Nonlinear Normal Modes in an Intrinsic Theory of Anisotropic Beams

Accepted versio

Nonlinear modes of clarinet-like musical instruments

The concept of nonlinear modes is applied in order to analyze the behavior of a model of woodwind reed instruments. Using a modal expansion of the impedance of the instrument, and by projecting the equation for the acoustic pressure on the normal modes of the air column, a system of second order ordinary differential equations is obtained. The equations are coupled through the nonlinear relation describing the volume flow of air through the reed channel in response to the pressure difference across the reed. The system is treated using an amplitude-phase formulation for nonlinear modes, where the frequency and damping functions, as well as the invariant manifolds in the phase space, are unknowns to be determined. The formulation gives, without explicit integration of the underlying ordinary differential equation, access to the transient, the limit cycle, its period and stability. The process is illustrated for a model reduced to three normal modes of the air column

Nonlinear Modal Analysis of Structural Systems Using Multi-Mode Invariant Manifolds

In this paper, an invariant manifold approach is introduced for the generationof reduced-order models for nonlinear vibrations of multi-degrees-of-freedomsystems. In particular, the invariant manifold approach for defining andconstructing nonlinear normal modes of vibration is extended to the case ofmulti-mode manifolds. The dynamic models obtained from this technique capture the essential coupling between modes of interest, while avoiding coupling fromother modes. Such an approach is useful for modeling complex systemresponses, and is essential when internal resonances exist between modes.The basic theory and a general, constructive methodology for the method arepresented. It is then applied to two example problems, one analytical andthe other finite-element based. Numerical simulation results are obtainedfor the full model and various types of reduced-order models, including theusual projection onto a set of linear modes, and the invariant manifoldapproach developed herein. The results show that the method is capable ofaccurately representing the nonlinear system dynamics with relatively fewdegrees of freedom over a range of vibration amplitudes.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/43322/1/11071_2004_Article_281266.pd

Construction de modèles réduits de systèmes non linéaires par modes non linéaires et variétés invariantes

In many applications, it is advantageous to achieve a thorough understanding of the dynamics of a complex structure. These structures may take many forms, including rotorcraft, buildings, bridges, vehicles, and aircraft. Modern design tools, such as Finite Element Analysis, have greatly expanded the modeling detail available for such structures. However, these techniques are limited in their abilities, especially when the structure dynamics enter a nonlinear regime. This limitation is often countered by sacrificing either time | through a large, expensive computer model, or accuracy | through the elimination of possibly significant in uences. As it is of continual interest to expand the performance envelope of such structures, they are becoming lighter, more exible and, consequently, more nonlinear. This trend points out a need for effcient, analytically rigorous, widely applicable analysis techniques for nonlinear structural systems. The primary goal of this dissertation is to address this need through the further development and implementation of model reduction through nonlinear normal modes. This work extends the invariant manifold approach, with the primary goal of obtaining accurate reduced order models of nonlinear structural systems.Dans de nombreux domaines, il est primordial de bien comprendre la dynamique de structures complexes. Les outils numériques modernes, comme la méthode des éléments finis, ont grandement amélioré le niveau de modélisation mais sont souvent limitées au cadre linéaire. Pourtant l'évolution des structures, plus légères et plus flexibles, vers un comportement non linéaire suggère la mise en œuvre de méthodes adaptées, couplant rapidité et précision. C'est dans cet esprit que ce travail de recherche développe une méthode de réduction de modèles non linéaires basée sur les variétés invariantes

Reduced order modeling of nonlinear structural systems using nonlinear normal modes and invariant manifolds.

The generation of reduced-order models of nonlinear systems is particularly difficult, due to the complex interactions of the system components. This work applies the invariant manifold formulation for nonlinear normal modes to create rigorous reduced-order models of a wide variety of nonlinear structures, including discrete, finite element, and continuous dynamic systems. This is accomplished through two types of expansion-based solutions for the invariant manifolds which govern the nonlinear normal modes of the structure. The first expansion is polynomial-based and produces analytic, third-order, invariant manifolds which are asymptotically accurate. The solution obtained is applicable to a subclass of weakly nonlinear structural systems with quadratic and cubic nonlinearities in displacement. The second method uses a Galerkin projection and numerical solver to determine the invariant manifold over a chosen domain. This approach is shown to be accurate for strong nonlinear effects as well as being more adaptable than the polynomial-based approach. Both methods are applied to various nonlinear structural systems, and the results indicate that, in general, the high accuracy of the Galerkin-based solution compensates for the additional computational effort. One field in which nonlinear interactions play a critical role, and are difficult to capture, is rotorcraft dynamics. In particular, blade simulations are cumbersome due to the large models which have been necessary to achieve accurate results. Equations of motion are developed for a uniform nonlinear Euler-Bernoulli beam, rotating at constant velocity, and constrained to move in only the transverse and axial directions. In the interest of improving the rotorcraft design process, the above reduction methods were applied to this simplified blade model. The results indicate that, although both methods capture the critical nonlinear coupling terms at low amplitudes, the Galerkin-based solution achieves excellent results, allowing accurate analysis for tip deflections as large as one meter (peak-to-peak), for a nine meter blade. However, the polynomial-based solutions remain applicable, as they allow investigations of internal resonances which are currently not possible using the present Galerkin-based formulation.Ph.D.Applied SciencesMechanical engineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/132449/2/9963874.pd