Longitudinal vibration analysis of non-uniform rods of any shapes based on differential transforms

A differential transforms based computational method is developed for the free longitudinal vibration analysis of non-uniform rods with smoothly varying cross-sections.The objective is to obtain a numerically stable formulation that accommodates changes in the cross-sectional profile without problem-specific re-derivation.Assuming the area variation is described by a differentiable function A(x), the governing variable-coefficient eigenvalue problem is mapped into the transform domain and expressed through recurrence relations, from which a characteristic polynomial in the frequency parameter is constructed to determine the natural frequencies.The required geometric input is provided solely through the differential spectrum of A(x).The method is validated against a broad set of benchmark problems from the literature, including polynomial, trigonometric, and exponential cross-sectional variations, and reproduces reference eigenfrequencies with agreement up to at least five significant digits, while requiring significantly fewer degrees of freedom than finite element discretizations for comparable accuracy.In addition, the formulation resolves previously reported inconsistencies and yields physically consistent fundamental modes across the examined boundary configurations.The approach is further demonstrated on composite crosssectional variations beyond standard benchmark profiles.