Free and Forced Nonlinear Vibrations of a Functionally Graded Porous Cantilever Beam with an Intermediate Spring Subjected to a Harmonic Tip Force

Purpose: This study investigates the free and forced vibrations of a functionally graded (FG) porous Euler–Bernoulli standing cantilever beam subjected to a harmonic tip force. The focus is on understanding how material gradation, porosity, and other parameters influence the beam’s dynamic response. Methods: The material properties of the beam are modeled using a power-law distribution across its length. The integro-differential equation of motion is derived using Hamilton’s principle, and the Galerkin method is applied to simplify it into a nonlinear ordinary differential equation. The method of multiple scales (MMS) is then used to obtain approximate solutions for nonlinear natural frequencies and primary resonance response curves. The results are validated through numerical integration. Results: The study reveals that the FG index, porosity factor, initial vibration displacement, material properties, spring stiffness, gravity parameter, and the distance of the spring from the fixed end significantly influence the natural frequencies and steady-state amplitude. Conclusion: The findings demonstrate that the dynamic response of FG porous cantilever beams is highly sensitive to various design and material parameters. This highlights the importance of these parameters in the design and analysis of FG porous beams for practical applications.