By using multi⁃scales method

the first primary resonance of the human arm and vibration mechanical coupling system under random narrow⁃band excitation was analyzed

and the two⁃degree⁃of⁃freedom nonlinear dynamic equation of the arm system was established.The first primary resonance amplitude frequency response equation and Ito stochastic differential equation of the system were solved in turn.In order to study the distribution law of variables under random vibration

the approximate expressions of the expectation and mean square value of the first primary resonance random mean square response were calculated using the moment estimation method.The influence of the parameters of the human arm system on the primary resonance was obtained through numerical analysis.The results show that the mean square response curves of the first primary resonance have the same topological structure as their amplitude⁃frequency response curves

and the mean square value of the primary resonance amplitudes and the resonance region increase with the increase of the amplitude of the external excitation

while the increase of the system damping will reduce the mean square value and the resonance region.The change of nonlinear stiffness coefficient has little effect on the system.