Analysis of the tip interference in low gear ratio internal spur gears with profile modification

This paper presents a discussion on the influence of the tip relief on the tip interference and contact ratio, as well as a methodology to select proper values of the teeth heights, center distances, rack shift coefficients, and tip reliefs to ensure maximum values of the contact ratio while avoiding tip interference, root interference and pitch interference. The influence of the quasi-static transmission error on the tip interference is also considered in the analysis.


Introduction
Internal gear pairs are employed to transmit motion between two parallel shafts rotating in the same direction.An internal spur gear pair is made up of a cylindrical ring gear (also known as a ring gear, annulus gear, or simply an annulus) with internal teeth and a cylindrical gear with external teeth.
External and internal gears have very similar geometry and design methods.To ensure effective tooth action in internal gears, interferences must be avoided, just like with external gears, but the internal gear drives have several additional limitations, besides all those that apply to external gears.
Deep studies in design and geometry for external spur gears have been conducted over the past few decades, taking all potential interferences into account [1][2][3][4].Therefore, the same concepts described for external spur gears can be applied to internal spur gears and add some specific design requirements [5][6][7][8].
The first one is the theoretical interference, primary interference, involute interference, or root interference.In internal spur gears is equivalent to that in external spur gears.In addition to the primary interference between the active flank profiles, in the situation where the pinion outside diameter is quite close to that of the annulus, a very specific interference, known as secondary interference or tip interference, can also occur in the internal spur gears.In fact, the tips of the pinion and ring teeth can interfere outside the line of action, causing the secondary interference.This may occur when the number of teeth of pinion and ring are relatively similar.
The design conditions frequently consider strict limits on the pinion tooth number in order to reduce the size of gear drives for mechanical power transmission systems and, consequently, their weight and cost.In these situations, the internal gear pair design decisions must account for the requirement to eliminate the dual risk of primary and secondary interferences.
Other recent studies have focused on interference problems that can appear in modern machining and generation processes [9][10][11][12].
This paper presents an analysis of the influence of the tip relief on the tip interference and contact ratio.A methodology to select proper values of the teeth heights, center distances, shift coefficients, and tip reliefs to ensure maximum values of the contact ratio while avoiding tip interference, is presented.Internal spur teeth generated by pinion cutter have been considered.Teeth pointing, root interference, and pitch interference restrictions have been regarded.The influence of the quasi-static transmission error on the tip interference is also considered in the analysis.Although the geometrical calculation of the tip interference is included in modern gear calculation programs, as RIKOR© or Kiss-Soft©, the proposed method allows the analysis of the interferences to be performed by means of simple, analytical equations, which allows considering all the geometrical restrictions simultaneously, and includes the influence of the elastic deformation of the contacting teeth.

Geometry of internal spur gears
Internal spur gears can be generated by pinion cutter.The geometry of the cutter-number of teeth, shift coefficient, tip radius, etc.-and generating conditions-generating center distance-will obviously have influence on the geometry of the generated internal tooth gear.Specifically, the relationships among generating center distance Cg, pinion cutter shift coefficient x0, and internal gear shift coefficient x2 are given by: where αn is the normal pressure angle, αg the generating pressure angle, Z the number of teeth, rb the base radius, and subscripts 0 and 2 denote the pinion cutter and the internal gear, respectively.Eqs. 1 and 2 are simplified if the internal gear is generated by a pinion cutter with equal shift coefficient, x 0 = x 2 , as follows: where rp is the pitch radius.The assumption of x 0 = x 2 is not restrictive.The generated involute profile does not depend on the cutter shift coefficient, and the generated root profile depends very slightly.This assumption will be made in this paper and Eqs. 3 and 4 will be used, although the general equations expressed as a function of Cg will be also given.
Similarly, the number of teeth on pinion cutter has no influence on the generated involute profile, and very small influence on the root profile.Of course, the obvious constraint Z 0 < Z 2 should be regarded.In addition, the greater the cutter tooth number, the smaller the risk of interference at internal gear tooth root.In this paper the following cutter tooth number will be considered: in which int denotes the integer part function and subscript 1 the external tooth pinion.Accordingly, the root radii of external tooth pinion and internal tooth wheel rd1 and rd2 will be: being m the module, ha0 the tool addendum coefficient (the gears dedendum coefficient), and ro0 the cutter outside radius.The outside radii of pinion and ring will be given by: where ha is the addendum coefficient.The intersection point of the involute profile and the root trochoid of the internal tooth wheel E2 is generated by the intersection point of the Fig. 1 Generation of the intersection point of the involute profile and the root trochoid of internal tooth wheel involute profile and the tooth tip circumference E0 of the pinion cutter.According to Fig. 1: The parameter θ describes the involute profile point as follows: The parameter of point E0 at the pinion cutter profile, also from Fig. 1, is given by: where rf is the tool tip radios coefficient and, from Fig. 1: The parameter of the intersection point of the involute profile and root trochoid of the external tooth pinion, assuming non undercut teeth, can be computed from: Finally, the tooth angular thickness at the involute region of the pinion and ring can be expressed as: 3 Operating constraints Some constraints should be regarded for the proper operation of the tooth pair.Firstly, to avoid tooth pointing the tooth thickness at the outside circle should not be smaller than 30% the module: where: In addition, the outside radius of the ring should be greater than the base radius, which results in: Pitch interference occurs when the angular pitch is smaller than the sum of the angular thickness of pinion and ring at the operating pitch radii.Considering that the parameter of the operating pitch point θr is: ˛0 n being the operating pressure angle, from Eq. 13 the condition of no pitch interference is: where mh is the backlash.The operating pressure angle depends on the operating center distance C: The center distance is usually taken to keep the radial allowance between the root circle of one of the gears and the outside circle of the other gear m(ha0-ha), and therefore: Tooth root interference occurs when the tip of the mating tooth tries to mesh beyond the intersection point of the involute and root-trochoid profiles.The meshing equation is similar to the generating condition expressed in Eq. 8, and is given by: Accordingly, the parameters of the inner point of contact of pinion and ring θinn1 and θinn2, which mesh with the outer points of the mating profiles θo2 and θ01, respectively, will be given by: and conditions for no root interference are expressed as: Finally, tip interference occurs when both pinion and ring tooth tips impact beyond the end of contact.Point I in Fig. 2 corresponds to the intersection between both outside circles.Obviously, if the pinion outer point reaches point I before ring outer point, impact and therefore tip interference exists.Form Fig. 2, the angle rotated by the pinion from the outer point of contact to point I is: K and therefore, the angle rotated by the ring in this interval is: The angle to be rotated to reach point I is: In consequence, the condition to avoid tip interference is:

Analysis of the constraints
All the above constraints should be verified for the proper operation of the gear drive.And all of them are influenced by the geometrical parameters.It is therefore convenient to know the influence of these geometrical parameters on each operating constraint separately.For this preliminary analysis, internal gear pairs with x 1 = x 2 = 0, h a = 1, and outside radii and center distance according to Eqs. 7 and 20 will be considered.With these values of shift coefficients and center distance, the backlash obtained from Eq. 18 is always equal to 0, and the pitch interference constraint is verified.However, both positive and negative values of x1 or x2 result in pitch interference.Only equal values of the shift coefficients keep the backlash equal to 0. Accordingly, the analyses of the influence of the shift coefficients will be restricted to the case of Similarly, increases in center distance result in pitch interference, so that only center distances smaller than one given by Eq. 20 can be considered.According to Eq. 16, the minimum number of teeth on the ring will be: Equation 23 expresses the condition to avoid root interference.Table 1 presents the minimum number of teeth on the ring to avoid interference at pinion root-which is much more frequent than that at ring root-for several values of the tool tip radius and pressure angle, considering dedendum coefficient h a0 = 1.25.It can be observed that the number of teeth on the ring gear can be reduced by reducing the tool tip radius; however, this will result in higher tooth root stresses, which not always will be an acceptable solution.Root interference can also be avoided by increasing the dedendum coefficient, but the tooth root stress will increase again.Positive values of the shift coefficients x 1 = x 2 > 0 provides higher values of (θinn1-θE1), but reduce noticeably the contact ratio εα.Finally, the teeth height shortening will reduce the root interference, but the contact ratio will be drastically reduced.

Study of the tip interference
To study the tip interference, a new parameter called tip interference parameter TI is introduced, which is defined as follows: According to Eq. 27, the condition for no tip interference is the tip interference parameter to be greater than 0: It can be checked that the tip parameter TI increases as the difference of tooth number on ring and pinion increases.Table 2 shows the minimum number of teeth on the ring gear for no tip interference as a function of the number of teeth on the pinion and the pressure angle, for x 1 = x 2 = 0 and h a = 1.Obviously, the dedendum coeffi- K cient ha0 and tool tip radius rf has influence on the tip interference.
From Table 1, for low contact ratio internal gears, the required number of teeth on the pinion to avoid root interference, is very high.For example, for the standard geometry (˛n = 20 ı , r f = 0.3, h a0 = 1.25, h a = 1, x 1 = x 2 = 0, and ro and C according to Eqs. 7 and 20) and gear ratio u = 1.1, the minimum pinion tooth number would be Z 1min = 147; for ˛n = 25 ı the minimum number of teeth is Z 1min = 65.This means that the tooth root interference and the tip interference should be studied jointly.To analyze the interference at pinion tooth root, the root interference parameter is defined as follows: which should be greater than 0 for no root interference.
To avoid tip interference, the tip interference parameter can be increased in several ways, but all of them have inconveniences, which should be assessed: Positive (and equal) increments of the shift coefficients increase the tip interference parameter and the root interference parameter, but the contact ratio decreases.Variations in center distance cannot improve the tip interference parameter, because greater center distance results in pitch interference, while shorter center distance results in smaller TI (this result can be easily checked although it is quite unintuitive).Accordingly, the variations in center distance will not be considered in the analysis.Reductions on the teeth height increase TI and RI .However, produce a strong reduction in the contact ratio.Finally, as seen in Fig. 2, tip relief on pinion and gear profiles increases TI and does not affect RI .However, the contact ratio decreases and the tooth thickness at the outside circle decreases, which may result in pointed teeth.
The three possible solutions-increasing the shift coefficients, decreasing the teeth height, and introducing tip relief-will be compared to find the optimal one.Next in this section, the first two solutions will be compared.In the next section the influence of tip relief will be discussed and compared/combined with the better of the above two.The comparative analysis will be performed in three steps: 1.The variation of the tip interference parameter and the contact ratio for a small variation of each parameter x1=2 and ha will be calculated, from which the variation of the contact ratio with the tip interference parameter [Δεα=Δ(ΔξTI)] will be assessed.2. The required value of each parameter to avoid root interference and the corresponding variation of εα and TI , will be computed.3. The required value of each parameter to avoid tip interference (if possible) and the corresponding variation of εα and TI , will also be computed.
Table 3 shows the results of the analysis.It is observed that for small variation of the shift coefficients x 1 = x 2 = 0.01, the ratio OE" ˛= .TI / = −27.5 is much smaller than that for small variation of the ring tooth height h a2 = 0.99, which is OE" ˛= .TI / = −11.4.This means that reductions of the ring tooth height produce an improved tip interference parameter with smaller reduction of the contact ratio.In fact, the required value of the shift coefficients for no root interference x 1 = x 2 = 0.2316 results in contact ratio " ˛= 1.9234 and tip interference parameter TI = −0.0390,while the required value of the ring teeth height h a2 = 0.9464 results in " ˛= 1.9674 and TI = −0.0386,both of them greater than the previous ones.To avoid the tip interference, a strong reduction of the ring tooth height h a2 = 0.5528 is required, which results in a reduced contact ratio " ˛= 1.4806.Nevertheless, avoiding tip interference is not possible by increasing the shift coefficients.As shown in Table 3, very high shift coefficients as x 1 = x 2 = 1.200only provides 57% of the required increase of the tip interference parameter.
In conclusion, reducing the ring tooth height is more efficient than increasing the shift coefficients to avoid the tip interference in internal spur gear drives.

K 6 Influence of the tip relief on the tip interference
An amount of modification of R1 at the pinion tip or R2 at the ring tip correspond to a delay angle of the tooth Â R1=R2 given by: This delay influences the tip interference.From Fig. 2, the angle rotated by the pinion from the outer point of contact to point I is: the angle rotated by the ring gear in this interval is: and the angle to be rotated to reach point I is: In consequence, the condition to avoid tip interference with tip relief is: which can be also expressed as: According to Eq. 37, the tip relief improves the tip interference parameter in the amount .R1 + R2 /, the sum of the depth of relief in pinion teeth and ring teeth.
In addition, the contact ratio is reduced in the same amount .R1 + R2 /, so that the corresponding ratio OE" ˛= .TI / = −1.0 is much greater than those of the previous cases.The problem is that the depths of relief R1 and R2 cannot be too big because they are limited by the tooth pointing constraint.Additionally, tooth root interference cannot be avoided by means of tip relief.
The amount of relief R1=R2 induces a delay angle which is given by Eq. 32.This delay angle is coincident with the reduction of the angular thickness of the teeth at the outside circle: For the example of the previous section, the required value of the amount of relief to avoid tip interference is .R1 + R2 / = 0.0452.This means a reduction of the contact ratio in the same amount, and therefore " ˛= 1.9948, which is much greater than " ˛= 1.4806 obtained by reducing the ring tooth height.The reduction of the angular thickness is not dangerous.If assumed R1 = R2 = 0.0226, the angular thickness decreases are o1 = 0.0023 rad and o2 = 0.0022 rad.The resulting angular thickness are o1 = 0.0230 rad and o2 = 0.0262 rad, both greater than the minimum thickness for pointing o1min = 0.0097 rad and o2min = 0.0094 rad.The above values of R1 and R2 correspond to depth of relief of 73 μm and 69 μm for pinion and ring, respectively.
Although this tip reliefs avoid tip interference providing a very small reduction of the contact ratio-from 2.0400 to 1.9948-, the solution is not suitable because tooth root interference occurs at pinion root.The definitive solution is performed in two steps: first, reduce the ring tooth height until avoiding the root interference; and second, introduce a tip relief until avoiding the tip interference.
As discussed above, to avoid root interference the ring tooth height should be reduced up to h a2 = 0.9464, which results in " ˛= 1.9674, TI = −0.0386,and o2 = 0.0293.Now, to avoid tip interference, a tip relief of .R1 + R2 / = 0.0386 is required, which results in a new contact ratio " ˛= 1.9288.Assuming R1 = R2 = 0.0193, the angular thicknesses are reduced by o1 = 0.0020 rad and o2 = 0.0018 rad, which result in final angular thicknesses at the outside circle of o1 = 0.0233 rad and o2 = 0.0274 rad, both far from the limit values of pointing constraint.The depth of relief corresponding to R1 = R2 = 0.0193 are 63 μm for the pinion and 59 μm for the ring.

Influence of the teeth deflections
The teeth deflections under load induces the mesh-in impact and quasi-static transmission error (QSTE).To avoid the mesh-in impact the amount of relief at the wheel tooth should be equal to (or greater than) the QSTE at the theoretical inner point of contact δ(ξinn).The QSTE describes a delay of the wheel respect to the pinion which increases the tip interference, hence the condition for no tip interference of loaded teeth would be: However, the QSTE is not uniform along the meshing cycle and its value is maximum at the inner point of contact, as presented in Fig. 3. From Fig. 2, the additional delay at point I δ(ξI) may therefore be smaller than δ(ξinn), and therefore the condition for no tip interference is less restrictive and can be expressed as follows: From Fig. 2, the angle rotated by the pinion from the inner point of contact to point I is: from which the parameter describing the meshing position corresponding to the pinion tip at point I is: Since δ(ξ) is a periodic function of period = 1, it is verified: where frc function denotes the fractional part.From Fig. 3, if ξI is located outside the interval of relief, the value of δ(ξI) can be up to 40% smaller than the value of δ(ξinn) (20% smaller in the case of contact ratio greater than 2).These percentages have been obtained by considering the bending deflections, contact deflections, and gear body deflections.If housing deflections were considered, percentages might be slightly different.The depth of relief to avoid the mesh-in impact depends on the applied torque and the elastic properties of the steel.For normal values of the modulus of elasticity (2.07 GPa) and Poisson ratio (0.3), for an applied torque of 200 N • m the depth of relief for avoiding mesh-in impact (i.e., the transmission error at the inner point of contact) is ı .inn / = 6μm.If lengths of relief R−inn = R−inn = 0.2, regarding condition (45), are chosen, the condition for no tip interference will be: In this case, .R1 + R2 / is the additional depth of relief required to avoid tip interference.The total depth of relief will be: Doing R1−T = R2−T = 0.0199, the decreases of the angular thickness of the teeth are o1 = 0.0021 rad and o2 = 0.0019 rad, which result in final angular thicknesses at the outside circle of o1 = 0.0232 rad and o2 = 0.0274 rad, greater than the required values for no pointing.The corresponding depths of relief are 65 μm and 61 μm, respectively.
Finally, it should be noted that depth of relief required to avoid the mesh-in impact moves the inner point of contact to its theoretical location, and therefore does not reduce K the theoretical contact ratio.And the same occurs with the mesh-out push and the depth of relief required to absorb the QSTE at the outer point of contact δ(ξo).Consequently, the reduction of the contact ratio respect to the theoretical value will be: For the considered example, " ˛= −0.0358, resulting in an effective contact ratio of " ˛= 1.9317.

Conclusions
In this paper the tip interference, root interference, and pitch interference in internal spur gears have been investigated.The influence of the teeth height, center distance, shift coefficients, and tip reliefs have been analyzed.The influence of the tooth pair deflections has also been considered.The following conclusions can be drawn: Both increases and decreases of the shift coefficients of the pinion or ring teeth produce pitch interference, except for equal variation on both gears.However, negative shift factors on pinion and wheel results in higher tip interference and higher root interference, and hence only positive values of the shift coefficients should be considered.Increases of the center distance result in pitch interference, while decreases result in higher root and tip interference.Decreases of the teeth height result in smaller root interference and tip interference, and do not affect the pitch interference.Tip relief reduces the tip interference and does not affect the root interference or the pitch interference but reduces the tooth thickness at the tip.
All these parameters have also influence on the contact ratio, which should be reduced as less as possible.The conclusions of the influence analysis are the following: Increases of the shift coefficients of pinion and wheel reduces the contact ratio.Decreases of the teeth height reduces the contact ratio as well.For a given decrease of the contact ratio, the required shift coefficients produce an improvement on the root interference and tip interference which is smaller than that produced by the corresponding required decrease of the teeth height.The tip relief reduces the contact ratio, but for a given decrease of the contact ratio, the required tip relief produce an improvement on the tip interference which is much greater than that produced by the corresponding required decrease of the teeth height.
From the above considerations, the following procedure is proposed for avoiding interferences with the minimum possible reduction of the contact ratio: 1. First, reduce the tooth height just to avoid tooth interference.2. Then introduce tip reliefs on pinion and ring to avoid tip interference.
If the relative position of the point of tip interference falls inside the two pair tooth contact interval, the lower transmission error respect to that at the inner point of contact may reduce the required values of the reliefs or tooth height reductions to avoid tip interference, which will increase the effective contact ratio.
Funding The authors express their gratitude to the Spanish Council for Scientific and Technological Research for the support of the project PID2019-110996RB-I00 "Simulation and control of transmission error of cylindric gears", as well as the School of Engineering of UNED for the support of the action 2023-ETSII-UNED-04, "Gears".

Fig. 3
Fig. 3 Quasi-static transmission error: theoretical (dotted line) and with profile modification (solid line) If R−inn and R−o are the length of relief at the ring and pinion tips, respectively, the portion of the contact interval without profile modification is described by:inn + R−inn < < o − R−o(44) If point I should belong to this no modification interval, the length of modification should verify: R−inn < frc .I − inn / R−o < frc .o − I / (45) For the above example, after the reduction of the ring tooth height, from Eq. 42 it is obtained I = 15.6070, and considering that inn = 2.4006 and o = 4.3680, the length of relief should verify R−inn < 0.2064 and R−o < 0.7610.

Table 1
Minimum number of teeth on the ring gear to avoid interference at pinion root h a0 = 1.25

Table 2
Minimum number of teeth on the ring gear for no tip

Table 3
Results of the analysis