Damping Ratio - Derivation

Derivation

Using the natural frequency of the simple harmonic oscillator and the definition of the damping ratio above, we can rewrite this as:

This equation can be solved with the approach.

where C and s are both complex constants. That approach assumes a solution that is oscillatory and/or decaying exponentially. Using it in the ODE gives a condition on the frequency of the damped oscillations,

• Undamped: Is the case where corresponds to the undamped simple harmonic oscillator, and in that case the solution looks like, as expected.

• Underdamped: If s is a complex number, then the solution is a decaying exponential combined with an oscillatory portion that looks like . This case occurs for, and is referred to as underdamped.

• Overdamped: If s is a real number, then the solution is simply a decaying exponential with no oscillation. This case occurs for, and is referred to as overdamped.

• Critically damped:The case where is the border between the overdamped and underdamped cases, and is referred to as critically damped. This turns out to be a desirable outcome in many cases where engineering design of a damped oscillator is required (e.g., a door closing mechanism).