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

[+] Author and Article Information
Bin Tang1

Institute of Internal Combustion Engine, Dalian University of Technology, Dalian 116023, Chinabtang@dlut.edu.cn

M. J. Brennan

Department of Mechanical Engineering, UNESP, Ilha Solteira, SP15385-000, Brazil

1

Corresponding author.

J. Vib. Acoust 134(2), 024501 (Jan 13, 2012) (5 pages) doi:10.1115/1.4005010 History: Received September 22, 2010; Accepted July 12, 2011; Published January 13, 2012; Online January 13, 2012

## Abstract

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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## Figures

Figure 4

Time histories for the free vibration of the nonlinear systems, when the initial conditions are y (0) = 0.2, y' (0) = 0 for (a)-(c); y (0) = 0.0, y' (0) = 0.2 for (d)-(f). Solid line —, analytical results from the method of multiple scales or averaging; circles •, numerical results from the Runge-Kutta method with the step size control algorithm; dashed line ---, envelope of the approximate results. (a), (d) nonlinear system of Fig. 1a with ζ1=10; (b), (e) nonlinear system with quadratic damping ζ2=1; (c), (f) nonlinear system with cubic damping ζ3=10/3.

Figure 3

Normalized Envelopes for the free vibration of linear and nonlinear systems when the initial conditions are y (0) = 0.2, y' (0) = 0; dashed-dotted line –·–, linear system with ζ  = 0.4; solid line —, nonlinear system of Fig. 1a with ζ1=10; dotted line ·····, nonlinear system with quadratic damping ζ2=1; dashed line ---, nonlinear system with cubic damping ζ3=10/3

Figure 2

Ratio of the equivalent damping ratio to the damping ratio of a linear SDOF system; dashed line ---, exact result; solid line —, first order approximate result

Figure 1

Single degree-of-freedom systems with (a) a horizontal linear damper. (b) quadratic or cubic damping.

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