# How to Calculate Damping

Written by doug leenhouts
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An oscillating system such as a spring or pendulum, once set in motion, will reduce the amplitude of each cycle when a damping force is applied. The damping coefficient determines how quickly or slowly the harmonic system returns to rest. Systems can be overdamped, meaning the system slowly returns to equilibrium without oscillating; critically damped, where the system returns to equilibrium without oscillating as quickly as possible; or underdamped, where the system slowly oscillates to rest.

Skill level:
Easy

• Scale
• Stopwatch
• Ruler
• Calculator

## Instructions

1. 1

Measure the mass of the oscillatory object, be it a weight attached to a pendulum or a spring.

2. 2

Set the oscillator in motion and measure the period of the oscillations. Use the stopwatch to time how long it takes for the object to reach five peaks, then divide that time by 5. Dividing 1 by this time will give you the angular frequency of the system, w.

3. 3

Measure the amplitude of two successive oscillations by marking the two heights on a background and subtracting them from the equilibrium point where the oscillator is at rest. Take the natural log, n, of the ratio of the two amplitudes.

4. 4

Square n. This will equal 4pi^2d^2/(1-d^2). Solving for d, this will give you the damping ratio. For example, successive amplitude measurements of 6 and 2 inches would yield a damping factor by In(6/2) = 1.0986. Squaring this yields 1.2069 = 39.487d^2/(1-d^2). 1-d^2 = (39.487/1.2069)d^2. (1/33.7177) = d^2, and finally d = (1/33.7177)^(1/2) = 0.1722.

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