Optimization for the natural frequencies of functionally graded beams using metaheuristic algorithms

The thesis describes optimizing the lowest natural frequencies of functionally graded beams, which have varying material characteristics based on a power-law across thickness, using classical boundary conditions. The Euler-Bernoulli beam concept is expanded to include power-law distributions in each layer, resulting in spatially variable stiffness and mass density. Frequencies of free vibration expressed in closed form. To navigate the very non-convex design space defined by volume fraction gradation parameters and layer thickness ratios, MATLAB employs two metaheuristic techniques. The first is the Tug of War Optimization (TWO) algorithm, and the second is the Constrained Particle Algorithm (CPA), which uses swarminspired updates coupled with adaptive penalty functions to enforce mass targets and manufacturability constraints in the non-convex design space defined by layer thickness ratios and volume fraction gradation parameters. Extensive numerical studies in clamped-free, clamped-clamped, and simply supported cases show that CPA converges faster to high-quality solutions, achieving up to a 20% increase in the first natural frequency in preliminary runs, whereas TWO achieves even greater frequency enhancements (up to 25%) at the expense of longer runtimes. Sensitivity studies reveal that higher material gradients disproportionately increase stiffness, providing an effective tuning lever. Overall, CPA is suggested for quick design iterations and massor time-sensitive applications, whereas TWO is preferred when peak vibrational performance trumps computing cost. Future work will expand these algorithms to three-dimensional plates and shells, incorporate gradient-based local adjustments, and test optimal gradation profiles empirically using 3D-printed specimens. Key Words: Functionally Graded Beams, Natural Frequency Optimization, Tug of War Optimization (TWO), Constrained Particle Algorithm (CPA), Euler–Bernoulli Beam Theory, Metaheuristic Algorithms, Volume Fraction Gradation