# Critical Damping Calculator

Created by Mehjabin Abdurrazaque
Reviewed by Wojciech Sas, PhD and Rijk de Wet
Last updated: Jun 05, 2023

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.

🙋 If you want to learn more about friction, the free fall with air resistance calculator is another clear example of friction opposing a movement.

## What is the definition of damping?

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.

## 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:

$F=−c \frac{\text{d}x}{\text{d}t},$

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.

## What are the three degrees of damping?

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.

## What is the critical damping coefficient? The damping coefficient formula

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:

$c_\text{c} = 2\sqrt{k\cdot m},$

where:

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

## How to use the critical damping calculator

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 (ωn) and critical damping coefficient (cc) of the oscillator.

✅ You can enter the mass and natural frequency to find out the other two variables, or you can enter the critical damping coefficient and one other variable to estimate the remainders. In short, enter any two variables and obtain the other two variables.

## FAQ

### What is simple harmonic motion?

An oscillating body executes simple harmonic motion (SHM) if:

1. The magnitude of the forces acting on it is directly proportional to the magnitude of its displacement from the mean position; and
2. The restoring force always acts towards the mean position.

### What are the differences between overdamping, underdamping, and critical damping?

There are three damping systems:

• An overdamped system returns to the equilibrium position slowly without oscillating.
• An underdamped system moves to the equilibrium position quickly and oscillates before coming to rest.
• A critically damped system moves as quickly as possible toward equilibrium without oscillating about the equilibrium.

### How do I find the critical damping coefficient using the natural frequency?

The critical damping coefficient (cc) is twice the product of the mass (m) and natural frequency (ωn) of the oscillating object: cc = 2mωn.

So, find the product of the mass and natural frequency and multiply it by 2 to obtain the critical damping coefficient.

### How do I calculate the stiffness of an oscillator with mass and natural frequency?

To find the stiffness of an oscillator, equate the two formulas to estimate the critical damping coefficient:

1. cc = 2mωn, where m is the mass and ωn is the oscillating object's natural frequency; and
2. cc = 2√(k×m), where k is the stiffness.

Thus, stiffness is the product of the oscillator's mass and square of the natural frequency: k = mωn2.

### How do I find the critical damping coefficient of an oscillator of mass 2 kg and stiffness 2 N/m?

The formula for calculating the critical damping coefficient (cc) using the oscillator's mass (m) and stiffness (k) is:

• cc = 2√(k×m).

So, the critical damping coefficient of an oscillator of mass 2 kg and stiffness 2 N/m is 2√(2 × 2) = 4 Ns/m.

### How do I calculate the stiffness of an oscillator of mass 5 kg and critical damping coefficient 1 Ns/m?

The formula to calculate stiffness is:

• k = cc2/2m,

where k is the stiffness, m is the mass, and ωn is the natural frequency of the oscillating object. Thus, an oscillator of mass 5 kg and critical damping coefficient 1 Ns/m has a stiffness of 12 / (2 × 5) = 0.05 N/m.

Mehjabin Abdurrazaque
Stiffness (k)
N/m
Mass (m)
kg
Natural circular frequency (ω)
Hz
Critical damping coefficient (c꜀)
lbf·sec/in
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