Mathematical model and geometrical design method of noncircular face gear with intersecting axes

In order to establish the design method of a noncircular face gear (NFG) with intersecting axes, the meshing theory of this gear is investigated based on the principle of space gear meshing. A generalized approach for designing closed pitch curve of the NFG with intersecting axes was proposed based on Fourier series. The mathematical model of the NFG generated by a shaper cutter was established. The fundamental design parameters of the gears were defined, with the principle for determining their values discussed. The prototype of a NFG was machined by 5-axis CNC milling and the motion rule was tested. Experimental results verify the feasibility of the innovative transmission mechanism and the correctness of the mathematical model of NFG with intersecting axes.


Introduction
Noncircular gears integrate the features of the gear and cam, which can transmit accurately non-uniform motion between two shafts. As a simple function generator, they can be used to replace the expensive servo system and sensors in many mechanical devices to achieve specific motion law, such as the generator of the pulsating blood flow, 1 soft robot joint actuation mechanism, 2 differential velocity vane pump, 3 indexing mechanism, 4 potted vegetable seeding transplanting mechanism with punching hole, 5 speed variator, 6 differential of off-road vehicles, 7 etc. The use of noncircular gear in equipment not only significantly reduces the system cost, but also greatly improves the reliability of the products in bad operating environment. Noncircular gears have broad application prospect in mechanical motion control fields. It is of great significance to study the meshing theory of noncircular gear for its application.
Much work has been performed on the meshing theory of planar noncircular gears. Scholars utilized the ellipse, eccentric circle, limacon 8 and some complex function, such as spline curve 9 and Fourier series curve, 10 to construct pitch curves of noncircular gears with different transmission rules. Furthermore, the deformed and high-order non-circular gears are investigated to achieve more complicated motion rules. [11][12][13] In actual application, it is hoped that the pitch curve of non-circular gear can be reversely calculated in accordance with the required transmission ratio. However, the transmission ratio of non-circular gear must satisfy the harsh closed conditions. Thus, it is generally hard to construct the closed pitch curve according to the given transmission ratio. For this, Liu and Ren 14 proposed a compensation method for designing the closed noncircular gears satisfying drive rule in a certain part. In addition to the synthesis of the pitch curves, the generation of the tooth surface is another key problem of the noncircular gears. Litvin et al. 15 expounded systematically geometry of elliptical gears with straight and helical teeth generated by the rack, shaper and hob respectively, which could provide a theoretical basis for the design and manufacture of all the planar noncircular gear. The meshing theory of planar noncircular gears is relatively perfect.
With the development of process technology and the increase of the requirement on intersecting shafts transmission, the space noncircular gears become a hot research topic gradually. Noncircular bevel gear is the earliest spatial gears transmitting variable rotational speed. Xia et al. 16 presented the meshing theory in polar coordinates and established the mathematical model of the tooth profile of the gears generated by a pair of conjugated crown racks. Further, they deduced the tooth profile equation of noncircular bevel gears with concave pitch curve generated by a bevel gear cutter 17 For the high-order deformed elliptic bevel gears, Lin et al. 18 paid considerable attention to the transmission mode and the kinematic characteristics, while Figliolini and Angeles 19 focused on the synthesis of the pitch cones. On the basis of the geometric model, Lin and He 20 simulated the meshing behaviors of the noncircular bevel gears to acquire the contact features and transmission errors. It can be seen from the above studies that the mathematical model of the tooth profile of the noncircular bevel gears is complicated and it is hard to process the conjugated space tooth surfaces of the gears, which restricts their application to some extent.
By replacing the pinion by a planar noncircular gear in a face gear mechanism, Lin et al. proposed a new gear mechanism comprised of a planar noncircular gear and a conjugated face gear called curve face gear. They carried out a series of fruitful studies on the meshing theory of the new gears, including the transmission mode, 21 geometric model, 22 compound motion, 23 contact features 24 and so on. For the curve face gear mechanism, the axial installation error of the planar noncircular gear has no effect on the transmission accuracy. Hence, the difficulties of installation and manufacture of the curve face gear are reduced greatly relative to the noncircular bevel gears.
To further reduce the design and processing difficulty of the space noncircular gear, we presented a new gear called noncircular face gear (NFG), which is conjugated with an ordinary cylindrical gear. The transmission principle of the NFG drive, the construction approach of its pitch curves and the generation of tooth profile are demonstrated in literature. 25 Moreover, considering the convenience of the eccentric wheel as the wheel blank of the new face gear, the meshing theory of the eccentric face gear is investigated, with the touching trajectory on the tooth surface, the contact ratio and transmission characteristics discussed. 26 Since there is only one noncircular gear in the new gear pair, the closure condition of the pitch curve can be simplified greatly. In theory, a closed pitch curve of the noncircular face gear can be constructed by any given periodic transmission ratio. On the other hand, failure will occur more easily on the pinion due to the less teeth. So the noncircular face gear mechanism has better interchangeability than the noncircular bevel gear or the curve face gear mechanism does.
However, the existing investigations only focused on the noncircular face gear with perpendicular shafts, which can not apply to those with arbitrary intersecting shafts. Thereby, the emphasis in this article is paid on the meshing theory of noncircular face gears with intersecting axes to build the generalized mathematical models. The pitch curves and the transmission ratio function of the noncircular face gear with intersecting axes are presented, with the tooth surface model established. The fundamental design parameters of the gear are defined and the design procedure of the parameters is given. Finally, through the simulation and experiment, the correctness of the mathematical model and the design method of the noncircular face gear is validated.

Pitch surface and pitch curve
The noncircular face gear (NFG) drive with intersecting axes is composed by a cylindrical gear and a noncircular face gear as shown in Figure 1(a). The cylindrical gear is a straight involute gear, while the NFG has non-constant rotational radius and its teeth distribute on the end face of the gear. In Figure 1(b), k is referred to as the angle between the gear axes. Generally, the cylindrical gear is the driving gear. When it rotates at a constant speed, the NFG runs with a periodically time-varying velocity.
The working pitch surface of the cylindrical gear is its reference cylinder in the case without an installation error. Correspondingly, the pitch surface of the NFG is tangent to that of the cylindrical gear. When k ¼ 90 � , the pitch surface of the NFG is a plane. When k 6 ¼ 90 � , it is a conical surface. Figure 2 illustrates the pitch cone of the NFG and the pitch cylinder in a plane with k 6 ¼ 90 � , in which the triangle and rectangle are the projections of the pitch cone and the pitch cylinder. Coordinate systems S m and S k are both fixed on the frame. The origin of S m , O m , lies at the vertex of the pitch cone. z m is the rotational axis of the NFG. The rotational axis of the cylindrical gear, z k , intersects with z m at O k , which is the origin of coordinate system S k . Axis y k and axis y m both perpendicular to plane O m x m y m . The pitch cylinder is tangent to the pitch cone on their generatrix, O m M, which is known as the pitch line. Line O k I is the instantaneous rotation axis of the gears. The cross point P between lines O m M and O k I is specified pitch point. At point P, the two gears keep pure rolling. When the gears run, trajectories of point P generate the pitch curves on the pitch surface of the two gears. The pitch curve of the NFG is a space noncircular curve on the pitch cone, while the pitch curve of the cylindrical gear is a spiral line on the pitch cylinder with non-constant helical pitch.
Coordinate system S 2 is fixed on the NFG. Figure 3 illustrates the relationships of coordinate system S 2 and S m . Origin O 2 and axis z 2 of coordinate systems S 2 coincide with origin O m and axis z m of coordinate system S m respectively. / 2 denotes the rotational angle of the NFG. The projection of the pitch curve of the NFG on plane O 2 x 2 y 2 is a noncircular curve, whose radius vector and polar angle are r 2 and u 2. When the gear starts to rotation at u 2 ¼ 0, the polar angle is equal to the rotational angle of the NFG. For easy description, they are equivalent in the following content. Let the pitch circle radius of the cylindrical gear be R. The transmission ratio between the cylindrical gear and the NFG can be represented by where x 1 and x 2 denote rotational velocities of the cylindrical gear and the NFG respectively. Since r 2 depends on the rotation angle of the NFG, the NFG drive can transmit non-constant motion between two shafts.

Construction of pitch curve
In equation (1), R is a constant. The variable transmission ratio is determined by the pitch curve of the NFG uniquely, which is different from the ordinary noncircular gears. In the design of the NFG drive, we only need to construct the pitch curve of the NFG. When the rotational angle of the NFG is used as the independent variable, it will be very easy to construct a closed pitch curve of the NFG to obtain continuous transmission. The closure condition of the NFG is that the period of the transmission ratio is 2p/m, where m represents the number of the NFG order. The transmission ratio of the NFG drive can be expressed as where i j ¼ Z 2 /Z 1 , Z 1 and Z 2 represent the number of teeth of the cylinder gear and the NFG respectively, m denotes the order of the pitch curve of the NFG, a n and b n are the coefficients of the transmission ratio. Substituting equation (2) into equation (1) could lead to the mathematical expression of the radius vector r 2 .
Then the formulae of the spatial pitch curve of the NFG can be represented as follows in coordinate system S 2   If the independent variable of the transmission ratio of the NFG drive is the rotational angle of the cylindrical gear, u 1 . The transmission ratio function can be expressed as follows For getting a closed pitch curve of the NFG, we only need to guarantee i j ¼ z 2 /z 1 in equation (2) and equation (4). The other parameters, m, a n and b n , can be assigned any data. Equations (2) and (4) could be used as generalized transmission ratio formulae for designing the closed pitch curve of the NFG.

Mathematical model of tooth surface of NFG
Enveloping theorem of NFG Figure 4 illustrates the generation of the NFG by a shaper cutter. Coordinate system S 2 is fixed on the NFG, which rotates around z 2 . The rotational axis of the shaper is z n . It intersects with axis z 2 at point O n , which is the origin of the coordinate system S n fixed on the frame. The angle between axis z 2 and axis z n is k, the shaft angle of the NFG drive. The shaper makes a reciprocating motion along axis z n . Meanwhile the shaper and the NFG rotate with angular velocity x s and x 2 , which satisfy the following relationship where i s2 is the transmission ratio between the shaper and the NFG and q is the radius of the pitch circle of the shaper. If the number of teeth of the shaper is equal to that of the cylindrical gear, the NFG will contact with the cylindrical gear on lines in the engagement. It will result in incorrect engagement easily due to the installation errors. Using for reference of the manufacture method of the circular face gear, we let the number of the shaper teeth be 1�3 more than that of the cylindrical gears. Then the NFG drive will make a point contact, which could reduce the adverse effect caused by the installation errors.
In Figure 5, the shaper and the cylindrical gear make an imaginary inner engagement and they mesh with the NFG simultaneously. Coordinates system S s and S 1 are fixed on the shaper and the cylindrical gear. R represents the pitch circle radius of the cylindrical gear. The tangent line between the pitch cylinder of the shaper and that of the cylindrical gear coincides with the pitch line O m M in Figure 2 and passes through the pitch point P on the pitch cone of the NFG. Figure 6 shows a tooth of the shaper which meshes with the cylindrical gear and the NFG simultaneously. The contact line between the shaper and the cylindrical gear is a line parallel to their rotational axes, L s1 . The curve L s2 on the tooth surface of the shaper is the meshing line between the shaper and the NFG. The intersecting point Q between L s1 and L s2 is the contact point between the cylindrical gear and the NFG in practical engagement.

Tooth surface model of the shaper
The tooth profile of the shaper is straight involute. In ¼   Here, u s is the coordinate of the tooth profiles on axis z s , and h q ¼ pðN s �1Þ 2Z s þ p 2Z s � inva 0 , where Z s represents the number of teeth of the shaper, a 0 is the pressure angle of the shaper, and N s denotes the tooth number with N s ¼1, 2. . .Z s . In addition, the upper and lower signs in equation (7) represent the left and right tooth profiles of the shaper respectively. According to the differential geometry theory, the unit normal vector on the tooth profile of the shaper can be expressed bỹ where the upper and lower signs correspond to the left and right profiles respectively. Introduce another assistant coordinates fixed on the frame, S c . There are five coordinate systems used in the derivation of the tooth surface formula of the NFG in total, in which S 2 and S s are moveable coordinate systems fixed on the NFG and the cylindrical gear respectively, S m , S c and S n are fixed on the frame. Figure 8(a) illustrates the pitch cone of the NFG, the pitch cylinder of the shaper and the three fixed coordinate systems in a plane. Axis z m and axis z n are the rotational axes of the NFG and the shaper, which intersect at point O n . The angle between axis z m and axis z n is k, which is the same as the angle between shafts of the cylindrical gear and the NFG. Axis x m is parallel to axis z c and the distance between them is q /sink. Figure 3 and Figure 8 (b) to (d) show the relationships of the five coordinate systems in three-dimensional space.

Meshing equation
According to the principle of gear meshing, the relative velocity between the cutting contact points on the tooth surfaces of two meshed gears is perpendicular to the common normal line on the points. The general formula of the meshing equation of gears is expressed by f ¼Ñ ðsÞ �ṽ ðs2Þ ¼0 whereÑ ðsÞ is the normal of the shaper tooth surface andṽ ðs2Þ is the relative velocity between the cutting contact points on the tooth surfaces. In the engagement of the cylindrical gear and the NFG, they still yield to the principle of gear meshing. We deduce the meshing equation of the NFG based on equation (9) below. The meshing equation of the NFG is deduced in coordinate system S s , whereÑ ðsÞ can be represented by the unit normal vector,ñ s . Let the anticlockwise rotation velocity be positive. The rotation speed vector of the cylindrical gear in S s is written bỹ x s s ¼½0 0 x s � T . The rotation speed vector of the NFG in S c is written byx 2 c ¼½0 0 � x 2 � T . By coordinate transformation we can get the rotation speed of the NFG in S s as follows where M sc is the coordinate transformation matrix from S c to S s , and / s is the rotational angle of the shaper. The relative velocity between the tooth surfaces of the shaper and the NFG in S s can be derived by the following equatioñ   Substituting equations (8) and (11) where the upper and lower sign correspond to the left and right tooth profile of the shaper.

Tooth surface model of the NFG
According to equation (12), we can get the function of parameter u s as follows Substituting equation (13) into equation (7) can result in the expression of the meshing line between the shaper and the NFG in coordinate system S s as the following equationr Transforming the coordinates of the meshing lines from S s to S 2 leads to the formula of the working tooth surface of the NFG as follows where M 2s is the coordinate transformation matrix from S s to S 2 , with In equation (16), the rotation angles / 2 and / s satisfy the following equation The fillets of the NFG are enveloped by the tooth tips of the shaper. In coordinate system S s , the tooth tip can be expressed byr s (u s , h s *), where Here, r as is the radius of the tooth top circle of the shaper. The fillets of the NFG can be derived by transformingr s (u s , h s * ) from S s to S 2.

Basic design parameters
For the NFG drive, there are five basic design parameters like ordinary involute cylindrical gears, the module, number of teeth, pressure angle, addendum coefficient and tip clearance coefficient. The module of the NFG on the pitch curve is equal to that of the cylindrical gear on the pitch circle. So we define the module of the cylindrical gear as the module of the NFG pair, which can be selected in the standard module series of involute gears. Z 1 and Z 2 are specified the number of the teeth of the cylindrical gear and the NFG, which satisfy the equation i j ¼ Z 2 /Z 1 . Generally, the transmission ratio of the NFG drive will be constructed by equations (2) and (7) firstly. Then according to the installation space and the operating requirement, we decide the number of teeth of the cylindrical gear. Finally, the number of teeth of the NFG can be derived by Z 2 ¼ Z 1 i j . Using the universal transmission ratio of equation (2) or (7), we can obtain a closed NFG as long as Z 1 and Z 2 are integers.
For involute cylindrical gears, the pressure angle is that on the pitch circle of the gears. The pressure angle of the NFG on the pitch cone is equal to that of the cylindrical gear. So the pressure angles of the two gears a 0 can be chosen from the standard pressure angle of involute gears.
The addendum coefficient and tip clearance coefficient of the NFG drive, h a * and c * , are the same to those of the ordinary gears. For the NFG, the addendum is represented by h a ¼m h a * , while the dedendum is expressed by h f ¼ m (h a * þ c * ). The addendum surface and dedendum surface are the normal isometric surfaces of the pitch cone one.

Tooth width
The pitch curve of the NFG, C 2 , is a space curve on the pitch cone as shown in Figure 9. The curves C 2n and C 2w are the inner and outer curve of the NFG teeth on the pitch cone. The distance between the inner curve and the pitch curve along the generator of the pitch cone is L n , while the distance between the outer curve and the pitch curve is L w . The tooth width of the NFG is expressed by B 2 ¼ L w þ L n . In the coordinate systems S 2 , the inner and outer curve can be represented by where the upper signs n and w correspond to the inner and outer curve of the teeth. As we know, the undercutting and tip-cutting will appear on the tooth of the circular face gear if the tooth width exceeds a certain value. The same phenomenon occurs on the NFG. For circular face gear, the academic criterion of the undercutting and tip-cutting was presented in literature, 27 with the maximal tooth width discussed. However, variable radius vector of the NFG brings large difficulty to deduce the academic criterion of the tooth shape imperfections. From machining simulation, we found that undercutting and tip-cutting appear more easily on the tooth where the radius vector is minimal. 26 Studies on circular face gear state that the limit width of the gears decreases as the face gear radius decreases too. So, it can be possible to conclude that the limit width of the NFG is decided by the minimum of the radius vector of its pitch curve, r 2min . Near r 2min , the transmission ratio of the NFG does not change much, where the tooth shape is similar to that of a circular face gear with r 2min as the radius of its pitch circle. Thereby, the circular face gear with radius r 2min is defined as the equivalent gear of the NFG, which is used to judge the undercutting and tip-cutting and calculate the limit width of the NFG. Literature 27 gives a derivation of the limit width of the circular face gear in detail, which will not be repeated in this article.
It can be seen from Figure 9 that the teeth of the NFG distribute along the pitch curve. For guaranteeing normal engagement of the NFG drive, the cylindrical gear must have the long enough tooth width. The minimum of the tooth width of the cylindrical gear can be written by where r 2max and r 2min are the maximum and minimum of r 2 .
The process for designing the NFG drive is illustrated in Figure 10. Firstly, the basic parameters of the cylindrical gear need to be determined. Parameters a 0 , h a * and c * are assigned with the standard values of the involute spur gears. The crossed axis angle, l, is determined by the operating requirement of the gearing. m and Z 1 would be estimated with consideration of the installation space, empirical values and so on. Then the number of the teeth of the NFG is obtained by Z 2 ¼Z 1 i j . Otherwise, the pitch curve of the NFG is not closed. The minimum radius vector of the pitch curve of the NFG, r 2min , is gained by the transmission ratio and pitch circle radius of the cylindrical gear of the NFG drive. According to the judgement rules of the undercutting and tip-cutting, the limit width of a circular face gear with radius r 2min can be calculated, which is used as the limit width of the NFG. Then, using equation (22), the minimum of the tooth width of the cylindrical gear can be derived. Finally, the contact and bending stress of the NFG drive should be calculated to verify that the gears satisfy the strength requirement. If the requirement is not satisfied, we need to adjust parameters m and Z 1 , and recalculate others according to the design procedure in Figure 10.

Results and discussion
In this section, a design example of NFG with intersecting axes is given. An analysis of the transmission ratio characteristics of the NFG is made, with the effects of the design parameters on the transmission ratio discussed. Then the simulation models of the NFG are built by the proposed mathematical model. Using the simulation model, the NFG is machined by a 5-axis CNC machine center, with the practical transmission ratio tested. The first goal here is to demonstrate the transmission ratio rules, which can be used to determine or adjust the value of parameters in realistic application. The second goal is the validation of the mathematical model and design method of the NFG with intersecting axes by the experiment.
In the example, the first term harmonic in equation (7) is selected as the transmission ratio of a NFG drive. Setting a 1 ¼ e /i j , and b 1 ¼ 0, we can get the simplest expression of the transmission ratio function as where e represent the eccentric ratio of the NFG. The transmission ratio parameters are determined based on the application requirement. Here, set i j ¼3.7, e ¼0.5 and m ¼ 1.The basic design parameters of the NFG are given as shown in Table 1, where the shaper cutter has one more teeth than the cylindrical gear does. Following the design flow in Figure 10, the maximum tooth width parameters of the NFG is calculated as L wmax ¼10.1111 mm and L nmax ¼3.8158 mm. Take L w ¼10 mm and L n ¼3.5 mm. The tooth width of the NFG is given by B 2 ¼ 13.5 mm. Using equation (22), we calculated that the minimal tooth width of the cylindrical gear is 41.287 mm. Eventually, we take the tooth width of the cylindrical gear as B 1 ¼ 45 mm.

Transmission ratio characteristics
It is known from equation (23), the transmission ratio i 21 comprises of the reduction ratio i j and the variable transmission ratio i b ¼1þe cos(mu 1 /i j ). Hence, the NFG drive is equivalent to a serial gear mechanism composed of a pair of bevel or face gears and a pair of noncircular gears as shown in Figure 11(a), where the reduction ratio of bevel gear is 3.7 and the noncircular gears behave according to the transmission ratio i b shown in Figure 11(b). In practice, the serial gear mechanism can be replaced by the NFG drive to reduce the weight and installation space of the transmission system greatly. This is quite appropriate for the equipment which has strict demand on the weight and size, such as robots, instruments, aircraft and so on.
In equation (23), there are three parameters, i j , e and m, to determine the transmission ratio of the NFG. Adjusting one of the three parameters independently and keeping the other parameters in Table 1 invariant, we draw the transmission ratio curves varying with u 2 to analyze the effects of the three parameters on the transmission rules in Figure 12. Set the transmission range of the gears as the ratio of the maximum to the minimum of the transmission ratio. The reduction ratio, transmission range and number of transmission ratio period are used to describe features of the transmission rules of the NFG.
Set Z 1 as 17, 18 and 19 respectively. Correspondingly, the values of the i j are 4.1, 3.9 and 3.7. With the decrease of i j , the transmission rule curves move down along the vertical axis integrally in Figure 12(a). However, the transmission range and number of transmission ratio period do not change with i j . Similarly, when e or m varies, one of the three transmission features of the NFG changes accordingly. The transmission range expands as e increases in Figure 12(b), while the period of the transmission ratio linearly increases with an increment of m in Figure 12(c).
Thereby, it can be concluded that the reduction ratio, transmission range and number of the transmission ratio period of the NFG drive can be controlled by i j , e and m independently. However, for ordinary closed noncircular gears, the reduction ratio (the ratio of the number of teeth between the two noncircular gears) must be equal to the ratio of orders of their pitch curves. In this case, the number of the transmission ratio period is associated with the reduction ratio. For example, the reduction ratio of the noncircular gears is 3/2 in Figure 13, which is equal to the ratio of orders of the two gears. Correspondingly, the number of the transmission ratio period is 3 when the driven gear rotates a revolution. If the number of the transmission ratio period in Figure 13 is changed to 4, then the reduction ratio of the noncircular gears will be 2. By comparison, the features of the transmission ratio of the NFG are decided by specific parameters independently, which provides a huge advantage of flexibility over other noncircular gears in design.

Simulation modeling and manufacture
The NFG is machined by a 5-axis CNC milling. The 3-D simulation model should be constructed by the presented mathematical model. Using equations (15) and (19), we obtain the working surface and fillet (a) (b) Figure 11. Equivalent transmission mechanism of the NFG drive: (a) two-stage gear mechanism; (b) variable transmission ratio i b . surface data of the NFG with design parameters in Table 1. Then import those data into a 3-D graphics software to generate the tooth surfaces by the surface fitting function. A tooth model of the NFG is given in Figure 14, where the solid curves on the left and right side of the tooth are the meshing lines between the NFG and the shaper, and the dashed curves are the intersecting line between the working and fillet surfaces. Due to the noncircular shape of the NFG, the tooth is different from each other on the gear. So the data of every tooth must be calculated. Figure 15(a) and (b) show the pitch curve on the pitch surface of the noncircular face gear. The 3-D simulation model of the NFG is constructed as shown in Figure 15(c). Using this model, the manufacturing process of the gear by a 5-axis CNC milling machine is made, with the tooth path planed.
The process includes roughing, grooving, semifinishing and finishing. Firstly, a slightly larger diameter cylindrical milling cutter is used to coarsely mill the tooth groove. Secondly the basic shape of tooth profile was obtained by further milling the tooth groove with a small diameter cylindrical milling cutter. Then a slightly larger diameter ball milling cutter is used for semi-precision milling of tooth surface as shown in Figure 15(d). Finally, a small diameter spherical milling cutter is used for precision milling of tooth surface and we obtain the NFG sample as shown in Figure 15(e).

Transmission test
A NFG transmission platform is built as shown in Figure 16, in which the prime mover is a variable frequency motor, the load is a magnetic powder, and the NFG and the cylindrical gear are installed at an angle of 80 .
Let the cylinder gear rotate at a 200 r/min, 600 r/ min and 1000 r/min respectively. The load on the NFG keeps zero. The practical rotation speed of the NFG is tested by an optical encoder which is installed on the shaft of the NFG. Through the numerical difference, the rotational velocity of the NFG can be calculated. Then the practical transmission ratio of the NFG drive is obtained, which is compared with the theoretical transmission ratio in Figure 17.
It can be observed in Figure 17 that the theoretical and experimental results agree with each other  basically under different input speed. The NFG mechanism could transmit periodically variable motion in equation (23), which verifies the feasibility of transmission of the NFG drive with intersecting axes and the correctness of its design method.
However, there exists a definite error between the theoretical and experimental transmission ratio. It may result from the errors of simulation model, manufacture, installation and test. Since we applied the ball bearing with a seat in the platform, the accuracy of (a) (b) Figure 14. A tooth of the NFG generated by the tooth surface data: (a) left tooth surface; (b) right tooth surface. the axis angle of the gears is hard to be controlled precisely during the installation process. The axis angle error would be the major factor resulting in the transmission error. So the transmission precision of the NFG in Figure 16 could be improved by reducing the installation error. Considering that there are offset error and axial error in the NFG mechanism, it is necessary to further make an analysis of the tooth contact of the gears and reveal the influence rule of different installation errors on meshing behaviors to control the transmission accuracy effectively.

Conclusion
Noncircular face gear drive breaks the matching pattern of the traditional noncircular gears, which is a new type of variable transmission ratio gear mechanism composed of a cylindrical gear and a noncircular face gear. Aiming at the NFG with intersecting shafts, we investigate the construction of the pitch curves, the generation of the tooth surfaces and the features of the transmission rules and obtain the following conclusions: 1. The generalized transmission ratio functions of the noncircular face gear are proposed based on Fourier series, which can be used to construct the closed pitch curves of noncircular face gears with intersecting shafts conveniently. 2. The mathematical model of the tooth surfaces of the noncircular face gear with intersecting shafts is established by spatial meshing theory, which is verified by the transmission test. The model lays the theoretical foundation for the further contact analysis, the dynamic simulation and the manufacture of the noncircular face gear. 3. The noncircular face gear drive can replace a serial gear mechanism with a pair of circular gears and a pair of noncircular gears in practice to light weight and reduce installation space. 4. The reduction ratio, transmission range and number of transmission ratio period of the noncircular face gear drive can be controlled independently, which lead to great flexibility in design relative to the traditional noncircular gear.

Declaration of Conflicting Interests
The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.