A couple months ago, I posted about endurance, and when it is best to stay the course vs when it’s best to cut bait. One of the examples I gave was my thesis. I wasn’t sure if staying the course made sense, or if I should change topics. After a lot of deliberation, last month I determined that staying the course was the best route, and have been slogging away at the thing (which is a big part of why blogging has skimped out lately). I’m starting to get some real traction in several areas, having found some very useful papers, and making some progress on the modeling side of things.
So I have a question for any hardcore math nerds.
In my model for radial vibration of a piezoelectric crystal, I get an equation for the radial displacement as a function of the radial location (at rest), and time:
ur(r,t) = [A * J0 (k * r) + B * Y0 (k *r)] * cos (omega * t)
Where J0 and Y0 are Bessel functions of the first and second kind (0th order) respectively, k is omega/vp, omega is the driving frequency for the piezoelectric crystal, and vp is the speed of sound of the crystal in I think the thickness direction. Vp = sqrt (c11/rho), where c11 is the stiffness in the thickness direction, and rho is density. The problem is that in order to model losses, c11 is really a complex number, with the real part being analogous to the static stiffness, and the imaginary part being the “loss modulus”.
Anyhow, the problem I have is that the complex number is inside the Bessel Function, and I was having a hard time figuring out how to handle that for my model. I’ve came up with an idea, but wanted to run it past some more eyes.
Ok, so since the stiffness, c11, is complex, the velocity is also complex, so you can sub in vp = vr – i * vi, where vr is the real component of velocity, and vi is the imaginary component. So the term inside the Bessel function becomes:
(omega * r) / (vr – i * vi)
My thought was, what happens if you multiply the top and bottom by (vr + i * vi)? If I’m doing the math right, I get:
(omega * r * (vr + i * vi))/(vr^2 + vi^2)
The bottom term is just the square of the magnitude of the complex velocity, ie |vp|^2. Splitting things up a bit, and subbing in term |k| = omega/|vp| you get:
|k| * r * vr/|vp| + i * |k| * r * vi/|vp|
Basically, the left hand side looks just like the “k * r” term in the original formula multiplied by a term that is just the magnitude of the real over the magnitude of the complex velocities. The right hand side is just the imaginary version of the same. Now, |k|, vr, vi, and |vp| are all just constants based on the density, the static stiffness, the quality factor of the material, and the driving frequency of the system.
So, the Bessel functions appear to be in a form like this:
J0 ((A + i * B) *r)
The two big questions I have (other than have I made any obvious math errors so far?) are:
1. Are Bessel Functions distributive?
Ie, does J0 ((A + i * B) *r) = J0 (A*r) + J0 (i*B*r)?
2. Since the Modified Bessel Function (I), is represented as In(x)=(-1)^n * Jn(ix), can I then rewrite this all as: J0((A+i*B) * r) = J0 (A*r) + I0(B*r)?
If both of those are true, I may finally have figured this out. Can anyone shed any light on that?
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