Abstract
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 profile 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 finite 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.
1. 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. [2–4] 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 [14–16] studied the theory and performance of pure rolling single arc gear. Based on the conjugate curve method, Liang et al. [17–20] 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.
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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.
2. 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 C1, C2, C3, and C4, respectively. The teeth surfaces formed by the four teeth profiles are represented as Σ1, Σ2, Σ3, and Σ4, respectively, as shown in Figure 3(b). When the gear 1 rotates counterclockwise, Σ1 and Σ2 are the conjugate teeth surfaces. When the gear 1 rotates clockwise, Σ3 and Σ4 are the conjugate teeth surfaces.
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2.1. Tooth Surface Equations
The formation of the tooth surface Σ1 is taken as an example. The coordinate system is established, as shown in Figure 4 [23]. S1(O1-x1y1z1) 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 z1 axis. Sp1(Op1-xp1yp1zp1) is a coordinate system fixedly connected with the rack of the tooth profile C1, xp1 axis is parallel to x-axis, and zp1 axis is parallel to z-axis. Ss1(Os1-xs1ys1zs1) is a coordinate system where the section of the tooth profile C1 is located. o1 is the center of the convex tooth profile. β is the spiral angle of the gear 1. R1 is the radius of the pitch cylinder of the gear 1. φ1 is the gear 1 rotation angle. ω1 is the gear 1 angular rotational velocity. Suppose P is a helical rack, and its convex tooth surface ΣP1 is formed by scanning along the zs1 axis using the tooth profile C1. 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 ω1, the pitch plane of the helical rack P is shifted along the y-axis at the speed of ω1R1. In this generating motion, the tooth surface Σ1 is formed by the tooth surface ΣP1.
The convex tooth profile C1 equation in the coordinate system:where , r1 is the initial circle radius of the convex tooth profile C1, t is the logarithmic helical coefficient, θi1 is the angle of any point on the tooth profile C1, , f1 and w1 are the offset distance of the tooth profile C1 center point o1 relative to the xs1 and ys1 axes, respectively.
Based on the coordinate transformation principle, Equation (1) is transformed from the coordinate system Ss1(Os1-xs1ys1zs1) to the coordinate system Sp1(Op1-xp1yp1zp1). The transformation matrix equation is expressed as follows:
The equation of the tooth profile C1 in the coordinate system Sp1(Op1-xp1yp1zp1) can be obtained.
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 S1(O1-x1y1z1); then, the result is combined with Equation (4) to obtain the convex tooth surface Σ1 equation.
The coordinate systems in Figure 5 are established to derive the concave tooth surface Σ2 equation. S(O-xyz) is a fixed coordinate system of the gear 2 with z-axis parallel to z2 axis. The subscript of the expression symbol in Figure 5 is changed from 1 to 2.
The concave tooth profile C2 equation in the coordinate system Ss2(Os2-xs2ys2zs2) is obtained as follows:where , r2 is the initial circle radius of the concave tooth profile C2, θi2 is the angle of any point on the tooth profile C2, , f2 and w2 are the offset distance of the tooth profile C2 center point o2 relative to the xs2 and ys2 axes, respectively.
Similarly, the tooth surface Σ2 equation can be obtained as follows:
As shown in Figure 6, the tooth profile C3 can be obtained by transforming the position of the tooth profile C2 in the coordinate system Ss(Os-xsyszs). The tooth profile C2 is symmetric along the ys axis and then is moved to the tooth profile C3 with distances of c and d along the xs and ys axes, respectively. The contact points of the tooth profiles C2 and C3 are k1 and k2, respectively. The line intersects xs axis at point p. The length of is the same as the length of .
The equation of the concave tooth profile C3 in the coordinate system Ss(Os-xsyszs) can be obtained as follows:where .
The tooth surface Σ3 equation can be obtained by referring to the formation of the tooth surface Σ1.
Similarly, the tooth surface Σ4 equation can be obtained by referring to the formation of the tooth surface Σ2.
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.
The tooth surface Σ1 meshing with the tooth surface Σ2 is taken as an example. As shown in Figure 4, the tooth profile C1 in the coordinate system Sp1(Op1-xp1yp1zp1) is moved to the coordinate system S(O-xyz) with a distance of R1φ1 along the y-axis. Then, Equation (4) can be used to obtain the meshing surface equation. In the meshing surface equation, when θi = θ0, the equation of the contact line of the tooth surface Σ1 can be obtained as follows:where .
Similarly, the equation of the contact line of the surface Σ2 can be obtained.where .
In Equation (5), when θi = θ0, the equation of the tooth surface working line of the tooth surface Σ1 is obtained as follows:
In Equation (7), when θi = θ0, the equation of the tooth surface working line of the tooth surface Σ2 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, . 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.
2.3. Tooth Profile Parameters
Figures 8(a) and 8(b) show the meshing diagram of the tooth profiles C1 and C2. j is the side clearance of the gear. hg1 and hg2 are the vertical distances from the transition points ka and kb to the nodal line, respectively. ha and hf are the tooth addendum and dedendum, respectively. In the coordinate systems SG1(OG1-xG1yG1zG1) and SG2(OG2-xG2yG2zG2), OG1 is the center point of the gear tooth thickness, OG2 is the center point of the gear tooth groove width, and xG1 and xG2 coincide with the pitch lines of the gears 1 and 2, respectively. o1 and o2 are the center points of the convex and concave tooth profiles, respectively. θ0 and θ3 are the pressure angles of the convex and concave tooth profiles, respectively. θ0 is equal to θ3. θ1 and θ4 are the process angles of the convex and concave tooth profiles, respectively. θ2 and θ5 are the terminal angles of the convex and concave tooth profiles, respectively.
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According to the geometric relationship, the calculation formulas of the tooth profile parameters are shown in Table 1.
The parameters r1, r2, and t of the logarithmic helical can be selected in an infinite number of ways, which lead to the uncertainty of the angles θ1 and θ5. 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.
2.4. Sliding Ratio
It is assumed that the initial contact points of the tooth surfaces Σ1 and Σ2 are K1 and K2, respectively, as shown in Figure 9. After m time, the tooth surfaces Σ1 and Σ2 are tangent at K. The arc length of the contact point passing through the tooth surface Σ1 is S1. The arc length of the contact point K passing through the tooth surface Σ2 is S2.
Then, the slip rates of the tooth surfaces Σ1 and Σ2 can be calculated, respectively, as follows:where .
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), can be obtained. The sliding rates of the two meshing teeth surfaces are zero, which verify the gear pure rolling contact.
3. 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 θ1 and θ5. Then, the overlapping interference of the end face profile is examined.
3.1. Geometric Interference of Tooth Profile
As shown in Figure 8, when the concave and convex tooth profiles are meshing, for avoiding geometric interference, hg1 and hg2 should be greater than ha. The equations of the geometric interference condition are derived.
In the gear design process, when the basic parameters ha, hf, r1, r2, t, θ0, and θ3 are given in Table 1, the angles θ2 and θ4 can be solved. By Equation (20), the ranges of the angles θ1 and θ5 without geometric interference are calculated.
3.2. Root Cutting
The tangent direction of any point K on the surface along any curve L on the surface is expressed as , 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, θi2 should be replaced by –θi2.
The solution of Equation (22) can be obtained by Newton iteration [23].
3.3. Profile Overlapping Interference
As shown in Figure 11(a), the coordinate system Sg(O1-xgyg) is setup. The gear 1 center point O1 is taken as the origin of the coordinate system Sg(O1-xgyg). xg axis coincides with the centerline of the end face tooth thickness of tooth a. yg axis is perpendicular to xg 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 AB1 encloses curve σ1. If this curve σ1 intrudes into the tooth of a convex tooth profile, the interference will occur. As shown in Figure 12, it is assumed that the polar coordinate of any point on the convex tooth profile of tooth a is (RP1, Φ1). The polar coordinate of any point on the curve σ1 with the same radius is (RP1, θσ1). If θσ1 < Φ1. profile overlapping interference occurs.
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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 θσ2 < Φ2, profile overlapping interference occurs.
3.3.1. Polar Coordinate Equation of the Tooth a Profiles of the Gear 1 on the End Face
The equations for ys1 in Equations (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: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.
3.3.2. Polar Coordinate Equations of the Curves σ1 and σ2
As shown in Figure 13, X1 coincides with the centerline of the end face tooth thickness of tooth a and X2 coincides with the centerline of the end face tooth thickness of tooth b. The position of the centerlines X1 and X2 is denoted by the angles δ1 and δ2, 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.
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According to the law of sine and cosine and the gear meshing principle [25], the gear angular position relation can be known. Then, θσ1 and θσ2 can be expressed as follows:where a is the center distance of the gear pair. The parameters RB1, RB2, θB1, and θB2 can be known from the geometry of the gear 2, as shown in Figure 14.
4. Modeling and Stress Analysis
4.1. Basic Parameters
The basic parameters of the asymmetric logarithmic spiral gear are shown in 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 r1 and r2 can be calculated. Referring to the pure rolling single arc gear [16], r1 = 1.4 mn is selected. In order to reduce the tooth surface contact stress, the difference between r1 and r2 should be kept as small as possible without interference. Using the method of judging the profile overlapping interference, r2 = 2r1 is selected. The other parameters of the tooth profiles can be obtained by Table 1 and Equation (15). θ1 and θ5 are still unknown. Therefore, the tooth profile parameters are selected as described in Table 4.
4.2. Determination of Unknown Parameters
According to Equations (20)–(22), the ranges of the angles θ1 and θ5 without geometric interference and root cutting can be obtained, as shown in Table 5.
In Table 5, θ1 = 5° and θ5 = 37° are chosen for modeling and interference analysis.
The above parameters are substituted into Equations (23)–(25). Then, the change of the angles θσ1 and Φ1 with the convex tooth profile angle are plotted, as shown in Figure 15(a). The change of the angles θσ2 and Φ2 with the concave tooth profile angle are plotted, as shown in Figure 15(b). It is known that the angles θσ1 and θσ2 are greater than the angles Φ1 and Φ2, respectively. Thus, the tooth profiles on the end face of the gears do not interfere.
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4.3. 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.
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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.
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4.3.2. Verification of the Pure Rolling Meshing
As shown in 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 ΔS1 = ΔS2 = 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.
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4.4. 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 MPa and Poisson’s ratio . The input torque is set as 100 Nm.
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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.
4.4.2. 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.
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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 k1 are smaller than that of the pure rolling single arc gears at the same pressure angle θ0. 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.
4.4.3. 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.
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As can be seen from Figure 23, the maximum bending stress of the asymmetric logarithmic spiral gear is 162.82 MPa and that of the pure rolling single arc gear is 169.88 MPa. The 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.
5. 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.
Authors’ Contributions
Zenghuang He carried out the construction of the model logic and writing. Xu Gong build model derivation and simulation model. Shengping Fu checked the model and revised the paper. Shanming Luo proposed ideal and thesis revision. Jingyu Mo respond to thesis revisions.
Acknowledgments
This work was partially supported by the National Natural Science Foundation of China (grant nos. 51405410, 52205055, and 51975499) and the Natural Science Foundation of Fujian Province, China (grant no. 2019J01862). The authors would like to express gratitude to their financial support.