DTM APPLIED TO THE BEAM EQUATION WITH VARIABLE INERTIA: RECURRENCE, COEFFICIENTS, AND CONVERGENCE TO THE ANALYTICAL SOLUTION

Abstract

This paper presents a comparative analysis between the exact analytical solution and the Differential Transformation Method (DTM) for studying the mechanical behavior of a cantilever beam with a circular planar cross-section whose diameter varies linearly along its length. The beam is loaded at the free end by a concentrated force, and the differential equation of the deformed neutral axis (Euler-Bernoulli model) has variable coefficients due to the position-dependent axial moment of inertia. The exact analytical solution is obtained through successive integration, leading to a closed-form expression for the beam deflection at the free end. The DTM transforms the differential equation into an algebraic recurrence relation for the coefficients of the Taylor series of the solution, allowing a systematic approximation without direct integration. Numerical results demonstrate the rapid convergence of the DTM: for 6 terms, the relative error of the deflection at the free end is 4.34%, and for 11 terms, the error decreases to 0.19%. An extended analysis over the entire domain shows that the maximum error occurs in the upper-middle region (x̄ ≈ 0.8), and for M=11 terms, this error is below 0.95%. The exponential convergence and linear complexity of the algorithm recommend DTM as an efficient and accurate alternative for the analysis of beams with variable cross-section, especially in cases where exact analytical solutions become impossible to obtain