Free vibration analysis of functionally graded porous beams using the differential transform method under Euler–Bernoulli and Timoshenko theories

This study investigates the free vibration behavior of functionally graded porous (FGP) beams using the Differential Transform Method (DTM) within the frameworks of Euler–Bernoulli and Timoshenko beam theories. Governing equations of the beams are reduced to fourth-order ordinary differential equations in the coordinate system and subsequently transformed via DTM. The proposed DTM approach demonstrates reliable convergence, requiring a moderate number of terms for accurate frequency prediction. Two distinct porosity distribution models, symmetric and asymmetric, are considered to represent open-cell metal foam cores. The governing equations and boundary conditions are solved through DTM for clamped, simply supported, and free edge conditions. A convergence and validation study confirms the efficiency and accuracy of the approach against available analytical results. The results highlight the strong influence of porosity coefficient, distribution type, boundary conditions, and slenderness ratio (L/h) on the natural frequencies of FGP beams. The novelty of this work lies in simultaneously applying DTM to both Euler–Bernoulli and Timoshenko porous beams under diverse boundary conditions, providing a systematic comparison between symmetric and asymmetric porosity effects on vibrational behavior. The findings not only enhance the understanding of porous beam dynamics but also establish a foundation for lightweight, vibration-resistant structural design in aerospace and mechanical applications.