This paper investigates the vibrational behavior of the connecting rod in a crank-slider mechanism by accounting for its structural flexibility and the effect of external constraint forces. Unlike idealized rigid-body models, the approach adopted here models the connecting rod as a slender elastic beam undergoing planar motion. The mathematical formulation includes both longitudinal and transverse vibrations and is derived through an iterative procedure based on Laplace and finite Fourier transforms. The resulting system of equations enables the computation of displacement fields at increasing levels of approximation. Numerical simulations are performed for different geometric and inertial configurations, with a focus on free vibrations. The results, obtained under the assumption of small deformations and linear elastic behavior, demonstrate that transverse displacements dominate the dynamic response, while longitudinal components remain negligible. The proposed method provides a high-fidelity analytical framework for modeling the elastodynamic behavior of flexible links in planar mechanisms, with direct relevance to high-speed or precision engineering applications.