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This paper presents a case study in which the finite element model for a curved circular plate is calibrated to reproduce both the linear and nonlinear dynamic response measured from two nominally identical samples. The linear dynamic response is described with the linear natural frequencies and mode shapes identified with a roving hammer test. Due to the uncertainty in the stiffness characteristics from the manufactured perforations, the linear natural frequencies are used to update the effective modulus of elasticity of the full order finite element model (FEM). The nonlinear dynamic response is described with nonlinear normal modes (NNMs) measured using force appropriation and high speed 3D digital image correlation (3D-DIC). The measured NNMs are used to update the boundary conditions of the full order FEM through comparison with NNMs calculated from a nonlinear reduced order model (NLROM). This comparison revealed that the nonlinear behavior could not be captured without accounting for the small curvature of the plate from manufacturing as confirmed in literature. So, 3D-DIC was also used to identify the initial static curvature of each plate and the resulting curvature was included in the full order FEM. The updated models are then used to understand how the stress distribution changes at large response amplitudes providing a possible explanation of failures observed during testing. [ABSTRACT FROM AUTHOR]

The ability to model nonlinear structures subject to random excitation is of key importance in designing hypersonic aircraft and other advanced aerospace vehicles. When a structure is linear, superposition can be used to construct its response to a known spectrum in terms of its linear modes. Superposition does not hold for a nonlinear system, but several works have shown that a system's dynamics can still be understood qualitatively in terms of its nonlinear normal modes (NNMs). This work investigates the connection between a structure's undamped nonlinear normal modes and the spectrum of its response to high amplitude random forcing. Two examples are investigated: a spring-mass system and a clamped-clamped beam modeled within a geometrically nonlinear finite element package. In both cases, an intimate connection is observed between the smeared peaks in the response spectrum and the frequency-energy dependence of the nonlinear normal modes. In order to understand the role of coupling between the underlying linear modes, reduced order models with and without modal coupling terms are used to separate the effect of each NNM's backbone from the nonlinear couplings that give rise to internal resonances. In the cases shown here, uncoupled, single-degree-of-freedom nonlinear models are found to predict major features in the response with reasonable accuracy; a highly inexpensive approximation such as this could be useful in design and optimization studies. More importantly, the results show that a reduced order model can be expected to give accurate results only if it is also capable of accurately predicting the frequency-energy dependence of the nonlinear modes that are excited. [ABSTRACT FROM AUTHOR]

Finite element models can be used to model and predict the hysteresis and energy dissipation exhibited by nonlinear joints in structures. As a result of the nonlinearity, the frequency and damping of a mode is dependent on excitation amplitude, and when the modes remain uncoupled, quasi-static modal analysis has been shown to efficiently predict this behavior. However, in some cases the modes have been observed to couple such that the frequency and damping of one mode is dependent on the amplitude of other modes. To model the interactions between modes, one must integrate the dynamic equations in time, which is several orders of magnitude more expensive than quasi-static analysis. This work explores an alternative where quasi-static forces are applied in the shapes of two or more modes of vibration simultaneously, and the resulting load–displacement curves are used to deduce the effect of other modes on the effective frequency and damping of the mode in question. This methodology is demonstrated on a simple 2D cantilever beam structure with a single bolted joint which exhibits micro-slip nonlinearity over a range of vibration amplitudes. The predicted frequency and damping are compared with those extracted from a few expensive dynamic simulations of the structure, showing that the quasi-static approach produces reasonable albeit highly conservative bounds on the observed dynamics. This framework is also demonstrated on a 3D structure where dynamic simulations are infeasible. [ABSTRACT FROM AUTHOR]

Reduced order models, such as Hollkamp and Gordon's Implicit Condensation and Expansion (ICE) model, are a highly efficient alternative to full-order finite element models (FEM) of geometrically nonlinear structures. However, a reduced order model (ROM) is typically only valid for one FEM. It does not capture how each ROM coefficient changes due to variations in the FEM (e.g., design parameters or uncertainties), so if the FEM is updated then the ROM needs to be re-computed with a new set of static load–displacement data. This study presents a data-driven reduced order modeling approach that creates a single ROM that incorporates design variations in FEM. The proposed method applies Gaussian Process Regression (GPR) to the ICE approach, making each coefficient in an ICE ROM a regression model with respect to a collection of FEMs with varying material properties or geometric parameters. Once the GPR ROM has been identified, one can immediately produce an ICE ROM for a set of FEM parameters without a need to solve any static load–displacement cases on the full FEM. This dramatically enhances the computational efficiency and could be helpful when model uncertainty needs to be considered or when seeking to update a model to correlate with measurements. Additionally, the coefficients of a ROM can often change considerably if the scale on the load–displacement data changes, so it can be difficult to know whether the scaling that was chosen has really identified an accurate ROM. The proposed GPR ROM estimates the mean ROM coefficients for a range of load scaling as well as the uncertainty on each ROM coefficient with respect to the load level. This can be used to gauge the success of the ROM identification and to eliminate ROM coefficients that are unimportant and hence highly variable. The proposed GPR ROM approach is evaluated by applying it to flat and curved beam structures, revealing that the advantages outlined above can be realized with a relatively modest increase in cost relative to a traditional ICE ROM. [ABSTRACT FROM AUTHOR]

The efficiency of a modal substructuring method depends on the component modes used to reduce each subcomponent model. Methods such as Craig-Bampton have been used extensively to reduce linear finite-element models with thousands or even millions of degrees of freedom down orders of magnitude while maintaining acceptable accuracy. A novel reduction method is proposed here for geometrically nonlinear finite-element models using the fixed-interface and constraint modes of the linearized system to reduce each subcomponent model. The geometric nonlinearity requires an additional cubic and quadratic polynomial function in the modal equations, and the nonlinear stiffness coefficients are determined by applying a series of static loads and using the finite-element code to compute the response. The geometrically nonlinear, reduced modal equations for each subcomponent are then coupled by satisfying compatibility and force equilibrium. This modal substructuring approach is an extension of the Craig-Bampton method and is readily applied to geometrically nonlinear models built directly within commercial finite-element packages. The efficiency of this new approach is demonstrated on two example problems: one that couples two geometrically nonlinear beams at a shared rotational degree of freedom, and another that couples an axial spring element to the axial degree of freedom of a geometrically nonlinear beam. The nonlinear normal modes of the assembled models are compared with those of a truth model to assess the accuracy of the novel modal substructuring approach. [ABSTRACT FROM AUTHOR]

Measurements are presented from a two-beam structure with several bolted interfaces in order to characterize the nonlinear damping introduced by the joints. The measurements (all at force levels below macroslip) reveal that each underlying mode of the structure is well approximated by a single degree-of-freedom (SDOF) system with a nonlinear mechanical joint. At low enough force levels, the measurements show dissipation that scales as the second power of the applied force, agreeing with theory for a linear viscously damped system. This is attributed to linear viscous behavior of the material and/or damping provided by the support structure. At larger force levels, the damping is observed to behave nonlinearly, suggesting that damping from the mechanical joints is dominant. A model is presented that captures these effects, consisting of a spring and viscous damping element in parallel with a four-parameter Iwan model The parameters of this model are identified for each mode of the structure and comparisons suggest that the model captures the stiffness and damping accurately over a range of forcing levels. [ABSTRACT FROM AUTHOR]

Abstract: The nonlinear normal modes of a dynamical system provide a modal framework in which the dynamics of a structure can be readily understood. Current numerical approaches use continuation to find a nonlinear normal mode branch that initiates at a low energy, linearized mode. The predictor-corrector based approach follows the periodic solutions as the response amplitude increases, forming the nonlinear normal mode. This method uses the Jacobian of the shooting function in a Newton–Raphson algorithm to find the initial conditions and integration period that result in a periodic response of the conservative equations of motion. Large scale finite element models require that the Jacobian be computed using finite differences since the closed form equations are not explicitly available. The Jacobian must be computed with respect to all of the states, making the algorithm prohibitively expensive for models with many degrees-of-freedom. In this paper, the initial conditions of each periodic solution are determined based on a subset of the linear modes of a geometrically nonlinear finite element model. The first approach, termed enforced modal displacement, sets the initial conditions as a linear combination of linear mode shapes. The second approach, here called the applied modal force method, applies a static load to the structure in a combination of applied forces that would excite a single linear mode, computes the static response to that load, and uses that to set the initial conditions. Both of these algorithms greatly reduce the number of variables that are iterated on during continuation. As a result, the cost of computing each solution along the nonlinear normal mode is only on the order of ten times the cost required to integrate the finite element model over one period of the response. The algorithm is initiated with only one linear mode and additional modes are added in a systematic way as they become important to the periodic solutions along the nonlinear mode branch. The approach is demonstrated on two geometrically nonlinear finite element models, showing a dramatic reduction in the computational cost required to obtain the nonlinear normal mode. [Copyright &y& Elsevier]

Abstract: The industry standard method used to validate finite element models involves correlation of test and analysis mode shapes using reduced Test-Analysis Models (TAMs). Some organizations even require this model validation approach. Considerable effort is required to choose sensor locations and to create a suitable TAM so that the test and analysis mode shapes will be orthogonal to within the required tolerance. This work uses a probabilistic framework to understand and quantify the effect of small errors in the test mode shapes on test-analysis orthogonality. Using the proposed framework, test-orthogonality is a probabilistic metric and the problem becomes one of choosing sensor placement and TAM generation techniques that assure that the orthogonality has a high probability of being within an acceptable range if the model is correct, even though the test measurements are contaminated with random errors. A simple analytical metric is derived that is shown to give a good estimate of the sensitivity of a TAM to errors in the test mode shapes for a certain noise model. These ideas are then applied to a generic satellite system, using TAMs generated by the Static, Modal and Improved Reduced System (IRS) reduction methods. Experimental errors are simulated for a set of mode shapes and Monte Carlo simulation is used to estimate the probability that the orthogonality metric exceeds a threshold due to experimental error alone. For the satellite system considered here, the orthogonality calculation is highly sensitive to experimental errors, so a set of noisy mode shapes has a small probability of passing the orthogonality criteria for some of the TAMs. A number of sensor placement techniques are used in this study, and the comparison reveals that, for this system, the Modal TAM is twice as sensitive to errors on the test mode shapes when it is created on a sensor set optimized for the Static TAM rather than one that was optimized specifically for the Modal TAM. These findings are evaluated in light of previously published studies of TAM sensitivity, and special attention is given to Gordis''s theory, which suggest that TAM sensitivity is related to the natural frequencies of the structure when all measurement points are fixed. Some aspects of TAM sensitivity are problem dependent, so this one work cannot achieve a conclusive ranking of all of the available methodologies. Instead, this work focuses on presenting a set of tools and a probabilistic framework that can be used to correctly quantify TAM sensitivity and demonstrating the approach for one dynamic system and for a particular probabilistic model for the errors contaminating the test mode shapes. [Copyright &y& Elsevier]

• Experimental measurements show coupling between modes due to friction in bolted joints. • One-way coupling is observed where Mode 2 affects Mode 1 but not the reverse. • Micro-scale geometry is carefully characterized and included in finite element models. • Nonlinear contact simulations predict natural frequency versus preload. The goal of this paper is to present a set of measurements from a benchmark structure containing two bolted joints to support future efforts to predict the damping due to the joints and to model nonlinear coupling between the first two elastic modes. Bolted joints introduce nonlinearities in structures, typically causing a softening in the natural frequency and an increase in damping because of frictional slip between the contact interfaces within the joint. These nonlinearities pose significant challenges when characterizing the response of the structure under a large range of load amplitudes, especially when the modal responses become coupled, causing the effective damping and natural frequency to not only depend on the excitation amplitude of the targeted mode, but also the relative amplitudes of other modes. In this work, two nominally identical benchmark structures, known in some prior works as the S4 beam, are tested to characterize their nonlinear properties for the first two elastic modes. Detailed surface measurements are presented and validated through finite element analysis and reveal distinct contact interactions between the two sets of beams. The free-free test structures are excited with an impact hammer and the transient response is analyzed to extract the damping and frequency backbone curves. A range of impact amplitudes and drive points are used to isolate a single mode or to excite both modes simultaneously. Differences in the nonlinear response correlate with the relative strength of the modes that are excited, allowing one to characterize mode coupling. Each of the beams shows different nonlinear properties for each mode, which is attributed to the different contact pressure distributions between the parts, although the mode coupling relationship is found to be consistent between the two. The test data key finding are presented in this paper and the supporting data is available on a public repository for interested researchers. [ABSTRACT FROM AUTHOR]

The use of shear wave propagation to noninvasively measure material properties and loading in tendons and ligaments is a growing area of interest in biomechanics. Prior models and experiments suggest that shear wave speed primarily depends on the apparent shear modulus (i.e., shear modulus accounting for contributions from all constituents) at low loads, and then increases with axial stress when axially loaded. However, differences in the magnitudes of shear wave speeds between ligaments and tendons, which have different substructures, suggest that the tissue's composition and fiber alignment may also affect shear wave propagation. Accordingly, the objectives of this study were to (1) characterize changes in the apparent shear modulus induced by variations in constitutive properties and fiber alignment, and (2) determine the sensitivity of the shear wave speed-stress relationship to variations in constitutive properties and fiber alignment. To enable systematic variations of both constitutive properties and fiber alignment, we developed a finite element model that represented an isotropic ground matrix with an embedded fiber distribution. Using this model, we performed dynamic simulations of shear wave propagation at axial strains from 0% to 10%. We characterized the shear wave speed-stress relationship using a simple linear regression between shear wave speed squared and axial stress, which is based on an analytical relationship derived from a tensioned beam model. We found that predicted shear wave speeds were both in-range with shear wave speeds in previous in vivo and ex vivo studies, and strongly correlated with the axial stress (R2 = 0.99). The slope of the squared shear wave speed-axial stress relationship was highly sensitive to changes in tissue density. Both the intercept of this relationship and the apparent shear modulus were sensitive to both the shear modulus of the ground matrix and the stiffness of the fibers' toe-region when the fibers were less well-aligned to the loading direction. We also determined that the tensioned beam model overpredicted the axial tissue stress with increasing load when the model had less well-aligned fibers. This indicates that the shear wave speed increases likely in response to a load-dependent increase in the apparent shear modulus. Our findings suggest that researchers may need to consider both the material and structural properties (i.e., fiber alignment) of tendon and ligament when measuring shear wave speeds in pathological tissues or tissues with less well-aligned fibers. [Display omitted] • Our computational model captures shear wave propagation in fibrous soft tissues. • Axial stress, density, and apparent shear modulus modulate shear wave speeds. • Less well aligned fibers increase the apparent shear modulus. • The shear wave speed-stress relationship is sensitive to constitutive properties. • The shear wave speed-stress relationship is sensitive to fiber alignment. [ABSTRACT FROM AUTHOR]

Bolted joints not only provide critical connections in assembled structures but are also a significant source of damping. The damping induced by joints is often nonlinear, which is currently not captured by typical modeling practices. A method that was recently proposed by Festjens, Chevallier, and Dion addresses this by using a quasi-static loading to infer the effective stiffness and damping of each mode of a structure. Allen and Lacayo further developed this method, demonstrated its efficacy for structures where the joints were represented with discrete Iwan elements, and dubbed the method quasi-static modal analysis. This work expands on those studies by applying quasi-static modal analysis to 2-D and 3-D FE models in which the geometry, contact pressure, and friction in the joint are modeled in detail. Coulomb friction is assumed between the nominally flat contacting surfaces. In order to obtain confidence in the predictions, the mesh density, contact laws, and other solver settings are explored to understand what is needed to obtain accurate results for this type of problem. The studies are culminated with a first ever comparison between measurements and quasi-static predictions of the amplitude dependent natural frequency and damping ratio of a mode of a real structure. The results to date are promising but highlight areas in which further research is needed. [ABSTRACT FROM AUTHOR]

Structures with mechanical joints are difficult to model accurately. Even if the natural frequencies of the system remain essentially constant, the damping introduced by the joints is often observed to change dramatically with amplitude. Although models for individual joints have been employed with some success, accurately modeling a structure with many joints remains a significant obstacle. To this end, Segalman proposed a modal Iwan model, which simplifies the analysis by modeling a system with a linear superposition of weakly-nonlinear, uncoupled single degree-of-freedom systems or modes. Given a simulation model with discrete joints, one can identify the model for each mode by selectively exciting each mode one at a time and observing how the transient response decays. However, in the environment of interest several modes may be excited simultaneously, such as in an experiment when an impulse is applied at a discrete point. In this work, the modal Iwan model framework is assessed numerically to understand how well it captures the dynamic response of typical structures with joints when they are excited with impulsive forces applied at point locations. This is done by comparing the effective natural frequency and modal damping of the uncoupled modal models with those of truth models that include nonlinear modal coupling. These concepts are explored for two structures, a simple spring-mass system and a finite element model of a beam, both of which contain physical Iwan elements to model joint nonlinearity. The results show that modal Iwan models can effectively capture the variations in frequency and damping with amplitude, which, for damping, can increase by as much as two orders of magnitude in the microslip regime. However, even in the microslip regime the accuracy of a modal Iwan model is found to depend on whether the mode in question is dominant in the response; in some cases the effective damping that the uncoupled model predicts is found to be in error by tens of percent. Nonetheless, the modal model captures the response qualitatively and is still far superior to a linear model. [ABSTRACT FROM AUTHOR]

Bolted joints are a significant source of damping and nonlinearity in built up structures. In the micro-slip regime (i.e. when the bolts do not slip completely) the log of the damping tends to increase linearly with the log of the vibration amplitude, the so-called power-law behavior. The natural frequency also tends to decrease slightly with vibration amplitude. While many works have successfully tuned phenomenological models to capture these behaviors, far fewer have sought to predict the nonlinearity in a bolted joint and validated the predictions with measurements over a range of bolt preloads and vibration amplitudes. This work presents a step towards such a prediction, using a detailed finite element model of a structure, including the preload forces in the bolts and Coulomb friction in the contact, to seek to predict the nonlinear damping and stiffness of the structure and how they vary with preload. The structure studied consists of two beams bolted together at their free ends, the so-called S4 Beam. While the contact interfaces were machined to be nominally flat, the micron-scale curvature in the contact interface is included in the model, approximated in two different ways, to seek to understand what fidelity is needed at the contact interface to successfully capture its dynamic behavior. The recently presented Quasi-Static Modal Analysis (QSMA) method is employed, where the amplitude dependent damping and natural frequency can be predicted from a single quasi-static simulation, avoiding the expense of numerical integration. The predicted damping and natural frequency are compared with measurements at various preloads, showing reasonable agreement if a Coulomb friction coefficient of 0.2 is used for all simulations. The simulations also revealed that, while the micron-scale curvature of the interfaces was important, the results were not very sensitive to how it was approximated. • Joint modeled with a detailed Finite Element model and Coulomb friction. • Effective natural frequency and damping ratio predicted for two modes. • Predictions compared to measurements at various joint preloads. • Linear natural frequencies predicted as a function of joint preload. • Two modes are studied, involving shear and torsion of the joint. [ABSTRACT FROM AUTHOR]

Several reduced-order modeling strategies have been developed to create low-order models of geometrically nonlinear structures from detailed finite element models, allowing one to compute the dynamic response of the structure at a dramatically reduced cost. However, the parameters of these reduced-order models are estimated by applying a series of static loads to the finite element model, and the quality of the reduced-order model can be highly sensitive to the amplitudes of the static load cases used and to the type/number of modes used in the basis. This paper proposes to combine reduced-order modeling and numerical continuation to estimate the nonlinear normal modes of geometrically nonlinear finite element models. Not only does this make it possible to compute the nonlinear normal modes far more quickly than existing approaches, but the nonlinear normal modes are also shown to be an excellent metric by which the quality of the reduced-order model can be assessed. Hence, the second contribution of this work is to demonstrate how nonlinear normal modes can be used as a metric by which nonlinear reduced-order models can be compared. Various reduced-order models with hardening nonlinearities are compared for two different structures to demonstrate these concepts: a clamped-clamped beam model, and a more complicated finite element model of an exhaust panel cover. [ABSTRACT FROM AUTHOR]