# Find the displacement of the underdamped driven mass.

1. Apr 11, 2012

### Aesteus

1. The problem statement, all variables and given/known data

A mass "m" is oscillating on a spring in one direction. And the mass has a dampening constant ψ. The right end of the spring is attached to the mass, and the left end is driven by a force. For t<0, the spring end is at rest, but for t>0 the end oscillates with amplitude Acos(w(driving)*t). Find the displacement of the mass for t>0.

2. Relevant equations

x(0)=0
v(0)=0
x = Re[A(r) + iA(i)](cos(wt) + isin(wt))] ...... A(r) and A(i) are just variables

3. The attempt at a solution

Starting with x = Re[A(r) + iA(i)](cos(wt) + isin(wt))] we get x(real) = A(r)cos(wt) -A(i)sin(wt).

Now ma = -kx - bv = Acos(w(driving)*t)
and (β^2)*a + (β)*γ*v + ω^2 = Acos(w(driving)*t) .... where γ = ψ/m and ω^2 = k/m

Now, I want to add the solutions for the homogeneous and non-homogeneous equations, and then satisfy all of my initial conditions. But, I'm not sure how to go about doing this.