Damping Ratio Calculator

The damping ratio calculator will help you analyze damped oscillatory systems. There are three ways to calculate the damping factor, so don't worry — we've got you covered no matter what variables you know (or what you need to find — our calculator also works in reverse!).

Have you ever wondered why you need to pump a swing to keep it moving? Read on to learn what damping is, see the equations behind the damping ratio, and understand why it is useful. We have also included a damping ratio calculation example, so without a doubt, you will know all the answers by the end of this article!

To get the gist of what damping is, consider an oscillating object such as a pendulum. If you leave it to itself, it will eventually stop its harmonic motion. The reason is that its mechanical energy (a sum of its potential and kinetic energy) will be dissipated by frictional forces with the air and at its pivot. You can observe this as with every swing, the pendulum's amplitude becomes smaller and smaller. Such motion is said to be damped, and this kind of pendulum is a damped harmonic oscillator.

Feel free to check out friction calculator if you need a short refresher before continuing.

Hence, there are four cases of oscillatory systems:

• Underdamped — damping is small enough to make the amplitude decrease with time. This is the case when the damped harmonic oscillator (e.g., pendulum) is left on its own.

• Overdamped — the object fails to complete even a single oscillation and its velocity approaches zero as it approaches the equilibrium (rest) position. An example would be a pendulum submerged in a viscous liquid, such as honey.

• Critically damped — this case is similar to the overdamped one, but the object reaches its equilibrium faster. The damping is just right and so is the minimum value to provide non-oscillatory motion. Any less and it would be considered underdamped.

• Undergoing simple harmonic motion — damping does not occur. The oscillator keeps on moving forever, constantly exchanging its kinetic and potential energy (which you can read more about in the potential energy calculator).

The system's damping factor determines how quickly the mechanical energy is dissipated and the object returns to rest. How do you find the damping coefficient? It is an intrinsic property of the spring, usually found by measuring the amplitude decay rate and performing computations or deriving it from other properties.

The damping ratio helps us establish which case of the damped motion we are considering. The table presented below will help you interpret the results obtained from the damping ratio calculator.

There are three useful formulae used in our damping ratio calculator. We will present each of them with an explanation below.

1.

where:

• $c$ — damping coefficient (in Ns/m); and

• $c_c$ — critical damping coefficient (in Ns/m).

This is the most basic formula. If the damping coefficient is any less than the critical one, the result will be less than 1 and imply that the system is underdamped, as expected.

2.

where:

• $m$ — suspended mass (in kg); and
• $k$ — spring constant, as seen in Hooke's law (in N/m).

This damping ratio formula is similar to the first one, but we used the fact that $c_c = 2\sqrt{mk}$.

3.

where:

• $\omega_0$ — natural angular frequency. This is the magnitude of the angular velocity of the system when it undergoes the simple harmonic motion (in rad/s). If you'd like to read more about it, visit the angular frequency calculator.

In this case, we rearranged the formulae further, using $\omega_0 = \sqrt{\frac{k}{m}}$. Click here to see the derivation of the damping ratio and natural angular frequency formulae.

Notice that if the values of the other quantities are established, you also know how to find the damping coefficient. This method is much simpler and more approachable than the ones we mentioned in the first section.

As you may have noticed in the first formula, the critical damping coefficient and the damping coefficient units are the same. It means that the damping ratio is dimensionless.

Although the pendulum is the most common and straightforward example, it does not seem overly applicable to most of us — at least not initially. After all, not many people own a classical standing clock, and even fewer use devices such as seismometers or gravitometers. However, there are other examples of oscillators that you may encounter; some of them are listed below:

• Friction pendulums, which are used as seismic isolators. In other words, they protect buildings from earthquakes. The motion of the ground causes the pendulum to sway, reducing the risk of structural damage. If there were no damping, the building would continue to move long after the seismic event. Its residents may not want to live in an enormous cradle, though.

• Door dampers are used to slow down the door, thereby preventing it from slamming and making noise. This is an overdamped system. Notice that if the door is massive (e.g., burglar-proof), the damper needs to be more robust — as can be deduced from equations 2 and 3.

• Car speedometers, which may be surprising. After all, it's a dial with a needle pointing toward the instantaneous speed of the vehicle — where are the oscillations? It turns out that this is an example of critical damping. Otherwise, the pointer would either vibrate or move too slowly to reach the actual value. Now, if you ever find yourself having such issues, you will know the cause!

• Swings, as we now know, are simply pendulums but more fun. A sophisticated variation would be some amusement park rides that can also sway vertically. In both cases, you need some driving force to cause the body to oscillate. If, after that, you stop pumping, the swing will eventually stop moving. Of course, you can affect how soon that will happen by playing with the pendulum's properties.

It is time we showed an example. Consider a swing from the local playground. Suppose that the damping coefficient is 180 Ns/m, the suspended mass is 60 kg, and you use a stopwatch to measure the period. This allowed you to estimate the value of the natural angular frequency to be 1.7 rad/s. Inputting all of this data into the damping ratio calculator gives the value of 0.882. Comparing this number with the table shows that the swing is, in fact, an example of the underdamped system.

Now you know not only how to calculate the damping ratio but also how to use the result!