Equation of Motion for a weighted vertical spring

Question

A spring-mass system (below) is initially at rest. If mass m is excited

by a sinusoidal force p(t) = P sin ωt, what is the response x(t), t ≥ 0? Assume that m = 1 kg, k = 100 N/m,

P = 5 N, and ω = 2 rad/s.

I think I need to look at this as a static problem and then as a dynamic one.

I've been given that the result x(t) = 5/96[sin2t-1/5(sin10t)], t ≥ 0, but I'm quite sure how to get there.

Thank you

Answer 1

There are two forces being applied to the mass: the spring force and the sinusoidal force. when both forces are added together, they equal mass times acceleration by Newton's Second Law. Therefore, we have

m y^''(t) = -k y(t) + P sin(ωt) [in Newtons]. 〈1〉

Substituting known values of k, P, ω and m in〈1〉and moving terms around yields the second-order linear non-homogeneous equation with constant coefficients

y''(t) + 100 y(t) = 5 sin(2t). 〈2〉

First, observe the homogeneous version of〈2〉which is

y''(t) + 100 y(t) = 0. 〈3〉

It has solutions of the form e^{rt} [for r complex]. Substituting e^{rt} for y(t) in 〈3〉 gives

[r^2 + 100] e^{rt} = 0. 〈4〉

This means that r^2 + 100 = 0, thus r = ±10i. Imaginary roots signify that cos(10t) and sin(10t) are homogeneous solutions to〈3〉. We set y_h(t) = A cos(10t) + B sin(10t).

Now to find a particular solution. We use the method of undetermined coefficients. To obtain 5 sin(2t) on the right hand side, we must have a particular solution of the form y_p(t) = C cos(2t) + D sin(2t). Computing y_p^''(t) and substituting for y(t) in 〈2〉results in

96 C cos(2t) + 96 D sin(2t) = 0 cos(2t) + 5 sin(2t), so C = 0 and D = 5/96, meaning that

y_p(t) = 5/96 * sin(2t). 〈5〉

Our complete solution has the form y(t) = y_h(t) + y_p(t) = A cos(10t) + B sin(10t) + 5/96 * sin(2t).

Since the mass was initially at rest, we know that y(0)=0 (that is, the spring was at equilibrium) and y'(0)=0 (an object at rest has no velocity). Computing y'(t) and substituting t=0 into y and y' yields that

A=0 and B= -1/96, meaning that the solution to the problem is

y(t) = -1/96 * sin(10t) + 5/96 * sin(2t).