Non‐Central Wishart Random Matrices in Structural Dynamics: A Modal‐Basis Approach

ABSTRACT This article develops a non‐parametric stochastic computational model for linear structural dynamics based on a reduced non‐central Wishart (NCW) random matrix formulation. The objective is to represent uncertainty in the mass and stiffness matrices while preserving the matrix properties required for modal and frequency‐domain analysis. Although the non‐central Wishart distribution provides a more flexible random matrix representation than the central Wishart (CW) model, its direct identification in physical finite‐element coordinates is limited by the storage and computational cost associated with full matrix‐covariance information. To address this limitation, a reduced modal‐basis calibration of the non‐central Wishart random matrix model is proposed. The method uses a reference parametric ensemble to identify variance information in the reduced system matrices and provides a direct sampling procedure for forward uncertainty propagation. The formulation is demonstrated using the forced vibration response of a clamped aluminum plate with spatially correlated uncertainty in material and thickness properties. A parametric random‐field ensemble provides the reference matrix statistics and frequency‐response uncertainty bands and the proposed model is compared with a central Wishart baseline through driving‐point and cross frequency response functions. The response statistics indicate that, for the plate problem considered, the dominant contribution to the frequency‐response uncertainty can be represented by the mode‐indexed diagonal covariance content of the reduced system matrices. The proposed formulation therefore provides a computationally tractable non‐central Wishart model for forward uncertainty propagation, while identifying the treatment of off‐diagonal inter‐modal covariance as a key direction for future development.