<p>This paper presents a novel approach to analyzing the free vibration of columns by simultaneously considering several critical factors that influence structural behavior: axially functionally graded materials, gravitational effects due to self-weight, tapering profiles, regular polygon cross sections, constraints of constant volume and various end constraints. The explicit stiffness properties of the column are mathematically formulated, encompassing axial rigidity, flexural rigidity, mass per unit length and mass moment of inertia. These stiffness parameters are then applied to the free vibration analysis of the column subjected to axial compressive loading. The governing differential equations and their corresponding boundary conditions are derived, and numerical solution methods are developed to compute the natural frequencies and mode shapes. The natural frequencies obtained from this study demonstrate strong agreement with previously reported results in the literature. Additionally, numerical results are presented in tabular form and illustrated with figures, accompanied by an extensive discussion of the parametric study of natural frequencies. The parametric study yields three significant findings: (a) the constraint at the ends of the column critically impacts natural frequency calculations; (b) the optimal cross section for the natural frequency, under constant volume constraints, is identified as the equilateral triangle; and (c) tapering effects, while maintaining constant volume, lead to optimized natural frequencies.</p>