Normal Modes of Mechanical Systems - Oscillations and Instabilities

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.

Give a brief introduction to finding normal modes from the potential energy surface of an N degree of freedom system, and the three types of modes mentioned above..
Example 2 degree of freedom system. Two masses connected by springs to walls and to each other. We analytically find the natural frequencies of the system near equilibrium and corresponding normal mode shapes..
General motion near equilibrium is made up of a sum of normal modes (via the superposition principle), which gives rise to Lissajous figures..
Geometric interpretation of the equations of motion near equilibrium in terms of a "force field" for the case of a positive-definite potential energy matrix. For the eigendirections, we have what looks like Hooke's law for a spring-mass system, and simple harmonic motion..
Normal coordinates: Using the normal modes as new generalized coordinates for the Lagrangian dynamics. The dynamics in the normal modes becomes decoupled and we consider the interpretation of quasiperiodic motion on tori parametrized by energy in each mode. For more information, see my video on "action-angle" variables https://youtu.be/z-dGZgq-6jg.
Why find normal modes? Because if have a mechanical system which is "forced", that is, has some oscillatory driving force that has a forcing frequency close to a natural frequency, we can get a large response, which could lead to failure. We illustrate this with a 1 degree of freedom model and show the frequency-response curve. For an extended discussion of the frequency response curves and resonance, see my video on "Frequency response curves for linear and nonlinear oscillators" https://youtu.be/eRnM5zVQsMc.