The lumped mass method is used to establish the nonlinear vibration model of single degree of freedom gear transmission as shown in Fig. 1
Where: igtj is the moment of inertia of the jth gear, kg · m2, j = 1,2; θ gtj is the angular displacement of the jth gear, RAD; rgtj is the base circle radius of the jth gear, m; CGT is damping, n · s · M-1; EGT (T) is comprehensive transmission error, m, t is time variable; KGT (T) is time-varying meshing stiffness, n · M-1; TD is driving torque, n · m; TL is load torque, n · m; F [rgt1 θ gt1-rgt2 θ GT2 EGT (T)] is a nonlinear backlash function, m; t is time, S.
The nonlinear backlash equation f (xnon) is defined as:
Where: xnon is dimensionless generalized transmission error; B is dimensionless backlash; B and bnom are actual backlash and nominal backlash respectively.
The dimensionless time τ and the equivalent mass me: of gear pair are introduced
Where: ω n is the natural vibration frequency of gear pair; kgta is the average meshing stiffness.
The equation can be further expressed in dimensionless form
Where: λ is dimensionless damping; κ is dimensionless meshing stiffness; ω G is gear meshing frequency, Hz; φ G is gear meshing phase angle, RAD; fgae is average equivalent external load, N; FGA is external load amplitude, n.
The dimensionless form of the nonlinear vibration motion equation of single degree of freedom gear transmission is expressed as a form which is easy to be analyzed and solved, as shown in the formula.