Vibration of Harmonically Excited Oscillators With Asymmetric Constraints

1992 ◽  
Vol 59 (2S) ◽  
pp. S284-S290 ◽  
Author(s):  
S. Natsiavas ◽  
H. Gonzalez
Keyword(s):  
Piecewise Linear ◽  
Periodic Motions ◽  
Van Der Pol ◽  
System Parameters ◽  
State Response ◽  
Damping Coefficients ◽  
Restoring Forces ◽  
Van Der Pol Oscillators ◽  
The Stability ◽  
Stiffness And Damping

investigation is carried out for a class of piecewise linear oscillators with asymmetric characteristics. The damping and restoring forces are general trilinear functions of the system velocity and displacement, respectively, while the excitation is harmonic in time. First, an analysis is presented which determines harmonic and subharmonic steady-state response. Then, a special formulation is employed in examining the stability of located periodic motions. Finally, numerical results are presented for several representative sets of the system parameters. Effects of asymmetries in the response due to unequal gaps as well as unequal stiffness and damping coefficients are analyzed in detail. Asymmetric response of a system with symmetric technical characteristics is also investigated. The behavior of the systems examined resembles response of similar nonlinear systems with continuous characteristics, like the response of the Duffing and van der Pol oscillators. Complicated nonperiodic response is also encountered and analyzed.

Dynamics of Two Van Der Pol Oscillators Coupled via a Bath

2003 ◽  
Author(s):  
Erika Camacho ◽  
Richard Rand ◽  
Howard Howland
Keyword(s):  
Software Package ◽  
Bifurcation Theory ◽  
Floquet Theory ◽  
Van Der Pol Oscillator ◽  
Direct Connection ◽  
Periodic Motions ◽  
Van Der Pol ◽  
Existence And Stability ◽  
Van Der Pol Oscillators ◽  
The Stability

In this work we study a system of two van der Pol oscillators, x and y, coupled via a “bath” z: x¨−ε(1−x2)x˙+x=k(z−x)y¨−ε(1−y2)y˙+y=k(z−y)z˙=k(x−z)+k(y−z) We investigate the existence and stability of the in-phase and out-of-phase modes for parameters ε > 0 and k > 0. To this end we use Floquet theory and numerical integration. Surprisingly, our results show that the out-of-phase mode exists and is stable for a wider range of parameters than is the in-phase mode. This behavior is compared to that of two directly coupled van der Pol oscillators, and it is shown that the effect of the bath is to reduce the stability of the in-phase mode. We also investigate the occurrence of other periodic motions by using bifurcation theory and the AUTO bifurcation and continuation software package. Our motivation for studying this system comes from the presence of circadian rhythms in the chemistry of the eyes. We present a simplified model of a circadian oscillator which shows that it can be modeled as a van der Pol oscillator. Although there is no direct connection between the two eyes, they can influence each other by affecting the concentration of melatonin in the bloodstream, which is represented by the bath in our model.

Dynamics of a Continuous System With Clearance and Motion Limiting Stops

1999 ◽  
Author(s):  
P. Metallidis ◽  
S. Natsiavas
Keyword(s):  
Piecewise Linear ◽  
Research Work ◽  
Continuous System ◽  
Continuous Model ◽  
Equation Of Motion ◽  
Model System ◽  
Periodic Motions ◽  
System Parameters ◽  
Rigid Rods ◽  
The Stability

Abstract The present study generalises previous research work on the dynamics of discrete oscillators with piecewise linear characteristics and investigates the response of a continuous model system with clearance and motion-limiting constraints. More specifically, in the first part of this work, an analysis is presented for determining exact periodic response of a periodically excited deformable rod, whose motion is constrained by a flexible obstacle. This methodology is based on the exact solution form obtained within response intervals where the system parameters remain constant and its behavior is governed by a linear equation of motion. The unknowns of the problem are subsequently determined by imposing an appropriate set of periodicity and matching conditions. The analytical part is complemented by a suitable method for determining the stability properties of the located periodic motions. In the second part of the study, the analysis is applied to several cases in order to investigate the effect of the system parameters on its dynamics. Special emphasis is placed on comparing these results with results obtained for similar but rigid rods. Finally, direct integration of the equation of motion in selected areas reveals the existence of motions, which are more complicated than the periodic motions determined analytically.

scholarly journals Study on dynamic characteristics of gas films of spherical spiral groove hybrid gas bearings

2019 ◽  
Vol 233 (8) ◽  
pp. 1169-1181
Author(s):  
Chenhui Jia ◽  
Haijiang Zhang ◽  
Shijun Guo ◽  
Ming Qiu ◽  
Wensuo Ma ◽  
...  
Keyword(s):  
Pressure Effect ◽  
Dynamic Pressure ◽  
Iteration Algorithm ◽  
Unstable State ◽  
Gas Bearings ◽  
Damping Coefficients ◽  
The Stability ◽  
Stiffness And Damping ◽  
Film Stiffness ◽  
Gas Bearing

According to the gas film force variation law, when the bearing axis is slightly displaced from the static equilibrium position, displacement and velocity disturbance relation expressions for the gas film force increment are constructed. Moreover, combined with the bearing rotor system motion equation, calculation model equations for the gas film stiffness and damping coefficients are established. The axial and radial vibration and velocity of the gas bearings during operation are collected. The instantaneous stiffness and damping coefficients of the gas film are calculated by the rolling iteration algorithm using MATLAB. The dynamic changes in the gas film stiffness and damping under different motion states are analyzed, and the mechanism of the gas film vortex and oscillation is studied. The results demonstrate the following: (1) When the gas bearing is running in the linear steady state in cycle 1, the dynamic pressure effect is enhanced and the stability is improved by increasing the eccentricity; when the gas supply pressure is increased, the static pressure effect is enhanced and the gas film vortex is reduced, but the oscillation is strengthened. (2) With the increase in rotational speed, the gas film vortex force gradually exceeds the gas film damping force, and the stability gradually worsens, causing a fluctuation in the gas film stiffness and damping, following which singularity occurs and a half-speed vortex is formed. Meanwhile, the gas film oscillation is intensified, and the rotor enters the nonlinear stable cycle 2 state operation. (3) As the fluctuation of the film force increases, the instantaneous stiffness and damping oscillation of the film intensifies, most of the stiffness and damping coefficients exhibit distortion, and the rotor operation will enter a chaotic or unstable state. Therefore, the gas bearing stiffness and damping variation characteristics can be used to study and predict the gas bearing operating state. Finally, measures for reducing the vortex and oscillation of the gas film and improving the stability of the gas bearing operation are proposed.

Moment Stability of Stochastic Regenerative Cutting Process

2010 ◽  
Vol 97-101 ◽  
pp. 3038-3041
Author(s):  
Xiao Qin Zhou ◽  
Wen Cai Wang ◽  
Hong Wei Zhao
Keyword(s):  
Stability Analysis ◽  
Stability Boundary ◽  
Cutting Process ◽  
Damping Coefficients ◽  
Cutting Stability ◽  
Stationary Stochastic Processes ◽  
The Stability ◽  
Stiffness And Damping ◽  
Moment Stability ◽  
Stochastic Uncertainties

The stochastic uncertainties of regenerative cutting process (RCP) are taken into consideration, and both cutting stiffness and damping coefficients are modeled as two stationary stochastic processes. The eigenvalue equations are established for the stability analysis of stochastic RCP, corresponding to the differential equations of the first and second order moments. Thus the stability analysis of stochastic RCP is transformed into that of the first two order moments. The influence of stochastic uncertainties on the cutting stability of RCP is discussed. The numerical experiments have verified that with the increase of stochastic uncertainties, the cutting stability boundary was shifted downwards significantly, and the number of lobes was also multiplied.

Periodic Motions of a Semi-Active Suspension System With MR Damping.

2006 ◽  
Author(s):  
Albert C. J. Luo ◽  
Arun Rajendran
Keyword(s):  
Piecewise Linear ◽  
Active Suspension ◽  
Suspension System ◽  
Mapping Technique ◽  
Periodic Motions ◽  
Mapping Structures ◽  
The Stability ◽  
Magneto Rheological ◽  
Mr Damping ◽  
Full Nonlinearity

Periodic motions in a hysteretically damped, semi-active suspension system are investigated. The Magneto-Rheological damping varying with relative velocity is modeled through a piecewise-linear model. The theory for discontinuous dynamical systems is employed to determine the grazing motions in such a system, and the mapping technique is used to develop the mapping structures of periodic motions. The periodic motions are predicted analytically and verified numerically. The stability and bifurcation analyses of such periodic motions are performed, and the parameters for all possible motions are developed. This model is applicable for the semi-active suspension system with the Magneto-Rheological damper in automobiles. The further investigation on the Magneto-Rheological damping with full nonlinearity should be completed.

Dynamics of Two Coupled Van der Pol Oscillators With Delay Coupling

1997 ◽  
Author(s):  
Stephen Wirkus ◽  
Richard Rand
Keyword(s):  
Three Dimensions ◽  
Slow Flow ◽  
Method Of Averaging ◽  
Van Der Pol ◽  
Laser Oscillators ◽  
Van Der Pol Oscillators ◽  
The Stability

Abstract We investigate the dynamics of a system of two van der Pol oscillators with delayed velocity coupling. We use the method of averaging to reduce the problem to the study of a slow-flow in three dimensions. In particular we study the stability of the in-phase and out-of-phase modes, and the bifurcations associated with changes in their stability. Our interest in this system is due to its relevance to coupled laser oscillators.

Stability of Multirecess Hybrid-Operating Oil Journal Bearings

1985 ◽  
Vol 107 (1) ◽  
pp. 116-121 ◽  
Author(s):  
Y. S. Chen ◽  
H. Y. Wu ◽  
P. L. Xie
Keyword(s):  
Finite Difference Method ◽  
Dynamic Performance ◽  
Journal Bearings ◽  
Design Parameters ◽  
Stability Threshold ◽  
Damping Coefficients ◽  
Hybrid Bearing ◽  
The Stability ◽  
Stiffness And Damping ◽  
Bearing System

An analysis and a numerical solution using finite difference method to predict the dynamic performance of multirecess hybrid-operating oil journal bearings are presented. The linearized stiffness and damping coefficients of a typical capillary-compensated bearing with four recesses are computed for various design parameters. The corresponding stiffness and the stability threshold of the bearing are then obtained, and the opposite influences of the hydrodynamic action on them are demonstrated. The effect of rotor flexibility on the onset of self-excited whirl is discussed, and a method is given to determine the stability threshold of a rotor-hybrid bearing system.

scholarly journals Stability of a Rigid Rotor Supported on Flexible Oil Journal Bearings

1988 ◽  
Vol 110 (1) ◽  
pp. 181-187 ◽  
Author(s):  
B. C. Majumdar ◽  
D. E. Brewe ◽  
M. M. Khonsari
Keyword(s):  
Steady State ◽  
Journal Bearings ◽  
Rigid Rotor ◽  
Elasticity Parameter ◽  
Damping Coefficients ◽  
Pressure Dependent Viscosity ◽  
The Stability ◽  
Dependent Viscosity ◽  
Stiffness And Damping

This investigation deals with the stability characteristics of oil journal bearings, including the effect of elastic distortions in the bearing liner. Graphical results are presented for (1) steady-state load, (2) stiffness and damping coefficients, and (3) the stability. These results are given for various slenderness ratios, eccentricity ratios, and elasticity parameters. The lubricant is first assumed to be isoviscous. The analysis is then extended to the case of a pressure-dependent viscosity. It has been found that stability decreases with increase of the elasticity parameter of the bearing liner for heavily loaded bearings.

Period-m Motions in a Periodically Forced van der Pol Oscillator

2013 ◽  
Author(s):  
Albert C. J. Luo ◽  
Arash Baghaei Lakeh
Keyword(s):  
Fourier Series ◽  
Periodic Motion ◽  
Approximate Solutions ◽  
Van Der Pol Oscillator ◽  
Periodic Motions ◽  
Van Der Pol ◽  
Stable Period ◽  
Stability And Bifurcation Analysis ◽  
Series Expression ◽  
The Stability

Period-m motions in a periodically forced, van der Pol oscillator are investigated through the Fourier series expression, and the stability and bifurcation analysis of such periodic motions are carried out. To verify the approximate solutions of period-m motions, numerical illustrations are given. Period-m motions are separated by quasi-periodic motion or chaos, and the stable period-m motions are in independent periodic motion windows.

Modal Analysis of Rotor on Piecewise Linear Journal Bearings Under Seismic Excitation

1999 ◽  
Vol 121 (2) ◽  
pp. 190-196 ◽  
Author(s):  
B. J. Gaganis ◽  
A. K. Zisimopoulos ◽  
P. G. Nikolakopoulos ◽  
C. A. Papadopoulos
Keyword(s):  
Modal Analysis ◽  
Dynamic Properties ◽  
Piecewise Linear ◽  
Computer Time ◽  
Seismic Excitation ◽  
Damping Coefficients ◽  
Highly Nonlinear ◽  
Rotor Bearing ◽  
Stiffness And Damping ◽  
Bearing System

A rotor bearing system is expected to exhibit large vibration amplitudes when subjected to a large seismic excitation. It is possible that these vibrations can lead to large values the eccentricity of the bearings. Then the bearing is operated in highly nonlinear region because the stiffness and the damping coefficients are nonlinear as functions of the eccentricity. To solve this problem numerical integration must be performed with high cost in computer time. The idea of this paper was to divide the nonlinear area into more areas where the stiffness and damping coefficients could be considered to be constants. In other words the bearing coefficients are considered to be piecewise constant. The excitation due to the earthquake is modelled as a movement of the base of the bearings using the El Centro data for the acceleration. Then a simplified modal analysis for each of these piecewise linear regions can be performed. The equation of motion of the rotor contains rotational speed depended terms, known as gyroscopic terms, and terms due to base excitation. The response and the variation of the dynamic properties of this complicated rotor bearing system are investigated and presented.

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