Obtain the differential equation of forced oscillation.

Explanation

The force act on oscillator to sustain the oscillations external periodic force
`F(t)= F_(0) cos omega_(d) t`
Here, F(t) is the amplitude `F_(0)` of external periodic force dependently on time.
Three forces act on oscillator
(1) Restoring force `" "F(x) = -kx(t)`
(2) Resistive force `" "F_(S) = -bv(t)`
External periodic force `F_(d) = F_(0) cos omega_(d)t`
Where `F_(0) cos omega_(d) t` is external periodic force and `omega_(d)= (2pi)/(T) implies T= (2pi)/(omega_d)`
Period of external periodic force depends on the angular frequency of driving force.
From Newton.s second law of motion, if the net force is `F(t)= ma(t)` then
`F(t) = F(x) +F_(S)+F_(d)" ""......."(1)`
`ma(t) = -kx(t) -bv(t) +F_(0)cos omega_(d)t`
but `a(t) = (d^(2)x(t))/(dt^2), v(t) = (dx(t))/(dt)`
From equation (1)
`m(d^(2)x(t))/(dt^2)= -b(dx(t))/(dt) -kx(t) -bv(t) +F_(0)cos omega_(d)t`
`therefore m(d^(2)x(t))/(dt^2)+ b(dx(t))/(dt) +kx(t) = F_(0)cos omega_(d)t`
`therefore (d^(2)x(t))/(dt^2)+(b)/(m)*(dx(t))/(dt) +(k)/(m) x(t)= (F_0)/(m) cos omega_(d)t`
is the differential equation of forced oscillation.
m is the mass of oscillator.
This oscillator initially oscillates with its natural frequency `omega`. When we apply the external periodic force, the oscillations with the natural frequency die out, and then the body oscillates with the angular frequency of the external force.
Its displacement, after the natural oscillations die out, is given by `x(t) = A cos (omega_(d)t + phi)` where t is the time measured from the moment when the periodic force is applied.
The amplitude of forced oscillation, `A= (F_0)/([m^(2)(omega^(2)-omega_(d)^(2))^(2)+(omega_(d)^(2)b^(2))^(1/2)])` and phase `phi = tan^(-1) ((v_0)/(omega_(d)x_(0)))`
Where `v_(0)` is the velocity of the particle at time `t=0" and "x_(0)` is the displacement of the particle at time t=0.