Damped and forced harmonic motion

Harmonic motion essentially describes the movement of a system where the restoring force is directly proportional to the displacement.
In the context of mechanisms and systems, damping refers to the reduction of vibrational amplitude due to factors such as resistance or friction. When this operations occur, it causes what is known as damped harmonic motion.
The damped harmonic motion can be mathematically represented using the second-order differential equation: mx’’ + γx’ + kx = 0, where m is the mass, γ is the damping coefficient, k is the stiffness constant, and x is the distance from the equilibrium position.
It can be observed that if γ = 0, the equation reduces to the one representing simple harmonic motion.
Damped motion can be classified into over-damped, critically damped, and under-damped based on the value of γ. The ratio γ/2m is often described as the damping ratio.
Over-damping occurs when the damping ratio is greater than 1. The mechanism returns to equilibrium without any overshooting, but very slowly. The solution to the damped harmonic equation would be in exponential form in this case.
Under-damping is observed when the damping ratio is less than 1. The system oscillates, with each oscillation having less amplitude than the previous due to the damping effect.
For a damping ratio of exactly 1, the system is said to be critically damped. The system returns to equilibrium as quickly as possible without any oscillations.
Forced harmonic motion refers to a scenario where an external force is applied to the system, causing it to oscillate with the same frequency as the external force.
The inclusion of an external force gives the differential equation the form mx’’ + γx’ + kx = Fcos(wt), where F is the amplitude of the external force and w is the frequency.
In the presence of damping and external forcing, a phenomenon known as resonance occurs when the frequency of the external force equals the natural frequency of the system. This may cause the amplitude of the oscillations to increase dramatically.