Nanomechanical resonators have recently been studied as high-sensitivity probes of fluid dynamics in superfluid helium. When turbulence is introduced to such a system, quantum vortices may form and become trapped by the resonator, either completely surrounding the oscillating beam (a fully trapped vortex) or surrounding only part of it (a partially trapped vortex). Fully trapped vortices have been studied using the model of a linear harmonic oscillator behaviour with great success. However, in the presence of a partially trapped vortex, the oscillators have been found to behave non-linearly. We analysed the non-linear response of a doubly clamped nanobeam resonator submerged in helium-4 at 10 mK in the presence of a partially trapped vortex. Our analysis demonstrates that the observed non-linearities are caused by the dynamics of the vortex line. N. M. acknowledges the financial support of the US-UK Fulbright Program.