Damped Harmonic Motion: Have you ever come up with a pendulum clock? When the clock strikes an hour, the pendulum begins to ring. The pendulum swings back and forth about its mean position as it rings. The swinging of this pendulum slows and ultimately comes to rest. Why is that? Due to friction between air and bob, the pendulum slows and ultimately comes to rest but what it is called?

Have you ever tried swinging? The ones that move back and forth as you exert force on them with your limbs. But what happens when you stop pushing forward? The swing swings back and forth for a while before coming to a halt. Why is that? When we swing a pendulum, we know that it will ultimately come to rest due to air pressure and friction at the support. This motion is described as damped harmonic motion. The amplitude of a damped simple harmonic oscillator gradually decreases. Read this article to know the definition, examples, and expressions of damped simple harmonic motion.

Learn Concepts of Simple Harmonic Motion

What is Simple Harmonic Motion?

A simple harmonic motion is the oscillatory motion of the simplest type. The motion of an oscillating system is said to be a simple harmonic when the force acting on it is directly proportional to the displacement of the oscillating object from its mean position. At any point during the simple harmonic motion, the force acting on an object will act towards its mean position.

The oscillating systems eventually come to a halt due to the damping and dissipative forces. The systems can be forced to remain oscillating with the help of some external periodic agency.

What is Damping?

Damping can be defined as energy dissipation by restraining the vibratory motion like mechanical oscillations, alternating electric current, and noise. Damping can be understood as the resistance offered to the oscillation of a body. In automobiles, shock absorbers and carpet pads act like damping devices. Damping can be of two types:

1. Natural Damping
2. Artificial Damping

Damped Simple Harmonic Motion

A simple harmonic motion whose amplitude goes on decreasing with time is known as damped harmonic motion. These oscillations fade with time as the energy of the system is dissipated continuously. If the damping applied to the system is relatively small, then its motion remains almost periodic. Frictional forces generally act as dissipative forces.

Expression for Damped Simple Harmonic Motion

Let us understand the effect of a damping force on the motion of a harmonic oscillator. Suppose we have a block of mass mm connected to an elastic spring having a spring constant k that is oscillating vertically. Now, this block is pushed down slightly and released immediately. The spring block system will start oscillating. Let the angular frequency of the oscillation be ω.

Then, \(\omega = \sqrt {\left({\frac{k}{m}} \right)} \)

Although in reality, a damping force will be exerted by the surrounding medium, i.e. air on the block’s motion. Due to this, the mechanical energy of the spring-block system will decrease. The loss in the system’s energy will appear in the form of heat energy in the surroundings. The magnitude and the type of damping forces acting on the system will depend on the nature of the surrounding medium. When the spring block system is immersed in fluid instead of air, due to the larger magnitude of the damping thus, the dissipation of energy will be much faster.

Damped Harmonic Oscillator Derivation

We know from the stokes law that the damping force, in general, is directly proportional to the velocity. It acts in a direction opposite to the direction of velocity.

Let the damping force acting on the spring-block system be \({F_d},\) then
\({F_d} = – bv – – – \left( 1 \right)\)
Where \(v\): is the velocity of the block
\(b\): it is a positive constant whose value depends on the properties of the medium, for example, viscosity and the shape and size of the block.
The equation \(\left( 1 \right)\) is only valid for small values of the velocity. When the mass \(m\) is hung vertically attached to the spring, as shown in the figure, when the block is released from rest, the spring will elongate a little, and the mass attached to the spring will ultimately settle at a height. The point \(O\) represents the position of equilibrium of the mass.
When the mass is pushed up or pulled down by some distance,
The restoring force acting on the block due to the spring,
\({F_s} = – kx\)
Where \(x\) is the displacement of the mass from its equilibrium position
\(k\) is the spring constant
Thus, at any time \(t,\) the total force acting on the mass, \(F = – kx – bv – – – \left( 2 \right)\)
Let \(a\left( t \right)\) is the acceleration of the mass in time \(t.\)
By applying Newton’s laws of motion along the direction of motion,
The force due to the motion, \(F = ma\left(t \right) – – – \left( 3 \right)\)
Using the equations \(\left( 2 \right)\) and \(\left( 3 \right),\) we get:
\( – kx\left(t \right) – bv\left( t \right) = ma\left( t \right) – – – \left( 4 \right)\)
Since we are only concerned with the one-dimensional motion of the mass, we can drop the vector notation. We know,
Velocity can be determined by the first derivative of displacement, \(v\left( t \right) = \frac{{dx}}{{dt}}\)
Acceleration can be determined by the second derivative of displacement, \(a\left( t \right) = \frac{{{d^2}x}}{{d{t^2}}}\)
Substituting the values from above into equation \(\left( 4 \right),\) we get:
\(m = \frac{{{d^2}x}}{{d{t^2}}} = – kx – b\frac{{dx}}{{dt}}\)
\(m\frac{{{d^2}x}}{{d{t^2}}} + b\frac{{dx}}{{dt}} + kx = 0\)
The block’s motion under the effect of a damping force, which is proportional to the velocity, can be determined by the solution of the above equation. The solution of this equation is of the form:
\(x\left(t \right) = A{e^{ – bt/2m}}\,\cos \,\left({\omega ‘t + \phi }\right) – – – \left( 5 \right)\,\)
In this equation, \(A\) is the amplitude, \(\omega ‘\) is the angular frequency of the damped oscillator, \(\phi \) is the phase constant, and the cosine function has a period of \(\frac{{2\pi }}{{\omega ‘}}\)
The angular frequency can be given as:
\(\omega ‘ = \sqrt {\frac{k}{m} – \frac{{{b^2}}}{{4{m^2}}}} \)
The function \(x\left( t \right)\) in equation \(\left( 5 \right)\) is not exactly periodic because of the presence of the factor \({e^{ – bt/2m}},\) which as we can see, decreases continuously with time. However, if the decrease in one time period is small, the motion of the mass can be taken to be periodic.
The equation \(\left( 5 \right)\) can be graphically represented as:
It is a cosine function whose amplitude \(A{e^{ – bt/2m}}\) is gradually decreasing with time.
We know the expression for the mechanical energy of an undamped oscillator can be given as, \(E = \frac{1}{2}k{A^2}\)
When the applied damping is small, the amplitude is not constant, but it depends on the time. Thus, for the given damped oscillator, the expression for mechanical energy can be given as:
\(E\left( t \right) = \frac{1}{2}k{\left({A{e^{ – bt/2m}}} \right)^2}\)
\(E\left( t \right) = \frac{1}{2}kA{e^{ – bt/m}} – – – \left( 6 \right)\)
From equation \(\left( 6 \right),\) we can see that the total mechanical energy decreases exponentially with time.
Note: For small damping, the dimensionless ratio \(b/\sqrt {{\text{km}}} \) is less than one. For \(b = 0,\) the damped oscillator will behave like an undamped oscillator.

Damped Harmonic Motion Examples

1. The shock absorbers in a car
2. Strings of a guitar
3. A sound wave striking our eardrum
4. The motion of a swing
5. A simple pendulum
6. Cradle of a baby
7. Bungee jumping rings for kids

Summary

A simple harmonic motion whose amplitude goes on decreasing with time is known as damped harmonic motion. These oscillations fade with time as the energy of the system is dissipated continuously.
The expression for a given damped oscillator is: \(x\left( t \right) = A{e^{ – bt/2m}}\,\cos \,\left({\omega ‘ t + \phi } \right)\)
In this equation, \(A\) is the amplitude, \(\omega ‘\) is the angular frequency of the damped oscillator, \(\phi \) is the phase constant, and the cosine function has a period of \(\frac{{2\pi }}{{\omega ‘}}.\) When the applied damping is small, the amplitude is not constant, but it depends on the time.
For the given damped oscillator, the expression for mechanical energy can be given as,
\(E\left( t \right) = \frac{1}{2}kA{e^{ – bt/m}}\)

Learn All About Simple Pendulums

FAQs on Damped Harmonic Motion

Here we have listed out some of the commonly asked questions related to Damped Simple Harmonic Motion:

Q.1. What is damped harmonic motion?
Ans: The simple harmonic motion in which the oscillation’s amplitude decreases continuously with time due to the presence of dissipative forces like friction is called damped harmonic motion.

Q.2. Define simple harmonic motion.
Ans: S.H.M or simple harmonic motion can be defined as the motion in which restoring force is directly proportional to the displacement of the body from its equilibrium position.

Q.3. What will happen to the damping of a block when it is immersed in a liquid?
Ans: The magnitude of damping will increase.

Q.4. Give the expression of energy for a damped harmonic oscillator.
Ans: The expression for the energy of a damped harmonic oscillator: \(E\left( t \right) = \frac{1}{2}kA{e^{ – bt/m}}\)

Q.5. Write the expression for damping force acting on a mass acting at any time \(t.\)
Ans: The damping force acting on a mass acting at any time \(t,\)
\(F = – kx – bv\)

Q.6. Is the amplitude of a damped simple harmonic oscillator increase or decrease?
Ans: When an oscillator’s motion is reduced due to an external force, the oscillator and its motion are damped. Hence, the amplitude of a damped simple harmonic oscillator gradually decreases.