Equation of motion for a coil around a mass, attached with a spring driven by magnetic field?
First of all, thanks for reading and hello! First time poster on Physics Stack Exchange. I'm an electronics engineer (so forgive my poor maths below!) and tend to hang out there.
I have a questions that's been killing me for a few days.
I need to design some electronics to control a system which I think looks like the above. It is a linear motor connected to a spring. The motor is of type "moving coil", that is, the mass has a coil wrapped around it, within a permeant magnetic field. Some electronics (that I will design) will drive AC current through the coil to make it oscillate at it's resonant frequency.
I understand how to model the electronic side (a coil in an inductor, with a series resistance and a back EMF). 780
Now, the mass spring is not too hard, there's lots of literature. But, this (I believe) is different - it has an additional 'friction' which is the back EMF generated from <PERSON>'s law (negative of rate of change of flux).
So, I tried to model this system (a coil driving a mass spring system). For me, my end goal is a transfer function, in the laplace domain, that is the input (IE, current through the coil, or the voltage across it). As a side note, this is the reason for the modelling, I'd like to see the implications, if any, about controlling this via a sinusoidal voltage (much easier in electronics to deal with) or a sinusoidal current; this can still be done but the electronics is a little more involved.
Equations of motion
So, I think I start of with the <PERSON>'s second law (assuming there's no resistance in the system, other than the back EMF):
$ \sum F_x = m \cdot a_x $
$ \sum F_x = F_{spring} + F_{BEMF} - F_{Motor} $
$ \sum F_x = -k \cdot x(t) + \frac{\delta \phi}{\delta t} - B \cdot i_{coil}(t) \cdot L $
I think the back emf term can be rewritten using the systems coord frame (x pointing right and y point up):
$ \frac{\delta \phi}{\delta t} = \frac{\delta B \cdot A}{\delta t} = B \cdot \frac{\delta ( x \cdot y)}{\delta t}$
Since the magnetic flux is constant, and so is the area of flux enclosed by the coil in the y coordinate, I took these out of the differential. That leaves us with the x coordinate. Lets put a constraint on the system whereby the coil/mass will never be displaced by more than it's full length, $l_x$. 780