We report a detailed experimental study of the flow of isotopically-pure He II, generated by a vibrating grid. Our measurements span a wide range of temperatures (50 mK < T < 1.37 K) and pressures (2 bar < p < 15 bar). The response of the grid was found to be of a Lorentzian form up to a sharply-defined threshold value. This threshold value does not change appreciably with pressure; the form of the resonant response of the grid is qualitatively the same for all temperatures while the threshold value is a monotonically increasing function of temperature. We discuss the measured variation of the resonant frequency of the grid as a function of applied pressure (density) of He II and relate this to a hydrodynamic effective mass of the grid. These measurements extend our previously reported studies [Nichol et al, Phys. Rev. E 70, 056307 (2004)] and form an integral part of a series of experiments aimed at providing a better understanding of classical and quantum turbulence.