Consider our model for the displacement of a mass m attached

Consider our model for the displacement of a mass m, attached to a spring with stiffness k. acted on by a driving force f(t) = F_0 cos (omega t), my\" + dy\' + ky = F_0 cos (omega t). If we assume that there is no damping in the system (i.e d = 0) then we reduce the above equation to simply y\" + omega^2 _0 y = F_0/m cos (omega t) where omega_0 = Squareroot k/m. Show that this equation has a general solution of the form y(t) = c_1 cos (omega_0 t) + c_2 sin (omega_0 t) + F_0/m(omega^2 _0 - omega^2) cos (omega t) which is the sum of two periodic motions with different frequencies omega notequalto omega_0. We call the first two terms the transient solution y_T = c_1 cos (omega_0 t) + c_2 sin(omega_0 t), which depends on the initial conditions of the system. In the case where damping occurs the transient solution always satisfies the following limit lim_t rightarrow infinity y^T(t) rightarrow 0. Thus, the motion of the mass governed by y_T eventually will die out. When there is no damping, then lim_t rightarrow infinity y_T notequalto 0 and the motion governed by y_T is purely oscillatory and always remains present as time evolves. The last term we call the steady state solution y_ss(t) = F_0/m(omega^2 _0 - omega^2) cos(omega t) which depends solely on the forcing function f(t). This solution satisfies the limit lim_t rightarrow infinity y(t) rightarrow y_ss(t) with the assumption that d > 0. Eventually the motion of the mass should reach its steady state.

Solution

Given Differential equation is y\'\' + w₀²y = F₀/m * cos ( wt)

Auxillary equation is given by D² + w₀² = 0 where D= d/dt

Root of above equation is D= 0 + jw₀ and D= 0 -  jw₀

Solution of above Auxillary equation is Complementry function which is given by

C.F. = c₁ cos ( w₀t)  +c₂ sin ( w₀t) which is correspond to the transient part

Now perticular Integration (P.I.) = 1/( D² + w₀² ) * F₀/m * cos ( wt)

=  F₀/m *1/( D² + w₀² ) *cos ( wt)

=  F₀/m *cos ( wt) /( -w²+ w₀² )   

Total solution = CF + PI =  c₁ cos ( w₀t)  +c₂ sin ( w₀t) +  F₀/m *cos ( wt) /( w₀^{2 -} -w² )