Nonlinear dynamics and control in macro/micro-mechanics: some computational issues

Computational issues in the global dynamics of two systems in micro- and macro-mechanics, with different dimensionality, are addressed. Attention is focused on calculation of integrity measures, determination of saddle manifolds undergoing global bifurcations, implementation of a control procedure for delaying basins erosion, selection of 2D cross-sections of multidimensional basins of attraction for understanding the role of transient dynamics in the global scenario of coupled steady responses

Response robustness and safety against jump to contact in AFMs controlled via different techniques

The role of a global dynamics analysis to assess a system robustness and actual safety in operating conditions is investigated by studying the effect of different local and global control techniques on the nonlinear behavior of a noncontact AFM via dynamical integrity concepts and tools

Space-time numerical simulation and validation of analytical predictions for nonlinear forced dynamics of suspended cables

This paper presents space-time numerical simulation and validation of analytical predictions for the finite-amplitude forced dynamics of suspended cables. The main goal is to complement analytical and numerical solutions, accomplishing overall quantitative/qualitative comparisons of nonlinear response characteristics. By relying on an approximate, kinematically non-condensed, planar modeling, a simply supported horizontal cable subject to a primary external resonance and a 1:1, or 1:1 vs. 2:1, internal resonance is analyzed. To obtain analytical solution, a second-order multiple scales approach is applied to a complete eigenfunction-based series of nonlinear ordinary-differential equations of cable damped forced motion. Accounting for both quadratic/cubic geometric nonlinearities and multiple modal contributions, local scenarios of cable uncoupled/coupled responses and associated stability are predicted, based on chosen reduced-order models. As a cross-checking tool, numerical simulation of the associated nonlinear partial-differential equations describing the dynamics of the actual infinite-dimensional system is carried out using a finite difference technique employing a hybrid explicit-implicit integration scheme. Based on system control parameters and initial conditions, cable amplitude, displacement and tension responses are numerically assessed, thoroughly validating the analytically predicted solutions as regards the actual existence, the meaningful role and the predominating internal resonance of coexisting/competing dynamics. Some methodological aspects are noticed, along with a discussion on the kinematically approximate versus exact, as well as planar versus non-planar, cable modeling

Optimization of a pseudoelastic absorber for vibration mitigation

Damic vibration absorbers (DVAs) have received special attention in recent years due to their capability to reduce structural vibrations of a primary structure. In this work, a DVA of the Tuned Mass Damper type based on a Shape Memory Alloy (SMA) element with pseudoelastic behavior is considered. Owing to their rich thermomechanical response, SMAs can exhibit hysteresis loops with rather different features in terms of overall energy dissipation and of pseudoelastic stiffness. As a first step towards the comprehensive evaluation of the performances of such a device, the optimization of a TMD based on SMA devices with different features is studied. Numerical simulations show that the size and the shape of the pseudoelastic loops can influence in a significant way the performances of the DVA

Nonlinear vibrations of symmetric cross-ply laminates via thermomechanically coupled reduced order models

Thermomechanically coupled, geometrically nonlinear, laminated plates are addressed through a unified 2D formulation, by considering classical and third-order shear-deformable von Karman models, along with correspondingly consistent linear and cubic variations of the temperature along the thickness. Minimal dimension reduction of the mechanical problem is accomplished for symmetric cross-ply laminates, ending up for both models to a coupled three-mode reduced model with terms and coefficients of variable nature depending on the variety of mechanical and/or thermal excitations. Nonlinear vibrations of the classical model are investigated in conditions of thermal dynamics either passively entrained by the harmonically varying transverse load via the existing coupling terms, or also playing some active role owed to a temperature difference with respect to the surrounding medium

analytical investigation of single and double neimark sacker bifurcations

The analytical investigation of bifurcations is a very challenging task for many applied scientists and engineers. Often, numerical simulations cannot clarify the complicated dynamics of mechanical systems, in this cases, preprogrammed softwares can be of valid help during the investigation. Also, in the literature, methodology to study bifurcations are presented for most of the cases. However, the presented procedures, are often very hard to be understood from applied scientists with low mathematical background. In this paper we present in details the typical procedure to analyze single and double Neimark-Sacker bifurcations. Especially regarding the double Neimark-Sacker bifurcations of maps, very few sources can be found in the literature, although this kind of bifurcation is very common in many dynamical systems

Modelling and Nonlinear Dynamics of Third-Order Thermomechanically Coupled Laminated Plates

Thermomechanically coupled, geometrically nonlinear, laminated plates are addressed through a unified 2D formulation that integrates mechanical and thermal aspects and consistently accounts for cubic variations along the thickness of both in-plane displacement components and temperature. It allows to address a variety of thermal boundary conditions on the plate upper and lower surfaces. Minimal dimension reduction of the problem is pursued for symmetric cross-ply laminates. A numerical case study provides hints on the potential of the reduced model for the analysis of thermomechanical coupling effects on the system nonlinear dynamics

Modeling, Dimension Reduction, and Nonlinear Vibrations of Thermomechanically Coupled Laminated Plates☆

Abstract A unified formulation of thermomechanical, geometrically nonlinear, laminated plates that integrates mechanical and thermal aspects is presented. It allows for constructing and comparing a variety of continuous models of different mechanical richness and with full thermoelastic coupling embedded, as well as for deriving minimal reduced order models suitable to provide useful information on fundamental thermomechanical phenomena occurring in the system nonlinear and complex dynamics. Comparative numerical investigations of free and forced vibrations can be carried out through both models of three, fully coupled, ordinary differential equations and simplified, partially coupled, models of two, or even one, ODEs, with the aim to unveil the actual importance of accounting for the various terms to reliably describe the most important thermomechanical effects on the system response

Two-to-one resonant multi-modal dynamics of horizontal/inclined cables. Part II : internal resonance activation, reduced-order models and nonlinear normal modes

Resonant multi-modal dynamics due to planar 2:1 internal resonances in the nonlinear, finite-amplitude, free vibrations of horizontal/inclined cables are parametrically investigated based on the second-order multiple scales solution in Part I [1]. The already validated kinematically non-condensed cable model accounts for the effects of both non-linear dynamic extensibility and system asymmetry due to inclined sagged configurations. Actual activation of 2:1 resonances is discussed, enlightening on a remarkable qualitative difference of horizontal/inclined cables as regards non-linear orthogonality properties of normal modes. Based on the analysis of modal contribution and solution convergence of various resonant cables, hints are obtained on proper reduced-order model selections from the asymptotic solution accounting for higher-order effects of quadratic nonlinearities. The dependence of resonant dynamics on coupled vibration amplitudes, and the significant effects of cable sag, inclination and extensibility on system non-linear behavior are highlighted, along with meaningful contributions of longitudinal dynamics. The spatio-temporal variation of non-linear dynamic configurations and dynamic tensions associated with 2:1 resonant non-linear normal modes is illustrated. Overall, the analytical predictions are validated by finite difference-based numerical investigations of the original partial-differential equations of motion

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