Abstract
Transient solution of a fractional stochastic dynamical system under wide-band noise excitation is investigated. Generalized Harmonic Balance technique is firstly used to approximate restoring force of the given system as an amplitude-dependent form. In this way, stochastic averaging method then can be applied to transform the system into an Ito differential equation. Furthermore, the fractional derivative in the integral-differential form can be equivalent to a combination of periodic functions after the averaging procedure. As the following, Galerkin method therein is utilized to obtain the transient probability density functions by solving associated Fokker-Planck-Kolmogorov (FPK) equation. As an example, the Rayleigh oscillator is studied to illustrate the efficiency and accuracy of the proposed approaches. Numerical results show that exact stationary solution and transient solution derived from Galerkin method are in good agreement with those from Monte Carlo Simulation.
Highlights
Engineering structures such as high buildings and huge bridges often vibrate if they are subjected to random excitation, such as earthquake or strong wind
As for transient response and more practically wide-band noise, due to their instantaneous feature with the time change and different forms of power spectrums, analysis of them is very rare and valuable. Different from these works, the present paper aims to extend Galerkin method often used in deterministic systems to study transient response for those systems under wideband noise excitation in which damping or restoring force is of fractional order
Stochastic averaging method was initially proposed by Stratonovich in solving the response of stochastic dynamical systems; its basic idea is to approximate the system response by a Fokker-Planck-Kolmogorov equation, which is derived from a diffusive Markov equation with transient probability density governed by a partial differential equation; more relative investigations and mathematical foundation can be found in [18, 19]
Summary
Engineering structures such as high buildings and huge bridges often vibrate if they are subjected to random excitation, such as earthquake or strong wind. The vibration behavior of these engineering structures is usually characterized by a dynamical system under noise excitation in mathematical framework. How to determine the solution or response, that is, vibrating displacement and velocity, of engineering structural system is an important issue in the field of structural dynamics and mechanical engineering. The other is transient response which shows the dynamical evolution of system solution with respect to each time instant. Transient response is more important due to its comprehensive expression of system evolution with the time than stationary response and more difficult to get owing to time involved
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