Maybe this isn't the ideal place for this, but I'm completely stuck and this is due tomorrow.
So I'm working on a problem for physics, but I keep going around in circles...
I need to derive and solve the equation of motion for a driven damped harmonic oscillator, where the Force has the form Fcos(wt).
I know I can write this as the non-homogenous linear second order differential equation;
m(d^2x/dt^2) + b(dx/dt) + kx = Fcos(wt)
What I don't understand is how to solve this. I haven't actually learnt how to solve this kind of equation yet (I'm doing an advanced physics unit at uni, and unfortunately our lecturer seems to think we've done more maths than we have).
The other information I'm given; Fcos(wt) = F/2[e^(iwt) + e^(-iwt)] = F+(t) + F-(t) (my notation here is kinda dodgy, but I mean that the Force is a function of time, and there's a positive and negative case for it).
The particular solution is given by solving x+(t) with F+(t) plus solving x-(t) with F-(t). I do not really understand how to do this.
I'm also told to assume for the particular solution that x(t) has the form Ae^(+/-iwt).
Finally, I'm told that the total solution is the sum of the homogenous solution and the particular solution.
For one thing, the homogenous equation has THREE separate solutions for different cases (namely the damped, underdamped and critically damped cases), so how can I add the homogenous solution?
But mostly I don't understand how to find the particular solutions.
I don't want someone to work this out for me, but I do need advice on how to solve it.
So I'm working on a problem for physics, but I keep going around in circles...
I need to derive and solve the equation of motion for a driven damped harmonic oscillator, where the Force has the form Fcos(wt).
I know I can write this as the non-homogenous linear second order differential equation;
m(d^2x/dt^2) + b(dx/dt) + kx = Fcos(wt)
What I don't understand is how to solve this. I haven't actually learnt how to solve this kind of equation yet (I'm doing an advanced physics unit at uni, and unfortunately our lecturer seems to think we've done more maths than we have).
The other information I'm given; Fcos(wt) = F/2[e^(iwt) + e^(-iwt)] = F+(t) + F-(t) (my notation here is kinda dodgy, but I mean that the Force is a function of time, and there's a positive and negative case for it).
The particular solution is given by solving x+(t) with F+(t) plus solving x-(t) with F-(t). I do not really understand how to do this.
I'm also told to assume for the particular solution that x(t) has the form Ae^(+/-iwt).
Finally, I'm told that the total solution is the sum of the homogenous solution and the particular solution.
For one thing, the homogenous equation has THREE separate solutions for different cases (namely the damped, underdamped and critically damped cases), so how can I add the homogenous solution?
But mostly I don't understand how to find the particular solutions.
I don't want someone to work this out for me, but I do need advice on how to solve it.