Design and Investigation of an Asymmetric Logarithmic Spiral Gear Drive

. Sliding on gear teeth working surfaces has negative effects on the performances of gears, such as tooth surface wear, pitting, etc. In order to reduce the gear sliding during high speed and heavy load, a new type of pure rolling gear, named as an asymmetric logarithmic spiral gear, is designed referring to the characteristics of the Issus planthopper gear. To explore the meshing principle of this kind of gear, the equations of the teeth surfaces, their working lines, and contact lines are all derived. Then, the tooth pro ﬁ le parameters and slip rate are calculated. To ensure accurate gear engagement, the gear interferences are analyzed to build the gear models. Subsequently, the gear is performed to simulate its working condition by the ﬁ nite element method. Furthermore, the results are compared with that of the pure rolling single arc gear. As a result, the asymmetric logarithmic spiral gear behaviors less contact and bending stresses than the pure rolling single arc gear under the same work condition.


Introduction
Gear has the advantages of compact structure, smooth transmission, and constant transmission ratio, so it is widely used in various fields. When most gears transmit power, rolling and sliding occur simultaneously on the working surfaces of the gear teeth. Sliding may cause wear, gluing, and plastic deformation. Especially under the condition of high speed and heavy load, it reduces the transmission efficiency, increases the energy consumption, shortens the life of gear, and produces the larger vibration and noise. The lubricants are usually used to mitigate these problems, leading to food contamination in the food machinery industry.
In order to reduce the slip between gear teeth surfaces, scholars have done a lot of explorations and researches on pure rolling gears. Tan et al. [1] derived a simplified equation for the bevel gear to satisfy continuous pure rolling contact. Chen et al. [2][3][4] introduced the geometric design, meshing performance, and mechanical property of a new pure rolling gear. Xiao et al. [5] designed a pure rolling noncircular gear with fewer teeth. He et al. [6] proposed a pure rolling gear with divisible center distance. Wagner et al. [7] proposed the gear profiles of pure rolling contact for parallel axis transmission. Tao et al. [8] studied the meshing principle and characteristics of a new type of bihemispheric rolling gear and applied it to engineering. Song et al. [9] proposed a new pure rolling cycloid gear and applied it to the reducer, which could achieve higher transmission efficiency. Xue et al. [10] proposed a new cycloid gear transmission, which could realize pure rolling and improve the contact ratio and the service life. Tan et al. [11] studied the geometric principle and processing process of the pure rolling cycloidal gear and completed the performance experiment. X. Huang and A. Huang [12] established the constraint conditions of the pure rolling contact gear and found a variety of simple and practical tooth profile curves of pure rolling gear. Chen et al. [13] designed a new type of pure circular arc rolling helical gear and made a prototype for test. It was concluded that this kind of gear has pure rolling contact, high coincidence ratio, and large comprehensive strength. Geng, Zhou, and Zhao [14][15][16] studied the theory and performance of pure rolling single arc gear. Based on the conjugate curve method, Liang et al. [17][18][19][20] proposed a circular arc gear with a sliding rate close to zero, which is approximate to pure rolling contact.
To sum up, pure rolling theory has been applied to several kinds of gears, such as bevel gear, noncircular gear, cycloidal gear, and arc gear. The pure rolling bevel gear is mainly used in the occasion where two axes intersect. The pure rolling noncircular gear is mainly applied in the occasion of low speed and special periodic motion. The pure rolling constraint equations of these two gears are obtained by controlling the contact points velocity to zero. The pure rolling cycloid gear realizes pure rolling contact under the condition that the contact point is the velocity instantaneous center. It is mainly used in planetary mechanism of the reducer. The pure rolling single arc gear can realize pure rolling by limiting the contact points to the pitch line. It is mainly used in high speed and heavy load transmission with two parallel shafts. However, the machining of the pure rolling single circular arc gear requires two cutting tools, so its universality is poor.
Burrows and Sutton [21] found a pair of gears in the hip joint of the planthopper, as shown in Figure 1. The working characteristics of the Issus planthopper gear are high speed and heavy load. Referring to the tooth shape of the Issus planthopper gear and the shortcoming of the pure rolling single arc gear, a new pure rolling gear is proposed in this paper, which is designed to be used in high speed and heavy load transmission with parallel shafts.
As shown in Figure 2, the new gear tooth shape is asymmetrical with a convex tooth profile on one side and a concave tooth profile on the other side based on the shape of the Issus planthopper gear. Since the tooth profiles of the Issus planthopper gear are naturally generated and the logarithmic helix is also a natural curve, the tooth profiles of the Issus planthopper gear are assumed to be logarithmic helix. Wang [22] proved the feasibility of logarithmic helices as gear tooth profile. Therefore, the logarithmic helix is taken as the working tooth profiles of the new gear, which is named as an asymmetric logarithmic spiral gear.
The rest of this paper is organized as follows: Section 2 explores the gear meshing principle, including the equations of the tooth surface, working line, contact line, tooth profile parameters, and sliding ratio. Section 3 analyzes the gear interferences, including the geometric interference, root cutting, and tooth profile overlapping interference. In Section 4, the gear models are built to ensure the accurate gear engagement and the gear stress distributions are simulated and analyzed.

Meshing Mechanism
The asymmetric logarithmic spiral gear is a kind of helical gear and can realize bidirectional transmission. To ensure the correct transmission, it is necessary to meet the contact ratio greater than or equal to 1, the pressure angles of the meshing teeth profiles are equal, and the helical angles of the driving and driven gears are equal in magnitude and opposite direction. Figure 3(a) shows the meshing teeth profiles. The four working teeth profiles are represented as C 1 , C 2 , C 3 , and C 4 , respectively. The teeth surfaces formed by the four teeth profiles are represented as Σ 1 , Σ 2 , Σ 3 , and Σ 4 , respectively, as shown in Figure 3(b). When the gear 1 rotates counterclockwise, Σ 1 and Σ 2 are the conjugate teeth surfaces. When the gear 1 rotates clockwise, Σ 3 and Σ 4 are the conjugate teeth surfaces.
2.1. Tooth Surface Equations. The formation of the tooth surface Σ 1 is taken as an example. The coordinate system is established, as shown in Figure 4 [23]. S 1 (O 1 -x 1 y 1 z 1 ) is a moving coordinate fixedly connected with the gear 1. S(O-xyz) is a fixed coordinate system of the gear 1 with z-axis parallel to z 1 axis. S p1 (O p1 -x p1 y p1 z p1 ) is a coordinate system fixedly connected with the rack of the tooth profile C 1 , x p1 axis is parallel to x-axis, and z p1 axis is parallel to z-axis. S s1 (O s1 -x s1 y s1 z s1 ) is a coordinate system where the section of the tooth profile C 1 is located. o 1 is the center of   Applied Bionics and Biomechanics the convex tooth profile. β is the spiral angle of the gear 1. R 1 is the radius of the pitch cylinder of the gear 1. φ 1 is the gear 1 rotation angle. ω 1 is the gear 1 angular rotational velocity. Suppose P is a helical rack, and its convex tooth surface Σ P1 is formed by scanning along the z s1 axis using the tooth profile C 1 . The pitch plane of the helical rack P is tangent to the gear 1 pitch cylinder. When the gear 1 rotates clockwise at an angular velocity of ω 1 , the pitch plane of the helical rack P is shifted along the y-axis at the speed of ω 1 R 1 . In this generating motion, the tooth surface Σ 1 is formed by the tooth surface Σ P1 .
The convex tooth profile C 1 equation in the coordinate system: where a 1 ¼ r 1 e t × θ i1 sin θ i1 , r 1 is the initial circle radius of the convex tooth profile C 1 , t is the logarithmic helical coefficient, θ i1 is the angle of any point on the tooth profile C 1 , b 1 ¼ r 1 e t × θ i1 cos θ i1 , f 1 and w 1 are the offset distance of the tooth profile C 1 center point o 1 relative to the x s1 and y s1 axes, respectively.
Based on the coordinate transformation principle, Equation (1) is transformed from the coordinate system S s1 (O s1 -x s1 y s1 z s1 ) to the coordinate system S p1 (O p1 -x p1 y p1 z p1 ). The transformation matrix equation is expressed as follows: The equation of the tooth profile C 1 in the coordinate system S p1 (O p1 -x p1 y p1 z p1 ) can be obtained.
ð Þcos β þ μ 1 sin β; The expression of the contact condition can be derived by using the instantaneous rotation axis method [23].
Equation (3) is transformed into the coordinate system S 1 (O 1 -x 1 y 1 z 1 ); then, the result is combined with Equation (4) to obtain the convex tooth surface Σ 1 equation.
cos φ 1 ; The coordinate systems in Figure 5 are established to derive the concave tooth surface Σ 2 equation. S(O-xyz) is a fixed coordinate system of the gear 2 with z-axis parallel The pitch plane of the oblique rack P Auxiliary rack direction to z 2 axis. The subscript of the expression symbol in Figure 5 is changed from 1 to 2.
The concave tooth profile C 2 equation in the coordinate system S s2 (O s2 -x s2 y s2 z s2 ) is obtained as follows: where a 2 ¼ r 2 e t × θ i2 sin θ i2 , r 2 is the initial circle radius of the concave tooth profile C 2 , θ i2 is the angle of any point on the tooth profile C 2 , b 2 ¼ r 2 e t × θ i2 cos θ i2 , f 2 and w 2 are the offset distance of the tooth profile C 2 center point o 2 relative to the x s2 and y s2 axes, respectively. Similarly, the tooth surface Σ 2 equation can be obtained as follows: As shown in Figure 6, the tooth profile C 3 can be obtained by transforming the position of the tooth profile C 2 in the coordinate system S s (O s -x s y s z s ). The tooth profile C 2 is symmetric along the y s axis and then is moved to the tooth profile C 3 with distances of c and d along the x s and y s axes, respectively. The contact points of the tooth profiles C 2 and C 3 are k 1 and k 2 , respectively. The line k 1 k 2 intersects x s axis at point p. The length of k 1 p is the same as the length of k 2 p.
The equation of the concave tooth profile C 3 in the coordinate system S s (O s -x s y s z s ) can be obtained as follows: The tooth surface Σ 3 equation can be obtained by referring to the formation of the tooth surface Σ 1 .
Similarly, the tooth surface Σ 4 equation can be obtained by referring to the formation of the tooth surface Σ 2 .
2.2. Contact Line and Tooth Surface Working Line. The contact line of the asymmetric logarithmic spiral gear is the trajectory of the instantaneous contact points in the fixed coordinate system S(O-xyz). The tooth surface working line is the collection of the theoretical contact points on the teeth surfaces in the transmission process. By deducing the equations of the contact line and tooth surface working line, the pure rolling conditions can be obtained.
Te pitch plane of the oblique rack P Auxiliary rack direction : Schematic diagram of the gear 2 concave tooth surface formation.
x s y s : Coordinate transformation of concave tooth profile.

Applied Bionics and Biomechanics
The tooth surface Σ 1 meshing with the tooth surface Σ 2 is taken as an example. As shown in Figure 4, the tooth profile C 1 in the coordinate system S p1 (O p1 -x p1 y p1 z p1 ) is moved to the coordinate system S(O-xyz) with a distance of R 1 φ 1 along the y-axis. Then, Equation (4) can be used to obtain the meshing surface equation. In the meshing surface equation, when θ i = θ 0 , the equation of the contact line of the tooth surface Σ 1 can be obtained as follows: where Similarly, the equation of the contact line of the surface Σ 2 can be obtained.
where a 3 ¼ r 2 e t×θ 0 sin θ 0 ; b 3 ¼ r 2 e t×θ 0 cos θ 0 . In Equation (5), when θ i = θ 0 , the equation of the tooth surface working line of the tooth surface Σ 1 is obtained as follows: In Equation (7), when θ i = θ 0 , the equation of the tooth surface working line of the tooth surface Σ 2 is obtained as follows: According to the characteristics of the pure rolling point contact [14], the contact points are limited to the nodal line, which means that the distances between the contact lines and the nodal line are zero, that is, l ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi x 2 þ y 2 p ¼ 0. By combining Equations (11) and (12), the pure rolling constraint equations can be obtained as follows: Equation (15) is substituted into Equations (11)- (14). It can be known that the teeth surfaces working lines of the driving and driven gears are both helices on the teeth surfaces. The contact lines of the driving and driven gears are both straight lines parallel to the rotation axis of the gear pair, as shown in Figure 7.
is the center point of the gear tooth thickness, O G2 is the center point of the gear tooth groove width, and x G1 and x G2 coincide with the pitch lines of the gears 1 and 2, respectively. o 1 and o 2 are the center points of the convex and concave tooth profiles, respectively. θ 0 and θ 3 are the pressure angles of the convex and concave tooth profiles, respectively. θ 0 is equal to θ 3 . θ 1 and θ 4 are the process angles of the convex and concave tooth profiles,  respectively. θ 2 and θ 5 are the terminal angles of the convex and concave tooth profiles, respectively. According to the geometric relationship, the calculation formulas of the tooth profile parameters are shown in Table 1.
The parameters r 1 , r 2 , and t of the logarithmic helical can be selected in an infinite number of ways, which lead to the uncertainty of the angles θ 1 and θ 5 . So, it is important to determine the logarithmic helical parameters. According to the design principle of gear tooth profile [24], the logarithmic helical parameter ranges can be calculated as in Table 2.

Sliding Ratio.
It is assumed that the initial contact points of the tooth surfaces Σ 1 and Σ 2 are K 1 and K 2 , respectively, as shown in Figure 9. After m time, the tooth surfaces Σ 1 and Σ 2 are tangent at K. The arc length of the contact point passing through the tooth surface Σ 1 is S 1 . The arc length of the contact point K passing through the tooth sur-

Parameter
Computational formula Process angle of convex tooth profile θ 1 e ¼ s þ j Arc radius at the bottom of convex tooth profile r g1 r g1 ¼ r 1 e t × θ 1 sin θ 1 þ h f − f 1 Arc radius at the bottom of concave tooth profile r g2

Parameter
Computational formula Minimum initial circle radius of the concave profile r 2 min r 2 min ¼ h a =e t × θ 3 sin θ 3 Maximum initial circle radius of the concave profile r 2 max r 2 max ¼ 0:645m n = e t × θ 4 cos θ 4 − e t × θ 3 cos θ 3 ð Þ Minimum initial circle radius of the convex profile r 1 min r 1 min ¼ f 2 − r 2 e t × θ 4 sin θ 4 ð Þ =e t × θ 0 sin θ 0 Maximum initial circle radius of the convex profile r 1 max r 1 max ¼ 0:895m n = e t × θ 2 cos θ 2 − e t × θ 0 cos θ 0 ð Þ : Slide diagram of tooth surfaces [20]. 6 Applied Bionics and Biomechanics Then, the slip rates of the tooth surfaces Σ 1 and Σ 2 can be calculated, respectively, as follows: The values of the gear slip rate can be obtained from the tooth surface working line equations [20]. Then, Equations (18) and (19) can be obtained by Equations (13) and (14).
By combining Equations (18) and (19) with Equation (15), dS 1 ¼ dS 2 can be obtained. The sliding rates of the two meshing teeth surfaces are zero, which verify the gear pure rolling contact.

Interference Analysis
When designing an asymmetric logarithmic spiral gear, it is easy to interfere if the tooth profile parameters are determined without rules. Although it is possible to establish the gear model to distinguish the gear without interference, a great deal of work will be done in this way. Therefore, the geometric interference and root cutting conditions of the gear are derived to obtain the allowed range of θ 1 and θ 5 . Then, the overlapping interference of the end face profile is examined. Figure 8, when the concave and convex tooth profiles are meshing, for avoiding geometric interference, h g1 and h g2 should be greater than h a . The equations of the geometric interference condition are derived.

Geometric Interference of Tooth Profile. As shown in
In the gear design process, when the basic parameters h a , h f , r 1 , r 2 , t, θ 0 , and θ 3 are given in Table 1, the angles θ 2 and θ 4 can be solved. By Equation (20), the ranges of the angles θ 1 and θ 5 without geometric interference are calculated.

Root
Cutting. The tangent direction of any point K on the surface along any curve L on the surface is expressed as dr k =dm ¼ ∂r u =∂u du=dm ð Þþ∂r v =∂v ∂v=dm ð Þ, as shown in Figure 10. If the guide vector is zero, the conditional formula of the singular point is established. The boundary equations of gear root cutting are obtained by excluding the singular points on the tooth surface.
The following boundary equations can be derived by referring to the root cutting conditions of the circular arc gear [23]: where the subscript x(1, 2) in the formulas represents the convex and concave tooth profiles of the asymmetric logarithmic spiral gear. Aiming at the concave surface, θ i2 should be replaced by -θ i2 . The solution of Equation (22) can be obtained by Newton iteration [23]. Figure 11(a), the coordinate system S g (O 1 -x g y g ) is setup. The gear 1 center point O 1 is taken as the origin of the coordinate system S g (O 1 -x g y g ). x g axis coincides with the centerline of the end face tooth thickness of tooth a. y g axis is perpendicular to x g axis. It is assumed that the gear 1 is fixed and the gear 2 rotates counterclockwise around the gear 1. According to the principle of gear relative motion, the tooth b endpoint A B1 encloses curve σ 1 . If this curve σ 1 intrudes into the tooth of a convex tooth profile, the interference will occur. As shown in Applied Bionics and Biomechanics Figure 12, it is assumed that the polar coordinate of any point on the convex tooth profile of tooth a is (R P1 , Φ 1 ). The polar coordinate of any point on the curve σ 1 with the same radius is (R P1 , θ σ1 ). If θ σ1 < Φ 1 . profile overlapping interference occurs. Figure 11(b) shows the clockwise rotation of the gear 2 around the gear 1. When Figure 11(b) is compared with Figure 11(a), the subscript of the parameters changes from 1 to 2. Similarly, if θ σ2 < Φ 2 , profile overlapping interference occurs.  (1) and (8) are divided by cos β, then Equations (1) and (8) become the end face tooth profile equations. The polar coordinate expressions of the convex and concave tooth profiles of tooth a on the end face can be obtained as follows:

Profile Overlapping Interference. As shown in
where the subscript x(1, 2) in the formula represents the convex and concave tooth profiles. When x = 1, the sign is positive and when x = 2, the sign is negative.

Polar Coordinate
Equations of the Curves σ 1 and σ 2 . As shown in Figure 13, X 1 coincides with the centerline of the end face tooth thickness of tooth a and X 2 coincides with the centerline of the end face tooth thickness of tooth b. The position of the centerlines X 1 and X 2 is denoted by the angles δ 1 and δ 2 , respectively. Figure 13(a) shows the state of the gear 2 rotating counterclockwise. Figure 13(b) shows the state of the gear 2 rotating clockwise. According to the law of sine and cosine and the gear meshing principle [25], the gear angular position relation can be known. Then, θ σ1 and θ σ2 can be expressed as follows: where a is the center distance of the gear pair. The parameters R B1 , R B2 , θ B1 , and θ B2 can be known from the geometry of the gear 2, as shown in Figure 14.  Table 3.
To avoid interference, the logarithmic helical coefficient should be less than zero. So, choose the logarithmic helical coefficient of −0.0025. By substituting the known parameters into the calculation formulas in Table 2, the value ranges of the parameters r 1 and r 2 can be calculated. Referring to the pure rolling single arc gear [16], r 1 = 1.4 m n is selected. In order to reduce the tooth surface contact stress, the difference between r 1 and r 2 should be kept as small as possible without interference. Using the method of judging the profile overlapping interference, r 2 = 2r 1 is selected. The other parameters of the tooth profiles can be obtained by Table 1 and Equation (15). θ 1 and θ 5 are still unknown. Therefore, the tooth profile parameters are selected as described in Table 4.

Determination of Unknown Parameters.
According to Equations (20)- (22), the ranges of the angles θ 1 and θ 5 without geometric interference and root cutting can be obtained, as shown in Table 5.
The above parameters are substituted into Equations (23)- (25). Then, the change of the angles θ σ1 and Φ 1 with the convex tooth profile angle are plotted, as shown in Figure 15(a). The change of the angles θ σ2 and Φ 2 with the concave tooth profile angle are plotted, as shown in Figure 15(b). It is known that the angles θ σ1 and θ σ2 are greater than the angles Φ 1 and Φ 2 , respectively. Thus, the tooth profiles on the end face of the gears do not interfere.

Establishment of the Gear Models.
The single-tooth model is obtained by sweeping the tooth profiles of both end faces along the helical lines, as shown in Figures 16(a)  and 16(b). The complete gear model can be established by array method.
4.3.1. Verification of the Noninterference Model. Several meshing teeth are selected for interference detection under 100 Nm load. As can be seen from Figure 17, there are two contact points between the two gears and no interference occurs elsewhere. Therefore, the above selected gear parameters meet the requirements.   Figure 18, several meshing teeth are selected to verify the pure rolling meshing without load. The tooth surface working lines and contact lines are generated according to Equations (11)- (14). It can be obtained from the measurement command that ΔS 1 = ΔS 2 = 45.8143 mm, so the gears sliding rates of two meshing surfaces are both zero according to Equations (16) and (17). Therefore, the gear belongs to pure rolling meshing.

Simulation Stress Analysis
4.4.1. Simulation Setup. The models established above are imported into the finite element software. In grid division, hexahedral grid is adopted, the grid of the gears is set as   1 mm, and the grid of the contact teeth surfaces is separately encrypted, as shown in Figure 19(a). In load application, torque is applied to the driving gear. The driven gear is fixed, as shown in Figure 19(b). The material of the gear is steel with Young's modulus E ¼ 2:09 × 10 5 MPa and Poisson's ratio μ ¼ 0:3. The input torque is set as 100 Nm. The grid of the contact teeth surfaces is separately encrypted as 0.4, 0.3, 0.2, and 0.1 mm. The corresponding maximum contact stresses are obtained by simulation, as shown in Table 6.
As can be seen from Table 6, when the grid of the contact teeth surfaces changes from 0.15 to 0.1 mm, the maximum contact stresses of the teeth surfaces change from 763.02 to 763.11 MPa, and the change rate of the maximum contact stress is 0.09%. Therefore, the grid of the contact teeth surfaces is separately encrypted as 0.15 mm.

Contact Stress.
The pure rolling single arc gears with the same size as the asymmetric logarithmic spiral gears are selected and the same simulation parameters are chosen. Then, the gears stress distributions at the tooth width center are obtained, as shown in Figure 20.
As can be seen from Figure 20, the maximum contact stress of the asymmetric logarithmic spiral gear is 757.28 MPa, which occurs on the driven gear. The maximum    contact stress of the pure rolling single arc gear is 809.25 MPa, which occurs on the driving gear. In order to understand the change of the maximum contact stress during the gears rotation, finite element analysis is carried out on some positions of the gears rotation. The image drawn from the results is shown in Figure 21.
It can be seen from Figure 21 that the maximum contact stress of the asymmetric logarithmic spiral gear is lower than that of the pure rolling single arc gear. When the rotation angle is zero, the gear contacts at the end face, and the maximum contact stress is larger than that of the other rotation angles of the gears.
As shown in Figure 22, the teeth profiles slope of the asymmetric logarithmic spiral gears contact point k 1 are smaller than that of the pure rolling single arc gears at the same pressure angle θ 0 . On the premise of noninterference, the difference of the meshing teeth profiles curvature radius of the asymmetric logarithmic spiral gears is smaller, so the maximum contact stress of the asymmetric logarithmic spiral gear is lower than that of the pure rolling single arc gear.

Bending Stress.
The bending stresses of these two gears are extracted by path selection when the gears contact at the center of tooth width, as shown in Figure 23.
As can be seen from Figure 23, the maximum bending stress of the asymmetric logarithmic spiral gear is 162.

12
Applied Bionics and Biomechanics maximum bending stress of these two gears occurs on the driving gear. Similarly, in order to understand the change of the maximum bending stress of the gears rotation, finite element analysis is carried out on some positions of the gears rotation. The image drawn from the results is shown in Figure 24.
It can be seen from Figure 24 that the maximum bending stress of the asymmetric logarithmic spiral gear is lower than that of the pure rolling single arc gear.
As shown in Figure 25, comparing with the pure rolling single arc gear, the root thickness of the asymmetric logarithmic spiral gear increases, so the maximum bending stress is lower than that of the pure rolling single arc gear.
In summary, the asymmetric logarithmic spiral gear shows lower maximum contact and bending stresses than that of the pure rolling single arc gear.

Conclusions
In this paper, a new type of pure rolling gear, named as an asymmetric logarithmic spiral gear, is proposed by referring to the tooth shape of the Issus planthopper gear. The main conclusions of this paper can be drawn as follows:  (1) To explore the meshing principle of this kind of gear, the equations of the tooth surfaces, working line, and contact line are derived. The conditions of the pure rolling gear are determined. The calculation formulas of the tooth profile parameters and slip rates are obtained. The results show that the slip rates of the meshing teeth surfaces are zero, which verify that it is a kind of pure rolling gear in theory. (2) To ensure accurate gear engagement, the geometric interference, root cutting, and tooth profile interference are analyzed theoretically. The appropriate parameters are selected to establish three-dimensional solid models to simulate the gear transmission process. The results show that the noninterference gear pair can be successfully designed using the proposed method. (3) The stress distributions of the asymmetric logarithmic spiral gear and pure rolling single arc gear are analyzed by finite element method. The results show that the asymmetric logarithmic spiral gear has lower contact stress and bending stress than the pure rolling single arc gear. Furthermore, the asymmetric logarithmic spiral gear can be machined only with one tool. So, the performance and manufacturability of the asymmetric logarithmic spiral gear are better than that of the pure rolling single arc gear.
The related investigations on this novel type of gear pair, which include: (1) tooth profile modification considering deformation and error; (2) fatigue life of the gear; and (3) processing and prototype performance test of the gear, are being carried out or would be the next step of work by the authors. Efforts putting this drive forward into practical application are also needed in the near future.

Data Availability
Data can be obtained from Zenghuang He upon request.

Conflicts of Interest
The authors declare that they have no conflicts of interest.