A Rapid Design Method for Bidirectional Transmission Parallel-Axis External Line Gears

Abstract

To address the limited design flexibility in meshing equations, complex modeling processes, and a lack of systematic research on bidirectional transmission in traditional parallel-axis line gears, this paper proposes a rapid design method for bidirectional transmission parallel-axis external line gears (BTPELG). Firstly, a multi-coordinate system is established, and homogeneous transformation matrices are derived. Secondly, the meshing equation is extended with an angular parameter to obtain a widely applicable conjugate condition. Then, two pairs of conjugate contact curves and center guide curves are constructed, and the offset vector for a circular-arc tooth profile is derived. Subsequently, taking the equidistant cylindrical helix as the driving contact curve, a 3D solid model is built via the sweeping method. Finally, kinematic simulations and experiments on 3D-printed prototypes verify the transmission ratios under forward and reverse conditions. Results show stable bidirectional transmission with average ratios matching theory and small fluctuations, confirming the method’s feasibility and providing a reference for rapid design and engineering applications.

1. Introduction

Line gear (LG) is a novel gear mechanism based on the spatial curve meshing theory [1]. Unlike traditional gears based on spatial surface meshing (such as involute gears, circular-arc gears, etc.), line gears transmit motion and power through continuous point contact of a pair of spatial conjugate curves—namely, the driving contact curve and the driven contact curve. This unique meshing principle endows line gears with a range of advantages: compact structure, small size, light weight, capability for large transmission ratios, a minimum tooth number of one, design flexibility, etc., making them particularly suitable for applications with strict space and weight constraints, such as micro-mechanisms, robotic joints, and medical devices [2,3].

The spatial curve meshing equation is the cornerstone of line gear theory. Chen et al. [1] first established the basic form of the spatial curve meshing equation in 2007: ${\mathit{v}}_{12}\cdot \mathit{\beta }=0$, where ${\mathit{v}}_{12}$ is the relative velocity of the driving and driven gears at the contact point, and $\mathit{\beta }$ is the principal normal vector of the driving contact curve. This equation ensures continuous contact of the conjugate curves during meshing process by constraining the dot product of the relative velocity and the principal normal vector to zero. Subsequently, the meshing equation was revised to account for the influence of the gear tooth diameter, improving the accuracy of the design [4]. Then, the meshing equation was further extended to intersecting axes at arbitrary angles and skew axes [5], establishing a unified theoretical framework applicable to various transmission configurations. Chen et al. further expanded this theory by proposing a pure rolling gear design method based on the active design of meshing curve functions, achieving active control over transmission errors and contact performance [6,7]. In terms of the solution and application of meshing equation, researchers have developed a simplified design method based on screw theory [8] and an improved design method based on conjugate curve constraints [9], significantly streamlining the design process of line gears.

The construction of geometric models for line gears is a key link connecting theory and application. Traditional methods typically require solving complex tooth surface equations first, followed by generating solid models through sweeping or surface fitting. This process is cumbersome and demands high expertise from designers [10,11,12]. To simplify the modeling process, researchers have explored direct modeling methods based on the “center guide curve” (the trajectory of the tooth profile center), as well as modeling strategies employing different tooth profile forms such as circular and elliptical arcs.

Bidirectional transmission is a key function for the engineering application of line gears. Li et al. [13] proposed a novel line tooth structure with bilateral contact curves, achieving backlash-free transmission in both forward and reverse directions through conjugate curve matching, thereby laying a structural foundation for bidirectional transmission. He et al. [14] designed line gears with concave-convex circular-arc tooth profiles, optimized the non-interference condition of the tooth surfaces, improved the load-carrying capacity of bidirectional meshing, and provided support for optimizing the smoothness and efficiency of bidirectional transmission.

However, in the aforementioned studies, the normal direction in the meshing equations is defined as the principal normal vector at the point of contact on the contact curve. While this constraint ensures the simplicity of the equation form, it also limits design flexibility. When it is necessary to optimize transmission performance by adjusting the morphology of the contact curve, the fixed normal direction cannot accommodate more flexible design requirements. In addition, traditional methods for modeling line gears typically require first solving complex tooth surface equations and then constructing a solid model through sweeping or surface fitting. This process is relatively cumbersome and does not facilitate rapid design verification in the early stages. Furthermore, the existing design methods for bidirectional transmission line gears involve numerous geometric parameters and complex constraint conditions, making the design process relatively cumbersome and unfavorable for rapid iteration and application in engineering practice. Moreover, systematic research on the rapid design methods for BTPELG is still lacking.

To address the above issues, this paper aims to propose a rapid design method for BTPELG. This method extends the traditional meshing equations, providing theoretical flexibility for optimizing transmission performance by defining two contact curves on a single gear tooth, corresponding to forward and reverse transmissions, respectively. Based on the meshing equations and coordinate transformations, the contact curves and center guide curves required for solid modeling are derived. Subsequently, the solid model of the line gear is created directly using the sweeping method, thereby avoiding the derivation of complex tooth surface equations. Finally, the validity of the proposed method is verified through kinematic simulations of examples and kinematic experiments on 3D-printed prototypes. This study provides a theoretical foundation and technical support for the engineering application of line gears in mechanical systems requiring bidirectional transmission.

2. Design of the BTPELG

2.1. Coordinate System and Coordinate Transformation

The meshing coordinate system of parallel-axis line gears is shown in Figure 1. In the spatial four-coordinate system, two fixed Cartesian coordinate systems $S_0(o_0\text{-}{x}_0{y}_0{z}_0)$ and $S_{\mathrm{p}}(o_{\mathrm{p}}\text{-}{x}_{\mathrm{p}}{y}_{\mathrm{p}}{z}_{\mathrm{p}})$ and two rotating Cartesian coordinate systems $S_1(o_1\text{-}{x}_1{y}_1{z}_1)$ and $S_2(o_2\text{-}{x}_2{y}_2{z}_2)$ are given, where coordinate system $S_1$ is fixed to the driving gear and coordinate system $S_2$ is fixed to the driven gear. At the initial meshing position, coordinate systems $S_1$ and $S_2$ coincide with $S_0$ and $S_{\mathrm{p}}$, respectively. The driving gear rotates about the $z$-axis of coordinate system $S_0$ with an angular velocity ${\omega }_1$, and the driven gear rotates about the $z_{\mathrm{p}}$-axis of coordinate system $S_{\mathrm{p}}$ with an angular velocity ${\omega }_2$. The rotation angles over a period of time t are ${\varphi }_1$ and ${\varphi }_2$, respectively. $a$ is the distance between the $z$-axis of coordinate systems $S_0$ and $S_{\mathrm{p}}$, and $b$ is the width of the gear. $Q_g^{(1)}$ and $Q_n^{(2)}$ represent two meshing conjugate curves, where $Q_g^{(1)}$ is the driving contact curve and $Q_n^{(2)}$ is the driven contact curve, with the superscripts indicating the coordinate system. $M$ denotes the meshing point of the driving contact curve $Q_g^{(1)}$ and the driven contact curve $Q_n^{(2)}$ at a certain instant. According to the definition of the transmission ratio, the relationship between the transmission ratio $i_{12}$, the angular velocities and the rotation angles is shown in Equation (1).

$$i_{12}={\displaystyle \frac{{\omega }_1}{{\omega }_2}}={\displaystyle \frac{{\varphi }_1}{{\varphi }_2}}$$

(1)

The coordinate transformation matrix $M_{01}$ between coordinate systems $S_0$ and $S_1$ is shown in Equation (2).

$$M_{01}=\left[\begin{array}{cccc}\cos {\varphi }_1& \sin {\varphi }_1& 0& 0\\ -\sin {\varphi }_1& \cos {\varphi }_1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$$

(2)

The coordinate transformation matrix $M_{2\mathrm{p}}$ between coordinate systems $S_2$ and $S_{\mathrm{p}}$ is shown in Equation (3).

$$M_{2\mathrm{p}}=\left[\begin{array}{cccc}\cos {\varphi }_2& \sin {\varphi }_2& 0& 0\\ -\sin {\varphi }_2& \cos {\varphi }_2& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$$

(3)

The coordinate transformation matrix $M_{\mathrm{p}0}$ between coordinate systems $S_{\mathrm{p}}$ and $S_0$ is shown in Equation (4).

$$M_{\mathrm{p}0}=\left[\begin{array}{cccc}1& 0& 0& a\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$$

(4)

From Equations (2) to (4), the transformation matrix $M_{21}$ between coordinate systems $S_1$ and $S_2$ can be obtained as shown in Equation (5).

$$M_{21}=M_{2\mathrm{p}}\cdot M_{\mathrm{p}0}\cdot M_{01}=\left[\begin{array}{cccc}\cos ({\varphi }_1+{\varphi }_2)& \sin ({\varphi }_1+{\varphi }_2)& 0& a\cos {\varphi }_2\\ -\sin ({\varphi }_1+{\varphi }_2)& \cos ({\varphi }_1+{\varphi }_2)& 0& -a\sin {\varphi }_2\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$$

(5)

2.2. Meshing Equation

In traditional parallel-axis line gear design, an equation for the driving contact curve is typically specified first. The equation for the driven contact curve is then derived using coordinate transformations and the meshing equation ${\mathit{v}}_{12}\cdot \mathit{\beta }=0$, followed by modeling via a sweeping method. Here, ${\mathit{v}}_{12}$ is the relative velocity of the driving and driven gears at the point of contact, and $\mathit{\beta }$ is the principal normal vector of the driving contact curve in the fixed coordinate system $S_0$. While maintaining design simplicity, the above meshing equation also provides effective support for the preliminary configuration of line gears. However, to optimize the contact path and improve overall meshing performance, the meshing equation is extended to the form shown in Equation (6).

$$\{\begin{array}{l}{\mathit{v}}_{12}^{(0)}\cdot {\mathit{n}}^{(0)}=0\\ {\mathit{n}}^{(1)}={\mathit{\beta }}^{(1)}\cos \phi +{\mathit{\gamma }}^{(1)}\sin \phi \end{array}$$

(6)

where ${\mathit{v}}_{12}^{(0)}$ is the relative velocity of the contact point in coordinate system $S_0$. ${\mathit{n}}^{(0)}$ and ${\mathit{n}}^{(1)}$ are the unit common normal vectors of the two contact curves at the contact point in coordinate systems $S_0$ and $S_1$, respectively. And ${\mathit{n}}^{(0)}$ can be obtained through a coordinate transformation from ${\mathit{n}}^{(1)}$. ${\mathit{\beta }}^{(1)}$ and ${\mathit{\gamma }}^{(1)}$ are the principal and binormal vectors of the contact curve in coordinate system $S_1$. $\phi$ is an angular parameter whose value lies in the range [0, π/2]. By introducing the angular parameter, the direction of the common normal is defined within the complete normal plane spanned by ${\mathit{\beta }}^{(1)}$ and ${\mathit{\gamma }}^{(1)}$, thereby providing the necessary theoretical freedom for subsequent active design of transmission errors and optimization of contact performance by adjusting $\phi$.

Given that the parametric equation of the driving contact curve $Q_g^{(1)}$ in coordinate system $S_1$ is shown in Equation (7).

$$Q_g^{(1)}={(x^{(1)},y^{(1)},z^{(1)})}^T$$

(7)

Then, the unit tangent vector, principal normal vector, and binormal vector of the driving contact curve can be obtained as shown in Equation (8).

$$\{\begin{array}{l}{\mathit{\alpha }}^{(1)}={\dot{Q}}_g^{(1)}/\left|{\dot{Q}}_g^{(1)}\right|\\ {\mathit{\beta }}^{(1)}={\dot{\mathit{\alpha }}}^{(1)}/\left|{\dot{\mathit{\alpha }}}^{(1)}\right|\\ {\mathit{\gamma }}^{(1)}={\mathit{\alpha }}^{(1)}\times {\mathit{\beta }}^{(1)}\end{array}$$

(8)

From Equations (2), (6) and (8), the common normal vector ${\mathit{n}}^{\left(0\right)}$ in coordinate system $S_0$ can be obtained.

In Figure 1, let the coordinates of the meshing point $M$ in coordinate system $S_0$ be $(x_M^{(0)},y_M^{(0)},z_M^{(0)})$, let ${\mathit{q}}^{(0)}$ be the position vector of the driving contact curve, and ${\mathit{q}}^{(\mathrm{p})}$ be the position vector of the driven contact curve. Let $\overrightarrow{O_{\mathrm{P}}O}=a\mathit{i}$, then

$$\{\begin{array}{l}{\mathit{q}}^{(0)}=\overrightarrow{OM}=(x_M^{(0)},y_M^{(0)},z_M^{(0)})\\ {\mathit{q}}^{(\mathrm{p})}=\overrightarrow{O_{\mathrm{p}}M}=(x_M^{(0)}+a,y_M^{(0)},z_M^{(0)})\end{array}$$

(9)

Then the velocity ${\mathit{v}}_1$ of the meshing point $M$ when moving with the driving gear and the velocity ${\mathit{v}}_2$ when moving with the driven gear are shown in Equation (10).

$$\{\begin{array}{l}{\mathit{v}}_1={\mathit{\varpi }}_1\times \overrightarrow{OM}={\mathit{\varpi }}_1\times {\mathit{q}}^{(0)}\\ {\mathit{v}}_2={\mathit{\varpi }}_2\times \overrightarrow{O_{\mathrm{p}}M}={\mathit{\varpi }}_2\times {\mathit{q}}^{(\mathrm{p})}\\ {\mathit{\varpi }}_1=(0,0,-{\omega }_1{)}^T\\ {\mathit{\varpi }}_2=(0,0,{\omega }_2{)}^T\end{array}$$

(10)

From Equations (9) and (10), ${\mathit{v}}_1$ and ${\mathit{v}}_2$ can be obtained as shown in Equation (11).

$$\{\begin{array}{l}{\mathit{v}}_1=({\omega }_{1}y,-{\omega }_{1}x,0)\\ {\mathit{v}}_2=(-{\omega }_{2}y,{\omega }_2(x+a),0)\end{array}$$

(11)

Thus, the relative velocity ${\mathit{v}}_{12}^{(0)}$ of the two line gears at the meshing point is obtained as shown in Equation (12).

$${\mathit{v}}_{12}^{(0)}={\mathit{v}}_1-{\mathit{v}}_2=\left[\begin{array}{l}({\omega }_1+{\omega }_2)y\\ -({\omega }_1+{\omega }_2)x-a{\omega }_2\\ 0\end{array}\right]$$

(12)

From Equations (2) to (6) and Equation (12), the relationship between ${\varphi }_1$ and $t$ can be obtained, and then the equations for the contact curves and the center guide curves can then be derived.

2.3. Solution of the Contact Curve Equation and the Center Guide Curve Equation

2.3.1. Contact Curve Equation

The transmission of a line gear pair is realized through the meshing of two contact curves located on the driving line gear and the driven line gear, respectively. To achieve bidirectional transmission (forward and reverse rotation) of the line gear, each single tooth must have one contact curve distributed on each of its two tooth faces. That is, a single tooth of the driving gear has driving contact curve 1 and driving contact curve 2, and a single tooth of the driven gear correspondingly has driven contact curve 1 and driven contact curve 2. Driving contact curve 1 is generally predetermined.

As shown in Figure 2, assuming that during forward rotation of the line gear pair, driving contact curve 1 meshes conjugately with driven contact curve 1, then driven contact curve 1 can be obtained from driving contact curve 1 through coordinate transformation. Driving contact curve 2 is obtained by rotating driving contact curve 1 by an angle ${\theta }_1$. For driven contact curve 2, when the line gear pair rotates in reverse, it is not driven contact curve 2 that meshes directly with driving contact curve 2, but rather a driven contact curve 3 on the adjacent tooth surface. Driven contact curve 3 can be obtained from driving contact curve 2 through coordinate transformation. Since the tooth profiles are evenly distributed circumferentially on the gear body, the driven contact curve 2 used for modeling can be obtained either by rotating driven contact curve 3 around the driven gear axis by an angle $2{\theta }_2$, or by rotating driven contact curve 1 around the gear axis by ${\theta }_2$. The values of the rotation angles ${\theta }_1$ and ${\theta }_2$ are determined by the number of teeth $N_1$ on the driving gear and the transmission ratio, where ${\theta }_1=\pi /N_1$ and ${\theta }_2={\theta }_1/i_{12}$.

Given the equation of driving contact curve 1 in coordinate system $S_1$ as shown in Equation (13).

$${{\displaystyle Q}}_{g1}^{(1)}:\{\begin{array}{l}{x}_{g1}^{(1)}=x_1^{(1)}(t)\\ y_{g1}^{(1)}=y_1^{(1)}(t)\\ z_{g1}^{(1)}=z_1^{(1)}(t)\end{array}\hspace{1em}(t_s\le t\le t_e)$$

(13)

In Equation (13), $t$ is the independent variable of the constant-pitch cylindrical helix, where $t_{\mathrm{s}}$ is the variable value at the starting point of and $t_{\mathrm{e}}$ is the variable value at the ending point.

Based on Equations (5) and (13), the parametric equation of driven contact curve 1 corresponding to driving contact curve 1 in coordinate system $S_2(o_2\text{-}{x}_2{y}_2{z}_2)$ can be obtained as shown in Equation (14).

$${{\displaystyle Q}}_{n1}^{(2)}:\{\begin{array}{l}{x}_{n1}^{(2)}=x_1^{(1)}\cos ({\varphi }_1+{\varphi }_2)+y_1^{(1)}\sin ({\varphi }_1+{\varphi }_2)+a\cos {\varphi }_2\\ y_{n1}^{(2)}=y_1^{(1)}\cos ({\varphi }_1+{\varphi }_2)-x_1^{(1)}\sin ({\varphi }_1+{\varphi }_2)-a\sin {\varphi }_2\\ z_{n1}^{(2)}=z_1^{(1)}\end{array}$$

(14)

Rotating the driving contact curve 1 by an angle ${\theta }_1$ yields the parametric equation of the driving contact curve 2 in coordinate system $S_1(o_1\text{-}{x}_1{y}_1{z}_1)$ as shown in Equation (15).

$${{\displaystyle Q}}_{g2}^{(1)}:\{\begin{array}{l}{x}_{g2}^{(1)}=x_1^{(1)}\cos {\theta }_1+y_1^{(1)}\sin {\theta }_1\\ y_{g2}^{(1)}=y_1^{(1)}\cos {\theta }_1-x_1^{(1)}\sin {\theta }_1\\ z_{g2}^{(1)}=z_1^{(1)}\end{array}$$

(15)

Similarly, in coordinate system $S_2(o_2\text{-}{x}_2{y}_2{z}_2)$, the parametric equation of the driven contact curve 2 can be obtained by rotating the driven contact line 1 around the $z_{\mathrm{p}}$-axis by ${\theta }_2$, as shown in Equation (16).

$${{\displaystyle Q}}_{n2}^{(2)}:\{\begin{array}{l}{x}_{n2}^{(2)}=x_1^{(1)}\cos ({\varphi }_1+{\varphi }_2+{\theta }_2)+y_1^{(1)}\sin ({\varphi }_1+{\varphi }_2+{\theta }_2)+a\cos ({\varphi }_2+{\theta }_2)\\ y_{n2}^{(2)}=y_1^{(1)}\cos ({\varphi }_1+{\varphi }_2+{\theta }_2)-x_1^{(1)}\sin ({\varphi }_1+{\varphi }_2+{\theta }_2)-a\sin ({\varphi }_2+{\theta }_2)\\ z_{n2}^{(2)}=z_1^{(1)}\end{array}$$

(16)

2.3.2. Center Guide Curve Equation

To obtain an accurate geometric model of the line gear tooth using the sweeping method and to avoid interference during meshing, it is required that, during the modeling process, the normal tooth profile of the line gear tooth not only sweeps along the contact curve but also requires an auxiliary curve to guide the twisting amplitude of the generated tooth surface. The tooth profile of the line gear tooth can be designed as any cross-sectional shape, such as a circular arc, elliptical arc, involute, cycloid, etc. In this paper, a circular-arc tooth profile is selected for analysis. The sweeping auxiliary curve can be regarded as the trajectory formed by the center of the circular-arc tooth profile, and is therefore referred to as the center guide curve for short. The schematic diagram of its offset is shown in Figure 3.

In Figure 3, the center guide curve $C_i^{(j)}$ (It is shown with dashed line for better visu-alization) can be obtained by offsetting the contact curve $Q_i^{(j)}$ along the direction of the vector ${\mathit{n}}_i^{(j)}$ by a distance $r_j$, where $r_j$ is the radius of the aforementioned circular-arc tooth profile. Specifically, the driving center guide curve is obtained by offsetting the driving contact curve along the vector ${\mathit{n}}_i^{(1)}$ by an equidistant offset $r_1$, and the driven center guide curve is obtained by offsetting the driven contact curve opposite to the direction of the vector ${\mathit{n}}_i^{(2)}$ by an equidistant offset $r_2$, as shown in Equation (17). In the equation, the vectors ${\mathit{\alpha }}_i^{(j)}$, ${\mathit{\beta }}_i^{(j)}$, ${\mathit{\gamma }}_i^{(j)}$ are the unit tangent vector, principal normal vector, and binormal vector of the contact curve, and ${\phi }_i$ is the angle between the vectors ${\mathit{n}}_i^{(j)}$ and ${\mathit{\beta }}_i^{(j)}$.

$$\{\begin{array}{l}{C}_i^{(j)}=Q_i^{(j)}\pm r_j\cdot {\mathit{n}}_i^{(j)}\\ {\mathit{n}}_i^{(j)}={\left[{\mathit{\alpha }}_i^{(j)},{\mathit{\beta }}_i^{(j)},{\mathit{\gamma }}_i^{(j)}\right]}^T\cdot {\left(0,\cos {\phi }_i,\sin {\phi }_i\right)}^T\\ {\mathit{\alpha }}_i^{(j)}={\displaystyle \frac{d{Q}_i^{(j)}/dt}{\left|d{Q}_i^{(j)}/dt\right|}}\\ {\mathit{\beta }}_i^{(j)}={\displaystyle \frac{d{\alpha }_i^{(j)}/dt}{\left|d{\alpha }_i^{(j)}/dt\right|}}\\ {\mathit{\gamma }}_i^{(j)}={\mathit{\alpha }}_i^{(j)}\times {\mathit{\beta }}_i^{(j)}\\ i=1,2\\ j=1,2\\ {\phi }_1=-{\phi }_2\end{array}$$

(17)

More specifically, the driving center guide curve 1 ($C_{Q_{g1}}^{(1)}$) is obtained by offsetting the driving contact curve 1 shown in Equation (13) along the direction of vector ${\mathit{n}}_1^{(1)}$ by a distance $r_1$. The driving center guide curve 2 ($C_{Q_{g2}}^{(1)}$) is obtained by offsetting the driving contact curve 2 shown in Equation (15) along the direction of vector ${\mathit{n}}_2^{(1)}$ by a distance $r_1$. The driven center guide curve 1 ($C_{Q_{n1}}^{(2)}$) is obtained by offsetting the driven contact curve 1 shown in Equation (14) opposite to the direction of vector ${\mathit{n}}_1^{(2)}$ by a distance $r_2$. And the driven center guide curve 2 ($C_{Q_{n2}}^{(2)}$) is obtained by offsetting the driven contact curve 2 shown in Equation (16) opposite to the direction of vector ${\mathit{n}}_2^{(2)}$ by a distance $r_2$.

3. A Design Example

3.1. Design Parameters

The cylindrical helix is the most commonly used spatial curve for line gear meshing. In this paper, the meshing curve of the line gear is illustrated using an equidistant cylindrical helix as an example.

The parametric equation of the driving contact curve 1 in the coordinate system is given by Equation (18).

$${{\displaystyle Q}}_{g1}^{(1)}:\{\begin{array}{l}{x}_{g1}^{(1)}=m\cos t\\ y_{g1}^{(1)}=m\sin t\\ z_{g1}^{(1)}=nt\end{array}\hspace{1em}(t_s\le t\le t_e)$$

(18)

where $m$ is the helix radius of the cylindrical helix, $n$ is the pitch coefficient, and $t$ is the independent variable for the constant-pitch cylindrical helix, with $t_s$ being the variable value at the starting point of meshing and $t_e$ being the variable value at the ending point of meshing. The specific design parameters for the line gears are shown in

Table 1. Design parameters for parallel-axis external line gear pairs.

Parameters	Value
Helix radius of cylindrical helix $m$ (mm)	10
Pitch coefficient of cylindrical helix $n$ (mm)	60/π
Independent variable of cylindrical helix $t$	[0, π/2]
Number of teeth of driving gear $N_1$	8
Transmission ratio $i_{12}$	3
Center distance $a$ (mm)	40
Tooth width $b$ (mm)	30
Normal tooth profile arc radius of driving gear $r_1$ (mm)	4
Normal tooth profile arc radius of driven gear $r_2$ (mm)	4
Angular parameter of meshing equation $\phi$ (rad)	π/3

.

From Equations (2) and (18), the parametric equation of driving contact curve 1 in coordinate system $S_0$ denoted as $Q_{g1}^{(0)}$, can be obtained as shown in Equation (19).

$${{\displaystyle Q}}_{g1}^{(0)}=\{\begin{array}{l}{x}_{g1}^{(0)}=m\cos (t-{\varphi }_1)\\ y_{g1}^{(0)}=m\sin (t-{\varphi }_1)\\ z_{g1}^{(0)}=nt\end{array}\hspace{1em}(0\le t\le \pi /2)$$

(19)

From Equations (2), (6), (8) and (19), the unit common normal vector ${\mathit{n}}^{(0)}$ of driving contact curve 1 at the contact point in coordinate system $S_0$ can be obtained as shown in Equation (20).

$${\mathit{n}}^{(0)}=\left[\begin{array}{l}({\displaystyle \frac{n\cos {\varphi }_1\sin \phi }{\sqrt{m^2+n^2}}}-\cos \phi \sin {\varphi }_1)\sin t-(\cos \phi \cos {\varphi }_1+{\displaystyle \frac{n\sin \phi \sin {\varphi }_1}{\sqrt{m^2+n^2}}})\cos t\\ -{\displaystyle \frac{n\sin \phi \cos (t-{\varphi }_1)}{\sqrt{m^2+n^2}}}-\cos \phi \sin (t-{\varphi }_1)\\ {\displaystyle \frac{m\sin \phi }{\sqrt{m^2+n^2}}}\end{array}\right]$$

(20)

Combining Equations (6), (12), (19) and (20), the meshing equation is obtained as shown in Equation (21).

$${\mathit{v}}_{12}^{(0)}\cdot {\mathit{n}}^{(0)}={\displaystyle \frac{n{\omega }_2(m+m{i}_{12}+a\cos (t-{\varphi }_1))\sin \phi }{\sqrt{m^2+n^2}}}+a{\omega }_2\cos \phi \sin (t-{\varphi }_1)=0$$

(21)

Thus, the relationship between ${\varphi }_1$ and $t$ is obtained as shown in Equation (22).

$${\varphi }_1=t+\pi$$

(22)

3.2. 3D Solid Model

Substituting Equation (22) and the parameters in Table 1 into Equations (13)–(16), the driving contact curve 1, driven contact curve 1, driving contact curve 2, and driven contact curve 2 are obtained as shown in Equations (23), (24), (25) and (26), respectively.

$${{\displaystyle Q}}_{g1}^{(1)}=\{\begin{array}{l}{x}_{g1}^{(1)}=10\cos t\\ y_{g1}^{(1)}=10\sin t\\ z_{g1}^{(1)}={\displaystyle \frac{60}{\pi }}t\end{array}$$

(23)

$${{\displaystyle Q}}_{n1}^{(2)}=\{\begin{array}{l}{x}_{n1}^{(2)}=30\cos ({\displaystyle \frac{\pi +t}{3}})\\ y_{n1}^{(2)}=-30\sin ({\displaystyle \frac{\pi +t}{3}})\\ z_{n1}^{(2)}={\displaystyle \frac{60}{\pi }}t\end{array}$$

(24)

$${{\displaystyle Q}}_{g2}^{(1)}=\{\begin{array}{l}{x}_{g2}^{(1)}=10\cos ({\displaystyle \frac{\pi -8t}{8}})\\ y_{g2}^{(1)}=-10\sin ({\displaystyle \frac{\pi -8t}{8}})\\ z_{g2}^{(1)}={\displaystyle \frac{60}{\pi }}t\end{array}$$

(25)

$${{\displaystyle Q}}_{n2}^{(2)}=\{\begin{array}{l}{x}_{n2}^{(2)}=30\sin ({\displaystyle \frac{\pi }{8}}-{\displaystyle \frac{t}{3}})\\ y_{n2}^{(2)}=-30\cos ({\displaystyle \frac{\pi }{8}}-{\displaystyle \frac{t}{3}})\\ z_{n2}^{(2)}={\displaystyle \frac{60t}{\pi }}\end{array}$$

(26)

Similarly, substituting the parameters in Table 1 into Equation (17), the center guide curves corresponding to the four contact curves can be obtained, as shown in Equations (27)–(30).

$$C_{g1}^{(1)}=\{\begin{array}{l}{x}_{Q_{g1}}^{(1)}=8\cos t+12\sqrt{{\displaystyle \frac{3}{36+{\pi }^2}}}\sin t\\ y_{Q_{g1}}^{(1)}=8\sin t-12\sqrt{{\displaystyle \frac{3}{36+{\pi }^2}}}\cos t\\ z_{Q_{g1}}^{(1)}=2\pi \sqrt{{\displaystyle \frac{3}{36+{\pi }^2}}}+{\displaystyle \frac{60t}{\pi }}\end{array}$$

(27)

$$C_{n1}^{(2)}=\{\begin{array}{l}{x}_{Q_{n1}}^{(2)}=28\cos ({\displaystyle \frac{\pi +t}{3}})-12\sqrt{{\displaystyle \frac{3}{36+{\pi }^2}}}\sin ({\displaystyle \frac{\pi +t}{3}})\\ y_{Q_{n1}}^{(2)}=-28\sin ({\displaystyle \frac{\pi +t}{3}})-12\sqrt{{\displaystyle \frac{3}{36+{\pi }^2}}}\cos ({\displaystyle \frac{\pi +t}{3}})\\ z_{Q_{n1}}^{(2)}=-2\pi \sqrt{{\displaystyle \frac{3}{36+{\pi }^2}}}+{\displaystyle \frac{60t}{\pi }}\end{array}$$

(28)

$$C_{g2}^{(1)}=\{\begin{array}{l}{x}_{Q_{g2}}^{(1)}=8\cos ({\displaystyle \frac{\pi }{8}}-t)+12\sqrt{{\displaystyle \frac{3}{36+{\pi }^2}}}\sin ({\displaystyle \frac{\pi }{8}}-t)\\ y_{Q_{g2}}^{(1)}=-8\sin ({\displaystyle \frac{\pi }{8}}-t)+12\sqrt{{\displaystyle \frac{3}{36+{\pi }^2}}}\cos ({\displaystyle \frac{\pi }{8}}-t)\\ z_{Q_{g2}}^{(1)}=-2\pi \sqrt{{\displaystyle \frac{3}{36+{\pi }^2}}}+{\displaystyle \frac{60t}{\pi }}\end{array}$$

(29)

$$C_{n2}^{(2)}=\{\begin{array}{l}{x}_{Q_{n2}}^{(2)}=28\sin ({\displaystyle \frac{\pi }{8}}-{\displaystyle \frac{t}{3}})+12\sqrt{{\displaystyle \frac{3}{36+{\pi }^2}}}\cos ({\displaystyle \frac{\pi }{8}}-{\displaystyle \frac{t}{3}})\\ y_{Q_{n2}}^{(2)}=-28\cos ({\displaystyle \frac{\pi }{8}}-{\displaystyle \frac{t}{3}})+12\sqrt{{\displaystyle \frac{3}{36+{\pi }^2}}}\sin ({\displaystyle \frac{\pi }{8}}-{\displaystyle \frac{t}{3}})\\ z_{Q_{n2}}^{(2)}=2\pi \sqrt{{\displaystyle \frac{3}{36+{\pi }^2}}}+{\displaystyle \frac{60t}{\pi }}\end{array}$$

(30)

Based on the contact curve and center guide curve equations obtained above, a three-dimensional solid model of the line gear is created using SolidWorks software. Taking the driving gear as an example, the modeling process is shown in Figure 4: (a) Based on the driving contact curve 1 and the driving center guide curve 1, the tooth profile and tooth surface on one side of a single tooth are constructed using the sweeping method. The specific steps are as follows: First, using the 3D sketch function, draw the corresponding curves based on the equations of contact curve 1 and center guide curve 1, respectively. Second, establish a reference normal plane at one endpoint of contact curve 1; this plane is also the normal plane of the endpoint of center guide curve 1. Then, sketch a sector arc on the reference plane: the central angle of the sector is determined by the actual conditions and is set to 80° in this paper; the center of the arc segment is located at the endpoint of the driving center guide curve 1, the endpoint of the driving contact curve 1 is placed at the midpoint of the arc segment, and the arc radius equals the aforementioned offset distance r_j. Finally, generate a single-sided tooth profile using the sweep feature, setting the sector arc as the sweep profile, contact curve 1 as the sweep path, and center guide curve 1 as the guide line. The profile orientation varies with the path, and the profile twist varies with both the path and the guide line. (b) Using the same method described above, construct the tooth profile and tooth surface on the other side of the single tooth based on driving contact curve 2 and driving center guide curve 2; then, perform a revolved cut on both ends according to the tooth width requirement to obtain a complete single tooth model. (c) Perform the circular pattern on the single tooth model to generate all teeth. (d) Use the extrude feature to create the central solid part of the line gear and create the gear shaft hole as required.

The sweeping modeling methods described in Figure 4a,b above are the core of line gear modeling. Since this modeling process does not require deriving tooth surface equations, the line gear modeling method proposed in this paper is referred to as the direct sweeping modeling method.

Similarly, the 3D model of the driven gear can be obtained; the 3D solid model of the entire line gear pair is shown in Figure 5.

4. Simulation and Experimental Verification

4.1. Kinematics Simulation

Kinematic simulation analysis of forward and reverse motion was performed on the 3D model of the line gear pair described above by using the Basic Motion functions of SolidWorks 2020. First, in the SolidWorks assembly environment, use a Concentric Mate to align the centerlines of the driving gear and driven gear with the assembly’s reference axis. Simultaneously, use a Coincident Mate to restrict axial degrees of freedom, ensuring that both gears have only rotational freedom about their respective axes. After establishing the assembly mates, set the simulation conditions in Basic Motion: apply a Rotary Motor to the driving gear and set it to Constant Speed mode with a value of 600 deg/s to simulate the effect of motor drive; simultaneously, add Solid Body Contact to the tooth surfaces of both the driving and driven gears to prevent them from intersecting during motion. Finally, according to the motion continuity requirements to be verified in this paper, the frames per second (FPS) are set to 100, and both the geometry precision and the 3D contact resolution are set to Medium to ensure a balance between simulation accuracy and computational efficiency.

To provide a direct and quantitative assessment of the transmission performance, the output rotational speeds of driven gear and instantaneous transmission ratios were benchmarked against their theoretical ideals, as illustrated in Figure 6 and Figure 7. As shown in Figure 6, the actual output speed of the driven gear closely follows the theoretical ideal of 200 deg/s and remains strictly within the ±1.5% tolerance band (197 deg/s to 203 deg/s) throughout the entire process in both directions.

The instantaneous transmission ratios for forward and reverse rotations are presented in Figure 7a. During forward rotation, the average transmission ratio is 3.0014 (relative error 0.0467%), with an absolute standard deviation (σ) of 0.0546. During reverse rotation, the average transmission ratio is 3.0013 (relative error 0.0433%), with an absolute standard deviation of 0.0471. Figure 7b shows the statistical distribution of the transmission ratio error. The dashed vertical lines indicate the 3σ limits, which contain approximately 99.7% of the data under a normal distribution assumption. The fact that nearly all instantaneous errors lie within this narrow band quantitatively proves that the transmission fluctuations are highly concentrated and the motion is exceptionally stable. The relative standard deviations are strictly constrained to 1.80% and 1.65% for forward and reverse motions, respectively.

The kinematic simulation results clearly indicate that the proposed line gear pair can accurately maintain the designed transmission ratio in both directions, with relative standard deviations below 2% and all data points strictly bounded by the predefined tolerance limits. This demonstrates a reliable and symmetric transmission capability.

4.2. Kinematics Experiment

To further verify the transmission accuracy and stability of the designed line gear pair under forward and reverse conditions, a pair of line gear prototypes was fabricated using stereolithography (SLA) 3D printing (material: ABS-like photosensitive resin, printing accuracy ±0.1 mm), and kinematic experiments were conducted on the experimental platform shown in Figure 8. The main components and parameters of the experimental platform are as follows: A three-phase asynchronous motor (rated power 2.2 kW, rated speed 2875 rpm) serves as the drive motor for the driving gear; a variable frequency drive (VFD, power 2.2 kW, output frequency 0–600 Hz) regulates the motor speed; two incremental rotary encoders (resolution 5000 pulses/rev) are coaxially mounted on the input and output shafts to measure angular displacement and rotational speed; an encoder reader module using the Modbus-RTU protocol acquires encoder signals and transmits them to a computer; a data acquisition card (NI USB-6210, sampling frequency 100 Hz) synchronously records the encoder signals; and the computer communicates with the reading module via a USB-to-RS485 interface to simultaneously record the angular data of the input and output shafts.

To eliminate the unstable effects during the motor start-up and stop phases, operational data from the 3rd to the 15th second of the experiment were selected for analysis. Based on the rotational speed signals collected from the input and output shaft encoders, the instantaneous transmission ratio was calculated, and the transmission ratio over time is presented in Figure 9.

In Figure 9, the raw experimental data (plotted as scattered points with transparency) exhibit high-frequency fluctuations. These instantaneous scattering effects are inevitable in physical prototypes and are primarily attributed to real-world factors, including the inherent noise of the dynamic sensors, slight micro-clearances during assembly, and vibrations from the motor drive system. To extract the underlying kinematic characteristics of the line gear pair, a Savitzky–Golay smoothing filter (implemented in Python 3.11) was applied to the raw signals. The resulting smoothed trends (thick solid lines) effectively represent the macroscopic transmission baseline, filtering out environmental and sensor-induced artifacts.

A direct, quantitative comparison reveals the core relationship between the raw measurements and the theoretical design. While the raw instantaneous data naturally oscillate, the extracted smoothed trends for both forward and reverse motions converge tightly around the theoretical ideal ratio i = 3.0 (black dashed line). More importantly, as visually confirmed in Figure 9, the macroscopic transmission trends under both low-speed (≈115 rpm) and high-speed (≈175 rpm) conditions are strictly bounded within the predefined engineering tolerance band of ±0.05 (gray shaded area, i.e., 3.0 ± 0.05), which corresponds to a relative error of approximately ±1.67%.

Based on the above transmission ratio data, statistical indicators such as the average transmission ratio, standard deviation, and range were further calculated for different operating conditions, and error analysis was performed, as shown in

Table 2. Kinematic performance of the line gear pair under different rotation directions and driving speeds.

	Low-Speed Condition (≈115 rpm)		High-Speed Condition (≈175 rpm)
Parameter	Forward 114 rpm	Reverse 116 rpm	Forward 174 rpm	Reverse 176 rpm
Average transmission ratio	3.0191	3.0172	3.0194	3.0152
Transmission ratio error (%)	0.637	0.573	0.647	0.507
Minimum transmission ratio error (%)	–1.313	–0.849	–0.996	–0.805
Maximum transmission ratio error (%)	2.283	2.116	1.98	2.428
Standard deviation	0.0269	0.019	0.0225	0.0166
Relative standard deviation (%)	0.89	0.629	0.743	0.552
Range	0.108	0.089	0.097	0.0893
Relative Range (%)	3.574	2.948	3.212	2.961

. The rotational speeds in Figure 9 and Table 2 are all experimentally measured speeds of the driving gear.

The experimental results indicate that under both forward and reverse rotations, the measured average transmission ratios deviate from the theoretical value of 3 by less than 0.65% across all tested cases. For the low-speed condition (≈115 rpm), the average ratios are 3.0191 (forward) and 3.0172 (reverse), with relative standard deviations (RSD) of 0.89% and 0.629%, respectively. At the high-speed condition (≈175 rpm), the average ratios are 3.0194 (forward) and 3.0152 (reverse), with RSD values of 0.743% and 0.552%. As summarized in Table 2, the overall standard deviations range from 0.0166 to 0.0269, the RSD from 0.552% to 0.89%, and the relative range from 2.948% to 3.574%. These statistical metrics confirm that the instantaneous transmission ratios exhibit only minor fluctuations around their respective means. Moreover, the fluctuations tend to decrease slightly with increasing speed—an effect likely attributable to inertial smoothing of meshing impacts—and remain consistent and symmetric between the two directions. These observations indicate that the proposed line gear pair achieves stable bidirectional transmission.

5. Discussion

A comparison of the kinematic simulation results with the experimental data reveals consistent trends between the two in terms of the accuracy and stability of the transmission ratio. Compared with the experimental results, the simulation results have a smaller average transmission ratio error but a larger relative standard deviation. This may be because the simulation is based on an ideal geometric model (free of manufacturing errors and assembly deviations), and the contact stiffness and friction coefficient were set to the software’s default values.

The transmission ratios observed in the experimental measurements exhibit fluctuations and deviations from theoretical values. The primary reasons may include: (1) Prototype manufacturing errors: The line gear pair was fabricated using SLA 3D printing with a dimensional tolerance of ±0.1 mm. Deviations in tooth profile, pitch, and surface roughness can lead to transmission errors and rotational speed fluctuations during meshing [15,16,17]. (2) Errors caused by material properties: Compared with metallic materials, resin materials have a lower elastic modulus, which can cause minor elastic deformation during transmission, leading to fluctuations in the instantaneous transmission ratio [18,19]. Additionally, the coefficient of friction between resin materials is relatively high, and this coefficient varies with rotational speed and load, further increasing the instability of the instantaneous transmission ratio [20,21]. (3) Installation and coaxiality errors: The motor, encoder, gear pair, and load are connected via an elastic coupling. Unavoidable coaxiality deviations during installation introduce periodic fluctuation errors in speed measurement [22,23]. (4) Encoder counting errors and inherent errors of the data acquisition system: The encoder has a resolution of 5000 pulses per revolution, a theoretical angular resolution of 0.072°, and an expanded uncertainty in angular measurement of approximately 0.07°. This uncertainty contributes less than 0.2% to the transmission ratio error and is not a primary source of error.

Despite the aforementioned error factors, the average transmission ratio error (<0.65%) and fluctuation range (relative range < 3.6%) in this experiment both meet the requirements for verifying kinematic principles. The parallel-axis external line gear pair designed using the method described in this paper can achieve continuous, accurate, and smooth power transmission, with the transmission ratio meeting design requirements.

Research on line gears has primarily focused on the modification and extension of the theory of conjugate meshing of spatial curves [4,14], analysis of friction and meshing efficiency [3], CNC machining processes [24], and design of pure rolling [6,7]. While these references demonstrate progress and achievements in the design of line gears, none of the studies provide quantitative statistics for bidirectional transmission. Compared to the aforementioned literature, the characteristics and advantages of this work are as follows: First, existing literature lacks systematic bidirectional transmission experiments for parallel-axis external line gears. This paper provides, for the first time, complete transmission ratio data for both forward and reverse directions, with highly symmetrical performance. In terms of the key indicator of bidirectional transmission symmetry, our work is significantly superior to that of reference [25] (although the performance metrics studied are different), which indirectly validates the feasibility of using the direct sweeping method proposed in this paper to construct line gears for bidirectional transmission. Second, traditional line gear design methods require deriving complex tooth surface equations from the meshing equations [10,11,12], whereas the direct sweeping method proposed in this paper constructs the 3D model of the line gear directly from the contact curves and center guide curves, thereby reducing modeling steps, improving design efficiency, and facilitating the learning of line gears for beginners.

Of course, considering the existing research on factors affecting the transmission performance of other types of gears, this study still has the following limitations: (1) The design example is limited, which to some extent restricts the generalizability of the conclusions. (2) The kinematic experiments were conducted under no-load and low-speed conditions, without considering the effects of temperature rise and wear on transmission performance during long-term operation [15,23]. (3) The mechanical properties of resin materials differ from those of metals, so the generalization of the experimental results must account for the influence of material characteristics [15,21]. (4) The impact of lubrication conditions on transmission performance was not considered [26,27,28].

Therefore, with reference to the progress and directions of research on other types of gears, the line gears described in this paper can be further studied in the following aspects in the future: (1) Investigate the optimization patterns of transmission performance under different tooth profile parameters or shapes (such as arc radius, offset, elliptical arcs, etc.) [29,30,31,32]. (2) CNC-machine line gear pairs made of metal materials to verify their transmission performance under medium-to-high speed and loaded conditions. (3) Conduct long-term operational experiments to analyze the effects of temperature rise and wear on transmission errors. (4) Develop an automated design program based on the direct sweeping method described in this paper to accelerate the engineering application of line gears [33,34].

6. Conclusions

This paper proposes a design method for BTPELG, covering the extension of the meshing equation, the solution of contact curves and center guide curves, 3D solid modeling, as well as simulation and experimental verification. The main conclusions are as follows:

(1)

A generalized spatial curve meshing equation incorporating angular parameters is proposed. This extends the direction of the common normal in the traditional meshing equation to the complete normal plane spanned by the principal and secondary normal vectors. This extension provides the necessary theoretical flexibility for subsequent active design of transmission errors and optimization of contact performance through the adjustment of angular parameters.

(2)

A direct sweeping method is proposed that does not require solving complex tooth surface equations. Starting from a given driving contact curve, other contact curves and their corresponding center guide lines are determined using the generalized meshing equation and coordinate transformations, and the line gear teeth are generated by combining this with direct sweeping of the tooth profile. This method bypasses the complex derivation of tooth surface equations, simplifies the design process, and provides a new approach for the rapid design and engineering validation of line gears, while also helping beginners quickly learn modeling and get started with line gears.

(3)

The feasibility and smoothness of bidirectional transmission parallel-axis external line gears are verified. Kinematic simulations and experiments are performed on an example line gear pair, and the results indicate that the bidirectional line gear pair designed in this paper can achieve continuous, accurate, and smooth transmission, meeting the design expectations. This provides a technical foundation for the subsequent optimization of line gear design.