Natural Frequency of Solid Spheres

[YMU]

Dear Physics Forum community,

I am posting here as a last resort, so any guidance/references would be much appreciated.

As a small part of my project, I need to calculate the natural frequency of metallic solid spheres. All I have been able to find on the web is the Schummann Resonance, which may or may not be very accurate in my case. I've also read somewhere that it can only be solvable through the application of FEM models with 2nd order differential equation boundary conditions.

ANY input is much appreciated.

Thank you

 

[Mech_Engineer]

It seems to me a simple unconstrained modal analysis on the sphere in any standard FEA package would do the trick, no exotic boundary conditions needed.

What kind of modes are you hoping to solve for?

 

[AlephZero]

Google gave this as the first hit:
Horace Lamb, "On the Vibrations of an Elastic Sphere ", Proc. London Math. Soc. (1881) s1-13(1): 189-212

Lamb wrote textbooks on the theory of elasticity, dynamics, and acoustics that are sitill in print. IIRC the radial vibration of a sphere is of some interest in acoustics.

Just thinking about the equations of motion in spherical coordiates, there will be many diferent families of vibration modes. You probably need to be more specific about what you really want to know.

 

[Studiot]

For a uniform elastic sphere, radius a and density ρ vibrating radially under no external forces the radial displacement U satisfies


$$\left( {\lambda + 2\mu } \right)\left( {\frac{{{\partial ^2}U}}{{\partial {r^2}}} + \frac{2}{r}\frac{{\partial U}}{{\partial r}} - \frac{{2U}}{{{r^2}}}} \right) = \rho \frac{{{\partial ^2}U}}{{\partial {t^2}}}$$

Where λ & $\mu$ are elastic constants

The radial stress is


$${\sigma _r} = \left( {\lambda + 2\mu } \right)\frac{{\partial U}}{{\partial r}} + 2\lambda \frac{U}{r}$$

The periods of the normal modes of vibration are given by


$$\frac{{2\pi a}}{{{c_1}\xi }}$$

Where



$$\begin{array}{l}<br /> c_1^2 = \frac{{\left( {\lambda + 2\mu } \right)}}{\rho } \\ <br /> \xi = positive\;roots\;of\quad 4\xi \cot \xi = 4 - {\beta ^2}{\xi ^2} \\ <br /> {\beta ^2} = \frac{{\left( {\lambda + 2\mu } \right)}}{\mu } \\ <br /> \end{array}$$