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Critical stress and load conditions for pitting calculations of involute spur and helical gear teeth
Mechanism and Machine Theory 46 (2011) 425–437
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Mechanism and Machine Theory j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / m e c h m t
Critical stress and load conditions for pitting calculations of involute spur and helical gear teeth José I. Pedrero a,⁎, Miguel Pleguezuelos a, Marta Muñoz b a b
UNED, Departamento de Mecánica, Juan del Rosal 12, 28040 Madrid, Spain UNED, Departamento de Ingeniería Energética, Juan del Rosal 12, 28040 Madrid, Spain
a r t i c l e
i n f o
Article history: Received 25 January 2010 Received in revised form 7 September 2010 Accepted 1 December 2010 Available online 5 January 2011 Keywords: Cylindrical gears Contact pressure Load distribution Pitting resistance
a b s t r a c t Calculation methods of spur and helical gear drives for preliminary designs or standardization purposes available in technical literature, use the Hertz equation to evaluate the contact stress, assuming the load to be uniformly distributed along the line of contact. However, this model presents some discrepancies with experimental results because the changing rigidity of the pair of teeth along the path of contact produces a non-uniform load distribution, which implies that some load distribution factors are required to compute the contact stress. In this paper, a non-uniform model of load distribution along the line of contact, recently developed, obtained from the minimum elastic potential criterion, has been used. This model combined with the Hertz equation yields more accurate values of the contact stress. As the load per unit of length at any point of the line of contact and any position of the meshing cycle has been described by a very simple analytic equation, a complete study of the location and value of the critical contact stress has been carried out. From this study, a recommendation for the calculation of the pitting load capacity of spur and helical gears is proposed. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction To compute the pitting load capacity of spur and helical involute gears, the most widely used standards as AGMA [1,2] or ISO [3] employ the Hertz equation to evaluate the contact pressure. The load is assumed to be uniformly distributed along the line of contact, though it is known that the load distribution depends on the meshing stiffness of the pair of teeth, which is different at any contact point. This means that the load per unit of length is also different at any point of the line of contact, and therefore several influence factors for load distribution are required to correct the calculated values of the bending and contact stresses [1,4]. Some studies on the load distribution along the line of contact can be found in technical literature [5–13], but all of them provide results obtained by numerical techniques or the finite element method (FEM). Those methods allow to obtain some conclusions regarding the considered gear pair, but make it very difficult to extract general conclusions, valid for any gear pair. In previous works [14–19], the authors obtained a new load distribution model from the minimum elastic potential energy criterion. The elastic potential energy of a pair of teeth was calculated and expressed as a function of the contact point and the normal load. The load sharing among several pairs of contacting teeth in spur gears was obtained by solving the variational problem of minimizing the total potential energy (equal to the addition of the potential energy of each pair at its respective contact point) taking into account the restriction of the total load to be equal to the sum of the load at each pair. The same approach was used for helical gear teeth by dividing each helical pair in infinite slices, perpendicular to the gear axis, assuming each slice to be equivalent to a spur gear with differential face width, and extending the integrals to the complete line of contact. This approach allowed the value of the load per unit of length at any point of the line of contact and at any position of the meshing cycle to be known. Through this model, some preliminary studies on the location of the points of critical stress and the determinant load ⁎ Corresponding author. Tel.: + 34 913986430; fax: + 34 913986536. E-mail address: [email protected] (J.I. Pedrero). 0094-114X/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmachtheory.2010.12.001
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conditions were carried out for bending and wear, resulting in good agreement with experimental data [15,17]. However, the numerical method used for integration of the elastic potential energy, which yields numerical values of the load per unit of length at discrete contact points and meshing positions, presented the same problem of no generality of the obtained results. Recently, the authors [20] presented an approximate analytic equation for the inverse unitary potential, 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 now has an analytic expression. This will make possible general conclusions about the critical load conditions to be obtained, and consequently to study the load capacity, efficiency, etc., allowing to make proposals for calculation methods for preliminary designs or standardization purposes. In this paper, a complete study of the location and value of the critical contact stress on involute spur and helical gear teeth with a transverse contact ratio between 1 and 2 has been carried out, and according to this a proposal for pitting calculations is established. 2. Load distribution model Reference [20] presents in detail the model of load distribution of minimum elastic potential energy. In general terms, that model was obtained by computing the total elastic potential energy from the equations of the theory of elasticity, considering all 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). It has been demonstrated that the load per unit of length depends on the inverse unitary potential v(ξ), which is defined as the inverse of the elastic potential for unitary load and face width. Obviously, the inverse unitary potential depends on the contact point, which is described by the ξ parameter of the contact point at the pinion profile as: z ξ= 1 2π
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 rC1 −1 2 rb1
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). This parameter ξ has a physical meaning: it is the ratio of the curvature radius at the load point C and the circular base pitch; it is also the ratio between the rolling angle of point C and the angular pitch 2π/z. For spur gears, the load on each pair of teeth in simultaneous contact can be expressed as a function of the inverse unitary potential as [20]: Fi ðξi Þ =
vi ðξi Þ z1 −1
∑ vj ξ j
ð2Þ
F
j=0
where Fi(ξi) and vi(ξi) are the load and the inverse unitary potential of tooth 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) would be: Ri ðξi Þ =
Fi ðξi Þ = F
vi ðξi Þ z1 −1
∑ vj ξ j
ð3Þ
j=0
and the load per unit of length f, for spur gears, can be expressed as:
fi ðξi Þ =
F R ðξ Þ b i i
ð4Þ
being b the effective face width. Note that, at a given meshing position, the parameters ξ corresponding to teeth i and j are related by: ξj = ξi + ðj−iÞ:
ð5Þ
Similarly, for helical gears, the load per unit of length at a point of the line of contact described by ξ, at the meshing position corresponding to a reference transverse section contacting at a point described by ξ0, is given by [20]: f ðξ;ξ0 Þ =
εβ cosβb vðξÞ F b Iv ðξ0 Þ
ð6Þ
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Fig. 1. Typical shape of the graph of the inverse unitary potential v(ξ).
where εβ is the axial contact ratio, βb the base helix angle and function Iv(ξ0) can be derived from: z1 −1
Iv ðξ0 Þ = ∫ vðξÞdξ = ∑ lc
ξ0 + j
∫ vðξÞdξ:
ð7Þ
j = 0 ξ + j−ε 0 β
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 a higher contact point. The inverse unitary potential v(ξ) is described very accurately by [20]: vðξÞ = cos½b0 ðξ−ξm Þ
ð8Þ
where: ε ξm = ξinn + α 2 −1 = 2 1 ε 2 1 + α −1 b0 = 2 2
ð9Þ
being ξinn the involute parameter of the inner point of contact of the pinion and εα the transverse contact ratio. Fig. 1 shows the typical aspect of function v(ξ) for non-undercut teeth. According to this, the load sharing ratio for spur gears can be obtained by replacing Eq. (8) in Eq. (3) and accounting Eq. (5), which yields the following result for the transverse contact ratio between 1 and 2: RðξÞ =
vðξÞ vðξ−1Þ + vðξÞ + vðξ + 1Þ
ð10Þ
which has been represented in Fig. 2. The ordinates of singular points are always very close to 0.33 and 0.67, so that the load sharing ratio given by Eq. (10) can be also computed from: RðξÞ =
1 ξ−ξinn 1+ 3 εα −1
RðξÞ = 1
1 ξ + εα −ξ 1 + inn RðξÞ = 3 εa −1
for
ξinn ≤ξ≤ξinn + εα −1;
for
ξinn + εα −1≤ξ ≤ξinn + 1; and
for
ξinn + 1≤ξ≤ξinn + εα :
ð11Þ
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. The shape of function Iv(ξ0) is similar to the shape of the evolution of the length of contact along the mesh cycle. 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, function Iv(ξ0) takes different shapes, as represented in Fig. 3. Note that the influence of dα + dβ is the same as that on the length of contact. Since Iv(ξ0) is the integral of v(ξ) along the line of contact, Iv(ξ0) increases in the intervals in which the length of contact increases, decreases in the intervals in which the length of contact decreases, and is more or less constant in the intervals in which the length of contact is constant.
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Fig. 2. Load sharing ratio for spur gears with a transverse contact ratio between 1 and 2.
Profile modifications as tip relief are not explicitly taken into account in this model of load distribution. This is due to that the integrals for the elastic potential energy –Eq. (1) in [20] – do not change significantly so the inverse unitary potential v(ξ), and consequently the load sharing ratio R(ξ), are not affected. Moreover, profile modifications are many times calculated to absorb misalignments or manufacturing deviations in order to ensure a quasi-conjugate contact between meshing profiles. Neither ISO 6336-2 [3] nor AGMA 908-B89 [2] considers profile modifications in their equations for contact stress calculations (all of them are obtained from the equations of the involute); however these equations are valid for modified profiles, for this reason. 3. Contact stress For the contact between involute gear teeth, Hertz's equation may be written as: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f 1 1 σH = ZE + cosβb ρ1 ρ2
ð12Þ
where ZE is the elasticity factor [3], f the load per unit of length of the line of contact and ρ the curvature radius of the section perpendicular to the line of contact. This section forms with the transverse section an angle equal to the base helix angle βb, so according to Euler's theorem and taking into account that the curvature radius of the transverse section at the contact point is equal to the distance between the contact point and the tangency point of the pressure line and the base circumference, we have: ρt1 πmn cosαt = ξ cosβb cosβcosβb ρ πmn cosαt z1 + z2 tanα′t −ξ ρ2 = t2 = cosβb cosβcosβb 2π
ρ1 =
ð13Þ
being ρt the curvature radius of the transverse section at the contact point, mn the normal module, αt the transverse pressure angle, β the standard helix angle and α′t the operating transverse pressure angle. The goal is to find the critical contact stress and load conditions, i.e., the value of the maximum contact stress and the location of the contact point where this maximum arises.
Fig. 3. Typical shapes of the graphs of function Iv(ξ0), the integral of the inverse unitary potential.
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Fig. 4. Evolution of the contact stress along the line of action of spur gears.
3.1. Critical stress and load conditions for spur gears According to Eqs. (12), (4) and (13), the critical contact stress for spur gears will be given by the maximum of function: RðξÞ ξ λξ −ξ
ð14Þ
where: λξ =
z1 + z2 tanα′t : 2π
ð15Þ
From Fig. 2, R(ξ) increases in the interval [ξinn, ξinn + εα − 1]. On the other hand, function [ξ (λξ − ξ)]− 1 decreases in the interval [0, λξ / 2] and increases in [λξ / 2, λξ]. Finally, considering that λξ / 2 is always greater than ξinn + εα / 2 (the midpoint of the interval of contact) we conclude that the point of maximum contact stress will be necessarily located inside the interval [ξinn, ξinn + εα − 1]. It can be also demonstrated that if function [R(ξ) / ξ (λξ − ξ)] has a point with a horizontal tangent inside this interval, this local extreme is always a minimum. Consequently, the critical contact stress will be necessarily located at one of the interval limits: either at ξ = ξinn for which, according to Eq. (11), R(ξ) = 0.33, or at ξ = ξinn + εα − 1 for which R(ξ) = 1. Depending on the transmission geometry, the critical contact point may be the inner limit of the interval of contact or the inner point of single pair tooth contact, as shown in Fig. 4. The easiest way to know which limit is critical is to compute the contact stress at both points, and select the highest one. In conclusion, the nominal contact stress for spur gears will be given by:
σH0
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv u u λ F 0:33 1 u ξ @ ; A = ZE tmax b πmn cosαn λ −ξ ðξ + ε −1Þ λ −ðξ + ε −1Þ ξ ξ
inn
inn
inn
α
ξ
inn
ð16Þ
α
where αn is the standard normal pressure angle. Usually, contact stress will be critical at the inner point of single pair tooth contact, but the critical contact point will be the inner point of contact for small values of ξinn, as in Fig. 4 (left). In these cases, contact may occur near to the base circle which could be not a recommended design. These conditions are not usual but can be given for a small number of teeth on pinion and high gear ratio. Note that ISO 6336-2 [3] predicts the critical stress at the inner point of contact in more cases, as it considers a load sharing ratio equal to 0.5 instead of 0.33. 3.2. Critical stress and load conditions for helical gears For helical gears, Eq. (12) can be written as: σH = ZE
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffivffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u F εβ cosβcosβb ucos½b0 ðξ−ξm Þ 1 : λξ t b πmn cosαt Iv ðξ0 Þ ξ λ −ξ
ð17Þ
ξ
The critical load conditions will correspond to the values of ξ and ξ0 for which is the maximum of the function: ΦðξÞ Iv ðξ0 Þ
ð18Þ
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where: ΦðξÞ =
cos½b0 ðξ−ξm Þ : ξ λξ −ξ
ð19Þ
The study 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. 3.2.1. Maximum of Φ(ξ) To find the maximum of function Φ(ξ), note that it will be necessarily located inside the interval [ξinn, ξm], as cos [b0(ξ − ξm)] increases in this interval and is symmetric with respect to ξ = ξm (where, of course, it reaches a maximum), and [ξ (λξ − ξ)]− 1 decreases in the interval [0, λξ / 2] and is symmetric with respect to ξ = λξ / 2 (where, of course, it reaches a minimum). Moreover, as λξ / 2 is greater than ξm, Φ(ξ) always decreases at ξ = ξm. Consequently, if a local extreme of Φ(ξ) exists in the interval [ξinn, ξm], it will be a local maximum, as shown in Fig. 5. To find it, the condition of the derivative equal to 0 yields the equation: tan½b0 ðξ−ξm Þ +
1 λξ −2ξ = 0: b0 ξ λ −ξ ξ
ð20Þ
Eq. (20) is a strongly non-linear equation which does not allow to obtain an analytic expression for the local maximum, ξmaxL. Nevertheless, a good approximation can be obtained by the following method: Eq. (20) may be written as: 2 3 λξ −2ξ 1 1 5: ξm = ξ + ð21Þ arctan4 b0 b0 ξ λ −ξ ξ
This equation allows to know the required value of ξm to locate the maximum at a given value of ξmax. Then, the required values of ξm to locate the maximum at ξinn and at ξm are, respectively, 2 3 λξ −2ξinn 1 1 5 ξm;inn = ξinn + arctan4 b0 b0 ξ λ −ξ 2
ξm;m
inn
ξ
inn
3 1 1 λξ −2ξm 5 4 : = ξm + arctan b0 b0 ξ λ −ξ m ξ m
ð22Þ
The local maximum of Φ(ξ), ξmaxL, can be estimated by assuming that ξinn, ξm and ξmaxL are in the same ratio than ξm,inn, ξm,m and ξm, so: ðξm −ξinn Þ ξm −ξm;inn : ð23Þ ξmaxL = ξinn + ξm;m −ξm;inn This local maximum will be the absolute maximum ξmaxR if two conditions are verified: (i) interpolation limits are not crossed, i.e., ξm,inn b ξm,m, and (ii) local maximum ξmaxL is located inside the interval [ξinn, ξm]. Otherwise the absolute maximum will be located at one of the interval limits, ξinn or ξm. Since conditions (i) and (ii) may be expressed as: ξm;inn ≤ξm ≤ξm;m
ð24Þ
Fig. 5. Typical shapes of the graphs of function Φ(ξ).
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431
the absolute maximum of Φ(ξ), ξmaxR, can be computed from:
If ξm;inn ≤ξm ≤ξm;m ⇒ξmaxR
if not ξm;inn ≤ξm ≤ξm;m ⇒
ðξm −ξinn Þ ξm −ξm;inn ; = ξinn + ξm;m −ξm;inn ! ξmaxR = ξinn if Φðξinn Þ N Φðξm Þ ξmaxR = ξm
if Φðξinn Þ b Φðξm Þ
ð25Þ :
The next section presents a complete study on the accuracy of Eq. (25) to compute the maximum of function Φ(ξ).
Fig. 6. Contact map for dα + dβ b 1 and dα N dβ.
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3.2.2. Minimum of Iv(ξ0) According to Eqs. (7) and (8), function Iv(ξ0) can be written as: z1 −1
Iv ðξ0 Þ = ∑
ξ0 + i
i = 0 ξ + i−ε 0 β
=
Eγ ξi;sup
∫ vðξÞdξ = ∑ ∫ cos ½b0 ðξ−ξm Þdξ i=0 ξ
ð26Þ
i;inf
E i h ii 1 γ h h ∑ sin b0 ξi;sup −ξm −sin b0 ξi;inf −ξm b0 i = 0
where Eγ is the integer part of the total contact ratio εγ = εα + εβ, and the integration limits are given by: ξi sup = ðξ0 + iÞ + ðξinn + εα Þ−minðξ0 + i;0Þ−maxðξ0 + i;ξinn + εα Þ ξi inf = ξ0 + i−εβ + ðξinn + εα Þ−min ξ0 + i−εβ ;0 −max ξ0 + i−εβ ;ξinn + εα :
ð27Þ
As shown in Fig. 3, both for dα + dβ greater or less than 1, there is an interval of ξ0 in which Iv(ξ0) takes quite uniform values, very close to the minimum one. These intervals are [ξinn + dα + dβ, ξinn + 1] for dα + dβ b 1, and [ξinn, ξinn + dα + dβ − 1] for dα + dβ N 1. We can consider Iv(ξ0) to be constant (and minimum) along the entire intervals, and Ivmin the value for ξ0 = ξinn, because the inner point of contact always belongs to the interval of minimum Iv, as seen in Fig. 3. Consequently: Iv min = Iv ðξinn Þ
ð28Þ
which can be computed with Eqs. (26) and (27). 3.2.3. Simultaneity condition It is clear that, if there is a contact point with ξ = ξmaxR during the interval of minimum Iv, we will have:
ΦðξÞ Φ ΦðξmaxR Þ : = max = Ivmin Iv ðξ0 Þ max Iv ðξinn Þ
ð29Þ
It is also clear that simultaneity is ensured if εβ N 1 or ξmaxR = ξinn. In fact, if εβ N 1 we can find a contact point with a given value of ξ (inside the interval of contact [ξinn, ξinn + εα]) at any position of the meshing cycle, and in particular during the interval of Ivmin. Simultaneity is also ensured if ξmaxR = ξinn because ξinn always belongs to the interval of minimum Iv. For the cases in which εβ b 1 and ξmaxR ≠ ξinn, simultaneity can be studied with the help of the contact map of the transmission. Fig. 6 shows the contact map of a helical pair with dα + dβ b 1 and dα N dβ. The horizontal axis represents the profile parameter of the contact point of the reference transverse section, ξ0. The vertical axis represents the profile parameters of the points of the line of contact, ξ. The points of the dotted zones with abscissa ξ0 represent the points of the line of contact at the meshing position described by ξ0. As shown in Fig. 6, for the case of dα + dβ b 1, simultaneity is ensured if ξmaxR is contained in the interval [ξinn + dα, ξm], because ξmaxR cannot be greater than ξm, as discussed above, and ξm, which is equal to ξinn + εα / 2, is always smaller than ξinn + 1. Similarly, Fig. 7 shows that, for the case of dα + dβ N 1, simultaneity is ensured if ξmaxR is contained in any of the intervals [ξinn, ξinn + dα + dβ − 1] or [ξinn + 1 − dβ, ξm], because ξinn + εα / 2 is greater than ξinn + 1 − εβ. Note that if 1 – dβ b dα + dβ − 1, simultaneity is ensured. More complicated is the case in which simultaneity is not given. The mathematical problem is not easy to solve, but an approximate solution may be suitable for standardization purposes or preliminary designs. In this sense, it is important to consider that for no-simultaneity cases, 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., |ξmaxR − ξ| small – Φ(ξ) is not excessively different from the maximum value Φ(ξmaxR). Then, we will be able 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, if Eq. (29) is also used for the case of no-simultaneity, induced error will be small and in the sense of safety. In conclusion, for helical gears the critical contact stress can be computed from: σH = ZE
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffivffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u F εβ cosβcosβb ucos½b0 ðξmaxR −ξm Þ 1 λξ t b πmn cosαt Iv ðξinn Þ λ −ξ ξ maxR
ξ
ð30Þ
maxR
regardless of simultaneity considerations. 4. Accuracy of the method The previous section stated that Eq. (25) is accurate enough for evaluating the maximum of function Φ(ξ), as well as Eq. (29) to evaluate the maximum of function Φ(ξ) / Iv(ξ0) for the cases in which simultaneity is not given. This section presents a complete study of the accuracy of both methods.
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ζ
ξ ξinn+εα
v/ρ(ξ)
433
ζinn+εα
ξinn+2-dmax
ξinn +1
ξ=ξm
ζinn +1
ξinn+dmax ξinn +dmin
ξinn+1-dmax
ξinn
ζ0
ζ =0 ζ =0
dα+dβ >1
ξinn
ξ=0
ξ0
ξ=0 Iv max
Iv (ξ 0)
Iv min
ξ0 ξinn
ξinn+dα+dβ -1
ξinn +dmin ξinn+dmax
ξinn+1
Fig. 7. Contact map for dα + dβ N 1 and dα b dβ.
4.1. Estimation of the maximum of Φ(ξ) Eq. (19) shows that only three parameters are required to define function Φ(ξ): ξinn, εα and λξ. The range of variation of each one for this study was: 1:25≤λξ ≤20:00 1:01≤εα ≤1:99 0:01≤ξinn ≤0:8λξ −
εα : 2
ð31Þ
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The upper limit of ξinn is obtained by accounting that the midpoint of the interval of contact should be located between the inner point of contact and the operating pitch point, so that: εα z1 ≤ λ 2 z1 + z2 ξ u ε λ − α ξinn ≤ 1+u ξ 2
ξm = ξinn +
ð32Þ
where u is the gear ratio, whose value rarely exceeds 8. An exhaustive study has been carried out considering 70 values of λξ, 70 values of ξinn and 70 values of εα, resulting in 343,000 different combinations. 69 of these cases were discarded because the limits of ξinn were inverted: 0:8λξ −
εα b0:01 2
ð33Þ
while 17,428 cases were discarded because they did not verify the compulsory condition: λξ N ξinn + εα :
ð34Þ
For the 325,503 remaining cases, the value of ξmax corresponding with the maximum of Φ(ξ) was computed by numerical techniques and compared with the value of ξmaxR, computed with Eq. (25). Deviation was represented by the relative shift factor δ, defined as: jξ −ξmax j 100: δð%Þ = 2 maxR εα
ð35Þ
Note that εα / 2 is the length of the interval in which ξmax is necessarily located. In 182,433 cases ξmax was located at one of the limits of this interval, ξinn or ξm; in all of them ξmaxR took the same value as ξmax, which means that Eq. (24) discriminates the maximum to be a local maximum or to be located at the limits on the interval, with a 100% of effectiveness. Obviously, in all these 182,433 cases the result δ = 0% was obtained. In the other 143,070 cases, with local maximum, typical values of δ oscillate between 1% and 8%, though in some particular cases, the relative shift factor reached values of 25% and higher. But these error levels, which are probably acceptable, are drastically reduced when expressed in terms of the contact stress. In fact, expressing the error in the estimation of function Φ(ξ) as: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ΦðξmaxR Þ− Φðξmax Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi eΦ ð%Þ = 100 Φðξmax Þ
ð36Þ
the average error level of the mentioned 143,070 cases was 0.037%, while the maximum one was 5%. Moreover, the highest levels were obtained for very small values of the transverse contact ratio, between 1.01 and 1.05, values which are not interesting for industrial applications. If we consider gear pairs with a transverse contact ratio greater than 1.1, the number of cases of local maximum decreases to 139,286, the average error level to 0.031%, and the maximum one to 2.4%. 4.2. Estimation of the maximum of Φ(ξ) / Iv(ξ0) With the assumption of (Φ / Iv)max = Φmax / Ivmin an error is obviously induced if the simultaneity condition discussed in Section 3.2.3 is not given. As presented in that section, simultaneity is ensured either for ξmaxR = ξinn or ξmaxR = ξm, consequently this no-simultaneity error can only be given if ξmaxR is a local maximum, just as in the previous case, in which an error in the estimation of the value of Φmax arose when ξmaxR was a local maximum. Therefore, if the maximum is a local maximum we will have two errors (one in the estimation of the maximum and other due to the no-simultaneity), but if the maximum is located at the inner point of contact or at the midpoint of the interval of contact, the method will be completely accurate (no errors). The effect of both errors on the final value of (Φ / Iv)max in the case of local maximum may be additive or subtractive, depending on the cases. But the important result is total error, which is what has been evaluated in this study, including consequently the error in the estimation of the maximum of Φ(ξ), considered in the previous subsection. A new variable has been included in the analysis: the axial contact ratio εβ, whose value varies between 0.050 and 0.995. The study considered 30 values of λξ, 30 values of ξinn, 30 values of εα and 20 values of εβ, resulting in 208,700 different combinations with local maximum. Cases of high error levels in the estimation of Φmax are not affected by this no-simultaneity error because these cases corresponded with small transverse contact ratios, very close to 1.1, for which dα is very small and simultaneity is almost guaranteed, as shown in Figs. 5 and 6. Nevertheless, higher errors can be found for a high transverse contact ratio, around 1.9, due to the no-simultaneity effect. The maximum error of the 208,700 cases studied was slightly greater than 5%, corresponding to εα = 1.875 and εβ = 0.1. Average errors increased a little to reach 0.8%.
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5. Comparison with the ISO rating method ISO 6336-2 [3] uses Eq. (12) to compute the contact stress, assuming the total load F to be evenly distributed along a virtual contact face width bvir, which is given by: bvir =
b Zε2
ð37Þ
where Zε is the contact ratio factor [3]. According to this, the ISO contact stress can be computed from: σH;ISO = ZE Zε
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi F 1 1 + : bcosβb ρ1 ρ2
ð38Þ
The curvature radii should be computed (i) for spur gears, at the inner point of single pair tooth contact or the operating pitch point, (ii) for helical gears with an axial contact ratio greater than 1, at the operating pitch point and (iii) for helical gears with an axial contact ratio between 0 and 1, at an interpolated value between two above. From the minimum elastic potential (MEP) load distribution presented herein, the contact stress can be computed with Eq. (12), considering the load per unit of length given by Eqs. (4) and (6). Iv should be computed at the inner point of contact, and the inverse unitary potential v and the curvature radii ρ1 and ρ2 at the point ξmaxR, as discussed in the previous section. For spur gears, 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 with (or the contact ratio factor) considered by ISO. Discrepancies depend on the values of the transverse contact ratio, but typically the ISO contact stress is around 10% smaller than the MEP contact stress. Only 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 a gear ratio greater than 3. For example, for Z1 = 21, Z2 = 40, αn = 25° and nominal outside radii and center distance, the MEP contact stress is 8.9% greater than the ISO contact stress, and both methods compute the critical contact stress at the inner point of single pair tooth contact. However, for Z1 = 18, Z2 = 55 and αn = 20°, the MEP contact stress is 34.8% greater than the ISO contact stress, and each method considers a different critical point (the inner point of single pair tooth contact ISO, and the inner point of contact MEP). For helical gears, the MEP contact stress is always smaller than the ISO contact stress. The MEP critical contact point – ξmaxR – is always closer to the inner point of contact than the ISO critical contact point — the operating pitch point, or the interpolated value if εβ b 1. Discrepancies are similar to those for spur gears: usually around 10%, up to 30% for special cases. To give an idea, let us consider the case above Z1 = 21 and Z2 = 40, combined with two normal pressure angles (20° and 25°) and three helix angles (15°, 20° and 25°). A face width to module ratio b/mn = 10 is considered to get εβ less than 1 for β = 15°, and greater than 1 for the other two helix angles. Discrepancies oscillate from 2.0% to 28.5%, and always the MEP contact stress is smaller. 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, whose value is determinant for evaluating the load capacity of the gear set. The study has been restricted to gears with a transverse contact ratio between 1 and 2, with non-undercut teeth. The results can be expressed as a recommendation to be considered for standardization purposes or preliminary calculations, which may be enunciated as follows: • For spur gears, the critical contact stress will arise at one of the following two points: (i) at the inner point of contact of the pinion, loaded with 33% of the total load, or (ii) at the inner point of single pair tooth contact of the pinion, loaded with 100% of the total load. • For 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 some contact position. It has been demonstrated that induced error is very small, and in the sense of safety, if coincidence does not occur. The location of the maximum of function Φ(ξ), ξmaxR, should 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 inner point of contact of the pinion or at the midpoint of the interval 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 has been checked, and according to the obtained results, is good enough for strength models. This makes the above recommendation to be suitable for preliminary design calculations or standardization purposes. Nomenclature b Face width, mm Virtual face width, mm bvir
436
dα dβ Eγ eΦ F f mn R rb rC u v ZE Zε z αn αt α′t β βb δ εα εβ εγ ρ ρt σH ξ
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Fractional part of εα Fractional part of εβ Integer part of εγ Error in the estimation of the maximum of function Φ Load, N Load per unit of length, N/mm Normal module, mm Load sharing ratio Base radius, mm Contact point radius, mm Gear ratio Inverse unitary potential, N/mm2 Elasticity factor (ISO 6336-2), MPa1/2 Contact ratio factor (ISO 6336-2) Number of teeth Standard normal pressure angle Transverse pressure angle Operating transverse pressure angle (pressure angle at the pitch cylinder) Standard helix angle Base helix angle Relative shift factor Transverse contact ratio Axial contact ratio Total contact ratio Curvature radius, mm Curvature radius at the transverse section, mm Contact stress, MPa Involute profile parameter
Acknowledgements Thanks are expressed to the Spanish Council for Scientific and Technological Research for the support of the project DPI200805787, “Calculation Models for Special Cylindrical Gears”. Special thanks to Raquel Martín for some keys for the approximate resolution of the maximization problem of function Φ. 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] AGMA Information Sheet 908-B89, Geometry Factors for Determining the Pitting Resistance and Bending Strength of Spur, Helical and Herringbone Gear Teeth, American Gear Manufacturers Association, Alexandria, VA, 1989. [3] ISO Standard 6336-2:2003, Calculation of Load Capacity of Spur and Helical Gears — Part 2: Calculation of Surface Durability (Pitting), International Organization for Standardization, Geneva, Switzerland, 2003. [4] ISO Standard 6336-1:2003, Calculation of Load Capacity of Spur and Helical Gears — Part 1: Basic Principles, Introduction and General Influence Factors, International Organization for Standardization, Geneva, Switzerland, 2003. [5] K. Hayashi, Load distribution on the contact line of helical gear teeth, JSME Bulletin 22 (1963). [6] H. Winter, T. Placzek, Load distribution and topological flank modification of helical and double helical gears, European Journal of Mechanical Engineering 36 (3) (1991). [7] Y. Zhang, Z. Fang, Analysis of tooth contact and load distribution of helical gears with crossed axes, Mechanism and Machine Theory 34 (1) (1999). [8] J. Boerner, Very efficient calculation of the load distribution on external gear sets — the method and applications of the program LVR, Proc. 7th International Power Transmission and Gearing Conference, San Diego, CA, 1996. [9] M. Ajmi, P. Velex, A model for simulating the quasi-static and dynamic behavior of solid wide-faced spur and helical gears, Mechanism and Machine Theory 40 (2005). [10] S. Li, Effect of addendum on contact strength, bending strength and basic performance parameters of a pair of spur gears, Mechanism and Machine Theory 43 (2008). [11] M.H. Arafa, M.M. Megahed, Evaluation of spur gear mesh compliance using the finite element method, Proceedings of the Institution of Mechanical Engineers, Part C 213 (1999). [12] M. Pimsarn, K. Kazerounian, Efficient evaluation of spur gear tooth mesh load using pseudo-interference stiffness estimation method, Mechanism and Machine Theory 37 (2002). [13] L. Vedmar, “On the Design of External Involute Helical Gears”, Ph. D. Thesis, Lund Technical University, Lund, Sweden, 1981. [14] J.I. Pedrero, M. Artés, A. Fuentes, Modelo de distribución de carga en engranajes cilíndricos de perfil de evolvente, Revista Iberoamericana de Ingeniería Mecánica 3 (1) (1999). [15] J.I. 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). [16] J.I. Pedrero, M. Estrems, A. Fuentes, Determination of the efficiency of cylindric gear sets, Proc. IV World Congress on Gearing and Power Transmissions, Paris, France, vol. 1, 1999. [17] M. Pleguezuelos, “Modelo de distribución de carga en engranajes cilíndricos de perfil de evolvente”, Ph.D. Thesis, UNED, Madrid, Spain, 2006.
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[18] J.I. Pedrero, I.I. Vallejo, Determinant bending load conditions on high transverse contact ratio spur and helical gear drives, Journal of Mechanical Engineering Design 9 (1) (2006). [19] 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). [20] 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).
Contents lists available at ScienceDirect
Mechanism and Machine Theory j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / m e c h m t
Critical stress and load conditions for pitting calculations of involute spur and helical gear teeth José I. Pedrero a,⁎, Miguel Pleguezuelos a, Marta Muñoz b a b
UNED, Departamento de Mecánica, Juan del Rosal 12, 28040 Madrid, Spain UNED, Departamento de Ingeniería Energética, Juan del Rosal 12, 28040 Madrid, Spain
a r t i c l e
i n f o
Article history: Received 25 January 2010 Received in revised form 7 September 2010 Accepted 1 December 2010 Available online 5 January 2011 Keywords: Cylindrical gears Contact pressure Load distribution Pitting resistance
a b s t r a c t Calculation methods of spur and helical gear drives for preliminary designs or standardization purposes available in technical literature, use the Hertz equation to evaluate the contact stress, assuming the load to be uniformly distributed along the line of contact. However, this model presents some discrepancies with experimental results because the changing rigidity of the pair of teeth along the path of contact produces a non-uniform load distribution, which implies that some load distribution factors are required to compute the contact stress. In this paper, a non-uniform model of load distribution along the line of contact, recently developed, obtained from the minimum elastic potential criterion, has been used. This model combined with the Hertz equation yields more accurate values of the contact stress. As the load per unit of length at any point of the line of contact and any position of the meshing cycle has been described by a very simple analytic equation, a complete study of the location and value of the critical contact stress has been carried out. From this study, a recommendation for the calculation of the pitting load capacity of spur and helical gears is proposed. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction To compute the pitting load capacity of spur and helical involute gears, the most widely used standards as AGMA [1,2] or ISO [3] employ the Hertz equation to evaluate the contact pressure. The load is assumed to be uniformly distributed along the line of contact, though it is known that the load distribution depends on the meshing stiffness of the pair of teeth, which is different at any contact point. This means that the load per unit of length is also different at any point of the line of contact, and therefore several influence factors for load distribution are required to correct the calculated values of the bending and contact stresses [1,4]. Some studies on the load distribution along the line of contact can be found in technical literature [5–13], but all of them provide results obtained by numerical techniques or the finite element method (FEM). Those methods allow to obtain some conclusions regarding the considered gear pair, but make it very difficult to extract general conclusions, valid for any gear pair. In previous works [14–19], the authors obtained a new load distribution model from the minimum elastic potential energy criterion. The elastic potential energy of a pair of teeth was calculated and expressed as a function of the contact point and the normal load. The load sharing among several pairs of contacting teeth in spur gears was obtained by solving the variational problem of minimizing the total potential energy (equal to the addition of the potential energy of each pair at its respective contact point) taking into account the restriction of the total load to be equal to the sum of the load at each pair. The same approach was used for helical gear teeth by dividing each helical pair in infinite slices, perpendicular to the gear axis, assuming each slice to be equivalent to a spur gear with differential face width, and extending the integrals to the complete line of contact. This approach allowed the value of the load per unit of length at any point of the line of contact and at any position of the meshing cycle to be known. Through this model, some preliminary studies on the location of the points of critical stress and the determinant load ⁎ Corresponding author. Tel.: + 34 913986430; fax: + 34 913986536. E-mail address: [email protected] (J.I. Pedrero). 0094-114X/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmachtheory.2010.12.001
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conditions were carried out for bending and wear, resulting in good agreement with experimental data [15,17]. However, the numerical method used for integration of the elastic potential energy, which yields numerical values of the load per unit of length at discrete contact points and meshing positions, presented the same problem of no generality of the obtained results. Recently, the authors [20] presented an approximate analytic equation for the inverse unitary potential, 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 now has an analytic expression. This will make possible general conclusions about the critical load conditions to be obtained, and consequently to study the load capacity, efficiency, etc., allowing to make proposals for calculation methods for preliminary designs or standardization purposes. In this paper, a complete study of the location and value of the critical contact stress on involute spur and helical gear teeth with a transverse contact ratio between 1 and 2 has been carried out, and according to this a proposal for pitting calculations is established. 2. Load distribution model Reference [20] presents in detail the model of load distribution of minimum elastic potential energy. In general terms, that model was obtained by computing the total elastic potential energy from the equations of the theory of elasticity, considering all 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). It has been demonstrated that the load per unit of length depends on the inverse unitary potential v(ξ), which is defined as the inverse of the elastic potential for unitary load and face width. Obviously, the inverse unitary potential depends on the contact point, which is described by the ξ parameter of the contact point at the pinion profile as: z ξ= 1 2π
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 rC1 −1 2 rb1
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). This parameter ξ has a physical meaning: it is the ratio of the curvature radius at the load point C and the circular base pitch; it is also the ratio between the rolling angle of point C and the angular pitch 2π/z. For spur gears, the load on each pair of teeth in simultaneous contact can be expressed as a function of the inverse unitary potential as [20]: Fi ðξi Þ =
vi ðξi Þ z1 −1
∑ vj ξ j
ð2Þ
F
j=0
where Fi(ξi) and vi(ξi) are the load and the inverse unitary potential of tooth 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) would be: Ri ðξi Þ =
Fi ðξi Þ = F
vi ðξi Þ z1 −1
∑ vj ξ j
ð3Þ
j=0
and the load per unit of length f, for spur gears, can be expressed as:
fi ðξi Þ =
F R ðξ Þ b i i
ð4Þ
being b the effective face width. Note that, at a given meshing position, the parameters ξ corresponding to teeth i and j are related by: ξj = ξi + ðj−iÞ:
ð5Þ
Similarly, for helical gears, the load per unit of length at a point of the line of contact described by ξ, at the meshing position corresponding to a reference transverse section contacting at a point described by ξ0, is given by [20]: f ðξ;ξ0 Þ =
εβ cosβb vðξÞ F b Iv ðξ0 Þ
ð6Þ
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Fig. 1. Typical shape of the graph of the inverse unitary potential v(ξ).
where εβ is the axial contact ratio, βb the base helix angle and function Iv(ξ0) can be derived from: z1 −1
Iv ðξ0 Þ = ∫ vðξÞdξ = ∑ lc
ξ0 + j
∫ vðξÞdξ:
ð7Þ
j = 0 ξ + j−ε 0 β
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 a higher contact point. The inverse unitary potential v(ξ) is described very accurately by [20]: vðξÞ = cos½b0 ðξ−ξm Þ
ð8Þ
where: ε ξm = ξinn + α 2 −1 = 2 1 ε 2 1 + α −1 b0 = 2 2
ð9Þ
being ξinn the involute parameter of the inner point of contact of the pinion and εα the transverse contact ratio. Fig. 1 shows the typical aspect of function v(ξ) for non-undercut teeth. According to this, the load sharing ratio for spur gears can be obtained by replacing Eq. (8) in Eq. (3) and accounting Eq. (5), which yields the following result for the transverse contact ratio between 1 and 2: RðξÞ =
vðξÞ vðξ−1Þ + vðξÞ + vðξ + 1Þ
ð10Þ
which has been represented in Fig. 2. The ordinates of singular points are always very close to 0.33 and 0.67, so that the load sharing ratio given by Eq. (10) can be also computed from: RðξÞ =
1 ξ−ξinn 1+ 3 εα −1
RðξÞ = 1
1 ξ + εα −ξ 1 + inn RðξÞ = 3 εa −1
for
ξinn ≤ξ≤ξinn + εα −1;
for
ξinn + εα −1≤ξ ≤ξinn + 1; and
for
ξinn + 1≤ξ≤ξinn + εα :
ð11Þ
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. The shape of function Iv(ξ0) is similar to the shape of the evolution of the length of contact along the mesh cycle. 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, function Iv(ξ0) takes different shapes, as represented in Fig. 3. Note that the influence of dα + dβ is the same as that on the length of contact. Since Iv(ξ0) is the integral of v(ξ) along the line of contact, Iv(ξ0) increases in the intervals in which the length of contact increases, decreases in the intervals in which the length of contact decreases, and is more or less constant in the intervals in which the length of contact is constant.
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Fig. 2. Load sharing ratio for spur gears with a transverse contact ratio between 1 and 2.
Profile modifications as tip relief are not explicitly taken into account in this model of load distribution. This is due to that the integrals for the elastic potential energy –Eq. (1) in [20] – do not change significantly so the inverse unitary potential v(ξ), and consequently the load sharing ratio R(ξ), are not affected. Moreover, profile modifications are many times calculated to absorb misalignments or manufacturing deviations in order to ensure a quasi-conjugate contact between meshing profiles. Neither ISO 6336-2 [3] nor AGMA 908-B89 [2] considers profile modifications in their equations for contact stress calculations (all of them are obtained from the equations of the involute); however these equations are valid for modified profiles, for this reason. 3. Contact stress For the contact between involute gear teeth, Hertz's equation may be written as: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f 1 1 σH = ZE + cosβb ρ1 ρ2
ð12Þ
where ZE is the elasticity factor [3], f the load per unit of length of the line of contact and ρ the curvature radius of the section perpendicular to the line of contact. This section forms with the transverse section an angle equal to the base helix angle βb, so according to Euler's theorem and taking into account that the curvature radius of the transverse section at the contact point is equal to the distance between the contact point and the tangency point of the pressure line and the base circumference, we have: ρt1 πmn cosαt = ξ cosβb cosβcosβb ρ πmn cosαt z1 + z2 tanα′t −ξ ρ2 = t2 = cosβb cosβcosβb 2π
ρ1 =
ð13Þ
being ρt the curvature radius of the transverse section at the contact point, mn the normal module, αt the transverse pressure angle, β the standard helix angle and α′t the operating transverse pressure angle. The goal is to find the critical contact stress and load conditions, i.e., the value of the maximum contact stress and the location of the contact point where this maximum arises.
Fig. 3. Typical shapes of the graphs of function Iv(ξ0), the integral of the inverse unitary potential.
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Fig. 4. Evolution of the contact stress along the line of action of spur gears.
3.1. Critical stress and load conditions for spur gears According to Eqs. (12), (4) and (13), the critical contact stress for spur gears will be given by the maximum of function: RðξÞ ξ λξ −ξ
ð14Þ
where: λξ =
z1 + z2 tanα′t : 2π
ð15Þ
From Fig. 2, R(ξ) increases in the interval [ξinn, ξinn + εα − 1]. On the other hand, function [ξ (λξ − ξ)]− 1 decreases in the interval [0, λξ / 2] and increases in [λξ / 2, λξ]. Finally, considering that λξ / 2 is always greater than ξinn + εα / 2 (the midpoint of the interval of contact) we conclude that the point of maximum contact stress will be necessarily located inside the interval [ξinn, ξinn + εα − 1]. It can be also demonstrated that if function [R(ξ) / ξ (λξ − ξ)] has a point with a horizontal tangent inside this interval, this local extreme is always a minimum. Consequently, the critical contact stress will be necessarily located at one of the interval limits: either at ξ = ξinn for which, according to Eq. (11), R(ξ) = 0.33, or at ξ = ξinn + εα − 1 for which R(ξ) = 1. Depending on the transmission geometry, the critical contact point may be the inner limit of the interval of contact or the inner point of single pair tooth contact, as shown in Fig. 4. The easiest way to know which limit is critical is to compute the contact stress at both points, and select the highest one. In conclusion, the nominal contact stress for spur gears will be given by:
σH0
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv u u λ F 0:33 1 u ξ @ ; A = ZE tmax b πmn cosαn λ −ξ ðξ + ε −1Þ λ −ðξ + ε −1Þ ξ ξ
inn
inn
inn
α
ξ
inn
ð16Þ
α
where αn is the standard normal pressure angle. Usually, contact stress will be critical at the inner point of single pair tooth contact, but the critical contact point will be the inner point of contact for small values of ξinn, as in Fig. 4 (left). In these cases, contact may occur near to the base circle which could be not a recommended design. These conditions are not usual but can be given for a small number of teeth on pinion and high gear ratio. Note that ISO 6336-2 [3] predicts the critical stress at the inner point of contact in more cases, as it considers a load sharing ratio equal to 0.5 instead of 0.33. 3.2. Critical stress and load conditions for helical gears For helical gears, Eq. (12) can be written as: σH = ZE
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffivffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u F εβ cosβcosβb ucos½b0 ðξ−ξm Þ 1 : λξ t b πmn cosαt Iv ðξ0 Þ ξ λ −ξ
ð17Þ
ξ
The critical load conditions will correspond to the values of ξ and ξ0 for which is the maximum of the function: ΦðξÞ Iv ðξ0 Þ
ð18Þ
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where: ΦðξÞ =
cos½b0 ðξ−ξm Þ : ξ λξ −ξ
ð19Þ
The study 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. 3.2.1. Maximum of Φ(ξ) To find the maximum of function Φ(ξ), note that it will be necessarily located inside the interval [ξinn, ξm], as cos [b0(ξ − ξm)] increases in this interval and is symmetric with respect to ξ = ξm (where, of course, it reaches a maximum), and [ξ (λξ − ξ)]− 1 decreases in the interval [0, λξ / 2] and is symmetric with respect to ξ = λξ / 2 (where, of course, it reaches a minimum). Moreover, as λξ / 2 is greater than ξm, Φ(ξ) always decreases at ξ = ξm. Consequently, if a local extreme of Φ(ξ) exists in the interval [ξinn, ξm], it will be a local maximum, as shown in Fig. 5. To find it, the condition of the derivative equal to 0 yields the equation: tan½b0 ðξ−ξm Þ +
1 λξ −2ξ = 0: b0 ξ λ −ξ ξ
ð20Þ
Eq. (20) is a strongly non-linear equation which does not allow to obtain an analytic expression for the local maximum, ξmaxL. Nevertheless, a good approximation can be obtained by the following method: Eq. (20) may be written as: 2 3 λξ −2ξ 1 1 5: ξm = ξ + ð21Þ arctan4 b0 b0 ξ λ −ξ ξ
This equation allows to know the required value of ξm to locate the maximum at a given value of ξmax. Then, the required values of ξm to locate the maximum at ξinn and at ξm are, respectively, 2 3 λξ −2ξinn 1 1 5 ξm;inn = ξinn + arctan4 b0 b0 ξ λ −ξ 2
ξm;m
inn
ξ
inn
3 1 1 λξ −2ξm 5 4 : = ξm + arctan b0 b0 ξ λ −ξ m ξ m
ð22Þ
The local maximum of Φ(ξ), ξmaxL, can be estimated by assuming that ξinn, ξm and ξmaxL are in the same ratio than ξm,inn, ξm,m and ξm, so: ðξm −ξinn Þ ξm −ξm;inn : ð23Þ ξmaxL = ξinn + ξm;m −ξm;inn This local maximum will be the absolute maximum ξmaxR if two conditions are verified: (i) interpolation limits are not crossed, i.e., ξm,inn b ξm,m, and (ii) local maximum ξmaxL is located inside the interval [ξinn, ξm]. Otherwise the absolute maximum will be located at one of the interval limits, ξinn or ξm. Since conditions (i) and (ii) may be expressed as: ξm;inn ≤ξm ≤ξm;m
ð24Þ
Fig. 5. Typical shapes of the graphs of function Φ(ξ).
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the absolute maximum of Φ(ξ), ξmaxR, can be computed from:
If ξm;inn ≤ξm ≤ξm;m ⇒ξmaxR
if not ξm;inn ≤ξm ≤ξm;m ⇒
ðξm −ξinn Þ ξm −ξm;inn ; = ξinn + ξm;m −ξm;inn ! ξmaxR = ξinn if Φðξinn Þ N Φðξm Þ ξmaxR = ξm
if Φðξinn Þ b Φðξm Þ
ð25Þ :
The next section presents a complete study on the accuracy of Eq. (25) to compute the maximum of function Φ(ξ).
Fig. 6. Contact map for dα + dβ b 1 and dα N dβ.
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3.2.2. Minimum of Iv(ξ0) According to Eqs. (7) and (8), function Iv(ξ0) can be written as: z1 −1
Iv ðξ0 Þ = ∑
ξ0 + i
i = 0 ξ + i−ε 0 β
=
Eγ ξi;sup
∫ vðξÞdξ = ∑ ∫ cos ½b0 ðξ−ξm Þdξ i=0 ξ
ð26Þ
i;inf
E i h ii 1 γ h h ∑ sin b0 ξi;sup −ξm −sin b0 ξi;inf −ξm b0 i = 0
where Eγ is the integer part of the total contact ratio εγ = εα + εβ, and the integration limits are given by: ξi sup = ðξ0 + iÞ + ðξinn + εα Þ−minðξ0 + i;0Þ−maxðξ0 + i;ξinn + εα Þ ξi inf = ξ0 + i−εβ + ðξinn + εα Þ−min ξ0 + i−εβ ;0 −max ξ0 + i−εβ ;ξinn + εα :
ð27Þ
As shown in Fig. 3, both for dα + dβ greater or less than 1, there is an interval of ξ0 in which Iv(ξ0) takes quite uniform values, very close to the minimum one. These intervals are [ξinn + dα + dβ, ξinn + 1] for dα + dβ b 1, and [ξinn, ξinn + dα + dβ − 1] for dα + dβ N 1. We can consider Iv(ξ0) to be constant (and minimum) along the entire intervals, and Ivmin the value for ξ0 = ξinn, because the inner point of contact always belongs to the interval of minimum Iv, as seen in Fig. 3. Consequently: Iv min = Iv ðξinn Þ
ð28Þ
which can be computed with Eqs. (26) and (27). 3.2.3. Simultaneity condition It is clear that, if there is a contact point with ξ = ξmaxR during the interval of minimum Iv, we will have:
ΦðξÞ Φ ΦðξmaxR Þ : = max = Ivmin Iv ðξ0 Þ max Iv ðξinn Þ
ð29Þ
It is also clear that simultaneity is ensured if εβ N 1 or ξmaxR = ξinn. In fact, if εβ N 1 we can find a contact point with a given value of ξ (inside the interval of contact [ξinn, ξinn + εα]) at any position of the meshing cycle, and in particular during the interval of Ivmin. Simultaneity is also ensured if ξmaxR = ξinn because ξinn always belongs to the interval of minimum Iv. For the cases in which εβ b 1 and ξmaxR ≠ ξinn, simultaneity can be studied with the help of the contact map of the transmission. Fig. 6 shows the contact map of a helical pair with dα + dβ b 1 and dα N dβ. The horizontal axis represents the profile parameter of the contact point of the reference transverse section, ξ0. The vertical axis represents the profile parameters of the points of the line of contact, ξ. The points of the dotted zones with abscissa ξ0 represent the points of the line of contact at the meshing position described by ξ0. As shown in Fig. 6, for the case of dα + dβ b 1, simultaneity is ensured if ξmaxR is contained in the interval [ξinn + dα, ξm], because ξmaxR cannot be greater than ξm, as discussed above, and ξm, which is equal to ξinn + εα / 2, is always smaller than ξinn + 1. Similarly, Fig. 7 shows that, for the case of dα + dβ N 1, simultaneity is ensured if ξmaxR is contained in any of the intervals [ξinn, ξinn + dα + dβ − 1] or [ξinn + 1 − dβ, ξm], because ξinn + εα / 2 is greater than ξinn + 1 − εβ. Note that if 1 – dβ b dα + dβ − 1, simultaneity is ensured. More complicated is the case in which simultaneity is not given. The mathematical problem is not easy to solve, but an approximate solution may be suitable for standardization purposes or preliminary designs. In this sense, it is important to consider that for no-simultaneity cases, 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., |ξmaxR − ξ| small – Φ(ξ) is not excessively different from the maximum value Φ(ξmaxR). Then, we will be able 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, if Eq. (29) is also used for the case of no-simultaneity, induced error will be small and in the sense of safety. In conclusion, for helical gears the critical contact stress can be computed from: σH = ZE
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffivffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u F εβ cosβcosβb ucos½b0 ðξmaxR −ξm Þ 1 λξ t b πmn cosαt Iv ðξinn Þ λ −ξ ξ maxR
ξ
ð30Þ
maxR
regardless of simultaneity considerations. 4. Accuracy of the method The previous section stated that Eq. (25) is accurate enough for evaluating the maximum of function Φ(ξ), as well as Eq. (29) to evaluate the maximum of function Φ(ξ) / Iv(ξ0) for the cases in which simultaneity is not given. This section presents a complete study of the accuracy of both methods.
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ζ
ξ ξinn+εα
v/ρ(ξ)
433
ζinn+εα
ξinn+2-dmax
ξinn +1
ξ=ξm
ζinn +1
ξinn+dmax ξinn +dmin
ξinn+1-dmax
ξinn
ζ0
ζ =0 ζ =0
dα+dβ >1
ξinn
ξ=0
ξ0
ξ=0 Iv max
Iv (ξ 0)
Iv min
ξ0 ξinn
ξinn+dα+dβ -1
ξinn +dmin ξinn+dmax
ξinn+1
Fig. 7. Contact map for dα + dβ N 1 and dα b dβ.
4.1. Estimation of the maximum of Φ(ξ) Eq. (19) shows that only three parameters are required to define function Φ(ξ): ξinn, εα and λξ. The range of variation of each one for this study was: 1:25≤λξ ≤20:00 1:01≤εα ≤1:99 0:01≤ξinn ≤0:8λξ −
εα : 2
ð31Þ
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The upper limit of ξinn is obtained by accounting that the midpoint of the interval of contact should be located between the inner point of contact and the operating pitch point, so that: εα z1 ≤ λ 2 z1 + z2 ξ u ε λ − α ξinn ≤ 1+u ξ 2
ξm = ξinn +
ð32Þ
where u is the gear ratio, whose value rarely exceeds 8. An exhaustive study has been carried out considering 70 values of λξ, 70 values of ξinn and 70 values of εα, resulting in 343,000 different combinations. 69 of these cases were discarded because the limits of ξinn were inverted: 0:8λξ −
εα b0:01 2
ð33Þ
while 17,428 cases were discarded because they did not verify the compulsory condition: λξ N ξinn + εα :
ð34Þ
For the 325,503 remaining cases, the value of ξmax corresponding with the maximum of Φ(ξ) was computed by numerical techniques and compared with the value of ξmaxR, computed with Eq. (25). Deviation was represented by the relative shift factor δ, defined as: jξ −ξmax j 100: δð%Þ = 2 maxR εα
ð35Þ
Note that εα / 2 is the length of the interval in which ξmax is necessarily located. In 182,433 cases ξmax was located at one of the limits of this interval, ξinn or ξm; in all of them ξmaxR took the same value as ξmax, which means that Eq. (24) discriminates the maximum to be a local maximum or to be located at the limits on the interval, with a 100% of effectiveness. Obviously, in all these 182,433 cases the result δ = 0% was obtained. In the other 143,070 cases, with local maximum, typical values of δ oscillate between 1% and 8%, though in some particular cases, the relative shift factor reached values of 25% and higher. But these error levels, which are probably acceptable, are drastically reduced when expressed in terms of the contact stress. In fact, expressing the error in the estimation of function Φ(ξ) as: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ΦðξmaxR Þ− Φðξmax Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi eΦ ð%Þ = 100 Φðξmax Þ
ð36Þ
the average error level of the mentioned 143,070 cases was 0.037%, while the maximum one was 5%. Moreover, the highest levels were obtained for very small values of the transverse contact ratio, between 1.01 and 1.05, values which are not interesting for industrial applications. If we consider gear pairs with a transverse contact ratio greater than 1.1, the number of cases of local maximum decreases to 139,286, the average error level to 0.031%, and the maximum one to 2.4%. 4.2. Estimation of the maximum of Φ(ξ) / Iv(ξ0) With the assumption of (Φ / Iv)max = Φmax / Ivmin an error is obviously induced if the simultaneity condition discussed in Section 3.2.3 is not given. As presented in that section, simultaneity is ensured either for ξmaxR = ξinn or ξmaxR = ξm, consequently this no-simultaneity error can only be given if ξmaxR is a local maximum, just as in the previous case, in which an error in the estimation of the value of Φmax arose when ξmaxR was a local maximum. Therefore, if the maximum is a local maximum we will have two errors (one in the estimation of the maximum and other due to the no-simultaneity), but if the maximum is located at the inner point of contact or at the midpoint of the interval of contact, the method will be completely accurate (no errors). The effect of both errors on the final value of (Φ / Iv)max in the case of local maximum may be additive or subtractive, depending on the cases. But the important result is total error, which is what has been evaluated in this study, including consequently the error in the estimation of the maximum of Φ(ξ), considered in the previous subsection. A new variable has been included in the analysis: the axial contact ratio εβ, whose value varies between 0.050 and 0.995. The study considered 30 values of λξ, 30 values of ξinn, 30 values of εα and 20 values of εβ, resulting in 208,700 different combinations with local maximum. Cases of high error levels in the estimation of Φmax are not affected by this no-simultaneity error because these cases corresponded with small transverse contact ratios, very close to 1.1, for which dα is very small and simultaneity is almost guaranteed, as shown in Figs. 5 and 6. Nevertheless, higher errors can be found for a high transverse contact ratio, around 1.9, due to the no-simultaneity effect. The maximum error of the 208,700 cases studied was slightly greater than 5%, corresponding to εα = 1.875 and εβ = 0.1. Average errors increased a little to reach 0.8%.
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5. Comparison with the ISO rating method ISO 6336-2 [3] uses Eq. (12) to compute the contact stress, assuming the total load F to be evenly distributed along a virtual contact face width bvir, which is given by: bvir =
b Zε2
ð37Þ
where Zε is the contact ratio factor [3]. According to this, the ISO contact stress can be computed from: σH;ISO = ZE Zε
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi F 1 1 + : bcosβb ρ1 ρ2
ð38Þ
The curvature radii should be computed (i) for spur gears, at the inner point of single pair tooth contact or the operating pitch point, (ii) for helical gears with an axial contact ratio greater than 1, at the operating pitch point and (iii) for helical gears with an axial contact ratio between 0 and 1, at an interpolated value between two above. From the minimum elastic potential (MEP) load distribution presented herein, the contact stress can be computed with Eq. (12), considering the load per unit of length given by Eqs. (4) and (6). Iv should be computed at the inner point of contact, and the inverse unitary potential v and the curvature radii ρ1 and ρ2 at the point ξmaxR, as discussed in the previous section. For spur gears, 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 with (or the contact ratio factor) considered by ISO. Discrepancies depend on the values of the transverse contact ratio, but typically the ISO contact stress is around 10% smaller than the MEP contact stress. Only 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 a gear ratio greater than 3. For example, for Z1 = 21, Z2 = 40, αn = 25° and nominal outside radii and center distance, the MEP contact stress is 8.9% greater than the ISO contact stress, and both methods compute the critical contact stress at the inner point of single pair tooth contact. However, for Z1 = 18, Z2 = 55 and αn = 20°, the MEP contact stress is 34.8% greater than the ISO contact stress, and each method considers a different critical point (the inner point of single pair tooth contact ISO, and the inner point of contact MEP). For helical gears, the MEP contact stress is always smaller than the ISO contact stress. The MEP critical contact point – ξmaxR – is always closer to the inner point of contact than the ISO critical contact point — the operating pitch point, or the interpolated value if εβ b 1. Discrepancies are similar to those for spur gears: usually around 10%, up to 30% for special cases. To give an idea, let us consider the case above Z1 = 21 and Z2 = 40, combined with two normal pressure angles (20° and 25°) and three helix angles (15°, 20° and 25°). A face width to module ratio b/mn = 10 is considered to get εβ less than 1 for β = 15°, and greater than 1 for the other two helix angles. Discrepancies oscillate from 2.0% to 28.5%, and always the MEP contact stress is smaller. 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, whose value is determinant for evaluating the load capacity of the gear set. The study has been restricted to gears with a transverse contact ratio between 1 and 2, with non-undercut teeth. The results can be expressed as a recommendation to be considered for standardization purposes or preliminary calculations, which may be enunciated as follows: • For spur gears, the critical contact stress will arise at one of the following two points: (i) at the inner point of contact of the pinion, loaded with 33% of the total load, or (ii) at the inner point of single pair tooth contact of the pinion, loaded with 100% of the total load. • For 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 some contact position. It has been demonstrated that induced error is very small, and in the sense of safety, if coincidence does not occur. The location of the maximum of function Φ(ξ), ξmaxR, should 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 inner point of contact of the pinion or at the midpoint of the interval 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 has been checked, and according to the obtained results, is good enough for strength models. This makes the above recommendation to be suitable for preliminary design calculations or standardization purposes. Nomenclature b Face width, mm Virtual face width, mm bvir
436
dα dβ Eγ eΦ F f mn R rb rC u v ZE Zε z αn αt α′t β βb δ εα εβ εγ ρ ρt σH ξ
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Fractional part of εα Fractional part of εβ Integer part of εγ Error in the estimation of the maximum of function Φ Load, N Load per unit of length, N/mm Normal module, mm Load sharing ratio Base radius, mm Contact point radius, mm Gear ratio Inverse unitary potential, N/mm2 Elasticity factor (ISO 6336-2), MPa1/2 Contact ratio factor (ISO 6336-2) Number of teeth Standard normal pressure angle Transverse pressure angle Operating transverse pressure angle (pressure angle at the pitch cylinder) Standard helix angle Base helix angle Relative shift factor Transverse contact ratio Axial contact ratio Total contact ratio Curvature radius, mm Curvature radius at the transverse section, mm Contact stress, MPa Involute profile parameter
Acknowledgements Thanks are expressed to the Spanish Council for Scientific and Technological Research for the support of the project DPI200805787, “Calculation Models for Special Cylindrical Gears”. Special thanks to Raquel Martín for some keys for the approximate resolution of the maximization problem of function Φ. 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] AGMA Information Sheet 908-B89, Geometry Factors for Determining the Pitting Resistance and Bending Strength of Spur, Helical and Herringbone Gear Teeth, American Gear Manufacturers Association, Alexandria, VA, 1989. [3] ISO Standard 6336-2:2003, Calculation of Load Capacity of Spur and Helical Gears — Part 2: Calculation of Surface Durability (Pitting), International Organization for Standardization, Geneva, Switzerland, 2003. [4] ISO Standard 6336-1:2003, Calculation of Load Capacity of Spur and Helical Gears — Part 1: Basic Principles, Introduction and General Influence Factors, International Organization for Standardization, Geneva, Switzerland, 2003. [5] K. Hayashi, Load distribution on the contact line of helical gear teeth, JSME Bulletin 22 (1963). [6] H. Winter, T. Placzek, Load distribution and topological flank modification of helical and double helical gears, European Journal of Mechanical Engineering 36 (3) (1991). [7] Y. Zhang, Z. Fang, Analysis of tooth contact and load distribution of helical gears with crossed axes, Mechanism and Machine Theory 34 (1) (1999). [8] J. Boerner, Very efficient calculation of the load distribution on external gear sets — the method and applications of the program LVR, Proc. 7th International Power Transmission and Gearing Conference, San Diego, CA, 1996. [9] M. Ajmi, P. Velex, A model for simulating the quasi-static and dynamic behavior of solid wide-faced spur and helical gears, Mechanism and Machine Theory 40 (2005). [10] S. Li, Effect of addendum on contact strength, bending strength and basic performance parameters of a pair of spur gears, Mechanism and Machine Theory 43 (2008). [11] M.H. Arafa, M.M. Megahed, Evaluation of spur gear mesh compliance using the finite element method, Proceedings of the Institution of Mechanical Engineers, Part C 213 (1999). [12] M. Pimsarn, K. Kazerounian, Efficient evaluation of spur gear tooth mesh load using pseudo-interference stiffness estimation method, Mechanism and Machine Theory 37 (2002). [13] L. Vedmar, “On the Design of External Involute Helical Gears”, Ph. D. Thesis, Lund Technical University, Lund, Sweden, 1981. [14] J.I. Pedrero, M. Artés, A. Fuentes, Modelo de distribución de carga en engranajes cilíndricos de perfil de evolvente, Revista Iberoamericana de Ingeniería Mecánica 3 (1) (1999). [15] J.I. 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). [16] J.I. Pedrero, M. Estrems, A. Fuentes, Determination of the efficiency of cylindric gear sets, Proc. IV World Congress on Gearing and Power Transmissions, Paris, France, vol. 1, 1999. [17] M. Pleguezuelos, “Modelo de distribución de carga en engranajes cilíndricos de perfil de evolvente”, Ph.D. Thesis, UNED, Madrid, Spain, 2006.
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