Higher-order Melnikov functions for single-DOF mechanical oscillators: theoretical treatment and applications

A Melnikov analysis of single-degree-of-freedom (DOF) oscillators is performed by taking into account the first (classical) and higher-order Melnikov functions, by considering Poincaré sections nonorthogonal to the flux, and by explicitly determining both the distance between perturbed and unperturbed manifolds ("one-half" Melnikov functions) and the distance between perturbed stable and unstable manifolds ("full" Melnikov function). The analysis is developed in an abstract framework, and a recursive formula for computing the Melnikov functions is obtained. These results are then applied to various mechanical systems. Softening versus hardening stiffness and homoclinic versus heteroclinic bifurcations are considered, and the influence of higher-order terms is investigated in depth. It is shown that the classical (first-order) Melnikov analysis is practically inaccurate at least for small and large excitation frequencies, in correspondence to degenerate homo/heteroclinic bifurcations, and in the case of generic periodic excitations

consequences of different definitions of bending curvature on nonlinear dynamics of beams

Abstract Beam theories may be grouped in two broad categories, namely induced or intrinsic theories. In the former, beam models are obtained as exact consequences of three-dimensional theory, making use either of asymptotic expansions in a slenderness parameter or projections of three-dimensional elasticity on certain function spaces, while beams are inherently one-dimensional bodies in the latter category. Although induced theories show a clear connection between three- and one-dimensional representations, they are often more demanding with respect to intrinsic ones, in which a finite number of strain parameters, depending on just one space variable, characterizes the motion of beams in an internally consistent way and without a direct linkage to three-dimensional material properties. Hence, as a consequence, intrinsic theories do not provide any structure for constitutive equations and, at least in principle, different choices can be allowed. A typical example of this fact is represented by the one-dimensional relationship between the bending moment and the beam curvature, since for this latter two notions are admissible. Indeed, both are adopted in the literature and, apparently, preferring one to the other is only related to the predictive capability of the ensuing model. The arising question is about possible differences in both static and dynamic responses of beams, when one or the other definition of curvature is selected

Nonlinear oscillations, transition to chaos and escape in the Duffing system with non-classical damping

We investigate the power of a ripping head in the process of concrete cutting. Using nonlinear embedding methods we study the corresponding time series obtained during the cutting process. The calculated maximal Lyapunov exponent indicates the exponential divergence typical for chaotic or stochastic systems. The recurrence plots technique has been used to get nonlinear process statistics for identification and description of nonlinear dynamics, lying behind the cutting process

Chaos theory and applications : a retrospective on lessons learned and missed or new opportunities

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Comparing Nonlinear Free Vibrations of Timoshenko Beams with Mechanical or Geometric Curvature Definition

Abstract The nonlinear free oscillations of a planar, initially straight Timoshenko beam are investigated by means of the asymptotic development method. Attention is focus on the difference in considering the "mechanical" vs the "geometric" curvature of the axis of the beam, which are different for extensible beams. A comparison of the results obtained by the two models is proposed, and it is shown when they are equivalent and when they give different nonlinear behaviours. A parametric analysis showing the effects of the slenderness (arbitrary, not necessarily large) and of the stiffness of the right-end axial spring is performed

Jump and pull-in dynamics of an electrically actuated bistable MEMS device

This study analyzes a theoretical bistable MEMS device, which exhibits a considerable versatility of behavior. After exploring the coexistence of attractors, we focus on each rest position, and investigate the final outcome, when the electrodynamic voltage is suddenly applied. Our aim is to describe the parameter range where each attractor may practically be observed under realistic conditions, when an electric load is suddenly applied. Since disturbances are inevitably encountered in experiments and practice, a dynamical integrity analysis is performed in order to take them into account. We build the integrity charts, which examine the practical vulnerability of each attractor. A small integrity enhances the sensitivity of the system to disturbances, leading in practice either to jump or to dynamic pull-in. Accordingly, the parameter range where the device, subjected to a suddenly applied load, can operate in safe conditions with a certain attractor is smaller, and sometimes considerably smaller, than in the theoretical predictions. While we refer to a particular case-study, the approach is very general

Modelling of Low-velocity Impacts on Composite Beams in Large Displacement

The paper provides an evaluation of the nonlinear dynamic response of a cantilever beam made of composite material subjected to low-velocity impacts. The structure is assumed to respond in a quasi-static manner and modelled by a continuous beam in large displacement with a lumped mass attached. First, an analytical model was developed to study the free vibrations of a beam, taking into account the nonlinearities due to large displacements and inertia. Then, the analytical findings were compared with experimental test data. The vibration of a real composite beam has been acquired through high-speed imaging technique. The displacements of the beam were extracted by digital image analysis; then, the nonlinear parameters of the analytical model were determined by the Fitting Time History technique. The results obtained by the analytical model and the experimental test are compared with numerical analysis. The validated analytical model was adapted to study a low-velocity impact; the lumped mass was associated with a rigid projectile, whose initial speed represents the impact velocity

influence of the mechanics of escape on the instability of von mises truss and its control

Abstract The elastic von Mises truss model is a prototype for bi-stable structures. It allows a deep understanding of the static and dynamic buckling of several planar and spatial truss systems and shallow lattice shell structures, including the geodesic dome, and has a theoretical and practical interest. This structure has a highly nonlinear response in the presence of static and dynamic loads. The geometric nonlinearity is particularly significant even at low load levels when this structure is shallow. This paper presents an exact nonlinear formulation, which is used to investigate the mechanics of erosion and escape from the safe pre-buckling well. As the static pre-load increases, the probability of escape increases in a nonlinear manner. Permanent and transient escapes, as well as the influence of random noise, on the dynamic buckling load are investigated. To increase the load carrying capacity, a method for controlling the global nonlinear dynamics for the elastic von Mises truss is employed. The control method consists of the (optimal) elimination of homoclinic intersection by properly adding superharmonic terms to a given harmonic excitation. Permanent and transient basins of attraction are obtained. The results highlight the complex nonlinear dynamics of this class of structures and the effectiveness of the control method in increasing the integrity of the basins of attraction of the system and its practical safety