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?
Jonathan Goff
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does J0 ((A + i * B) *r) = J0 (A*r) + J0 (i*B*r)?
That would only be true for a linear function. So, no.
Ok, that would be true of a nonlinear function that takes the real and imaginary components and deals with them seperately, but that doesn’t include Bessel functions.
Simon,
Yeah, I figured that out shortly after posting this. If you look at the series expansion definition of the Bessel function J0(x), there’s a X^2n term in there, which means that you’d have (Ar + iBr)^2n, which is most definitely not (Ar)^2n + (iBr)^2n….
So, it looks like there is no clean and easy way, and I’m stuck with the brutal and ugly way. C’est la vie.
~Jon
What are you trying to solve?
I don’t see other thesis posts. Point them out if you have them and I’ll read on. 8)
Are you limited to disk-like crystals?
If I recall correctly (and a web search indicates that I may not be recalling correctly), the Bessel functions do satisfy an additive law much as the trigonometric and hyperbolic functions do, but it is more complex. I recommend looking at something “thick and soviet” for more details (that reference is the best one I know of and probably would have more identies and the like than anything but some research papers, but unfortunately, I don’t have my copy at hand). Bessel functions can easily accept complex arguments, so I see nothing unusual about your request.
Here is the addition formula for Bessel Functions:
http://www.math.sfu.ca/~cbm/aands/page_363.htm
–Carl Feynman
Please see below:
COLLOQUIUM
Department of Mathematics
University of Louisiana at Lafayette
Thursday, March 27, 2008 at 3:30 in MDD 208
Professor Vladimir Varlamov
Department of Mathemtics, University of Texas – Pan American,
Edinburg, Texas 78541, USA
Eigenfunction expansion method for the damped
Boussinesq equation in a disc
Abstract: Solutions of semi-linear evolution equations in bounded domains can be constructed by the method of eigenfunction expansions. In contrast to
Galerkin’s method, the projection is made onto the infinite-dimensional space
spanned by the set of eigenfunctions of the main elliptic operator. Of particular
interest is a problem of excitation of a circular elastic membrane by an incident
acoustic wave. Membrane oscillations are governed by the 2D damped Boussinesq
equation. The solution in question is represented by a series of eigenfunctions of
the Laplace operator in a disc. Proving decay of the eigenfunction expansion coefficients leads to an appearance of a new family of special functions, convolutions
of Rayleigh functions with respect to the Bessel index. Rayleigh functions appear in classical linear problems of vibrating drumheads, heat conduction in cylinders and Fraunhofer diffraction through circular apertures. They are defined as a series
l(m) =
1 Xn=1
1
2l
m, n
,
where m, n are positive zeros of the Bessel function Jm(x), m = ±1, ±2, … and
l, n = 1, 2, 3, …. Convolutions of such sums with respect to the Bessel function index
form a new family of special functions. A general representation for this family
is obtained and asymptotic expansions as |m| ! 1 are computed for practically
important cases.
—————————————————————————
Refreshments will be served in MDD 208 at 3:15 pm