Fluid$-$Structure Interaction in Vibrational Dynamics: Delve into the theoretical aspects of fluid$-$structure interaction $($FSI$)$ in vibrational dynamics and its importance in mechanical engineering. Discuss the mathematical formulations used to model the coupling between fluid flow and structural vibrations, encompassing phenomena like added mass, damping, and hydroelastic effects. Evaluate the challenges and methodologies involved in analyzing FSI problems, emphasizing their relevance in designing structures subjected to fluid$-$induced vibrations, such as offshore platforms or aircraft wings.