Dynamic properties of piecewise linear systems with fractional time-delay feedback

Jianchao Zhang,Jiangchuan Niu,Yufei Hu,Jun Wang

doi:10.1177/14613484211007633

Jianchao Zhang, Jiangchuan Niu + Show 2 more

Open Access

https://doi.org/10.1177/14613484211007633

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Abstract

The forced vibration of a single-degree-of-freedom piecewise linear system containing fractional time-delay feedback was investigated. The approximate analytical solution of the system was obtained by employing an averaging method. A frequency response equation containing time delay was obtained by studying a steady-state solution. The stability conditions of the steady-state solution, the amplitude–frequency results, and the numerical solutions of the system under different time-delay parameters were compared. Comparison results indicated a favorable goodness of fit between the two parameters and revealed the correctness of the analytical solution. The effects of the time-delay and fractional parameters, piecewise stiffness, and piecewise gap on the principal resonance and bifurcation of the system were emphasized. Results showed that fractional time delay occurring in the form of equivalent linear dampness and stiffness under periodic variations in the system and influenced the vibration characteristic of the system. Moreover, piecewise stiffness and gap induced the nonlinear characteristic of the system under certain parameters.

Highlights

As a branch of nonlinear dynamics, nonsmooth systems extensively exist in engineering applications, including circuits,[1,2] vehicle suspensions,[3,4] and vibration isolators.[5]
We considered the weakening effect of the equivalent stiffness generated by the fractional time-delay feedback term on system stiffness and the resulting periodic change
The forced vibration of an SDOF piecewise linear system with fractional time-delay feedback was investigated in this paper

Summary

Introduction

As a branch of nonlinear dynamics, nonsmooth systems extensively exist in engineering applications, including circuits,[1,2] vehicle suspensions,[3,4] and vibration isolators.[5]. Where m is the oscillator mass; k1x is the linear spring force; gðxÞ is the piecewise spring force; c is the damping coefficient; K1DP1⁄2xðt À sÞ is the fractional time-delay feedback term; K1 is the coefficient of fractional time-delay feedback; p is the fractional order whose physical meaning and geometric interpretation have been explained in References[35,36,37]; s is the time delay of the system; F is the excitation force; and x is the excitation frequency Many analytical methods, such as fractional complex transform,[38] variational iteration,[39] homotopy perturbation method,[40] and He’s frequency method,[21] could be used for the fractional differential equation as shown in equation (1). The following equation is obtained in accordance with the Caputo fractional differential definition.[41]

Cð1 À dÞ t

The set of transition points is

Findings

Conclusion