A spring and dashpot system is to be designed for a 32lb weight so that the overall system is critically damped.
(a) How must the damping constant γ and spring constant k be related?
(b) Assume the system is to be designed so that the mass, when given an initial velocity of 4 ft/sec from its rest position, will have a maximum displacement of 6 inches. What values of damping constant γ and spring constant k required?
The Attempt at a Solution
For the system to be critically damped, γ^(2) = 4km
(a) γ = sqrt(4km)
k = γ^(2)/4m
(b) the IVP is... my'' + λγ' + ky = 0, y(0) = 6, y'(0) = 4
Since γ^2 = 4km,
The solution of the IVP is... y(t) = c1e^(λt) + c2te^(λt)
Imposing the initial conditions, I got c1 = 6
and c2 = 4 + 12 k^(1/2)*m^(-1/2)
Now, how do I solve for the damping constant γ and the spring constant k?