Everybody has seen a vibrating tuning fork or a swinging pendulum. Their motion eventually slows down and ceases if they are left alone. However, why does this occur? This is where damping enters the picture.
The oscillations in presence of dissipative force where amplitude decrease gradually with the passage of time are called damped oscillations. In actual practice, most of the oscillations occur in viscous media, such as air, water, etc.
Damped Simple Harmonic Motion
A part of energy of the oscillating system is lost in the form of heat, in overcoming these resistive forces. As a result the amplitude of such oscillations decreases exponentially with time. Eventually these oscillations die out.
In the oscillations the amplitude of the oscillation decreases exponentially due to damping forces like frictional force, viscous force, etc. Due to decrease in amplitude, the energy of the oscillator also goes on decreasing exponentially.
The force producing a resistance to the oscillation is called damping force.
What is Simple Harmonic Motion (SHM)?
In order to understand damping, let’s quickly review SHM.
Simple harmonic motion is a kind of periodic motion in which the restoring force always works in the direction of the mean position and is directly proportionate to the displacement from it.
Example:
A mass that is fastened to a spring. It oscillates around its equilibrium position when tugged and released. It is SHM here.
Why Does Motion Slow Down in Real Life?
No system in actual life continues to function indefinitely. This is due to forces that oppose the motion, such as air resistance or friction. Over time, these forces reduce the system’s energy, which results in a progressive decrease in the motion’s amplitude (maximum displacement).
This gradual loss of energy leads to damped motion.
What is Damped Simple Harmonic Motion?
Damped Simple Harmonic Motion (damped SHM) is a kind of SHM in which the system’s energy is gradually reduced by an external force (such as air resistance or friction), causing the oscillation’s amplitude to weaken over time.
The oscillations get smaller and eventually stop, yet the system is still oscillating.
How Does Damping Work?
We think a pendulum that is swinging in midair. Every swing gets smaller than the one before it. This is because energy is being lost due to air resistance, a dampening force.
This dampening force works in the opposite direction of motion and is often proportionate to the object’s velocity. The damping force increases with object velocity.
Mathematically, the damping force is often written as:
F < sub > damping < / sub > = –b·v
where:
b is the damping constant (how strong the damping is)
v is the velocity of the object
Types of Damping
There are three types of damping based on how quickly the system returns to rest:
1. Underdamped:
Oscillations continue but with decreasing amplitude.
Most common case in real life (e.g., swinging pendulum).
2. Critically Damped:
System returns to rest quickly without oscillating.
Used in designing doors, shock absorbers, etc.
3. Overdamped:Returns to rest very slowly without oscillating.
More damping than needed.
Equation of Damped SHM
For damped motion, the equation looks like this:
x(t) = A·e < sup > –bt/2m < / sup > · cos(ω’t + φ)
Where:
A is the initial amplitude
b is the damping constant
m is the mass of the object
ω’ is the damped angular frequency
φ is the phase angle
e < sup > –bt/2m < /sup > shows how the amplitude decays over time
The exponential part (e < sup > –bt/2m < /sup >) is what causes the amplitude to shrink with time.
Damped SHM
When we plot displacement against time, it appears as a wave that progressively shrinks, much like a spring that bounces less and less until it stops.
Applications of Damped SHM
Shock absorbers in cars (to prevent bouncy rides)
Building design (to handle earthquakes and vibrations)
Musical instruments (control the sound decay).
Seismographs (recording earthquake waves without extra noise on this graph)
Summary
Damped Real-world oscillations that gradually lose energy are the focus of simple harmonic motion. While dampening makes sure that the motion eventually fades out, optimal SHM never stops.
Designing systems that withstand natural vibrations while being stable, secure, and effective requires an understanding of damping.
Thus, keep in mind that damping is the reason why a swing slows down or an automobile smoothes out a bump.
A form of oscillatory motion known as damped simple harmonic motion occurs when an external factor, such as air resistance or friction, causes the oscillation’s amplitude to progressively reduce over time. Eventually, the motion stops.
Resistive factors that drain energy from the system, like air resistance or friction, are what produce damping. The total mechanical energy is decreased by these forces, which oppose motion.
In damped SHM, the amplitude gradually drops as a result of energy loss, and the motion finally stops, whereas in undamped SHM, the oscillations continue with a constant amplitude indefinitely (the ideal situation).
The damping force is a resistive force that opposes the motion and is usually proportional to the velocity of the object. It is given by:
F < sub > damping < /sub > = –b·v,
where b is the damping constant and v is velocity.
There are three types of damping:
Underdamped: Oscillations occur with gradually decreasing amplitude.
Critically damped: System returns to equilibrium quickly without oscillating.
Overdamped: System returns to equilibrium slowly, also without oscillating.
The displacement in damped SHM is given by:
x(t) = A·e < sup > –bt/2m < /sup > · cos(ω’t + φ)
where
A is the initial amplitude,
b is the damping constant,
m is mass,
ω’ is the damped angular frequency, and
φ is the phase angle.
Examples from real life are:
The shock absorber system of an automobile;
A swinging pendulum in midair;
The progressive loss of sound in musical instruments;
Building earthquake dampers; and
Vibration control in industry.