Geometric design and kinematics analysis of coplanar double internal meshing non-circular planetary gear train

Abstract

A new type of non-circular planetary gear train is proposed, in which the planetary gear is internally meshed with the sun gear and the ring gear at the same time. The structural form and transmission principle are analyzed, and the design method of non-circular gear pitch curves and related parameters of the gear train are discussed. The input–output relationship under various working conditions of the gear train is deduced. The precise tooth profile of the non-circular gears is obtained by the Boolean operation of MATLAB, then the virtual prototype model of the planetary gear train is established, and the kinematics simulation is carried out in the ADAMS software. The simulation results verify the correctness of the transmission principle and the theoretical analysis of the motion law.

Introduction

The non-circular gear is a mechanism that can realize the non-uniform speed ratio transmission between the driving and driven gears. ¹ Compared with the linkages, cams, and automatic control technology, non-circular gears have the advantages of low economic cost, compact structure, accurate transmission ratio, and good dynamic characteristics. Due to the precise transmission ratio, non-circular gears and their combined mechanisms have an irreplaceable role in many occasions with special motion and force requirements. Therefore, many scholars have done a lot of research in related design theory and engineering applications.

Non-circular gears can output a specific motion law, and thus can be used as a function generating mechanism, which has the advantages of simple structure and good transmission characteristics.^2,3 D Mundo et al. ⁴ studied the five-bar linkages with non-circular gears and proposed an optimization of the mechanism by defining the objective function on the basis of proper dimensional and kinematic criteria. JY Liu et al. ⁵ designed a pair of elliptical gears to substitute the anti-parallel four-bar linkage and studied the specific trajectory curves generated by elliptical gears. D Barkah et al. ⁶ proposed the theory of using non-circular gears to balance the swinging force and torque of the linkage mechanism and JY Han ⁷ applied the theory to the space linkage mechanism and explained the design method of the non-circular gear for realizing the complete dynamic balance of the mechanism. O Karpov et al. ⁸ investigated the applicability of non-circular gears for preventing resonance oscillations in gear mechanisms. AA Prikhodko et al. ⁹ and Zheng et al. ¹⁰ studied the application of non-circular gears in intermittent motion mechanisms. JH Kerr and colleagues^11,12 and Hebbale et al. ¹³ studied the application of non-circular gears in infinitely variable transmission. M Addomine et al. ¹⁴ discussed the landmark contributions of the Italian clockmaker Giovanni Dondi to the mechanical design of astronomical clockwork which demonstrates the unique application of non-circular gears in astronomy. Breaking through the traditional mode of non-circular gears, D Liu et al. ¹⁵ presented a new gear mechanism composed of a cylinder involute gear and an eccentric face gear to implement a synthetic motion; F Zheng et al. ¹⁶ proposed the definition of variable center distance gear, which is a new form of non-uniform transmission.

Regarding the generation of non-circular gears, relevant scholars have conducted in-depth research on the processing theory and manufacturing process of non-circular gears in recent years. I Zarębski and T Sałaciński ¹⁷ presented methods for designing of non-circular gears, including internal and external gears with spur or helical teeth and proposed the context analysis method to find non-circular gears’ flank. Y Liu ¹⁸ researched the gear shaping strategy for internal helical non-circular gears and conducted comparative analysis of four linkage models. F Zheng and colleagues^19,20 proposed a 3-linkage model with an equal arc-length cutting method involving feed and investigated curvilinear non-circular gear generation based on face-milling method. K Shi et al. ²¹ proposed a method for processing non-circular bevel gear with concave pitch curve using bevel gear cutter enveloping.

Non-circular planetary gear train (PGT) absorbs the advantages of traditional PGT and non-circular gear mechanism and has the characteristics of compact structure, large transmission ratio, and variable transmission ratio, so scholars have conducted a lot of research on it. Yokoyama et al. ²² studied the design method of the inner and outer pitch curves of the non-circular PGT and proposed the conditions for avoiding interference. GY Volkov et al. ²³ proposed a new approach to the geometric synthesis of non-circular gears for a rotary hydraulic machine. D Mundo ²⁴ proposed a new type of non-circular PGT and studied its pitch curve design and tooth profile generation method. LS Guo et al. ²⁵ developed the kinematic parametric equation for an eccentric PGT and studied a rice transplanting mechanism with the gear train.

As shown in Figure 1, there are mainly four types of traditional non-circular PGTs: N-G-W, N-W, W-W, and N-N.^{9,10,14,22–25} Among them, the N-G-W type has a single row with smaller axial dimension, and its radial dimension is slightly larger than the latter three types. N-W, W-W, and N-N are in the form of double rows, although the radial dimension can be reduced, the axial dimension is inevitably increased. In addition, these four types of traditional non-circular PGTs generally use several planetary gears to share the load, so there is a possibility of uneven loading. ²⁶ In this article, a new type of coplanar double internal meshing non-circular planetary gear train (CDIMN-PGT) is proposed, which has the characteristics of compact structure, high power density, and the ability to output a specific regular rotational motion. Compared with traditional non-circular PGTs, CDIMN-PGT has only one planetary gear, thus avoiding the uneven load problem that may occur with multiple planetary gears. Due to the structural specialty, the modulus of the sun gear and the ring gear can be different, which reduces the difficulty of teeth matching. In addition to the internal meshing between the planetary gear and the ring gear, the planetary gear and the sun gear are also internally meshed, that is, double internal meshing, and the internal meshing is advantageous for improving the contact ratio of the gear transmission. ²⁷ The design method of the pitch curves of the gear train based on internal high-order non-circular gears is proposed, and the boundary conditions of the design parameters are analyzed. The theoretical analysis of the transmission ratio of the gear train under different working conditions was carried out and kinematic simulation was accomplished in the ADAMS software.

Figure 1.

Several traditional non-circular planetary gear trains: (a) N-G-W type, (b) N-W type, (c) W-W type, and (d) N-N type.

Transmission principle

The main structure of CDIMN-PGT is shown in Figure 2. The gear train is mainly composed of the sun gear, the planetary gear, the ring gear, and the planet carrier. The sun gear, the planetary gear, and the ring gear are all non-circular gears. The planetary gear has both inner and outer teeth, respectively, meshing with the sun gear and the ring gear, thereby transmitting motion and power. The planet gear makes a planetary motion that revolves around its own axis $o_2(o_3)$ and revolves around the axis $o_1(o_4)$ of the planet carrier. The sun gear, the ring gear, and the planet carrier rotate around its own axis. The rotation axis of the sun gear, the ring gear, and the planet carrier coincide with each other and the rotation axis of the planetary gear is offset from the rotation axis of the sun gear by a distance $o_1-o_2$ , as shown in Figure 2(b). CDIMN-PGT actually has two pairs of internal meshing non-circular gears, and the sun gear and inner gear of the planetary gear can be called non-circular gear pair $G_{12}$ , and the ring gear and the outer gear of the planetary gear can be called non-circular gear pair $G_{34}$ . Due to the variable transmission ratio characteristic of the non-circular gear, under the condition that the gear train obtains a constant input, an output that changes according to a certain motion law can be obtained.

Figure 2.

Main structure of CDIMN-PGT: (a) kinematic sketch and (b) pitch curves.

Pitch curves design and analysis

Non-circular gears are a key component of CDIMN-PGT. Since the non-circular gear pairs $G_{12}$ and $G_{34}$ are both internally meshed and need to meet the condition of equal center distance, the design of the pitch curves becomes particularly complicated. Based on the design of the internal meshing non-circular gear pitch curve, the boundary conditions of the parameters of CDIMN-PGT are analyzed, and the matching methods of the parameters such as the major axis radius and eccentricity of non-circular gears are proposed.

Internal non-circular gear pair

Unlike the external non-circular gear pair, the internal non-circular gear pair is composed of an external non-circular gear and an internal non-circular gear. Assume that the external gear is the driving gear and the internal gear is the driven gear. Due to the particularity of the internal non-circular gear, the order $n_1$ of the driving gear is generally smaller than the order $n_2$ of the driven gear. The internal high-order non-circular gear pitch curve is shown in Figure 3. The driving gear rotates counterclockwise at the instantaneous angular velocity ${\omega }_1$ , and the rotation angle is ${\varphi }_1$ ; the driven gear rotates counterclockwise at the instantaneous angular velocity ${\omega }_2$ , and the rotation angle is ${\varphi }_2$ . The pitch curve equation of the driving gear is as follows

$$r_1({\varphi }_1)=\frac{a_1(1-k_1^2)}{1-k_1\cos n_1{\varphi }_1}=\frac{p_1}{1-k_1\cos n_1{\varphi }_1}$$

(1)

where $a_1$ is the major axis radius, $k_1$ is the eccentricity, $n_1$ is the order, and $r_1$ is the polar diameter.

Figure 3.

Pitch curves of internal non-circular gear pair.

In order to make the teeth evenly distributed on the pitch curve, the following conditions must be met

$$L=\underset{0}{\overset{2\pi }{\int }}\sqrt{r_1^2({\varphi }_1)+{\left(\frac{d{r}_1({\varphi }_1)}{d{\varphi }_1}\right)}^2}d{\varphi }_1=\pi m{z}_1$$

(2)

where L is the circumference of the driving gear pitch curve, m is the modulus, and $z_1$ is the number of teeth of the driving gear.

Transmission ratio of internal non-circular gear pair is

$$\begin{array}{l}{i}_{12}({\varphi }_1)=\frac{d{\varphi }_1}{d{\varphi }_2}=\frac{r_2}{r_1}=\frac{A+r_1({\varphi }_1)}{r_1({\varphi }_1)}\\ =\frac{A-A{k}_1\cos n_1{\varphi }_1+p_1}{p_1}\end{array}$$

(3)

where A is the center distance.

According to the pitch curve closure condition, ¹ the following equation can be obtained

$$\frac{2\pi }{n_2}=\underset{0}{\overset{\frac{2\pi }{n_1}}{\int }}\frac{1}{i_{12}({\varphi }_1)}d{\varphi }_1=\frac{2\pi p_1}{n_1\sqrt{{(A+p_1)}^2-A^2{k}_1^2}}$$

(4)

Let $n=n_2/n_1$ , then $n>1$ . According to equation (4), the center distance A can be expressed as follows

$$A=a_1[\sqrt{1+(1-k_1^2)(n^2-1)}-1]$$

(5)

It can be seen from equation (5) that the center distance A is negatively correlated with the eccentricity and positively correlated with n.

The pitch curve equation of the driven gear is as follows

$$\{\begin{array}{c}{r}_2({\varphi }_1)=r_1({\varphi }_1)+A\hfill \\ \multicolumn{1}{c}{{\varphi }_2({\varphi }_1)=\underset{0}{\overset{{\varphi }_1}{\int }}\frac{1}{i_{12}({\varphi }_1)}d{\varphi }_1}\end{array}$$

(6)

namely

$$\{\begin{array}{c}{r}_2=\frac{p_2}{1-k_2\cos n_2{\varphi }_2}\hfill \\ \multicolumn{1}{c}{p_2=\frac{n^2{p}_1}{\sqrt{n^2-k_1^2(n^2-1)}}}\\ \multicolumn{1}{c}{k_2=\frac{k_1}{\sqrt{n^2-k_1^2(n^2-1)}}}\end{array}$$

(7)

According to equations (1), (3), and (5), the transmission ratio of internal non-circular gear can be expressed as follows

$$i_{12}({\varphi }_1)=\frac{\sqrt{n^2-k_1^2\left(n^2-1\right)}\left(1-k_1\cos n_1{\varphi }_1\right)-k_1^2+k_1\cos n_1{\varphi }_1}{1-k_1^2}$$

(8)

Center distance equal condition

Ensuring that the center distances of the two pairs of non-circular gears G₁₂ and G₃₄ are equal is the basic condition for achieving correct transmission. And the center distance is determined by the relevant parameters of the pitch curve. Therefore, when designing CDIMN-PGT, the parameters of the two pairs of non-circular gear pitch curves are usually coordinated according to the condition of the equal center distance. The center distance equal condition for the non-circular gear pair $G_{12}$ and $G_{34}$ is as follows

$$A_{12}=A_{34}$$

(9)

It can be known from equation (5) that the following relationship exists at this point

$$a_1\left[\sqrt{n_{12}-k_1^2\left(n_{12}^2-1\right)}-1\right]=a_3\left[\sqrt{n_{34}-k_3^2\left(n_{34}^2-1\right)}-1\right]$$

(10)

where $n_{12}=n_2/n_1$ and $n_{34}=n_4/n_3$ .

It is known from equation (8) that the transmission ratio of the internal non-circular gear pair is related to the order and eccentricity of the driving and driven gear. Therefore, the center distance equal condition can be satisfied by adjusting only the major axis radius ai (i =1,3) without changing the gear ratio. In the case where the order of the driving and driven gear is determined, let the eccentricity $k_1=k_3$ , then $a_3$ can be expressed as follows

$$a_3=\frac{a_1[\sqrt{n_{12}-k_1^2(n_{12}^2-1)}-1]}{\sqrt{n_{34}-k_1^2(n_{34}^2-1)}-1}$$

(11)

Major axis radius $a_3$ can be obtained by equation (11) under the condition that the center distance and the pitch curve equation of the non-circular gear pair $G_{12}$ are determined, and then the pitch curve equation of the non-circular gear pair $G_{34}$ can be determined.

Boundary conditions without interference

The planetary gear of CDIMN-PGT can be regarded as being composed of two coaxial internal non-circular gears, and the inner and outer teeth are, respectively, meshed with the sun gear and the planetary gear. When designing the planetary gears, the pitch curves of the inner and outer non-circular gears should not be interfered. Only in terms of geometric relationship, the maximum value of the inner pitch curve’s polar radius should be less than the minimum value of the outer pitch curve’s polar radius. However, considering the teeth matching and the installation method of the planetary gear, the above conditions cannot meet the requirements. Therefore, it is necessary to leave a certain gap $\mathrm{\Delta }r$ between the inner and outer pitch curves according to the actual situation, as shown in Figure 4. At this point, the following relationship exists between the inner and outer pitch curve of the planetary gear

$$r_2^{\max }+\mathrm{\Delta }r\le r_3^{\min }$$

(12)

Figure 4.

Limit position of the inner and outer pitch curves of the planetary gear.

According to equations (1), (7), and (11), $r_2^{\max }$ and $r_3^{\min }$ can be expressed as follows

$$\{\begin{array}{c}{r}_2^{\max }=\frac{p_2}{1-k_2}=\frac{a_1(1-k_1^2)n_{12}^2}{\sqrt{n_{12}^2-k_1^2(n_{12}^2-1)}-k_1}\hfill \\ \multicolumn{1}{c}{r_3^{\min }=\frac{a_3(1-k_3^2)}{1+k_3}=\frac{a_1(1-k_1)[\sqrt{n_{12}^2-k_1^2(n_{12}^2-1)}-1]}{[\sqrt{n_{34}^2-k_1^2(n_{34}^2-1)}-1]}}\end{array}$$

(13)

To simplify the analysis, $\mathrm{\Delta }r=0$ is assumed to analyze the boundary conditions. Under the condition of eccentricity $k_1=k_3$ , $n_{12}=2/3$ , $n_{34}=4/5$ , the following relationship can be obtained by combining equations (12) and (13)

$$\begin{array}{c}\multicolumn{1}{c}{\frac{r_2^{\max }}{a_1(1-k_1)}=\frac{(1+k_1)n_{12}^2}{\sqrt{n_{12}^2-k_1^2(n_{12}^2-1)}-k_1}}\\ \multicolumn{1}{c}{\le \frac{\sqrt{n_{12}^2-k_1^2(n_{12}^2-1)}-1}{\sqrt{n_{34}^2-k_1^2(n_{34}^2-1)}-1}=\frac{r_3^{\min }}{a_1(1-k_1)}}\end{array}$$

(14)

By solving equation (14), it can be obtained that $k_1\le 0.1719$ .

Kinematic analysis

As shown in Figure 2, the rotational speeds of the gears in the non-circular gear pairs $G_{12}$ and $G_{34}$ are ${\omega }_1,{\omega }_2,{\omega }_3,{\omega }_4$ , respectively, and the planet carrier rotational speed is ${\omega }_H$ . According to the relative motion principle, the kinematic relationship between the gears of CDIMN- PGT can be expressed as

$$\{\begin{array}{c}\frac{{\omega }_1-{\omega }_H}{{\omega }_2-{\omega }_H}=i_{12}({\varphi }_1)\hfill \\ \multicolumn{1}{c}{\frac{{\omega }_3-{\omega }_H}{{\omega }_4-{\omega }_H}=i_{34}({\varphi }_3)}\end{array}$$

(15)

Since the internal teeth of the planetary gear are fixed to the external teeth, the following constraint can be obtained

$${\omega }_2={\omega }_3$$

(16)

Multiplying the left and right sides of the two subtypes of equation (15), the following equation can be obtained

$$\frac{{\omega }_1-{\omega }_H}{{\omega }_4-{\omega }_H}=i_{12}({\varphi }_1)i_{34}({\varphi }_3)$$

(17)

The specific rotational speed relationship between the sun gear, the ring gear, and the planet carrier can be obtained by deforming equation (17) as follows

$${\omega }_1-i_{12}({\varphi }_1)i_{34}({\varphi }_3){\omega }_4=[1-i_{12}({\varphi }_1)i_{34}({\varphi }_3)]{\omega }_H$$

(18)

What is discussed above is only a special case. In fact, there may be a phase angle $\psi$ between the inner pitch curve and the outer pitch curve of the planetary gear, as shown in Figure 5. When the value of $\psi$ changes, the transmission ratio of the whole gear train changes accordingly. It is assumed that the inner pitch curve of the planetary gear is fixed, when the outer pitch curve of the planetary gear rotates counterclockwise by a certain angle with respect to the inner pitch curve of the planetary gear, $\psi$ takes a positive value, whereas $\psi$ takes a negative value. According to the above analysis, the transmission ratio of the non-circular gear pair $G_{34}$ is represented as $i_{34}({\varphi }_3+\psi )$ at this point. Therefore, combined with equation (18), the output rotational speed of CDIMN-PGT under different operating states can be obtained, as shown in Table 1.

Figure 5.

Phase angle $\psi$ of the inner and outer pitch curves of the planetary gear.

Table 1.

Working state and output rotational speed of CDIMN-PGT.

Workingstatus	Fixedlink	Degreeof freedom	Input link	Output link	Output rotational speed
I	None	2	Sun gear, ringgear	Planet carrier	${\omega }_H=\frac{{\omega }_1-i_{12}({\varphi }_1)i_{34}({\varphi }_3+\psi ){\omega }_4}{1-i_{12}({\varphi }_1)i_{34}({\varphi }_3+\psi )}$
II	None	2	Sun gear, planet carrier	Ring gear	${\omega }_4=\frac{{\omega }_1+[i_{12}({\varphi }_1)i_{34}({\varphi }_3+\psi )-1]{\omega }_H}{i_{12}({\varphi }_1)i_{34}({\varphi }_3+\psi )}$
III	None	2	Ring gear, planet carrier	Sun gear	${\omega }_1=i_{12}({\varphi }_1)i_{34}({\varphi }_3+\psi ){\omega }_4-[i_{12}({\varphi }_1)i_{34}({\varphi }_3+\psi )-1]{\omega }_H$
IV	Sun gear	1	Ring gear	Planet carrier	${\omega }_H=\frac{i_{12}({\varphi }_1)i_{34}({\varphi }_3+\psi ){\omega }_4}{i_{12}({\varphi }_1)i_{34}({\varphi }_3+\psi )-1}$
V	Sun gear	1	Planet carrier	Ring gear	${\omega }_4=\frac{[i_{12}({\varphi }_1)i_{34}({\varphi }_3+\psi )-1]{\omega }_H}{i_{12}({\varphi }_1)i_{34}({\varphi }_3+\psi )}$
VI	Ring gear	1	Sun gear	Planet carrier	${\omega }_H=\frac{{\omega }_1}{1-i_{12}({\varphi }_1)i_{34}({\varphi }_3+\psi )}$
VII	Ring gear	1	Planet carrier	Sun gear	${\omega }_1=[1-i_{12}({\varphi }_1)i_{34}({\varphi }_3+\psi )]{\omega }_H$

In order to facilitate the description of the kinematics of CDIMN-PGT, a set of design examples was analyzed. Through the design method of CDIMN-PGT pitch curve described in the previous chapter, the specific design parameters of each non-circular gear are obtained, as shown in Table 2.

Table 2.

Design parameters of non-circular gears.

	Center distance (mm)	Number of teeth	Order	Modulus (mm)	Eccentricity
Non-circular gear pair $G_{12}$	$A_{12}=A_{34}=29.6025$	$z_1=30$ $z_2=45$	2 3	$m=4.0020$	$k_1=0.0500$ $k_2=0.0334$
Non-circular gear pair $G_{34}$		$z_3=60$ $z_4=75$	4 5	$m=4.0324$	$k_3=0.0500$ $k_4=0.0334$

Through Table 2, the transmission ratio function of the non-circular gear pairs $G_{12}$ and $G_{34}$ can be obtained. Combined with Table 1, the transmission ratio curve of the CDIMN-PGT in the single input-output state can be plotted, as shown in Figure 6.

Figure 6.

Transmission ratio curve under different working status: (a) Working status IV, (b) Working status V, (c) Working status VI, and (d) Working status VII.

Influence of planetary gear phase angle $\psi$ on transmission ratio

Taking the working state IV as an example, the relationship between the transmission ratio of the gear train and the phase angle $\psi$ is specifically analyzed. From the equation in Table 1, the following relationship exists between the transmission ratio $i_{4H}$ of the gear train and the phase angle $\psi$

$$i_{4H}=\frac{{\omega }_4}{{\omega }_H}=\frac{i_{12}({\varphi }_1)i_{34}({\varphi }_3+\psi )-1}{i_{12}({\varphi }_1)i_{34}({\varphi }_3+\psi )}$$

(19)

As can be seen from Figure 7, the transmission ratio $i_{4H}$ in the working state IV is a waveform function with a period of π, and the period of the transmission ratio $i_{4H}$ is independent of the value of the phase angle $\psi$ . When $\psi =0$ , $i_{4H}$ has two peaks of equal amplitude in one cycle. As the value of $\psi$ increases, the transmission ratio $i_{4H}$ changes to a single peak gradually, and the amplitude increases and shifts to the right gradually.

Figure 7.

Relationship between phase angle $\psi$ and transmission ratio $i_{4H}$ .

Influence of eccentricity k₁ on transmission ratio

Taking the working state VII as an example, the relationship between the transmission ratio of the gear train and the eccentricity $k_1$ is specifically analyzed. In the case of the phase angle $\psi =0$ , the transmission ratio $i_{H1}$ of the gear train can be obtained according to Table 1, as follows

$$i_{H1}=\frac{{\omega }_H}{{\omega }_1}=\frac{1}{1-i_{12}({\varphi }_1)i_{34}({\varphi }_3)}$$

(20)

From equations (8) and (20), the variation of $i_{H1}$ at $k_1=0.05$ , $k_1=0.1$ , and $k_1=0.15$ can be plotted. As shown in Figure 8, the variation range of the transmission ratio $i_{H1}$ increases as $k_1$ increases and the period remains unchanged.

Figure 8.

Relationship between eccentricity $k_1$ and transmission ratio $i_{H1}$ .

Non-circular gear tooth profile generation

In this article, a MATLAB Boolean operation method based on the envelope principle is proposed to obtain an accurate tooth profile of non-circular gears. This algorithm continually performs Boolean subtraction operations between a region enclosed by the tooth profile of the shaper cutter and the closed region of the processed non-circular gear blank contour. In this process, the region enclosed by the blank is continuously cut according to the envelope principle, so the blank contour curve is also continuously iterated, and finally the complete non-circular gear tooth profile curve is obtained. Since there are few studies on the profile generation of internal non-circular gears, the algorithm is verified by the internal non-circular gear profile generation.

Coordinate transformation

The pitch curve of the non-circular gear can be expressed as $r(\varphi )$ , and I represents the contact point between the pitch curve of the non-circular gear and the pitch circle of the shaper cutter. The coordinates of the tangent point I can be expressed as

$$\{\begin{array}{c}{x}_I=r(\varphi )\cos (\varphi )\hfill \\ \multicolumn{1}{c}{y_I=r(\varphi )\sin (\varphi )}\end{array}$$

(21)

The unit tangent vector t of the pitch curve can be expressed as ²⁸

$$\mathbf{t}(\varphi )=\left[\begin{array}{c}{t}_x(\varphi )\hfill \\ \multicolumn{1}{c}{t_y(\varphi )}\end{array}\right]=\left[\begin{array}{c}\frac{r\prime (\varphi )\cos \varphi -r(\varphi )\sin \varphi }{\sqrt{r\prime {(\varphi )}^2+r{(\varphi )}^2}}\hfill \\ \multicolumn{1}{c}{\frac{r\prime (\varphi )\cos \varphi +r(\varphi )\sin \varphi }{\sqrt{r\prime {(\varphi )}^2+r{(\varphi )}^2}}}\end{array}\right]$$

(22)

where $r\prime (\varphi )=\mathrm{dr}(\varphi )/d\varphi$ .

Since the unit normal vector n of the pitch curve is perpendicular to the unit tangent vector t, it can be known from $\mathbf{n}·\mathbf{t}=0$ that the unit vector n can be expressed as

$$\mathbf{n}(\varphi )=\left[\begin{array}{c}{n}_x(\varphi )\hfill \\ \multicolumn{1}{c}{n_y(\varphi )}\end{array}\right]=\left[\begin{array}{c}{t}_y(\varphi )\hfill \\ \multicolumn{1}{c}{-t_x(\varphi )}\end{array}\right]$$

(23)

The basic principle of the non-circular gear shaping is to ensure that the pitch circle of the pinion shaper cutter is purely rolling on the pitch curve of the non-circular gear. As shown in Figure 9, it is assumed that the non-circular gear is fixed on the plane, and the coordinate system $S_0(o_0-x_0{y}_0)$ is fixed on the non-circular gear. A follow-up coordinate system $S_1(o_1-x_1{y}_1)$ is established at the center of the shaper cutter, and its x₁-axial and y₁-axial directions are the same as vector n and t, respectively. The coordinate system $S_2(o_2-x_2{y}_2)$ is fixed on the shaper cutter, and its rotation angle $\theta$ with respect to the coordinate system $S_1(o_1-x_1{y}_1)$ is the rotation angle of the shaper cutter, and the pitch radius of the shaper cutter is $r_2$ .

Figure 9.

Illustration of coordinate systems $s_0,s_1,s_2$ .

According to the pure rolling relationship between the pitch curve and the pitch circle, it can be seen that when the polar angle of the non-circular gear is $\varphi$ , the rotation angle $\theta$ of the shaper cutter can be expressed as

$$\theta =\frac{S(\varphi )}{r_2}=\frac{{\int }_0^{\phi }\sqrt{r\prime {(\varphi )}^2+r{(\varphi )}^2}d\varphi }{r_2}$$

(24)

where $S(\varphi )$ is the arc length that rolled over.

According to the above derivation and geometric relationship, the coordinates of the rotation center $o_2$ of the shaper cutter in the coordinate system $S_0$ can be obtained from the normal equidistant as follows

$$\{\begin{array}{c}{x}_{02}=r(\varphi )\cos \varphi -r_2{t}_y(\varphi )\hfill \\ \multicolumn{1}{c}{y_{02}=r(\varphi )\sin \varphi +r_2{t}_x(\varphi )}\end{array}$$

(25)

According to the basic coordinate transformation relationship, the angle between the two coordinate systems $S_2$ and $S_1$ on the shaper cutter is $\theta$ , so the coordinate transformation relationship can be expressed as

$${\mathbf{M}}_{12}=\left[\begin{array}{ccc}\cos \theta \hfill & -\sin \theta \hfill & 0\hfill \\ \multicolumn{1}{c}{\sin \theta }& \cos \theta \hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 1\hfill \end{array}\right]$$

(26)

The coordinate transformation relationship between coordinate systems $S_1$ and $S_0$ can be expressed as

$$\begin{array}{c}{\mathbf{M}}_{01}=\left[\begin{array}{ccc}{n}_x(\varphi )\hfill & t_x(\varphi )\hfill & x_{02}\hfill \\ \multicolumn{1}{c}{n_y(\varphi )}& t_y(\varphi )\hfill & y_{02}\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 1\hfill \end{array}\right]\hfill \\ =\left[\begin{array}{ccc}{t}_y(\varphi )\hfill & t_x(\varphi )\hfill & x_{02}\hfill \\ \multicolumn{1}{c}{-t_x(\varphi )}& t_y(\varphi )\hfill & y_{02}\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 1\hfill \end{array}\right]\hfill \end{array}$$

(27)

According to the coordinate transformation relationship derived above, assuming that the tooth profile equation of the shaper cutter is ${\mathbf{r}}_2(t)=[x_2(t)y_2(t)1{]}^T$ , tooth profile of the non-circular gear can be presented as the envelope curves of the tooth profile of the shape cutter, as follows

$$\mathbf{r}={\mathbf{M}}_{01}{\mathbf{M}}_{12}{\mathbf{r}}_2$$

(28)

Tooth profile extraction

According to the coordinate transformation method described in the previous section, it is easy to obtain the envelope curves of the non-circular gear tooth profile and then the complete tooth profile curve of the non-circular gear can be obtained by extracting the boundary of the envelope curves. The extraction process mainly relies on MATLAB’s Boolean operation on two-dimensional closed regions. As shown in Figures 10 and 11, it is assumed that the region surrounded by the contour curve of the gear blank in the plane is ${\mathrm{\Omega }}^{(k)}$ , and the region surrounded by the tooth profile of the shaper cutter at a certain coordinate position is ${\mathrm{\Omega }}^{(k+1)}$ . The position of the shaper cutter after each coordinate transformation can be obtained by equation (28). As the coordinate position of the shaper cutter changes, the region enclosed by the shaper cutter becomes ${\mathrm{\Omega }}^{(k+2)}$ , ${\mathrm{\Omega }}^{(k+3)}$ , ${\mathrm{\Omega }}^{(k+4)}$ , … To obtain the tooth profile of the non-circular gear, it is only necessary to perform Boolean subtraction on the closed regions ${\mathrm{\Omega }}^{(k)}$ and ${\mathrm{\Omega }}^{(k+i)}(i=1,2,3,\dots )$ . In the actual operation of MATLAB, Boolean subtraction is performed on only two closed regions at a time, and ${\mathrm{\Omega }}^{(k)}$ is iterated with the transformation of the coordinate position of the shaper cutter until the last coordinate position. The iterative formula is as follows

$${\mathrm{\Omega }}^{(k)}={\mathrm{\Omega }}^{(k)}-{\mathrm{\Omega }}^{(k+i)}i=1,2,3\dots$$

(29)

Figure 10.

Shaper cutter profile.

Figure 11.

MATLAB Boolean operation to extract the tooth profile curve.

The tooth profile and basic parameters of the shaper cutter are shown in Figure 10, where m is the modulus. According to the basic parameters of each non-circular gear in Table 2, the tooth profile curve of the CDIMN-PGT is obtained using the tooth profile extraction algorithm described above, as shown in Figure 12.

Figure 12.

Tooth profile of the CDIMN-PGT.

Virtual prototype and simulation

The non-circular gear tooth profile obtained in the previous chapter is actually composed of a large number of two-dimensional coordinate points. Therefore, after obtaining the tooth profile points, the non-circular gear pairs $G_{12}$ and $G_{34}$ can be easily modeled by the three-dimensional (3D) modeling software SolidWorks. Then combined with the planet carrier, a simplified virtual prototype 3D model of the CDIMN-PGT is obtained, as shown in Figure 13. Finally, the model is imported into the ADAMS software for simulation analysis.

Figure 13.

Virtual prototype model of the CDIMN-PGT.

The transmission ratio of the CDIMN-PGT in the single input-output state is simulated. From the input–output relationship of the gear train in the last four working states of Table 1 and Figure 6, it can be seen that the transmission ratio of the gear train in working state IV and V is reciprocal, and the transmission ratio of working states VI and VII is also the same, so only the transmission ratio under working states IV and V is simulated. The simulation results are shown in Figures 14 and 15, respectively.

Figure 14.

Working state IV transmission ratio simulation result.

Figure 15.

Working state VI transmission ratio simulation result.

It can be seen from Figures 14 and 15 that the two sets of simulation values have the same period as the theoretical value, the amplitudes are relatively close, and the simulated values at the peaks and troughs deviate significantly from the theoretical values. The error sources mainly include the tooth profile accuracy of the non-circular gears and the inertial force generated by the dynamic imbalance of the non-circular gears. Among them, the influence of the tooth profile accuracy on the simulation results is reflected in the slight fluctuation of the simulation value near the theoretical value, and the influence of the inertial force of the non-circular gears on the simulation result is manifested in the difference between the simulated value and the theoretical value at some positions. These positions are mainly at the peaks and troughs of the transmission ratio curve.

Conclusion

The pitch curve design method of CDIMN-PGT is studied. The boundary conditions of the planetary gear design parameters are derived from the geometric relationship.

The transmission characteristics of the gear train are analyzed from the kinematics. The input–output relationship of the gear train under various working conditions is derived. The influence of the eccentricity of the non-circular gear and the phase angle of the planetary gear on the overall transmission ratio of the gear train is analyzed.

Combined with the specific design example, the precise tooth profile of the non-circular gear is obtained by the Boolean operation function of MATLAB software. This method is simple and efficient. Based on this, the virtual prototype model of the gear train was established and its kinematics simulation was carried out in the ADAMS software. The simulation results verify the correctness of the gear train design method and kinematics theory analysis.