A Simple Discussion for Undamped Duffing Impact Oscillator

2015 ◽  
pp. 676-688 ◽  
Author(s):  
Jixiong Fei ◽  
Bin Lin ◽  
Shuai Yan ◽  
Xiaofeng Zhang
Keyword(s):  
Impact Oscillator

scholarly journals Robust Position Control of a Two-Sided 1-DoF Impacting Mechanical Oscillator Subject to an External Persistent Disturbance by Means of a State-Feedback Controller

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-14 ◽  
Author(s):  
Firas Turki ◽  
Hassène Gritli ◽  
Safya Belghith
Keyword(s):  
State Feedback ◽  
Position Control ◽  
External Disturbance ◽  
Feedback Controller ◽  
Linear Matrix ◽  
Stability Conditions ◽  
Mechanical Oscillator ◽  
Impact Oscillator ◽  
State Feedback Controller ◽  
The Impact

This paper proposes a state-feedback controller using the linear matrix inequality (LMI) approach for the robust position control of a 1-DoF, periodically forced, impact mechanical oscillator subject to asymmetric two-sided rigid end-stops. The periodic forcing input is considered as a persistent external disturbance. The motion of the impacting oscillator is modeled by an impulsive hybrid dynamics. Thus, the control problem of the impact oscillator is recast as a problem of the robust control of such disturbed impulsive hybrid system. To synthesize stability conditions, we introduce the S-procedure and the Finsler lemmas by only considering the region within which the state evolves. We show that the stability conditions are first expressed in terms of bilinear matrix inequalities (BMIs). Using some technical lemmas, we convert these BMIs into LMIs. Finally, some numerical results and simulations are given. We show the effectiveness of the designed state-feedback controller in the robust stabilization of the position of the impact mechanical oscillator under the disturbance.

scholarly journals Complex response analysis of a non-smooth oscillator under harmonic and random excitations

2021 ◽  
Vol 42 (5) ◽  
pp. 641-648
Author(s):  
Shichao Ma ◽  
Xin Ning ◽  
Liang Wang ◽  
Wantao Jia ◽  
Wei Xu
Keyword(s):  
Excitation Frequency ◽  
System Stability ◽  
Response Analysis ◽  
External Excitation ◽  
Stochastic Response ◽  
Impact Oscillator ◽  
Nonlinear Parameters ◽  
Bifurcation Phenomena ◽  
Mc Simulation ◽  
Smooth System

AbstractIt is well-known that practical vibro-impact systems are often influenced by random perturbations and external excitation forces, making it challenging to carry out the research of this category of complex systems with non-smooth characteristics. To address this problem, by adequately utilizing the stochastic response analysis approach and performing the stochastic response for the considered non-smooth system with the external excitation force and white noise excitation, a modified conducting process has proposed. Taking the multiple nonlinear parameters, the non-smooth parameters, and the external excitation frequency into consideration, the steady-state stochastic P-bifurcation phenomena of an elastic impact oscillator are discussed. It can be found that the system parameters can make the system stability topology change. The effectiveness of the proposed method is verified and demonstrated by the Monte Carlo (MC) simulation. Consequently, the conclusions show that the process can be applied to stochastic non-autonomous and non-smooth systems.

Global Analysis of the Double Impact Oscillator Dynamics

1999 ◽  
Author(s):  
František Peterka
Keyword(s):  
Global Analysis ◽  
Impact Oscillator ◽  
System Parameters ◽  
Impact Oscillators ◽  
Real Value ◽  
Simulation Results ◽  
The Impact ◽  
Impact Motion ◽  
Double Impact ◽  
Phase Motion

Abstract The double impact oscillator represents two symmetrically arranged single impact oscillators. It is the model of a forming machine, which does not spread the impact impulses into its neighbourhood. The anti-phase impact motion of this system has the identical dynamics as the single system. The in-phase motion and the influence of asymmetries of the system parameters are studied using numerical simulations. Theoretical and simulation results are verified experimentally and the real value of the restitution coefficient is determined by this method.

Numerical Study of an Impact Oscillator

1995 ◽  
Author(s):  
Kannan Marudachalam ◽  
Faruk H. Bursal
Keyword(s):  
Dynamical Systems ◽  
Numerical Algorithm ◽  
Initial Conditions ◽  
Numerical Study ◽  
Harmonic Excitation ◽  
General Purpose ◽  
Constrained Systems ◽  
Global Dynamics ◽  
Impact Oscillator ◽  
Stick Slip

Abstract Systems with discontinuous dynamics can be found in diverse disciplines. Meshing gears with backlash, impact dampers, relative motion of components that exhibit stick-slip phenomena axe but a few examples from mechanical systems. These form a class of dynamical systems where the nonlinearity is so severe that analysis becomes formidable, especially when global behavior needs to be known. Only recently have researchers attempted to investigate such systems in terms of modern dynamical systems theory. In this work, an impact oscillator with two-sided rigid constraints is used as a paradigm for studying the characteristics of discontinuous dynamical systems. The oscillator has zero stiffness and is subjected to harmonic excitation. The system is linear without impacts. However, the impacts introduce nonlinearity and dissipation (assuming inelastic impacts). A numerical algorithm is developed for studying the global dynamics of the system. A peculiar type of solution in which the trajectories in phase space from a certain set of initial conditions merge in finite time, making the dynamics non-invertible, is investigated. Also, the effect of “grazing,” a behavior common to constrained systems, on the dynamics of the system is studied. Based on the experience gained in studying this system, the need for an efficient general-purpose numerical algorithm for solving discontinuous dynamical systems is motivated. Investigation of stress, vibration, wear, noise, etc. that are associated with impact phenomena can benefit greatly from such an algorithm.

scholarly journals Analysis of moored ship oscillations in waves

2020 ◽  
Author(s):  
Д.П. Ковалев ◽  
П.Д. Ковалев ◽  
А.С. Борисов
Keyword(s):  
Dynamic System ◽  
Sea Level ◽  
Order Differential Equation ◽  
External Excitation ◽  
Impact Oscillator ◽  
Mooring Lines ◽  
Sea Level Fluctuations ◽  
Port Area ◽  
Sea Level Fluctuation ◽  
The Impact

В работе рассмотрены особенности колебаний пришвартованного судна для основных портов Сахалинской области, поскольку качка судна у причала может представлять опасность и приводить к повреждению судна или швартовых линий. По данным натурных измерений морского волнения в портовых бухтах рассчитаны спектры колебаний уровня и определены периоды существующих в них волн для диапазона периодов от 2 с до 30 минут. Произведен расчет периодов собственных колебаний (качки) двух типов судов, преимущественно швартующихся в портах. С учетом полученных результатов выполнено моделирование движения судов при волнении как динамической с системы внешним возбуждающим воздействием на основе дифференциального уравнения второго порядка. Показано влияние коэффициента вязкого демпфирования и жесткости швартовых на реакцию динамической системы без удара о причал и для режима ударного осциллятора. Установлено, что в случае прихода в район порта Корсаков длинноволновой зыби движения судна могут переходить в хаотические. The paper considers the peculiarities of moored vessel oscillations for the main ports of the Sakhalin region, since the pitching of the vessel at the berth can be dangerous and lead to damages of the vessel or mooring lines. Spectra of sea level fluctuations and periods of waves in port bays were calculated using sea level fluctuation measurements obtained in the range from 2 seconds to 30 minutes. Calculations of resonance periods (pitching) of two types of vessels mainly moored in ports were done. Taking into consideration these results the simulation of the vessel movement in waves as a dynamic system with an external excitation was performed on the base of second-order differential equation. The influence of viscous damping coefficient and mooring stiffness on the response of the dynamic system is shown for two cases: for system without impact and for the impact oscillator mode. It is established that in the event of a long-wave swell coming to the Korsakov port area, the vessels movements may become chaotic.

scholarly journals Экспериментальное и теоретическое исследование осциллятора с соударениями

2020 ◽  
Vol 90 (10) ◽  
pp. 1672
Author(s):  
В.В. Нарожнов
Keyword(s):  
Theoretical Model ◽  
Equations Of Motion ◽  
Spectral Characteristics ◽  
Contact Theory ◽  
Hertz Contact ◽  
Mechanical Oscillator ◽  
Impact Oscillator ◽  
Fourier Filtering ◽  
Nonlinear Mechanical ◽  
The Impact

The results of a study of a nonlinear mechanical oscillator with elastic impacts are presented. The experiment was carried out using an electromechanical impact oscillator. The theoretical model is based on the equations of motion, taking into account the elastic force, which is calculated under the Hertz contact theory. It is shown that bifurcations and attractors of the “stable focus” and “limit cycle” types can occur for the impact oscillator. Fourier filtering was used to analyze the spectral characteristics of the signals.

Non-Smooth Dynamics and Non-Classical Bifurcations in Impulse-Impact Oscillators

1999 ◽  
Author(s):  
Stefano Lenci ◽  
Giuseppe Rega
Keyword(s):  
Local Level ◽  
Basins Of Attraction ◽  
Dynamical Regime ◽  
Impact Oscillator ◽  
Stable And Unstable Manifolds ◽  
Impact Oscillators ◽  
Unstable Manifolds ◽  
Local And Global Bifurcations ◽  
Initial Description ◽  
Smooth Dynamics

Abstract Some aspects of the nonlinear dynamics of an impulse-impact oscillator are investigated. After an initial description of the prototype mechanical model used to illustrate the results, attention is paid to the classical local and global bifurcations which are at the base of the changes of dynamical regime. Some non-classical phenomena due to the particular nature of the investigated system are then considered. At a local level, it is shown that periodic solutions may appear (or disappear) through a non-classical bifurcation which involves synchronization of impulses and impacts. Similarities and differences with the classical bifurcations are discussed. At a global level, the effects of the non-continuity of the orbits in the phase space on the basins of attraction topology are investigated. It is shown how this property is at the base of a non-classical homoclinic bifurcation where the homoclinic points disappear after the first touch between the stable and unstable manifolds.

scholarly journals Dynamics of an impact oscillator near a degenerate graze

Nonlinearity ◽  
2010 ◽  
Vol 23 (11) ◽  
pp. 2723-2748 ◽  
Author(s):  
D R J Chillingworth
Keyword(s):  
Impact Oscillator
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