Dynamical analysis of Mathieu equation with two kinds of van der Pol fractional-order terms

Wen, Shaofang; Shen, Yongjun; Li, Xianghong; Yang, Shaopu

doi:10.1016/j.ijnonlinmec.2016.05.001

Cited by 21 publications

(8 citation statements)

References 27 publications

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“…To analyze the stability of the stationary solution, we substitute a ¼ a þ Da and h ¼ h þ Dh into equation (17), where Da and Dh are two small perturbations. Then…”

Section: Stability Conditionsmentioning

confidence: 99%

“…Shen et al [12][13][14][15] studied the vibration of several oscillators with fractional-order derivative, and found that the fractional-order derivatives had equivalent damping and equivalent stiffness effects on the system response. Wen et al 16,17 investigated the dynamical analysis of different Mathieu equation with a fractional-order derivative. The research on piecewise linear oscillators with fractional derivatives is little.…”

Section: Introductionmentioning

confidence: 99%

“…Wen et al. 16,17 investigated the dynamical analysis of different Mathieu equation with a fractional-order derivative. The research on piecewise linear oscillators with fractional derivatives is little.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Dynamic response of a piecewise linear single-degree-of-freedom oscillator with fractional-order derivative

Wang

Shen

Yang

et al. 2019

Journal of Low Frequency Noise, Vibration and Active Control

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In this paper, the dynamic response of a piecewise linear single-degree-of-freedom oscillator with fractional-order derivative is studied. First, a mathematical model of the single-degree-of-freedom system is established, and the approximate steady-state solution associated with the amplitude-frequency equation is obtained based on the averaging method. Then, the amplitude-frequency response equations are used for stability analysis, and the stability condition is founded. To validate the correctness and precision, the approximate solutions determined by the analytical method are compared with the solutions based on the numerical integration method. It is found that the approximate solutions and the numerical solutions are in excellent agreement. Finally, the effects of system parameters, such as fractional-order coefficient, order, clearance, and piecewise stiffness, on the complex dynamical behaviors of the piecewise linear singledegree-of-freedom oscillator are studied. The results show that the system parameters not only influence resonance amplitude and resonance frequency but also affect the size of the unstable region.

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“…To analyze the stability of the stationary solution, we substitute a ¼ a þ Da and h ¼ h þ Dh into equation (17), where Da and Dh are two small perturbations. Then…”

Section: Stability Conditionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Dynamic response of a piecewise linear single-degree-of-freedom oscillator with fractional-order derivative

Wang

Shen

Yang

et al. 2019

Journal of Low Frequency Noise, Vibration and Active Control

Self Cite

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show abstract

“…Najafi et al have used the generalized differential transform method [18]. The dynamics of Mathieu equation with two kinds of van der Pol (VDP) fractional-order terms is investigated by Wen et al [19]. Recently the Block-Pulse wavelets approximation method [20] was used by Pirmohabbati et al for numerical solution of fractional Mathieu equation.…”

Section: Introductionmentioning

confidence: 99%

Numerical Study of Fractional Mathieu Differential Equation Using Radial Basis Functions

Ghorbani¹,

Mahmoudi²,

Saei³

2020

MMEP

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In this paper, we introduce a method based on Radial Basis Functions (RBFs) for the numerical approximation of Mathieu differential equation with two fractional derivatives in the Caputo sense. For this, we suggest a numerical integration method for approximating the improper integrals with a singularity point at the right end of the integration domain, which appear in the fractional computations. We study numerically the affects of characteristic parameters and damping factor on the behavior of solution for fractional Mathieu differential equation. Some examples are presented to illustrate applicability and accuracy of the proposed method. The fractional derivatives order and the parameters of the Mathieu equation are changed to study the convergency of the numerical solutions.

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“…A few researchers studied the fractional-order complex nonlinear system. For example, Wen et al [28,29] investigated the dynamical characteristics of Mathieu-Duffing oscillator with the fractional-order delayed feedback, and found that the fractional-order delayed feedback had the characteristics of both delayed velocity feedback and displacement feedback. Xu et al [30] analyzed the dynamical responses of fractional-order Duffing-Rayleigh oscillator with the Gauss white noise excitation.…”

Section: Introductionmentioning

confidence: 99%

Dynamical response of fractional-order nonlinear system with combined parametric and forcing excitation

Xing¹,

Xiao²,

Song³

et al. 2018

J. vibroeng.

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A dynamical analysis of a Mathieu-van der Pol-Duffing nonlinear system with fractional-order derivative under combined parametric and forcing excitation is studied in this paper. The approximate analytical solution is researched for 1/2 sub harmonic resonance coupled with primary parametric resonance based on the improved averaging approach. The steady-state periodic solution including its stability condition is established. The equivalent linear stiffness coefficient (ELDC) and equivalent linear damping coefficient (ELSC) for this nonlinear fractional-order oscillator are defined. Then, the numerical simulations are presented in three typical cases by iterative algorithms. The time history, phase portrait, FFT spectrum and Poincare maps are shown to explain the system response. Some different responses, such as quasi-periodic, multi-periodic and single periodic behaviors are observed and investigated. The results of comparisons between the numerical solutions and the approximate analytical solutions in three typical cases show the correctness of the analytical solutions. The influences of the fractional-order parameters on the system dynamical response are researched based on the ELDC and ELSC. Through analysis, it could be found that the increase of the fractional-order coefficient would result in the rightward and downward movements of the amplitude-frequency curves. The increase of the fractional-order coefficient will also move the bifurcation point rightwards and will make the existing range of steady-state solution larger. It could also be found that the ELSC will become larger and ELDC smaller when the fractional order is closer to zero, so that the decrease of the fractional order would make the response amplitude larger. At last, the detailed conclusions are summarized, which is beneficial to design and control this kind of fractional-order nonlinear system.

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Dynamical analysis of Mathieu equation with two kinds of van der Pol fractional-order terms

Cited by 21 publications

References 27 publications

Dynamic response of a piecewise linear single-degree-of-freedom oscillator with fractional-order derivative

Dynamic response of a piecewise linear single-degree-of-freedom oscillator with fractional-order derivative

Numerical Study of Fractional Mathieu Differential Equation Using Radial Basis Functions

Dynamical response of fractional-order nonlinear system with combined parametric and forcing excitation

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