I'm trying to model a Euler-Bernoulli beam to gather the total angular torque it will provide on a hub on which it is anchored. The beam is a cantilever, and I'm using the standard deflection equations which represent behavior to an applied force on the tip.
You can solve for the force needed to get a specific deflection by using the delta max function with a known displacement. This, however, isn't my question.
The Euler-Bernoulli equation to show the shape of this beam (equation shown below) only takes into account the change along the y-axis. This is because the equation is made for rectangular coordinates. What I want is to also show the change in the tip position relative to the mounting location (it should stay roughly the same, and eventually get shorter with large deflections). I should add that I need to do this for the entire equation, not just the tip. The end goal is to model the torque provided by the beam moving under its own power (piezoelectric) so I need to model the relative positions of all infinitely small positions along the beam.
The current equation only changes y, and thus makes the beam longer to account for the deflection. I've included a screenshot of a plot to illustrate this:
Is anyone aware of a modified form of this equation that will account for this? I want to say it's as easy as changing from rectangular to polar coordinates, but I don't think so. Since the beam actually develops an arc shape, this would change depending on where the position being observed on the beam correct?
Any input would be greatly appreciated!