Mechanisms for the nonreciprocal transmission of acoustic energy and the construction of non-reciprocal vibro-acoustic systems have been subjects of intense research in recent years. An experimental study of acoustic nonreciprocal transmission in an experimental system with a nonlinear membrane and two acoustic cavities of different sizes is reported. The membrane can be simplified into a Duffing oscillator, and the asymmetry of the frequency response function of this oscillator is used to realize the non-reciprocal transmission of acoustic energy. The asymptotic solution of the frequency response function of the nonlinear membrane is obtained by the complexification-averaging method. The theoretical simulation results accord well with the experimental results. The results show that the experimental system has a maximum non-reciprocal quantity of 9.1 times in theory, 4.3 times in the experiment, and the normalized frequency bandwidth of the jump phenomenon region is up to 0.56. The research results reveal the mechanism of non-reciprocal transfer of acoustic energy in the experimental system and demonstrate a new way to realize the asymmetric transfer of acoustic energy in an acoustic system with an air medium.