Overdamped Vs Critically Damped: Comparative Analysis

By knowing the concept of damping, we must understand the difference between overdamped vs critically damped oscillations.

To understand overdamped vs critically damped, one can say that a system that is overdamped goes slowly toward equilibrium, whereas a system that is critically damped moves as swiftly as possible toward equilibrium without fluctuating about it.

difference between overdamped and critically damped oscillations

Now let us see a table below where all information has summarize to make a comparative analysis of overdamped vs critically damped oscillations.

Overdamped vs Critically Damped Oscillation:

Overdamped Critically damped
Overdamping occurs when oscillations come to a halt after a significant period of time has passed since the resistive force was applied. In oscillatory system, the oscillations come to a halt as soon as critical damping is reached.
If a system responds to a step-change input by taking up a new position, it can either fluctuate around the final position before settling to the new value, or it can gradually approach the new value over time. At a given level of damping, the system does not actually oscillate; however, it may slightly exceed before returning to the final value.
By solving damped harmonic oscillator, the case of overdamping is given by, b2>4mk By solving damped harmonic oscillator, the case of critical damping is given by, b2=4mk
In the case of Overdamping b is comparatively large than m and k In the case of Critical damping b is just between over and underdamping
The roots of overdamping are real and distinct. Because the roots are real, overdamping is the simplest situation to solve mathematically. The roots of critically damped oscillator are real and same.
The characteristic roots can be given as, -b+√(b2-4mk)/2m
r2=-b-√(b2-4mk)/2m
The characteristic roots of critical damping are given as, -b/2m, -b/2m.

The general solution for a critically damped oscillation can be given as follows:

 x(t) = (C_1 + C_2 t)e^{-\gamma t}

Where:

  •  x(t) is the displacement at time ( t ).
  •  C_1 and  C_2 are constants determined by the initial conditions of the system.
  •  \gamma is the damping coefficient.
  • ( e ) is the base of the natural logarithm.

This is the detailed comparative analysis of overdamped vs critically damped oscillation.

Overdamped Vs Critically Damped
Best example of swing illustrating Overdamped Vs Critically damped

Image Credits: Image by Goran Horvat from Pixabay 

Before understanding overdamped vs critically damped oscillations, let us begin with overview of damping oscillation.

We all are familiar with damping and we also know damping oscillation examples in our surrounding.

If a system responds to a step-change input by taking up a new position, it can either fluctuate around the final position, finally settling to the new value, or it can steadily approach the new value, taking its time.

The system doesn’t truly oscillate at a certain level of damping; however, it may slightly overshoot before immediately returning to the final value. This is critical dampening, and it’s typically the goal.

Overdamped Vs Critically Damped
Overdamped Vs Critically Damped

Damped Oscillator:

We know the damped harmonic oscillator equation can be given as:

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…..(1)

With m > 0, b ≥ 0 and k > 0. It has characteristic equation

ms2+bs+k=0………. (2)

With characteristic roots

image 77

Depending on the sign of the term under the square root, there are three possibilities:

  • b2 < 4mk (This is the case of Underdamping as b is comparatively small than m and k)
  • b2 > 4mk (This is the case of Overdamping as b is comparatively large than m and k)
  • b2 = 4mk(This is the case of Critical damping as b is just between over and underdamping)

Overdamping is the simplest situation to solve mathematically since the roots are real. Most people, however, perceive the oscillatory behaviour of a damped oscillator.

Read more about why critical damping is faster than overdamping.

Here we will see the case of Overdamping and critical damping as we have to do comparative analysis of overdamped vs critically damped oscillation.

Overdamping (real and distinct roots):

When b2 > 4mk , then the value under the square root will be positive and the characteristic roots will be real and distinct. In case of b2 > 4mk the damping constant b should be comparatively large.

One thing to remember is that in this situation, the roots are both negative. You can know this by looking at equation (2). Because the quantity under the square root is assumed to be positive, the roots are real.

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By using these roots to solve the equation (1),

The characteristic roots are:

image 78

Exponential solutions are:

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Therefore, the general solution can be given as:

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Let’s take a look at this from a physical point of view. When the damping is high, the frictional force is so high that the system cannot oscillate. Unusually, an unforced overdamped harmonic oscillator does not oscillate. Because both exponents are negative, any solution in this situation approaches x = 0 asymptotically.

Many doors have a spring at the top that closes them automatically. The spring is damped to control the rate at which the door closes. If the damper is powerful enough to overdampen the spring, the door will simply settle back to its mean position (i.e., closed) without oscillating, which is normally what is desired in this situation.

Critical Damping (real and same roots):

When b2 = 4mk, then the value under the square root becomes 0 and the characteristic polynomials has same roots -b/2m , -b/2m.

Now by using the roots to solve equation (1) in this situation. Because we only have one exponential answer, we must multiply it by t to obtain the second.

Therefore, basic solutions are:

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And the general solutions can be given as:

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This does not fluctuate like the overdamped situation. It’s worth mentioning that picking b as the critical damping value for a fixed m and k results in the quickest return of the system to its equilibrium state.

This is frequently a desired feature in engineering design. This can be observed by verifying the roots, but we won’t go over the algebra that illustrates it.

Read more about detailed insights of critical damping application

So, in this article you have learned the comparative analysis of overdamped vs critically damped oscillations.


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