Using vibrating wire in non-linear regime as a thermometer in superfluid $^3$He-B

Vibrating wires are common temperature probes in $^3$He experiments. By measuring mechanical resonance of a wire driven by AC current in magnetic field one can directly obtain temperature-dependent viscous damping. This is easy to do in a linear regime where wire velocity is small enough and damping force is proportional to velocity. At lowest temperatures in superfluid $^3$He-B a strong non-linear damping appears and linear regime shrinks to a very small velocity range. Expanding measurements to the non-linear area can significantly improve sensitivity. In this note I describe some technical details useful for analyzing such temperature measurements.