Damped Oscillations

Fundamental Concepts

Damped oscillations refer to oscillatory motion where the amplitude gradually decreases due to an external force, commonly friction or resistance. This leads to a loss of mechanical energy over time.
These oscillations can be modelled by second-order differential equations of the form m * d²x/dt² + γ * dx/dt + kx = 0, where m is the mass, k is the spring/elasticity constant, γ is the damping coefficient and x is the displacement function.
The type of damping experienced can be distinguished into three different categories based on the damping factor b = (γ^2) / (4mk): underdamped (b < 1), critically damped (b = 1), overdamped (b > 1).

Underdamped systems (b < 1) will perform oscillations with a diminishing amplitude. These still exhibit oscillatory behaviour, but with the amplitude of oscillation gradually decreasing till it comes to a standstill.
In critically damped systems (b = 1), the system returns to equilibrium as quick as possible without oscillating. This type of damping is often desirable in systems like car suspension or door closers.
Overdamped systems (b > 1) take the longest to return to equilibrium and does not oscillate. It takes the longest time among these three systems to reach equilibrium.

For underdamped systems, the general solution can be of the form x(t) = e^(-(γt)/2m) [A cos(ω’t) + B sin(ω’t)], where ω’ = √[(4mk - γ^2) / 4m^2].
For critically damped systems, the general solution can be of the form x(t) = e^(-(γt)/2m) (At + B).
For overdamped systems, the general solution would be of the form x(t) = e^(-(γt)/2m) [A e^(λt) + B e^(-λt)], where λ = √[(γ^2 - 4mk) / 4m^2].

Damped oscillations find numerous applications in various real-world systems including electrical circuits, mechanical systems, acoustics, among others.
Understanding the behaviour of these oscillations helps in designing systems with desired properties, such as adjusting dampers in a car’s suspension to enhance ride comfort, or tuning an electrical circuit for optimal performance.
The key takeaway here is to appreciate the intricate balance between damping and restoring forces in producing the observed motion in a damped oscillatory system.