How Do You Calculate Oscillation Frequency in a Nonlinear System?

[elias123]

Homework Statement

I am struggling to figure out how one would find the oscillation frequency for a nonlinear system that's experiencing a force such as F = 4x^-2 -2x.

Homework Equations

The Attempt at a Solution

Im really not sure how to approach it. Obviously you can't use methods for simple harmonic motion, and it doesn't seem like you would be able to model it as a sin wave because its not symmetric. Any tips would be greatly appreciated. Thanks.

 

[king vitamin]

I think you're supposed to approximate the system as simple harmonic close to its stable equilibrium point (otherwise this is an incredibly difficult problem). There's only one equilibrium here, x = 2^(1/3), and at that point, since the force is going from positive to negative, the particle is being pushed back to that point on either side, so it would oscillate about that point. Approximating the system as linear about that point, the force constant would just be k = -F'(2^(1/3)) and your linear force close to equilibrium is F ~ -kx. Now SHM applies and the frequency easily found.