(A) A damped oscillator is described by the equation
m x′′ = −b x′− kx .
What is the condition for critical damping? Assume this condition is satisfied.
(B) For t < 0 the mass is at rest at x = 0. The mass is set in motion by a sharp impulsive force at t = 0, so that the velocity is v0 at time t = 0. Determine the position x(t) for t > 0.
(C) Suppose k/m = (2π rad/s)2 and v0=10 m/s. Plot, by hand, an accurate graph of x(t). Use graph paper. Use an appropriate range of t.
For critically damped, β2 = w02
where β = b/(2m) and w0 = √(k/m)
The Attempt at a Solution
Ok, for this problem, what I did initially was find the general form of position for a critically damped oscillator, which is:
x(t) = (A + B*t)*e-β*t
and the velocity function is:
v(t) = -Aβe-βt + (Be-βt - Bβte-βt)
Using the conditions given, I found:
x(0) = A (obviously) which we don't know x(0)
B = v0 + Aβ
and x(t) can be rewritten as:
x(t) = A(e-βt + βte-βt) + v0te-βt
This is where I run into a wall. I can't seem to solve for A. I believe that x(0) should also be the max displacement since there is no driver for the impulse force, so A should be the max displacement, but this doesn't seem to get me anywhere. Any help on solving for A? I know how to do the rest other than that.