KAHIBARO
Table of Contents
Energy Loss in Oscillations
In real life, oscillations do not continue forever with exactly the same amplitude. A swinging pendulum gradually slows down, a vibrating guitar string becomes quieter, and a mass on a spring eventually comes to rest. This gradual decrease of motion is called damping.
Damping happens because energy is transferred from the oscillating system to its surroundings. Usually this energy becomes thermal energy because of friction, air resistance, or internal resistance in the material itself. The oscillation still repeats back and forth, but each cycle is a little smaller than the previous one.
Damping is the effect that causes the amplitude of an oscillation to decrease with time because energy is lost from the system.
Physical Origin of Damping
A perfectly ideal oscillator has no energy loss. In practice, this ideal situation is almost never reached. Several common effects produce damping.
Friction at contact surfaces can oppose motion. Air resistance can slow moving objects. Fluids can strongly resist motion when an object moves through them. Even solid materials can lose energy internally because different parts of the material rub or deform slightly as they move.
The important idea is that the damping force acts in a direction that opposes the motion. For many simple cases, especially when speeds are not too large, the damping force is approximately proportional to velocity.
This is written as
$$
F_d = -bv
$$
where $b$ is the damping constant, $v$ is the velocity, and the minus sign shows that the force opposes the motion.
For simple damped motion, a common model for the damping force is
$$
F_d = -bv
$$
The minus sign means the force acts opposite to the velocity.
Damping in a Mass-Spring System
Consider a mass attached to a spring, moving horizontally. Without damping, the spring force alone produces simple harmonic motion. With damping, there is also a resistive force.
The forces are the spring force and the damping force, so Newton's second law gives
$$
m\frac{d^2x}{dt^2} = -kx - b\frac{dx}{dt}
$$
or equivalently,
$$
m\frac{d^2x}{dt^2} + b\frac{dx}{dt} + kx = 0
$$
This is the standard equation for a damped oscillator.
Here, $m$ is the mass, $k$ is the spring constant, $b$ is the damping constant, and $x$ is the displacement from equilibrium.
The extra term involving velocity is what makes the motion lose energy over time.
The equation of motion for a damped mass-spring oscillator is
$$
m\frac{d^2x}{dt^2} + b\frac{dx}{dt} + kx = 0
$$
The term $b\frac{dx}{dt}$ represents damping.
What Damping Does to the Motion
In an undamped oscillator, the amplitude stays constant forever. In a damped oscillator, the object still oscillates for a while, but the amplitude decreases with time. The peaks get smaller and smaller until the motion becomes negligible.
The displacement often has the form of an oscillation multiplied by a decaying factor. A common form is
$$
x(t) = A_0 e^{-\frac{b}{2m}t}\cos(\omega t + \phi)
$$
for the case where oscillation continues while shrinking in size. The exact angular frequency depends on the amount of damping, but the key feature here is the factor
$$
e^{-\frac{b}{2m}t}
$$
which causes the amplitude to decrease exponentially.
This means the amplitude does not drop by equal amounts in equal times. Instead, it drops by the same fraction in equal times.
In damped oscillatory motion, the amplitude commonly decreases exponentially:
$$
A(t) = A_0 e^{-\frac{b}{2m}t}
$$
So damping causes the amplitude to shrink continuously with time.
Energy and Damping
Because damping removes energy from the system, the total mechanical energy decreases with time. Since the energy of an oscillator depends on the square of the amplitude, energy falls even faster than amplitude.
If amplitude decreases exponentially, then energy also decreases exponentially. The system does not lose all its energy at once. Instead, it gradually transfers energy to the surroundings.
This is why a damped oscillator slows down smoothly rather than stopping instantly.
Light, Moderate, and Strong Damping
Different amounts of damping produce different kinds of behavior. The details of the motion are discussed more fully in related sections, but the main idea is simple. If damping is small, the system continues oscillating for many cycles while the amplitude slowly decreases. If damping is very large, the system may return to equilibrium without oscillating at all.
The table below summarizes the general effect.
| Damping amount | Behavior |
|---|---|
| Small damping | Oscillates, amplitude slowly decreases |
| Moderate damping | Returns to equilibrium quickly, may barely oscillate |
| Large damping | Does not oscillate, motion dies out slowly or quickly depending on the case |
These different cases are important because many physical systems are designed to have a particular amount of damping. A car suspension, for example, should not bounce for a long time after a bump.
Everyday Examples
A playground swing slows down because of air resistance and friction at the support. A pendulum clock would stop if it were not given energy repeatedly. A door closer uses damping so the door does not slam. Shock absorbers in vehicles are designed to damp oscillations after the wheels pass over uneven ground.
In each case, the common pattern is the same. Motion continues for a while, but the amplitude becomes smaller because energy is lost.
Visualizing Damping
A damped oscillator still crosses the equilibrium position again and again, but the maximum displacement on each side gets smaller. The curve is trapped inside two shrinking envelopes.
The blue curve shows the oscillation. The dashed red curves show the decreasing amplitude envelope.
Key Features to Remember
Damping is not a separate kind of oscillation by itself, but an effect that changes oscillatory motion. It makes the amplitude shrink, removes mechanical energy from the system, and usually comes from resistive forces such as friction or drag.
Important facts about damping:
$$
F_d = -bv
$$
$$
m\frac{d^2x}{dt^2} + b\frac{dx}{dt} + kx = 0
$$
Damping opposes motion, reduces amplitude, and causes the oscillator to lose energy with time.