The structure and working principle of a two-cylinder four-stroke single-piston hydraulic free piston engine(HFPE) were introduced. The basic vibration equation of free piston assembly(FPA) was established based upon the energy conversion between the injected fuel and the friction together with the load. Both the theoretical and numerical results show that the vibration system of FPA is a nonlinear conservative autonomous system in one cycle. The FPA vibration is symmetric with constant amplitude when FPA is only driven by the compression pressure in the compression accumulator and that in the combustion chamber. When considering the friction and load, FPA could still achieve a stable vibration after a few cycles' adjustment whether the input energy is equal to the consumed energy or not. The vibration characteristics are different when FPA vibrates in the compression stroke and the expansion stroke, which is the unique feature of the single-piston HFPE.
