Study of Singularities in Nonsmooth Dynamical Systems via Singular Perturbation

In this article we describe some qualitative and geometric aspects of nonsmooth dynamical systems theory around typical singularities. We also establish an interaction between nonsmooth systems and geometric singular perturbation theory. Such systems are represented by discontinuous vector fields on R(l), l >= 2, where their discontinuity set is a codimension one algebraic variety. By means of a regularization process proceeded by a blow-up technique we are able to bring about some results that bridge the space between discontinuous systems and singularly perturbed smooth systems. We also present an analysis of a subclass of discontinuous vector fields that present transient behavior in the 2-dimensional case, and we dedicate a section to providing sufficient conditions in order for our systems to have local asymptotic stability.

On non-ideal and non-linear portal frame dynamics analysis using Bogoliubov averaging method

We apply the Bogoliubov Averaging Method to the study of the vibrations of an elastic foundation, forced by a Non-ideal energy source. The considered model consists of a portal plane frame with quadratic nonlinearities, with internal resonance 1:2, supporting a direct current motor with limited power. The non-ideal excitation is in primary resonance in the order of one-half with the second mode frequency. The results of the averaging method, plotted in time evolution curve and phase diagrams are compared to those obtained by numerically integrating of the original differential equations. The presence of the saturation phenomenon is verified by analytical procedures.

Lyapounov stability for impulsive dynamical systems

This article addresses the problem of stability of impulsive control systems whose dynamics are given by measure driven differential inclusions. One important feature concerns the adopted solution which allows the consideration of systems whose singular dynamics do not satisfy the so-called Frobenius condition. After extending the conventional notion of control Lyapounov pair for impulsive systems, some stability conditions of the Lyapounov type are given. Some conclusions follow the outline of the proof of the main result.

Quenching in a non-ideal mechanical system and the Averaging Method

In this paper, for the first time, a quenching result in a non-ideal system is rigorously obtained. In order to do this a new mechanical hypothesis is assumed, it means that the moment of inertia of the rotating parts of the energy source is big. From this is possible to use the Averaging Method. © 2012 American Institute of Physics.

Generic bifurcation of refracted systems

In this article we discuss some qualitative and geometric aspects of non-smooth dynamical systems theory. Our goal is to study the diagram bifurcation of typical singularities that occur generically in one parameter families of certain piecewise smooth vector fields named Refracted Systems. Such systems has a codimension-one submanifold as its discontinuity set. © 2012 Elsevier Ltd.

The non-ideal problem behavior using a dynamic vibration absorber with nonlinear essential stiffness and timedependent damping properties

The purpose of this study is to develop a dynamic vibration absorber using viscoelastic material with nonlinear essential stiffness and time-dependent damping properties for a non-ideal vibrating system with Sommerfeld effect, resonance capture, and jump phenomenon. The absorber is a mass-bar subsystem that consists of a viscoelastic bar with memory attached to mass, in which the internal dissipative forces depend on current, deformations, and its operational frequency varies with limited temperature. The non-ideal vibrating system consists of a linear (nonlinear) oscillator (plane frame structure) under excitation, via spring connector, of a DC-motor with limited power supply. A viscoelastic dynamic absorber modeled with elastic stiffness essentially nonlinearities was developed to further reduce the Sommerfeld effect and the response of the structure. The numerical results show the performance of the absorber on the non-ideal system response through the resonance curves, time histories, and Poincarésections. Furthermore, the structure responses using the viscoelastic damper with and without memory were studied. © IMechE 2012.

A note on non-linear phenomena in a non-ideal oscillator, with a snap-through truss absorber, including parameter uncertainties

This work considers the vibrating system that consists of a snap-through truss absorber coupled to an oscillator under excitation of an electric motor with an eccentricity and limited power, characterizing a non-ideal oscillator. It is aimed to use the non-linearity and quasi-zero stiffness of absorber (snap-through truss absorber) to obtain a significantly attenuation the jump phenomenon. There is also an interest to exhibit the reduction of Sommerfeld effect, to confirm the saturation phenomenon occurrence and show the power transfer in a non-linear structure, evidencing the pumping energy. As shown by simulations in this work, this absorber allows the energy pumping before and during the jump phenomenon, decreasing the higher amplitudes of considered system. Additionally, the occurrence of saturation phenomenon due use of snap-through truss absorber is verified. The analysis of parameter uncertainties was introduced. Sensitivity of system with parametric errors demonstrated a trustable system. © IMechE 2012.

Control design applied to a non-ideal structural system with behavior chaotic

In this paper we study the behavior of a structure vulnerable to excessive vibrations caused by an non-ideal power source. To perform this study, the mathematical model is proposed, derive the equations of motion for a simple plane frame excited by an unbalanced rotating machine with limited power (non-ideal motor). The non-linear and non-ideal dynamics in system is demonstrated with a chaotic behavior. We use a State-Dependent Riccati Equation Control technique for regulate the chaotic behavior, in order to obtain a periodic orbit small and to decrease its amplitude. The simulation results show the identification by State-Dependent Riccati Equation Control is very effective. © 2013 Academic Publications, Ltd.

A non-ideal portal frame energy harvester controlled using a pendulum

A model of energy harvester based on a simple portal frame structure is presented. The system is considered to be non-ideal system (NIS) due to interaction with the energy source, a DC motor with limited power supply and the system structure. The nonlinearities present in the piezoelectric material are considered in the piezoelectric coupling mathematical model. The system is a bi-stable Duffing oscillator presenting a chaotic behavior. Analyzing the average power variation, and bifurcation diagrams, the value of the control variable that optimizes power or average value that stabilizes the chaotic system in the periodic orbit is determined. The control sensitivity is determined to parametric errors in the damping and stiffness parameters of the portal frame. The proposed passive control technique uses a simple pendulum to tuned to the vibration of the structure to improve the energy harvesting. The results show that with the implementation of the control strategy it is possible to eliminate the need for active or semi active control, usually more complex. The control also provides a way to regulate the energy captured to a desired operating frequency. © 2013 EDP Sciences and Springer.

Influence of ideal and non-ideal excitation sources on the dynamics of a nonlinear vibro-impact system

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

The dynamic behavior of a cantilever beam coupled to a non-ideal unbalanced motor through numerical and experimental analysis

An excitation force that is not influenced by the system state is said to be an ideal energy source. In real situations, a direct and feedback coupling between the excitation source and the system must always exist at a certain level. This manifestation of the law of conservation of energy is known as the Sommerfeld effect. In the case of obtaining a mathematical model for such a system, additional equations are usually necessary to describe the vibration sources with limited power and its coupling with the mechanical system. In this work, a cantilever beam and a non-ideal DC motor fixed to its free end are analyzed. The motor has an unbalanced mass that provides excitation to the system which is proportional to the current applied to the motor. During the coast up operation of the motor, if the drive power is increased slowly, making the excitation frequency pass through the first natural frequency of the beam, the DC motor speed will remain the same until it suddenly jumps to a much higher value (simultaneously its amplitude jumps to a much lower value) upon exceeding a critical input power. It was found that the Sommerfeld effect depends on some system parameters and the motor operational procedures. These parameters are explored to avoid the resonance capture in the Sommerfeld effect. Numerical simulations and experimental tests are used to help gather insight of this dynamic behavior. (C) 2014 Elsevier Ltd. All rights reserved.