New Insights Into the Minimum Teeth Number and Modification Coefficient of Standard Involute Gears Without Undercutting


 The calculation equations for the minimum teeth number and the minimum modification coefficient of standard involute gears without undercutting are inaccurately described and researched in many studies. These errors are knowledge blind spots in mechanical engineering that have been introduced into the design and manufacturing of standard involute gears. Furthermore, these errors are continually imparted to tens of thousands of students in colleges every yearly. Racks have been incorrectly regarded as the rack-shaped cutters in the derivation of the calculation equations. The theoretical research, numerical simulation and experiments of the undercutting phenomenon of standard involute gears manufactured using rack-shaped cutters are presented. Firstly, theoretical analysis and calculation of the minimum teeth number and the minimum modification coefficient of standard involute gears without undercutting are performed. Subsequently, the correct minimum teeth number and the minimum modification coefficient without undercutting are deduced. The computer simulations of the theoretical calculations are then examined considering the necessary cases. Experiments of standard involute spur gears with z = 10, 12, 17 and 22 are conducted, thereby validating the research. This study addresses the long-standing errors regarding the traditional undercutting phenomenon and provides an effective theoretical guide for gear design and manufacturing.


Introduction
Gear design optimization has become more of a necessity than ever before [1]. Involute gear transmission has been widely used because of its unique technical advantages. Research on stress distribution [2,3], tolerance analysis [4] noise [5], vibration [6] and thermal analysis [7] in geared systems have been centred on gear-specific components. However, for the meshing of specific teeth number in a gear pair, an interference exists between the tip of the teeth on the pinion and the fillet or root of the teeth on the gear [8]. Undercutting at the pinion root may affect the load distribution and meshing character amongst pairs of involute gear teeth in simultaneous contact [9]. Therefore, the conditions for avoiding undercutting have been extensively studied and discussed in many technical literature references, such as Litvin [10], Chao [11] and Ye [12]. The calculation equations for the minimum teeth number without undercutting zmin and the minimum modification coefficient without undercutting xmin of standard involute gears are inaccurately described in many textbooks wroldwide, such as the Design of machine elements, Mechanical Engineering Design, and Foundation of Machine Design. These errors have also been introduced into the design and manufacture of involute gears [13,14]. Thus, gears with undercutting defects have been used in a variety of transmission systems. Furthermore, this information is continually imparted to tens of thousands of students in universities yearly. A certain degree of undercutting can improve the meshing environment and the stability of gear pairs. However, the meshing region length of the tooth surface decreases when the undercutting of tooth exceeds a certain level whilst the strength of the roots decrease, which may cause the tooth to break. Therefore, studies on the undercutting -2 -phenomenon of gears are significant to its design, production and application.
The involute gear is extensively used in many kinds of mechanisms and machines, from can openers to aircraft carriers. Involute gears can be manufactured using finger cutters, rack cutters, gear shapers, and hobs. In traditional theory, the meshing conditions of involute gear (generated using a rack cutter) are well-defined when considering the meshing of the rectilinear profile of the rack cutter with an involute tooth gear [15,16]. In the involute gear mesh, undercutting and meshing interferences occur when the teeth number is smaller than a certain value. Instead of defining the geometry of the gear to manufacture, designers often define the geometry of the rack conjugated to the gear, that is, the basic rack. This approach makes it easy for the manufacturer to find the geometry of the tool to cut the gear [17]. However, in the derivation process of the minimum teeth number without undercutting, the rack cutter fillet has an evident influence on the result, which has been omitted in many mechanical engineering textbooks for many years. The addendum of the standard involute rack-shaped cutter is higher than that of the rack because a clearance 'c' exists at the top of the cutter.
Considering top clearance c in the article, the theoretical z min of standard involute gear is specified in this article. In addition, x min with fewer teeth (fewer than z min ) is analyzed. The minimum teeth number and the modification coefficients of standard involute gears are simulated using a specific computer program to verify the validity of the theoretical studies. The experiments of undercutting phenomenon of standard involute spur gears are also conducted.

Minimum teeth number of involute gears without undercutting in the literature
Interference and its attendant undercutting can be prevented by avoiding gears with few teeth. The teeth profile enlarge as its number decreases for a fixed diameter gear. Then, the dedendum will exceed the radial distance between the base circle and the pitch circle in the manufacturing process, and tooth undercutting will occur.
Regarding the undercutting issue, one of the earlier books that referred to gear manufacturing was compiled by the Former Soviet Union (FSU) mechanical expert I. I. Semenchenko in 1944 [18]. The precise definition of gear undercutting and the z min of the standard involute gear are provided. The noninterference minimum teeth of the meshing gear with the rack is z 1min =17 when the pressure angle α=20° and h a * =1 1 ; and if the rack is replaced by a rack-shaped cutter, then z 1min =20.52 [18]. The z min of the rack-shaped cutter is larger than that of the rack because the clearance coefficient c * was added into the equation. However, c * =0.2 in his study, which is inconsistent with the modern standard involute gears.
However, in many mechanical engineering textbooks, rack cutter is mistakenly regarded as rack when the minimum teeth number without undercutting is deduced, thereby causing the misconception of the undercutting phenomenon. Undercutting is also discussed in another book by FSU professor лнтвин, in which the undercutting phenomenon was reported to occur when the spur gear teeth number is less than 17 [19]. In the Design of Machine Elements book compiled by professor Spotts in the Northwestern University of America, undercut not only weakens the tooth, but also removes a small portion of involute adjacent to the base circle, and the loss of involute curves may cause a serious reduction of the contact length. His study also showed that the undercutting phenomenon disappears when the z min is larger than or equal to 17 [20,21].
In the literature Mechanical Engineering Design written by Edward and Mischke, the conditions for avoiding of the traditional undercutting of gears are defined; and they found that the smallest number of pinion teeth was 18 teeth to a α=20° full-depth teeth [22]. In Gears compiled by Houghton in England, the z min of standard involute spur gear is 17 under the pressure angle α=20° [23]. In the Gear Handbook of America, Dudley reported that the undercutting of standard involute spur gears can be avoided when the number of teeth is larger than or equal to 18 [24]. In the Machine Element -3 -in Mechanical Design by Mott, the responsibility of designers is to ensure that interference does not occur in a given application; and the best way to avoid undercutting is to control the z min of the standard spur gear (pressure angle is 20°) to larger than or equal to 18 [25,26]. Budynas also proved that the smallest number of pinion teeth to mesh with a rack is 18 for a α=20° full-depth tooth [27].
A generalized approach for defining the undercutting phenomenon of involute teeth was proposed by Alipiev. In the proposed approach, apart from the traditional undercutting, type I, two additional undercutting cases, types IIa and IIb, were described. An additional boundary condition for avoiding types IIa and IIb undercutting was drawn, and the maximum radius value of the rack cutter fillet, wherein no undercutting occurred, was specified [28].
Undercutting phenomenon is defined in the book of Design of Machinery compiled by Pu in China [29]. Fig. 1 shows that if the intersection point of the tool top line and the mesh line surpasses the limit mesh point N1, then gear profile of involutes will be removed by the cutting tool. A similar definition is also described by Sun [30] and Ye [31]. The minimum numbers of teeth of standard involute spur gear without undercutting in the two books are both 17. The academic group of the essay is dedicated to the gear design and manufacturing research, and some preliminary results are acquired, including the minimum number of teeth of standard involute gears without undercutting and the undercutting of involute spur gear with different pressure angles. During the study of the minimum number of teeth of standard involute gear teeth without undercutting, the concept of 'rack cutter has been mistakenly regarded as rack in the study of the minimum number of teeth without undercutting in many textbooks' is promoted for the first time [32] by the researchers. In the studies of involute spur gear undercutting with different pressure angles based on the VB programme and the neural network method, undercutting of involute gear with different pressure angles, namely, 14.5°, 22.5°, and 25°, are discussed [33][34][35][36][37]. Researchers from all over the world are dedicated to the gear design and manufacturing research and a significant amount of important findings have been acquired. For instance, a gear tooth mathematical model is developed to investigate the tooth undercutting of curvilinear-tooth gears via differential geometry and the numerical method [38]. Mimmi gives a contribution to the internal -4 -gear theory by analytically determining the limits on the design parameters for avoiding undercutting by using the limit curve concept [39]. Chiang proposes a geometric approach to determining the grinding wheel profile and the conditions to avoid undercutting [40]. However, many problems remain unsolved. As previously mentioned, in many textbooks, rack cutter has been mistakenly regarded as rack when minimum teeth without undercutting are developed, which causes the misconception of the undercutting phenomenon. Though the undercutting phenomenon of the involute gear cut by rack-shaped cutter was mentioned by Semenchenko, the top clearance coefficient c* used in his study did not conform to modern gears manufacturing. Furthermore, modified involute gears are required in certain situations. However, reports on the modification coefficient of the gear manufactured by rack cutter remain unsolved. Therefore, in the following sections, the minimum number of teeth and the modification coefficient of the standard involute gear undercutting phenomenon are comprehensively discussed. The tip-line of the rack should be under the ultimate contact point of teeth N ( Fig.2) to avoid undercutting. In other words, the length of PB should be shorter than that of PN, which means that the condition for non-undercutting can be expressed by the inequality PB≤PN.
Notably, PB and PN belong to triangles PBM and OPN, respectively, and they can be calculated as follows: where r is the radius of reference circle, and m is the module of gear. Considering r=mz/2 (z is the tooth number of gear) and integrating the Eqs. (1) and (2), the traditional condition for nonundercutting is written as follows: Given the default values h a * = 1 and α=20°, the minimum tooth number zmin=17.097 (rounded to 17). That is, the minimum number of teeth for spur gears without undercutting is 17.

z min of involute helical gear and bevel gears
For standard involute helical gears, the relation between the tooth numbers of helical gear z h and the tooth numbers of its equivalent spur gear z vh is written as follows [31]: where β is the helix angle of the gear. Thus, the minimum teeth number for the helical gear z hmin is z hmin =z vhmin cos 3 β.
As mentioned in Section 3.1.1, the minimum number of teeth of equivalent spur gear without undercutting is also 17 (z vh =17). Hence, considering β=15 and h an * = 1 (h an * is the coefficient of addendum height in normal direction), the minimum teeth number of helical gears without undercutting is z hmin =15.32≈16 based on the Eq. (5).
Straight bevel gears are the most widely used because they are easy to design and manufacture. Hence, only straight bevel gears are discussed in this study. The back cone of the bevel gear can be unfolded imaginarily into a sector gear, which can be filled up to the full gear. The new full gear is called the equivalent spur gear of the original bevel gear. The tooth profile of the virtual gear is almost the same as that of the bevel gear at the large end, and its module and pressure angle are the same as those of the bevel gear. However, the c * of bevel gear is 0.2(m≥1), which is different from that of the spur gear. The tooth number of virtual gear z vb is associated with the tooth number z b and the reference cone angle δ of the original bevel gear as follows: The minimum tooth number without undercutting for the equivalent gear of the bevel gear is equal to the z min of the spur gear. Thus, the minimum teeth number for the bevel gear without undercutting z bmin is z bmin =z vbmin cosδ. (7) Substituting z vbmin =17 and δ=35° into formula (7) yields z bmin =13.93 (rounded to 14).
The minimum tooth numbers of the helical and bevel gears without undercutting are both less than those of the spur gear. However, in the analysis process of the minimum tooth numbers without undercutting based on the traditional theory, the c of rack-shaped cutter is ignored in the calculation, which has a remarkably effect on the result. Thus, the effect of c is discussed in the next section.

z min of involute gear manufactured rank-shaped cutters considering the effect of c
Rack-shaped cutter is widely used in the actual manufacturing process of involute gears. In traditional theories, undercutting occurs when the rack addendum goes beyond the mesh limit point, but the actual gear machining process is ignored. In practice, the c (the height fillet portion, c= c * m) must be added on the cutter top to generate the radial clearance portion [36]. Therefore, in the actual manufacturing process of involute gears, the limit meshing point is not the intersection point of the meshing line and the addendum line of the cutter, but the crossover point of the meshing line and the top line of the fillet portion of the rack-shaped cutter, as shown in Fig. 3. The cutting edge between the tip and the addendum lines is not a straight line, and the corresponding profile cut is not involute. Therefore, if the minimum teeth number without undercutting gained from the traditional theory is used to guide the manufacture of involute gears, then involute portion removed in reality will be more than that and the degree of undercutting will be correspondingly worse. Therefore, if standard involute spur gears are manufactured with a rack-shaped cutter, then z min without undercutting can be calculated by the inequality PB≤PN, as shown in Fig. 3, with the following result: z min =2(h a * +c * )/sin 2 α.  22). That is, the z min of the spur gear manufactured with a rack-shaped cutter is 22.
As previously mentioned, the minimum tooth number for the helical gear without undercutting is also 22 (z vh =22). Considering β=15°, z vhmin =22 and h an * =1, the minimum teeth number of the helical gear without undercutting is z hmin =19.8 (rounded to 20) based on the Eq. (5). The minimum tooth number for the equivalent gear of the bevel gear without undercutting is equal to the minimum teeth number of the spur gear without undercutting, which means that z vbmin =22. For bevel gear manufactured with a rack-shaped cutter with a bottom clearance, z bmin (rounded to 18) is obtained by substituting z vbmin =22 and δ=35° in formula (7), as shown in Table 1. Considering the bottom clearance c of the rack-shaped cutter, the minimum teeth numbers of the three kinds of involute gears are high. The results are strikingly different from those of the traditional theories and are in accordance with the actual situation employing the rack-shaped cutter. This phenomenon necessitates a theoretical conference on involute gears manufacturing.

x min of the standard involute gear in conventional theory
Owing to their perfect interchangeability, standard involute gears are widely used in many kinds of machines. However, these gears also have some disadvantages; for example, the number of teeth should not be less than z min to prevent cutter interference, and a compact construction cannot be achieved with large z values. Moreover, the curvature radius of the tooth profile and the tooth thickness of the pinion at the root are less than those of the gear. Therefore, the strength of the pinion is much lower than that of the gear.
-7 -Addendum modification is employed to improve the performance of gears. Corrected gears are those with addendum modification, and the modified value is expressed by the distance between the pitch line and the centerline of the cutter. The distance is defined as xm, where x is addendum modification coefficient and m is also the module of the gear. If the cutter is set away from the position for cutting a standard gear, then x and the modification are considered positive. Otherwise, x and the modification are negative.
In traditional theories, the modification distance xm of the rack-shaped cutter should be calculated to prevent cutting, as shown in Fig. 1. The condition for preventing the undercutting of the root is xm≥(h a * m-N 1 G). The result is x≥h a * - from which the minimal modification coefficient x min of the spur gear can be derived using Eq. (3): Evidently, x min >0 if z<z min . When z>z min , the gear can be negatively modified as x min <0.

x min of the involute gear considering the effect of bottom clearance c
The bottom clearance c is also ignored in the deduction process of x min in traditional theories. However, c should be considered in practice, as shown in Fig. 4. Then, the condition for preventing undercutting of the root is xm≥(h a * m +c * m-N 1 G). The result is x≥(h a * +c * - from which the minimal modification coefficient x min of the spur gear can be derived using Eq. The minimum teeth numbers for the spur, helical, and bevel gears in practice are provided in Sections 3.1.1 and 3.1.2, and the results are shown in Table 1. The minimum modification coefficient can be calculated by substituting h a * = 1 and c * =0.25 (for bevel gear, c * =0.2) and inserting the results in column Ⅱ of Table 2 into Eq. (12), and the results are shown in column Ⅱ.

Numerical simulation of the undercutting of standard involute spur gear
A specialized computer software is used to simulate and analyze the cutter interference of involute spur gears manufactured by a rack cutter, to illustrate the undercutting phenomenon in the traditional theory and in the research in this paper. In the following simulations, the values of used parameters are set as h a * =1, c * =0.25, α=20°, m=4, and width of gear B=20 mm.

Simulation of spur gear with z=10
As mentioned in the main text, undercutting at the pinion root may affect the load distribution and meshing character amongst pairs of involute gear teeth in simultaneous contact. Hence, the undercutting phenomenon of standard involute spur gear with z=10 is simulated as follows. Comparison of the three types of situations in Fig. 5 shows the following: 1) heavy undercutting appears in the gear without modification; 2) undercutting occurs in the spur gear manufactured with xmin calculated following the conventional theory Eq. (10); 3) no cutter interference emerges when xmin is calculated based on Eq. (12) and when the effect of bottom clearance is considered. Therefore, the bottom clearance c of the rack-cutter must be considered during the manufacturing of involute gears, and the c is a crucial factor for the studies on undercutting phenomenon and minimum teeth number of standard involute gears.

Simulation of spur gear with z=17
The drawing of spur gear with z=17 in conventional theory is simulated as shown in Fig.7. Meanwhile, Fig.7 (B) indicates that the top of the rack-cutter enters the profile of the gear and some parts of the involute profile near the root portion are removed. Cutter interference has already occurred in its manufacturing process when z=17. Using Eq. (12), the xmin of standard involute spur gear with z=17 can be obtained as 0.28. The simulation of corrected spur gear of z=17 with xmin=0.28 is shown in Figs.7 (C) and (D). The simulation results showed that no cutter interference occurs in the corrected spur gear of z=17, thereby proving the accuracy of the research in Section 4.2.

Experiment of undercutting phenomenon of standard involute spur gear
The undercutting of gear with fewer teeth is directly explored by conducting machining experiments for the standard involute spur gears with teeth numbers of 10, 12, 17, and 22. Support was provided by Kunming Machine Tool Co., Ltd. of Shenyang Machine Tool Group.

Material
For convenience of gear machining and subsequent antirust processing, aluminium alloy bar model 6061-T651 is selected as the raw material for gear machining. Magnesium and silicon are the main alloy elements, and that form the Mg2Si phase. Via pre-tension technology processing, the bar processes good quality, such as excellent processing performance, fine welding characteristics and electroplating, excellent corrosion resistance, high toughness and high resistance to deformation, dense material without defects, and easy polishing.

Processing equipment
The Y3180H General-purpose Gear Hobbing Machine manufactured by Chongqing Machine Tool (Group) Co., Ltd. is adopted in the manufacturing process of gears. The model is applicable to cut spur and helical gears, worm wheels, sprockets, and short spline shafts in jobbing, small, or batch production. The maximum allowable machining diameter of the workpiece is 800 mm, and the maximum allowed machining module is 8 mm. The machine is characterized by good rigidity, high working accuracy, easy operation, and simple setup procedure. Either conventional or climb method can be used in the machine.

Manufactured gears
In testing and verifying the undercutting phenomenon in the gear machining process, five types of standard involute spur gears with teeth numbers z=10, 12, 14, 17 and 22 are manufactured, and the addendum modifications are considered in the manufacturing process. The detailed parameters of the gears are shown in Table 3.

No undercutting
A B -13 -

Standard involute spur gears of z=10
The tooth profiles of z=10 standard involute spur gears with and without modification are intuitively investigated by comparing the two processed gears, as shown in Fig. 9. Figs Fig.9 (E) shows that the gear tooth profile of z=10 with modification coefficient of xmin=0.68 presents no undercut, and the entire involute line is sufficiently smooth such that gears can mesh regularly.

Standard involute spur gears of z=12
As previously mentioned, the tooth profiles of z=12 standard involute spur gears with and without  Fig. 10 (D) shows that part of the gear tooth profile of z=12 without modification is cut off (yellow shading), indicating the occurrence of heavy undercut phenomenon and the abnormal meshing of gear. Fig. 10 (E) shows that the gear tooth profile of z=12 with modification coefficient of xmin=0.57 exhibits no undercut, and the entire involute line is sufficiently smooth such that gears can mesh regularly.

Standard involute spur gears of z=17
For standard involute spur gears, the noninterference minimum tooth of the meshing gear with rack is zmin=17 in traditional theory when the pressure angle α=20° and h a * =1. However, the current study concludes that the root cutting will continue to occur when z=17. The teeth changes of standard involute spur gears of z=17 with and without modification are directly determined by comparing the two processed gears, as shown in  Fig. 11 (D) shows that a tiny part of the gear tooth profile of z=17 without modification is cut off (yellow shading), indicating the occurrence of a slight undercut phenomenon and the abnormal meshing of gear. Fig.  11 (E) shows that the gear tooth profile of z=17 with modification coefficient of xmin=0.28 exhibit no undercutting, and the entire involute line is sufficiently smooth such that gears can mesh regularly.

Involute spur gears of z=22
The overall results show that the correct noninterference minimum tooth of the meshing standard involute spur gear with rack is zmin=22. The tooth profile of a correspondingly manufactured gear is analyzed, shown in Fig.12. Figs. 12 (A) and (B) show that no changes occur in the tooth profile of z=22 without modification. To further observe the gear tooth profile changes, the profiles of gear without modification in Fig.12 (A) are enlarged and then shown in Fig.12 (C). No changes clearly occur in the gear tooth profile of z=22 without modification, indicating no interference in the gear processing. In addition, the entire involute line of tooth is sufficiently smooth such that gears can mesh regularly. These results are consistent with the theoretical and numerical results.

Conclusions
The gear tooth undercutting issues are important in gear design and manufacture. In many textbooks on mechanical engineering, rack has been mistakenly regarded as rack cutter. Thus the minimum teeth number without undercutting has been deduced, causing the misunderstanding of the undercut phenomenon for many years. Meanwhile, this misconception is continually imparted to tens of thousands of students in universities yearly. In the present study, the theoretical analysis and calculation of the z min of standard involute gears, were performed, where the rack in conventional theories was changed into the rack cutter. The calculation values showed that the minimum teeth numbers of the three most frequently used kinds of involute gears should be large (as Table1). The results are strikingly different from those of traditional theories, which necessitate a theoretical conference on involute gear manufacturing.
For standard involute gears, the number of teeth should not be less than z min to prevent cutter interference, and a more compact construction cannot be achieved with large z values. Moreover, the curvature radii of the tooth profile and the tooth thickness of the pinion at the root are less than those of the gear. Addendum modification is usually employed to satisfy the fewer teeth requirement and improve the performance of gears. In the present study, the x min of the standard involute gear 16 -in conventional theories and the effect of the bottom clearance were analysed. The x min of the three kinds of standard involute gears was calculated on basis of deduction formula (12). The results showed that the x min increased when the bottom clearance was considered.
The spur gear was taken as an example, where the minimum teeth number and the modification coefficient in conventional theory and those in the current research were simulated to visually illustrate the theoretical research on the z min and x min of standard involute gears. Meanwhile, the experiments were developed correspondingly. The simulation and experiment results proved the validity of the theoretical analysis. Based on the theoretical analysis and simulations, the bottom clearance c of the rack-shaped cutter must be considered during the manufacturing of involute gears, which is a crucial factor for the studies on the undercutting phenomenon and the minimum teeth number of involute gears.

Declarations 1) Availability of data and materials
All data generated or analyzed during this study are included in the article.

2) Competing interests
No conflict of interest exits in the submission of this manuscript, and manuscript is approved by all authors for publication.