# Differential Equation, Forced Undamped Motion

A mass m is attached to a helical spring whose spring constant is k. At t=0, it is brought to rest and constant forcing function f(t) = (1/b ) Ib is impressed on the system. After b sec, the force is removed. (a) find the position of y of the mass as a function of time. Hint. First solve with f(t)= 1/b. Find y(b) and y'(b). Now solve the equation with f(t) = o and initial conditions t=b, y= y(b), dy/dt = y'(b) remark: after the input or forcing function /b is removed, note that it still is possible to have an output y(t). Note, too, that if b is small, say 1/50, then f(t) = 50 Ib, but that it acts for only 1/50 sec. it is as if the mass were given a sudden blow by a force that was immediately removed. finally note that the output or response function y(t) is continues for t>or= 0, even thought the input or forcing the function f(t) is discontinuous. The latter can be written as f(t) -> (1) 1/b , 0b.if b =0 , f(t) DOES NOT EXIST. In engineering circles, however, the fictitious forcing function f(t) which results when b = 0, is called a unit impulse. (b) show that as b -> 0, the solution y(t) – let us call it y0(t) – approaches y0(t) = (1/k) sqrt(k/m) sin sqrt(k/n)tHint, use the fact that lim when theta goes to 0 sin(theta/theta) = 1. Now prove that y0(t) satisfies the equation m(d^2y/dt^2) +ky = 0 with initial conditions t = 0, y=0, dy/dt = 1/m. The Function y0(t) is called the impulsive response or the response of the system to a unit impulse. ( The solution should be typed )

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