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Active vibration control of a dynamical system via negative linear velocity feedback

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An Erratum to this article was published on 29 March 2014

Abstract

In this paper, a negative velocity feedback is added to a dynamical system which is represented by second-order nonlinear differential equations having quadratic coupling, quadratic, and cubic nonlinearities. The system describes the vibration of the system subjected to multi-parametric excitation forces. The method of multiple scale perturbation technique is applied to obtain the response equation near the simultaneous internal and super-harmonic resonance case of this system. The stability to the system is investigated applying frequency response equations. The numerical solution and the effects of some parameters on the vibrating system are investigated and reported. The simulation results are achieved using MATLAB 7.0 program. A comparison is made with the available published work.

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Abbreviations

\(x_{1}\) and \(x_{2}\) :

The two mode amplitudes

\(a_{i} (i= 1,2)\) :

Steady-state amplitudes of the system

\(D_{0}\) and \(D_{1}\) :

Differential operators

\(T_{0}\) and \(T_{1}\) :

Fast and slow time scales, respectively (\(T_{n}=\varepsilon ^{n}t), \; n=0,1\)

t :

Time

\(\omega _{1}\), \(\omega _{2}\) :

The natural angular frequencies of the modes

\(\mu _{i} (i~= 1,2)\) :

The linear damping coefficients

\(\alpha _{1}, \beta _{1}\) :

The cubic nonlinear coefficients

\(\alpha _{2}, \beta _{2}\) :

The quadratic nonlinear coefficients

\(\alpha _{3}\) :

The coupling quadratic nonlinear coefficient

\(F_{j}, P_{j} \, (j = 1,\) :

The excitation force amplitudes

\(2, \ldots , { N})\) :

of the modes

\(G_{i} (i = 1,2)\) :

Positive constants (gains)

\(\Omega \) :

The excitation frequency

\(\varepsilon \) :

A small perturbation parameter

\(\dot{x}_i ,\ddot{x}_i (i=1,2)\) :

The derivatives with respect to t

\(\lambda \) :

Eigenvalue

\(\sigma _{i} (i = 1, 2)\) :

The detuning parameters

\(A_{n0} (n =1, 2)\) :

Functions of \(T_{1}\)

\(\gamma _{i} (i = 1, 2)\) :

Phase of the motion

\(\theta _{i} (i = 1,2)\) :

Phase of the motion

\(p_{i}, q_{i} (i=1, 2)\) :

Real parameters

\(r_{i} (i= 1,2, \ldots , 4)\) :

Constants

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Acknowledgments

The author would like to thank the reviewers for their valuable comments and suggestions for improving the quality of this paper.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Zagazig University, Zagazig, 44519, Egypt

    H. S. Bauomy

  2. Department of Mathematics, College of Arts and Science in Wadi Addawasir, Salman Bin Abdulaziz University, P. O. Box 54, Wadi Addawasir, 11991, Saudi Arabia

    H. S. Bauomy

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Bauomy, H.S. Active vibration control of a dynamical system via negative linear velocity feedback. Nonlinear Dyn 77, 413–423 (2014). https://doi.org/10.1007/s11071-014-1306-4

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