# 8.3: Damping and Resonance

- Page ID
- 16408

# Damping

If an oscillating system experiences a non-conservative force, then naturally some of its mechanical energy is converted to thermal energy. Since the energy in an oscillating system is proportional to the square of the amplitude, this loss of mechanical energy will manifest itself as a decaying amplitude. A common damping force to account for is one for which the force is proportional to the velocity of the oscillating mass, and in the opposite direction of its motion (naturally – it has to do negative work to take out mechanical energy). Air resistance (and other fluid drag) behaves like this when an object moves through the medium at low relative speeds. Adding this force into our Newton's second law equation changes Equation 8.1.1 to:

\[ \dfrac{d^2x}{dt^2}+\beta\dfrac{dx}{dt} + \dfrac{k}{m}x = 0 \]

The constant \(\beta\) determines the contribution of the acceleration due to the drag force on the object. It is beyond the scope of this work to discuss how such differential equations are solved, but the solution will be given, and the reader is encouraged to plug the solution back into the differential equation to confirm that it works (actually, guessing-and-confirming is pretty much how such differential equations are solved!):

\[ x\left(t\right) = Ae^{-\frac{1}{2}\beta t}\sin\left(\omega t + \phi\right),\;\;\;\;\;\; where:\;\; \omega \equiv \sqrt{\dfrac{k}{m} - \frac{1}{4}\beta^2} \]