Cubic Lienard Equations with Quadratic Damping (II)

In this paper, the global analysis of a Liénard equation with quadratic damping is studied. There are 22 different global phase portraits in the Poincaré disc. Every global phase portrait is given as well as the complete global bifurcation diagram. Firstly, the equilibria at finite and infinite of the Liénard system are discussed. The properties of the equilibria are studied. Then, the sufficient and necessary conditions of the system with closed orbits are obtained. The degenerate Bogdanov-Takens bifurcation is studied and the bifurcation diagrams of the system are given.

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A Comparison of the Effects of Nonlinear Damping on the Free Vibration of a Single-Degree-of-Freedom System

Journal of Vibration and Acoustics ◽

10.1115/1.4005010 ◽

2012 ◽

Vol 134 (2) ◽

Cited By ~ 19

Author(s):

Bin Tang ◽

M. J. Brennan

Keyword(s):

Free Vibration ◽

Equations Of Motion ◽

Nonlinear Damping ◽

Degree Of Freedom ◽

Damping Force ◽

Single Degree Of Freedom ◽

Sdof System ◽

Single Degree ◽

Dependent Term ◽

Quadratic Damping

This article concerns the free vibration of a single-degree-of-freedom (SDOF) system with three types of nonlinear damping. One system considered is where the spring and the damper are connected to the mass so that they are orthogonal, and the vibration is in the direction of the spring. It is shown that, provided the displacement is small, this system behaves in a similar way to the conventional SDOF system with cubic damping, in which the spring and the damper are connected so they act in the same direction. For completeness, these systems are compared with a conventional SDOF system with quadratic damping. By transforming all the equations of motion of the systems so that the damping force is proportional to the product of a displacement dependent term and velocity, then all the systems can be directly compared. It is seen that the system with cubic damping is worse than that with quadratic damping for the attenuation of free vibration.

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The Measurement of Nonlinear Damping in Single-Degree-of-Freedom Systems

Journal of Vibration and Acoustics ◽

10.1115/1.2930147 ◽

1991 ◽

Vol 113 (1) ◽

pp. 132-140 ◽

Cited By ~ 27

Author(s):

H. J. Rice ◽

J. A. Fitzpatrick

Keyword(s):

Nonlinear Distortion ◽

Nonlinear Damping ◽

Degree Of Freedom ◽

Crucial Importance ◽

Single Degree Of Freedom ◽

Single Degree ◽

Wide Range ◽

Impulse Excitation ◽

Excitation Techniques ◽

Quadratic Damping

The measurement and correct modelling of damping is of crucial importance in the prediction of the dynamical performance of systems for a wide range of engineering applications. In most cases, however, the experimental methods used to measure damping coefficients are extremely basic and, in general, poorly reported. This paper shows that damping is a deceptive parameter which is prone to subtle nonlinear distortion which often appears to satisfy general linear criteria. An efficient experimental method which provides for the measurement of both the linear and nonlinear damping for a single-degree-of-freedom system is proposed. The results from a numerical simulation study of a model with “drag” type quadratic damping are shown to give reliable estimates of parameters of the system when both random and impulse excitation techniques are used.

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