How to Calculate Damping Ratio

In a perfect world, a pendulum or any other oscillating system will swing back and forth indefinitely. The real world is far from perfect, and the forces of friction in its many forms will conspire to slow down the oscillation and dampen the motion. This dampening can be succinctly described using a simple ratio, called the damping ratio.

Difficulty: Easy

Instructions

Things You'll Need:

• Oscillating system you need the damping ratio for
• Scale (for mass measurement)
• Timer
• Ruler

1. Determine what you know. In an oscillating system, there are three primary quantities used to describe the differential equation: mass m, spring constant k and damping constant c. Each of these needs to be known to determine the damping ratio, which is given as: Z = c / (2*√(k*m)).

2. Measure the mass. Using your scale, weigh the object to get the mass in grams. If your scale measures in pounds, divide by 2.2 to convert pounds to kilograms (for increased accuracy, use 2.20462262).

3. Measure the frequency of oscillation. To determine k, we need the frequency of oscillation, or cycles per second. The best way to get that is to measure the period of oscillation, or how long it takes to return to the same position. Good practice will involve taking several measurements and averaging the results. Once you have a period (T) in seconds, invert it (1/T) to get the frequency w.

4. Convert frequency to wave number. In an oscillating system, the frequency is related to the mass and spring constant by the equation w²=k/m, or to rearrange terms, k=w²*m. Given the frequency and mass, you can compute the spring constant.

5. Calculate the logarithmic decrement d. This is the natural logarithm of the extent of two oscillations of the system. If your system is oscillating about a point x=0, using the ruler, measure the distance from 0 of two successive oscillations, x1 and x2. Take the natural logarithm of their ratio: d = ln(x1/x2). Do this a number of times to ensure you get an accurate result.

6. Calculate the damping ratio. If you know k, m and c, you can use the above equation Z = c / (2*√(k*m)). If you do not know c, however, you can use the logarithmic decrement d by the equation Z = d/√((2*pi)^2+d^2). Now you have the damping ratio.

Tips & Warnings

• Good measurement practice is to repeat measurements a number of times, usually 10 or so, and average the results to get a final answer. This eliminates random noise and ensures an added level of accuracy.