You will find a good application of Bessel functions in transient heat transfer by conduction in cylindrical coordinates, I have put here a link to that. Good luck!

http://www.ewp.rpi.edu/hartford/~ernesto/S2006/CHT/Notes/ch03.pdf

Any problem in circular cylindrical coordinates:

i'm sure you can come up with more!

Bessel's equation has frequent usage in cylindrical shapes. Heat conduction in cylindirical material, electromagnetic waves in a cylindrical wave guide is some areas of application where Bessel's function is invoked for solution of the problems.

Check out the closed form solution of a vibrating membrane under Dirichlet boundary conditions. (drum)

Just look at these

1. Vibration of a circular membrane( Leonard Meirovitch, Analytical methods in vibration, Vibrations of Continous systems, Membrane).

2. Transverse vibration of a beam which it's cross section is a complete trianlgle ( I attach a file for this).

3. Heat conduction in cylinder. (in fact almost any flux in a circular surface will lead to a bessel solution).

Good Luck!

Bessel functions are usually found as the solutions to the Bessel equations. These come up in various problems (electromagnetism, vibrations of membranes and plates, heat conduction) with cylindrical or spherical symmetry, when the original Partial Differential Equations are solved using separation of variables. For an application in Mechanical Engineering, look up the modal decomposition of a circular plate. You'll find a lot of Bessel functions there!

Boundry value problems

You can refer Analytical methods in conduction heat transfer by Glen E Myers. However it contains only heat transfer related problems. But It's a classic.

It appears in analytical solution of transient viscous flow in a pipe. You can check 

for example the book: "Viscous Flow", written by: Frank White.

When modelling a circular plate for any mechanical system using  polar coordinates, you might end up in Bessel functions when you analytically solve the differential equation.  This is true for modelling a circular cylindrical vibrating membrane as well.

here is a link that would help

http://www.crcpress.com/product/isbn/9780415281300

Heat conduction is one of the most important application of Bessel functions

- Solution of alternating flow of fluids in a cylindrical pipe is based on Bessel functions

from p50 ...:

https://www.dropbox.com/s/r1xyhr95vv2zfej/HDR%20Jean-Luc%20Dion.pdf?dl=0

- Spectrum of Frequency modulation (rotating machine) is built with Bessel functions.

http://en.wikipedia.org/wiki/Frequency_modulation

One of the most interesting applications of Bessel functions is to solve the 2-D modeling of rounded-shape piezoelectric wafer active sensors (polar coordinate) in structural health monitoring. See "Structural Health Monitoring with piezoelectric wafer active sensors", Victor Giurgiutiu, Academic Press Publication (imprint of Elsevier), 2008

Bessel functions are widely used in Analytical solving of partial differential equations. PDE's in polar coordinate need Bessel's functions to be solved. Therefore, I propose you to identify the famous second order PDE's which are used in modeling of mechanical phenomena like heat transfer, diffusion, traffic equation and etc in polar systems. I also suggest you to have a look at Chapter 7 of  "Applied Differential Equations with Fourier Series and Boundary Value Problems" by Proff. Richard Hibermann, Bessel's function and its usage is explained in this book.