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

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.


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. 1,2uring 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.[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. 6nder 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][8][9][10][11][12] higher-order elliptical curves, [13][14][15][16][17][18][19] Pascal's limac xon, 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. 26nder the condition of transmission ratio known, by scaling the theoretical pitch curve proportionally, at the expense of changing the center distance of the gear pair 27 ; 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][29][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.
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.
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 noncircular 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 noncircular gear pair.

The working conditions model of the variable center distance non-circular gear pair
With the driving gear's rotation angle f 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 PC A and PC 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.
Assume that the time required for PC A and PC B meshing a cycle is constant, and the rotation angle f PCA t i of the driving gear at any meshing instant t i are equal to Df PCA .Then take f PCA as the abscissa, and take the length of the instantaneous meshing pole radius R PCA u PCA À Á and R PCB u PCB À Á on PC A and PC B as the ordinate, the PC A and PC B can be converted into the instantaneous meshing pole radius curves as shown in Figures 9 and 10.
At last, takes f PCA as the abscissa, and respectively takes a AB , i AB , R PCA u PCA À Á , and R PCB u PCB À Á 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.

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 PC A and PC B , the ratio of L PCA and L PCB is M : N ; during the meshing process, the time for PC A and PC B to make one revolution is t A and t B , and the time for PC A and PC B to mesh a round is t AB ; the instantaneous meshing arc Arc PCA t a on PC A meshing with the instantaneous meshing arc Arc PCB t a on PC B with at t a .During the meshing process of the gear pair, it can be seen that the same instantaneous meshing arc Arc PCA , and its meshing partner are the different instantaneous meshing arcs Arc PCB   Based on the definition above, the associated instantaneous meshing arcs of the instantaneous meshing arc pair Arc PCA t i and Arc PCB t i 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 ArcSet t i based on the time sequence.

And the meshing partner of each instantaneous meshing arc
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 PC A to make three revolutions which is also the time required for PC B to make four revolutions.Assuming that a&iSet 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&iSet t i , 13 categories of a&iSet t i can be found as shown in Table 1.
Afterwards, according to the categories of a&i t i in a&iSet t i , the geometric shape of the instantaneous meshing arcs in ArcSet t i , and the rounding algorithm adaptability of the instantaneous meshing arcs in ArcSet 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 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 PCA t a and Arc PCB t a 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][29][30] that means, after rounding process of Arc PCA t a and Arc PCB t a the center distance and the transmission ratio obtained by them will 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 PCA and Arc PCB t b , 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 f PCA t i of PC A before rounding is equal to the rotation angle f ÃPCA t i of PC ÃA after rounding at any instant t i , as shown as: Before the rounding process of PC A and PC B , as shown in Figure 13, R PCA u PCA À Á and R PCB u PCB À Á are the length of the instantaneous meshing pole radius on PC A and PC B , Arc PCA t i and Arc PCB t i are the instantaneous meshing arcs of PC A and PC B which meshing at point P PCAB distance a AB t i and the instantaneous transmission ratio i AB t i at the instant t i before the rounding process can be expressed as: Based on the meshing theory,  can be present as: When instant t i is small enough, the instantaneous meshing angle u ÃPCA  and can be expressed as: It can be seen from Figure 13 and formula (8) that the angle u ÃPCA t i between the instantaneous meshing pole radius R ÃPCA u ÃPCA t i on PC ÃA and the x-axis can be expressed as: Based on formulas from ( 6) to (10) ) are the dif- is the angle formed by R ÃPCB u ÃPCB t i and the x-axis.So, the following equations can be obtained: And based on Figure 14 and formula (9), u ÃPCB t i can be expressed as: Suppose that, l ÃPCA , and they can be expressed as: And based on the meshing theory, it can be known that: Finally, a ÃPCAB t i and i ÃPCAB t i , the center distance and transmission ratio of the variable center distance noncircular gear pair at t i after rounding, can be expressed as: Constant center distance rounding algorithm.Based on the meshing theory, 1 after rounding, l ÃPCA , and g ÃPCB t i satisfy formulas ( 6)-( 9), then l ÃPCA t i and l ÃPCB t i can be expressed as: In order to make sure the instantaneous center distance unchanged before and after rounding, thus, a PCAB t i equals to a ÃPCAB t i , it can be known that: Then, u ÃPCA t i and u ÃPCB t i , as described in formulas ( 10) and ( 14), can be expressed as: The slope k ÃPCAB t i of the tangent line at point P ÃPCAB t i can be expressed as: Also, k ÃPCAB t i can be expressed as: With the help of formulas from (20) to (26), 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: 1.In formula (13) Based on the above assumptions, according to the formulas from (1), ( 6), ( 7), ( 8), ( 9), ( 10), (13), and ( 14 on PC ÃA , PC ÃB (red part) before and after the constant transmission ratio rounding process at any instant t i can be shown in Figure 16.
There must be a proportionality coefficient Dq ÃPCAB t i .0,so that: 10 Advances in Mechanical Engineering Then according to the formulas ( 15) and ( 16), it can be known that: Combining this equation with formulas (30) and (31), it can be seen that l ÃPCA , 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): Which can prove that the instantaneous transmission ratio remains unchanged during this constant transmission ratio rounding process, therefore, the abovementioned 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 ÃPCAB t i can be expressed as: 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.
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 DL PCA and DL PCB are the total rounding amount of PC A and PC B ; DL poolA 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 PCA t a and RAFlag PCA t a 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 r ÃPCA  should not be greater than the maximum value r PCB max of the curvature radius on PC B before rounding.is the pole radius length of P ÃPCA t aÀ1 , then, x ÃPCA t aÀ1 , t a and y ÃPCA t aÀ1 , t a , the coordinates of P ÃPCA t aÀ1 at t aÀ1 , can be expressed as: According to formula (1), the rotation angle from t aÀ1 to t a is Df PCA , after rounding, P ÃPCA t a is the meshing point on PC ÃA meshing at t a , u ÃPCA t a is the meshing angle of is the pole radius length of , and y ÃPCA t a , the coordinates of , can be expressed as: Assume that, k ÃPCA t aÀ1 is the slope of the tangent line at P ÃPCA t aÀ1 , k ÃPCA t aÀ1 , t a the slope of the tangent line at P ÃPCA t aÀ1 when PC ÃA rotated from t aÀ1 to t a , P ÃPCA t aÀ1 rotated to P ÃPCA t aÀ1 , t a , the slope k ÃPCA t aÀ1 , t a of the tangent line at P ÃPCA t aÀ1 , t a can be expressed as: Then, the common tangent circle of P ÃPCA t aÀ1 , t a and P ÃPCA t a can be obtained, and the coordinates of the center point x O t a and y O t a of the circle can be solved by formulas ( 36)- (40) with the following equations: And the curvature radius r ÃPCA t a of Arc ÃPCA t a can be solved by the equations below: In the same way, the curvature radius r ÃPCB t a of Arc ÃPCB t a can also be obtained.If r ÃPCA t a and r ÃPCB t a 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 PCA  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 parallelaxis 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][35][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 noncircular gear pair As a special case of the variable center distance noncircular gear pair, the conditions for the circular and non-circular gear pair composed of a circular gear and a non-circular gear [38][39][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 circularnon-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 u PCA À Á of the driving gear's theoretical pitch PC A is: And PC A is shown in Figure 18.
The pitch curve length L PCA of PC A can be calculated by the following formula as below: Figure 18.Theoretical pitch curve of driving gear.
The polar coordinate function R PCB u PCB À Á of the driven gear's theoretical pitch curve PC B is: And PC B is shown in Figure 19.
The pitch curve length L PCB of PC B can also be calculated by the following formula below: When the gear pair's modulus m G = 2, tooth pitch p G = 2p, 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 PC A and PC 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.
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.
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.
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 noncircular 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.
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.
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        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.
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.
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 SINUM ERIK 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 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.
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.
The center distance difference from the actual center distance to the theoretical center distance is shown in Figure 40.
And the transmission ratio difference curve from the actual transmission ratio to the theoretical transmission ratio is shown in Figure 41.
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 mm, 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: 1. 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 nonworking curves, and the gear pair's theoretical pitch curves.2. 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.3. 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. 4. 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. 5. Carried out a variable center distance noncircular 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: 1.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.2. 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.3. 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.

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.

Figure 2 .
Figure 2. i AB of the variable center distance non-circular gear pair based on working conditions.

Figure 1 .
Figure 1. a AB of the variable center distance non-circular gear pair based on working conditions.

Figure 3 .
Figure 3.The working conditions model of the variable center distance non-circular gear pair.

Figure 4 .
Figure 4.The classification and analysis of the instantaneous meshing arcs in the working conditions model.

Figure 5 .
Figure 5.The meshing model and the rounding algorithm of the instantaneous meshing arc pair.

Figure 6 .
Figure 6.Dynamic allocation strategy of rounding amount.

Figure 7 .
Figure 7. Theoretical pitch curve of driving gear.

Figure 9 .
Figure 9. Instantaneous meshing pole radius curve of the driving gear.

Figure 10 .
Figure 10.Instantaneous meshing pole radius curve of the driven gear.

Figure 8 .
Figure 8. Theoretical pitch curve of driven gear.

Figure 11 .
Figure 11.The working conditions model of the variable center distance non-circular gear pair.

Figure 12 .
Figure 12.An example of the associated instantaneous meshing arc (a) and the associated instant (b).
t i and u ÃPCB t i after rounding are same as u PCA

Figure 14 .
Figure 14.General meshing model of the instantaneous meshing arc pair after rounding.

i
and DL poolB i are the total rounding amount of the rounding amount pool of PC A and PC 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 PCA IV and L PCB IV are the total instantaneous meshing arc length of all instantaneous meshing arcs in ArcSet 4 on PC A and PC B ; l PCA t a and l PCB t a are the length of any the

Figure 15 .
Figure 15.Instantaneous meshing arcs before and after the constant center distance rounding algorithm.Figure16.Instantaneous meshing arcs before and after the constant transmission ratio rounding algorithm.

Figure 16 .
Figure 15.Instantaneous meshing arcs before and after the constant center distance rounding algorithm.Figure16.Instantaneous meshing arcs before and after the constant transmission ratio rounding algorithm. t

Figure 17 .
Figure 17.Flow chart of the dynamic allocation strategy of rounding amount.

to 1 .
If r ÃPCA

Figure 19 .
Figure 19.Theoretical pitch curve of driven gear.

Figure 20 .
Figure 20.The theoretical center distance curve of the variable center distance non-circular gear pair based on working conditions.

Figure 21 .
Figure 21.The theoretical transmission ratio curve of the variable center distance non-circular gear pair based on working conditions.

Figure 22 .
Figure 22.The working conditions model of the variable center distance non-circular gear pair.

Figure 27 .
Figure 27.The rounded pitch curves of the driving gear.

Figure 28 .
Figure 28.The rounded pitch curves of the driven gear.

Figure 29 .
Figure 29.The radial difference curve of the driving gear's pitch curve before and after rounding.

Figure 30 .
Figure 30.The radial difference curve of the driven gear's pitch curve before and after rounding.

Figure 31 .
Figure 31.The center distance curve of the variable center distance non-circular gear pair after rounding.

Figure 32 .
Figure 32.The transmission ratio curve of the variable center distance non-circular gear pair after rounding.

Figure 33 .
Figure 33.The difference curve of the center distance curve before and after rounding.

Figure 34 .
Figure 34.The difference curve of the transmission ratio curve before and after rounding.

Figure 37 .
Figure 37. Variable center distance non-circular gear pair.

Figure 38 .
Figure 38.The real center distance curve of the rounded variable center distance non-circular gear pair.

Figure 39 .
Figure 39.The real transmission ratio curve of the rounded variable center distance non-circular gear pair.

Figure 40 .
Figure 40.The transmission ratio difference curve from the actual transmission ratio to the theoretical transmission ratio of the gear pair.

Figure 41 .
Figure 41.The transmission ratio difference curve from the actual transmission ratio to the theoretical transmission ratio of the gear pair.
And the associated instant of t a is t a +iÃt A ð Þ %t AB %t B +jÃt B or t a +jÃt B ð Þ %t AB %t A +iÃt A , t a also has MÃN associated instants including itself.
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 PCA413 and Arc PCB 413 will be classified into ArcSet 13 .And that result is consistent with the above definition of the associated instantaneous meshing arc and the associated instant.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 Figures1 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 AB t i and i AB t i at t i belongs to the non-working curve; aw&in means that a AB t i belongs to the working curve, and i AB t i belongs to the non-working curve; an&iw means that a AB , 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 PCA , i AB t f 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 Table1, 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 PCA

Table 1 .
Thirteen categories of a&iSet ti .There are a&i t a belongs to an&iw, and a&i t b belongs to aw&in, obtained by Arc PCA 1definitely change.So, there is no rounding algorithm described in this paper can round Arc PCA t a and Arc PCB t a without change their working center distance and the transmission ratio curves.
1the arc length l PCA i can be built, as shown in the red part in Figure14.It can be seen that at instant t i , PC ÃA and PC ÃB meshing at P ÃPCAB , the general meshing model of Arc ÃPCA t i and Arc ÃPCB t i meshing with each other at t on PC ÃA , PC ÃB (red part) before and after the constant center distance rounding process at any instant t i can be shown in Figure15.
, the value of Du ÃPCB ArcSet 1, ArcSet 2, or ArcSet 3 on PC A and PC B meshing at any instant t a , RAFlag PCA And the curvature radius r ÃPCA can be obtained by the formulas below.Assume that f ÃPCA t aÀ1 is the rotation angle of PC ÃA at t aÀ1 , P ÃPCA t aÀ1 is the meshing point on PC ÃA meshing with PC ÃB at t aÀ1 , u ÃPCA