Non-stationary response statistics of nonlinear oscillators with fractional derivative elements under evolutionary stochastic excitation

Abstract

An approximate analytical technique is developed for determining the non-stationary response amplitude probability density function (PDF) of nonlinear/hysteretic oscillators endowed with fractional derivative elements and subjected to evolutionary stochastic excitation. Specifically, resorting to stochastic averaging/linearization leads to a dimension reduction of the governing equation of motion and to a first-order stochastic differential equation (SDE) for the oscillator response amplitude. Associated with this first-order SDE is a Fokker–Planck partial differential equation governing the evolution in time of the non-stationary response amplitude PDF. Next, assuming an appropriately chosen time-dependent PDF form of the Rayleigh kind for the response amplitude, and substituting into the Fokker–Planck equation, yields a deterministic first-order nonlinear ordinary differential equation for the time-dependent PDF coefficient. This can be readily solved numerically via standard deterministic integration schemes. Thus, the non-stationary response amplitude PDF is approximately determined in closed-form in a computationally efficient manner. The technique can account for arbitrary excitation evolutionary power spectrum forms, even of the non-separable kind. A hardening Duffing and a bilinear hysteretic nonlinear oscillators with fractional derivative elements are considered in the numerical examples section. To assess the accuracy of the developed technique, the analytical results are compared with pertinent Monte Carlo simulation data.

Appendix

In this appendix, more details on the derivation of Eqs. (12) and (13) are included for completeness, while the interested reader is also directed to Refs [32, 33]. In this regard, Eqs. (10) and (11) are rewritten, equivalently, in the form

$$\begin{aligned} \beta (A) = \frac{S(A)}{A\omega (A)} - \beta \frac{S_\alpha (A)}{\pi A\omega (A)} - \beta _0 \end{aligned}$$

(52)

and

$$\begin{aligned} \omega ^2(A) = \frac{F(A)}{A} + \beta \frac{F_\alpha (A)}{\pi A} \end{aligned}$$

(53)

where S(A) and F(A) are given by Eqs. (14) and (15), respectively. Further,

$$\begin{aligned} S_\alpha (A) = \int _{0}^{2\pi } \mathscr {D}^{\alpha }_{0,t}(A\cos \phi )\sin \phi \mathrm{d}\phi \end{aligned}$$

(54)

and

$$\begin{aligned} F_\alpha (A) = \int _{0}^{2\pi } \mathscr {D}^{\alpha }_{0,t}(A\cos \phi )\cos \phi \mathrm{d}\phi \end{aligned}$$

(55)

In general, the fractional derivative of order $\alpha$ for the cosine function is given by

$$\begin{aligned} \mathscr {D}^{\alpha }_{0,t}\mathrm{cos}(bx) = b^\alpha \mathrm{cos}\left( \frac{\alpha \pi }{2} + bx\right) \end{aligned}$$

(56)

where $0<\alpha <1$. Determining analytically the integrals defined in Eqs. (54) and (55) by employing Eq. (56) is not straightforward, as it involves the evaluation of rather complex integral forms. Nevertheless, appropriately approximating Eqs. (54) and (55) facilitates significantly the related computations. In particular, assuming that the time parameter $\tau$ takes small values, Eq. (4) becomes [32, 33]

$$\begin{aligned} \dot{x}(t-\tau ) \approx \dot{x}(t)\cos (\omega (A)\tau )+x(t)\omega (A)\sin (\omega (A)\tau )\nonumber \\ \end{aligned}$$

(57)

Next, combining Eq. (57) with Eq. (2), the Caputo derivative defined in Eq. (2) takes the form

$$\begin{aligned} \mathscr {D}^{\alpha }_{0,t}x(t)= & {} \frac{1}{\varGamma (1-\alpha )} \left\{ \dot{x}(t) \int _{0}^{t} \frac{\cos (\omega (A)\tau )}{\tau ^\alpha } \mathrm{d}\tau \right. \nonumber \\&+\, \left. x(t)\omega (A) \int _{0}^{t} \frac{\sin (\omega (A)\tau )}{\tau ^\alpha } \mathrm{d}\tau \right\} \end{aligned}$$

(58)

Further, utilizing the integrals [33]

$$\begin{aligned} \int _{0}^{t} \frac{\cos (\omega (A)t)\mathrm{d}\tau }{\tau ^\alpha }= & {} \omega ^{\alpha -1}(A) \left[ \varGamma (1-\alpha )\sin \left( \frac{\alpha \pi }{2}\right) \right. \nonumber \\&+\, \frac{\sin (\omega (A) t)}{(\omega (A) t)^\alpha } + \left. O(\omega (A) t)^{-\alpha } \right] \nonumber \\ \end{aligned}$$

(59)

and

$$\begin{aligned} \int _{0}^{t} \frac{\sin (\omega (A)t)\mathrm{d}\tau }{\tau ^\alpha }= & {} \omega ^{\alpha -1}(A)\left[ \varGamma (1-\alpha )\cos \left( \frac{\alpha \pi }{2}\right) \right. \nonumber \\&-\, \frac{\cos (\omega (A) t)}{(\omega (A) t)^\alpha } + \left. O\left( \omega (A) t\right) ^{-\alpha } \right] \nonumber \\ \end{aligned}$$

(60)

Equation (58) becomes

$$\begin{aligned} \mathscr {D}^{\alpha }_{0,t}x(t)= & {} \omega ^{\alpha -1}(A) \left[ \dot{x}(t) \sin \left( \frac{\alpha \pi }{2}\right) \right. \nonumber \\&+\, \left. x(t) \omega (A) \cos \left( \frac{\alpha \pi }{2}\right) \right] + \frac{\omega ^{\alpha -1}(A)}{\varGamma (1-\alpha )} \nonumber \\&\times \, \frac{\dot{x}(t)\sin (\omega (A) t) - x(t) \omega (A)\cos (\omega (A) t)}{(\omega (A) t)^\alpha } \nonumber \\&+\, O\left( \omega (A) t\right) ^{-\alpha -1} \end{aligned}$$

(61)

Equation (61) constitutes an approximate expression that facilitates the determination of fractional derivatives of order $0<\alpha <1$, and thus, it is utilized in simplifying the integrands of Eqs. (54) and (55), which become

$$\begin{aligned} S_\alpha (A) = - \pi A\omega ^\alpha (A)\sin \left( \frac{\alpha \pi }{2}\right) \end{aligned}$$

(62)

and

$$\begin{aligned} F_\alpha (A) = \pi A\omega ^\alpha (A)\cos \left( \frac{\alpha \pi }{2}\right) \end{aligned}$$

(63)

respectively. Finally, Eqs. (12) and (13) are derived by considering Eqs. (62) and (63).