The structure of double-side meshing nutation drive is shown in Fig. 1. It is mainly composed of four double circular arc spiral bevel gears, input horizontal axis and nutation eccentric sleeve. The input horizontal axis and nutation eccentric sleeve are fixed together and the angle θ between the central axis is nutation angle. When the mechanism moves, the motor drives the input horizontal axis and the nutation eccentric sleeve to rotate, thereby driving the internal bevel gears 1 and 3 which are fixed together to perform the nutating motion. The nutation internal bevel gears 1 and 3 are respectively meshed with the fixed external bevel gear 2 and output external bevel gear 4 on the left and right sides, and the power is output by the external bevel gear 4. The double-sided meshing nutation drive can be regarded as a planetary gear transmission with less tooth difference.

In the dynamic modeling of double-sided meshing nutation drive, the system model is simplified to some extent, and the following assumptions are made: (1) The friction of tooth surface during gear meshing is not considered; (2) The system damping is regarded as general viscous damping; (3) The influence of the error between axes on the system is neglected; (4) The deformation of shaft and bearings is not considered, and the bearing clearance is neglected; (5) When nutation gear 1 and nutation gear 3 are fixed together, they are regarded as a whole, gear 1 and gear 3 have the same degree of freedom, the vibration displacement of the two gears is the same, and there is no mutual torsion between the two gears; (6) The torsional pendulum vibration has less influence on the system, ignoring the torsional pendulum vibration.

Based on the above assumptions, the bending-torsion-axis coupling analysis model of the double-sided meshing nutation drive system is established by using the lumped parameter method, as shown in Fig. 2. The gears are regarded as lumped mass and lumped inertia, the supporting shaft is regarded as a rigid body without mass, and the elastic support of the bearing is simulated by spring and damper.

In Fig. 2, O–xyz is a space meshing coordinate system, O–x0y0z0 is a follow-up coordinate system which rotates with nutation gears 1 and 3. The origins of the two coordinate systems coincide with the conical points of each bevel gear. T1 and T4 are input torque and load torque respectively. xi, yi and zi are the translational vibration displacements of the gears in the x, y and z directions respectively. The subscript i=13, 2, 4 respectively represent the nutation gears 1 and 3, fixed external bevel gear 2 and output external bevel gear 4. θi is the angular vibration displacement of each gear in the z direction. kik and Cik (i=13, 2, 4; k=x, y, z) are the support stiffness and support damping of each gear in three coordinate directions respectively. kmj, Cmj and enj(t) are the meshing stiffness, meshing damping and static transmission error of each gear pair. The subscript j=1, 2 respectively denote the gear pair 1 which is meshed by the fixed bevel gear 2 and the nutation bevel gear 1, and the gear pair 2 which is meshed by the output bevel gear 4 and the nutation bevel gear 3.

The generalized displacement array of the double-sided nutation drive system can be expressed as: 1X=x2,y2,z2,θ2,x13,y13,z13,θ13,x4,y4,z4,θ4T

The relative displacements λn1 and λn2 along the normal direction of the meshing point due to vibration and error between the meshing points of the two bevel gears in the gear pairs 1 and 2 are represented as: 2λn1=xn1-xn2-en1(t)=a11x13+a12y13+a13z13+a12θ13rm1-(a21x2+a22y2+a23z2+a22θ2rm2)-en1(t) 3λn2=xn3-xn4-en2(t)=a31x13+a32y13+a33z13+a32θ13rm3-(a41x4+a42y4+a43z4+a42θ4rm4)-en2(t) In Eqs. (2) and (3), 4a11=-cos⁡θ(cos⁡δ1sin⁡αn+sin⁡δ1sin⁡βcos⁡αn)-sin⁡θ(cos⁡δ1sin⁡δ1cos⁡αn-sin⁡δ1sin⁡αn)a12=a22=a32=a42=cos⁡βcos⁡αna13=sin⁡θ(cos⁡δ1sin⁡αn+sin⁡δ1sin⁡βcos⁡αn)-cos⁡θ(cos⁡δ1sin⁡δ1cos⁡αn-sin⁡δ1sin⁡αn)a21=-(cos⁡δ2sin⁡αn+sin⁡δ2sin⁡βcos⁡αn)a23=sin⁡δ2sin⁡αn-cos⁡δ2sin⁡βcos⁡αna31=-cos⁡θ(cos⁡δ3sin⁡αn+sin⁡δ3sin⁡βcos⁡αn)-sin⁡θ(cos⁡δ3sin⁡δ3cos⁡αn-sin⁡δ3sin⁡αn)a33=sin⁡θ(cos⁡δ3sin⁡αn+sin⁡δ3sin⁡βcos⁡αn)-cos⁡θ(cos⁡δ3sin⁡δ3cos⁡αn-sin⁡δ3sin⁡αn)a41=-(cos⁡δ4sin⁡αn+sin⁡δ4sin⁡βcos⁡αn)a43=sin⁡δ4sin⁡αn-cos⁡δ4sin⁡βcos⁡αn, where, δi (i=1, 2, 3, 4) is the pitch cone angle of each bevel gear, β is the helical angle, αn is the normal pressure angle, rmi (i=1, 2, 3, 4) is the radius at the meshing point of each gear.

The normal dynamic load of gear pair in meshing can be expressed as, 5Fnj=kmj(t)fλnj+Cmjλ˙nj, where, j=1, 2 represent the gear pairs 1 and 2, kmj(t) represent time-varying meshing stiffness of gear pairs, f(λnj) represent clearance function.

The clearance function is as follows: 6f(λnj)=λnj-bj/2λnj>bj/20λnj≤bjλnj+bj/2λnj<-bj/2, where, bj (j=1, 2) is the tooth backlash of each gear pair.

Considering the time-varying meshing stiffness, transmission error, meshing damping and tooth backlash, the dynamic differential equations of double-sided meshing nutation drive system are obtained as, 7m2x¨2+C2xx˙2+k2xx2+a21Fn1=0m2y¨2+C2yy˙2+k2yy2+a22Fn1=0m2z¨2+C2zz˙2+k2zz2+a23Fn1=0J2θ¨2+a22Fn1rm2=T1m13x¨13+2C13xx˙13+2k13xx13-a21Fn1+a41Fn2=0m13y¨13+2C13yy˙13+2k13yy13-a22Fn1+a42Fn2=L1/rm1+L3/rm3m13z¨13+2C13zz˙13+2k13zz13-a23Fn1+a43Fn2=0J13θ¨13-a22Fn1rm1+a42Fn2rm3=0m4x¨4+C4xx˙4+k4xx4-a41Fn2=0m4y¨4+C4yy˙4+k4yy4-a42Fn2=0m4z¨4+C4zz˙4+k4zz4-a43Fn2=0J4θ¨4-a42Fn2rm4=-T4, where, mi and Ji (i=13, 2, 4) are the lumped mass and moment of inertia of each gear; L1 and L3 are the gyroscopic moments generated by nutation motion of two nutation bevel gears, and the direction is along the axis O–y.

In the meshing process of double circular arc spiral bevel gears, meshing stiffness varies with the number of meshing points of convex and concave teeth, and the meshing stiffness has time-varying. By using the finite element contact analysis, the contact force and deformation of the gear teeth in the meshing period are obtained. Then the meshing stiffness values of gear pairs at different times are calculated, and these discrete points are connected by straight lines, as shown in Fig. 3.

In order to transform the time-varying meshing stiffness broken line into a continuous function curve with time as independent variable, the time-varying meshing stiffness kmj(t) is fitted into Fourier progression form. 8kmj(t)=kmj+∑r=1lkrjcos⁡(rωmjt+φrj), where, kmj is the average meshing stiffness of the gear pair, r (r=1, 2, …, l) is the order of Fourier expansion harmonic term, krj is the stiffness amplitude of the r order harmonic component of the gear pair, ωmj is the meshing angular frequency, φrj is the meshing phase of the r order harmonic component.

To simplify the calculation, the curve of time-varying meshing stiffness is obtained by fitting the 7th order Fourier progression as shown by the red dotted line in Fig. 3. And the average meshing stiffness of the gear pair 1 is calculated as km1=2.1168×108 N m-1, and the average meshing stiffness of the gear pair 2 is km2=2.2229×108 N m-1.

Theoretical modal analysis solves the inherent characteristics of the system, i.e. the natural frequency and the modal shape. Therefore, the influence of external loads can be neglected, and the damping term can also be neglected because the influence of damping is small. Thus, the free equation of motion of double-sided meshing nutation drive can be established. 9MX¨+KX=0, where, the mass matrix M=diag (m2, m2, m2, J2, m13, m13, m13, J13, m4, m4, m4, J4), the stiffness matrix K is a 12 by 12 matrix, which is not listed in the text due to limited space.

The intrinsic characteristics of the nutation drive system are solved in the MATLAB software, and the eig command is used to solve the problem. In solving the free equation of motion of nutation drive system, the average meshing stiffness of two pairs of gears is used to calculate the natural frequencies and modal shapes of nutation drive system. The parameters of bevel gears of nutation drive solved in this paper are shown in Table 1. The values of support stiffness and support damping of nutation drive system are shown in Table 2.

The natural frequencies of the double-sided meshing nutation drive are calculated as shown in Table 3, and the corresponding modal shapes are shown in Fig. 4. The degrees of freedom 1 to 4 are the degrees of freedom of the fixed external bevel gear 2, the degrees of freedom of 5 to 8 are the degrees of freedom of the nutation bevel gears 1 and 3, and the degrees of freedom of 9 to 12 are the degrees of freedom of the output external bevel gear 4. The natural frequency of the first 5th order of the nutation drive system is small, the natural frequency after the sixth stage is large, and the natural frequencies span of the system is also large. In general, the natural frequencies of the first to tenth order have a greater impact on the system. There is a natural frequency double root solution in the system, that is, the natural frequency values of the 4th and 5th order both are 7880 Hz.

In order to analyze the influence of meshing stiffness on modal, the natural frequencies of nutation drive system are calculated by taking the average meshing stiffness of two pairs of gears as 150 %, 200 % and 250 % of the original average meshing stiffness kmj (j=1, 2). The natural frequencies of nutation system with different average meshing stiffness are shown in Fig. 5.

Figure 5 shows that the average meshing stiffness has a great influence on the high order natural frequencies, and mainly affects the natural frequencies of the 10th, 11th and 12th order. The average meshing stiffness has little effect on the lower order, only the natural frequencies of the 5th and 7th order increase slightly. The lower order of natural frequencies have great influence on the system, so the change of average meshing stiffness has little influence on the nutation drive system.

Considering that there are two nutation gears in nutation drive system to make the nutation movement, the natural frequencies of nutation drive system are calculated by taking the bearing support stiffness of nutation gears as 150 %, 200 % and 250 % of the original support stiffness. The comparison of natural frequencies of nutation system under different bearing supporting stiffness of nutation gears is shown in Fig. 6.

It can be seen from Fig. 6 that the bearing support stiffness of nutation gears has a great influence on the natural frequency of intermediate order, especially on the sixth and seventh order. However, the bearing support stiffness of the nutation gears has little influence on the low-order and high-order, and the natural frequencies of the 10th to 12th order remain unchanged with different bearing supporting stiffness of nutation gears.