State-vector equation method for the frequency domain spectral element modeling of non-uniform one-dimensional structures

This paper presents a state-vector equation method (SVEM)-based spectral element method (SEM) to formulate frequency domain spectral element models for non-uniform 1-D structures. In the SVEM-based SEM, the frequency domain homogeneous governing differential equations for non-uniform 1-D structures are first transformed into state-vector equations. The system matrix and state-vector are represented with power series. These power series representations are then substituted into the state-vector equation, and the differential transformation method is used to efficiently derive a recurrence formula for the coefficients of the power series solution of the state-vector equation. By computing these coefficients from the recurrence formula, the transfer function that relates the state-vector at an arbitrary position to that at the initial position of a finite element is derived. The transfer matrix is then obtained from the transfer function. Finally, the spectral element matrix or exact dynamic stiffness matrix is obtained from the transfer matrix. The accuracy and computational efficiency of the proposed SVEM-based SEM are verified in due course through comparisons with other solution methods for non-uniform axial rods, Bernoulli-Euler beams, and Timoshenko beams with different spatial variations.