Robert J. Kuether

Nonlinear normal modes, modal interactions and isolated resonance curves

Abstract The objective of the present study is to explore the connection between the nonlinear normal modes of an undamped and unforced nonlinear system and the isolated resonance curves that may appear in the damped response of the forced system. To this end, an energy balance technique is used to predict the amplitude of the harmonic forcing that is necessary to excite a specific nonlinear normal mode. A cantilever beam with a nonlinear spring at its tip serves to illustrate the developments. The practical implications of isolated resonance curves are also discussed by computing the beam response to sine sweep excitations of increasing amplitudes.

Evaluation of Geometrically Nonlinear Reduced-Order Models with Nonlinear Normal Modes

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...

Nonlinear Modal Substructuring of Systems with Geometric Nonlinearities

The analysis of large, complicated structures can be simplified and made more computationally efficient if smaller, simpler subcomponents can be treated and assembled. Modal substructuring methods allow one to reduce the order of the model at the subcomponent level. Modes are also an intrinsic property of the subcomponent, so they lead to certain physical insights. While modal substructuring is relatively well developed for linear systems, its counterpart has not yet been developed for nonlinear subcomponent models. This work presents two modal substructuring techniques that can be used to predict the nonlinear dynamic behavior of an assembly. The first method uses the nonlinear normal modes of each subcomponent in a quasi-linear model to estimate the nonlinear modes of the assembly. In the second approach, a small number of linear modes are used to create a nonlinear reduced order model of each substructure, and the reduced models are assembled to build the nonlinear equations of motion of the assembly. Each approach is compatible with the finite element method, allowing for analysis of realistic engineering structures with global nonlinearities. The two methods are validated by using them to predict the nonlinear modes of a simple assembly of geometrically nonlinear beams, and both are found to perform well.

Computing Nonlinear Normal Modes Using Numerical Continuation and Force Appropriation

Many structures can behave nonlinearly, exhibiting behavior that is not captured by linear vibration theory such as localization and frequency-energy dependence. The nonlinear normal mode (NNM) concept, developed over the last few decades, can be quite helpful in characterizing a structure’s nonlinear response. In the definition of interest, an NNM is a periodic solution to the conservative nonlinear equations of motion. Several approaches have been suggested for computing NNMs and some have been quite successful even for systems with hundreds of degrees of freedom. However, existing methods are still too expensive to employ on realistic nonlinear finite element models, especially when the Jacobian of the equations of motion is not available analytically. This work presents a new approach for numerically calculating nonlinear normal modes by combining force appropriation, numerical integration and continuation techniques. This method does not require gradients, is found to compute the NNMs accurately up to moderate response amplitudes, and could be readily extended to experimentally characterize nonlinear structures. The method is demonstrated on a nonlinear mass-spring-damper system, computing its NNMs up to a 35% shift in frequency. The results are compared with those from a gradient based algorithm and the relative merits of each method are discussed.© 2012 ASME

Modal Substructuring of Geometrically Nonlinear Finite-Element Models

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 ...

Evaluating Convergence of Reduced Order Models Using Nonlinear Normal Modes

It is often prohibitively expensive to integrate the response of a high order nonlinear system, such as a finite element model of a nonlinear structure, so a set of linear eigenvectors is often used as a basis in order to create a reduced order model (ROM). By augmenting the linear basis with a small set of discontinuous basis functions, ROMs of systems with local nonlinearities have been shown to compare well with the corresponding full order models. When evaluating the quality of a ROM, it is common to compare the time response of the model to that of the full order system, but the time response is a complicated function that depends on a predetermined set of initial conditions or external force. This is difficult to use as a metric to measure convergence of a ROM, particularly for systems with strong, non-smooth nonlinearities, for two reasons: (1) the accuracy of the response depends directly on the amplitude of the load/initial conditions, and (2) small differences between two signals can become large over time. Here, a validation metric is proposed that is based solely on the ROM’s equations of motion. The nonlinear normal modes (NNMs) of the ROMs are computed and tracked as modes are added to the basis set. The NNMs are expected to converge to the true NNMs of the full order system with a sufficient set of basis vectors. This comparison captures the effect of the nonlinearity through a range of amplitudes of the system, and is akin to comparing natural frequencies and mode shapes for a linear structure. In this research, the convergence metric is evaluated on a simply supported beam with a contacting nonlinearity modeled as a unilateral piecewise-linear function. Various time responses are compared to show that the NNMs provide a good measure of the accuracy of the ROM. The results suggest the feasibility of using NNMs as a convergence metric for reduced order modeling of systems with various types of nonlinearities.

Validation of Nonlinear Reduced Order Models with Time Integration Targeted at Nonlinear Normal Modes

Recently, nonlinear reduced order models (ROMs) of large scale finite element models have been used to approximate the nonlinear normal modes (NNMs) of detailed structures with geometric nonlinearity distributed throughout all of its elements. The ROMs provide a low order representation of the full model, and are readily used with numerical continuation algorithms to compute the NNMs of the system. In this work, the NNMs computed from the reduced equations serve as candidate periodic solutions for the full order model. A subset of these are used to define a set of initial conditions and integration periods for the full order model and then the full model is integrated to check the quality of the NNM estimated from the ROM. If the resulting solution is not periodic, then the initial conditions can be iteratively adjusted using a shooting algorithm and a Newton–Raphson approach. These converged solutions give the true NNM of the finite element model, as they satisfy the full order equations, and they can be compared to the ROM predictions to validate the ROM at selected points along the NNM branch. This gives a load-independent metric that may provide confidence in the accuracy of the ROM while avoiding the excessive cost of computing the complete NNM of the full order model. This approach is demonstrated on two models with geometric nonlinearity: a beam with clamped-clamped boundary conditions, and a cantilevered plate used to study fatigue and crack propagation.

Craig-Bampton Substructuring for Geometrically Nonlinear Subcomponents

The efficiency of a modal substructuring method depends on the component modes used to reduce the subcomponent models. Methods such as Craig-Bampton (CB) and Craig-Chang have been used extensively to reduce linear finite element models with thousands or even millions of degrees-of-freedom down to a few tens or hundreds. The greatest advantage to these approaches is that they can obtain acceptable accuracy with a small number of component modes. Currently, these modal substructuring methods only apply to linear substructures. A new reduction method is proposed for geometrically nonlinear finite element models using the fixed-interface and constraint modes (collectively termed CB modes) of the linearized system. The reduced model for this system is written in terms of cubic and quadratic polynomials of the modal coordinates, and the coefficients of the polynomials are determined using a series of static loads/responses. This reduction is a nonlinear extension to the Craig-Bampton model for linear systems, and is readily applied to systems built directly in a commercial FEA package. The nonlinear, reduced modal equations for each subcomponent are then coupled by satisfying compatibility and force equilibrium. The efficiency of this new substructuring approach is demonstrated on an example problem that couples two geometrically nonlinear beams at a shared rotational degree-of-freedom. The nonlinear normal modes of the assembled models are compared with those of a truth model to validate the accuracy of the approach.

A Numerical Round Robin for the Prediction of the Dynamics of Jointed Structures

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.

Substructuring with Nonlinear Reduced Order Models and Interface Reduction with Characteristic Constraint Modes

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 fixedinterface and characteristic constraint modes. A non-intrusive method to fit the nonlinear reduced order model (NLROM) is used to identify the low order equations directly from a finite element model built within a commercial software package. The NLROMs 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 (DOF), and the nonlinear normal modes (NNMs) of the assembled equations are computed to validate the equations. The proposed substructuring approach reduces a 12,861 DOF nonlinear finite element model down to only 22 DOF, while maintaining the accuracy to compute the first three NNMs of the full order model.