APPLICATIONS OF DAMPED HARMONIC

OSCILLATIONS

Damped Harmonic Motion

In real systems, masses on springs don't continue to oscillate forever at the same amplitude;

eventually the oscillations die away and the object stops. This is due to the fact that springs

are not very efficient at storing and releasing energy. Much of the energy is dissipated as heat

due to friction within the spring. In order to better model these kinds of systems we can talk

about the damped harmonic oscillator, which is the solution to the differential equation:

Where we have taken the differential equation for the simple harmonic oscillator and added a

damping term, where is called the damping constant or drag coefficient. Since this damping

term acts in the opposite direction of motion and is proportional to velocity, it causes objects

with high velocity to slow down quickly. We can solve the damped harmonic oscillator

equation by using techniques that you will learn if you take a differential equations course.

The solutions are of the form:

Where

And

Observe Damped Harmonic Motion

In real systems, masses on springs don't continue to oscillate forever at the same amplitude;

eventually the oscillations die away and the object stops. This is due to the fact that springs

are not very efficient at storing and releasing energy. Much of the energy is dissipated as heat

due to friction within the spring. In order to better model these kinds of systems we can talk

about the damped harmonic oscillator, which is the solution to the differential equation:

Where we have taken the differential equation for the simple harmonic oscillator and added a

damping term, where is called the damping constant or drag coefficient. Since this damping

term acts in the opposite direction of motion and is proportional to velocity, it causes objects

with high velocity to slow down quickly. We can solve the damped harmonic oscillator

equation by using techniques that you will learn if you take a differential equations course.

The solutions are of the form:

Where

And

Observe that is just the frequency of oscillation of a simple harmonic oscillator. Thus we can

see that if we add damping to a simple harmonic oscillator, the frequency will change and the

amplitude of the oscillations will exponentially decay with time.

That is just the frequency of oscillation of a simple harmonic oscillator. Thus we can see that

if we add damping to a simple harmonic oscillator, the frequency will change and the

amplitude of the oscillations will exponentially decay with time.

Figure 2Damped Harmonic Oscillation Amplitude vs. Time Graph

Application-01 Shock Absorbers in automobiles

Shock absorbers in automobiles follows the principals of Damped Harmonic Oscillation. One

widely used application of damped harmonic motion is in the suspension system of an

automobile. A shock absorber is designed to introduce damping forces, which reduce the

vibrations associated with a bumpy ride. A shock absorber consists of a piston in a reservoir

of oil. When the piston moves in response to a bump in the road, holes in the piston head

permit the piston to pass through the oil. Viscous forces that arise during this movement

cause the damping.

In damped harmonic motion, the chassis oscillates with decreasing amplitude and it eventually

comes to rest. When the suspension system uses this shock absorbers, the damping get

smoothen. Typical automobile shock absorbers are designed to produce underdamped motion

Application-02 Pendulums in mechanical environments

Figure 1Shock Absorber

Pendulums are just like swings in a playground. When it oscillating in a mechanical

environment-environment with external resistivity (air resistance) it will oscillates in along

with the damped harmonic oscillation.