The Dynamics of an Impact Print Hammer

1988 ◽  
Vol 110 (2) ◽  
pp. 193-200 ◽  
Author(s):  
P. C. Tung ◽  
S. W. Shaw
Keyword(s):  
Period Doubling ◽  
Cycle Period ◽  
Flexible Beam ◽  
Qualitative Behavior ◽  
Periodic Motions ◽  
Chaotic Motions ◽  
Linear Spring ◽  
The Stability ◽  
Energy Type ◽  
The Impact

A mathematical model is developed to describe the characteristic behavior of an impact print hammer of the stored energy type. The armature of the impact print hammer is represented by a rigid mass held against a backstop by a preloaded linear spring with negative stiffness which characterizes the net effect of a permanent magnet and a prestressed flexible beam acting on the armature. Periodic sine pulses are adopted to represent currents which release the armature to strike the ribbon and paper which is represented by a linear spring and a linear viscous dashpot. A coefficient of restitution is employed to characterize the instantaneous behavior of impact and rebound at the backstop. In this paper, periodic motions with n impacts against the backstop per forcing cycle, period doubling bifurcations, and chaotic motions are found. The stability of the periodic motions is investigated as is the influence of various parameters on the performance of the impact print hammer. With this simple model we can predict much of the qualitative behavior of the actual physical system.

scholarly journals Mathematical analysis of an age structured epidemic model with a quarantine class

2021 ◽  
Author(s):  
Bedreddine AINSEBA ◽  
Tarik Touaoula ◽  
Zakia Sari
Keyword(s):  
Reproduction Number ◽  
Asymptotic Behavior Of Solutions ◽  
Model Parameters ◽  
Qualitative Behavior ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Age Structured ◽  
The Stability ◽  
The Impact

In this paper, an age structured epidemic Susceptible-Infected-Quarantined-Recovered-Infected (SIQRI) model is proposed, where we will focus on the role of individuals that leave their class of quarantine before being completely recovered and thus will participate again to the transmission of the disease. We investigate the asymptotic behavior of solutions by studying the stability of both trivial and positive equilibria. In order to see the impact of the different model parameters like the relapse rate on the qualitative behavior of our system, we firstly, give the explicit expression of the epidemic reproduction number $R_{0}.$ This number is a combination of the classical epidemic reproduction number for the SIQR model and a new epidemic reproduction number corresponding to the individuals infected by a relapsed person from the R-class. It is shown that, if $R_{0}\leq 1$, the disease free equilibrium is globally asymptotically stable and becomes unstable for $R_{0}>1$. Secondly, while $R_{0}>1$, a suitable Lyapunov functional is constructed to prove that the unique endemic equilibrium is globally asymptotically stable on some subset $\Omega_{0}.$

Switching Bifurcation and Chaos in an Extended Fermi-Acceleration Oscillator

2008 ◽  
Author(s):  
Albert C. J. Luo ◽  
Yu Guo
Keyword(s):  
Local Stability ◽  
Period Doubling ◽  
Time History ◽  
Fermi Acceleration ◽  
Periodic Motions ◽  
Bifurcation And Chaos ◽  
Chaotic Motions ◽  
Numerical Predictions ◽  
Global View ◽  
Acceleration Model

The Fermi acceleration oscillator is extensively used to interpret many physical and mechanical phenomena. To understand dynamic behaviors of a particle (or a bouncing ball) in such a Fermi oscillator, a generalized Fermi acceleration model is developed. This model consists of a particle moving vertically between a fixed wall and the piston in a vibrating oscillator. The motion switching bifurcation of a particle in such a generalized Fermi oscillator is investigated through the theory of discontinuous dynamical systems. The analytical conditions for the motion switching are developed for numerical predictions. Thus, periodic motions in the Fermi-acceleration oscillator are given and the corresponding local stability and bifurcation are presented. Periodic and chaotic motions in such an oscillator are presented via the displacement time-history. From switching bifurcation and period-doubling bifurcation, parameter maps of periodic and chaotic motions will be developed for a global view of motions in the Fermi acceleration oscillator. To illustrate motion switching phenomena, the acceleration responses of the particle and base in the Fermi oscillator are presented. Poincare mapping sections are also used to illustrate chaos, and energy dissipation in chaotic motions can be evaluated.

ANALYTICAL DYNAMICS OF PERIOD-m FLOWS AND CHAOS IN NONLINEAR SYSTEMS

2012 ◽  
Vol 22 (04) ◽  
pp. 1250093 ◽  
Author(s):  
ALBERT C. J. LUO ◽  
JIANZHE HUANG
Keyword(s):  
Dynamical Systems ◽  
Analytical Solutions ◽  
Nonlinear Dynamical Systems ◽  
Duffing Oscillator ◽  
Harmonic Balance Method ◽  
Period Doubling ◽  
Nonlinear Damping ◽  
Periodic Motions ◽  
Chaotic Motions ◽  
Nonlinear Dynamical

In this paper, the analytical solutions for period-m flows and chaos in nonlinear dynamical systems are presented through the generalized harmonic balance method. The nonlinear damping, periodically forced, Duffing oscillator was investigated as an example to demonstrate the analytical solutions of periodic motions and chaos. Through this investigation, the mechanism for a period-m motion jumping to another period-n motion in numerical computation is found. In this problem, the Hopf bifurcation of periodic motions is equivalent to the period-doubling bifurcation via Poincare mappings of dynamical systems. The stable and unstable period-m motions can be obtained analytically. Even more, the stable and unstable chaotic motions can be achieved analytically. The methodology presented in this paper can be applied to other nonlinear vibration systems, which is independent of small parameters.

scholarly journals The Stick-Slip Vibration and Bifurcation of a Vibro-Impact System with Dry Friction

2014 ◽  
Vol 8 (1) ◽  
pp. 308-313 ◽  
Author(s):  
Quanfu Gao ◽  
Xingxiao Cao
Keyword(s):  
Periodic Motion ◽  
Dry Friction ◽  
Low Frequency ◽  
Period Doubling ◽  
Periodic Motions ◽  
Vibratory System ◽  
Stick Slip ◽  
The Impact ◽  
Two Degree Of Freedom ◽  
Impact Motion

In this paper, the periodic motion, bifurcation and chatter of two-degree-of-freedom vibratory system with dry friction and clearance were studied. Slip-stick motion and the impact of system motions were analyzed and numerical simulations were also carried out. The results showed that the system possesses rich dynamics characterized by periodic motion, stick-slip-impact motion, quasi-periodic motion and chaotic attractors, and the routs from periodic motions to chaos observed via Hof bifurcation or period-doubling bifurcation. Furthermore, it was found that there exists the chatter phenomena induced by dry friction in low frequency, and the windows of chaotic motion are broadened in the area of higher excitation frequencies as the dry friction increases.

scholarly journals Bifurcation Analysis of a Rigid Impact Oscillator with Bilinear Damping

2018 ◽  
Vol 2018 ◽  
pp. 1-10 ◽  
Author(s):  
Liping Zhang ◽  
Haibo Jiang ◽  
Yang Liu
Keyword(s):  
Mechanical Model ◽  
Period Doubling ◽  
External Excitation ◽  
Coexisting Attractors ◽  
Impact Oscillator ◽  
Grazing Period ◽  
The Stability ◽  
Complex Phenomena ◽  
Impulsive Switched System ◽  
The Impact

This paper studies a rigid impact oscillator with bilinear damping developed as the mechanical model of an impulsive switched system. The stability and the bifurcation of periodic orbits in the impact oscillator are determined by using the mapping methods. One-parameter bifurcation analyses under variation of forcing frequency and amplitude of external excitation are carried out. Coexisting attractors and various types of bifurcations, such as grazing, period-doubling, and saddle-node, are observed, which show the complex phenomena inhered in this impact oscillator.

Dynamics of a Flexible Beam Contacting a Linear Spring at Low Frequency Excitation: Experiment and Analysis

2002 ◽  
Vol 124 (2) ◽  
pp. 237-249 ◽  
Author(s):  
Karen J. L. Fegelman ◽  
Karl Grosh
Keyword(s):  
Low Frequency ◽  
Theoretical Models ◽  
Model System ◽  
Degree Of Freedom ◽  
Flexible Beam ◽  
High Frequencies ◽  
Linear Spring ◽  
Frequency Excitation ◽  
The Impact ◽  
Period Motion

A detailed study of one-, two-, and three-impact per period motion of a vibro-impacting, pinned beam is presented involving experimental results, as well as one- and multi-degree-of-freedom theoretical models. The details of the impact event are examined and correlated to the qualitative appearance of the frequency response. In addition, it is noted that the multi-degree-of-freedom model is necessary in order to predict response at high frequencies. This study is unique in that the model system includes a pinned boundary condition, the forcing frequency is considerably lower than the fundamental in-contact natural frequency, and the frequency analysis extends into a range important for acoustic predictions.

Stability and Bifurcation for the Equispaced, Periodic Motion of a Horizontal Impact Damper

2001 ◽  
Author(s):  
Albert C. J. Luo
Keyword(s):  
Periodic Motion ◽  
Period Doubling ◽  
Period Doubling Bifurcation ◽  
Periodic Excitation ◽  
Periodic Motions ◽  
Chaotic Motions ◽  
Impact Damper ◽  
Stability And Bifurcation ◽  
Period 2 ◽  
Good Agreement

Abstract Stability and bifurcation conditions for the asymmetric, periodic motion of a horizontal impact damper under a periodic excitation are developed through four mappings for two switch-planes relative to discontinuities. Period-doubling bifurcation for equispaced motion does not occur, but the asymmetric period-1 motions change to the asymmetric, period-2 ones through a period doubling bifurcation. A numerical prediction for equispaced to chaotic motions is completed. The numerical and analytical predictions of the periodic motion are in very good agreement. The asymmetric, periodic motions are also simulated.

Global Chaos in a Periodically Forced, Linear System With a Dead-Zone Restoring Force

2003 ◽  
Author(s):  
Albert C. J. Luo ◽  
Santhosh Menon
Keyword(s):  
Linear System ◽  
Periodic Motion ◽  
Dead Zone ◽  
Poincaré Mapping ◽  
Periodic Motions ◽  
Restoring Force ◽  
Chaotic Motions ◽  
Mapping Structures ◽  
Poincare Mapping ◽  
The Stability

The Poincare mapping and the corresponding mapping sections for global motions in a linear system possessing a dead-zone restoring force are developed through the switching planes pertaining to the two constraints. The global periodic motions based on the Poincare mapping are determined, and the analysis for the stability and bifurcation of periodic motion is carried out. From the global periodic motions, the global chaos in such a system is investigated numerically. The bifurcation scenario with varying parameters was presented. The mapping structures of periodic and chaotic motions are discussed. The Poincare mapping sections for global chaos are given for illustration. The grazing phenomenon embedded in chaotic motion is observed.

PARAMETRIC ANALYSIS OF BIFURCATION AND CHAOS IN A PERIODICALLY DRIVEN HORIZONTAL IMPACT PAIR

2012 ◽  
Vol 22 (11) ◽  
pp. 1250268 ◽  
Author(s):  
YU GUO ◽  
ALBERT C. J. LUO
Keyword(s):  
Dynamical Systems ◽  
Parametric Analysis ◽  
Periodic Motions ◽  
Bifurcation And Chaos ◽  
Chaotic Motions ◽  
Periodically Driven ◽  
Mapping Structures ◽  
Stability And Bifurcation Analysis ◽  
Different Types ◽  
The Impact

In this paper, complex motions and chaos in the periodically driven horizontal impact pair are investigated using the theory of switchability for discontinuous dynamical systems. Domains and boundaries are defined due to the discontinuity caused by impacts. Analytical conditions for switching of stick and grazing motions are derived in detail. Generic mappings are introduced to describe different periodic motions and to identify the mapping structures of chaos. The periodic motions in such impact pair are analytically predicted, and the corresponding stability and bifurcation analysis of periodic motions are carried out. Parameter maps with different types of motions are developed. Periodic and chaotic motions with different mapping structures are illustrated numerically for a better understanding of physics of ball motions in the impact pair.

Nonlinear Response of Flexible Robotic Manipulators Performing Repetitive Tasks

1989 ◽  
Vol 111 (3) ◽  
pp. 470-479 ◽  
Author(s):  
D. A. Streit ◽  
C. M. Krousgrill ◽  
A. K. Bajaj
Keyword(s):  
Nonlinear Response ◽  
Period Doubling ◽  
Flexible Manipulator ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Periodic Motions ◽  
Dynamical Equations ◽  
Chaotic Motions ◽  
Compliant Motion ◽  
Repetitive Tasks

The dynamics of a flexible manipulator is described by two distinct types of variables, one describing the nominal motion and the other describing the compliant motion. For a manipulator programmed to perform repetitive tasks, the dynamical equations governing the compliant motion are parametrically excited. Nonlinear dynamics of a two-degree-of-freedom model is investigated in parameter regions where the nominal motion is predicted by the Floquet theory to be unstable. Multiple time scales technique is used to study the nonlinear response, and it is shown that the compliant coordinates can execute small but finite amplitude periodic motions. In one particular case, the amplitude of these periodic motions is shown to bifurcate to a periodic solution which subsequently undergoes period-doubling bifurcations leading to chaotic motions.

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