What is forced oscillations? Discuss the vibrations of a system executing damped simple harmonic motion when subjected to an external periodic force.

When an oscillator is made to oscillate by an external periodic force, whose frequency is different from the natural frequency of the free oscillator is defined as forced oscillations. 

In forced oscillations, the oscillator is under the influence of two forces simultaneously:
i) Restoring force of its own
ii) Force due to external periodic force.
Both these forces have a tendency to make the oscillator oscillate their own way.
Thus, for the first few, oscillator does not exhibit regular oscillations.
First, the restoring force is more effective and the oscillator oscillates with natural frequency and then external periodic force dominates and makes the oscillator oscillate with a frequency same as that of external periodic force. 

                    
Consider a mass m attached to a spring of constant k.

When an external periodic-force is applied on mass, it will oscillate under the influence of following forces:

(i) The restoring force which is proportional to displacement and directed towards equilibrium position.

i.e.,                           F₁= – ky

(ii) Resistive force of medium which is proportional to velocity and opposite to the direction of velocity.
i.e.,                             F₂ = – bv

(iii) The external periodic force F₃ = f_o cos ωt.
The net force acting on the load is, 
 
Therefore the equation of motion of load is,

This is the required differential equation of motion of damped oscillations.
The solution of the equation is,

From the above equation, we can deduce that  the amplitude of oscillation depends on the difference in the frequency of periodic force and natural frequency.
Smaller is the frequency difference greater is the magnitude of oscillation and vice versa.