Explore normal modes of mechanical systems in this comprehensive lecture from a course on analytical dynamics. Delve into the patterns of motion where all parts move together, focusing on oscillatory, steady drift, and unstable modes. Analyze a two-degree-of-freedom system with masses and springs to find natural frequencies and mode shapes. Examine general motion near equilibrium, geometric interpretations of equations of motion, and the concept of normal coordinates. Understand the importance of identifying normal modes in forced mechanical systems and their potential for large responses near natural frequencies. Gain insights into frequency-response curves, resonance, and the application of these concepts to both linear and nonlinear oscillators.