Critical Damping Calculator

Welcome to our critical damping calculator, which can help you estimate the critical damping coefficient of a damped oscillator.

If you used our simple pendulum calculator, you might have learned that a simple pendulum's motion is oscillatory, and the force describing that motion is proportional to the displacement of the bob from its mean position. However, in real situations, friction forces like air resistance also influence this motion and, in this case, oppose the pendulum's movement.

Read on to find more about these opposing frictional forces — the definition of damping force, degrees of damping, damping coefficient, and formula for estimating critical damping coefficient. We also explain underdamped, overdamped, and critically damped oscillations using figures and examples. After reading, you'll never be in doubt about how to calculate any critically damped system.

Damping is the process of dissipating energy from a vibrating structure, resulting in the reduction of the vibration's amplitude. Generally, it involves the conversion of the mechanical energy of the vibrating structure into thermal energy.

We know that a rolling ball comes to a stop because of air resistance and friction due to the surface on which it rolls. This way, any movement of an object is resisted by its surrounding objects. The resistance reduces the energy of the moving object with time, eventually slowing it down to a stop.

Now, let's apply this idea to a vibrating structure. The air surrounding such an object (or any medium in contact with it) tries to resist the vibrational motion and reduce its vibrational energy (mechanical energy). The higher the energy, the higher the vibration's amplitude. Thus, reducing the energy with time reduces the vibration's amplitude too.

Now that we've covered the mechanical theory let's jump to the next section to learn what is the damping coefficient.

The damping coefficient of the damping agent or damper tells us how effective the damper is in resisting an object's harmonic motion.

We calculate the resistive force exerted by a damper or the damping force as follows:

where:

• $c$ is the damping coefficient;
• $x$ is the displacement of the oscillating object from its mean position; and
• $\frac{\text{d}x}{\text{d}t}$ denotes the change in displacement ($x$) with time ($t$) — this is also known as the instantaneous velocity.

The damping coefficient $c$ helps estimate the energy dissipated due to frictional forces, which slow down oscillation.

Now it's high time to find out what is the critical damping coefficient.

Want to learn more about damping and damping coefficient? Take a look at our damping ratio calculator.

There are three degrees of damping that depend on how quickly the amplitude of the oscillations decreases:

1. Underdamping (light damping);
2. Critical damping; and
3. Overdamping (heavy damping).

In underdamped oscillations, the object comes to rest after a few oscillations with amplitude decreasing exponentially with time. Let's go back to the first oscillating object we have learned — a simple pendulum! It's an underdamped system.

In overdamping, the damper exerts a force that is what is required to prevent the oscillations. Hence, the object returns to its equilibrium position without oscillating. Have you noticed that the toilet flush handle gradually returns to its resting position without oscillating? It's an overdamped system. Another example is an automated door — it's overdamped by design to prevent damage to the glass.

In critical damping, the object reaches the equilibrium point as quickly as possible without oscillating. An automobile shock absorber is a great example of the practical application of critical damping.

The value of the damping coefficient at which critical damping occurs is the critical damping coefficient of a system. It's also equal to the undamped resonant frequency of the oscillator.

In terms of stiffness and mass of the oscillating object, we calculate the critical damping coefficient as follows:

where:

• $k$ is the stiffness of the oscillated body; and
• $m$ is the mass of the oscillating body.

It's simple and straightforward!

1. Select the units for each variable from the drop-down list beside them.
2. Enter the oscillator's mass in the input box for "Mass (m)."
3. Type the oscillator's stiffness in the input for "Stiffness (k)."

The calculator displays the natural frequency (ω_n) and critical damping coefficient (c_c) of the oscillator.