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<p>This paper considers the response to white noise excitation of nonlinear dynamic systems comprising fractional derivative elements. Specifically, it focuses on estimating the stationary power spectral densities of the response displacement and velocity for systems characterized by polynomial nonlinearities, and nonlinearities with memory effects. Various approximations for the spectral response are derived. The first approximation is based on the concept of the conditional spectrum. In doing this, a weighted averaging is performed over a set of surrogate spectral densities. Each surrogate density corresponds to the stationary random response of a linearized system, associated with the dynamics at specific response amplitude levels. These amplitude levels, assumed to vary slowly over time, are treated as constant within individual oscillation cycles. To refine the estimate, an enhanced formulation of the conditional spectrum is pursued. This improved approach introduces a corrective term that aligns the dynamics of the individual linearized systems with the expected response variance for the specific amplitude level at which they operate. For both the standard and the improved estimates, the stationary probability density function of the response amplitude is determined by solving an associated Fokker–Planck–Kolmogorov equation for the nonlinear system under consideration. Results for the stationary power spectral densities of the system’s displacement and velocity are compared to those obtained by numerically solving the covariance Lyapunov equation for the linearized system under stationary conditions. Specifically, the response-amplitude-dependent equivalent damping and stiffness are derived by approximating the fractional term using the Harmonic Balance method. The expected values of these equivalent quantities are then determined as functions of the response displacement variance by <i>apropos</i> averaging for the probability density function of the amplitude, which is determined by using the Stochastic Averaging technique. The reliability of the proposed approach is assessed by conducting ad hoc Monte Carlo simulations exploring various orders of the fractional derivative operator.</p>

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Stochastic response spectrum determination of nonlinear systems endowed with fractional derivative elements

  • 期刊论文

  • 发表日期：

    2025年3月10日

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  • Pol D. Spanos,
  • Beatrice Pomaro

摘要

This paper considers the response to white noise excitation of nonlinear dynamic systems comprising fractional derivative elements. Specifically, it focuses on estimating the stationary power spectral densities of the response displacement and velocity for systems characterized by polynomial nonlinearities, and nonlinearities with memory effects. Various approximations for the spectral response are derived. The first approximation is based on the concept of the conditional spectrum. In doing this, a weighted averaging is performed over a set of surrogate spectral densities. Each surrogate density corresponds to the stationary random response of a linearized system, associated with the dynamics at specific response amplitude levels. These amplitude levels, assumed to vary slowly over time, are treated as constant within individual oscillation cycles. To refine the estimate, an enhanced formulation of the conditional spectrum is pursued. This improved approach introduces a corrective term that aligns the dynamics of the individual linearized systems with the expected response variance for the specific amplitude level at which they operate. For both the standard and the improved estimates, the stationary probability density function of the response amplitude is determined by solving an associated Fokker–Planck–Kolmogorov equation for the nonlinear system under consideration. Results for the stationary power spectral densities of the system’s displacement and velocity are compared to those obtained by numerically solving the covariance Lyapunov equation for the linearized system under stationary conditions. Specifically, the response-amplitude-dependent equivalent damping and stiffness are derived by approximating the fractional term using the Harmonic Balance method. The expected values of these equivalent quantities are then determined as functions of the response displacement variance by apropos averaging for the probability density function of the amplitude, which is determined by using the Stochastic Averaging technique. The reliability of the proposed approach is assessed by conducting ad hoc Monte Carlo simulations exploring various orders of the fractional derivative operator.