Relation between stress and resonance

For an elastic rod under uniaxial stress is valid the relation for the longitudinal vibration:

$$frac{partial sigma_{xx}}{partial x} = rho a_x = rho frac{partial^2 u_x}{partial t^2}$$ where $rho$ is the density and $u_x(x,y,z)$ is the displacement at the point to the $x$ direction.

As $sigma_{xx} = Eepsilon_{xx} = Efrac{partial u_x}{partial x}$, we have a wave equation:

$$frac{partial^2 u_x}{partial x^2} = frac{rho}{E} frac{partial^2 u_x}{partial t^2}$$

with a solution of the type: $$u_x = Acosleft(kx - ksqrt{frac{E}{rho}}tright)$$

$$sigma_{xx} = -kEAsinleft(kx - ksqrt{frac{E}{rho}}t right)$$

$omega = ksqrt{frac{E}{rho}}$, so the stress is linearly proportional to the frequency, for the same material and geometry.

But it is only valid for an uniaxial case (and longitudinal vibration). In a generic 3D situation, and where the stress is not constant (a shrinking fit assembly for example) it may be hard to derive a relation frequency x stress state.