A mass of 1 slug is attached to a spring wh constant is 5 lb

A mass of 1 slug is attached to a spring wh constant is 5 lb/ft. Initially the mass is re- leased 1 foot below the equilibrium position ose with a downward velocity of 5 ft/s, and the with a downward velocity of 5 ft,/s, and subsequent motion takes place in a medium that offers a damping force numerically equal to 2 dx/dt. (a) Find the equation of motion if the mass is driven by an external force f(t) = 12 cos 2t + 3 sin 2t. (b) Find the transient and steady state solu- tions.

Solution

[a]

The equation of motion if the mass is driven by an external force is

F=-kx+ 2 dx/dt+f(t)

F=-kx+2dx/dt+ 12 cos2t+ 3sin2t

md²x/dt²= -kx+2dx/dt+ 12 cos2t+ 3sin2t

md²x/dt²-2dx/dt+kx-- 12 cos2t- 3sin2t -------------------equation { here we get the equation in quadratic form