Damped Simple Harmonic Motion

Consider when we swing a pendulum, which is an oscillating body, the pendulum keeps on moving to and fro about its mean position. After some time, we find that the displacement(amplitude) from its mean position keeps on decreasing, and eventually, it comes to rest. This explains that there is some dissipating force that is resisting the motion of the pendulum. As the resisting force increases, the oscillating body slows down and comes to rest at its equilibrium position. This dissipating force is called damping and the motion of the oscillating body is referred to as the damped simple harmonic motion.

Simple harmonic motion definition

In our day-to-day life, we see a lot of examples of periodic motion such as swinging on a swing, motion of the hands of a clock, etc. In these examples, the particles perform the same set of movements repeatedly in a periodic manner. Such a periodic oscillatory motion is defined as Simple Harmonic Motion(SHM).

Simple Harmonic Motion is a specific type of periodic oscillatory motion in which:

• The particles set to motion in the same direction in a straight line
• The particles oscillate to and fro about their mean position where the net force is zero i.e

F_net = 0

• The magnitude of acceleration is directly proportional to the displacement of the particle from the equilibrium position i.e

a ∝ -x

What is Damped Oscillation?

Damped oscillation refers to the following points:

• If in an oscillating body, the displacement or the amplitude keeps on decreasing with the increase in time, it is termed as damped oscillation

• There is a damping force like dissipating, pressure, viscous or friction force which opposes the motion of the oscillating body
• The amplitude of the oscillating body keeps on decreasing exponentially with respect to time
• The energy possessed by the damped simple harmonic oscillator also decreases exponentially

Impact of Damping force

The damping force is the one which opposes the motion of the oscillating body. But the oscillations remain approximately periodic for a small damping. It always acts in a direction opposite to the direction of motion or velocity. So, the magnitude of the damping force is directly proportional to the velocity of the oscillating body.

Thus, we can form the expression as given below:

F_d  = -bv

here F_d is the damping force

v is the velocity of the oscillating body, and -b is the damping constant which depends on the characteristics of the medium i.e, shape, size,viscosity etc.

The damping force is dependent on the nature of its surrounding medium. If the oscillating body is immersed in a liquid, the magnitude of the damping force would be much higher. As a result, the corresponding energy dissipation would also be much faster.

Equation for a Damped Simple Harmonic Oscillator

Let us again consider the example of an oscillating body such as a pendulum. When it is set in motion, it will start to and fro motion about its equilibrium position. The motion of the oscillating body then slows down due to some external forces. It will experience two external forces. One is the Restoring force and the other is the Damping force. The total net force then will be the summation of both the restoring and damping force.

Restoring force F_s = -k x

Damping force F_d = -b v

Total force Ftotal  = F_s + F_d = -kx – bv

Now, let the acceleration of the oscillating body at any time t be a(t).

Therefore, by Newton’s laws of motion applied along the direction of the motion, we have

F_Total = ma(t)

-kx-bv= md²x/dt²

md²x/dt²+kx+bv=0

md²x/dt²+kx+bdx/dt=0

d²x/dt²+bmdx/dt+kmx = 0

After solving the above differential equation, we get the equation of the Damped Simple Harmonic Oscillator.

x(t) = Ae^-bt/2mcos(‘t)

In the above expression, the damping is caused by the term e^-bt/2m

and the angular frequency is given by ‘.

So, mathematically, the angular frequency of the damped simple harmonic oscillator is given by the expression

‘=km-b24m2

where b is the damping constant, and

the time period of the damped simple harmonic oscillator is given by the expression

T =2𝛑km-b24m2

Now, let us consider that if b=0, that means there is no damping and becomes a Simple Harmonic Motion. It is given by the expression

x(t) = Acos(‘t)

Hence, all equations of a damped simple harmonic oscillator reduces to the corresponding undamped oscillator. For small damping, the dimensionless ratio bkm is much less than 1.

Graphical representation of a Damped Simple Harmonic Motion.

If we plot amplitude on the y-axis and time on the x-axis, then as the time progresses,the amplitude decreases with time.

There is an exponential decrease in the amplitude with respect to time. As a result, the energy of the damped simple harmonic oscillator decreases exponentially.

Hence, with decreasing amplitude of oscillation, the damped simple harmonic oscillator is approximately periodic. With more significant damping, the oscillations of the oscillating body die out faster.

Energy expression on damping

For a damped simple harmonic oscillator, the amplitude keeps on changing with time and is not constant. The total energy dissipation decreases exponentially with time and is given by the expression

E(t) = 1/2kAe^-bt/2m

A motion which repeats itself continuously is known as periodic motion. All periodic motions are not simple harmonic. Only periodic motion which is governed by Newton’s Law of Motion is Simple Harmonic Motion. In a real oscillating system, the mechanical energy decreases due to external forces which restricts the oscillations. Then, the energy transfer of mechanical energy to heat energy takes place. And, the motion of the oscillating system is said to be damped. Damping of oscillating bodies decreases amplitude with time. Damped simple harmonic motion is not purely simple harmonic. Only for the time intervals of 2m/b, where b is the damping constant, it is approximately simple harmonic.