# Design and Investigation of an Asymmetric Logarithmic Spiral Gear Drive

Authors: Zenghuang He; Xu Gong; Shengping Fu; Shanming Luo; Jingyu Mo
Journal: Applied Bionics and Biomechanics
 (2022)

---

## 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 theIssus 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.


---

## Body

## 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.Figure 1
Natural gear of theIssus planthopper [21].
(a)(b)As shown in Figure2, 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.Figure 2
Tooth profile of the asymmetric logarithmic spiral gear.The rest of this paper is organized as follows: Section2 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. Figure3(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.Figure 3
Introduction of the asymmetric logarithmic spiral gear tooth profile.
(a)(b)


### 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.Figure 4
Schematic diagram of the gear 1 convex tooth surface formation.When the gear 1 rotates clockwise at an angular velocity ofω1, 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 profileC1 equation in the coordinate system:(1)xs1θ=a1−f1,ys1θ=−b1−w1,zs1θ=0,where a1=r1et×θi1sinθi1, 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, b1=r1et×θi1cosθi1, 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:(2)xp1yp1zp11=10000cosβsinβμsinβ0−sinβcosβμcosβ1000xs1ys1zs11.The equation of the tooth profileC1 in the coordinate system Sp1(Op1-xp1yp1zp1) can be obtained.(3)xp1=a1−f1,yp1=−b1−w1cosβ+μ1sinβ,zp1=b1−w1sinβ+μ1cosβ.The expression of the contact condition can be derived by using the instantaneous rotation axis method [23].(4)b1−w1cosβ−μ1sinβ+R1φ1⋅−tb1+a1+a1−f1ta1+b1cosβ=0.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.(5)x1=a1−f1+R1cosφ1−a1−f1ta1+b1cosβtb1−a1sinφ1,y1=a1−f1+R1sinφ1+a1−f1ta1+b1cosβtb1−a1cosφ1,z1=b1−w1sinβ+cotβa1−f1ta1+b1cosβtb1−a1+b1−w1cosβ+R1φ1.The coordinate systems in Figure5 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.Figure 5
Schematic diagram of the gear 2 concave tooth surface formation.The concave tooth profileC2 equation in the coordinate system Ss2(Os2-xs2ys2zs2) is obtained as follows:(6)xs2θ=a2−f2,ys2θ=−b2+w2,zs2θ=0,where a2=r2et×θi2sinθi2, r2 is the initial circle radius of the concave tooth profile C2, θi2 is the angle of any point on the tooth profile C2, b2=r2et×θi2cosθi2, 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:(7)x2=a2−f2−R2cosφ2+a2−f2ta2+b2cosβtb2−a2sinφ2,y2=−a2−f2−R2sinφ2+a2−f2ta2+b2cosβtb2−a2cosφ2,z2=b2−w2sinβ+cotβa2−f2ta2+b2cosβtb2−a2+b2−w2cosβ+R2φ2.As shown in Figure6, 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 k1k2¯ intersects xs axis at point p. The length of k1p¯ is the same as the length of k2p¯.Figure 6
Coordinate transformation of concave tooth profile.The equation of the concave tooth profileC3 in the coordinate system Ss(Os-xsyszs) can be obtained as follows:(8)xs1θ=−a2−f2+c,ys1θ=−b2+w2+d,zs1θ=0,where c=2r1et×θ0sinθ0−f1,d=2r1et×θ0cosθ0−w1.The tooth surface Σ3 equation can be obtained by referring to the formation of the tooth surface Σ1.(9)x3=−a2+f2+c+R1cosφ1−−a2+f2+cta2+b2cosβtb2−a2sinφ1,y3=−a2+f2+c+R1sinφ1+−a2+f2+cta2+b2cosβtb2−a2cosφ1,z3=b2−w2+dsinβ+cotβ−a2+f2+cta1+b1cosβtb1−a1+b2−w2+dcosβ+R1φ1.Similarly, the tooth surface Σ4 equation can be obtained by referring to the formation of the tooth surface Σ2.(10)x4=−a1+f1+c−R2cosφ2+−a1+f1+cta1+b1cosβtb1−a1sinφ2,y4=−−a1+f1+c−R2sinφ2+−a1+f1+cta1+b1cosβtb1−a1cosφ2,z4=b1−w1+dsinβ+cotβa1−f1+cta1+b1cosβtb1−a1+b1−w1+dcosβ+R2φ2.


### 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 systemS(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:(11)xm1=a0−f1,ym1=a0−f1ta0+b0cosβtb0−a0,zm1=b0−w1sinβ+cotβa0−f1ta0+b0cosβtb0−a0+b0−w1cosβ+R1φ1,where a0=r1et×θ0sinθ0,b0=r1et×θ0cosθ0.Similarly, the equation of the contact line of the surface Σ2 can be obtained.(12)xm2=a3−f2,ym2=a3−f2ta3+b3cosβtb3−a3,zm2=b3−w3sinβ+cotβa3−f2ta3+b3cosβtb3−a3+b3−w2cosβ+R2φ2,where a3=r2et×θ0sinθ0,b3=r2et×θ0cosθ0.In Equation (5), when θi = θ0, the equation of the tooth surface working line of the tooth surface Σ1 is obtained as follows:(13)xw1=a0−f1+R1cosφ1−a0−f1ta0+b0cosβtb0−a0sinφ1,yw1=a0−f1+R1sinφ1+a0−f1ta0+b0cosβtb0−a0cosφ1,zw1=b0−w1sinβ+cotβa0−f1ta0+b0cosβtb0−a0+b0−w1cosβ+R1φ1.In Equation (7), when θi = θ0, the equation of the tooth surface working line of the tooth surface Σ2 is obtained as follows:(14)xw2=a3−f2−R2cosφ2+a3−f2ta3+b3cosβtb2−a2sinφ2,yw2=−a3−f2−R2sinφ2+a3−f2ta3+b3cosβtb3−a3cosφ2,zw2=b3−w2sinβ+cotβa3−f2ta3+b3cosβtb3−a3+b3−w2cosβ+R2φ2.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=x2+y2=0. By combining Equations (11) and (12), the pure rolling constraint equations can be obtained as follows:(15)f1=r1et×θ0sinθ0,f2=r2et×θ0sinθ0.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.Figure 7
Schematic diagram of the contact line and tooth surface working line.


### 2.3. Tooth Profile Parameters

Figures8(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.Figure 8
Meshing profiles of the asymmetric logarithmic spiral gear.
(a)(b)According to the geometric relationship, the calculation formulas of the tooth profile parameters are shown in Table1.Table 1
Calculation formulas of the tooth profile parameters.
ParameterComputational formulaProcess angle of convex tooth profileθ1θ1=arcsinr1et×θ0sinθ0−hg1/r1et×θ0Terminal angle of convex profileθ2θ2=arcsinha+f1/r1et×θ2Process angle of concave profileθ4θ4=arcsinr2et×θ3sinθ3−ha/r2et×θ4Terminal angle of concave profileθ5θ5=arcsinhg2+f2/r2et×θ5Gear tooth thicknessss=2r1et×θ0cosθ0−w1Space widthee=s+jArc radius at the bottom of convex tooth profilerg1rg1=r1et×θ1sinθ1+hf−f1Arc radius at the bottom of concave tooth profilerg2rg2=hf−r2et×θ5sinθ5+f2/1−cosπ/2−θ5The parametersr1, 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.Table 2
The logarithmic helical parameters range.
ParameterComputational formulaMinimum initial circle radius of the concave profiler2 minr2min=ha/et×θ3sinθ3Maximum initial circle radius of the concave profiler2 maxr2max=0.645mn/et×θ4cosθ4−et×θ3cosθ3Minimum initial circle radius of the convex profiler1 minr1min=f2−r2et×θ4sinθ4/et×θ0sinθ0Maximum initial circle radius of the convex profiler1 maxr1max=0.895mn/et×θ2cosθ2−et×θ0cosθ0


### 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.Figure 9
Slide diagram of tooth surfaces [20].Then, the slip rates of the tooth surfaces Σ1 and Σ2 can be calculated, respectively, as follows:(16)η1=limΔS1→0ΔS1−ΔS2ΔS1=dS1dm−dS2dmdS1dm=dS1−dS2dS1.(17)η2=limΔS2→0ΔS2−ΔS1ΔS2=dS2dm−dS1dmdS2dm=dS2−dS1dS2,where dS1=dx12+dy12+dz12dm,dS2=dx22+dy22+dz22dm.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).(18)dS1=a0−f1+R12+a0−f1ta0+b0cosβtb0−a02+R1cotβ2⋅ω1dm.(19)dS2=a3−f2−R22+a3−f2ta3+b3cosβtb2−a22+R2cotβ2⋅ω2dm.By combining Equations (18) and (19) with Equation (15), dS1=dS2 can be obtained. The sliding rates of the two meshing teeth surfaces are zero, which verify the gear pure rolling contact.


## 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.Figure 4
Schematic diagram of the gear 1 convex tooth surface formation.When the gear 1 rotates clockwise at an angular velocity ofω1, 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 profileC1 equation in the coordinate system:(1)xs1θ=a1−f1,ys1θ=−b1−w1,zs1θ=0,where a1=r1et×θi1sinθi1, 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, b1=r1et×θi1cosθi1, 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:(2)xp1yp1zp11=10000cosβsinβμsinβ0−sinβcosβμcosβ1000xs1ys1zs11.The equation of the tooth profileC1 in the coordinate system Sp1(Op1-xp1yp1zp1) can be obtained.(3)xp1=a1−f1,yp1=−b1−w1cosβ+μ1sinβ,zp1=b1−w1sinβ+μ1cosβ.The expression of the contact condition can be derived by using the instantaneous rotation axis method [23].(4)b1−w1cosβ−μ1sinβ+R1φ1⋅−tb1+a1+a1−f1ta1+b1cosβ=0.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.(5)x1=a1−f1+R1cosφ1−a1−f1ta1+b1cosβtb1−a1sinφ1,y1=a1−f1+R1sinφ1+a1−f1ta1+b1cosβtb1−a1cosφ1,z1=b1−w1sinβ+cotβa1−f1ta1+b1cosβtb1−a1+b1−w1cosβ+R1φ1.The coordinate systems in Figure5 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.Figure 5
Schematic diagram of the gear 2 concave tooth surface formation.The concave tooth profileC2 equation in the coordinate system Ss2(Os2-xs2ys2zs2) is obtained as follows:(6)xs2θ=a2−f2,ys2θ=−b2+w2,zs2θ=0,where a2=r2et×θi2sinθi2, r2 is the initial circle radius of the concave tooth profile C2, θi2 is the angle of any point on the tooth profile C2, b2=r2et×θi2cosθi2, 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:(7)x2=a2−f2−R2cosφ2+a2−f2ta2+b2cosβtb2−a2sinφ2,y2=−a2−f2−R2sinφ2+a2−f2ta2+b2cosβtb2−a2cosφ2,z2=b2−w2sinβ+cotβa2−f2ta2+b2cosβtb2−a2+b2−w2cosβ+R2φ2.As shown in Figure6, 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 k1k2¯ intersects xs axis at point p. The length of k1p¯ is the same as the length of k2p¯.Figure 6
Coordinate transformation of concave tooth profile.The equation of the concave tooth profileC3 in the coordinate system Ss(Os-xsyszs) can be obtained as follows:(8)xs1θ=−a2−f2+c,ys1θ=−b2+w2+d,zs1θ=0,where c=2r1et×θ0sinθ0−f1,d=2r1et×θ0cosθ0−w1.The tooth surface Σ3 equation can be obtained by referring to the formation of the tooth surface Σ1.(9)x3=−a2+f2+c+R1cosφ1−−a2+f2+cta2+b2cosβtb2−a2sinφ1,y3=−a2+f2+c+R1sinφ1+−a2+f2+cta2+b2cosβtb2−a2cosφ1,z3=b2−w2+dsinβ+cotβ−a2+f2+cta1+b1cosβtb1−a1+b2−w2+dcosβ+R1φ1.Similarly, the tooth surface Σ4 equation can be obtained by referring to the formation of the tooth surface Σ2.(10)x4=−a1+f1+c−R2cosφ2+−a1+f1+cta1+b1cosβtb1−a1sinφ2,y4=−−a1+f1+c−R2sinφ2+−a1+f1+cta1+b1cosβtb1−a1cosφ2,z4=b1−w1+dsinβ+cotβa1−f1+cta1+b1cosβtb1−a1+b1−w1+dcosβ+R2φ2.


## 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 systemS(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:(11)xm1=a0−f1,ym1=a0−f1ta0+b0cosβtb0−a0,zm1=b0−w1sinβ+cotβa0−f1ta0+b0cosβtb0−a0+b0−w1cosβ+R1φ1,where a0=r1et×θ0sinθ0,b0=r1et×θ0cosθ0.Similarly, the equation of the contact line of the surface Σ2 can be obtained.(12)xm2=a3−f2,ym2=a3−f2ta3+b3cosβtb3−a3,zm2=b3−w3sinβ+cotβa3−f2ta3+b3cosβtb3−a3+b3−w2cosβ+R2φ2,where a3=r2et×θ0sinθ0,b3=r2et×θ0cosθ0.In Equation (5), when θi = θ0, the equation of the tooth surface working line of the tooth surface Σ1 is obtained as follows:(13)xw1=a0−f1+R1cosφ1−a0−f1ta0+b0cosβtb0−a0sinφ1,yw1=a0−f1+R1sinφ1+a0−f1ta0+b0cosβtb0−a0cosφ1,zw1=b0−w1sinβ+cotβa0−f1ta0+b0cosβtb0−a0+b0−w1cosβ+R1φ1.In Equation (7), when θi = θ0, the equation of the tooth surface working line of the tooth surface Σ2 is obtained as follows:(14)xw2=a3−f2−R2cosφ2+a3−f2ta3+b3cosβtb2−a2sinφ2,yw2=−a3−f2−R2sinφ2+a3−f2ta3+b3cosβtb3−a3cosφ2,zw2=b3−w2sinβ+cotβa3−f2ta3+b3cosβtb3−a3+b3−w2cosβ+R2φ2.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=x2+y2=0. By combining Equations (11) and (12), the pure rolling constraint equations can be obtained as follows:(15)f1=r1et×θ0sinθ0,f2=r2et×θ0sinθ0.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.Figure 7
Schematic diagram of the contact line and tooth surface working line.


## 2.3. Tooth Profile Parameters

Figures8(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.Figure 8
Meshing profiles of the asymmetric logarithmic spiral gear.
(a)(b)According to the geometric relationship, the calculation formulas of the tooth profile parameters are shown in Table1.Table 1
Calculation formulas of the tooth profile parameters.
ParameterComputational formulaProcess angle of convex tooth profileθ1θ1=arcsinr1et×θ0sinθ0−hg1/r1et×θ0Terminal angle of convex profileθ2θ2=arcsinha+f1/r1et×θ2Process angle of concave profileθ4θ4=arcsinr2et×θ3sinθ3−ha/r2et×θ4Terminal angle of concave profileθ5θ5=arcsinhg2+f2/r2et×θ5Gear tooth thicknessss=2r1et×θ0cosθ0−w1Space widthee=s+jArc radius at the bottom of convex tooth profilerg1rg1=r1et×θ1sinθ1+hf−f1Arc radius at the bottom of concave tooth profilerg2rg2=hf−r2et×θ5sinθ5+f2/1−cosπ/2−θ5The parametersr1, 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.Table 2
The logarithmic helical parameters range.
ParameterComputational formulaMinimum initial circle radius of the concave profiler2 minr2min=ha/et×θ3sinθ3Maximum initial circle radius of the concave profiler2 maxr2max=0.645mn/et×θ4cosθ4−et×θ3cosθ3Minimum initial circle radius of the convex profiler1 minr1min=f2−r2et×θ4sinθ4/et×θ0sinθ0Maximum initial circle radius of the convex profiler1 maxr1max=0.895mn/et×θ2cosθ2−et×θ0cosθ0


## 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.Figure 9
Slide diagram of tooth surfaces [20].Then, the slip rates of the tooth surfaces Σ1 and Σ2 can be calculated, respectively, as follows:(16)η1=limΔS1→0ΔS1−ΔS2ΔS1=dS1dm−dS2dmdS1dm=dS1−dS2dS1.(17)η2=limΔS2→0ΔS2−ΔS1ΔS2=dS2dm−dS1dmdS2dm=dS2−dS1dS2,where dS1=dx12+dy12+dz12dm,dS2=dx22+dy22+dz22dm.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).(18)dS1=a0−f1+R12+a0−f1ta0+b0cosβtb0−a02+R1cotβ2⋅ω1dm.(19)dS2=a3−f2−R22+a3−f2ta3+b3cosβtb2−a22+R2cotβ2⋅ω2dm.By combining Equations (18) and (19) with Equation (15), dS1=dS2 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 ofθ1 and θ5. Then, the overlapping interference of the end face profile is examined.


### 3.1. Geometric Interference of Tooth Profile

As shown in Figure8, 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.(20)r2et×θ5×sinθ5−f2>r1et×θ2×sinθ2−f1,f1−r1et×θ1sinθ1>f2−r2et×θ4sinθ4.In the gear design process, when the basic parametersha, 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 pointK on the surface along any curve L on the surface is expressed as drk/dm=∂ru/∂udu/dm+∂rv/∂v∂v/dm, 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.Figure 10
The guide vector of the surface.The following boundary equations can be derived by referring to the root cutting conditions of the circular arc gear [23]:(21)cosθix=0x=1,2.(22)sin4θix+Rxrxet×θisin2β−fxrxet×θisin3θix−fx2rxet×θix2cot2β+fxrxet×θixcot2βsinθix=0x=1,2,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 Figure11(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.Figure 11
Interference diagram of the tooth profile end points.
(a)(b)Figure 12
Schematic diagram of polar coordinates of tooth a.Figure11(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 forys1 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:(23)φx=arctan⁡±rxetθix⋅cos⁡θix−rxetθ0⋅cos⁡θ0+s1/2R1+rxetθix⋅sin⁡θix−rxetθ0⋅sin⁡θ0⋅cos⁡βx=1,2.(24)RPx=±rxetθix⋅cosθix−rxetθ0⋅cosθ0+s1/2sinφx⋅cosβx=1,2,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 Curvesσ1 and σ2

As shown in Figure13, 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.Figure 13
Relative position of the tooth profiles during the gear transmission.
(a)(b)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:(25)θσx=arcsin⁡4a2RBx2−a2+RBx2−RPx222aRPx+z2z1⋅arccos⁡a2+RBx2−RPx22aRBx−θBx+s1+s2mnz1x=1,2,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.Figure 14
Geometric parameters of gear 2.


## 3.1. Geometric Interference of Tooth Profile

As shown in Figure8, 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.(20)r2et×θ5×sinθ5−f2>r1et×θ2×sinθ2−f1,f1−r1et×θ1sinθ1>f2−r2et×θ4sinθ4.In the gear design process, when the basic parametersha, 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 pointK on the surface along any curve L on the surface is expressed as drk/dm=∂ru/∂udu/dm+∂rv/∂v∂v/dm, 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.Figure 10
The guide vector of the surface.The following boundary equations can be derived by referring to the root cutting conditions of the circular arc gear [23]:(21)cosθix=0x=1,2.(22)sin4θix+Rxrxet×θisin2β−fxrxet×θisin3θix−fx2rxet×θix2cot2β+fxrxet×θixcot2βsinθix=0x=1,2,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 Figure11(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.Figure 11
Interference diagram of the tooth profile end points.
(a)(b)Figure 12
Schematic diagram of polar coordinates of tooth a.Figure11(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 forys1 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:(23)φx=arctan⁡±rxetθix⋅cos⁡θix−rxetθ0⋅cos⁡θ0+s1/2R1+rxetθix⋅sin⁡θix−rxetθ0⋅sin⁡θ0⋅cos⁡βx=1,2.(24)RPx=±rxetθix⋅cosθix−rxetθ0⋅cosθ0+s1/2sinφx⋅cosβx=1,2,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 Curvesσ1 and σ2

As shown in Figure13, 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.Figure 13
Relative position of the tooth profiles during the gear transmission.
(a)(b)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:(25)θσx=arcsin⁡4a2RBx2−a2+RBx2−RPx222aRPx+z2z1⋅arccos⁡a2+RBx2−RPx22aRBx−θBx+s1+s2mnz1x=1,2,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.Figure 14
Geometric parameters of gear 2.


## 3.3.1. Polar Coordinate Equation of the Tooth a Profiles of the Gear 1 on the End Face

The equations forys1 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:(23)φx=arctan⁡±rxetθix⋅cos⁡θix−rxetθ0⋅cos⁡θ0+s1/2R1+rxetθix⋅sin⁡θix−rxetθ0⋅sin⁡θ0⋅cos⁡βx=1,2.(24)RPx=±rxetθix⋅cosθix−rxetθ0⋅cosθ0+s1/2sinφx⋅cosβx=1,2,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 Curvesσ1 and σ2

As shown in Figure13, 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.Figure 13
Relative position of the tooth profiles during the gear transmission.
(a)(b)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:(25)θσx=arcsin⁡4a2RBx2−a2+RBx2−RPx222aRPx+z2z1⋅arccos⁡a2+RBx2−RPx22aRBx−θBx+s1+s2mnz1x=1,2,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.Figure 14
Geometric parameters of gear 2.


## 4. Modeling and Stress Analysis

### 4.1. Basic Parameters

The basic parameters of the asymmetric logarithmic spiral gear are shown in Table3.Table 3
Basic parameters of the asymmetric logarithmic spiral gear.
ParameterValueParameterValueNumber of teethz1/z242/42Gear tooth thicknesss (mm)4.62Normal modulemn (mm)3Space widthe (mm)4.8Tooth widthb (mm)44Standard pitch diameterd (mm)133.0008Spiral angleβ (°)18.67Tip diameterda (mm)135.4008Pressure angleθ0/θ3 (°)24Root diameterdf (mm)129.4008Center distancea (mm)133.0008Tooth addendumha (mm)1.2Side clearancej (mm)0.18Tooth dedendumhf (mm)1.8To 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 Table2, 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.Table 4
The parameters of the logarithmic spiral tooth profile.
ParameterNumerical value of convex tooth profileNumerical value of concave tooth profileConvex and concave tooth profile initial circle radiusr1/r2 (mm)4.28.4Logarithmic screw coefficientt−2.5 × 10−3−2.5 × 10−3Convex and concave tooth profile process angleθ1/θ4 (°)–14.4189Convex and concave profile termination angleθ2/θ5 (°)49.1265–Lateral offset of logarithmic spiral tooth profile centerw1/w2 (mm)1.30344.8269Longitudinal offset of logarithmic spiral tooth profile centerf1/f2 (mm)1.60883.2176


### 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.Table 5
The ranges of the anglesθ1 and θ5.
ParameterAngle rangeProcess angle convex tooth profileθ1 (°)(0, 5.6657)End angle of concave profileθ5 (°)(35.0327, 41.5016)In Table5, 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.Figure 15
Judgment of the end face tooth profile interference.
(a)(b)


### 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 Figures16(a) and 16(b). The complete gear model can be established by array method.Figure 16
Single tooth formation process of the asymmetric logarithmic spiral gear.
(a)(b)


#### 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 Figure17, there are two contact points between the two gears and no interference occurs elsewhere. Therefore, the above selected gear parameters meet the requirements.Figure 17
The gears at different turning angles under load: (a) 0°; (b) 10°.
(a)(b)


#### 4.3.2. Verification of the Pure Rolling Meshing

As shown in Figure18, 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.Figure 18
The gears at different turning angles without load: (a) 0°; (b) 10°.
(a)(b)


### 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 Figure19(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=2.09×105 MPa and Poisson’s ratio μ=0.3. The input torque is set as 100 Nm.Figure 19
The finite element setting of the asymmetric logarithmic spiral gear.
(a)(b)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 Table6.Table 6
The grid of the contact teeth surfaces and corresponding maximum contact stresses.
The grid of the contact teeth surfaces (mm)The maximum contact stress value (MPa)0.4577.370.3749.850.2760.330.15762.430.1763.11As can be seen from Table6, 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 Figure20.Figure 20
Contact stress: (a) and (b) are the driving and driven gears of the asymmetric logarithmic spiral gear, respectively; (c) and (d) are the driving and driven gears of the pure rolling single arc gear, respectively.
(a)(b)(c)(d)As can be seen from Figure20, 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 Figure21.Figure 21
Change of the maximum contact stress at different angles of the gears.It can be seen from Figure21 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 Figure22, 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.Figure 22
Comparison of the gears tooth shape.


#### 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 Figure23.Figure 23
Bending stress: (a) and (b) are the driving and driven gears of the asymmetric logarithmic spiral gear, respectively; (c) and (d) are the driving and driven gears of the pure rolling single arc gear, respectively.
(a)(b)(c)(d)As can be seen from Figure23, 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 Figure24.Figure 24
Change of the maximum bending stress at different angles of the gears.It can be seen from Figure24 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 Figure25, 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.Figure 25
Comparison of the gears tooth shape.In summary, the asymmetric logarithmic spiral gear shows lower maximum contact and bending stresses than that of the pure rolling single arc gear.


## 4.1. Basic Parameters

The basic parameters of the asymmetric logarithmic spiral gear are shown in Table3.Table 3
Basic parameters of the asymmetric logarithmic spiral gear.
ParameterValueParameterValueNumber of teethz1/z242/42Gear tooth thicknesss (mm)4.62Normal modulemn (mm)3Space widthe (mm)4.8Tooth widthb (mm)44Standard pitch diameterd (mm)133.0008Spiral angleβ (°)18.67Tip diameterda (mm)135.4008Pressure angleθ0/θ3 (°)24Root diameterdf (mm)129.4008Center distancea (mm)133.0008Tooth addendumha (mm)1.2Side clearancej (mm)0.18Tooth dedendumhf (mm)1.8To 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 Table2, 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.Table 4
The parameters of the logarithmic spiral tooth profile.
ParameterNumerical value of convex tooth profileNumerical value of concave tooth profileConvex and concave tooth profile initial circle radiusr1/r2 (mm)4.28.4Logarithmic screw coefficientt−2.5 × 10−3−2.5 × 10−3Convex and concave tooth profile process angleθ1/θ4 (°)–14.4189Convex and concave profile termination angleθ2/θ5 (°)49.1265–Lateral offset of logarithmic spiral tooth profile centerw1/w2 (mm)1.30344.8269Longitudinal offset of logarithmic spiral tooth profile centerf1/f2 (mm)1.60883.2176


## 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.Table 5
The ranges of the anglesθ1 and θ5.
ParameterAngle rangeProcess angle convex tooth profileθ1 (°)(0, 5.6657)End angle of concave profileθ5 (°)(35.0327, 41.5016)In Table5, 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.Figure 15
Judgment of the end face tooth profile interference.
(a)(b)


## 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 Figures16(a) and 16(b). The complete gear model can be established by array method.Figure 16
Single tooth formation process of the asymmetric logarithmic spiral gear.
(a)(b)


### 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 Figure17, there are two contact points between the two gears and no interference occurs elsewhere. Therefore, the above selected gear parameters meet the requirements.Figure 17
The gears at different turning angles under load: (a) 0°; (b) 10°.
(a)(b)


### 4.3.2. Verification of the Pure Rolling Meshing

As shown in Figure18, 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.Figure 18
The gears at different turning angles without load: (a) 0°; (b) 10°.
(a)(b)


## 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 Figure17, there are two contact points between the two gears and no interference occurs elsewhere. Therefore, the above selected gear parameters meet the requirements.Figure 17
The gears at different turning angles under load: (a) 0°; (b) 10°.
(a)(b)


## 4.3.2. Verification of the Pure Rolling Meshing

As shown in Figure18, 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.Figure 18
The gears at different turning angles without load: (a) 0°; (b) 10°.
(a)(b)


## 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 Figure19(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=2.09×105 MPa and Poisson’s ratio μ=0.3. The input torque is set as 100 Nm.Figure 19
The finite element setting of the asymmetric logarithmic spiral gear.
(a)(b)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 Table6.Table 6
The grid of the contact teeth surfaces and corresponding maximum contact stresses.
The grid of the contact teeth surfaces (mm)The maximum contact stress value (MPa)0.4577.370.3749.850.2760.330.15762.430.1763.11As can be seen from Table6, 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 Figure20.Figure 20
Contact stress: (a) and (b) are the driving and driven gears of the asymmetric logarithmic spiral gear, respectively; (c) and (d) are the driving and driven gears of the pure rolling single arc gear, respectively.
(a)(b)(c)(d)As can be seen from Figure20, 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 Figure21.Figure 21
Change of the maximum contact stress at different angles of the gears.It can be seen from Figure21 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 Figure22, 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.Figure 22
Comparison of the gears tooth shape.


### 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 Figure23.Figure 23
Bending stress: (a) and (b) are the driving and driven gears of the asymmetric logarithmic spiral gear, respectively; (c) and (d) are the driving and driven gears of the pure rolling single arc gear, respectively.
(a)(b)(c)(d)As can be seen from Figure23, 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 Figure24.Figure 24
Change of the maximum bending stress at different angles of the gears.It can be seen from Figure24 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 Figure25, 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.Figure 25
Comparison of the gears tooth shape.In summary, the asymmetric logarithmic spiral gear shows lower maximum contact and bending stresses than that of the pure rolling single arc gear.


## 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 Figure19(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=2.09×105 MPa and Poisson’s ratio μ=0.3. The input torque is set as 100 Nm.Figure 19
The finite element setting of the asymmetric logarithmic spiral gear.
(a)(b)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 Table6.Table 6
The grid of the contact teeth surfaces and corresponding maximum contact stresses.
The grid of the contact teeth surfaces (mm)The maximum contact stress value (MPa)0.4577.370.3749.850.2760.330.15762.430.1763.11As can be seen from Table6, 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 Figure20.Figure 20
Contact stress: (a) and (b) are the driving and driven gears of the asymmetric logarithmic spiral gear, respectively; (c) and (d) are the driving and driven gears of the pure rolling single arc gear, respectively.
(a)(b)(c)(d)As can be seen from Figure20, 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 Figure21.Figure 21
Change of the maximum contact stress at different angles of the gears.It can be seen from Figure21 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 Figure22, 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.Figure 22
Comparison of the gears tooth shape.


## 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 Figure23.Figure 23
Bending stress: (a) and (b) are the driving and driven gears of the asymmetric logarithmic spiral gear, respectively; (c) and (d) are the driving and driven gears of the pure rolling single arc gear, respectively.
(a)(b)(c)(d)As can be seen from Figure23, 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 Figure24.Figure 24
Change of the maximum bending stress at different angles of the gears.It can be seen from Figure24 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 Figure25, 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.Figure 25
Comparison of the gears tooth shape.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 theIssus 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.



---

*Source: 8550869-2022-12-26.xml*