How to Calculate Damping Coefficient

A spring has a damping factor that causes it to resist motion.

When a spring bounces up and down, it will gradually reduce the amplitude of each bounce until it comes to rest. The same thing happens to a pendulum. These systems are called simple harmonic oscillators, as they repeat a motion with a certain frequency. They eventually come to rest because each spring in nature has a damping coefficient, a factor that describes the frictional forces acting against the motion of the spring. The greater the coefficient, the quicker the spring will come to rest.

Things You'll Need

• Threaded hook
• Spring
• 1 kilogram weight
• Stopwatch
• Ruler

Instructions

1. Screw a threaded hook into the side of a wall, and tie one end of a spring to the hook. Tie the other end to a 1 kilogram weight.

2. Pull the weight down 3 inches, marking the height on the wall, and start a stopwatch the instant you release the weight. Allow the spring to complete 10 cycles, then stop the watch.

3. Divide the total time by 10 to determine the period of the oscillator. Divide 1 by the period to obtain the frequency, denoted w, of the oscillator. Square the frequency and multiply it by the mass of the weight to calculate the spring constant, k.

4. Stop the oscillator and place it at rest. Pull the weight down 3 inches and release it again. Measure the height of the bounce, in inches, and divide 3 inches by the height of the second bounce. Take the natural log of this ratio to calculate the logarithmic decrement, d.

5. Plug the logarithmic decrement into the equation to calculate the damping coefficient, Z = d/(4*pi² + d²)^(1/2). For example, if the first bounce is 2 inches, then the logarithmic decrement would be ln(3/2) = 0.405, and the damping coefficient is (0.405)/(4*pi² + 0.405²)^(1/2) = 0.064.

References

• Michigan Technological University: Damping
• Davidson University: Finding the Damping Coefficient of a Spring