Principal resonance responses of SDOF systems with small fractional derivative damping under narrow-band random parametric excitation

doi:10.1016/j.cnsns.2014.03.018

Communications in Nonlinear Science and Numerical Simulation

Volume 19, Issue 10, October 2014, Pages 3642-3652

https://doi.org/10.1016/j.cnsns.2014.03.018 Get rights and content

Highlights

•
The SDOF systems with fractional damping and the narrow-band random excitations.
•
Multiple scale method is developed to obtain the principal resonance responses.
•
Effects of the stability of trivial and non-trivial steady-state responses are discussed.
•
The stochastic jump is investigated for principal resonance responses.

Abstract

The principal resonance responses of nonlinear single-degree-of-freedom (SDOF) systems with lightly fractional derivative damping of order α (0 < α < 1) subject to the narrow-band random parametric excitation are investigated. The method of multiple scales is developed to derive two first order stochastic differential equation of amplitude and phase, and then to examine the influences of fractional order and intensity of random excitation on the first-order and second-order moment. As an example, the stochastic Duffing oscillator with fractional derivative damping is considered. The effects of detuning frequency parameter, the intensity of random excitation and the fractional order derivative damping on stability are studied through the largest Lyapunov exponent. The corresponding theoretical results are well verified through direct numerical simulations. In addition, the phenomenon of stochastic jump is analyzed for parametric principal resonance responses via finite differential method. The stochastic jump phenomena indicates that the most probable motion is around the larger non-trivial branch of the amplitude response when the intensity of excitation is very small, and the probable motion of amplitude responses will move from the larger non-trivial branch to trivial branch with the increasing of the intensity of excitation. Such stochastic jump can be considered as bifurcation.

Introduction

Fractional calculus is shown to be very suitable for descripting the constitutive relationship of materials with frequency-dependent damping behaviors. Gemant [1] provided a pioneering work of this field, which first use fractional calculus to study phenomenological constitutive equations for material behavior. Afterwards, Caputo and Mainardi [2] found good agreement with experimental results when using fractional derivatives for the description of the behavior of viscoelastic materials. In the beginning of 1980s, Bagley and Torvik [3], [4], [5] gave a physical justification for this concept, which is the theoretical basis for the application of fractional calculus to viscoelasticity. As a continuation of these works, many scholars, such as Koh and Kelly [6], Markris [7], [8], Pritz [9], Agrawal [10] and Mainardi [11] gave deep and detailed discusses on fractional calculus that are mostly used in mechanics fields. In addition, Rossikhin and Shitikova [12], [13] made two excellent reviews of development in this research field.

The analytical study is important for understanding dynamical behaviors of system involving fractional derivatives, and there are many analytical methods have already been used to investigate those deterministic dynamic systems. Qi and Xu [14] used the finite Fourier cosine and Laplace transforms to analysis the unsteady flow of viscoelastic fluid with the fractional-order derivative Maxwell model. Leung and Guo [15], [16], [17] proposed a residue harmonic balance method for autonomous, non-autonomous or delay systems with fractional derivative damping and derived accurately analyze results. Kovacic and Zukovic [18] considered the free oscillators with a power-form restoring force and fractional derivative damping based on the averaging method. Later, Shen et al. [19] investigated the primary resonance of Duffing oscillator with viscous damping and fractional derivative using standard averaging method. Also, another important analytical method, named multiple scales method, had been developed by Rossikhin and Shitikova [20], [21], [22] to discuss vibrations of nonlinear oscillations with fractional derivative.

It is generally known that stochastic perturbations are omnipresent and inevitable. Thus the researches of stochastic responses of stochastic systems with fractional derivative are necessary. Agrawal [23], [24], [25] used eigenvector expansion method and in combination with Laplace transforms to derive the analytical solution for linear stochastic system with fractional derivatives. Further, the system involving two fractional derivatives has been addressed by Huang et al. [26] and a Duhamel integral expression has been derived. Di Paola et al. [27] studied the stochastic responses of linear fractional viscoelastic systems subjected to stationary and non-stationary inputs. Based on the residue calculus method, Liu et al. [28] discussed the stochastic responses of an axially moving viscoelastic beam with fractional order constitutive relation under stationary and non-stationary random excitations. Alternatively, stochastic averaging method based on the generalized harmonic function has been explored to investigate the stochastic dynamics including fractional derivative by Huang and Jin [29]. As further, Chen and Zhu [30], [31], [32] used stochastic averaging to study the nonlinear system with fractional derivative damping under combined harmonic and white noise excitations. In Ref. [33], [34], Xu et al. proposed a new Lindstedt–Poincaré method to obtain the second-order approximate analytical solution of Fractional oscillators with random excitation. However, as a very important and effective analysis method multiple scales method has not been developed to investigate the principal resonance responses of stochastic system with fractional derivative and so we will do this work have.

In our paper, a multiple scales method is presented to research the SDOF stochastic system with fractional derivative. Firstly, we will get two closed-form stochastic differential equations of amplitude and phase. Secondly, the influence of the fractional order and the intensity of the noise on the steady-state response will be examined. Finally, as an example, Duffing oscillation is used to verify the results of theoretical analysis. The phenomenon of stochastic jump inducing by stochastic excitation is also considered.

This paper is organized as follows. In Section 2, the multiple time scales method is used to discuss the dynamic behaviors of SDOF systems with lightly fractional derivative damping subject to narrow-band random parametric excitations. Section 3 specializes on the Duffing system involving fractional order derivatives and narrow-band noise. Finally, some conclusions are drawn in Section 4.

Section snippets

Multiple time scales approximate solution

Consider a nonlinear SDOF oscillator with a lightly fractional derivative damping and subject to a random parametric fluctuation whose motion equation is $\frac{d^{2} x (t)}{{dt}^{2}} + ε δ D^{α} x (t) + ω_{0}^{2} x (t) + ε f (x (t), \dot{x} (t)) = ε x (t) ξ (t),$ where ɛ is a small positive parameter, δ is a constant coefficient, $f (x (t), \dot{x} (t))$ is a nonlinear odd function, and the random fluctuation ξ(t) is a narrow-band random process which can be described as the following form [35] $ξ (t) = h \cos (Ω t + \bar{γ} W (t)),$ where h > 0 is the deterministic amplitude of

Example: Duffing oscillator with fractional derivative damping term

In order to show a more thorough meticulous of the research above, the Duffing oscillator with fractional derivative damping term subject to narrow-band random excitation is considered as an example, which can be written as $\frac{d^{2} x (t)}{{dt}^{2}} + ε δ D^{α} x (t) + ω_{0}^{2} x (t) + ε β x^{3} = ε x (t) ξ (t) .$

From Eq. (16), the amplitude and phase equations of Eq. (24) yield $\begin{matrix} \dot{a} = - \frac{δ}{2} ω_{0}^{α - 1} a \sin (\frac{α π}{2}) + \frac{h}{4 ω_{0}} a \sin (θ), \\ a \dot{θ} = σ a - δ ω_{0}^{α - 1} a \cos (\frac{α π}{2}) - \frac{3 β}{ω_{0}} a^{3} + \frac{h}{2 ω_{0}} a \cos (θ) + γ a \dot{W}, \end{matrix}$ where θ = −2φ(T₁) + σT₁ + γW(T₁). Thus the first-order uniform expansion solution of Eq. (24) will

Conclusions

In this paper, the multiple scales method was successfully applied to research the responses of SDOF systems with small fractional derivative damping, which subject to a narrow-band random parametric excitation. We first derived the frequency–response relationship, and then the first-order and second-order steady-state moments of the non-trivial steady-state solution of SDOF systems about the fractional derivative of order α are given. As a special research example, the parameter principal

Acknowledgments

The project was supported by the National Natural Science Foundation of China (11372247, 11202160, 61303020), Program for New Century Excellent Talents in University, the Shaanxi Project for Young New Star in Science & Technology, NPU Foundation for Fundamental Research and New Faculties and Research Area Project, the Youth Science and Technology Research Fund of Shanxi Province, the Program of TYUST for Doctors’ scientific research (20122019) and the Program of Shanxi University (010951801005).

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Principal resonance responses of SDOF systems with small fractional derivative damping under narrow-band random parametric excitation

Highlights

Abstract

Introduction

Section snippets

Multiple time scales approximate solution

Example: Duffing oscillator with fractional derivative damping term

Conclusions

Acknowledgments

J Sound Vib

Mech Res Commun

Commun Nonlinear Sci Numer Simul

J Sound Vib

Commun Nonlinear Sci Numer Simul

Mech Res Commun

Commun Nonlinear Sci Numer Simul

J Sound Vib

Int J Non-Linear Mech

J Sound Vib

J Sound Vib

Int J Non-Linear Mech

Struct Safety

Mech Res Commun

On fractional differentials

Philos Mag

A new dissipation model based on memory mechanism

Pure Appl Geophys

A theoretical basis for the application of fractional calculus

J Rheol

Fractional calculus-a different approach to the analysis of viscoelastically damped structures

AIAA J

Fractional calculus in the transient analysis of viscoelastically damped structures

AIAA J