed to characterize the dynamic properties of jointed structures. In this paper, the nonlinear characteristics of a jointed beam is studied with the help of multiple identification methods, including the Hilbert Transform method, Peak Finding and Fitting method, Dynamic Mode Decomposition method, State-Space Spectral Submanifold, and Wavelet-Bounded Empirical Mode Decomposition method. The nonlinearities are identified by the responses that are measured via accelerometers in a series of experiments that consist of hammer testing, shaker ringdown testing, and response/force-control stepped sine testing. In addition to accelerometers, two high-speed cameras are used to capture the motion of the whole structure during the shaker ringdown testing. Digital Image Correlation (DIC) is then adopted to obtain the displacement responses and used to determine the mode shapes of the jointed beam. The accuracy of the DIC data is validated by the comparison between the identification results of acceleration and displacement signals. As enabled by full-field data, the energy-dependent characteristics of the structure are also presented. The setup of the different experiments is described in detail in Part I (Chen et al., 2021) of this research. The focus of this paper is to compare nonlinear system identification methods applied to different measurement techniques and to exploit the use of high spatial resolution data. • The results of several nonlinear system identification methods are compared. • Backbone and damping curves of a jointed beam are identified. • Free decay and forced responses data reveal consistent nonlinear characteristics. • The amplitude-dependent mode shape is reconstructed using digital image correlation. • Digital image correlation allows the computation of the kinetic energy of the beam. [ABSTRACT FROM AUTHOR]

Jointed structures are ubiquitous constituents of engineering systems; however, their dynamic properties (e.g., natural frequencies and damping ratios) are challenging to identify correctly due to the complex, nonlinear nature of interfaces. This research seeks to extend the efficacy of traditional experimental methods for linear system identification (such as impact testing, shaker ringdown testing, random excitation, and force or amplitude-control stepped sine testing) on nonlinear jointed systems, e.g., the half Brake–Reuß beam, by augmenting them with full-field data collected by high-speed videography. The full-field response is acquired using high-speed cameras combined with Digital Image Correlation (DIC), which enables studying the spatial–temporal dynamic characteristics of the system. As this is a video-based experiment, additional constraints are attached to the beam at the node points to remove the rigid body motion, which ensures that the beam is in the view of the camera during the entire test. The use of a video-based method introduces new sources of experimental error, such as noise from the high-speed camera's fan and electrical noise, and so the measurement accuracy of DIC is validated using accelerometer data. After validating the DIC data, the measurements are recorded for several types of excitation, including hammer testing, shaker ringdown testing, fixed sine testing, and stepped sine testing. Using the DIC data to augment standard nonlinear system identification techniques, modal coupling and the mode shapes' evolution are investigated. The suitability of videography methods for nonlinear system identification of nonlinear beams is explored for the first time in this paper, and recommendations for techniques to facilitate this process are made. This article focuses on developing an accurate data collection methodology as well as recommendations for nonlinear testing with DIC, which paves the way for video-based investigation of nonlinear system identification. In Part-II (Jin et al., 2021) of this work, the same data set is used for a rigorous assessment of nonlinear system identification with full-field DIC data. • Nonlinear system identification strategies are applied to a jointed structure. • The use of Digital Image Correlation for nonlinear system identification is assessed. • A novel shaker ringdown excitation method is proposed. • Hardening-softening behavior is observed in the frequency response of the structure. • Modal interactions between the first and third mode are observed. [ABSTRACT FROM AUTHOR]

• Model form error is considered for different model strategies for jointed structures. • Twenty-six models (ten friction models and two representations) are considered. • The suitability of the Masing assumptions for joint dynamics are evaluated. • In addition to frictional damping, material damping is necessary for quality fits. • Fits based on the Masing assumptions have additional errors for transient analyses. Hysteretic models are widely used to model frictional interactions in joints to recreate experimental behavior. However, it is unclear which models are best suited for fitting or predicting the responses of structures. The present study evaluates 26 friction model/interface representation combinations to quantify the model form error. A Quasi-Static Modal Analysis approach (termed Rayleigh Quotient Nonlinear Modal Analysis) is adopted to calculate the nonlinear system response, and a Multi-Objective Optimization is solved to fit experimental data of the first mode of the Brake-Reuß Beam. Optimized parameters from the first mode are applied to the second and third bending modes to quantify the predictive ability of the models. Formulations for both tracing full hysteresis loops and recreating hysteresis loops from a single loading curve (Masing assumptions) are considered. Smoothly varying models applied to a five patch representation showed the highest flexibility (for fitting mode 1) and good predictive potential (for modes 2 and 3). For a second formulation, which uses 152 frictional elements to represent the interface, the physically motivated spring in series with a Coulomb slip model (elastic dry friction) has high error for fitting mode 1 and performs near the middle for predicting higher modes. For both interface representation, the best fit models are not the most physical, but rather the ones with the most parameters (as expected); however, the more physical models perform somewhat better for predicting the higher modes. [ABSTRACT FROM AUTHOR]

• Hyper-reduced whole-joint approach improving previous non-stiffening surface coupling. • Initial efforts at development of consistent whole-joint contact constitutive models. • Hyper-reduced remeshing approach consistent with a reference FE representation. • Explorations of possible approaches for constructing reduced order representations. • Comparisons of the applicability of two different types of interface reduction. One strategy to develop both accurate and computationally tractable models of jointed structures is reduced order modeling through hyper-reduced representations of the interfaces in contact. Hyper reduction refers to reduction techniques that result in a Reduced Order Model (ROM) that is complete by itself, i.e., all displacements and forces are fully described in the ROM coordinates directly. Focusing primarily on applications involving small relative displacement contacts, two fundamentally different approaches are formulated and compared for merits and limitations in applicability. The first is an adaptation of the stiffness-preserving RBE3 constraint elements, and the second is an interpolation approach based on remeshing the interface. Although RBE3 is extensively used in the literature, the current formulation derives stiffness preserving elements that are specifically useful for contact dynamics applications. Transformations are developed to express force–displacement relationships in the ROM coordinates that are congruent (in the sense of using the same contact models) as well as consistent (in the sense of being faithful to the quantities involved) with a high-fidelity model of the same structure. These approaches are applied to study a three-bolted lap-joint structure (the Brake-Reuß Beam (BRB) benchmark) that has been observed to demonstrate characteristic contact non-linearities. Multiple strategies for the hyper reduction are evaluated, including graph partitioning, finite element coarsening, and homogenization of field objectives, some of which involve an extra step of remeshing/choosing patches based on a field objective (e.g., contact pressure). The performances of the ROMs are assessed by conducting nonlinear modal analysis and computing a posteriori error metrics. [ABSTRACT FROM AUTHOR]

• Two different nonlinear models are identified for two experimental test rigs. • The models are used to predict dynamic behavior under sine (-sweep) excitations. • The modal model is accurate for sine excitation close to resonance. • The modal model is sensitive for uncontrolled sweeps if the excitation force drops. • Polynomial nonlinear state-space models highly depend on the used training data. In the present paper, two existing nonlinear system identification methodologies are used to identify data-driven models. The first methodology focuses on identifying the system using steady-state excitations. To accomplish this, a phase-locked loop controller is implemented to acquire periodic oscillations near resonance and construct a nonlinear-mode model. This model is based on amplitude-dependent modal properties, i.e. does not require nonlinear basis functions. The second methodology exploits uncontrolled experiments with broadband random inputs to build polynomial nonlinear state-space models using advanced system identification tools. The methods are applied to two experimental test rigs, a magnetic cantilever beam and a free-free beam with a lap joint. The respective models obtained by either method for both specimens are then challenged to predict dynamic, near-resonant behavior observed under different sine and sine-sweep excitations. The vibration prediction of the nonlinear-mode and state-space models clearly highlight capabilities and limitations. The nonlinear-mode model, by design, yields a perfect match at resonance peaks and high accuracy in close vicinity. However, it is limited to well-spaced modes and sinusoidal excitation. The state-space model covers a wider dynamic range, including transient excitations. However, the real-life nonlinearities considered in this study can only be approximated by polynomial basis functions. Consequently, the identified state-space models are found to be highly input-dependent, in particular for sinusoidal excitations where they are found to lead to a low predictive capability. [ABSTRACT FROM AUTHOR]

• Digital image correlation is used to visualize the local kinematic behaviors of a jointed interface. • Test specimens include perturbations of the Brake-Reuß beam that have curved interfaces. • Results directly invalidate many assumptions common to the modeling of jointed structures. • In the first mode of the systems, both separation and microslip are observed. • For the second mode of all of the test specimens, macroslip is observed for a 5 N excitation. The nonlinear dynamic behavior and energy dissipation of a built-up structure are often attributable to the contact interfaces between the subcomponents of the structure. Due to the lack of direct measurement techniques for jointed interfaces, the local kinematic behaviors inside the interfaces are not well understood. Additionally, the part-to-part variability of the interface (such as surface curvature, surface roughness, geometric features, and residual stress) significantly affects the structural dynamics and, therefore, introduces uncertainty in the measured response of a jointed structure. In this paper, the local kinematic behavior of a jointed interface is measured by digital image correlation (DIC) during shaker excitation at the first two resonant frequencies for 10 different perturbations of the same assembled structure. For these perturbations, the interface curvature is varied to represent typical manufacturing variations. Experimental validation shows that the DIC method is able to measure slip and separation in the interface with a resolution of up to 0.23 μm for the reported experiments. Additionally, the nonlinear properties of the system are characterized via impact hammer test. For the first mode of the structure, separation (clapping) and microslip behaviors are observed; additionally, for the second mode, macroslip motions are observed. These results challenge multiple assumptions prevalent in the modeling of jointed structures. [ABSTRACT FROM AUTHOR]