Rounding theory and method for the pitch curves of the variable center distance non-circular gear pair based on working conditions

Abstract

Based on the working conditions of the variable center distance non-circular gear pair, this research provides the rounding theories and methods for the theoretical pitch curves of the gear pair, including establish the working conditions model of the variable center distance non-circular gear pair; group and classify the instantaneous working conditions and the instantaneous meshing arcs; design rounding algorithms for instantaneous meshing arc group category 1, 2, and 3; and propose the dynamic allocation strategy for distribute the total rounding amount on each pair of instantaneous meshing arcs. Finally, with the help of a variable center distance non-circular gear pair example, this research successfully proved the correctness and effectiveness of the above rounding theory and method, achieve the goal of rounding the theoretical pitch curves of the variable center distance non-circular gear pair.

Keywords

Variable center distance non-circular gear rounding theory working condition

Introduction

According to the meshing theory, in order to mesh properly each pair of gear driver can and must only have integer number of teeth arranged on both driving and driven gear’s pitch curves.^1,2

During the design process of the cylindrical gear, to make sure the length of the driving and driven gear’s pitch curves can be divisible by standard pitch, typically, the center distance of the cylindrical gear pair can be calculated by the module of the gear pair and the teeth number of the driving gear and the driven gear, after that the pitch curves of the driving gear and the driven gear can be calculated by the center distance mentioned above, and then the transmission ratio calculated by the teeth number of the driving gear and the driven gear. Clearly, these pitch curves can be divided evenly by standard pitch.^3–5

For non-circular gear pair, due to its changing transmission ratio during meshing, it is impossible to calculate the center distance of the gear pair by the module and the teeth number of the driving gear and the driven gear just like cylindrical gear. As a result, the theoretical pitch curves of the driving gear and the driven gear calculated by known conditions cannot be evenly divided by standard pitch in most cases. To address this issue, Xutang and Guihai suggest that designers should dynamically adjust the known conditions, such as the parameters that control the base curve used to fit the gear pair’s pitch curves, and the modulus that directly related to the pitch, until the pitch curve length for both driving gear and driven gear are multiple of the standard pitch.⁶

Under the condition of transmission ratio unknown, based on the aforementioned ideas, by adjusting the parameters in the base curves such as eccentric circular curves,^7–12 higher-order elliptical curves,^13–19 Pascal’s limaçon,^20,21 Hermite cubic parameter curves,²² epitrochoids,²³ Aronhold’s first principle regression curve,^24,25 etc., it is possible to design non-circular gear pair’s pitch curves which can be divisible by the pitch.²⁶

Under the condition of transmission ratio known, by scaling the theoretical pitch curve proportionally, at the expense of changing the center distance of the gear pair²⁷; or adjusting the gear modulus, at the expense of increasing the gear pair’s design and manufacturing costs dramatically, it is also possible to design pitch curves which can be divisible by the pitch and meshes same as the known transmission ratio.

However, the variable center distance non-circular gear pairs not only have variable transmission ratio but also have center distance during the meshing process, and they have one-to-one mapping relationship with the pitch curves of the gear pair,^28–30 which making it difficult to ensure the designed theoretical pitch curves can be evenly divided by the pitch through method mentioned above without adjusting the gear modulus, and that will highly increasing the design and manufacturing costs.

To address the aforementioned issues, after the research and analysis of the variable center distance non-circular gear pair’s meshing process, the theoretical center distance curve $a^{AB}$ and the theoretical transmission ratio curve $i^{AB}$ formed by this process can be divided into two parts based on working conditions: the working curves that used to describe the required center distance and transmission ratio at specific instants during the gear pair meshing; and the non-working curves that used to ensure the continuity and smoothness of $a^{AB}$ and $i^{AB}$ in their return or dwell periods during the gear pair meshing. This division is illustrated in Figures 1 and 2.

Figure 1.

$a^{AB}$ of the variable center distance non-circular gear pair based on working conditions.

Figure 2.

$i^{AB}$ of the variable center distance non-circular gear pair based on working conditions.

By combining the center distance curve, the transmission ratio curve, and the instantaneous meshing pole radius of the pitch curves of the driving gear and the driven gear, the working conditions model of the variable center distance non-circular gear pair can be established. Then, according to the grouping and classification method provided in this paper, the instantaneous meshing arcs in this model can be grouped and classified into four categories.

Based on the general meshing model of the instantaneous meshing arc pair establishes at the micro-level, this research provides rounding algorithms for instantaneous meshing arc group category 1, 2, and 3. After that, propose a dynamic allocation strategy for the total rounding amount at the macro level. Finally, the rounding theory and method for the pitch curves of variable center distance non-circular gear pair based on working conditions have been developed. And an example of variable center distance non-circular gear pair is provided to validate the correctness and effectiveness of these rounding theory and method.

Theory and method of pitch curve rounding for variable center distance non-circular gear pair based on working conditions

The theory and method of pitch curve rounding for variable center distance non-circular gear pair based on working conditions described in this section can be divided into four main parts as shown in Figures 3 to 6.

Figure 3.

The working conditions model of the variable center distance non-circular gear pair.

Figure 4.

The classification and analysis of the instantaneous meshing arcs in the working conditions model.

Figure 5.

The meshing model and the rounding algorithm of the instantaneous meshing arc pair.

Figure 6.

Dynamic allocation strategy of rounding amount.

As shown in Figure 3, the working conditions model of the variable center distance non-circular gear pair will be built in the first part of this research. The model is a coordinate system group, which takes the driving gear’s rotation angle as the abscissa, and respectively takes the instantaneous center distance, the instantaneous transmission ratio, and the length of the instantaneous meshing pole radius of the pitch curves of the driving gear and the driven gear as the ordinate. Based on the one-to-one relationship between the center distance curve, the transmission ratio curve and the theoretical pitch curves of the driving gear and the driven gear of the variable center distance non-circular gear pair, all of the center distance curve, the transmission ratio curve, and the curves made by the instantaneous meshing pole radius length of the theoretical pitch curves of the driving gear and the driven gear can be calculated and drawn on this coordinate system group.

As shown in Figure 4, the second part of this research is mainly focus on the classification and analysis of the instantaneous meshing arcs based on the above-mentioned model. In this part, the instantaneous meshing arcs that mesh at the instants associated with the current instant will be classified into same instantaneous meshing arc group. Then, according to the categories of the instantaneous working conditions connected with the instantaneous meshing arcs through the meshing instants, the geometric shape of instantaneous meshing arcs, and the rounding feasibility and rounding algorithm adaptability of the instantaneous meshing arcs in each instantaneous meshing arc group, these groups will be analyzed and classified into four categories in this part of the research.

As shown in Figure 5, the third part of this research will establish and analysis the general meshing model of any instantaneous meshing arc pair at the micro level. And propose the constant center distance rounding algorithm and the constant transmission rounding algorithm for instantaneous meshing arcs in instantaneous meshing arcs group category 1, 2, and 3.

As shown in Figure 6, the dynamic allocation strategy for rounding amount at the macro level will be proposed in the fourth part of this research which contain several rounding amount allocating rounds. In the beginning of this strategy, total rounding amount will be put into the rounding amount pool. Then, in each round, the rounding amount in the pool will be dynamic allocating on each pair of the instantaneous meshing arc, after the instantaneous meshing arc pairs will be rounded by suitable algorithms and verified, when the rounding result fails the verification, the rounding amount allocated on the arc pair will be reduced, the reduced part will be store in the pool, and round the arc pair again, until the rounding result can pass the verifications. After every arc pair passed the rounding process and the verifications, the start of a new rounding amount allocating round depends on whether the pool is not empty. When the pool is empty at the end of any round, the rounding amount allocating process is end, the rounding amount distribute on the pitch curves of the variable center distance non-circular gear pair and the rounded pitch curves of the variable center distance non-circular gear pair will be get, and finally achieve the goal of rounding the theoretical pitch curves of the variable center distance non-circular gear pair.

The working conditions model of the variable center distance non-circular gear pair

With the driving gear’s rotation angle $ϕ^{PCA}$ as the abscissa, the theoretical center distance curve $a^{AB}$ and the theoretical transmission ratio curve $i^{AB}$ of the variable center distance non-circular gear pair can be drawn as shown in Figures 1 and 2.

Based on the screw theory,³⁰ the theoretical pitch curves $P C^{A}$ and $P C^{B}$ of the variable center distance non-circular gear pair’s driving gear and driven gear can be calculated from $a^{AB}$ and $i^{AB}$ as shown in Figures 7 and 8.

Figure 7.

Theoretical pitch curve of driving gear.

Figure 8.

Theoretical pitch curve of driven gear.

Assume that the time required for $P C^{A}$ and $P C^{B}$ meshing a cycle is constant, and the rotation angle $ϕ_{t_{i}}^{PCA}$ of the driving gear at any meshing instant $t_{i}$ are equal to $Δ ϕ^{PCA}$ . Then take $ϕ^{PCA}$ as the abscissa, and take the length of the instantaneous meshing pole radius $R^{PCA} (φ^{PCA})$ and $R^{PCB} (φ^{PCB})$ on $P C^{A}$ and $P C^{B}$ as the ordinate, the $P C^{A}$ and $P C^{B}$ can be converted into the instantaneous meshing pole radius curves as shown in Figures 9 and 10.

Figure 9.

Instantaneous meshing pole radius curve of the driving gear.

Figure 10.

Instantaneous meshing pole radius curve of the driven gear.

At last, takes $ϕ^{PCA}$ as the abscissa, and respectively takes $a^{AB}$ , $i^{AB}$ , $R^{PCA} (φ^{PCA})$ , and $R^{PCB} (φ^{PCB})$ as the ordinate, the working conditions model of the variable center distance non-circular gear pair can be obtained by combining Figures 1, 2, 9, and 10 from top to bottom, as shown in the Figure 11.

Figure 11.

The working conditions model of the variable center distance non-circular gear pair.

Classification and analysis of instantaneous meshing arcs

Grouping of instantaneous meshing arcs

The definition of the associated instantaneous meshing arc and the associated instant are given as follows.

Assume that $L^{PCA}$ and $L^{PCB}$ are the length of $P C^{A}$ and $P C^{B}$ , the ratio of $L^{PCA}$ and $L^{PCB}$ is $M : N$ ; during the meshing process, the time for $P C^{A}$ and $P C^{B}$ to make one revolution is $t_{A}$ and $t_{B}$ , and the time for $P C^{A}$ and $P C^{B}$ to mesh a round is $t_{AB}$ ; the instantaneous meshing arc ${Arc}_{t_{a}}^{PCA}$ on $P C^{A}$ meshing with the instantaneous meshing arc ${Arc}_{t_{a}}^{PCB}$ on $P C^{B}$ with at $t_{a}$ . During the meshing process of the gear pair, it can be seen that the same instantaneous meshing arc ${Arc}_{(t_{a} + i * t_{A}) % t_{AB}}^{PCA}$ on $P C^{A}$ meshes at instant $(t_{a} + i * t_{A}) % t_{AB}$ , where $i = [0, N - 1]$ , and its meshing partner are the different instantaneous meshing arcs ${Arc}_{(t_{a} + i * t_{A}) % t_{AB}}^{PCB}$ on $P C^{B}$ . And the meshing partner of each instantaneous meshing arc ${Arc}_{(t_{a} + i * t_{A}) % t_{AB}}^{PCB}$ is ${Arc}_{(t_{a} + i * t_{A}) % t_{AB} % t_{B} + j * t_{B}}^{PCA}$ , where $j = [0, M - 1]$ . Then, ${Arc}_{(t_{a} + i * t_{A}) % t_{AB} % t_{B} + j * t_{B}}^{PCA}$ is the associated instantaneous meshing arc of ${Arc}_{t_{a}}^{PCA}$ . In the same way, ${Arc}_{(t_{a} + j * t_{B}) % t_{AB} % t_{A} + i * t_{A}}^{PCB}$ is the associated instantaneous meshing arc of ${Arc}_{t_{a}}^{PCB}$ , it can be seen that, $(t_{a} + j * t_{B}) % t_{AB} % t_{A} + i * t_{A}$ is equal to $(t_{a} + i * t_{A}) % t_{AB} % t_{B} + j * t_{B}$ , so ${Arc}_{(t_{a} + j * t_{B}) % t_{AB} % t_{A} + i * t_{A}}^{PCB}$ is the meshing partner of ${Arc}_{(t_{a} + i * t_{A}) % t_{AB} % t_{B} + j * t_{B}}^{PCA}$ , and both ${Arc}_{t_{a}}^{PCA}$ and ${Arc}_{t_{a}}^{PCB}$ has $M * N$ associated instantaneous meshing arcs including themselves.

And the associated instant of $t_{a}$ is $(t_{a} + i * t_{A}) % t_{AB} % t_{B} + j * t_{B}$ or $(t_{a} + j * t_{B}) % t_{AB} % t_{A} + i * t_{A}$ , $t_{a}$ also has $M * N$ associated instants including itself.

Based on the definition above, the associated instantaneous meshing arcs of the instantaneous meshing arc pair ${Arc}_{t_{i}}^{PCA}$ and ${Arc}_{t_{i}}^{PCB}$ meshing at any instant $t_{i}$ , thus, the instantaneous meshing arcs meshing at the associated instants of any instants $t_{i}$ , will be classified into the same instantaneous meshing arc group $ArcSe t_{t_{i}}$ based on the time sequence.

In order to further elucidate the relationship of any instant and the instantaneous meshing arcs meshing at that instant, to the associated instant and the associated instantaneous meshing arcs, another method used to find the associated instant and the associated instantaneous meshing arcs, apart from the definition above, is shown in the example below.

As shown in Figure 12, $t_{A} = 400$ , $t_{B} = 300$ , $t_{AB} = 1200$ , $M : N = 3 : 4$ , $t_{AB}$ is the time required for $P C^{A}$ to make three revolutions which is also the time required for $P C^{B}$ to make four revolutions. Assume that, at $t_{a} = 413$ , instantaneous meshing arcs ${Arc}_{413}^{PCA}$ and ${Arc}_{413}^{PCB}$ are meshing together, as shown in Figure 12(a). It can be seen that ${Arc}_{13}^{PCA}$ , ${Arc}_{413}^{PCA}$ , and ${Arc}_{813}^{PCA}$ are the same instantaneous meshing arc. And ${Arc}_{13}^{PCB}$ and ${Arc}_{813}^{PCB}$ , the meshing partner of ${Arc}_{13}^{PCA}$ and ${Arc}_{813}^{PCA}$ , have the same meshing partner as ${Arc}_{413}^{PCB}$ . So, in Figure 12(a) ${Arc}_{13}^{PCB}$ and ${Arc}_{813}^{PCB}$ are the associated instantaneous meshing arcs of ${Arc}_{413}^{PCB}$ . In the same way, ${Arc}_{113}^{PCA}$ , ${Arc}_{713}^{PCA}$ , and ${Arc}_{1013}^{PCA}$ are the associated instantaneous meshing arcs of ${Arc}_{413}^{PCA}$ in Figure 12(a). With the help of, ${Arc}_{113}^{PCA}$ , ${Arc}_{713}^{PCA}$ , ${Arc}_{1013}^{PCA}$ , ${Arc}_{13}^{PCB}$ , and ${Arc}_{813}^{PCB}$ , more associated instantaneous meshing arcs of ${Arc}_{413}^{PCA}$ and ${Arc}_{413}^{PCB}$ can be found, until there are no more associated instantaneous meshing arcs to find.

Figure 12.

An example of the associated instantaneous meshing arc (a) and the associated instant (b).

Finally, as shown in Figure 12(b), the associated instantaneous meshing arcs of ${Arc}_{413}^{PCA}$ are ${Arc}_{13}^{PCA}$ , ${Arc}_{113}^{PCA}$ , ${Arc}_{213}^{PCA}$ , ${Arc}_{313}^{PCA}$ , ${Arc}_{413}^{PCA}$ , ${Arc}_{513}^{PCA}$ , ${Arc}_{613}^{PCA}$ , ${Arc}_{713}^{PCA}$ , ${Arc}_{813}^{PCA}$ , ${Arc}_{913}^{PCA}$ , ${Arc}_{1013}^{PCA}$ , and ${Arc}_{1113}^{PCA}$ ; the associated instantaneous meshing arcs of ${Arc}_{413}^{PCB}$ are ${Arc}_{13}^{PCB}$ , ${Arc}_{113}^{PCB}$ , ${Arc}_{213}^{PCB}$ , ${Arc}_{313}^{PCB}$ , ${Arc}_{413}^{PCB}$ , ${Arc}_{513}^{PCB}$ , ${Arc}_{613}^{PCB}$ , ${Arc}_{713}^{PCB}$ , ${Arc}_{813}^{PCB}$ , ${Arc}_{913}^{PCB}$ , ${Arc}_{1013}^{PCB}$ , and ${Arc}_{1113}^{PCB}$ ; the associated instants of $t_{413}$ are $t_{13}$ , $t_{113}$ , $t_{213}$ , $t_{313}$ , $t_{413}$ , $t_{513}$ , $t_{613}$ , $t_{713}$ , $t_{813}$ , $t_{913}$ , $t_{1013}$ , $t_{1113}$ ; and all the associated instantaneous meshing arcs of ${Arc}_{413}^{PCA}$ and ${Arc}_{413}^{PCB}$ will be classified into $ArcSe t_{13}$ . And that result is consistent with the above definition of the associated instantaneous meshing arc and the associated instant.

Classification and analysis of instantaneous meshing arc group

According to Section “Introduction,” $a^{AB}$ and $i^{AB}$ of the variable center distance noncircular gear pair can be classified into two categories: working curve and non-working curve, as shown in Figures 1 and 2. Therefore, the instantaneous working conditions $a & i_{t_{i}}$ at any instant $t_{i}$ can be classified into four categories: $an & in$ means that both $a_{t_{i}}^{AB}$ and $i_{t_{i}}^{AB}$ at $t_{i}$ belongs to the non-working curve; $aw & in$ means that $a_{t_{i}}^{AB}$ belongs to the working curve, and $i_{t_{i}}^{AB}$ belongs to the non-working curve; $an & iw$ means that $a_{t_{i}}^{AB}$ belongs to the non-working curve, and $i_{t_{i}}^{AB}$ belongs to the working curve; $aw & iw$ means that $a_{t_{i}}^{AB}$ and $i_{t_{i}}^{AB}$ belong to working curves.

Assuming that $a & iSe t_{t_{i}}$ is a set of instantaneous working conditions includes $a & i_{t_{i}}$ at the associated instants of any instant $t_{i}$ , according to the possible categories of $a & i_{t_{i}}$ in $a & iSe t_{t_{i}}$ , 13 categories of $a & iSe t_{t_{i}}$ can be found as shown in Table 1.

Table 1.

Thirteen categories of $a & iSe t_{t_{i}}$ .

State	$an & in$	$aw & in$	$an & iw$	$aw & iw$
1	○	×	×	×
2	×	○	×	×
3	×	×	○	×
4	×	×	×	○
5	○	○	×	×
6	○	×	○	×
7	○	×	×	○
8	×	○	○	×
9	×	○	×	○
10	×	×	○	○
11	○	○	○	×
12	○	○	×	○
13	○	○	○	○

Afterwards, according to the categories of $a & i_{t_{i}}$ in $a & iSe t_{t_{i}}$ , the geometric shape of the instantaneous meshing arcs in $ArcSe t_{t_{i}}$ , and the rounding algorithm adaptability of the instantaneous meshing arcs in $ArcSe t_{t_{i}}$ , the instantaneous meshing arcs group will be analyzed and classified into four categories, $ArcSet_1$ , $ArcSet_2$ , $ArcSet_3$ , and $ArcSet_4$ :

(1) Among them, the $a & iSet$ of $ArcSet_1$ only contains $an & in$ as shown in the first row of Table 1. The characteristic of $an & in$ makes the rounding process of instantaneous meshing arcs in $ArcSet_1$ has nothing to do with the working curves of $a^{AB}$ and $i^{AB}$ . Therefore, almost any rounding algorithm can be used to round the instantaneous meshing arcs in $ArcSet_1$ .

(2) As shown in rows 2 and 5 of Table 1, the $a & iSet$ of $ArcSet_2$ must contains $aw & in$ , may contains $an & in$ , but never contains $an & iw$ and $aw & iw$ . And if any two working conditions $a & i_{t_{e}}$ , $a & i_{t_{f}}$ which obtained by ${Arc}_{t_{a}}^{PCA}$ and ${Arc}_{t_{b}}^{PCB}$ , ${Arc}_{t_{c}}^{PCA}$ and ${Arc}_{t_{d}}^{PCB}$ in $ArcSet_2$ belong to $aw & in$ , the geometric shape of ${Arc}_{t_{a}}^{PCA}$ , ${Arc}_{t_{c}}^{PCA}$ and ${Arc}_{t_{b}}^{PCB}$ , ${Arc}_{t_{d}}^{PCB}$ must be the same in pairs, $a_{t_{e}}^{AB}$ , $a_{t_{f}}^{AB}$ and $i_{t_{e}}^{AB}$ , $i_{t_{f}}^{AB}$ of $a & i_{t_{e}}$ , $a & i_{t_{f}}$ must also be the same in pairs. To ensure working curves of $a^{AB}$ related to $ArcSet_2$ remain unchanged during the rounding process, the rounding algorithm must make sure the center distance unchanged before and after rounding the instantaneous meshing arcs in $ArcSet_2$ .

(3) As shown in rows 3 and 6 of Table 1, the $a & iSet$ of $ArcSet_3$ must contains $an & iw$ , may contains $an & in$ , but never contains $aw & in$ and $aw & iw$ . And if any two working conditions $a & i_{t_{e}}$ , $a & i_{t_{f}}$ which obtained by ${Arc}_{t_{a}}^{PCA}$ and ${Arc}_{t_{b}}^{PCB}$ , ${Arc}_{t_{c}}^{PCA}$ and ${Arc}_{t_{d}}^{PCB}$ in $ArcSet_3$ $belong to an & iw$ , the geometric shape of ${Arc}_{t_{a}}^{PCA}$ , ${Arc}_{t_{c}}^{PCA}$ and ${Arc}_{t_{b}}^{PCB}$ , ${Arc}_{t_{d}}^{PCB}$ must be the same in pairs, $a_{t_{e}}^{AB}$ , $a_{t_{f}}^{AB}$ and $i_{t_{e}}^{AB}$ , $i_{t_{f}}^{AB}$ of $a & i_{t_{e}}$ , $a & i_{t_{f}}$ must also be the same in pairs. To ensure working curves of $i^{AB}$ related to $ArcSet_3$ remain unchanged during the rounding process, the rounding algorithm must make sure the transmission ratio unchanged before and after rounding the instantaneous meshing arcs in $ArcSet_3$ .

(4) All the instantaneous meshing arc groups which do not belong to $ArcSet_1$ , $ArcSet_2$ , and $ArcSet_3$ will be classified in $ArcSet_4$ . It can be seen from the analysis that the instantaneous meshing arcs in $ArcSet_4$ must have at least one of the following three characteristics which make them hard to be round by any rounding algorithm:

There is $a & i_{t_{a}}$ belongs to $aw & iw$ , obtained by ${Arc}_{t_{a}}^{PCA}$ and ${Arc}_{t_{a}}^{PCB}$ in $ArcSet_4$ meshing with each other. As mentioned in Section “Introduction,” the center distance curve and the transmission ratio curve of the variable center distance noncircular gear pair have one-to-one mapping relationship with the pitch curves of the gear pair,^28–30 that means, after rounding process of ${Arc}_{t_{a}}^{PCA}$ and ${Arc}_{t_{a}}^{PCB}$ the center distance and the transmission ratio obtained by them will definitely change. So, there is no rounding algorithm described in this paper can round ${Arc}_{t_{a}}^{PCA}$ and ${Arc}_{t_{a}}^{PCB}$ without change their working center distance and the transmission ratio curves.

There are $a & i_{t_{a}}$ belongs to $an & iw$ , and $a & i_{t_{b}}$ belongs to $aw & in$ , obtained by ${Arc}_{t_{a}}^{PCA}$ and ${Arc}_{t_{a}}^{PCB}$ , ${Arc}_{t_{b}}^{PCA}$ and ${Arc}_{t_{b}}^{PCB}$ in $ArcSet_4$ . With this characteristic, the constant transmission ratio rounding algorithm need to round ${Arc}_{t_{a}}^{PCA}$ and ${Arc}_{t_{a}}^{PCB}$ without change $i_{t_{a}}$ , and the constant center distance rounding algorithm need to round ${Arc}_{t_{b}}^{PCA}$ and ${Arc}_{t_{b}}^{PCB}$ without change $a_{t_{a}}$ . That not only requires the rounding algorithm to fully study the meshing process of the associated instantaneous meshing arcs of ${Arc}_{t_{a}}^{PCA}$ and ${Arc}_{t_{a}}^{PCB}$ , ${Arc}_{t_{b}}^{PCA}$ and ${Arc}_{t_{b}}^{PCB}$ , but also requires two algorithms do not interfere with each other in the rounding process, and the instantaneous meshing arcs in this $ArcSet$ also needs to meet stringent conditions. Otherwise, there may exist a pair of instantaneous meshing arcs ${Arc}_{t_{c}}^{PCA}$ and ${Arc}_{t_{c}}^{PCB}$ , in which ${Arc}_{t_{c}}^{PCA}$ rounded by constant center distance rounding algorithm and ${Arc}_{t_{c}}^{PCB}$ rounded by constant transmission ratio rounding algorithm, thus generating an error in rounding process. Because of the above conditions, in order to provide formalized theory and method, this research does not provide the rounding algorithm for the instantaneous meshing arc groups with this characteristic.

There are two identical working conditions $a & i_{t_{a}}$ and $a & i_{t_{b}}$ belongs to same instantaneous working conditions categories, $an & iw$ or $aw & in$ , which obtained by ${Arc}_{t_{a}}^{PCA}$ and ${Arc}_{t_{a}}^{PCB}$ , ${Arc}_{t_{b}}^{PCA}$ and ${Arc}_{t_{b}}^{PCB}$ , but the geometric shapes of ${Arc}_{t_{a}}^{PCA}$ and ${Arc}_{t_{b}}^{PCA}$ , ${Arc}_{t_{a}}^{PCB}$ and ${Arc}_{t_{b}}^{PCB}$ are not all the same. In order to round the instantaneous meshing arcs without change $a_{t_{a}}$ , $a_{t_{b}}$ or $i_{t_{a}}$ , $i_{t_{b}}$ , similar to the characteristic 2, this characteristic requires the rounding algorithm to fully study the meshing process of the associated instantaneous meshing arcs of ${Arc}_{t_{a}}^{PCA}$ and ${Arc}_{t_{a}}^{PCB}$ , ${Arc}_{t_{b}}^{PCA}$ and ${Arc}_{t_{b}}^{PCB}$ , and the instantaneous meshing arcs in this $ArcSet$ needs to meet stringent conditions. So, this research does not provide the rounding algorithm for the instantaneous meshing arc groups with this characteristic neither.

Here is the classification and analysis of the variable center distance noncircular gear pair’s instantaneous meshing arc group. Then, the rounding algorithms suitable for the instantaneous meshing arcs in $ArcSet_1$ , $ArcSet_2$ , and $ArcSet_3$ will be given in next section.

The meshing model and the rounding algorithm of the instantaneous meshing arc pair

In this section, the general meshing model of the instantaneous meshing arc pair on the variable center distance non-circular gear pair will be established and analysis, and the constant center distance rounding algorithm and the constant transmission ratio rounding algorithm suitable for the instantaneous meshing arcs in $ArcSet_1$ , $ArcSet_2$ , and $ArcSet_3$ will be proposed to achieved the goal of rounding the instantaneous meshing arcs in $ArcSet_1$ , $ArcSet_2$ , and $ArcSet_3$ without changing the instantaneous working center distance and the instantaneous working transmission ratio.

General meshing model of the instantaneous meshing arc pair

Assume that the rotation angle $ϕ_{t_{i}}^{PCA}$ of $P C^{A}$ before rounding is equal to the rotation angle $ϕ_{t_{i}}^{* PCA}$ of $P C^{* A}$ after rounding at any instant $t_{i}$ , as shown as:

ϕ_{t_{i}}^{PCA} = ϕ_{t_{i}}^{* PCA} = i * Δ ϕ^{PCA}

(1)

Before the rounding process of $P C^{A}$ and $P C^{B}$ , as shown in Figure 13, $R^{PCA} (φ^{PCA})$ and $R^{PCB} (φ^{PCB})$ are the length of the instantaneous meshing pole radius on $P C^{A}$ and $P C^{B}$ , ${Arc}_{t_{i}}^{PCA}$ and ${Arc}_{t_{i}}^{PCB}$ are the instantaneous meshing arcs of $P C^{A}$ and $P C^{B}$ which meshing at point $P_{t_{i}}^{PCAB}$ , $k_{t_{i}}^{PCAB}$ is the tangent slope of $P_{t_{i}}^{PCAB}$ , $φ_{t_{i}}^{PCA}$ and $φ_{t_{i}}^{PCB}$ are the instantaneous meshing angles of $P_{t_{i}}^{PCAB}$ on $P C^{A}$ and $P C^{B}$ , $R^{PCA} (φ_{t_{i}}^{PCA})$ and $R^{PCB} (φ_{t_{i}}^{PCB})$ are the length of the instantaneous meshing pole radius of $P_{t_{i}}^{PCAB}$ on $P C^{A}$ and $P C^{B}$ , $θ_{t_{i}}^{PCA}$ and $θ_{t_{i}}^{PCB}$ are the angles formed by $R^{PCA} (φ_{t_{i}}^{PCA})$ , $R^{PCB} (φ_{t_{i}}^{PCB})$ and the $x$ -axis, $γ_{t_{i}}^{PCA}$ and $γ_{t_{i}}^{PCB}$ are arc angles of ${Arc}_{t_{i}}^{PCA}$ and ${Arc}_{t_{i}}^{PCB}$ , $ϕ_{t_{i}}^{PCA}$ and $ϕ_{t_{i}}^{PCB}$ are the rotation angle of $P C^{A}$ and $P C^{B}$ at $t_{i}$ .

Figure 13.

General meshing model of the instantaneous meshing arc pair before rounding.

During the meshing simulation process of $P C^{A}$ and $P C^{B}$ , $R^{PCA} (φ_{t_{i}}^{PCA})$ , $R^{PCB} (φ_{t_{i}}^{PCB})$ , $φ_{t_{i}}^{PCA}$ , $φ_{t_{i}}^{PCB}$ , $θ_{t_{i}}^{PCA}$ , $θ_{t_{i}}^{PCB}$ , $γ_{t_{i}}^{PCA}$ , $γ_{t_{i}}^{PCB}$ , $ϕ_{t_{i}}^{PCA}$ , $ϕ_{t_{i}}^{PCB}$ , and the tangent slope $k_{t_{i}}^{PCAB}$ at $P_{t_{i}}^{PCAB}$ can be calculated, then the instantaneous center distance $a_{t_{i}}^{AB}$ and the instantaneous transmission ratio $i_{t_{i}}^{AB}$ at the instant $t_{i}$ before the rounding process can be expressed as:

\begin{matrix} a_{t_{i}}^{AB} = R^{PCA} (φ_{t_{i}}^{PCA}) * \cos (θ_{t_{i}}^{PCA}) + R^{PCB} (φ_{t_{i}}^{PCB}) * \cos (θ_{t_{i}}^{PCB}) \end{matrix}

(2)

i_{t_{i}}^{PCAB} = \frac{R^{PCB} (φ_{t_{i}}^{PCB}) * \cos (θ_{t_{i}}^{PCB}) + \frac{R^{PCA} (φ_{t_{i}}^{PCA}) * \sin (θ_{t_{i}}^{PCA})}{k_{t_{i}}^{PCAB}}}{R^{PCA} (φ_{t_{i}}^{PCA}) * \cos (θ_{t_{i}}^{PCA}) - \frac{R^{PCA} (φ_{t_{i}}^{PCA}) * \sin (θ_{t_{i}}^{PCA})}{k_{t_{i}}^{PCAB}}}

(3)

Based on the meshing theory,¹ the arc length $l_{t_{i}}^{PCA}$ and $l_{t_{i}}^{PCB}$ of ${Arc}_{t_{i}}^{PCA}$ and ${Arc}_{t_{i}}^{PCB}$ are equal to each other and can be expressed as:

l_{t_{i}}^{PCA} = \sqrt{{(R^{PCA} (φ_{t_{i}}^{PCA}))}^{2} + {([R^{PCA} (φ_{t_{i}}^{PCA})]')}^{2}} * γ_{t_{i}}^{PCA}

(4)

l_{t_{i}}^{PCB} = \sqrt{{(R^{PCB} (φ_{t_{i}}^{PCB}))}^{2} + {([R^{PCB} (φ_{t_{i}}^{PCB})]')}^{2}} * γ_{t_{i}}^{PCB}

(5)

By keeping $γ_{t_{i}}^{PCA}$ and $γ_{t_{i}}^{PCB}$ unchanged during the rounding process, after rounding, ${Arc}_{t_{i}}^{* PCA}$ and ${Arc}_{t_{i}}^{* PCB}$ , the rounding result of ${Arc}_{t_{i}}^{PCA}$ and ${Arc}_{t_{i}}^{PCB}$ , can meshing with each other correctly at $t_{i}$ as same as ${Arc}_{t_{i}}^{PCA}$ and ${Arc}_{t_{i}}^{PCB}$ . So, the instantaneous arc angles $γ_{t_{i}}^{* PCA}$ and $γ_{t_{i}}^{* PCB}$ can be present as:

γ_{t_{i}}^{* PCA} = γ_{t_{i}}^{PCA}

(6)

γ_{t_{i}}^{* PCB} = γ_{t_{i}}^{PCB}

(7)

When instant $t_{i}$ is small enough, the instantaneous meshing angle $φ_{t_{i}}^{* PCA}$ and $φ_{t_{i}}^{* PCB}$ after rounding are same as $φ_{t_{i}}^{PCA}$ and $φ_{t_{i}}^{PCB}$ and can be expressed as:

φ_{t_{i}}^{* PCA} = \sum_{k = 1}^{i - 1} γ_{t_{k}}^{PCA} + \frac{γ_{t_{i}}^{PCA}}{2} = φ_{t_{i}}^{PCA}

(8)

φ_{t_{i}}^{* PCB} = \sum_{k = 1}^{i - 1} γ_{t_{k}}^{PCB} + \frac{γ_{t_{i}}^{PCB}}{2} = φ_{t_{i}}^{PCB}

(9)

It can be seen from Figure 13 and formula (8) that the angle $θ_{t_{i}}^{* PCA}$ between the instantaneous meshing pole radius $R^{* PCA} (φ_{t_{i}}^{* PCA})$ on $P C^{* A}$ and the $x$ -axis can be expressed as:

\begin{matrix} θ_{t_{i}}^{* PCA} = θ_{t_{i}}^{PCA} = φ_{t_{i}}^{PCA} - ϕ_{t_{i}}^{PCA} = \sum_{k = 1}^{i - 1} γ_{t_{k}}^{PCA} + \frac{γ_{t_{i}}^{PCA}}{2} - ϕ_{t_{i}}^{PCA} \end{matrix}

(10)

Based on formulas from (6) to (10), the general meshing model of ${Arc}_{t_{i}}^{* PCA}$ and ${Arc}_{t_{i}}^{* PCB}$ meshing with each other at $t_{i}$ can be built, as shown in the red part in Figure 14. It can be seen that at instant $t_{i}$ , $P C^{* A}$ and $P C^{* B}$ meshing at $P_{t_{i}}^{* PCAB}$ ; $k_{t_{i}}^{* PCAB}$ is the slope of the tangent line at point $P_{t_{i}}^{* PCAB}$ ; $R^{* PCA} (φ_{t_{i}}^{* PCA})$ and $R^{* PCB} (φ_{t_{i}}^{* PCB})$ are the length of the instantaneous meshing pole radius on $P C^{* A}$ and $P C^{* B}$ at $t_{i}$ ; $Δ r_{t_{i}}^{* PCA} (φ_{t_{i}}^{* PCA})$ and $Δ r_{t_{i}}^{* PCB} (φ_{t_{i}}^{* PCB})$ are the difference from $R^{PCA} (φ_{t_{i}}^{PCA})$ and $R^{PCB} (φ_{t_{i}}^{PCB})$ to $R^{* PCA} (φ_{t_{i}}^{* PCA})$ and $R^{* PCB} (φ_{t_{i}}^{* PCB})$ ; $ϕ_{t_{i}}^{* PCB}$ is the rotation angle of $P C^{* B}$ ; $Δ ϕ_{t_{i}}^{* PCB}$ is the difference from $ϕ_{t_{i}}^{PCB}$ to $ϕ_{t_{i}}^{* PCB}$ ; $θ_{t_{i}}^{* PCB}$ is the angle formed by $R^{* PCB} (φ_{t_{i}}^{* PCB})$ and the $x$ -axis. So, the following equations can be obtained:

R^{PCA} (φ_{t_{i}}^{* PCA}) = R^{* PCA} (φ_{t_{i}}^{* PCA}) + Δ r_{t_{i}}^{* PCA} (φ_{t_{i}}^{* PCA})

(11)

R^{PCB} (φ_{t_{i}}^{* PCB}) = R^{* PCB} (φ_{t_{i}}^{* PCB}) + Δ r_{t_{i}}^{* PCB} (φ_{t_{i}}^{* PCB})

(12)

ϕ_{t_{i}}^{PCB} = ϕ_{t_{i}}^{* PCB} + Δ ϕ_{t_{i}}^{* PCB}

(13)

Figure 14.

General meshing model of the instantaneous meshing arc pair after rounding.

And based on Figure 14 and formula (9), $θ_{t_{i}}^{* PCB}$ can be expressed as:

θ_{t_{i}}^{* P C B} = φ_{t_{i}}^{* P C B} - ϕ_{t_{i}}^{* P C B} = \sum_{k = 1}^{i - 1} γ_{t_{k}}^{P C B} + \frac{γ_{t_{i}}^{P C B}}{2} - ϕ_{t_{i}}^{* P C B}

(14)

Suppose that, $l_{t_{i}}^{* PCA}$ and $l_{t_{i}}^{* PCB}$ are the arc length of ${Arc}_{t_{i}}^{* PCA}$ and ${Arc}_{t_{i}}^{* PCB}$ , and they can be expressed as:

l_{t_{i}}^{* PCA} = \sqrt{{(R^{* PCA} (φ_{t_{i}}^{* PCA}))}^{2} + {([R^{* PCA} (φ_{t_{i}}^{* PCA})]')}^{2}} * γ_{t_{i}}^{* PCA}

(15)

l_{t_{i}}^{* P C B} = \sqrt{{(R^{* P C B} (φ_{t_{i}}^{* P C B}))}^{2} + {({[R^{* P C B} (φ_{t_{i}}^{* P C B})]}^{'})}^{2}} * γ_{t_{i}}^{* P C B}

(16)

And based on the meshing theory, it can be known that:

l_{t_{i}}^{* PCA} = l_{t_{i}}^{* PCB}

(17)

Finally, $a_{t_{i}}^{* PCAB}$ and $i_{t_{i}}^{* PCAB}$ , the center distance and transmission ratio of the variable center distance non-circular gear pair at $t_{i}$ after rounding, can be expressed as:

\begin{matrix} a_{t_{i}}^{* PCAB} = R^{* PCA} (φ_{t_{i}}^{* PCA}) * \cos (θ_{t_{i}}^{* PCA}) \\ + R^{* PCB} (φ_{t_{i}}^{* PCB}) * \cos (θ_{t_{i}}^{* PCB}) \end{matrix}

(18)

\begin{matrix} i_{t_{i}}^{* PCAB} = \\ \frac{k_{t_{i}}^{* PCAB} * R^{* PCB} (φ_{t_{i}}^{* PCB}) * \cos (θ_{t_{i}}^{* PCB}) + R^{* PCA} (φ_{t_{i}}^{* PCA}) * \sin (θ_{t_{i}}^{* PCA})}{k_{t_{i}}^{* PCAB} * R^{* PCA} (φ_{t_{i}}^{* PCA}) * \cos (θ_{t_{i}}^{* PCA}) - R^{* PCA} (φ_{t_{i}}^{* PCA}) * \sin (θ_{t_{i}}^{* PCA})} \end{matrix}

(19)

Constant center distance rounding algorithm

Based on the meshing theory,¹ after rounding, $l_{t_{i}}^{* PCA}$ equals to $l_{t_{i}}^{* PCB}$ , $φ_{t_{i}}^{* PCA}$ , $φ_{t_{i}}^{* PCB}$ , $γ_{t_{i}}^{* PCA}$ , and $γ_{t_{i}}^{* PCB}$ satisfy formulas (6)–(9), then $l_{t_{i}}^{* PCA}$ and $l_{t_{i}}^{* PCB}$ can be expressed as:

l_{t_{i}}^{* PCA} = \sqrt{{(R^{* PCA} (φ_{t_{i}}^{* PCA}))}^{2} + {(R^{* PCA} (φ_{t_{i}}^{* PCA})')}^{2}} * γ_{t_{i}}^{* PCA}

(20)

l_{t_{i}}^{* PCB} = \sqrt{{(R^{* PCB} (φ_{t_{i}}^{* PCB}))}^{2} + {(R^{* PCB} (φ_{t_{i}}^{* PCB})')}^{2}} * γ_{t_{i}}^{* PCB}

(21)

In order to make sure the instantaneous center distance unchanged before and after rounding, thus, $a_{t_{i}}^{PCAB}$ equals to $a_{t_{i}}^{* PCAB}$ , it can be known that:

\begin{matrix} {(R^{* PCB} (φ_{t_{i}}^{* PCB}))}^{2} = \begin{matrix} {(a_{t_{i}}^{* PCAB} - R^{* PCA} (φ_{t_{i}}^{* PCA}) * \cos (θ_{t_{i}}^{* PCA}))}^{2} \\ + \\ {(R^{* PCB} (φ_{t_{i}}^{* PCB}) * \sin (θ_{t_{i}}^{* PCB}))}^{2} \end{matrix} \end{matrix}

(22)

Then, $θ_{t_{i}}^{* PCA}$ and $θ_{t_{i}}^{* PCB}$ , as described in formulas (10) and (14), can be expressed as:

θ_{t_{i}}^{* PCA} = \sum_{k = 1}^{i - 1} γ_{t_{k}}^{PCA} + \frac{γ_{t_{i}}^{PCA}}{2} - ϕ_{t_{i}}^{PCA}

(23)

θ_{t_{i}}^{* PCB} = \sum_{k = 1}^{i - 1} γ_{t_{k}}^{PCB} + \frac{γ_{t_{i}}^{PCB}}{2} - ϕ_{t_{i}}^{* PCB}

(24)

The slope $k_{t_{i}}^{* PCAB}$ of the tangent line at point $P_{t_{i}}^{* PCAB}$ can be expressed as:

\begin{matrix} k_{t_{i}}^{* PCAB} = \\ \frac{[R^{* PCA} (φ_{t_{i}}^{* PCA})]' * \cos (θ_{t_{i}}^{* PCA}) - R^{* PCA} (φ_{t_{i}}^{* PCA}) * [\sin (θ_{t_{i}}^{* PCA})]'}{[R^{* PCA} (φ_{t_{i}}^{* PCA})]' * \sin (θ_{t_{i}}^{* PCA}) - R^{* PCA} (φ_{t_{i}}^{* PCA}) * [\cos (θ_{t_{i}}^{* PCA})]'} \end{matrix}

(25)

Also, $k_{t_{i}}^{* PCAB}$ can be expressed as:

\begin{matrix} k_{t_{i}}^{* PCAB} = \\ \frac{[R^{* PCB} (φ_{t_{i}}^{* PCB})]' * \cos (θ_{t_{i}}^{* PCB}) - R^{* PCB} (φ_{t_{i}}^{* PCB}) * [\sin (θ_{t_{i}}^{* PCB})]'}{[R^{* PCB} (φ_{t_{i}}^{* PCB})]' * \sin (θ_{t_{i}}^{* PCB}) - R^{* PCB} (φ_{t_{i}}^{* PCB}) * [\cos (θ_{t_{i}}^{* PCB})]'} \end{matrix}

(26)

With the help of formulas from (20) to (26), $R^{* PCA} (φ_{t_{i}}^{* PCA})$ , the value of $R^{* PCA} (φ_{t_{i}}^{* PCA})'$ , $R^{* PCB} (φ_{t_{i}}^{* PCB})$ , $R^{* PCB} (φ_{t_{i}}^{* PCB})'$ , and $θ_{t_{i}}^{* PCB}$ can be solved, the meshing state of ${Arc}_{t_{i}}^{PCA}$ , ${Arc}_{t_{i}}^{PCB}$ on $P C^{A}$ , $P C^{B}$ (black part) and ${Arc}_{t_{i}}^{* PCA}$ , ${Arc}_{t_{i}}^{* PCB}$ on $P C^{* A}$ , $P C^{* B}$ (red part) before and after the constant center distance rounding process at any instant $t_{i}$ can be shown in Figure 15.

Figure 15.

Instantaneous meshing arcs before and after the constant center distance rounding algorithm.

By substituting $k_{t_{i}}^{* PCAB}$ , $R^{* PCA} (φ_{t_{i}}^{* PCA})$ , $R^{* PCB} (φ_{t_{i}}^{* PCB})$ , $θ_{t_{i}}^{* PCA}$ , and $θ_{t_{i}}^{* PCB}$ into formula (19), the instantaneous transmission ratio $i_{t_{i}}^{* PCAB}$ after rounding can be obtained as show in formula (27):

\begin{matrix} i_{t_{i}}^{* PCAB} = \\ \frac{k_{t_{i}}^{* PCAB} * R^{* PCB} (φ_{t_{i}}^{* PCB}) * \cos (θ_{t_{i}}^{* PCB}) + R^{* PCA} (φ_{t_{i}}^{* PCA}) * \sin (θ_{t_{i}}^{* PCA})}{k_{t_{i}}^{* PCAB} * R^{* PCA} (φ_{t_{i}}^{* PCA}) * \cos (θ_{t_{i}}^{* PCA}) - R^{* PCA} (φ_{t_{i}}^{* PCA}) * \sin (θ_{t_{i}}^{* PCA})} \end{matrix}

(27)

And that is the constant center distance rounding algorithm suitable for the instantaneous meshing arc group category 1 and 2.

Constant transmission ratio rounding algorithm

In order to make sure the instantaneous transmission ratio before and after rounding are equal, the following two assumptions can be made:

In formula (13), the value of $Δ θ_{t_{i}}^{* PCB}$ and $Δ ϕ_{t_{i}}^{* PCB}$ equal to zero;

$k_{t_{i}}^{* PCAB}$ equals to $k_{t_{i}}^{PCAB}$ .

Based on the above assumptions, according to the formulas from (1), (6), (7), (8), (9), (10), (13), and (14), it is easy to obtained that $ϕ_{t_{i}}^{PCA}$ , $ϕ_{t_{i}}^{PCB}$ , $γ_{t_{i}}^{PCA}$ , $γ_{t_{i}}^{PCB}$ , $φ_{t_{i}}^{PCA}$ , $φ_{t_{i}}^{PCB}$ , $θ_{t_{i}}^{PCA}$ , and $θ_{t_{i}}^{PCB}$ remain unchanged after rounding.

Then the meshing state of ${Arc}_{t_{i}}^{PCA}$ , ${Arc}_{t_{i}}^{PCB}$ on $P C^{A}$ , $P C^{B}$ (black part) and ${Arc}_{t_{i}}^{* PCA}$ , ${Arc}_{t_{i}}^{* PCB}$ on $P C^{* A}$ , $P C^{* B}$ (red part) before and after the constant transmission ratio rounding process at any instant $t_{i}$ can be shown in Figure 16.

Figure 16.

Instantaneous meshing arcs before and after the constant transmission ratio rounding algorithm.

There must be a proportionality coefficient $Δ q_{t_{i}}^{* PCAB}$ >0, so that:

Δ q_{t_{i}}^{* PCAB} * R^{* PCA} (φ_{t_{i}}^{* PCA}) = R^{PCA} (φ_{t_{i}}^{PCA})

(28)

Δ q_{t_{i}}^{* PCAB} * R^{* PCB} (φ_{t_{i}}^{* PCB}) = R^{PCB} (φ_{t_{i}}^{PCB})

(29)

Then according to the formulas (15) and (16), it can be known that:

\begin{matrix} l_{t_{i}}^{* PCA} = \sqrt{{(\frac{R^{PCA} (φ_{t_{i}}^{PCA})}{Δ q_{t_{i}}^{* PCAB}})}^{2} + {([\frac{R^{PCA} (φ_{t_{i}}^{PCA})}{Δ q_{t_{i}}^{* PCAB}}]')}^{2}} * \\ γ_{t_{i}}^{* PCA} = \frac{l_{t_{i}}^{PCA}}{Δ q_{t_{i}}^{* PCAB}} \end{matrix}

(30)

\begin{matrix} l_{t_{i}}^{* PCB} = \sqrt{{(\frac{R^{PCB} (φ_{t_{i}}^{PCB})}{Δ q_{t_{i}}^{* PCAB}})}^{2} + {([\frac{R^{PCB} (φ_{t_{i}}^{PCB})}{Δ q_{t_{i}}^{* PCAB}}]')}^{2}} * \\ γ_{t_{i}}^{* PCB} = \frac{l_{t_{i}}^{PCB}}{Δ q_{t_{i}}^{* PCAB}} \end{matrix}

(31)

Combining this equation with formulas (30) and (31), it can be seen that $l_{t_{i}}^{* PCA} = l_{t_{i}}^{* PCB}$ , which can prove the rounding result satisfy the meshing theory.

Put the formulas (28) and (29) into the formula (19), the result can be obtained as shown in formula (32):

\begin{matrix} i_{t_{i}}^{* PCAB} = i_{t_{i}}^{PCAB} = \\ \frac{k_{t_{i}}^{* PCAB} * \frac{R^{PCB} (φ_{t_{i}}^{PCB})}{Δ q_{t_{i}}^{* PCAB}} * \cos (θ_{t_{i}}^{* PCB}) + \frac{R^{PCA} (φ_{t_{i}}^{PCA})}{Δ q_{t_{i}}^{* PCAB}} * \sin (θ_{t_{i}}^{* PCA})}{k_{t_{i}}^{* PCAB} * \frac{R^{PCA} (φ_{t_{i}}^{PCA})}{Δ q_{t_{i}}^{* PCAB}} * \cos (θ_{t_{i}}^{* PCA}) - \frac{R^{PCA} (φ_{t_{i}}^{PCA})}{Δ q_{t_{i}}^{* PCAB}} * \sin (θ_{t_{i}}^{* PCA})} \end{matrix}

(32)

Which can prove that the instantaneous transmission ratio remains unchanged during this constant transmission ratio rounding process, therefore, the above-mentioned constant transmission ratio rounding algorithm can round the instantaneous meshing arc group category 1 and 3 without change the instantaneous transmission ratio of it.

While $a_{t_{i}}^{* PCAB}$ can be expressed as:

a_{t_{i}}^{* PCAB} = \frac{a_{t_{i}}^{PCAB}}{Δ q_{t_{i}}^{* PCAB}}

(33)

And that is the constant transmission ratio rounding algorithm suitable for the instantaneous meshing arc group category 1 and 3.

Dynamic allocation strategy of rounding amount

In order to get reasonable pitch curves which can be divided evenly by standard pitch after rounding, this section proposes a dynamic allocation strategy to distribute rounding amount on each instantaneous meshing arc pairs. And the flowchart of this strategy is shown in Figure 17.

Figure 17.

Flow chart of the dynamic allocation strategy of rounding amount.

The rounding amount allocating process under this strategy usually contents several rounds which depends on the rounding amount pool is empty or not. At the beginning of all rounds, the raflag of the meshing instantaneous arc pair in ArcSet_1, ArcSet_2, and ArcSet_3 will be set to 0, and the raflag of the meshing instantaneous arc pair in ArcSet_4 will be set to 1, total rounding amount will fill into the pool.

Then, in each round, the rounding amount allocating process starts from the initial rounding instantaneous meshing arc group which is the instantaneous meshing arc group in $ArcSet_1$ , $ArcSet_2$ , or $ArcSet_3$ , next to the first instantaneous meshing arc group in category 4 since the start meshing instant of the variable center distance non-circular gear pair.

Assume that $Δ L^{PCA}$ and $Δ L^{PCB}$ are the total rounding amount of $P C^{A}$ and $P C^{B}$ ; $Δ L_{i}^{poolA}$ and $Δ L_{i}^{poolB}$ are the total rounding amount of the rounding amount pool of $P C^{A}$ and $P C^{B}$ in any rounding amount allocating round $i$ , which contains total rounding amount at the beginning of round 1; $L^{PCA}$ and $L^{PCB}$ are the pitch curve length of the driving gear and the driven gear; $L_{IV}^{PCA}$ and $L_{IV}^{PCB}$ are the total instantaneous meshing arc length of all instantaneous meshing arcs in $ArcSet_4$ on $P C^{A}$ and $P C^{B}$ ; $l_{t_{a}}^{PCA}$ and $l_{t_{a}}^{PCB}$ are the length of any the instantaneous meshing arc pair ${Arc}_{t_{a}}^{PCA}$ and ${Arc}_{t_{a}}^{PCB}$ which belongs to $ArcSet_1$ , $ArcSet_2$ , or $ArcSet_3$ on $P C^{A}$ and $P C^{B}$ meshing at any instant $t_{a}$ , ${RAFlag}_{t_{a}}^{PCA}$ and ${RAFlag}_{t_{a}}^{PCB}$ are the flags used to record whether the rounding amount allocating process on ${Arc}_{t_{a}}^{PCA}$ and ${Arc}_{t_{a}}^{PCB}$ is complete (1) or not (0), the initial value of ${RAFlag}_{t_{a}}^{PCA}$ and ${RAFlag}_{t_{a}}^{PCB}$ of ${Arc}_{t_{a}}^{PCA}$ and ${Arc}_{t_{a}}^{PCB}$ belong to $ArcSet_1$ , $ArcSet_2$ , or $ArcSet_3$ are 0, and the initial value of ${RAFlag}_{t_{a}}^{PCA}$ and ${RAFlag}_{t_{a}}^{PCB}$ of ${Arc}_{t_{a}}^{PCA}$ and ${Arc}_{t_{a}}^{PCB}$ belong to $ArcSet_4$ are 1.

So, the rounding amount $Δ l_{i, t_{a}}^{PCA}$ and $Δ l_{i, t_{a}}^{PCB}$ distributed on ${Arc}_{t_{a}}^{PCA}$ and ${Arc}_{t_{a}}^{PCB}$ in any rounding amount allocating round $i$ can be present as

Δ l_{i, t_{a}}^{PCA} = {\begin{matrix} \frac{l_{t_{a}}^{PCA}}{L^{PCA} - L_{IV}^{PCA}} * Δ L^{PCA} & i = 1 \\ Δ l_{i - 1, t_{a}}^{PCA} + \frac{l_{t_{a}}^{PCA}}{L^{PCA} - L_{IV}^{PCA}} * Δ L_{i}^{poolA} & i > 1, {RAFlag}_{t_{a}}^{PCA} = 0 \\ Δ l_{i - 1, t_{a}}^{PCA} & i > 1, {RAFlag}_{t_{a}}^{PCA} = 1 \end{matrix}

(34)

Δ l_{i, t_{a}}^{PCB} = {\begin{matrix} \frac{l_{t_{a}}^{PCB}}{L^{PCB} - L_{IV}^{PCB}} * Δ L^{PCB} & i = 1 \\ Δ l_{i - 1, t_{a}}^{PCB} + \frac{l_{t_{a}}^{PCB}}{L^{PCB} - L_{IV}^{PCB}} * Δ L_{i}^{poolB} & i > 1, {RAFlag}_{t_{a}}^{PCB} = 0 \\ Δ l_{i - 1, t_{a}}^{PCB} & i > 1, {RAFlag}_{t_{a}}^{PCB} = 1 \end{matrix}

(35)

And the rounding amount distributed on the instantaneous meshing arcs which belongs to category 4 is zero, because their round amount flag are 1.

Then use the rounding algorithms mentioned above to round the instantaneous meshing arc pair belong to $ArcSet_1$ , $ArcSet_2$ , or $ArcSet_3$ .

When the instantaneous meshing arc pair belongs to $ArcSet_1$ or $ArcSet_3$ , it can be successfully rounded by the constant transmission ratio rounding algorithm described in Section “Constant transmission ratio rounding algorithm” with the rounding amount distributed by formulas (34) and (35).

When the instantaneous meshing arc pair belongs to $ArcSet_2$ , it can be known from formulas (20) to (26) that an unsuitable rounding amount will make the constant center distance round algorithm unsolvable. Therefore, after rounding a pair of instantaneous meshing arcs belongs to $ArcSet_2$ , the rounding result’s existence should be verified, if the rounding result is not existed, the rounding amount distributed on this arc pair should be reduced and store in the rounding amount pool until this pair of instantaneous meshing arcs has a rounding result after rounding, after that, set ${RAFlag}_{t_{a}}^{PCA}$ and ${RAFlag}_{t_{a}}^{PCA}$ to 1.

After that, use the curvature radius of the instantaneous meshing arcs as a necessary condition to verify whether the pitch curves are reasonable after rounding, the curvature radius $ρ_{t_{a}}^{* PCA}$ and $ρ_{t_{a}}^{* PCB}$ of ${Arc}_{t_{a}}^{* PCA}$ and ${Arc}_{t_{a}}^{* PCB}$ should satisfy the following three conditions:

The sign of $ρ_{t_{a}}^{* PCA}$ and $ρ_{t_{a}}^{* PCB}$ after rounding should be same as $ρ_{t_{a}}^{PCA}$ and $ρ_{t_{a}}^{PCB}$ of ${Arc}_{t_{a}}^{PCA}$ and ${Arc}_{t_{a}}^{PCB}$ before rounding;

When $ρ_{t_{a}}^{PCA} \geq 0$ , $ρ_{t_{a}}^{* PCA}$ should not be less than the minimum value $ρ_{\min}^{PCA}$ of the curvature radius on $P C^{A}$ before rounding; similarly, when $ρ_{t_{a}}^{PCB} \geq 0$ , $ρ_{t_{a}}^{* PCB}$ should not be less than the minimum value $ρ_{\min}^{PCB}$ of the curvature radius on $P C^{B}$ before rounding;

When $ρ_{t_{a}}^{PCA} < 0$ , $ρ_{t_{a}}^{* PCA}$ should not be greater than the maximum value $ρ_{\max}^{PCA}$ of the curvature radius on $P C^{A}$ before rounding; similarly, when $ρ_{t_{a}}^{PCB} < 0$ , $ρ_{t_{a}}^{* PCB}$ should not be greater than the maximum value $ρ_{\max}^{PCB}$ of the curvature radius on $P C^{B}$ before rounding.

And the curvature radius $ρ_{t_{a}}^{* PCA}$ and $ρ_{t_{a}}^{* PCB}$ of ${Arc}_{t_{a}}^{* PCA}$ and ${Arc}_{t_{a}}^{* PCB}$ can be obtained by the formulas below.

Assume that $ϕ_{t_{a - 1}}^{* PCA}$ is the rotation angle of $P C^{* A}$ at $t_{a - 1}$ , $P_{t_{a - 1}}^{* PCA}$ is the meshing point on $P C^{* A}$ meshing with $P C^{* B}$ at $t_{a - 1}$ , $φ_{t_{a - 1}}^{* PCA}$ is the meshing angle of $P_{t_{a - 1}}^{* PCA}$ , $R^{* PCA} (φ_{t_{a - 1}}^{* PCA})$ is the pole radius length of $P_{t_{a - 1}}^{* PCA}$ , then, $x_{t_{a - 1}, t_{a}}^{* PCA}$ and $y_{t_{a - 1}, t_{a}}^{* PCA}$ , the coordinates of $P_{t_{a - 1}}^{* PCA}$ at $t_{a - 1}$ , can be expressed as:

x_{t_{a - 1}, t_{a}}^{* PCA} = R^{* PCA} (φ_{t_{a - 1}}^{* PCA}) * \cos (φ_{t_{a - 1}}^{* PCA} - ϕ_{t_{a - 1}}^{* PCA} - Δ ϕ^{PCA})

(36)

y_{t_{a - 1}, t_{a}}^{* PCA} = R^{* PCA} (φ_{t_{a - 1}}^{* PCA}) * \sin (φ_{t_{a - 1}}^{* PCA} - ϕ_{t_{a - 1}}^{* PCA} - Δ ϕ^{PCA})

(37)

According to formula (1), the rotation angle from $t_{a - 1}$ to $t_{a}$ is $Δ ϕ^{PCA}$ , after rounding, $P_{t_{a}}^{* PCA}$ is the meshing point on $P C^{* A}$ meshing at $t_{a}$ , $φ_{t_{a}}^{* PCA}$ is the meshing angle of $P_{t_{a}}^{* PCA}$ , $R^{* PCA} (φ_{t_{a}}^{* PCA})$ is the pole radius length of $P_{t_{a}}^{* PCA}$ , $x_{t_{a}}^{* PCA}$ , and $y_{t_{a}}^{* PCA}$ , the coordinates of $P_{t_{a}}^{* PCA}$ , can be expressed as:

x_{t_{a}}^{* PCA} = R^{* PCA} (φ_{t_{a}}^{* PCA}) * \cos (φ_{t_{a}}^{* PCA} - ϕ_{t_{a - 1}}^{* PCA} - Δ ϕ^{PCA})

(38)

y_{t_{a}}^{* PCA} = R^{* PCA} (φ_{t_{a}}^{* PCA}) * \sin (φ_{t_{a}}^{* PCA} - ϕ_{t_{a - 1}}^{* PCA} - Δ ϕ^{PCA})

(39)

Assume that, $k_{t_{a - 1}}^{* PCA}$ is the slope of the tangent line at $P_{t_{a - 1}}^{* PCA}$ , $k_{t_{a - 1}, t_{a}}^{* PCA}$ the slope of the tangent line at $P_{t_{a - 1}}^{* PCA}$ when $P C^{* A}$ rotated from $t_{a - 1}$ to $t_{a}$ , $P_{t_{a - 1}}^{* PCA}$ rotated to $P_{t_{a - 1}, t_{a}}^{* PCA}$ , the slope $k_{t_{a - 1}, t_{a}}^{* PCA}$ of the tangent line at $P_{t_{a - 1}, t_{a}}^{* PCA}$ can be expressed as:

k_{t_{a - 1}, t_{a}}^{* PCA} = \frac{(k_{t_{a - 1}}^{* PCA} - \tan (Δ ϕ^{PCA}))}{(1 + k_{t_{a - 1}}^{* PCA} * \tan (Δ ϕ^{PCA}))}

(40)

Then, the common tangent circle of $P_{t_{a - 1}, t_{a}}^{* PCA}$ and $P_{t_{a}}^{* PCA}$ can be obtained, and the coordinates of the center point $x_{t_{a}}^{O}$ and $y_{t_{a}}^{O}$ of the circle can be solved by formulas (36)–(40) with the following equations:

(x_{t_{a - 1}, t_{a}}^{* PCA} - x_{t_{a}}^{O}) / k_{t_{a - 1}, t_{a}}^{* PCA} = y_{t_{a}}^{O} - y_{t_{a - 1}, t_{a}}^{* PCA}

(41)

\begin{matrix} (x_{t_{a}}^{O} - \frac{(x_{t_{a - 1}, t_{a}}^{* PCA} + x_{t_{a}}^{* PCA})}{2}) * \frac{(x_{t_{a}}^{* PCA} - x_{t_{a - 1}, t_{a}}^{* PCA})}{(y_{t_{a - 1}, t_{a}}^{* PCA} - y_{t_{a}}^{* PCB})} \\ = y_{t_{a}}^{O} - \frac{(y_{t_{a - 1}, t_{a}}^{* PCA} + y_{t_{a}}^{* PCA})}{2} \end{matrix}

(42)

And the curvature radius $ρ_{t_{a}}^{* PCA}$ of ${Arc}_{t_{a}}^{* PCA}$ can be solved by the equations below:

ρ_{t_{a}}^{* PCA} = \sqrt{{(x_{t_{a}}^{O} - x_{t_{a}}^{* PCA})}^{2} + {(y_{t_{a}}^{O} - y_{t_{a}}^{* PCA})}^{2}}

(43)

In the same way, the curvature radius $ρ_{t_{a}}^{* PCB}$ of ${Arc}_{t_{a}}^{* PCB}$ can also be obtained. If $ρ_{t_{a}}^{* PCA}$ and $ρ_{t_{a}}^{* PCB}$ cannot satisfy the above conditions, the rounding amount distributed on this arc pair should be reduced and store in the rounding amount pool until this pair of instantaneous meshing arcs has a rounding result after rounding and their curvature radius can satisfy the above conditions, then set ${RAFlag}_{t_{a}}^{PCA}$ and ${RAFlag}_{t_{a}}^{PCA}$ to 1. If $ρ_{t_{a}}^{* PCA}$ and $ρ_{t_{a}}^{* PCB}$ satisfied the above conditions, the rounding amount allocating process of ${Arc}_{t_{a}}^{PCA}$ and ${Arc}_{t_{a}}^{PCB}$ in round $i$ is complete.

After all instantaneous meshing arc pairs’ rounding amount allocating process is complete in round $i$ , and the round amount pool is not empty, a new round of rounding amount allocating process will be started in order to distribute the rounding amount left in the pool on the instantaneous meshing arc pairs with the round amount flags equal to 0. On the contrary, when the rounding amount allocating process of round $i$ is complete and the round amount pool is empty, the process of dynamic distribution of the rounding amount on the pitch curves of the variable center distance non-circular gear pair is complete, the rounding amount distribution on each pair of instantaneous meshing arc and the rounded pitch curves of the variable center distance non-circular gear pair can be obtained in this process.

Applicable scope

As mentioned in this section, whether a pair of parallel-axis variable center distance non-circular gear can be rounded by the method described in this research is usually depends on the instantaneous meshing arcs in the instantaneous meshing arc group category 1, 2, and 3 of the gear pair is enough for the total rounding amount to be completely and reasonably distributed on. So, in the research field of parallel-axis non-circular gear pair, this round method can mainly be used to round three types of the parallel-axis non-circular gear pair:

(1) Fixed center distance non-circular gear pair

In the research field of non-circular gear pair, most research and applications are focus on non-circular gear pair with fixed center distance, such as picking and planting mechanism,^31,32 silk reeling machine,³³ robot,^34–36 engine,³⁷ and etc. As a special case of the variable center distance non-circular gear pair, fixed center distance non-circular gear pair can surely be rounded by the method described in this research.

When its center distance can be fine-tuned in the design process, then the center distance curve of such gear pair can be seen as a non-working curve, and all instantaneous meshing arcs on the driving and driven gear’s pitch curves can be grouped and classified into $ArcSet_1$ or $ArcSet_3$ , rounded by the constant transmission ratio rounding algorithm. Therefore, such fixed center distance non-circular gear pair can be rounded by the method described in this research.

When the gear pair’s center distance is already given at the beginning of the design, then the center distance curve of such gear pair can be seen as a working curve, and all instantaneous meshing arcs on the driving and driven gear’s pitch curves can be grouped and classified into $ArcSet_2$ or $ArcSet_4$ , if the instantaneous meshing arcs in the instantaneous meshing arc group category 2 of the gear pair is enough for the total rounding amount to be completely and reasonably distributed on, then this gear pair can be rounded by the method described in this research.

(2) Variable center distance non-circular gear pair

For variable center distance non-circular gear pair composed of two non-circular gears,³⁰ the instantaneous meshing arcs on the driving and driven gear’s pitch curves can be grouped and classified into $ArcSet_1$ , $ArcSet_2$ , $ArcSet_3$ , or $ArcSet_4$ . So, as told before, this gear pair can be rounded by the method described in this research if the instantaneous meshing arcs in the instantaneous meshing arc group category 1, 2, and 3 of the gear pair is enough for the total rounding amount to be completely and reasonably distributed on.

(3) Variable center distance circular and non-circular gear pair

As a special case of the variable center distance non-circular gear pair, the conditions for the circular and non-circular gear pair composed of a circular gear and a non-circular gear^38–40 to be rounded by the method described in this research is different from the variable center distance non-circular gear pair mentioned in (2).

Since part of the circular and non-circular gear pair is a circular gear, during the rounding process, the rounding amount will be evenly distributed on the pitch curves of the circular gear as well as the non-circular gear, which means, every instantaneous meshing arc’s round amount on the gear pair’s pitch curves is directly proportional to its arc length, even if the round amount of the instantaneous meshing arc belongs to the instantaneous meshing arc group category 4 is not zero. However, the instantaneous meshing arcs belong to $ArcSet_4$ cannot be rounded according to the method described in this research. Therefore, if the circular-non-circular gear pair has instantaneous meshing arc group belongs to $ArcSet_4$ , the gear pair cannot be rounded by the rounding method described in this research; on the contrary, if the circular-non-circular gear pair only has instantaneous meshing arc groups belongs to $ArcSet_1$ , $ArcSet_2$ , or $ArcSet_3$ , then the gear pair can be rounded by the rounding method described in this research.

Example and analysis of the pitch curves’ rounding process of the variable center distance non-circular gear pair based on working conditions

In this section, a variable center distance non-circular gear pairs’ theoretical pitch curves will be taken as an example to be rounded and analyzed to prove the correctness and effectiveness of the theory and method mentioned above.

The polar coordinate function $R^{PCA} (φ^{PCA})$ of the driving gear’s theoretical pitch $P C^{A}$ is:

R^{PCA} (φ^{PCA}) = \frac{84.423}{2.02 + 0.2 * \cos (3 * (φ^{PCA} + π))} - 0.225

(44)

And $P C^{A}$ is shown in Figure 18.

Figure 18.

Theoretical pitch curve of driving gear.

The pitch curve length $L^{PCA}$ of $P C^{A}$ can be calculated by the following formula as below:

\begin{matrix} L^{PCA} = \begin{matrix} \sum_{i = 1}^{n - 1} \int_{φ_{t_{i}}^{PCA}}^{φ_{t_{i + 1}}^{PCA}} \sqrt{{(R^{PCA} (φ^{PCA}))}^{2} + {([R^{PCA} (φ^{PCA})]')}^{2}} d φ^{PCA} \\ + \\ \int_{φ_{t_{sum - 1}}^{PCA}}^{2 π} \sqrt{{(R^{PCA} (φ^{PCA}))}^{2} + {([R^{PCA} (φ^{PCA})]')}^{2}} d φ^{PCA} \end{matrix} \\ = 268.2949 mm \end{matrix}

(45)

The polar coordinate function $R^{PCB} (φ^{PCB})$ of the driven gear’s theoretical pitch curve $P C^{B}$ is:

R^{PCB} (φ_{t_{i}}^{PCB}) = \frac{84.423}{2.02 + 0.2 * \cos (3 * (φ^{PCA} + \frac{π}{4}))} - 0.225

(46)

And $P C^{B}$ is shown in Figure 19.

Figure 19.

Theoretical pitch curve of driven gear.

The pitch curve length $L^{PCB}$ of $P C^{B}$ can also be calculated by the following formula below:

\begin{matrix} L^{PCB} = \begin{matrix} \sum_{i = 1}^{n - 1} \int_{φ_{t_{i}}^{PCB}}^{φ_{t_{i + 1}}^{PCB}} \sqrt{{(R^{PCB} (φ^{PCB}))}^{2} + {([R^{PCB} (φ^{PCB})]')}^{2}} d φ^{PCB} \\ + \\ \int_{φ_{t_{sum - 1}}^{PCB}}^{2 π} \sqrt{{(R^{PCB} (φ^{PCB}))}^{2} + {([R^{PCB} (φ^{PCB})]')}^{2}} d φ^{PCB} \end{matrix} \\ = 268.2949 mm \end{matrix}

(47)

When the gear pair’s modulus $m^{G} = 2$ , tooth pitch $p^{G} = 2 π$ , according to the formulas (4) and (5), it can be proved that $L^{PCA}$ and $L^{PCA}$ cannot be divisible by $p^{G}$ , which means $P C^{A}$ and $P C^{B}$ need to be round.

Assuming that the teeth number of the driving gear and the driven gear after rounding is 42, and the theoretical center distance curve $a^{AB}$ and the theoretical transmission ratio curve $i^{AB}$ of the variable center distance non-circular gear pairs based on working conditions are as shown in Figures 20 and 21.

Figure 20.

The theoretical center distance curve of the variable center distance non-circular gear pair based on working conditions.

Figure 21.

The theoretical transmission ratio curve of the variable center distance non-circular gear pair based on working conditions.

Based on the theory and methods in Section “The working conditions model of the variable center distance non-circular gear pair,” the meshing condition of the variable center distance non-circular gear pairs can be established as shown in Figure 22.

Figure 22.

The working conditions model of the variable center distance non-circular gear pair.

By dividing the meshing time of the gear pair into 3600 meshing instants, based on the method in Section “Classification and analysis of instantaneous meshing arc group,” all the instantaneous meshing arcs meshing at same associated instant can be grouped and classified into four categories. Then, the meshing radius curves formed by the instantaneous meshing arcs in the instant meshing arc groups which belongs to $ArcSet_1$ , $ArcSet_2$ , $ArcSet_3$ , and $ArcSet_4$ respectively can be drawn and shown in Figures 23 to 26.

Figure 23.

Instantaneous meshing arc groups in $ArcSet_1$ .

Figure 24.

Instantaneous meshing arc groups in $ArcSet_2$ .

Figure 25.

Instantaneous meshing arc groups in $ArcSet_3$ .

Figure 26.

Instantaneous meshing arc groups in $ArcSet_4$ .

After calculation, it can be seen that the ratio of the total arc length of the gear pair’s instantaneous meshing arc group category 1, 2, and 3 to the gear pair’s pitch curves length is 71.42%. Therefore, it can be preliminarily judged that this variable center distance non-circular gear pair can be rounded by the method described in this research.

According to Section “Dynamic allocation strategy of rounding amount,” after the rounding amount allocating process, the rounding amount distribution on each pair of instantaneous meshing arc can be obtained, and so as the rounded pitch curves of the variable center distance non-circular gear pair as shown in the Figures 27 and 28. The theoretical pitch curves of the variable center distance non-circular gear pair also shown in these figures for comparison purpose.

Figure 27.

The rounded pitch curves of the driving gear.

Figure 28.

The rounded pitch curves of the driven gear.

And the radial difference curve of the pitch curves of the variable center distance non-circular gear pair before and after rounding are shown in Figures 29 and 30.

Figure 29.

The radial difference curve of the driving gear’s pitch curve before and after rounding.

Figure 30.

The radial difference curve of the driven gear’s pitch curve before and after rounding.

It can be seen from the radial difference curves that, the max radial difference of the driving gear before and after rounding is 2.09 mm, and the max radial difference of the driven gear before and after rounding is 1.93 mm.

Based on the rounded pitch curves in Figures 27 and 28, the center distance and transmission ratio curves after rounding can be obtained and shown in Figures 31 and 32 after the meshing simulation. For comparison purpose, the theoretical center distance curve and the theoretical transmission ratio curve of the variable center distance non-circular gear pair also shown in these figures.

Figure 31.

The center distance curve of the variable center distance non-circular gear pair after rounding.

Figure 32.

The transmission ratio curve of the variable center distance non-circular gear pair after rounding.

And the difference curves of the center distance curve and the transmission ratio curve of the variable center distance non-circular gear pair before and after rounding are shown in Figures 33 and 34.

Figure 33.

The difference curve of the center distance curve before and after rounding.

Figure 34.

The difference curve of the transmission ratio curve before and after rounding.

It can be seen from Figures 33 and 34 that, the working curve of the center distance and the transmission ratio remain unchanged after the rounding process, and the max difference of the center distance no-working curve after rounding is 3.88 mm, and the max difference of the transmission ratio no working curve after rounding is 0.00648.

Based on the rounded pitch curves of the gear pair, with the help of the CNC gear shaping machine (type YKS5132X3/297 equipped with SIEMENS SINUMERIK 802Dsl system) and the G-code used to control the CNC, the driving gear and the driven gear of the rounded variable center distance gear pair can be obtained and shown in Figures 35 and 36 after the shaping process

Figure 35.

Rounded driving gear.

Figure 36.

Rounded driven gear.

As shown in Figure 37, by testing the variable center distance non-circular gear pair on the double-flank gear tester mentioned in previous research,⁴¹ the gear pair’s meshing characteristics can be obtained.

Figure 37.

Variable center distance non-circular gear pair.

After the testing, the real center distance curve and the real transmission ratio curve of the rounded gear pair can be got as shown in Figures 38 and 39. For comparison purpose, the theoretical center distance curve and the theoretical transmission ratio curve of the variable center distance non-circular gear pair also shown in these two figures.

Figure 38.

The real center distance curve of the rounded variable center distance non-circular gear pair.

Figure 39.

The real transmission ratio curve of the rounded variable center distance non-circular gear pair.

The center distance difference curve from the actual center distance to the theoretical center distance is shown in Figure 40.

Figure 40.

The transmission ratio difference curve from the actual transmission ratio to the theoretical transmission ratio of the gear pair.

And the transmission ratio difference curve from the actual transmission ratio to the theoretical transmission ratio is shown in Figure 41.

Figure 41.

The transmission ratio difference curve from the actual transmission ratio to the theoretical transmission ratio of the gear pair.

It can be seen from the above Figures 38 to 41 that the working curves on the real center distance curve and the real transmission ratio curve of the variable center distance non-circular gear pair after round are almost consistent with the working curves on the theoretical center distance curve and the theoretical transmission ratio curve of the gear pair. And the max difference of the center distance working curve from the actual center distance to the theoretical center distance is 45.5 μm, the max difference of the transmission ratio working curve from the actual transmission ratio to the theoretical transmission ratio is 0.000136. By analogy to the accuracy standard of circular gears, it can be seen that the difference of the center distance working curve and the transmission ratio working curve are basically meet the accuracy requirements, thus proving the correctness and effectiveness of the above rounding theory and method.

Conclusions and outlook

Based on the working conditions of the variable center distance non-circular gear pair, this research provides the rounding theories and methods for the theoretical pitch curves of the gear pair to ensure the designed theoretical pitch curves can be evenly divided by the pitch without change its modulus, which can decrease the design and manufacturing costs of the variable center distance non-circular gear pair. Theoretically, there are four main contributions in this work:

Proposed a novel approach, which not only can divide the theoretical center distance curve and transmission ratio curve into working curves and non-working curves based on the working conditions, but also can build the working condition model of the gear pair to reveal the relationship between the working curves, the non-working curves, and the gear pair’s theoretical pitch curves.

A comprehensive method is proposed for group and classify all the instantaneous meshing arcs into four categories based on the geometric shape, the rounding algorithm adaptability, and the categories of instantaneous working conditions corresponding to them.

Developed a constant center distance rounding algorithm and a constant transmission ratio rounding algorithm, which can be used to round the instantaneous meshing arcs in arc group category 1, 2, and 3.

Proposed a dynamic allocation strategy for distribute the total rounding amount on the pitch curves, to reasonably distribute the rounding amount on the pitch curve.

Carried out a variable center distance non-circular gear pairs’ theoretical pitch curves as an example to be rounded and analyzed, the results successfully proved the correctness and effectiveness of the above rounding theory and method.

Although the above rounding theory and method achieve the goal of rounding the theoretical pitch curves of the variable center distance non-circular gear pair without change its modulus, there are still many works to do to improve and extend them:

Although cannot be rounded by the rounding theory and method mentioned in this research, the instantaneous meshing arc groups may be rounded by a complicated rounding process may still exist in ArcSet_4. So, in order to separate and round the arc groups that can be rounded from the arc groups cannot be rounded at all, method used to subdivide and round the instantaneous meshing arc groups in ArcSet_4 may be proposed in the further studies.

In each round of the rounding process, the rounding amount distributed on the instantaneous meshing arcs with RAflag = 0 at the first time is equal. When the rounding amount is relatively large, this rounding amount distribute strategy may cause the rounding strategy to spend more rounds as well as more time to round the current pitch curves. So, to improving the efficiency of the rounding amount dynamic allocation strategy in this research, a suitable rounding amount distribution function used to distribute the round amount more reasonable at the beginning of each round may be proposed.

The rounding theory and method mentioned in this research is applied in the parallel shaft transmission field, however, the above theories and methods can also be learned and improved in the field of non-parallel shaft transmission to solve their rounding problems.

Footnotes

Appendix

Handling Editor: Chenhui Liang

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Botao Li

References

Xutang

. Principle of gearing. Beijing: China Machine Press, 1982.

Litvin

Fuentes

. Gear geometry and applied theory. Cambridge: Cambridge University Press, 2004.

Dingfang

Jianyi

Jiajun

. Modern mechanical design handbook. Beijing: China Machine Press, 2014.

Kutz

. Mechanical engineers’ handbook, volume 1: materials and engineering mechanics. Hoboken, NJ: John Wiley & Sons, 2015.

Norton

. Design of machinery an introduction to the synthesis and analysis of mechanisms and machines. Maidenhead: McGraw-Hill, 1999.

Xutang

Guihai

. Non-circular gears and non-uniform ratio transmission. Beijing: China Machine Press, 1997.

Chen

Zhao

. Design and applications of generalized eccentric noncircular gears. Chin Mech Eng 2018; 29: 565.

Lin

Ran

. Compound motion characteristics analysis of eccentric herringbone curve-face gear pair. J Mech Sci Technol 2021; 35: 5571–5578.

Lin

Wang

, et al. Compound transmission principle analysis of eccentric curve-face gear pair with curvilinear-shaped teeth based on tooth contact analysis. Proc IMechE, Part C: J Mechanical Engineering Science 2022; 236: 4026–4037.

10.

Lin

. Calculation and characteristic analysis of tooth width of eccentric helical curve-face gear. Iran J Sci Technol Trans Mech Eng 2019; 43: 781–797.

11.

Zeng

Deng

, et al. Design and tests of a rotary plug seedling pick-up mechanism for vegetable automatic transplanter. Int J Agric Biol Eng 2020; 13: 70–78.

12.

Liu

, et al. Design and experimental research on seedling pick-up mechanism of planetary gear train with combined non-circular gear transmission. Chin J Mech Eng 2019; 32: 13–20.

13.

Yan

Guan

Chen

. Design system of the high order elliptic gear pair pitch curve based on the solidworks. J Mech Transm 2017; 41: 169–172.

14.

Zhang

Rao

Zheng

, et al. Design and kinematics analysis of non-circular gear. Mech Res Appl 2018; 31: 38–42.

15.

Lin

Xia

. Geometric design and kinematics analysis of coplanar double internal meshing non-circular planetary gear train. Adv Mech Eng 2018; 10: 1–12.

16.

Liu

, et al. Design of piecewise deformed elliptical gear with closed pitch curve and its conjugate pair. Sci Rep 2022; 12: 17156.

17.

Malakova

Urbanský

Fedorko

, et al. Design of geometrical parameters and kinematical characteristics of a non-circular gear transmission for given parameters. Appl Sci 2021; 11: 1–23.

18.

Liu

Zhu

. Research on non-circular helical gear NC hobbing linkage control scheme based on variable transmission ratio. Proc IMechE, Part B: J Engineering Manufacture 2022; 236: 1433–1442.

19.

Song

Zhou

. Design and extraction of tooth profile of denatured elliptical gear. Adv Mech Eng 2022; 14: 1–12.

20.

Liu

Wang

Xian

, et al. Design of the tooth profile of pascal curve gear. J Mech Transm 2015; 39: 50–52.

21.

Yongping

Xutang

Heqi

. Study on some common close convex pitch curves of noncircular gear pair. Trans Chin Soc Agric Mach 2007; 38: 143–146.

22.

Shenglin

Xianxiang

Xutang

. A study on carrying out pitch curve design of non-circular gear by the use of cubic spline. J Mach Des 1999; 16: 20–21.

23.

Fanghella

. Kinematic synthesis and design of non-circular gears through a symbolic-numeric modeling approach. In: International design engineering technical conferences and computers and information in engineering conference, Long Beach, CA, USA, 24–28 September 2005.

24.

Figliolini

Stachel

Angeles

. Synthesis of the base curves of non-circular gears via the return circle. In: International gear conference, Lyon, France, 26–28 August 2014, pp.540–550. Cambridge: Woodhead Publishing.

25.

Figliolini

Stachel

Angeles

. Base curves of involute cylindrical gears via Aronhold’s first theorem. Proc IMechE, Part C: J Mechanical Engineering Science 2016; 230: 1233–1242.

26.

Qinqin

Gang

Xingqiao

. Modification of the pitch curve design defect of non-circular gear based on spline interpolation method. J Mech Transm 2018; 42: 90–95.

27.

Tong

, et al. Design of a conjugate concave-convex non-circular gear mechanism. Chin Mech Eng 2016; 27: 2155.

28.

Dooner

. On the three laws of gearing. J Mech Des 2002; 124: 733–744.

29.

Dooner

Seireg

. The kinematic geometry of gearing: a concurrent engineering approach. Vol. 3. New York, NY: John Wiley & Sons, 1995.

30.

Zheng

Hua

Han

. The mathematical model and mechanical properties of variable center distance gears based on screw theory. Mech Mach Theory 2016; 101: 116–139.

31.

Wang

, et al. Optimization synthesis of hybrid six-bar mechanism with non-circular gear constraints. J Mech Des 2023; 145: 064502.

32.

Wang

Sun

, et al. Kinematic synthesis and optimization design of a rice pot seedling transplanting mechanism. Proc IMechE, Part C: J Mechanical Engineering Science 2023; 238: 1366–1381.

33.

Wang

Zhou

, et al. Study on improving the winding forming uniformity of highspeed silk reeling machine with large package. J Adv Mech Des Syst Manuf 2022; 16: JAMDSM0007.

34.

Lin

Wei

, et al. Characteristics analysis of supporting and locking mechanism based on the non-circular gear compound transmission. J Mech Sci Technol 2020; 34: 2561–2571.

35.

Liu

Jin

, et al. Dynamic characteristics of two-mass impact pipeline robot driven by non-circular gears. Adv Mech Eng 2022; 14: 1–15.

36.

Liu

Zhang

Cao

. Multi-joint bionic mechanism based on non-circular gear drive. Biomimetics 2023; 8: 272.

37.

Zhao

. Matching between mechanics and thermodynamics among 4 individual strokes in a 4-stroke engine by non-circular gear mechanism. J Cent South Univ 2022; 29: 2112–2126.

38.

Dooner

. Appendix C: noncircular gears. In: Dooner

(ed.) Kinematic geometry of gearing. Chichester: John Wiley & Sons, 2012, pp.409–424.

39.

Figliolini

Angeles

. The synthesis of elliptical gears generated by Shaper-Cutters. J Mech Des 2004; 125: 793–801.

40.

Litvin

Gonzalez-Perez

Fuentes

, et al. Design and investigation of gear drives with non-circular gears applied for speed variation and generation of functions. Comp Methods Appl Mech Eng 2008; 197: 3783–3802.

41.

Zheng

Guo

Zhang

. Non-uniform flank rolling measurement for shaped noncircular gears. Measurement 2018; 116: 207–215.

State	$an & in$	$aw & in$	$an & iw$	$aw & iw$
1	○	×	×	×
2	×	○	×	×
3	×	×	○	×
4	×	×	×	○
5	○	○	×	×
6	○	×	○	×
7	○	×	×	○
8	×	○	○	×
9	×	○	×	○
10	×	×	○	○
11	○	○	○	×
12	○	○	×	○
13	○	○	○	○

State	$an & in$	$aw & in$	$an & iw$	$aw & iw$
1	○	×	×	×
2	×	○	×	×
3	×	×	○	×
4	×	×	×	○
5	○	○	×	×
6	○	×	○	×
7	○	×	×	○
8	×	○	○	×
9	×	○	×	○
10	×	×	○	○
11	○	○	○	×
12	○	○	×	○
13	○	○	○	○

State	$an & in$	$aw & in$	$an & iw$	$aw & iw$
1	○	×	×	×
2	×	○	×	×
3	×	×	○	×
4	×	×	×	○
5	○	○	×	×
6	○	×	○	×
7	○	×	×	○
8	×	○	○	×
9	×	○	×	○
10	×	×	○	○
11	○	○	○	×
12	○	○	×	○
13	○	○	○	○