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We have discovered that the dc mass current through a superfluid ³He weak link is substantially increased when the Josephson frequency matches the resonant frequency of a coupled mechanical oscillator. The phenomenon is the result of homodyne mixing between the Josephson oscillations and the oscillating pressure field associated with the resonant system. The measured sizes of the current enhancements are in excellent agreement with calculations based on this homodyne model. Similar observations in superconducting junctions, in which microwave radiation changes the dc electronic current, were used for the first confirmation of the dynamics of the superconducting Josephson effect.