The Dynamic Response of a System With Preloaded Compliance

1988 ◽  
Vol 110 (3) ◽  
pp. 278-283 ◽  
Author(s):  
S. W. Shaw ◽  
P. C. Tung
Keyword(s):  
Dynamic Response ◽  
Piecewise Linear ◽  
Harmonic Excitation ◽  
Periodic Motions ◽  
Chaotic Motions ◽  
Linear Features ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Stable Periodic ◽  
Model Equations

We consider the dynamic response of a single degree of freedom system with preloaded, or “setup,” springs. This is a simple model for systems where preload is used to suppress vibrations. The springs are taken to be linear and harmonic excitation is applied; damping is assumed to be of linear viscous type. Using the piecewise linear features of the model equations we determine the amplitude and stability of the periodic responses and carry out a bifurcation analysis for these motions. Some parameter regions which contain no simple stable periodic motions are shown to possess chaotic motions.

The Dynamics of a Nonharmonically Excited System Having Rigid Amplitude Constraints

1992 ◽  
Vol 59 (3) ◽  
pp. 693-695 ◽  
Author(s):  
Pi-Cheng Tung
Keyword(s):  
Dynamic Response ◽  
Bifurcation Analysis ◽  
Piecewise Linear ◽  
Coefficient Of Restitution ◽  
Degree Of Freedom ◽  
Linear Oscillator ◽  
Chaotic Motions ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Excited System

We consider the dynamic response of a single-degree-of-freedom system having two-sided amplitude constraints. The model consists of a piecewise-linear oscillator subjected to nonharmonic excitation. A simple impact rule employing a coefficient of restitution is used to characterize the almost instantaneous behavior of impact at the constraints. In this paper periodic and chaotic motions are found. The amplitude and stability of the periodic responses are determined and bifurcation analysis for these motions is carried out. Chaotic motions are found to exist over ranges of forcing periods.

Periodic Motions in a Single-Degree-of-Freedom System Under Both an Aerodynamic Force and a Harmonic Excitation

2019 ◽  
Author(s):  
Bo Yu ◽  
Albert C. J. Luo
Keyword(s):  
Numerical Simulations ◽  
Analytical Approach ◽  
Aerodynamic Force ◽  
Excitation Frequency ◽  
Harmonic Excitation ◽  
Degree Of Freedom ◽  
Periodic Motions ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Stable Period

Abstract In this paper, a semi-analytical approach was used to predict periodic motions in a single-degree-of-freedom system under both aerodynamic force and harmonic excitation. Using the implicit mappings, the predictions of period-1 motions varying with excitation frequency are obtained. Stability of the period-1 motions are discussed, and the corresponding eigenvalues of period-1 motions are presented. Finally, numerical simulations of stable period-1 motions are illustrated.

A Periodically Forced Impact Oscillator With Large Dissipation

1983 ◽  
Vol 50 (4a) ◽  
pp. 849-857 ◽  
Author(s):  
S. W. Shaw ◽  
P. J. Holmes
Keyword(s):  
Harmonic Excitation ◽  
Coefficient Of Restitution ◽  
Periodic Motions ◽  
Impact Oscillator ◽  
Chaotic Motions ◽  
One Dimensional ◽  
Mapping Analysis ◽  
Stable Periodic ◽  
Almost All ◽  
Dimensional Mapping

We consider the simple harmonic oscillator with harmonic excitation and a constraint that restricts motions to one side of the equilibrium position. Thus, on the achievement of a specified displacement, the direction of motion is reversed using the simple impact rule. The coefficient of restitution for this impact, r, is taken to be small. For r = 0 the motions of the system can be studied using a one-dimensional mapping. Analysis of this map shows that stable periodic orbits exist at almost all forcing frequencies but that transient nonperiodic or chaotic motions can also occur. Moreover, over certain (narrow) frequency windows arbitrarily long stable periodic motions exist. These results are then extended to the case r ≠ 0, small.

The Dynamics of a Harmonically Excited System Having Rigid Amplitude Constraints, Part 1: Subharmonic Motions and Local Bifurcations

1985 ◽  
Vol 52 (2) ◽  
pp. 453-458 ◽  
Author(s):  
S. W. Shaw
Keyword(s):  
Simple Model ◽  
Piecewise Linear ◽  
Equation Of Motion ◽  
Harmonic Excitation ◽  
Coefficient Of Restitution ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Local Bifurcations ◽  
Excited System ◽  
Double Impact

A simple model for the response of mechanical systems having two-sided amplitude constraints is considered. The model consists of a piecewise-linear single degree-of-freedom oscillator subjected to harmonic excitation. Encounters with the constraints are modeled using a simple impact rule employing a coefficient of restitution, and excursions between the constraints are assumed to be governed by a linear equation of motion. Symmetric double-impact motions, both harmonic and subharmonic, are studied by means of a mapping that relates conditions at subsequent impacts. Stability and bifurcation analyses are carried out for these motions and regions are found in which no stable symmetric motions exist. The possible motions that can occur in such regions are discussed in the following paper, Part 2.

A Unique Formulation of Piecewise Exact Method to Analyze a Nonlinear Spring System under Harmonic Excitation

2014 ◽  
Vol 31 (3) ◽  
pp. 337-344
Author(s):  
P.-S. Xie ◽  
P.-J. Shih
Keyword(s):  
Exact Solution ◽  
Numerical Approximation ◽  
Piecewise Linear ◽  
Equilibrium Points ◽  
Harmonic Excitation ◽  
Exact Method ◽  
Nonlinear Stiffness ◽  
Time Error ◽  
Single Degree Of Freedom ◽  
Spring System

AbstractThis paper introduces a unique, efficient, and exact formulation for solving a single-degree-of-freedom system with nonlinear stiffness under a harmonic loading. This formulation is one kind of the piecewise exact method, and its benefit lies in providing the closed-form exact solution in each displacement segment. Since the exact solution is given in each segment, the continuity between two segments can be confirmed. Consequently, no instability errors affect the analysis. To determine the exact solutions in these segments, this research develops a technique that shifts the equilibrium points of the piecewise linear segments, which are discretized from a nonlinear stiffness curve, to new equilibrium points in order to satisfy the typical linear exact solution. Thus, positive- and negative-stiffness linear segments can be solved with this technique. This formulation saves roughly 60% of the calculation time (error < 10−10) as compared to the numerical approximation.

scholarly journals Shock Performance of Different Semiactive Damping Strategies

2010 ◽  
Vol 8 (02) ◽  
Author(s):  
D. F. Ledezma-Ramirez ◽  
N. Ferguson ◽  
M. Brennan
Keyword(s):  
Harmonic Excitation ◽  
Degree Of Freedom ◽  
Harmonic Vibration ◽  
Passive Stiffness ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Variable Damping ◽  
Stiffness And Damping ◽  
Shock Pulses

The problem of shock generated vibration is presented and analyzed. The fundamental background is explained based on the analysis of a single degree-of-freedom model with passive stiffness and damping. The advantages and limitations of such a shock mount are discussed. Afterwards, different semi-active strategies involving variable damping are presented. These strategies have been used for harmonic excitation but it is not clear how they will perform during a shock. This paper analyzes the different variable damping schemes already used for harmonic vibration in order to find any potential advantages or issues for theoretical shock pulses.

Bifurcation and Chaos in Friction-Induced Vibration

2002 ◽  
Author(s):  
Yong Li ◽  
Z. C. Feng
Keyword(s):  
Friction Model ◽  
Degree Of Freedom ◽  
Stable Limit ◽  
Mechanical Oscillator ◽  
Chaotic Motions ◽  
Sliding Speed ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Friction Models ◽  
Friction Induced Vibration

Friction-induced vibration is a phenomenon that has received extensive study by the dynamics community. This is because of the important industrial relevance and the evere-volving development of new friction models. In this paper, we report the result of bifurcation study of a single-degree-of-freedom mechanical oscillator sliding over a surface. The friction model we use is that developed by Canudas de Wit et al, a model that is receiving increasing acceptance from the mechanics community. Using this model, we find a stable limit cycle at intermediate sliding speed for a single-degree-of-freedom mechanical oscillator. Moreover, the mechanical oscillator can exhibit chaotic motions. For certain parameters, numerical simulation suggests the existence of a Silnikov homoclinic orbit. This is not expected in a single-degree-of-freedom system. The occurrence of chaos becomes possible because the friction model contains one internal variable. This demonstrates a unique characteristic of the friction model. Unlike most friction models, the present model is capable of simultaneously modeling self-excitation and predicting stick-slip at very low sliding speed as well.

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