If an undamped springmass system with a mass that weighs 6 l

If an undamped spring-mass system with a mass that weighs 6 lb and a spring constant 4 lb/in is suddenly set in motion at by an external force of 108cos 8t lb, determine the position of the mass at any time. Assume that g = 32 2. Solve for in feet

Solution

m = 6/32 = 3/16 lb.sec^2/ft,

k = 12 lb/ft,

thus the equation is : 3/16*u\"+12*u =108*cos(8*t)

with initial condition u(0) = 0, u\'(0) = 0

Rewrite the equation as: u\"+64*u=576*cos(8*t)

we see that w_o =sqrt(64) =8

w^2 =k/m =>m=k/w^2 =4lb/(1/12ft) =48lb/ft

=>u= Fo/(m(w_o^2-w^2) ) *(cos(w*t)-cos(w_o*t))=108/(1/16*(8^2-7^2))*(cos(7t)-cos(8t)) =1728/45*(cos(7t)-cos(8t))