The angle between the horizontal line and the mesh line is proposed to be an important reference angle. The solution method of the relative displacement along mesh line is summarized

while the reference angle locates in different quadrant. The dynamic mesh force of multi-mesh gear system can be gained conveniently according to this method. The dynamic model of complicated multi-stage gear-rotor-bearing system is derived

the effects of time-varying mesh stiffness

static transmission error

unbalance

bearing stiffness and flexible shaft are considered. The numerical integration is adopted to solve the dynamic response of the system. The results include rotational frequency

mesh frequencies and their sidebands

combinational frequencies and their sidebands. It can provide important basis for the vibration analysis and failure diagnosis of multi-stage gear rotor system used in engineering practice.