Contact stress calculation of high transverse contact ratio spur and helical gear teeth

Mechanism and Machine Theory 64 (2013) 93–110

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Contact stress calculation of high transverse contact ratio spur and helical gear teeth Miryam B. Sánchez, José I. Pedrero ⁎, Miguel Pleguezuelos UNED, Departamento de Mecánica, Juan del Rosal 12, 28040 Madrid, Spain

a r t i c l e

i n f o

Article history: Received 10 July 2012 Received in revised form 11 January 2013 Accepted 13 January 2013 Available online 27 February 2013 Keywords: Cylindrical gears Contact pressure Load distribution Pitting resistance High transverse contact ratio

a b s t r a c t For contact stress calculations of spur and helical gears, the Hertz equation can be used in combination with a model of load distribution along the line of contact. This load distribution is not uniform due to the changing rigidity of the pair of teeth along the path of contact and has decisive influence on the location and the value of the critical contact stress to consider for calculations. Moreover, the load distribution can be highly influenced for non-standard gearing conditions as the presence of undercut, enlarged tooth addendum or reduced center distance, often present in high transverse contact ratio gears. In this paper, a new calculation method of the contact stress of spur and helical gears with transverse contact ratio greater than 2 is developed. It is based on the Hertz equation and an enhanced model of load distribution, obtained from the minimum elastic potential criterion, suitable for non-standard gearing conditions. A complete study on the critical load conditions and the value of the critical contact stress has been carried out. As a result, a recommendation for pitting load capacity calculations is proposed. © 2013 Elsevier Ltd. All rights reserved.

1. Introduction Calculation methods of high transverse contact ratio (HTCR) spur and helical gears have not ever been rigorously developed, and cannot be found in technical literature. AGMA 2001-D04 International Standard [1] does not consider gear pairs with transverse contact ratio higher than 2. The scope of ISO Standard 6336-2 [2] includes cylindrical gear teeth with actual transverse contact ratio up to 2.5, however the background of the proposed calculation methods is not clearly established. Perhaps, the results provided by the simple models of the theory of elasticity are not in good agreement with experimental results when the load is assumed to be uniformly distributed along the line of contact [3]. Perhaps the high precision required to ensure triple pair tooth contact was not easy to achieve in the past. Anyway, neither designers nor researchers have ever been interested to study this kind of transmission in depth. However, modern manufacturing processes and mounting techniques provide such high precision, and some models of load distribution taking into account the changing rigidity of the pair of teeth along the path of contact are now available in literature [4–13]. This allows us to develop new accurate calculation methods for high transverse contact ratio gear drives, which will be suitable at least for preliminary calculations. Calculation methods of spur and helical gear drives available in technical literature [1,2] use the Hertz equation to evaluate the contact stress. The load is assumed to be uniformly distributed along the line of contact, and therefore several influence factors for load distribution are introduced to correct the calculated values of the contact stress. In fact, it is known that the load distribution depends on the meshing stiffness of the pair of teeth, which is different at any contact point, which means the load per unit of length is also different at any point of the line of contact.

⁎ Corresponding author. Tel.: +34 913986430; fax: +34 913986536. E-mail address: [email protected] (J.I. Pedrero). 0094-114X/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.mechmachtheory.2013.01.013

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The authors presented a model of load distribution based on the hypothesis of minimum elastic potential energy [14,15], which has been used to develop more accurate calculation methods of pitting load capacity [15,16] and efficiency [17–19]. A preliminary study of the load capacity of HTCR spur and helical gears was also developed [20]. However, the numerical method used for the integration of the elastic potential energy in all these studies provides numerical values of the load per unit of length at discrete contact points and meshing positions. This allows one to obtain some conclusions regarding the considered gear pairs, but makes it very difficult to extract general conclusions, valid for any gear pair. Afterwards, the authors [21] presented an approximate analytic equation for the inverse unitary potential (the inverse of the tooth-pair potential for unit load and face width), very simple and accurate, which depends exclusively on the transverse contact ratio. The load per unit of length can be computed from the inverse unitary potential and its integral along the complete line of contact, but this integral can be easily computed, as the inverse unitary potential has now an analytic expression. This made possible general studies on the pitting load capacity [22,23] and efficiency [24–26], allowing to make proposals for newer more accurate calculation methods. Recently, an enhanced model of load distribution for non-standard gears has been developed by the authors [27], which is valid for undercut teeth, non-standard tooth height and non-standard center distance. In this paper, this model, in combination with the Hertz model, is used to study the location and the value of the critical contact stress of involute spur and helical gear teeth with transverse contact ratio higher than 2. In fact, HTCR gears are often obtained by increasing the tooth addendum, by reducing the operating center distance (but keeping the tooth height unmodified) or by reducing the pressure angle, which increases the risk of undercut at the tooth root. With the results of this study a proposal for contact stress calculation of HTCR spur and helical gears is established, which may be suitable for preliminary designs or standardization purposes. 2. Load distribution model Reference [21] presents in detail the model of load distribution of minimum elastic potential energy. The elastic potential energy U is computed from the equations of the theory of elasticity and the teeth geometry. For calculations, a spur gear with unit load and face width is considered. Its elastic potential u – named unitary potential – and its inverse unitary potential v = u −1 are both dependent on the contact point, which is described by the ξ parameter of the contact point at the pinion profile as: z ξ¼ 1 2π

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 2C1 −1 r 2b1

ð1Þ

where z is the number of teeth, rC the radius of the contact point, rb the base radius and subscript 1 denotes the pinion (subscript 2 will denote the wheel). Note that the difference of ξ parameters corresponding to contact at the outer point of contact and at the inner point of contact is equal to the transverse contact ratio εα. Similarly, the difference of ξ parameters corresponding to two contiguous teeth in simultaneous contact is equal to 1. For spur gears, if the elastic potential energy is computed considering all the pairs of teeth in simultaneous contact, with an unknown fraction of the load acting on each one, and minimizing its value by means of variational techniques (Lagrange's method), the load at each pair results in [21]: F i ðξi Þ ¼

vi ðξi Þ zX  1 −1

vj ξj



ð2Þ

F

j¼0

where Fi (ξi) and vi (ξi) are the load and the inverse unitary potential of pair of teeth i when contact occurs at the point corresponding to ξi, F is the total transmitted load, and it is assumed vi(ξi) = 0 outside the interval of contact. According to this, the load sharing ratio R(ξ) (or the fraction of the load supported by the considered pair of teeth) is given by: Ri ðξi Þ ¼

F i ðξi Þ vi ðξi Þ vðξi Þ ¼ ¼ zX   zX F 1 −1 1 −1 vj ξj vðξi þ ðj−iÞÞ j¼0

ð3Þ

j¼0

while the load per unit of length f(ξ), for spur gears, can be expressed as: F f i ðξi Þ ¼ Ri ðξi Þ b

ð4Þ

being b the effective face width. The same approach may be used for helical gears by dividing the helical tooth in infinite slices, perpendicular to the gear axis. Each slice is equivalent to a spur gear with differential face width. In this case, the difference between the ξ parameters of two slices separated a distance dδ along the gear axis (or dl along the line of contact) is: dξ ¼

εβ εβ cos βb dδ ¼ dl b b

ð5Þ

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where εβ is the axial contact ratio (also known as overlap ratio [2]) and βb the base helix angle. By following a similar method than that described for spur gears, the load per unit of length of a helical gear at a point of the line of contact described by ξ, at the meshing position corresponding to a reference transverse section contacting at point described by ξ0, results in [21]: f ðξ; ξ0 Þ ¼

εβ cos βb vðξÞ F b Iv ðξ0 Þ

ð6Þ

where function Iv(ξ0) is given by: Iv ðξ0 Þ ¼ ∫ vðξÞdξ ¼

zX 1 −1

ξ0 þj

∫

vðξÞdξ:

ð7Þ

j¼0 ξ0 þj−ε β

lc

The reference transverse section can be any arbitrary transverse section of the helical tooth, however the expression for Iv(ξ0) depends on the chosen section. The reference transverse section corresponding to Eq. (7) is the end section of the tooth with higher contact point on pinion. 2.1. Load distribution model for standard teeth The inverse unitary potential v(ξ) for standard gears is described very accurately by the following approximate equation [21]: vðξÞ ¼ cos½b0 ðξ−ξm Þ

ð8Þ

where: ξm ¼ ξinn þ b0 ¼

εα 2

ð9aÞ

  −1=2 1 ε 2 1 þ α −1 2 2

ð9bÞ

being ξinn the involute parameter of the inner point of contact of the pinion. Fig. 1 shows the typical aspect of function v(ξ) for standard teeth. Note that, according to Eqs. (3), (6) and (7), the amplitude of v(ξ) has no influence on the load distribution, so a normalized function v(ξ), with a maximum value equal to 1, may be considered for calculations, as one given by Eq. (8) and represented in Fig. 1. According to this, the load sharing ratio for spur gears can be obtained by replacing Eq. (8) in Eq. (3). This yields the following result for transverse contact ratio εα between 1 and 2: RðξÞ ¼

vðξÞ vðξ−1Þ þ vðξÞ þ vðξ þ 1Þ

ð10Þ

which has been represented in Fig. 2 (remember that v(ξ)=0 outside the interval of contact ξinn ≤ξ≤ξinn +εα). The ordinates of singular points are always very close to 0.33 and 0.67, so the load sharing ratio given by Eq. (10) can be also computed from:   1 ξ−ξinn 1þ for ξinn ≤ ξ ≤ ξinn þ εα −1 3 εα −1 RðξÞ ¼ 1   for ξinn þ εα −1 ≤ ξ ≤ ξinn þ 1 1 ξinn þ εα −ξ 1þ for ξinn þ 1 ≤ ξ ≤ ξinn þ εα RðξÞ ¼ 3 εa −1 RðξÞ ¼

Fig. 1. Typical shape of the graph of inverse unitary potential v(ξ).

ð11Þ

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Fig. 2. Load sharing ratio for standard spur gears with transverse contact ratio between 1 and 2.

Similarly, the load sharing ratio for HTCR spur gears is given by: RðξÞ ¼

vðξÞ vðξ−2Þ þ vðξ−1Þ þ vðξÞ þ vðξ þ 1Þ þ vðξ þ 2Þ

ð12Þ

which has been represented in Fig. 3. In this case, the ordinates of singular points are not as unalterable as those with εα less than 2, and should be computed with Eqs. (8), (9a), (9b) and (12). However, intervals between singular points are quite linear also in this case. It is remarkable how these results are in very good agreement with results provided by other researchers, obtained by different ways. For example, Vedmar [5] calculated the load distribution on involute gear teeth from the contact and bending deformations, computed by finite element techniques. Obtained results presented in [5] are quite similar to the load sharing ratio obtained here, shown in Fig. 2, though the Vedmar load is not exactly uniform along the spur tooth face. Ajmi and Velex [12] obtained the deflection and the load distribution by the simultaneous solving of the equation of motion and the contact problem between teeth, and results [12] match accurately with Fig. 2 here. Li [13] combined mathematical programming methods with three-dimensional finite element methods to conduct loaded tooth contact analyses, deformations and stress calculations of spur gears. The tooth load-sharing rates obtained for low and high contact ratios [13] are identical to those represented in Figs. 2 and 3 here. For helical gears, the load per unit of length at any contact point (described by ξ) at any meshing position (described by ξ0) was given by Eq. (6), in which v(ξ) and Iv(ξ0) are given by Eqs. (8) and (7), respectively. Another equation for Iv(ξ0) which is more explicit than Eq. (7) can be found in [21]. Function Iv(ξ0) takes different shapes depending on whether the sum of the fractional parts of both transverse and axial contact ratios (dα and dβ, respectively) is less than 1 or not, as represented in Fig. 4. Note that for helical gears, the load distribution along the tooth face is described by the v(ξ) function, as each point of the line of contact has its own ξ parameter, and the load per unit of length at this point is proportional to v(ξ). Approximate values of the inverse unitary potential given by Eqs. (8), (9a) and (9b) are valid for a wide range of values of the geometrical parameters (parameters as the tooth number on pinion and wheel, the rack shift coefficients, the pressure angle, the helix angle, etc.) and the operating parameters (operating pressure angle or operating center distance), but only for tooth addendum equal to the normal module mn, tooth dedendum 1.25mn and outside diameters taken according to the operating center distance to keep the radial clearance equal to 0.25mn: r o1 ¼ C−r p2 −mn x2 þ mn ha r o2 ¼ C−r p1 −mn x1 þ mn ha

ð13Þ

Fig. 3. Load sharing ratio for HTCR spur gears.

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Fig. 4. Typical shapes of the graphs of function Iv(ξ0), the integral of the inverse unitary potential.

where ro is the outside radius, C the operating center distance, rp the standard pitch radius, x the rack shift coefficient and ha the addendum coefficient. For different values of the tooth height or operating center distance, Eqs. (8), (9a) and (9a) should be corrected. 2.2. Load distribution model for non-standard teeth Let us consider a non-standard gear pair with reduced addendum on wheel, ha2 b 1 (this means that the radial clearance at the pinion root is greater than 0.25mn). Let us also consider a standard pair with all the geometrical parameters equal to those of the non-standard pair, except h′a2 = 1. Obviously, the transverse contact ratio of the non-standard gear pair εα is smaller than the transverse contact ratio of the standard one ε′α. As the transverse contact ratio is equal to the difference of the ξ parameters of the outer point of contact and the inner point of contact, ε α ¼ ξo −ξinn

ð14Þ

the reduction of the contact ratio due to the reduced addendum on wheel can be expressed as: 0

0

ðΔε α Þinn ¼ ε α −εα ¼ ξinn −ξinn :

ð15Þ

The elastic potential of a pair of teeth when contact occurs at any point P of the profiles does not depend on the tooth profiles between point P and the outside profile point [21]. In other words, the meshing stiffness only depends on the tooth profiles between the contact point and the tooth roots. According to this, the inverse unitary potential curve of the reduced wheel addendum gear pair will be identical to that of the corresponding standard gear pair, except in the interval [ξ ′inn, ξinn], in which v(ξ) of the non-standard pair does not exist. Fig. 5 shows the shape of the inverse unitary potential of a reduced wheel addendum gear pair. Its values can be computed from Eqs. (8), (9a) and (9b) but considering the fictitious values of the inner point of contact parameter ξ ′inn and the transverse contact ratio ε′α, which are the values of the corresponding standard gear pair, i.e.: 

vðξÞ ¼ cos b00 ξ−ξ0m vðξÞ ¼ 0

for ξinn ≤ ξ ≤ ξinn þ ε α for ξ b ξinn or ξ > ξinn þ εα

ð16Þ

Fig. 5. Inverse unitary potential reduced wheel addendum gear pairs: (left) numerical integration of the equations of elasticity; (right) truncated approximate cosine function.

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with: 0

ε ξ0m ¼ ξ0inn þ α "  2  #−1=2 1 ε0 2 1 þ α −1 b00 ¼ 2 2

ð17Þ

where ε ' α = εα + (Δεα)inn = εα + (ξinn − ξ ' inn). Fig. 6 shows the load sharing ratio of a reduced wheel addendum spur gear, which can be computed from [27]:  0  1 ξ−ξ 1 þ 0 inn for 3 εα −1 RðξÞ ¼ 1 for   0 0 1 ξinn þ εα −ξ 1þ for RðξÞ ¼ 3 ε0a −1 RðξÞ ¼

ξinn ≤ ξ ≤ ξinn þ εα −1 ξinn þ εα −1 ≤ ξ ≤ ξinn þ 1 :

ð18Þ

ξinn þ 1 ≤ ξ ≤ ξinn þ εα

For wheel reduced addendum helical gears, the load distribution along the line of contact remains described by Eqs. (6) and (7). The inverse unitary potential v(ξ) can be approximated by Eqs. (16) and (17). Function Iv(ξ0) takes the same two different shapes as those represented in Fig. 4, according to the sum of the fractional parts of both transverse and axial contact ratios is greater or less than 1. For wheel reduced addendum helical gears, Iv(ξ0) can be computed from [23]: Iv ðξ0 Þ ¼

    Eγ  0  0  1X εα εα 0 0 − sin b sin b ζ − ζ − 0 0 i; sup i; inf b00 i¼0 2 2 0

ζ i; sup ¼ ξ0 þ i þ 2ξinn þ ε α −ξinn − minðξ0 þ i; ξinn Þ− maxðξ0 þ i; εα þ ξinn Þ     ζ i; inf ¼ ξ0 þ i−εβ þ 2ξinn þ εα −ξ0inn − min ξ0 þ i−εβ ; ξinn − max ξ0 þ i−εβ ; εα þ ξinn

ð19Þ

where Eγ is the integer part of the total contact ratio εγ = εα + εβ. Undercut by itself has no significant influence on the load distribution [21,27], and equations for standard gears presented in Section 2.1 are all suitable for undercut teeth if meshing conditions at the wheel tip are not affected by the presence of an undercut. However, if the undercut area is big enough, the outer point of the wheel profile may not find active profile on pinion to mesh with, as represented in Fig. 7. This is called vacuum gearing. Obviously, the effect of vacuum gearing at pinion root is the same as the effect of reduced addendum on wheel, and Eqs. (15) to (19) are all suitable also for this case. It may have strong interest for the study of HTCR gears because an easy way to get high values of the transverse contact ratio is to take small values of the generating pressure angle, but it is very well known that with small generating pressure angles undercut arises for lower tooth numbers. Similarly, for gears with reduced addendum on pinion, function v(ξ) will be equal to that for the corresponding standard pinion with h′a1 = 1, but truncated at the right side, as represented in Fig. 8. Moreover, enlarged addendum on pinion or wheel is represented by a v(ξ) function enlarged at the right or left side, respectively. Fig. 9 represents the inverse unitary potential for enlarged addendum on both gears. Equations for the inverse unitary potential and load sharing ratio for all these cases can be found in [27]. Finally, center distance modifications have the same effect on the load distribution as that of simultaneous modifications on the pinion and wheel addenda (enlarged addenda for shortened operating center distance and reduced addenda for enlarged operating center distance). But there is a little difference in the formulation of v(ξ) because in this case the profile parameter of the outer point of the pinion ξo1 does not change (as the outside radius does not change), while pinion addendum modifications induce changes in the value of ξo1. Slight reductions on the operating center distance may produce appreciable increases in the

Fig. 6. Load sharing ratio for reduced wheel addendum spur gears.

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Fig. 7. Vacuum gearing (SOI: start of involute).

transverse contact ratio, which can be useful to get HTCR gear sets. Fig. 10 shows the inverse unitary potential v(ξ) for gears with reduced operating center distance. Eqs. (16), (17) and (19) are suitable for calculating v(ξ) and Iv(ξ0) for all the above cases of non-standard tooth height. The only difference is the calculation of ξ ′inn and ε′α, which can be found in [27]. Similar equations for non-standard center distance can be also found in [27]. 3. Contact stress Reference [22] presents a complete study of the critical contact stress and the critical load conditions for non-undercut spur and helical gears. Reference [23] presents a similar study for undercut gears, which is also valid for reduced addendum gears, as discussed above. In this section, the study of the contact stress is done for HTCR spur and helical gears. As in [22] and [23], for the evaluation of the contact stress the Hertz's equation will be used – as done by AGMA and ISO standards [1,2] – which for the contact between involute teeth, may be written as: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   f 1 1 σ H ¼ ZE þ cosβb ρ1 ρ2

ð20Þ

where σH is the contact stress, ZE the elasticity factor [2], f the load per unit of length of the line of contact outlined above, and ρ the curvature radius of the section perpendicular to the line of contact, which can be expressed as: πmn cosα t ξ cosβ cosβb   πmn cosα t z1 þ z2 0 ρ2 ¼ tanα t −ξ cosβ cosβb 2π

ρ1 ¼

ð21Þ

Fig. 8. Reduced addendum on pinion.

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Fig. 9. Enlarged addendum on pinion and wheel.

with β being the standard helix angle, α t the transverse pressure angle and α ′t the operating transverse pressure angle. From the equations above, a complete study on the location and the value of the critical contact stress was carried out, resulting in a recommendation for pitting calculations for standard [22] and undercut [23] spur and helical gears, which may be considered for standardization purposes or preliminary calculations. In the present work a similar study for high transverse contact ratio gears is developed. 3.1. Critical stress and load conditions for HTCR spur gears The goal is to find the maximum value of the contact stress and the point at which it arises for HTCR spur gears. The procedure followed in references [22] and [23] are also suitable for this case. According to Eqs. (20), (4), (12) and (21), the problem is to find the maximum of the function: RðξÞ   ξ λξ −ξ

ð22Þ

where: λξ ¼

z1 þ z2 0 tanα t : 2π

ð23Þ

Fig. 11 represents both functions R(ξ) and [ξ (λξ − ξ)] −1 for a HTCR spur gear with standard tooth height and center distance, for which function R(ξ) is symmetric with respect to the midpoint of the interval of contact. Note that the x-axis represents, in terms of ξ, the interval of the line of action between the tangency points with both base circles. The interval of contact is limited by the intersection of the line of action and the outside circles of pinion and wheel, of radii ro1 and ro2, respectively (points T1 and T2 in Fig. 11). Taking into account the symmetry of both functions as well as their increasing and decreasing intervals, it can be concluded that the critical stress will be always located at one of the following three points (shown in Fig. 3): • The inner point of contact A3. • The inner point of two pair tooth contact B2. • The outer point of the inner interval of two pair tooth contact C2.

Fig. 10. Reduced operating center distance.

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Fig. 11. Evolution of the contact stress for HTCR spur gears.

In fact, a point of function [R(ξ)/ξ (λξ − ξ)] with horizontal tangent between points A3 and C2, if it exists, is always a minimum, as can be easily checked. This is true if the abscissa of the midpoint of the interval of contact ξm is smaller than the abscissa of the midpoint of the interval T1T2, i.e., is less than λξ/2. This is the same as the ξ parameter of the inner point of contact of pinion ξinn which is smaller than that of the inner point of contact of wheel ξinn2. Usually this condition is verified, but may be otherwise for a very low gear ratio. However, the discussion is valid even for these cases if we consider as “virtual pinion” the gear with smaller ξinn. Figs. 11 and 12 show the three different possibilities of the critical contact stress location, at points A3, B2 and C2, for HTCR spur gears with standard height and center distance (critical contact stress at point B2 in Fig. 11, at point C2 in Fig. 12-left and at point A3 in Fig. 12-right). For low gear ratio and high pinion tooth number, the critical contact stress is usually located at point C2, as derived from Fig. 12-left: the interval of contact is centered with respect to the interval T1T2, where function [ξ (λξ − ξ)] −1 is quite uniform. Only for low values of the number of teeth on pinion and the pressure angle is the critical contact stress shifted to points B2 or even A3. As the gear ratio increases, the interval of contact is shifted to the left (to point T1 in Figs. 11 and 12) and the critical contact stress can be located at points B2 or A3 more easily (for not so low values of the number of teeth on pinion and the pressure angle). For non-standard tooth height or center distance the discussion is quite similar. Any modification on the tooth height or the center distance means a displacement of the singular points A3, B2, C2, D2, E2 and F3 along their respective segments on the R diagram. More specifically: • Enlarged addendum on pinion produces points B2, D2 and F3 to be shifted to the right, while points A3, C2 and E2 remain fixed (Fig. 13). Reduced addendum on pinion or vacuum gearing at wheel root produce the opposite effect.

Fig. 12. Critical contact stress location HTCR spur gears.

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Fig. 13. Load sharing ratio for enlarged pinion HTCR spur gears.

• Enlarged addendum on wheel produces points A3, C2 and E2 to be shifted to the left, while points B2, D2 and F3 remain unalterable. Reduced addendum on wheel or vacuum gearing at pinion root produces the opposite effect. • Reduced operating center distance causes points B2, D2 and F3 to be shifted to the right and points A3, C2 and E2 to be shifted to the left. Enlarged operating center distance produces the opposite effect. It can be concluded that there is a new possible location of the critical contact stress at point D2, if point C2 is shifted to the left or if D2 is shifted to the left too. Consequently, for both standard and non-standard tooth heights and center distances, the critical contact stress is always located at one of the points A3, B2, C2 or D2. The easiest way to find the critical stress is to compute the stress at the four points and select the highest one:

σ H0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 0 1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv u u λξ ð Þ F R ξ u max@  A ¼ ZE t b πmn cosα n ξ λξ −ξ

ð24Þ

ξ¼ξA ;ξB ;ξC ;ξD

where αn is the standard normal pressure angle, ξA = ξinn, ξB = ξinn + εα − 2, ξC = ξinn + 1, ξD = ξinn + εα − 1 and R(ξ) is given by Eq. (12). According to Fig. 11, the critical contact stress will be located at point A3 (the inner point of contact) for small values of ξinn. In these cases, contact may occur near the base circle which could be not a recommended design. These conditions are not usual but can be given for small pressure angle and high gear ratio. 3.2. Critical stress and load conditions for HTCR helical gears For helical gears, Eq. (20) can be written as: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffivffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi  u F εβ cosβ cosβb u cos b00 ðξ−ξ0m Þ 1   σ H ¼ ZE : λξ t b πmn cosα t Iv ðξ0 Þ ξ λ −ξ

ð25Þ

ξ

The critical load conditions will correspond to the values of ξ and ξ0 maximizing the function: 2

3  0 0 4 cos b0 ξ−ξm 5⋅ 1 : Iv ðξ0 Þ ξ λξ −ξ

ð26Þ

For simplicity, we will do: 

cos b00 ξ−ξ0m   : ΦðξÞ ¼ ξ λξ −ξ

ð27Þ

As in [22] and [23], the study on the maximization of function [Φ(ξ)/Iv(ξ0)] will be performed in three steps: (i) find the maximum of Φ(ξ), (ii) find the minimum of Iv(ξ0) and (iii) study the conditions of both extremes to be given simultaneously at any position of the meshing cycle.

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3.2.1. Maximum of Φ(ξ) There are two different possibilities for the location of the maximum of Φ(ξ): • At a local maximum inside the interval [ξinn, ξ ′m], if it exists. • At the inner point of contact ξinn, if the local maximum does not exist. In fact, according to the symmetries and growth intervals of functions R(ξ) and [ξ (λξ − ξ)] −1, it can be immediately concluded that function Φ(ξ) always decreases at ξ ′m. Consequently, the maximum cannot be located here. On the other hand, the derivative of Φ(ξ) at ξ = ξinn may be positive or negative. If positive, there will necessarily be a local maximum inside the interval [ξinn, ξ ′m], as the derivative is negative at ξ = ξ′m. If negative, Φ(ξ) decreases in the whole interval [ξinn, ξ ′m], and the maximum will be located at ξinn (this is true for HTCR helical gears but not for gears with a transverse contact ratio less than 2: in this case a local minimum followed by a local maximum may occur [16]). As discussed in [22] and [23], the abscissa of the absolute maximum ξmaxR, can be computed from: 

 0 0 0 ξm −ξinn ξm −ξm;inn   ¼ ξinn þ ξ0m;m −ξ0m;inn

If

ξ0m;inn

If not

≤ ξ0m

0 ξm;inn

≤ ξ0m;m

0 ≤ ξm

0 ≤ ξm;m

⇒

ξ maxR

⇒

ξ maxR ¼ ξinn

ð28Þ

where: 2 3 1 1 λξ −2ξinn 5 4   ¼ ξinn þ 0 arctan 0 b0 b0 ξ inn λξ −ξinn 2 3 0 1 1 λξ −2ξm 5   : ¼ ξ0m þ 0 arctan4 0 0 b0 b0 ξ λ −ξ0

ξ0m;inn ξ0m;m

m

ξ

ð29Þ

m

Summarizing, the maximum of Φ(ξ) is: ½ΦðξÞ max ¼ Φðξ maxR Þ

ð30Þ

where ξmaxR can be computed from Eq. (28). 3.2.2. Minimum of Iv(ξ0) Function Iv(ξ0) for HTCR helical gears is described by Eq. (19), both for standard and non-standard tooth height and center distance. Its graphic representation is identical to that for standard helical gears, represented in Fig. 4. For dα +dβ greater or less than 1, there is an interval of minimum Iv, and for both cases the inner point of contact ξinn belongs to this interval. Consequently, also in this case it can be affirmed that: ½Iv ðξÞ min ¼ Iv ðξinn Þ

ð31Þ

which can be computed with Eq. (19). 3.2.3. Simultaneity condition The value [Φ(ξmaxR)/Iv(ξinn)] will correspond to the maximum contact stress if there is a contact point with ξ = ξmaxR during the interval of minimum Iv, but may be conservative if both conditions are not given simultaneously along the meshing interval. It can be checked [22] that simultaneity is ensured either if εβ > 1 or the absolute maximum ξmaxR is located at the inner point of contact ξinn. On the contrary, simultaneity is not ensured if ξmaxR is a local maximum, inside the meshing interval, and εβ b 1. Fig. 14 shows the contact map for a helical gear with dα + dβ b 1, dα > dβ, εβ b 1 and εα > 2. Gray zones represent the contribution of each tooth to the contact line at each position of the meshing cycle, represented by ξ0 at the x axis. It can be observed (dark gray zones) that simultaneity occurs if the local maximum ξmaxR verifies: ξinn þ dα ≤ ξ maxR ≤ ξinn þ 1 or ξinn þ 1 þ dα ≤ ξ maxR ≤ ξinn þ 2:

ð32Þ

Similarly, for dα + dβ > 1 simultaneity occurs if: ξinn þ 1−dβ ≤ ξ maxR ≤ ξinn þ dα þ dβ or ξinn þ 2−dβ ≤ ξ maxR ≤ ξinn þ 1 þ dα þ dβ as shown in the contact map represented in Fig. 15.

ð33Þ

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Fig. 14. Contact map for helical gear with dα + dβ b 1.

Nevertheless, in all the cases in which simultaneity is not given, ξmaxR is always a local maximum, in which the derivative of function Φ(ξ) is equal to 0. This means that for points not far from this local maximum – i.e., |ξmax R −ξ| small – Φ(ξ) is very close to the maximum value Φ(ξmaxR). Then, it will be possible to find a point of the contact domain whose ξ0 is included in the minimum Iv interval and whose ξ is close enough to the local maximum ξmaxR. At this point Iv =Ivmin and Φ≈Φmax. Consequently, computing the maximum contact stress

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105

Fig. 15. Contact map for helical gear with dα + dβ >1.

from the value [Φ(ξmaxR)/Iv(ξinn)] also for the case of no simultaneity, will induce a small error and in the sense of safety. In conclusion, for high transverse contact ratio helical gears the critical contact stress can be computed from: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffivffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  u F εβ cos β cos βb u cos b00 ðξ maxR −ξ0m Þ 1   λξ t σ H ¼ ZE b πmn cos α t Iv ðξinn Þ λ −ξ ξ maxR

ξ

maxR

ð34Þ

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regardless of simultaneity considerations. Note that (v/ρ)(ξ) is proportional to the square of the contact stress; consequently functions (v/ρ)(ξ) represented in Figs. 14 and 15 give an idea of how is the contact stress distribution along the line of contact on the tooth face. 4. Accuracy of the method A large number of simulations have been carried out to check the accuracy of the equations for the critical contact stress presented herein. As the paper presents a theoretical study it is important to cover a wide range of values for the transverse contact ratio, from values slightly greater than 2 to values as close as possible to 3. To get so high contact ratios is often necessary to consider gears with a high number of teeth, which may be not very realistic, but allows demonstrating that the proposed method is accurate enough even for these unrealistic cases. Obtained results are summarized in this section. 4.1. HTCR spur gears HTCR spur gears have been considered with design parameters contained in the following ranges: • • • •

Number of teeth on pinion between 40 and 150 Gear ratio between 1 and 3 Pressure angle between 10 and 18° Rack shift coefficient on pinion and wheel between − 0.2 and 0.2, for both gears

The proposed value of the critical contact stress was computed, according to the method described above, from Eq. (24) in which the load sharing ratio R(ξ) was calculated from the approximate, analytical equation of the inverse unitary potential v(ξ) given by Eq. (16). This value was compared with a reference value of the critical contact stress obtained by maximizing Eq. (20) by numerical techniques, in which the load per unit of length f was obtained from the values of the inverse unitary potential v(ξ) computed by numerical integration of the equations of the elasticity. For standard center distance (keeping the radial clearance) and tooth height (addendum coefficient equal to 1), typical errors were contained between 0.5% and 1%, while the maximum one was smaller than 2%. The critical contact points (A3, B2, C2 or D2) obtained from both methods were coincident in more than 95% of the cases. Similar values were obtained for non-standard center distance or tooth height, except for enlarged tooth on pinion or wheel, in which the maximum error was slightly greater than 3%. The critical contact points were coincident in 90% of the cases. It is remarkable that these maximum errors of 2%–3% do not correspond to cases with no coincident critical point location; in fact, the absence of coincidence occurs for similar values of the contact stress at both points, and errors are small (typically, under 1%). 4.2. HTCR helical gears For HTCR helical gears, the ranges of variation of the design parameters considered for the study were the following: • • • • •

Number of teeth on pinion between 45 and 100 Gear ratio between 1 and 4 Pressure angle between 10 and 18° Helix angle between 10 and 30° Rack shift coefficient on pinion and wheel between − 0.2 and 0.2, for both gears In this case, three errors have been evaluated:

• Error in the location of the abscissa of the maximum of function Φ(ξ), ξmaxR • Error in the estimation of the maximum of function Φ(ξ), Φ(ξmaxR) • Error for neglecting the non-simultaneity of the maximum of Φ(ξ) and the minimum of Iv(ξ0) According to the method proposed herein, the abscissa of the maximum of the function Φ(ξ), (ξmaxR)P, should be computed with Eq. (28). This value was compared with two reference values: • (ξmaxR)R1, obtained by numerical techniques from the function Φ(ξ) based on the inverse unitary potential v(ξ) computed by numerical integration of the equations of the elasticity. • (ξmaxR)R2, obtained by numerical techniques from the function Φ(ξ) based on the inverse unitary potential v(ξ) given by Eq. (16). To evaluate the error in the location, the relative shift coefficient δ is defined as the fraction of the interval of contact corresponding to the deviation:



ðξ maxR ÞP −ðξ maxR ÞR1ðR2Þ δ1ð2Þ ¼ : ð35Þ εα Both relative shift coefficients, δ1 and δ2, take quite similar values, typically between 0.00 and 0.05, while the maximum ones were never higher than 0.12. Obviously, δ1 = δ2 = 0 for the cases in which the maximum is located at the inner point of contact

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(i.e., there is no local maximum inside the interval of contact), except very few times in which Eq. (28) finds an inexistent local maximum, but always very close to the inner point of contact. Condition of Eq. (28), ξ ′m,inn ≤ ξ ′m ≤ ξ ′m,m (or not), discriminates the existence of a local maximum inside the interval of contact (or not) with a 99.98% of effectiveness [21]. On the contrary, errors in the estimation of the maximum of function Φ(ξ) by the proposed method – equal to Φ(ξmaxR)P – with respect to the first reference value Φ(ξmaxR)R1 are substantially different from those with respect to the second reference value Φ(ξmaxR)R2: while error with respect to reference value R2 is always smaller than 0.9%, the error with respect to reference value R1 grows up to 8%. Though this value seems to be a little high, the contact stress is proportional to the square root of the value of Φ; consequently maximum error in the estimation of the contact stress is around 4%, which can be considered accurate enough. Once again, high errors in the estimation of the maximum of Φ(ξ) do not correspond with high values of the relative shift coefficients (high errors in the estimation of the maximum location) because these high values of δ are obtained for cases in which function Φ(ξ) takes quite uniform values along the complete interval of contact, and errors are therefore small. Finally, a new error is induced with the assumption of (Φ/Iv)max = Φmax/Ivmin, if simultaneity condition discussed in Section 3.2.3 is not given (as in the case of Fig. 14). Once again, no error is induced if the critical contact point is located at the inner point of contact, because under these conditions simultaneity is always ensured. On the contrary, two errors are present when the maximum is a local maximum: one in the estimation of the maximum of Φ and another due to the absence of simultaneity. The effect of both errors on the final value of (Φ/Iv)max may be additive or subtractive, depending on the cases, but the important result is the total error. And total error is lower (subtractive effect occurs) in the cases in which error in the estimation of Φ is high, so that the maximum error is smaller. If the proposed value [Φ(ξmaxR)P/Iv(ξinn)] is compared with the maximum, computed by numerical techniques, of the function (Φ/Iv), Φ and Iv based on the inverse unitary potential v(ξ) computed by numerical integration of the equations of the elasticity, obtained errors are less than 4% for standard tooth height and center distance (when expressed in terms of contact stresses), and slightly greater (up to 6%) for non-standard tooth height or center distance. If proposed value [Φ(ξmaxR)P/Iv(ξinn)] is compared with the maximum, computed by numerical techniques, of the function (Φ/Iv), but Φ and Iv based on the analytic approximation of the inverse unitary potential v(ξ) given by Eq. (16), obtained errors in contact stress estimation are less than 1%. 5. Comparison with ISO rating method A complete study on the discrepancies between ISO rating methods and calculations based on the new model of load distribution of Minimum Elastic Potential (MEP) for standard gears can be found in [22]. A similar study for non-standard tooth height or center distance was presented in [27]. For this study the ISO rating method was adapted to non-standard dimensions by considering four possible determinant contact points (both limits of the interval of contact and both limits of the interval of one pair tooth contact), instead of the inner point of contact and the inner point on one pair tooth contact, exclusively. The results of both studies can be summarized as follows: • The critical contact stress is usually located at the inner point of single pair tooth contact, both for ISO and MEP methods. However, results are not identical due to the virtual face width (or the contact ratio factor [2]) considered by ISO. Discrepancies depend on the values of the transverse contact ratio, but typically the ISO contact stress is around 10% smaller than MEP contact stress. • For small values of the pinion tooth number, the MEP critical contact stress may be located at the inner point of contact, while the ISO one remains at the inner point of single pair tooth contact. In this case, discrepancies increase up to 35%, for gear ratio greater than 3. • For gears with non-standard tooth height or center distance, discrepancies between adapted ISO method and MEP method remain at the same level of around 10%, owing to the contact ratio factor [2] also in this case. Nevertheless, when ISO and MEP critical stresses are located at different contact points, discrepancies are significantly reduced, bellow 20%. For HTCR gears ISO computes the critical contact stress with half of the total load acting at the inner point of the inner interval of two pair tooth contact (point B2 in Fig. 3), described by ξ = ξinn + εα − 2. However, the scope is restricted to gears with transverse contact ratio up to 2.5. For HTCR spur gears, a study has been carried out considering values of the design parameters contained in the following ranges: • • • •

Number of teeth on pinion between 40 and 60 Gear ratio between 1 and 3 Pressure angle between 12 and 18° Rack shift coefficient on pinion and wheel between − 0.1 and 0.1, for both gears

The results of this study can be seen in Fig. 16. Lower aligned points correspond to cases in which MEP critical stress is located at point B2, as well as ISO stress. In these cases, discrepancies between ISO and MEP are due to the contact ratio factor [2] and to the different fraction of the load considered by each method. However, as the contact ratio factor increases with the transverse contact ratio, discrepancies are high for high contact ratio, reaching values higher than 25% for εα = 2.5. Upper, non-aligned points represent all the other cases, in which the source of the discrepancies are not only the contact ratio factor and the fraction of the load but also the different critical point – and relative curvature radius – considered by each method. Obviously, discrepancies are

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Fig. 16. Comparison between ISO and MEP critical contact stresses for spur gears.

even higher for these cases, but this increase is significantly smaller as the contact ratio increases. Consequently, maximum discrepancies grow very slightly, up to 28%. In all the cases MEP critical contact stress is higher than the ISO one. For HTCR helical gears, a similar study has been carried out taking the values of the design parameters contained in the following ranges: • • • • •

Number of teeth on pinion: 40, 50 and 60 Gear ratio between 1 and 3 Pressure angle between 12 and 16° Helix angle between 15 and 25° Rack shift coefficient on pinion and wheel between − 0.1 and 0.1, for both gears

The results of this study can be seen in Fig. 17. Three families of points can be distinguished, corresponding to three considered values of the pinion tooth number. Typically, differences are lower than 30%, but may grow up to 40% for low values of the profile parameter of the inner point of contact ξinn, which are given for number of teeth on pinion as small as possible (keeping the transverse contact ratio higher than 2) and high gear ratio. MEP critical contact stress is always higher than the ISO one, also for helical gears.

Fig. 17. Comparison between ISO and MEP critical contact stresses for helical gears.

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109

These discrepancies are due to the even load distribution considered by ISO (after load distribution corrections), and mainly due to the critical contact stress location at point B2, which is very weakly justified in the standard [2]. 6. Conclusions In this paper a model of non-uniform load distribution along the line of contact of spur and helical gear teeth, obtained from the minimum elastic potential energy criterion, has been applied to the determination of the critical contact stress of high transverse contact ratio gears. After a complete study on the value and location of the critical stress, the following recommendations may be done: • For HTCR spur gears, the critical contact stress will arise at one of the following points: (i) at the inner point of contact of the “virtual pinion”, (ii) at the inner point of the inner interval of two pair tooth contact, (iii) at the outer point of the inner interval of two pair tooth contact or (iv) at the inner point of the outer interval of two pair tooth contact. The “virtual pinion” is the gear with smaller ξinn, the pinion in the majority of the cases, but may be the wheel in transmissions with gear ratio close to 1 and significant amount of undercut at the pinion root or enlarged pinion tooth. For spur gears with standard center distance and tooth addendum coefficient equal to 1, critical contact stress at the inner point of the outer interval of two pair tooth contact (fourth possibility) is not possible. • For HTCR helical gears, the critical contact stress can be calculated assuming that conditions of maximum of function Φ(ξ) and minimum of function Iv(ξ0) are given simultaneously at any contact position. Induced error is very small if coincidence does not occur and reduces the maximum error in the estimation of the maximum of function Φ(ξ). The location of the maximum of function Φ(ξ), ξmaxR, can be computed by interpolation of ξm, if its value is inside the interpolation interval and the limits of this interval are not crossed; otherwise ξmaxR will be located at the pinion inner point of contact. The minimum of Iv(ξ0) is always located at the inner point of contact of the pinion. Equations are given for all the parameters and functions, allowing simple, analytic calculations of the contact stress. Accuracy of all these calculations is good enough for strength models. This makes the above recommendation suitable for preliminary design calculations or standardization purposes. Nomenclature b Face width, mm C Operating center distance, mm dα Fractional part of εα dβ Fractional part of εβ Eγ Integer part of εγ F Load, N f Load per unit of length, N/mm ha Addendum coefficient mn Normal module, mm R Load sharing ratio rb Base radius, mm rC Contact point radius, mm ro Outside radius, mm rp Standard pitch radius, mm U Elastic potential, N·mm u Unitary potential, mm 2/N v Inverse unitary potential, N/mm 2 x Rack shift coefficient ZE Elasticity factor (ISO 6336-2), MPa 1/2 z Number of teeth αn Standard normal pressure angle αt Transverse pressure angle α′t Operating transverse pressure angle (pressure angle at the pitch cylinder) β Standard helix angle βb Base helix angle δ Relative shift coefficient εα Transverse contact ratio ε′α Fictitious transverse contact ratio εβ Axial contact ratio (overlap ratio) εγ Total contact ratio ρ Curvature radius, mm σH Contact stress, MPa ξ Involute profile parameter

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Acknowledgments Thanks are expressed to the Spanish Council for Scientific and Technological Research for the support of the projects DPI2008-05787, “Calculation models for special cylindrical gears” and DPI2011-27661, “Advanced models for strength calculations and dynamic analysis of non-conventional cylindrical gears”. References [1] AGMA Standard 2001-D04, Fundamental Rating Factors and Calculation Methods for Involute Spur and Helical Gear Teeth, American Gear Manufacturers Association, Alexandria, VA, 2004. [2] ISO Standard 6336-2:2006, Calculation of Load Capacity of Spur and Helical Gears — Part 2: Calculation of Surface Durability (Pitting), International Organization for Standardization, Geneva, Switzerland, 2006. [3] A. Kawalec, J. Wiktor, D. Ceglarec, Comparative analysis of tooth-root strength using ISO and AGMA standards in spur and helical gears with FEM-based verification, Journal of Mechanical Design 128 (2006). [4] K. 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Pedrero, M. Artés, M. Pleguezuelos, C. García-Masiá, A. Fuentes, Theoretical model for load distribution on cylindric gears: application to contact stress analysis, AGMA Paper 99FTM15, 1999. [17] J.I. Pedrero, M. Estrems, A. Fuentes, Determination of the efficiency of cylindric gear sets, Proc. IV World Congress on Gearing and Power Transmissions, vol. 1, Ed. MCI, Paris, France, 1999. [18] M. Pleguezuelos, J.I. Pedrero, M.B. Sánchez, Model of efficiency of high transverse contact ratio spur gears, Proc. JSME International Conference on Motion and Power Transmissions, Ed. JSME, Sendai, Japan, 2009. [19] J.I. Pedrero, M. Pleguezuelos, M. Muñoz, Simplified calculation method for the efficiency of involute spur gears, Proc. ASME IDETC/CIE 2009, Ed. ASME, San Diego, California, 2009. [20] J.I. Pedrero, I.I. Vallejo, M. Pleguezuelos, Calculation of tooth bending strength and surface durability of high transverse contact ratio spur and helical gear drives, Journal of Mechanical Design 129 (1) (2007). [21] J.I. Pedrero, M. Pleguezuelos, M. Artés, J.A. Antona, Load distribution model along the line of contact for involute external gears, Mechanism and Machine Theory 45 (2010). [22] J.I. Pedrero, M. Pleguezuelos, M. Muñoz, Critical stress and load conditions for pitting calculations of spur and helical gear teeth, Mechanism and Machine Theory 46 (2011). [23] J.I. Pedrero, M. Pleguezuelos, M. Muñoz, Contact stress calculation of undercut spur and helical gear teeth, Mechanism and Machine Theory 46 (2011). [24] M. Pleguezuelos, J.I. Pedrero, M.B. Sánchez, Simplified calculation method for the efficiency of involute helical gears, New Trends in Mechanism Science, 2010. (Cluj-Napoca, Romania). [25] M. Pleguezuelos, J.I. Pedrero, M.B. Sánchez, Analytical expression of the efficiency of involute spur gears, Proc. 13th World Congress in Mechanism and Machine Science, Universidad de Guanajuato, Guanajuato, Mexico, 2011. [26] M. Pleguezuelos, J.I. Pedrero, M.B. Sánchez, Analytical model of the efficiency of spur gears: study of the influence of the design parameters, Proc. ASME International Power Transmission and Gearing Conference, Ed. ASME, Washington D.C., 2011 [27] M.B. Sánchez, M. Pleguezuelos, J.I. Pedrero, “Enhanced model of load distribution along the line of contact for non-standard involute external gears”, Meccanica, in press, http://dx.doi.org/10.1007/s11012-012-9612-8.