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Introduction to Engineering14 chapters | 123 lessons

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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.

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**.

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,

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.

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**.

Next consider a door in a modern building with a dashpot attached to the top corner. When this door is opened and released, it slowly returns to a closed position and will not oscillate back and forth. This system is called an **overdamped system**, and has a damping ratio of greater than 1.

In the middle, when the damping ratio is 1, the system is called **critically damped**. This would be like a door with no dashpot, slamming quickly to rest without oscillating.

Damping ratio can also be represented by the ratio of the actual damping coefficient to critical damping coefficient. So,

where,

Let's look at an example. An SDOF spring-mass-damper has the following characteristics: *m* = 10kg, *k* =100 N/*m*, and *c* = 1 Ns/*m*. (Recall that Ns is the Newton second, equivalent to a kilogram-meter per second (kg * m/s)).

What is its damping ratio?

Critical damping coefficient = 2 x the square root of (k x m) = 2 x the square root of (100 x 10) = 63.2 Ns/m

Since the actual damping coefficient is 1 Ns/*m*, the damping ratio = (1/63.2), which is much less than 1. So the system is underdamped and will oscillate back and forth before coming to rest.

When motion is oscillatory, the amplitude of the oscillation reduces over time due to friction, eventually going to zero. This type of an oscillation is called a **damped harmonic oscillation**. This can be understood using a **single degree of freedom (SDOF)** spring-mass-damper system consists of a spring, a mass, and a damper.

The defining equation for the motion of such a system is a second-order ordinary differential equation, with the behavior of the system dependent on the moving mass, spring constant, and damping coefficient.

A damped system returns to rest in different ways, which is determined by the **damping ratio**. A damping ratio:

- greater than 1 indicates an
**overdamped**system, which returns to rest slowly without oscillations. - less than 1 indicates an
**underdamped**system, which returns to rest in a oscillatory fashion. - equal to 1 is a
**critically damped**system, which returns to rest quickly without oscillating.

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Introduction to Engineering14 chapters | 123 lessons

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