All macroscopic oscillations are damped. The reason for this is the presence of various types of friction. In this section, we assume that the damping force is proportional to the velocity of the oscillator.
In this case, the term $-b\dot{x}(t)$ must be added to the equation of motion. This gives:
$$
m\ddot{x}(t) = -kx(t) - b\dot{x}(t)
$$
Dividing by the mass and bringing all terms to the left-hand side gives the standard form for damped oscillations:
$$
\ddot{x}(t) + 2\gamma\dot{x}(t) + \omega_0^2 x(t) = 0
$$
Here, $k/m$ has been replaced by the natural frequency $\omega_0$, and $b/m$ by twice the damping constant $\gamma$ (the factor 2 is for convenience in representation).
Solving this second-order differential equation requires familiarity with complex numbers and the exponential function, which will not be detailed here. Instead, we summarize three important special cases:
Underdamping ($\gamma < \omega_0$)
The motion is similar to harmonic oscillation, but with a decaying amplitude. The solution is:
$$
x(t) = A e^{-\gamma t} \sin(\omega t)
$$
The amplitude envelope follows an exponential decay. The actual angular frequency of the oscillator is:
$$
\omega = \sqrt{\omega_0^2 - \gamma^2}
$$
This frequency is slightly less than the undamped frequency $\omega_0$, and the shift increases with the damping constant $\gamma$.
The decay of amplitude can also be described using the logarithmic decrement $\delta$, defined as:
$$
\frac{x(t + T)}{x(t)} = e^{-\gamma T}
$$
Taking the natural logarithm of both sides:
$$
\ln\left[\frac{x(t)}{x(t+T)}\right] = \gamma T = \delta
$$
Note that the period $T$ refers to the damped system.
Overdamping ($\gamma > \omega_0$)
In this case, the system does not oscillate but instead returns slowly to equilibrium after a single displacement. This is also called the creep case. The solution is:
$$
x(t) = \frac{A}{\alpha} e^{-\gamma t} \left[ \alpha \cosh(\alpha t) + \gamma \sinh(\alpha t) \right]
$$
with $\alpha = \sqrt{\gamma^2 - \omega_0^2}$.
Critical Damping ($\gamma = \omega_0$)
This special case is particularly useful in engineering, such as in the design of shock absorbers, where oscillations should be suppressed as quickly as possible. The solution is:
$$
x(t) = A(1 + \gamma t) e^{-\gamma t}
$$
In all three damping types, the behavior of the displacement over time differs:
- Only in the underdamped case is there a visible sinusoidal oscillation.
- In the overdamped and critically damped cases, only an exponential return to equilibrium occurs.
- Although the amplitude in the overdamped case decays faster than in the underdamped case, the critically damped case results in the fastest return to rest without oscillation, making it the most effective damping.