Publications

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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.

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Thin beams and panels subjected to large loadings will behave nonlinearly due to membrane stretch effects as they approach deflections on the order of their thickness; this behavior can be efficiently and accurately modeled using nonlinear reduced order models based on the structure’s linear normal modes. However, the complexity of such reduced order models grows cubically with the number of linear modes in the basis set, making complicated geometries prohibitively expensive to compute. Component mode synthesis techniques may be used to reduce this cost by assembling a set of smaller nonlinear subcomponent models, each of which can be more quickly computed than a nonlinear model of the entire structure. Since geometric nonlinearity is heavily dependent on each structure’s boundary conditions, however, subcomponents of an assembly which are constrained only at their interfaces – such as panels mounted to an underlying frame – prove difficult to treat using existing nonlinear modeling techniques. This work uses Craig-Bampton dynamic substructuring combined with characteristic constraint modes for interface reduction to examine the challenges associated with panel and frame assemblies, with a simple example motivating a discussion of current solutions and future challenges.

Substructuring methods have been widely used in structural dynamics to divide large, complicated finite element models into smaller substructures. For linear systems, many methods have been developed to reduce the subcomponents down to a low-order set of equations using a special set of component modes, and these are then assembled to approximate the dynamics of a large-scale model. In this paper, a substructuring approach is developed for coupling geometrically nonlinear structures, where each subcomponent is drastically reduced to a low-order set of nonlinear equations using a truncated set of fixed-interface and characteristic constraint modes. The method used to extract the coefficients of the nonlinear reduced-order model is nonintrusive, in that it does not require any modification to the commercial finite element code but computes the reduced-order model from the results of several nonlinear static analyses. The nonlinear reduced-order models are then assembled to approximate the nonlinear differential equations of the global assembly. The method is demonstrated on the coupling of two geometrically nonlinear plates with simple supports at all edges. The plates are joined at a continuous interface through the rotational degrees of freedom, and the nonlinear normal modes of the assembled equations are computed to validate the models. The proposed substructuring approach reduces a 12,861-degree-of-freedom model down to only 23 degrees of freedom while still accurately reproducing the nonlinear normal modes.

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The Craig-Bampton approach for component mode synthesis in structural dynamics has been widely used to reduce the order of large, detailed finite element models made from linear elastic materials. This methodology separates the full order model into smaller subcomponents and reduces the equations of motion with a truncated set of fixed-interface modes and static constraint modes. A drawback of this approach is that the model has one constraint mode for every interface degree-of-freedom, which may result in a large and prohibitively costly superelement. Previous work has addressed this issue via characteristic constraint modes, which reduces the number of interface degrees-of-freedom by performing a secondary modal analysis on the interface partition. The current work extends the Craig-Bampton approach with interface reduction to include subcomponents with linear viscoelastic materials modeled using a Prony series. For substructures containing materials such as foams or polymers, the viscoelastic constitutive law more accurately represents the material energy dissipation compared to traditional viscous or modal damping. The new approach will be demonstrated on the assembly of two composite plates with fixed boundary conditions along one edge.

Nonlinear joints and interfaces modeled with a discrete four-parameter Iwan element are defined by parameters that are often unknown a priori or require calibration to get better agreement with test data. While this constitutive model has been validated experimentally, its drawback lies in the difficulty of identifying the correct coefficients. This work proposes a parameter estimation approach using a genetic algorithm to minimize the residual between experimental and model data. Global optimization schemes have the ability to find global minima/maxima of a broad parameter space but require a very large number of function evaluations. This research focuses on decreasing the computational cost of the optimization scheme by developing a simplified model of the structure of interest and defining the objective function with amplitude dependent frequencies and damping ratios. A recently developed quasi-static modal analysis technique is used to determine these amplitude dependent properties of the model at a significantly reduced cost in comparison to solutions obtained with numerical time integration. This technique is demonstrated on a structure termed the Ministack which contains a foam-to-metal interface held together with a press fit joint.

Structural dynamics models with localized nonlinearities can be reduced using Hurty/Craig-Bampton component mode synthesis methods. The interior degrees-of-freedom of the linear subcomponents are reduced with a set of dynamic fixedinterface modes while the static constraint modes preserve the physical coordinates at which the nonlinear restoring forces are applied. For finite element models with a highly refined mesh at the boundary, a secondary modal analysis can be performed to reduce the interface down to a truncated set of local-level characteristic constraint modes. In this research, the cost savings and accuracy of the interface reduction technique are evaluated on a simple example problem involving two elastic blocks coming into contact.

Structural dynamics models with localized nonlinearities can be reduced using Hurty/Craig-Bampton component mode synthesis methods. The interior degrees-of-freedom of the linear subcomponents are reduced with a set of dynamic fixedinterface modes while the static constraint modes preserve the physical coordinates at which the nonlinear restoring forces are applied. For finite element models with a highly refined mesh at the boundary, a secondary modal analysis can be performed to reduce the interface down to a truncated set of local-level characteristic constraint modes. In this research, the cost savings and accuracy of the interface reduction technique are evaluated on a simple example problem involving two elastic blocks coming into contact.

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Motivated by the current demands in high-performance structural analysis, and by a desire to better model systems with localized nonlinearities, analysts have developed a number of different approaches for modelling and simulating the dynamics of a bolted-joint structure. However, the types of conditions that make one approach more effective than the others remains poorly understood due to the fact that these approaches are developed from fundamentally and phenomenologically different concepts. To better grasp their similarities and differences, this research presents a numerical round robin that assesses how well three different approaches predict and simulate a mechanical joint. These approaches are applied to analyze a system comprised of two linear beam structures with a bolted joint interface, and their strengths and shortcomings are assessed in order to determine the optimal conditions for their use.

Broadband impact excitation in structural dynamics is a common technique used to detect and characterize nonlinearities in mechanical systems since it excites many frequencies of a structure at once. Non-stationary time signals from transient ring-down measurements require time-frequency analysis tools to observe variations in frequency and energy dissipation as the response evolves. This work uses the short-time Fourier transform to estimate the instantaneous parameters from measured or simulated data. By combining the discrete Fourier transform with an expanding or contracting window function that moves along the time axis, the resulting spectra are used to estimate the instantaneous frequencies, damping ratios and complex Fourier coefficients. This method is demonstrated on a multi-degree-of-freedom beam with a cubic spring attachment. The amplitude-frequency dependence in the damped response is compared to the undamped nonlinear normal modes. A second example shows the results from experimental ring-down measurements taken on a beam with a lap joint, revealing how the mechanical interface introduces nonlinear frequency and damping parameters.

Experimental dynamic substructuring is a means whereby a mathematical model for a substructure can be obtained experimentally and then coupled to a model for the rest of the assembly to predict the response. Recently, various methods have been proposed that use a transmission simulator to overcome sensitivity to measurement errors and to exercise the interface between the substructures; including the Craig-Bampton, Dual Craig-Bampton, and Craig-Mayes methods. This work compares the advantages and disadvantages of these reduced order modeling strategies for two dynamic substructuring problems. The methods are first used on an analytical beam model to validate the methodologies. Then they are used to obtain an experimental model for structure consisting of a cylinder with several components inside connected to the outside case by foam with uncertain properties. This represents an exceedingly difficult structure to model and so experimental substructuring could be an attractive way to obtain a model of the system.

The physical mechanisms of energy dissipation in foam to metal interfaces must be understood in order to develop predictive models of systems with foam packaging common to many aerospace and aeronautical applications. Experimental data was obtained from hardware termed “Ministack”, which has large, unbonded interfaces held under compressive preload. This setup has a solid aluminum mass placed into two foam cups which are then inserted into an aluminum can and fastened with a known preload. Ministack was tested on a shaker using upward sine sweep base acceleration excitations to estimate the linearized natural frequency and energy dissipation of the first axial mode. The experimental system was disassembled and reassembled before each series of tests in order to observe the effects of the assembly to assembly variability on the dynamics. There are some important findings in the measured data: there is significant assembly to assembly variability, the order in which the sine sweeps are performed influence the dynamic response, and the system exhibits nontrivial damping and stiffness nonlinearities that must be accounted for in modeling efforts. A Craig-Bampton model connected with a four-parameter Iwan element and piecewise linear springs is developed and calibrated using test data with the intention of capturing the nonlinear energy dissipation and loss of stiffness observed in experiment.

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