[putreyem] Harmonic motion on a nonlinear spring

Given a mass attached to a horizontal spring attached to a vertical moveable wall, how should the wall be moved to induce harmonic motion (sine wave) in the mass on the other end of the spring?  This is easy if the spring is linear, but what if there are nonnegligible higher order spring constant terms?  Or nonlinear damping?

Inspired by partner dancing, on the assumption that harmonic motion is aesthetically pleasing to achieve regardless of the underlying system.