Sensitivity and Hessian matrix analysis of power spectrum density function for non-classically damped systems subject to stationary stochastic excitations.

• The first and second derivatives for PSD function subject to stochastic excitations are derived. • The proposed methods are applicable to non-classically damped systems. • The highly efficient Pseudo-Excitation-Method (PEM) is adopted. • Two correction methods are developed without the need to calculate the eigen-sensitivities. • Computational accuracy is greatly improved by considering truncated high-order complex modes. Calculating the first and second derivatives of Power Spectrum Density (PSD) function with respect to various design variables is a prerequisite for random responses when gradient-based algorithms are adopted. This paper presented two numerical methods to capture the sensitivity and Hessian matrix of the PSD function for non-classically damped systems subject to stationary stochastic excitations. The direct differentiate method (DDM) is adopted to develop the design sensitivity analysis (DSA). By using Pseudo-Excitation Method (PEM), the governing equations of the non-classically damped system subject to stationary stochastic excitations are transformed into a corresponding deterministic harmonic response problem. Then, the first and second derivatives of the PSD function are given in detail using the DDM. Two numerical methods, namely PEM-modal displacement method (MDM) and PEM-hybrid expansion method (HEM), are proposed to compute the sensitivity and Hessian matrix of the PSD function. The computational accuracy and efficiency of both methods are discussed and compared theoretically and numerically by two illustrations. The results indicate that both methods are valid for the DSA of the PSD function for non-classically damped systems and the PEM-HEM is more suitable than the PEM-MDM with all computational considerations. [ABSTRACT FROM AUTHOR]