Methods and systems for a mechanically driven pump are provided. The pump is for lubrication and comprises: a cam lobe or an eccentric mounted on and driven by a shaft; a piston being in contact with the cam lobe or the eccentric; a spring acting on the piston; an inlet check valve positioned in an inlet port; and an outlet check valve positioned in an outlet port, where the inlet check valve and the outlet check valve do not comprise preload springs and are installed vertically in a normally closed position due to a force of gravity acting on internal balls of the check valves. Orifices fluidly coupled to the inlet check valve and the outlet check valve are position outside the piston stroke to prevent the piston from acting as a guillotine valve, which may increase volumetric efficiency of the mechanically driven pump.