Abstract
A method based on statistical linearization is proposed, for determining the response of single-degree-of-freedom hysteretic systems endowed with fractional derivative element and subjected to combined periodic and white/colored excitation. The method is developed by decomposing the system response into a combination of a periodic and of a zero-mean stochastic components. In this regard, first, the equation of motion is cast into two coupled fractional-order nonlinear differential equations with unknown deterministic and stochastic response components. Next, the harmonic balance method for the fractional-order deterministic equation and the statistical linearization for the stochastic equation are used, to obtain the Fourier coefficients of the deterministic response component and the variance of the stochastic response component, respectively. This yields two sets of coupled nonlinear algebraic equations which can be solved by appropriate standard numerical method. Pertinent numerical examples, including both softening and hardening Bouc–Wen hysteretic system endowed with different fractional-orders, are used to demonstrate the applicability and accuracy of the proposed method.
Highlights
Many mechanical and structural systems subjected to severe dynamic loads exhibit hysteretic behavior[1]
The harmonic balance method is utilized for solving the Fourier coefficients of the deterministic component, while the statistical linearization method is used for the variance/covariance of the stochastic component
A statistical linearization method has been proposed for the response determination of a SDOF hysteretic system endowed with fractional element and subjected to combined periodic and white/colored excitation
Summary
Many mechanical and structural systems subjected to severe dynamic loads exhibit hysteretic behavior[1]. A useful mathematical tools named fractional- In this paper, a method based on statistical linorder calculus has been widely used in various engineer- earization(SL) [29] is proposed, for determining response ing applications, especially for mechanical modeling of of the single-degree-of-freedom (SDOF) hysteretic sysvisco-elastic materials[16] In this context, one may ar- tem endowed with fractional element and subjected to gue that if visco-elastic relaxation test of the material combined periodic and white/colored excitation. The harmonic this regards, the response determination of a structural balance method and the SL method are utilized dynamic systems endowed with fractional derivatives for these sub-equations of motion, leading to a set of and subjected to random excitation arise naturally; see two non-linear algebraic equations. The harmonic balance method is utilized for solving the Fourier coefficients of the deterministic component, while the statistical linearization method is used for the variance/covariance of the stochastic component
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