A Grinding Method with an Innovative, Efficient, and Weight-Saving Design for Double Helical Gears

Abstract

Grinding technology exerts an enormous effect on the surface quality of double helical gears, which are subject to stringent requirements in the aviation industry. This study presents a novel gear-gap-space-borrowing grinding method that diverges from existing techniques. Unlike conventional approaches, this method mitigates the design–manufacturing conflict by exploiting the opposing gear gap space. By borrowing space, the grinding wheel’s size can be enlarged to enhance machinability, diminish the width of the undercut, and align with the lightweight design imperative of double helical gears. The relationship between the grinding wheel diameter and the design parameters of double helical gears is analytically established, leading to the formulation of design rules. These rules facilitate the rapid selection of appropriate grinding wheel sizes for the space-borrowing method. Additionally, through geometric modeling and experimental validation, the optimal gear undercut width and the maximum wheel diameter are determined. The validity and accuracy of the proposed method are verified by simulations and experimental investigations. The study quantitatively evaluates the benefits of this method, including the expansion of the grinding wheel size, the reduction in double helical gear weight, and enhancements in processing efficiency and quality, providing a comprehensive comparison and discussion of these improvements.

1. Introduction

Double helical gears are progressively being used in the aviation industry owing to their high load-bearing capacity, the smooth transmission characteristic of helical gears, and their ability to counteract the axial forces generated by such gear configurations [1,2,3]. In the aviation sector, double helical gears are crucial components that require machining to a high degree of precision to guarantee their performance under extreme service conditions, including high temperatures, high speeds, and heavy loads [4]. Therefore, machining technologies such as high-precision gear shaving [5] and milling [6] have been applied to machine double helical gears. However, the machining processes of aviation double helical gears frequently incorporates grinding as a complementary step to preserve mechanical strength and uphold the stringent surface quality requirements [7]. The diameter of the grinding wheel is a principal factor influencing both the quality and efficiency of the grinding process. Generally, a larger wheel diameter is associated higher grinding quality and efficiency [8]. Aviation transmission parts tend toward miniaturization and light weight in high-specific-power applications. However, the reduction in volume and space constraints limits the further extension of the grinding wheel diameter. The contradiction between smaller gears and a lager wheel diameter, particularly under the requirements of exacting surface dimensional integrity and high specific power, is becoming increasingly pronounced [9]. Therefore, research on the further extension of grinding wheel diameters in limited conditions is essential.

In the absence of alternative technologies for expanding wheel diameter, this paper will introduce the concept and method of gear-gap-space-borrowing grinding for the first time. This method is not suitable for batch production as the operator cannot accurately predict the size and space of the borrowing position. Consequently, there is a risk of touch damage, and the efficiency of implementation cannot be guaranteed. The contact issue involving multiple entity characteristics is crucial for the quantification and successful implementation of the gear-gap-borrowing grinding method. Contact simulation based on solid modeling is the most direct method for addressing this issue [10]. Also, custom model building for different double helical gears or secondary development based on solid modeling software is time-consuming and costly [11]. In addition, the debugging process during simulation may require human observation.

In recent years, the solid simulation method has been predominantly utilized for the force analysis of gears rather than for contact analysis. Patil and Ambhore [12] analyzed the stress and twisting of gears using CATIA V5 and ANSYS 10.0 software. Li [13] undertook the modeling and assembly of a double helical gear using UG 9.0 software. Meshing characteristics of the gear assembly model under static state were developed through the ANSYS software. The simulated double helical gear meshing force was used to verify the rationality of gear design. Liu [14] established the 3-DLTCA model of a double helical gear according to the differential geometry and mechanical analysis. Based on the multi-objective optimization method and this model, the effects of load, error, axial support stiffness of the pinion, and tooth surface modification on the 3-DLTE of double helical gears were studied through numerical examples.

The analytical method based on curve equations can quantitatively describe geometrical relationships [15]. However, constructing a mathematical model for gear tooth profiles and grinding wheels involves multiple piecewise functions, leading to a complex model after a couple of substitutions. Before solving the intersection point, identifying the parameter interval for possible intersecting piecewise function becomes necessary due to the complexity of models composed of piecewise functions, complicating the solution of the contact problem with uncertain intersection points. Feng et al. [16,17] developed a three-dimensional mathematical model to analyze the characteristics including the tooth contact area, the axial force on the pinion, and the contact ratio of double helical face gear drives. Ma et al. [18] employed accurate parameterized modeling and Pro/E 5.0 software to create a double helical gear with highly accurate involutes and tooth root transition curves. Finite element calculation and dynamic analysis confirmed the model’s accuracy for kinematics and dynamical analysis. Shen [19] used differential geometry and gear meshing theory to deduce the tooth surface of a spiroid face gear generated by shaving processing and modeled a computer-integrated shaving process for the gear on a five-axis CNC machine tool. Mo et al. [20] analytically developed a simplified gear tooth contact analysis model and refined the analysis with numerical methods. The application of an appropriate mathematical analysis method can also simplify simulation challenges. Liu et al. [21] effectively reduced the complexity of a gear contact simulation model based on the arbitrary Lagrangian Euler formula.

Although numerical and analytical methods have been utilized to quantitatively describe the geometry and contact characteristics of gears, there has been no investigation into the contact problems associated with the multiple entity characteristics of double helical gears. Consequently, this study proposes transforming the machining issue related to the expansion method of double helical gear space-borrowing grinding into a problem that combines analytical and simulation approaches. This transformation aims to circumvent the cumbersome simulation process and the application of piecewise functions. The model construction relies solely on the spatial geometric relationships between the gear and the grinding wheel, while solid contact simulation enhances the visibility and reference value of the quantity debugging process. Ultimately, the effectiveness and precision of the gear-gap-space-borrowing grinding method are confirmed through experimental verification.

2. Double Helical Gear-Gap-Space-Borrowing Grinding Method

2.1. The Quantitative Description of Gear-Gap-Space-Borrowing Grinding

Figure 1a illustrates that a double helical gear consists of two helical gears of opposite hands combined side by side. To fulfill processing requirements, an undercut is typically created between the gears, which simultaneously allows for some space for the extension of the grinding wheel’s diameter. Currently, the prevalent method for achieving wheel diameter extension involves utilizing a wider undercut width, leading to a proportional increase in both the mass and axial dimensions of the aviation components.

Figure 1b,c demonstrate the typical grinding machining mode for double helical gears that has become standardized, leaving no further possibilities for expanding the wheel diameter based on the grinding machine’s structure and motion pairs. Where B and C are the axis of rotation of gears and grinding wheels, respectively, A, X and Y are the rotational and translational degrees of freedom of the grinding machine, respectively.

Figure 2 reveals that, following extensive simulation research, an alternative method for extending the wheel diameter originates from the double helical gear itself. By adopting specific design parameters for undercut width and helical angle, it is feasible to utilize the space from the opposite or both the opposite and adjacent gear gaps to enhance the grinding wheel diameter, as depicted in Figure 2a,c. It is important to highlight that this approach is applicable solely to double helical gears without staggered teeth.

Therefore, this study aims to further expand the grinding wheel diameter through the method of gap space borrowing, necessitating a quantitative description of the aforementioned possibility. While contact simulation via secondary development offers a directly observable debugging process, it is both costly and time-consuming. Purely analytical methods present excessive complexity for resolution. After a period of exploration, a modeling approach based on geometrical relationships has been employed to delineate the correlation of key parameters, as shown in Figure 3. The feasibility of gap-space-borrowing grinding depends on the positional relationship between the grinding wheel and gear.

As shown in Figure 3, points a, b, and c are situated at the center of the tooth groove on the gear end face and are distributed along the pitch circle. The line segments connecting these points constitute the equilateral sides of the curved triangle Δabc. The values marked in the curved triangle Δabc are coplanar with the cylinder of pitch circle, where L is the undercut width and β is the helical angle. In the sector above the end face, R is the radius of the gear pitch circle.

Assuming that the grinding wheel is capable of utilizing the space from the opposite adjacent gap, the wheel originating from a point opposite point b must manage to penetrate the gap space at point a. This means the length of arc ab is an integer multiple of the gear pitch. Typically, the thickness of the grinding wheel is less than the width of the gap space, allowing these multiples some flexibility around whole numbers. Therefore, it is essential to discuss and ascertain the range of and permissible variation in these multiples, introducing parameter K for these multiples. Based on the geometric relationship described, the pitch can be articulated using the normal module m_n, the number of teeth z, and the radius of pitch circle R. Consequently, the arc ab can be represented in terms of L and β. Through substitution and simplification, a relationship between the multiples of pitch and arc ab can be established. The dimensions and meanings of all variable symbols in this article are described in Appendix A.

$$L=K\cdot \frac{\pi \cdot m_n}{\sin \beta }$$

(1)

The gap-borrowing grinding condition has been quantified as the index K, as shown in Figure 4, where distinct integer values of K signify varying extents of gap space borrowing. However, due to the geometry contact characteristics, the intersection between the grinding wheel and gear profile is feasible only within a certain range. As shown in Figure 4, the wheel progressively diverges from the gear with the increase in K. Thus, parameter K serves as a critical metric assessing the feasibility of gap space borrowing and the potential for interference.

Extensive testing and calculations have determined that when K exceeds 3, the profile of the grinding wheel and gear sufficiently diverges over the span of three teeth. In such cases, irrespective of the grinding wheel’s position and parameters relative to the gear, the grinding wheel typically does not interfere with the opposing tooth profile. Consequently, for aviation double helical gears, the value of K is only significant within the range (1, 3).

This stage merely sets the preliminary boundaries for the domain of K. Consequently, a separation function H, which depends on K, must be defined. To comprehend the implications of various K values, numerical simulations involving key parameters within their common ranges are essential and must be conducted for further exploration.

2.2. The Numerical Implications Study of Gap-Space-Borrowing Indicator

From the modeling perspective presented, parameter K serves as a crucial indicator for determining the feasibility of gap space borrowing. Figure 5 illustrates the numerical distribution of parameter K in a transverse projection, constructed based on the insights from Figure 4 through projection and transmission techniques.

Utilizing MATLAB software, extensive numerical and geometric simulations have been conducted in accordance with Formula (2). A critical finding is that the feasible interval for gap space borrowing always lies between K_a and K_b, which represent the intersection points of the gear involute profile and the pitch circle, as depicted in Figure 5. Section 3 will delve into a comprehensive analysis of how the thickness of the grinding wheel and the relative torsion of the gear profile impact the rule of K.

$$K=L\cdot \frac{\sin \beta }{\pi \cdot m_n}$$

(2)

As shown in Figure 5, the positions of integers and half integers, represented by the set {0, 0.5, 1, 1.5, 2} of parameter K, remain constant across any double helical gear, according to Formula (1). In the assemblies with a fixed gear center distance, a gear dressing operation is carried out to ensure the tooth side gaps [22,23]. The values of K_a and K_b are influenced by the tooth side gaps J_t, depending on the arc length of pitch S_p and tooth thickness S_m. In addition, considering the difference in the actual parts and the ideal model, a safety margin δ is also factored in. To articulate the specific implications of parameter K, it must be assigned a definite value. Initially, before addressing this, S_p is twice·S_m in a standard gear design, allowing the precise values of K_a and K_b to be identified as 0.75 and 1.25. Once the J_t is accounted for in the tooth thickness S_m and is removed, the K_a and K_b can be redefined as set (3).

$$\{\begin{array}{l}{K}_a=0.75-\frac{J_t}{S_p}+\delta \hfill \\ K_b=1.25+\frac{J_t}{S_p}-\delta \hfill \end{array}$$

(3)

In the design phase prior to gear dressing, extensive numerical simulations and tests have established that the relevant range for discussing parameter K is [0, 1.75). Should K exceed 1.75, the wheel and gear will disengage owing to the circular geometry. The gap-borrowing condition is particularly sensitive to variations in the helical angle, making the range of the helical angle a critical determinant in identifying the optimal value for K. The numerical distribution of K values is presented in

Table 1. Numerical distribution table of K.

K	Gap-Borrowing Mode	Implementation Assessment
[0, 0.25)	Opposite gap space	Infrequent; β is below usual range
[0.25, 0.75)	Interferes with opposite gear profile	Unavailable
[0.75, 1.25)	Opposite and adjacent gap space	Common; β ∈ (15°, 35°)
[1.25, 1.75)	Interferes with opposite and adjacent profile	Unavailable
[1.75, +∞)	No interface and no gap borrowing	Infrequent; β is out of usual range

.

In Table 1, the meaning of the K values is revealed. In a common range of key parameters, K in [0.25, 0.75) and [1.25, 1.75) is unavailable for gap-borrowing grinding. When K is in [1.75, +∞), the grinding wheel is completely separated from the profile of the gear. When it is in [0, 0.25), the value of β is too small to exploit the mechanical advantages of the helical gear. K = 0 means β = 0; neither of them are worth studying. In conclusion, the interval [0.75, 1.25) of K is the most significant range of discussion and application. At the same time, β ∈ (15°, 35°) is also commonly used in aviation helical gears. In addition, the sole integer of K in [0.75, 1.25) is K = 1, which is the optimal gap-borrowing situation in theory, but not in actual process due to the effects of tooth distortion and grinding wheel boundary tilt in the subsequent study.

The conditions for gap-borrowing grinding are encapsulated by the numerical value of K. This value represents the outcome of various combinations of key parameters. By analyzing the K value derived from these combinations, one can determine the conditions for gap-borrowing grinding. Conversely, by manipulating K, it is possible to optimize these parameter combinations to facilitate the design of double helical gears suitable for gap borrowing. A diagram illustrating the surface and divisions of key parameters when K equals 1 is provided in Figure 6.

All the key parameter combinations depicted on the map surface theoretically represent the optimal conditions for gap-borrowing grinding. Initial designs for double helical gears suitable for gap-borrowing grinding can be based on these parameter partitions. The visualization of the parameter design process is enhanced to a certain extent by Figure 6, which draws on Formula (1) and Table 1.

From the established quantitative relationships, the conditions for gap borrowing in existing double helical gears can be assessed. Moreover, this method facilitates the coordination of the design process to enable the creation of double helical gears that are compatible with gap-borrowing grinding, thus contributing to weight-saving objectives.

3. The Maximum Diameter Modeling of Gap-Borrowing Grinding Wheel

Prior to the grinding process of double helical gears, the tasks of conducting repeated grinding tests and dressing the grinding wheel to the appropriate size are time-consuming. The determination of wheel size largely relies on manual observation, compromising both accuracy and efficiency. Specifically, the diameter of the grinding wheel is a critical parameter affecting the efficiency and quality of the grinding process. Despite its significance, an accurate method for determining the diameter for gap-borrowing grinding, including for undercut-borrowing grinding, remains elusive.

As depicted in Figure 7, the determination of the wheel diameter (D) is conducted on the triangle Δabo. This process necessitates modeling an intermediate function that hinges on the contact relationship between the wheel and the gear.

Figure 8 shows the geometric contact relationship between the double helical gear and the radius of the grinding wheel in undercut-borrowing grinding. The grinding wheel diameter can be determined in Δdco’.

Given the triangular relationships in triangles Δabc, Δabo, and Δobc, the following sets of equations are deduced:

$$\{\begin{array}{l}ab=L\cdot \tan \beta \hfill \\ ac=L/\cos \beta \hfill \\ ao=\sqrt{{R_u}^2+a{b}^2}\hfill \\ co=\sqrt{{R_u}^2+L^2}\hfill \end{array}$$

(4)

where R_u is the radius of the addendum circle. Then, the angle of intersection between edges ac and ao is also determined by applying the cosine law to triangle Δaoc.

$$\cos \angle cao=\frac{a{c}^2+a{o}^2-c{o}^2}{2\cdot ac\cdot ao}$$

(5)

Similarly, the chord length of the wheel’s radius, represented as 2·cd, can be determined as follows:

$$cd=\sqrt{a{c}^2+a{d}^2-2\cdot \cos \angle cao\cdot ac\cdot ad}$$

(6)

where ad = ao − R_u. In order to solve the diameter D in Δdco’, the angle ∠dco’ must be determined firstly. Since line dg and line co’ are collinear, the angle ∠dco’ can be derived as follows:

$$\angle dco’=\frac{\pi }{2}+\angle gcd$$

(7)

where the angle ∠gcd in the above equation can be expressed as follows:

$$\angle gcd=a\sin (\frac{gd}{cd})$$

(8)

Based on the geometry relation, the lines gd can be calculated as follows:

$$\{\begin{array}{l}gd=ad\cdot \cos \angle gda\hfill \\ \angle gda=2\cdot \pi /Z\hfill \end{array}$$

(9)

According to the cosine law in Δdco’, the grinding wheel diameter D of undercut-space-borrowing grinding is finally solved as follows:

$$D=\frac{2\cdot \cos (\angle dco’)\cdot cd\cdot h+c{d}^2+h^2}{cd\cdot \cos (\angle dco’)+h}$$

(10)

It is important to emphasize that the meshing movement during the grinding process is constituted by the slight rotational motion of the gear and the grinding motion of the grinding wheel. This study constructs its model based on geometric relations without reliance on a coordinate system. Throughout the entire process, the positional relationship between the gear and grinding wheel is consistently described by the same geometric relationship. The determination of the maximum size of the grinding wheel corresponds to a specific positional relationship between the grinding wheel and the gear, thereby circumventing the need for boundary determination in analytical methods.

Based on the wheel diameter model and modeling method of undercut-borrowing grinding, the grinding wheel diameter D_Max of opposite and adjacent gap space borrowing can be determined in triangle Δfco’, as shown in Figure 9. However, there is a notable distinction in that lines cd and cf intersect at point c but are not collinear. Although the angle between lines cd and cf is minor, its value can vary significantly with changes in gear parameters. This variance introduces an error that cannot be overlooked when accurately determining the maximum diameter, D_Max.

In addition, the distortion of the helical gear on the base circle cylinder must be accounted for. As shown in Figure 9, the extended line of cd does not intersect with line ef. The plane on which triangle Δfco’ actually intersects at point f’ with gear top line f’e’ after distortion is shown.

In the expanded curved triangle Δd’ne’, the length of ee’ is a part of arc e’d’. Utilizing the relationships established within triangle geometry and the formula for arc length, the following equations can be derived to accurately determine this length and related parameters:

$$\{\begin{array}{l}ee’=w\cdot \tan \beta ·(\frac{R_u}{R_m}-1)\hfill \\ en=w/\cos \beta \hfill \\ e’n=\sqrt{e{n}^2+ee{’}^2+2\cdot en\cdot ee’\cdot \sin \beta }\hfill \end{array}$$

(11)

where w is the width of gear. In triangle Δf’de’, the absolute depth of the wheel borrowing gap space df’ is given by the following:

$$df’=\frac{\sin (\angle ne’e)\cdot e’d}{\cos (\beta -\angle ne’e)}$$

(12)

where the function expression of the intermediate substitution can be expressed as follows:

$$\{\begin{array}{l}\angle ne’e=arc\sin (\frac{w}{e’n})\hfill \\ e’d=ee’+ed\hfill \end{array}$$

(13)

The distortion of gear top line fe also results in a descension of point f to point f’ along the radius direction. The descending distance f_D is calculated by the following:

$$f_D=\sqrt{{(df\cdot \sin \beta )}^2+R_u^2}-R_u$$

(14)

Then, the angle of line cd and line cf can be expressed as follows:

$$\angle fcf’=\arctan (\frac{f_D}{df+cd})$$

(15)

To determine the D_Max within triangle Δfco’, it is necessary to first establish quantifiable relations for line df within the curved triangle Δdef. Alongside this, line ed can also be quantified by the following equation sets:

$$\{\begin{array}{l}ed=\frac{\pi \cdot R_u}{Z}-\frac{S_u}{2}\hfill \\ df=\frac{ed}{2\cdot \sin \beta }\hfill \end{array}$$

(16)

where R_u is the radius of the gear top circle. According to the calculation formula of the gear, the circular tooth thickness of the gear top circle S_u can be expressed as follows:

$$S_u=R_u\cdot (\frac{S_m}{R_m}+2\cdot inv\alpha -2\cdot inv{\alpha }_u)$$

(17)

where S_m and R_m are the circular tooth thickness and radius of the pitch circle, respectively, which can be expressed as follows:

$$\{\begin{array}{l}{S}_m=\frac{\pi \cdot m_n}{2\cdot \cos \beta }\hfill \\ R_m=\frac{m_n\cdot Z}{2\cdot \cos \beta }\hfill \end{array}$$

(18)

Substituting invα and invα_u into Equation (18) yields the following:

$$\{\begin{array}{l}inv\alpha =\tan \alpha -\alpha \hfill \\ inv{\alpha }_u=\tan (a\cos (\frac{R_m}{R_u}\cdot \cos \alpha ))-a\cos (\frac{R_m}{R_u}\cdot \cos \alpha )\hfill \end{array}$$

(19)

Finally, in triangle Δfco’, the grinding wheel diameter of opposite and adjacent gap borrowing is expressed as follows:

$$D_{Max}=\frac{2\cdot \cos (\angle f’co’)\cdot f’c\cdot h+f’c^2+h^2}{f’c\cdot \cos (\angle f’co’)+h}$$

(20)

where the substitution can be defined by the following:

$$\{\begin{array}{l}\angle f’co’=\angle fcf’+\angle fco’\hfill \\ f’c=\frac{df’+cd}{\cos (\angle fcf’)}\hfill \end{array}$$

(21)

The modeling process of D_Max includes assumptions and omissions. Model defects and errors need to be compensated and modified in the machining process.

4. Modification and Compensation of Gap-Space-Borrowing Grinding

4.1. Modification of Grinding Wheel Thickness and Side Contact

As depicted in Figure 10a, the value of K typically oscillates around 1, attributable to the rounding off of design parameters. The actual thickness of the grinding wheel is less than the gap width, as illustrated in Figure 10b. Simulations have demonstrated that fluctuations in parameters, the lateral space of the wheel, and the rotational profiles of adjacent and opposing gaps lead to complex contact and interference phenomena. These phenomena prove challenging to accurately describe using analytical methods. Through a series of contact simulations, the interaction or interference between the grinding wheel profile and the gear profile or tooth surface ultimately results in a sinusoidal curve, denoted as e’d’. This curve is generated by the projecting vertex of the grinding wheel and symbolizes the variation in the gap bridging effect, as depicted in Figure 10a.

The length of line f’c consists of line f’d and line cd; both of them change with the helical angle β. The curve consists of point e’, d, and d’, which are approximates for line e’d’; the variation in line cd on line e’d’ can be assumed as a linear function, as shown in Figure 10a. To avoid the analytical calculation of the complex contact or interference, this paper constructs the approximate coefficient function of line f’d and line cd according to the simulation results. After projection, the functions of curve e’d’ and line e’d’ are constructed as equation sets (22).

$$\{\begin{array}{l}U=\left|\cos (2\cdot K\cdot \pi )\right|\hfill \\ V=\frac{(K-1)\cdot {\sin }^2\beta }{L}+1\hfill \end{array}$$

(22)

As shown in Figure 11, the function types are consistent and their accuracies are also sufficient.

In addition, the modified line f’c can be expressed as follows:

$$FC=\frac{U\cdot df’+V\cdot cd}{\cos (\angle f’cf)}$$

(23)

Equation (20) can be updated as follows:

$$D_{Max}=\frac{2\cdot \cos (\angle f’co’)\cdot FC\cdot h+F{C}^2+h^2}{FC\cdot \cos (\angle f’co’)+h}$$

(24)

4.2. Compensation of Grinding Wheel Lifting Space

This study is focused on identifying the maximum diameter of the grinding wheel within the context of the machining process. It is essential to take into account the space required for the wheel’s elevation. As shown in Figure 12, the minimum space necessary for lifting includes two components: I_C for the wheel’s elevation and E_D for guaranteed comprehensive grinding of the gear surface, which needs axis c translate from point c to point c’ firstly.

The lengths of line c’c (E_D) and line c’s (E_L) can be expressed as follows:

$$\{\begin{array}{l}{E}_D=C_{cn}\cdot \sin \beta \hfill \\ E_L=C_{cn}\cdot \sin \beta \cdot \cos \beta \hfill \end{array}$$

(25)

where the half chord length of gap space C_cn is given by the following:

$$C_{cn}=R_u\cdot \sin (\frac{\pi }{z}-\frac{S_u}{2\cdot R_u})$$

(26)

Apart from E_D, there is another necessary space in lifting space I_C which originates from the arc grinding out a path for avoiding inertia collision, which is usually determined in the CNC programming process. When the compensation is applied in the determination of the wheel maximum diameter D and D_Max, they can be updated as equation sets (27).

$$\{\begin{array}{l}D=\frac{2\cdot \cos (\angle dco’)\cdot (cd-E_D-I_C)\cdot h+{(cd-E_D-I_C)}^2+h^2}{(cd-E_D-I_C)\cdot \cos (\angle dco’)+h}\hfill \\ D_{Max}=\frac{2\cdot \cos (\angle f’co’)\cdot (FC-E_D-I_C)\cdot h+{(FC-E_D-I_C)}^2+h^2}{(FC-E_D-I_C)\cdot \cos (\angle f’co’)+h}\hfill \end{array}$$

(27)

CNC programming research is not covered in this paper; the value of I_C in the later verification and discussion is ignored.

5. Verification

To ensure comprehensive verification, solid touch simulation is employed to validate intermediate variables within the model that are challenging to measure during actual processing. Subsequently, trial cutting and touch experiments are conducted using the Klingelnberg Rapid 1000 CNC grinder to validate the final variables of the model. The manufacturer of this equipment is Harding Machine Tools (Shanghai) Co., Ltd., and the place of production is Switzerland. This section encompasses the validation of all key variables within the model.

5.1. Verification Based on Simulation

5.1.1. Procedure of Simulation Verification

The study presented in this paper addresses various instances of interference between double helical gears and the grinding wheel, as illustrated in Figure 13. It approximates the aforementioned interference relationships through a fitting function, which requires verification.

Solid contact simulation, utilizing UG software, is conducted on twelve typical double helical gears to verify the accuracy of the mathematical model by examining the interference scenarios. This verification is independent of the actual grinding load conditions. Moreover, the grinding load is disregarded in this analysis, as it does not affect the contact and interference relationship between the grinding wheel and the gear.

5.1.2. Results of Simulation Verification

The lifting space distance of the wheel, denoted as E_D in Equation (25), serves as an intermediate variable that is not readily measurable or directly verifiable within the context of machining. However, its relationship to the model’s accuracy necessitates the preliminary verification of the accuracy of E_D through simulation methods.

Different parameters of the gear and lifting space distances of the wheel E_D are shown in

Table 2. Double helical gear parameters and lifting space distance evaluation.

Double Helical	mn	β (°)	α (°)	Z	w (mm)	L (mm)	Φu (mm)	Φd (mm)	ED (mm)	ED’ (mm)	Error (%)
Gear 1	3.8788	30	22.5	27	47	20	130.595	108.5	3.183	3.281	2.987
Gear 2	3.8788	30	22.5	118	47	20	537.42	515.317	2.771	3.0	7.633
Gear 3	2.514	30	22.5	27	30	20	85.305	71.15	2.181	2.168	0.600
Gear 4	2.514	30	22.5	106	30	20	312.68	298.55	1.711	1.911	10.466
Gear 5	2.75	30	22.5	33	38	20	110.265	97.81	1.987	2.073	4.149
Gear 6	2.75	30	22.5	115	38	20	370.62	358.2	1.862	1.954	4.708
Gear 7	4.25	30	22.5	31	61	20	160.605	141.41	3.088	3.221	4.129
Gear 8	4.25	30	22.5	130	61	20	646.42	627.25	2.872	3.012	4.648
Gear 9	4	30	22.5	34	48	20	165.8	147.8	3.005	3.04	1.151
Gear 10	4	30	22.5	31	48	20	151.183	133.183	2.911	3.031	3.959
Gear 11	1.56	30	22.5	36	16.6	15	68.075	60.98	1.137	1.174	3.152
Gear 12	1.56	30	22.5	48	16.6	15	89.425	82.33	1.077	1.143	5.774

, where the values of E_D and E_D’ are derived from model calculations and contact simulations.

The wheel diameter without gear gap borrowing D and the wheel diameter with gear gap borrowing D_Max obtained by model (10) and (24), respectively, can also be preliminarily verified by simulation methods. The grinding wheel interference simulation of Gear 1 to Gear 12 were completed by UG software, as shown in Figure 13.

The double helical gear-gap-borrowing condition is shown in

Table 3. Double helical gear-gap-borrowing condition and diameter evaluation.

Double Helical	K	(Ka, Kb)	Gap Borrowing	D (mm)	D’ (mm)	Error (%)	DMax (mm)	DMax’ (mm)	Error (%)
Gear 1	0.821	(0.746, 1.254)	Available	48.755	49.195	0.915	64.334	68	5.391
Gear 2	0.821	(0.746, 1.254)	Available	48.938	48.186	1.573	59.037	63	6.29
Gear 3	1.266	(0.745, 1.256)	Unavailable	80.705	86.609	6.817
Gear 4	1.266	(0.745, 1.255)	Unavailable	75.974	75.051	0.132
Gear 5	1.158	(0.746, 1.255)	Available	90.617	93.961	3.559	118.422	128.159	7.598
Gear 6	1.158	(0.745, 1.255)	Available	82.469	82.514	0.055	100.243	102.66	2.354
Gear 7	0.749	(0.747, 1.253)	Available	53.629	53.979	0.648	53.691	52	3.252
Gear 8	0.749	(0.747, 1.253)	Available	52.872	52.462	0.782	52.764	55	4.065
Gear 9	0.796	(0.747, 1.253)	Available	56.588	57.341	1.313	67.52	67	0.776
Gear 10	0.796	(0.747, 1.253)	Available	57.32	57.728	0.707	68.15	68	0.221
Gear 11	1.53	(0.741, 1.259)	Unavailable	103.596	111.467	7.061
Gear 12	1.53	(0.741, 1.259)	Unavailable	96.946	100.921	3.939

. D and D’ are the wheel diameter without gear gap borrowing, which come from model calculation and contact simulation, respectively. D_Max and D_Max’ are the wheel diameter with gear gap borrowing, which come from model calculation and contact simulation, respectively.

As shown in Table 2, the discrepancies between E_D and E_D’ are very small with an average relative error of 4.45%, which is consistent with the modeling requirements. As an intermediate variable, the accuracy of E_D has been verified, which in turn partially substantiates the overall accuracy of the model.

As shown in Table 3, the relative errors of D and D_Max are both under 10%, with average errors of 3.43% and 3.74%, respectively. The analysis reveals that the assumptions and approximations made during the modeling process result in an inward convergence error of 3.59%. This level of error is deemed sufficiently accurate and safe for both theoretical modeling and practical processing studies. The validity and precision of Equations (10), (24), and (27) have been initially corroborated through simulation. This validation further demonstrates that the approximation and hypothesis operations undertaken during the modeling process are both reasonable and appropriate.

5.2. Verification Based on Machining Process

5.2.1. Procedure of Machining Verification

Due to constraints imposed by the production schedule, validation experiments must be integrated flexibly within the production process. The grinding wheel diameters are standardized in a series. Furthermore, to mitigate the risk of damaging the tooth surface, it is not feasible to utilize the limit value D or D_max for grinding as part of the verification process. Instead, D_A is utilized as a comparative value against D and D_max for processing verification. These experiments are conducted on the Klingelnberg Rapid 1000 CNC grinder. Given that grinding parameters do not influence the outcomes of trial cutting and contact analysis, they are not specifically addressed in the experimental design. The Gear Production software, integrated with the grinder, calculates the necessary strokes to complete a tooth groove. These calculated values are then compared with the results from contact simulations and model calculations.

The validation experiment is carried out in conjunction with production processes. To prevent scratching the tooth surface of the finished gear during the grinding wheel contact test, a Bakelite grinding wheel is employed. This Bakelite wheel is specifically chosen due to its limited size, adequate strength, and moderate hardness, making it suitable for validating the proposed double helical gear-gap-space-borrowing grinding technology. Additionally, Bakelite’s ability to be cut and ground without melting or burning under the high temperatures generated during the grinding process further substantiates its suitability for this application (Figure 14).

The model grinding wheel used is 15 mm thick, with an outer diameter of 175 mm and a holding hole diameter of 10 mm. New gears, Gear 13 and Gear 14, are selected for the contact test. As illustrated in Figure 15, trimming and touching tests are conducted multiple times prior to the grinding tests. According to the retraction compensation space calculated by the Klingelnberg Rapid 1000 CNC grinder, the grinding wheel completes its cutting, retracts, and then touches the top of the opposing gear tooth precisely when the diameter of the model grinding wheel is reduced to DT = 150 mm.

Figure 16 illustrates the process of reducing the grinding wheel’s thickness from 15 mm to 13 mm. Following multiple rounds of trimming and touching, the wheel, with a reduced diameter of D_TM = 125 mm, navigates across the profile of the opposing adjacent tooth (tooth 1). It then precisely contacts the top of the subsequent adjacent tooth (tooth 2) after executing the space borrowing and retraction sequence.

For more precise validation, Gear 13 and Gear 14 undergo grinding with wheel diameters set at 149 mm and 54 mm, respectively. The procedure employs the Klingelnberg Rapid 1000 CNC gear grinding machine, with the grinding wheel’s linear velocity maintained at 30 m/s to ensure the quality of the grind. The fine grinding allowance for the tooth surface is established at 0.15 mm, aiming for high precision by setting the single grinding removal rate at 0.001 mm. To balance efficiency, the grinding stroke speed is configured at 2000 m/min, and each cog undergoes three grinding strokes. Consequently, the final grinding cycle is executed a total of five times to complete the process. The outcomes indicate that the grinding wheel does not interfere with the gear structure, and both the accuracy and the quality of the gear tooth surfaces meet the required standards.

5.2.2. Results of Machining Verification

In

Table 4. Comparison and verification of grinding wheel size D_A with D and D_max.

Double Helical	mn	β (°)	α (°)	Z	w (mm)	L (mm)	D (mm)	DMax (mm)	DA
Gear 1	3.8788	30	22.5	27	47	20	48.745	64.334	62
Gear 2	3.8788	30	22.5	118	47	20	48.938	59.037	55
Gear 3	2.514	30	22.5	27	30	20	80.705		67
Gear 4	2.514	30	22.5	106	30	20	75.974		65
Gear 5	2.75	30	22.5	33	38	20	90.617	118.422	68
Gear 6	2.75	30	22.5	115	38	20	82.469	100.243	69
Gear 7	4.25	30	22.5	31	61	20	53.629	53.691	55
Gear 8	4.25	30	22.5	130	61	20	52.76	52.872	53
Gear 9	4	30	22.5	34	48	20	56.588	67.52	67
Gear 10	4	30	22.5	31	48	20	57.32	68.15	67
Gear 11	1.56	30	22.5	36	16.6	15	103.596		65
Gear 12	1.56	30	22.5	48	16.6	15	96.946		65

, D_A represents the actual size of the grinding wheel used in production. When the error margin of 3.59% is factored into D and D_max, all instances of D_A are found to be smaller than either D or D_max, aligning with real-world processing outcomes. Notably, the grinding of Gear 7 and Gear 8 presents increased challenges due to the adoption of new high-strength gear materials. Despite the inefficiency in gap space borrowing, the selection of the grinding wheel size must ultimately adhere to D_max. Consequently, Gear 7 and Gear 8 serve as pertinent case studies for validating the accuracy of the gap-space-borrowing grinding model discussed in this article.

Not every instance of grinding wheel selection maximizes the potential of space-borrowing grinding, as seen with Gear 5, Gear 6, Gear 11, and Gear 12, which present significant opportunities for enhancing both grinding efficiency and quality. The methodology introduced in this study not only facilitates the enlargement of the grinding wheel’s diameter but also incorporates considerations for reducing weight, a crucial design criterion. These insights warrant further in-depth exploration.

Based on Equations (1) and (3), it is determined that Gear 13 cannot be processed using the gear-gap-space-borrowing method, whereas Gear 14 can be. Additionally, Equations (10) and (24) help calculate their maximum permissible grinding wheel diameters; D and D_Max are also shown in

Table 5. Accurate verification for the max diameter of grinding wheel.

Double Helical	mn	β (°)	α (°)	Z	w (mm)	L (mm)	K	Φu (mm)	Φd (mm)	D (mm)	DMax (mm)
Gear 13	1.75	30	22.5	31	15	18	1.64	66.442	58.492	149.8084
Gear 14	1.7	27	22.5	67	15	14	1.19	132.412	121.728	48.3082	54.7443

.

5.2.3. Conclusions of Machining Verification

Table 4 reveals that, after accounting for a 3.59% error margin, the average discrepancy between the D_max and D_A of Gear 7 and Gear 8 is only 2.23%. In addition, the grinding wheel diameters of Gear 1, Gear 2, Gear 9, and Gear 10 all fall within the range greater than D and less than D_max. These gears have been effectively processed using the gap-space-borrowing grinding method, underscoring its viability.

Table 5 prioritizes the validation of Gear 13, noted for its smaller diameter. The comparison reveals that the discrepancy between the theoretical diameter D_T and actual diameter D, recorded as 149.8084 mm, is merely 0.13%. This finding further validates the precision of Equation (10), demonstrating the robustness and accuracy of the underlying mathematical model.

As depicted in Figure 15, when compared to a model diameter (D_TM) of 57 mm, the actual maximum diameter (D_Max) of 54.7443 mm, as listed in Table 5, exhibits an error of only 3.96%. This comparison not only underscores the slight deviation but also confirms the validity of the gap-borrowing grinding method as formalized in Equation (24). This further reinforces the model’s applicability and accuracy in predicting the maximum permissible grinding wheel diameter for efficient gear machining.

Drawing from the simulation and experimental outcomes presented in Section 5.1 and Section 5.2, the methodological approach of this study, along with the derived model’s accuracy, has been comprehensively validated.

5.3. Discussion Based on Verification

The verification process, grounded in both simulations and experiments, has unveiled several phenomena meriting further discussion. These observations, particularly regarding the potential for gap borrowing in double helical gears and their subsequent improvements, are detailed in

Table 6. Efficiency evaluation of gear gap borrowing.

Double Helical	K	(Ka, Kb)	D (mm)	DMax (mm)	Improvements (%)
Gear 1	0.821	(0.746, 1.254)	48.745	64.334	31.981
Gear 2	0.821	(0.746, 1.254)	48.938	59.037	20.636
Gear 5	1.158	(0.746, 1.255)	90.617	118.422	30.684
Gear 6	1.158	(0.745, 1.255)	82.469	100.243	21.552
Gear 7	0.749	(0.747, 1.253)	53.629	53.691	0.116
Gear 8	0.749	(0.747, 1.253)	52.872	52.764	−0.204
Gear 9	0.796	(0.747, 1.253)	56.588	67.52	19.319
Gear 10	0.796	(0.747, 1.253)	57.32	68.15	18.894

.

As detailed in Table 6, not every double helical gear that could potentially utilize gap borrowing needs to undergo this process. Specifically, Gears 5 and 6 experience significant diameter enlargement through gap borrowing, yet their diameters without this technique are already adequate for high-volume grinding. The composition of current grinding wheels, made from abrasive grains bound by adhesives through pressing or sintering, implies that excessive diameter increases post-gap borrowing may compromise wheel strength, as noted in reference [24]. Conversely, this scenario highlights that Gears 5 and 6 have overly large undercut design widths, indicating opportunities for weight reduction. Considering the operational limits of typical grinding machines and wheel specifications, the maximum allowable diameter (D_A) for the wheel post-dressing is set at 78 mm. The resultant reductions in simulated undercut design widths (L) when constraining the wheel diameters of Gears 5 and 6 to 78 mm are tabulated in

Table 7. Reduction in L under actual grinding wheel diameter D_A.

Double Helical	Φu (mm)	L (mm)	DMax (mm)	DA (mm)	L’ (mm)	Reduction in L (%)
Gear 5	110.265	20	118.422	78	15.504	22.48
Gear 6	370.62	20	100.243	78	16.29	18.55

.

Table 7 illustrates that, with a focus on maintaining an adequate grinding wheel diameter, the undercut widths for Gears 5 and 6 can be decreased by 22.48% and 18.55%, respectively. For larger double helical gears, such reductions in undercut width can yield considerable advantages in terms of weight reduction and space efficiency, not only for individual parts or components but for entire systems as well. This emphasizes the potential for optimization in gear design that balances operational efficiency with structural integrity.

In addition, as shown in Table 6, not all the double helical gears that are available for gap borrowing are suitable for the operation; the gap-borrowing efficiencies of Gear 7 and Gear 8 are not significant and even result in negative effects due to the interference of wheel thickness. In Gear 7 and Gear 8, the gap-borrowing parameter K = 0.749 is too close to the boundary K_a = 0.747, so the safety margin δ in set (3) should be at least 0.002 to limit the gap-borrowing condition of Gear 7 and Gear 8. A new K_a = 0.749 will prevent Gear 7 and Gear 8 from borrowing gear gaps.

As shown in Figure 17, when the tooth side gaps J_t =0 and safety margin δ = 0, by gear gap borrowing, most double helical gears available for gear-gap-borrowing grinding have a significant effect on grinding wheel diameter expansion in theory. Every double helical gear available for gear gap borrowing has its own optimal space-borrowing interval of K; the efficiency of grinding wheel diameter expansion in the optimal interval is significantly higher. As shown in Figure 17, the D_max between K_a and K_b is significantly greater than D.

6. Application

The practical application of the conclusions drawn from the above analysis has shown noteworthy results. The gear-gap-space-borrowing grinding technology has been implemented in machining practices, notably for Gear 1 and Gear 2, as mentioned in Table 4 and

Table 8. Efficiency improvement based on gear-gap-space-borrowing grinding method.

Double Helical	D (mm)	DB (mm)	NWheel	PT (min)	DMax (mm)	DA (mm)	NWheel	PT (min)
Gear 1	48.745	45	1.5	210	64.334	62	1	150
Gear 2	48.938	45	6.5	1100	59.037	55	4.5	960
Sum	-	-	9	1310			5.5	1110

. Initially, without employing the gap-borrowing technique, the diameters of the grinding wheels (D_B) chosen for these gears are merely 45 mm, inadequate for sustaining the grinding wheel’s service life. A smaller wheel diameter leads to frequent replacements and subpar surface finishes, contributing to low processing efficiency and quality for Gear 1 and Gear 2—a longstanding issue that had not been effectively addressed.

By adopting the gap-borrowing grinding technology detailed in this study, the actual diameters of the grinding wheels (D_A) utilized for machining Gear 1 and Gear 2 are increased to 62 mm and 55 mm, respectively. This adjustment significantly enhances both the service life of the grinding wheels and the overall machining quality and efficiency, demonstrating the substantial impact of this technological application in machining practices.

The implementation of double helical gear-gap-borrowing grinding technology has shown significant benefits in machining efficiency and cost reduction, as illustrated in Table 8. By expanding the diameter of the grinding wheel, there is a notable decrease in both the number of wheels used (N_Wheel) and the processing time (P_T) for Gears 1 and 2. Specifically, the technology has led to a reduction of 3.5 in the number of wheels needed and has shortened the processing time by 200 min, translating to an efficiency increase of 15.3%.

This enhancement in efficiency and reduction in cost is accompanied by improvements in product quality. Figure 18 highlights the tangible differences in tooth surface roughness (R_a) when observed under a microscope, indicating a superior finish on the gears processed with the gap-borrowing grinding technique. This qualitative improvement is crucial for applications where gear performance and longevity are dependent on the precision and smoothness of the tooth surfaces. Thus, the application of gear-gap-space-borrowing grinding technology not only optimizes manufacturing processes but also elevates the quality of the final product [25,26].

The implementation of the double helical gear-gap-borrowing grinding technology has shown significant improvements in machining efficiency, cost-effectiveness, and product quality. The expansion of the grinding wheel diameter not only reduces the number of wheels required and shortens processing time but also enhances the surface quality of the gears. For instance, Gear 1 and Gear 2 experience a reduction of 3.5 in the number of grinding wheels used and a 200 min decrease in processing time, resulting in a 15.3% increase in efficiency. Furthermore, the surface roughness improvement by 0.05 μm due to using larger-diameter wheels indicates a direct correlation between wheel size and the glossiness of the gear teeth.

In terms of design efficiency, the redesign of Gear 5 and Gear 6 to decrease their undercut width based on gap-borrowing grinding theory not only contributes to significant weight savings and axial space conservation but also enhances the work weight ratio of parts, components, and entire systems.

Moreover, the gap-borrowing grinding model streamlines the commissioning process for grinding double helical gears with new parameters. By directly trimming the grinding wheel to the calculated optimal diameter, the method reduces the setup time from hours to just 30 min per gear, thereby improving efficiency by up to 90% without additional cost investments.

These outcomes underscore the transformative potential of the gear-gap-borrowing grinding method and its associated design approach for double helical gears. Despite the complexity of its computational process being a barrier to wider adoption, integrating this technology into user-friendly software could unlock its full potential, significantly impacting gear manufacturing processes by enhancing efficiency, reducing costs, and improving product quality. This direction for future research promises to make these advanced machining techniques more accessible and beneficial across the industry.

7. Conclusions

The main conclusion of the study can be drawn as follows:

• The gap-space-borrowing grinding method for double helical gears is proposed to expand the size of the grinding wheel and minimize the undercut width of double helical gears. This paper successfully develops and verifies techniques for evaluating the conditions necessary for gap-space-borrowing grinding, designing double helical gears based on undercut width specifically for this method, and analytically modeling the relationship between the undercut width of double helical gears and the maximum diameter of the grinding wheel.
• This technology, compared to traditional double helical gear grinding methods, offers an efficient solution for enlarging the diameter of a grinding wheel and achieving a weight-saving gear design within constrained spaces. The proposed method substantially enhances processing quality and efficiency, as well as the load-bearing capacity of the gears.
• Certain double helical gears can significantly extend the diameter of grinding wheels by reducing the undercut width based on the gap-borrowing grinding method.
• The reduction in the size of the undercut width of the double helical gear, as presented in this study, not only fulfills the requirements for weight reduction designs but also allows for the expansion of tooth width. This adjustment enhances the load-bearing capacity of the double helical gears.
• Each double helical gear suitable for gear gap borrowing possesses a distinct interval for efficient wheel diameter expansion, which plays a crucial role in facilitating weight reduction design efforts.
• The gap-borrowing coefficient K not only encapsulates the effect of space borrowing but also indicates the interference position of the grinding wheel and opposite tooth.