Design and Analysis of a Novel Single-input Multi-output Spatial Petal Tooth Nutation Drive

Single-input multi-output spatial petal gear is a new type of transmission, which combines the ancient lotus petal structure with the nutation drive principle, has many advantages such as small size, light weight, low noise, high bearing capacity, long life, high reliability, etc. and a broad application prospect, especially suitable for the reducer with high bearing capacity. The purpose of this paper is to clarify the basic structure and transmission principle of spatial petal gear, derive the tooth surface equation and meshing equation, analyze the boundary dimensions according to the meshing characteristics of petal gear. The basic parameters of petal gear are designed and calculated by using Matlab software, the 3D model is established by using Creo software, and the kinematics simulation is analyzed to verify the correctness of the innovative structure. The strength analysis are carried out on the tooth surface by finite element method, where the maximum stress occurs to verify the strength requirements of the gear and the rationality of structural design.


Introduction
Gear is the most common transmission component in modern mechanical transmission, with high transmission efficiency and accurate transmission ratio.Gear transmission is the most common transmission mode, which can transmit the rotation between parallel, intersecting and staggered shafts through the meshing between gears.With the development of modern science and technology, the traditional gear is no longer enough to meet the requirements of some special transmissions, such as the studied its contact condition.Zhao et al. (2008aZhao et al. ( , 2008b) made a detailed study on the meshing and other aspects of face gear, put forward some methods for gear modification through analyzing and sorting out data, designed and analyzed the geometric transmission error of face gear for the edge contact problem, and improved the continuity and stability of face gear.Zhu et al. (1999Zhu et al. ( , 2010) ) built a certain analysis system based on the theoretical analysis, mainly including the tooth surface formation method of face gear and the definition of maximum and minimum internal and external radius coefficients, studied the curvature and speed of face gear transmission, obtained several curvature distribution laws, and carried out the stress analysis and modal research of the face gear using finite element tools.Xu et al. (2010) carried out geometric 3D modeling and finite element analysis of the face gear, and the results recorded that the stress changes generated by its tooth surface when it contacted.He et al. (2017) deduced the tooth surface equation of face gear, and developed a face gear combination with more teeth, large modulus and large pressure angle by studying the relationship between the geometry of face gear and transmission factors.Spatial petal nutation face gear is a new type of spatial gear transmission developed on the basis of "face-face" gear pair and in combination with the principle of nutation gear transmission, which has many advantages, such as small size, light weight, low noise, high bearing capacity, long service life and high reliability.Because of the boundary conditions of the face gear tooth surface, the face gear tooth length (or tooth thickness) cannot be arbitrarily increased like the involute spur gear, so that the face gear transmission cannot provide greater bearing capacity in a limited space (Yang et al. 2020).The petal gear can have a multi-layer structure, which can make up for the lack of bearing capacity of the face gear, and has great research significance.
The purpose of this paper is to clarify the basic structure and transmission principle of spatial petal gear, derive the tooth surface equation and meshing equation, analyze the boundary dimensions according to the meshing characteristics of petal gear.The basic parameters of petal gear are designed and calculated by using Matlab software, the three-dimensional model is established by using Creo software, its mesh is refined by Hypermesh, and the kinematics simulation is analyzed to verify the correctness of the innovative structure.Finite element simulation is carried out for the transmission of petal gear, and the Mises stress distribution nephogram and overload analysis are obtained.According to the failure form of the gear, strength analysis is carried out on the tooth surface where the maximum stress occurs to verify the strength requirements of the gear.

Configuration analysis of petal nutation face gear
In nature, there are several important rules for the arrangement of lotus petals (shown in Fig. 1), which are multi-layer arrangement, each layer should be staggered at an angle, and that the closer the petals are to the core, the smaller the size is.Referring to the lotus arrangement rule, when the tool modulus changes, the gear pitch cone angle, the shaft angle and the tool pitch cone angle remain unchanged, only the meshing position changes, which provides a theoretical basis for petal face gear transmission.The forming principle of petal nutation face gear drive is shown in Fig. 2, in which the teeth numbers of tool 1 and tool 2 are the same, but the moduli are different.When the moduli of tool 1 and tool 2 change, the meshing position of the formed tool and the internal or external petal gear respectively changes, forming multiple layers of teeth, just like the lotus petal arrangement, so are tools 3 and 4. The petal nutation face gears are shown in Fig. 3. Petal teeth make full use of the end space of nutation face gear transmission, and increase the number of meshing teeth without increasing the space.Theoretically, it has greater bearing capacity, lower noise or smaller volume under the same working condition.The number of layers can be 2 or more.There is an angle difference between each layer, which can avoid the phenomenon of "in and out at the same time" when engaging.In theory, it has better vibration characteristics.
The basic structure of single-input multi-output petal gear transmission is shown in Fig. 4, in which the fixed petal gear disc 3 and the fixed sides of the external planetary disc 4 and the inner planetary disc 5 respectively form two face-to-face gear pairs, and the number of pairs can exceed two pairs as required; the external planetary disc 4, the rotating side of the inner planetary disc 5, the external rotary disc 13 and the external output shaft 16 form two face-to-face gear pairs, similarly, more than two pairs can be formed (Wang et al. 2021).The fixed petal gear disc 3, the external planetary disc 4, the internal planetary disc 5, the external rotary disc 13 and the external output shaft 16 can be made into a whole, or they can be made and assembled separately.The mating face-to-face gear pair involved in meshing is formed by internal and external cutting with the same forming cutter, the mating face-to-face gear pair of fixed petal gear disc 5, external planetary disc 4, internal planetary disc 5, external rotary disc 13, and external output shaft 16 can be either externally or internally cut.( ) where bs r is the base circle radius of the tool, 0 s  is the angle from the symmetrical line of the tool to the starting point of the involute, s  is the angle of any point on the involute of the tool, s u is the axial parameter of any point on the tool tooth surface, s x , s y and s z are the coordinates of any point on the tool on axes x , y and z respectively.Therefore where s  is the tool pressure angle, s Z is the number of cutter teeth and Relationship between coordinate systems According to Eq. ( 1), the unit normal vector s n  of the tool tooth surface is: As shown in Fig. 6, ( ) is a fixed coordinate system rigidly connected with the internal petal gear, ( ) , , , 0 is a fixed coordinate system rigidly connected with the tool, ( ) is a moving coordinate system rigidly connected with the internal petal gear, and ( ) is a moving coordinate system rigidly connected with the tool.The coordinate origins of the four coordinate systems mentioned above coincide, axis 20 z coincides with axis 2 z , and axis z coincides with axis S z .The included angle between axis z and axis 20 z is 2  , 2  is the instantaneous angle of the internal petal gear, S  is the instantaneous rotation angle of the tool, and  is the nutation angle.The spiral direction of the right hand is taken as the positive direction (Li 2020).So the tooth surface equation ( ) The tooth surface unit normal vector ( ) of the internal petal gear is: For a point P on the tool tooth surface, let its vector diameter s r  in coordinate system s S be: The speed 2 v of point P moving with coordinate system 2 S is: Then the relative speed of the contact between the tool and the tooth surface of the internal petal gear is: The tooth number ratio s q 2 of the cutter and the internal petal gear is: Through comprehensive arrangement, the following eqution can be obtained: According to the gear meshing principle, the meshing conditions of two gear tooth surfaces are: After sorting out, the tooth surface meshing equation between the imaginary tool and the internal petal gear is obtained that: )

Analysis of tooth surface limit size of petal gear
In the process of machining the internal petal gear, the phenomenon of root undercut and tooth groove sharpening will occur.In order to make the internal petal gear machining smooth, it is necessary to limit the design of its axial parameters.As shown in Fig. 7, After sorting out Eq. ( 15), the following equation can be gotten: Substituting Eqs. ( 12) and ( 14) into Eq.( 16), and then the following equation can be obtained:

zS z2
When the tooth groove of the internal face gear becomes sharp, the following relationships are obtained (Li 2020): ( ) where . Therefore, according to the basic parameters of the imaginary tool, the value of t  can be calculated, substituting t  into Eq. ( 19 and the value of s u can be obtained, which s u is   s u here.At the same time, as shown in Fig. 7, there is the following relationship when the tooth slot becomes sharp: ( ) where ps r is the pitch radius of the tool and g a is the tooth addendum height of the tool.According to the parameters obtained above, the maximum value From the above analysis, it can be seen that the external petal gear and the internal petal gear are cut by the same tool, so the design of inter-axle parameter s u needs to meet: The follow equation can be seen from the equation that the cutter meshes with the external and internal petal gears respectively:

Simulation and verification of petal gear transmission mechanism
Refer to Table 1 for design parameters and the 3D model can be established as shown in Fig. 8.
are set to describe the angular velocity of each component in Fig. 8, and 1  is the angular velocity of input shaft.The angular velocity of each component can be calculated by the following equations.According to the transformation mechanism method, the angular velocity 4  of the inner planetary petal gear can be calculated by the following equation: Similarly, the angular velocity 5  of the external planetary petal gear can be calculated by the following equation: The angular velocity 6  of the inner rotating petal gear can be directly calculated by the transmission ratio 16 i as: Similarly, the angular velocity 7  of the external rotating petal gear can be directly calculated by the transmission ratio 17 i as:   and external rotating petal gear angular velocity 7  is extracted, and the processed results are compared with the theoretical angular velocity as shown in Table 2.It can be seen that the simulation values are basically consistent with the theoretical calculation results, and the transmission error is also small.The simulation curve is shown in Fig. 10 that the inner and outer planetary petal gears generate different angular velocities during the movement of the mechanism, and transmit them to the inner and outer rotational gears respectively, finally achieving the effect of the same angular velocity input and different angular velocity output, which verifies the rationality of the transmission mechanism.

Finite element analysis of tooth surface contact stress
In order to further study the meshing characteristics of the petal teeth and the overload capacity of the tooth surface, and check the structural strength of the reducer gear, it is necessary to carry out the finite element simulation analysis of the tooth surface contact stress of the nutation petal gear.
The gear material on the prototype is 42CrMo, and its basic material properties are shown in Table 3.
Taking the meshing of fixed petal gear as an example (as shown in Fig. 11), the three-dimensional model of tooth surface is imported into ANSA for meshing.In order to obtain the ideal contact simulation results, the mesh of tooth surface needs to be fully refined, but at the same time, the computational efficiency of the computer should be taken into consideration, and part of the gears should be cut for meshing.The grid size is set to 0.4mm, the grid type is C3D8R, and there are 187524 grids in total.The mesh is imported into ABAQUS in the format of "inp", and the material properties are set according to Table 3.The general static analysis step and the face-to-face contact method are adopted, and the calculation can be carried out by setting different load conditions.
Apply torque 2031N.m to the fixed petal gear and constrain the planetary petal gear, the contact pressure on the fixed side of petal gear is shown in Fig. 12.At the same time, the contact stress on the fixed side of petal gear is shown in Fig. 13.

Results & Discussion
According to the failure form of the petal gear, the strength of the tooth surface where the maximum stress occurs has been analyzed to verify the strength requirements of the drive.At the same time, in order to study the overload capacity of the petal gear, increase the output torque to 6000 N.m, and the contact stress of the fixed side gear obtained at this time is shown in Fig. 16.It can be seen that the number of teeth engaged increases to 10, the maximum stress of the fixed petal gear is 803.7 MPa, and the rest is between 460 and 660 MPa.The maximum stress of planetary petal gear is 615 MPa, and the rest is between 360 and 500 MPa.With the increase of torque, the stress does not increase sharply because when the torque increases, more teeth participate in the meshing, and the tooth surfaces with small distance also participate in the meshing due to deformation, which is suitable for some heavy-load applications.

1Fig. 4 AFig. 5
Fig. 4 A single-input multi-output spatial petal tooth nutation drive the transformation matrix from coordinate system s S to coordinate system 2 S , the 3×3 submatrix of the s M 2 .

γ m 2 Fig. 7
Fig. 7 Boundary dimensions of internal petal gear directions x and z, respectively,

u
are the boundary dimensions of the external petal gear, are the boundary dimensions of the external petal gear, are the boundary dimensions of the internal petal gear.
Fig. 8 3D model of petal tooth nutation drive

1 =
Guo et al. 2017), the inner and outer fixed petal gears are rigidly connected with the frame, the frame and the horizontal shaft of the input shaft form a rotating pair, the inner planetary petal gear and the inclined shaft of the input shaft form a rotating pair, and the inner and outer planetary petal gears form a rotating pair.Both sides of the inner and outer planetary petal gear and the fixed and rotating petal gears of the inner and outer layers respectively form the inner and outer petal gear meshing pair.Set the input shaft angular velocity   100 rad/s, and the solution time is 0.01s, and after 2000 steps of calculation, the simulation results are shown in Fig.9.

Fig. 9
Fig. 9 Instantaneous position of petal tooth nutation drive

Fig. 10
Fig. 10 Speed curves of petal tooth nutation drive

Fig. 11
Fig. 11 Mesh of petal gears (a) Fixed petal gear: 1 increment (b) Fixed petal gear: 2 increments (c) Fixed petal gear: 3 increments (d) Planetary gear: 1 increment (e) Planetary gear: 2 increments (f) Planetary gear: 3 increments Fig. 12 Contact pressure on the fixed side of petal gear (a) Miss stress of the fixed petal gear (b) Miss stress of the fixed side of planetary petal gear Fig. 13 Contact stress on the fixed side of petal gear under 2031 N.m torque (a)Ratary petal gear: 1 increment (b)Ratary petal gear: 2 increments (c)Ratary petal gear: 3 increments (d) Planetary gear: 1 increment (e) Planetary gear: 2 increments (f) Planetary gear: 3 increments Fig. 14 Contact pressure on the rotating side of petal gear The results show that with the increase of the increment step, the number of teeth involved in meshing increases gradually, the contact area also increases gradually, and the number of teeth engaged reaches 8.The maximum stress of fixed petal gear is 391 MPa, with an average of 130~260 Mpa; the maximum stress of planetary petal gear is 333 MPa, with an average of 110~220 MPa, which meets the design requirements.(a) Miss stress of the rotary petal gear (b) Miss stress of the rotary side of planetary petal gear Fig. 15 Contact stress on the rotating side of petal gear under 2000 N.m torque In the same way, apply 2000 N.m torque to the rotating petal gear and constrain the planetary petal gear to obtain the contact pressure of the rotating petal gear as shown in Fig. 14.It can be seen that with the increase of the increment step, the number of teeth involved in meshing begins to increase and the contact area gradually increases.Under the action of load, the number of teeth engaged reached 10.At the same time, the contact stress of the gear tooth surface on the rotating side is obtained as shown in Fig. 15.The results show that the maximum stress of the rotary petal gear is 362 MPa, and the rest is between 150 and 240 MPa; the maximum stress of the planetary petal gear is 415 MPa, and the rest is 140~277 MPa, meeting the design requirements, which once again verifies the correctness of the theoretical calculation.

Fig. 16 the unit vector of coordinate system 2 S 2 v 2  2  1  the angular velocity of input shaft 4  the angular velocity of the inner planetary petal gear 5  the angular velocity of the external planetary petal gear 6  7 
Fig. 16 Contact stress on the fixed side of petal gear under 6000 N.m torque

Table 1
Basic parameters of petal tooth nutation drive

Table 2
Comparison between simulation results and theoretical results