Mechanical Vibrations is a cornerstone course that sets mechanical engineering students apart through its rigorous treatment of physical systems, providing them with an unparalleled depth of understanding compared to other engineering disciplines. This course serves as a fundamental bridge to advanced control subjects such as system modeling, control theory, and automatic control by developing students' mathematical and physical intuition of second-order mechanical systems. Students will study the theoretical foundations of second-order differential equations and their solutions, methods for quantifying vibration characteristics, distinctions between free and forced responses, and parameters that govern vibration behavior. The course emphasizes both analytical understanding and practical application, incorporating numerical methods for solving complex differential equations. Through self-directed MATLAB programming exercises, students independently develop computational skills essential for modern engineering practice, thereby acquiring both theoretical knowledge and hands-on problem-solving capabilities.