# Damping Ratio: Definition & Formula

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• 0:04 Damped Harmonic Oscillation
• 0:48 Equation
• 2:50 Under-, Over-, and…
• 4:08 Example
• 5:02 Lesson Summary

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Lesson Transcript
Instructor: Raghav Mahalingam

Raghav has a graduate degree in Engineering and 20 years of professional experience.

This lesson defines damping ratio for a single degree-of-freedom (SDOF) damped harmonic oscillation and describes a formula to calculate it. We'll also go over some specific examples.

## What Is a Damped Harmonic Oscillation?

As kids, many of us swung on playground swing sets. The back and forth motion of a swing is called an oscillation. We encounter oscillatory motion in many systems in daily life. Other than playground swings, common examples include the pendulum of a grandfather clock and the suspension of a car.

Now, imagine you're pushing someone on a swing. In the absence of friction, such as aerodynamic drag or friction in the couplings, a single push is enough to keep the swing going forever. However, in reality, the motion is oscillatory, but the amplitude of the oscillation reduces over time, eventually going to zero. This type of an oscillation is called a damped harmonic oscillation.

## Equation

For a mechanical system, it's easy to understand damped harmonic oscillations by studying a spring-mass-damper system. As the name suggests, a single degree of freedom (SDOF) spring-mass-damper system consists of a spring, a mass, and a damper. Motion is defined by just one independent coordinate, like time.

In this system, m denotes the moving mass, k denotes the spring constant, and c is the damping coefficient. The spring constant represents the force exerted by the spring when it is compressed for a unit length. The damping coefficient is the force exerted by the damper when the mass moves at unit speed.

The mass is free to move along one axis, but any time the mass moves, its motion is resisted by the spring and the damper. In the figure pictured, imagine now that the mass moves down a certain distance. It compresses the spring and moves the damper by the same distance. The spring stores and releases energy during one cycle. The damper only absorbs energy and doesn't release it back to the mass.

The equation for the system is called a second-order, ordinary differential equation and is:

Here,

And is called the natural frequency in radians, and

And is called the damping ratio.

The natural frequency is the frequency of oscillation of the system if it is disturbed (tapped or hit) from rest. Think about a tuning fork: if we tap it on a surface, the fork vibrates at a fixed frequency. This frequency is the tuning fork's natural frequency.

## Under-, Over-, and Critically Damped

The damping ratio determines the way the system oscillations go to zero. To understand damping ratio, lets use an analogy of doors.

First, let's consider swinging doors like the ones seen in restaurant kitchens or in old western movies. When someone pushes the door open and releases it, the doors swing back, and go past the rest point to the other direction. This back and forth motion happens several times before the doors come to a complete stop. This type of oscillatory behavior happens when the system has a damping ratio less than 1. Such systems are called underdamped system.

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