I have a damped forced vibration problem. Most problems I have dealt with have a constant amplitude sinusoidal force such as F_0sin(wt). Lets say you have an applied force that decreases each second at a constant rate over 11 seconds. It looks like this:
F=0.5*[(11-t)sin(2*pi*t)+(11-t)]; (0<= t >=11)
The graph of this force has been attached.
The system is also critically damped so the system's differential equation is:
m[d^2/dt^2](x) + c(dx/dt) +kx = 0.5*[(11-t)sin(2*pi*t)+(11-t)]
The Attempt at a Solution
From here i'm not too sure how to proceed. Normally with forced vibration the solution would include both a complimentary and particlular soluntion since it is a nonhomogenous linear second order differential equation. So
x = x_c + x_p
If the force had a constant magnitude (steady state vibration) and assuming critical damping x_c would look like:
but since the vibration is not steady state does this equation hold true? I assume it does only because the complimentary solution is found by setting the force to zero and assuming only free vibration. A and B are then found using the initial conditions where:
A= x_c(0) or is it A=x(0)
B=Dx(0)/dt + w_n*x(0)
Then x_p needs to be found and this im having trouble with. how would one solve this part of the differential?