The resonance frequency of a steel ball

In the general case you would need to solve the following eigenvalue problem:

$$(lambda + 2mu)nabla(nablacdotmathbf{u}) - munablatimesnablatimes mathbf{u} = -rhoomega^2 mathbf{u}, ,$$

and, as mentioned above, you could express the solution in terms of spherical harmonics.

Nevertheless, there are solution that have rotational symmetry. For those solutions

$$mathbf{u} = u_r(r) hat{mathbf{e}}_r, ,$$

and, after replacing in the equation above, we get

$$r^{2} frac{d^{2}}{d r^{2}} u_r(r) + 2 r frac{d}{d r} u_r (r) - 2 u_r(r) + r^2frac{rhoomega^{2}}{(lambda + 2mu)} u_r(r)= 0, ,$$

that has as (physically admissible) solution

$$u_r = frac{J_{3/2}left(kappa rright)}{sqrt{r}}, ,$$

with

$$kappa^2 = frac{rhoomega^2}{(lambda + 2mu)}, .$$

To find the values of $omega$ you would need to impose the traction-free boundary conditions and solve for the roots of the determinant.