Crossed helical gears—simulative studies and experimental results on non-involute geometries

Crossed helical gear units generally consist of a steel worm and a plastic helical gear with an axis crossing angle. Due to the high transmission ratio in a small installation space, they are used in auxiliary and actuating gears such as in window regulators, assembly lines, household appliances and positioning tasks. This paper deals with a calculation approach for crossed helical gears to include non-involute flank geometries for new and optimized gearboxes. The geometry derivation for general flank contours based on a polynomial and extends the geometric possibilities beyond the special case of the involute form. The use of principal curvatures and principal curvature directions to describe freely curved surface geometries enables the calculation of flank pressures, efficiency and sliding paths in the entire contact area. Previous studies confirmed the advantages in the mentioned load characteristics for geometries deviating from the involute. Furthermore, parameter studies show the influences of the individual design parameters and demonstrate the effects on operating behavior, with those a design recommendation for crossed helical gear units with general flank geometries can be created. ZC and ZC‑S geometries are used as calculation examples. With regard to the use of these new geometries in applications, manufacturing is a crucial aspect. Grinding disc parameters as well as a three-roll size to control the worm manufacturing for general flank geometries are necessary and described below. In addition to the theoretical calculation, a ZC crossed helical gear geometry was designed, manufactured and compared to a ZI reference gear in test rigs. This confirmed the advantages in load-carrying capacity and efficiency increase resulting from the calculation and proved the benefits.


Introduction
According to the current state of research, the calculations for crossed helical gears are based on Niemann and Winter [7].With a virtual counterpart rack arranged in space, the profiles of the two gears in the transverse sections were derived.The basic equations consider a steel-steel material pairing and are specified for involute flanks.For gears which correspond to the law of gearing, it is assumed that the pitch points in the transverse sections, between one gear and the rack, coincide at a spatial point which also intersects the z-axis.This is called the screw point criteria [7].
Based on Niemann and Winter [7], Pech [8] and Boehme [3] developed extensions and adaptations of the calculation procedure which, in addition to a calculation for the loadcarrying capacity with consideration of the plastic material [8], also consider deviations in the axis crossing angle and helix angles [3].Boehme demonstrated that the screw point criterion of Niemann and Winter is sufficient but not necessary.The pitch points in the transverse sections do not have to coincide spatially at one point and could exist as separate points on the spatial path of contact.Current calculation methods are based on simplifications and assumptions from involute geometry.The new approach removes these restrictions and considers freely curved surfaces for a general geometry description.

Calculation of general flank geometries
To derive the geometries, a virtual counterpart rack is defined, whose flank contour is described by a fourth-order polynomial with Eqs. 1 and 2. y q .u/= q 0 + q 1 u + q 2 u 2 + q 3 u 3 + q 4 u 4  (1) z q .u/= u mm (2) The points on the tooth flank are described with a parameter u which is zero at the reference diameter.The coordinate system of the worm, the wheel and the virtual counterpart rack are shown in the Fig. 1.The polynomial coefficients q0 to q4 are calculated with the basic gear data.The term q0 describes the tooth thickness at the reference diameter with the tooth thickness factor smx* at the worm and positions the flank contour in the coordinate system, q1 specifies the gradient of the flank at the same place, q2 enables a concave contour (ZC) in case of a positive value (also negative values for a convex circular contour are possible, but currently not considered), q3 leads to an optimization of the flank curvature at the contact beginning and end by an S-bend modification and using q4, further free-form geometries are possible.The algorithm leads to a tooth flank side generated in the first quadrant of the coordinate system.Based on these definition, the coefficient q0 must be positive and q1 negative.The sign of q3 and q4 can be selected according to the desired geometry.Studies show that due to the polynomial curve negative values for q3 and positive values for q4 leads to reasonable results [1].The whole tooth is created by mirroring.Depending on the K Fig. 1 Coordinate systems of the crossed helical gear in the normal section [2] geometry, also a part of the polynomial can be used.In case of the involute, a linear polynomial with the tooth thickness factor (q0) and the pressure angle (q1) is sufficient, while in case of the ZC geometry, the parameters of the grinding disc are added (q2).With the help of the tooth root fillet factors of the worm and the wheel, the fillet curve radii are designed at the rack.
One side each of the spatial couterpart rack generates the contour of the worm and the wheel, by using the general law of gearing.The root fillet curve is created by the rack tip and root fillet radii in the same way.The single tooth in the transverse section is multiplied along the circumference to the number of teeth and transferred to the spatial by screwing the transverse section contour around the axis of rotation.The geometry derivation in the transverse section is shown in Fig. 2. Literature [1] explains the equations in detail.
Fig. 2 Derivation of the transverse section geometry with a counterpart rack in space [1] In the transverse section, between the worm or the wheel and the virtual rack the start of contact is called A, the pitch points named C1 or C2 and the end of contact E. For the three-dimensional contact of the gears, both lines are projected into space and result in the path of contact of the crossed helical gear.The points on the path of contact are described as a function of the value u, like the points on the tooth flank before.
To calculate the flank pressure, efficiency and sliding path, both wheels should be considered in one coordinate system X, Y, Z which corresponds to the coordinate system of the worm.So the parameter from the wheel coordinate system need to be transformed and are calculated for each point along the path of contact as a function of the parameter u.For a detailed description of the calculation, please refer to the literature [1,2].

Influence of the polynomial coefficients
The fourth-order polynomial to describe the counterpart rack is formed by the parameter u and the polynomial coefficients q0, q1, q2, q3, and q4.Depending on the flank design different combinations are possible.The individuals have an influence on the contact ratio, flank pressure, efficiency and the sliding path, whereby the one on the plastic gear is preferably considered because it is the subject with higher stress.Calculated with the Hertzian theory, the flank pressure is obtained by taking into account the plastic material and the alternating number of teeth contact along the path of contact.Due to unknown specific properties, the friction coefficient μzm was assumed to be constant and comparable to similar materials.Accordingly, the efficiency results as a geometric quantity.A changing friction coefficient for a tribological consideration is part of further research work.The sliding path on the wheel is calculated with the position of the contact ellipse between the worm and the wheel, the tangential velocities and the contact time during the contact area [1].
The intensity of the parameter dependencies on the coefficients depends on different effects, for example the normal pressure angle αn.As a result of the ZC geometry smaller pressure angles up to 5°are possible which can lead to a significant increase in contact ratio.In general for all geometries, the contact ratio decreases with increasing pressure angle.The flank pressure and the sliding path on the plastic wheel partly behave in the opposite direction.Figure 3 shows the dependence of the above-mentioned variables using the example of a ZC geometry which was designed in [1] and based on a ZI reference gear from Pech [2].
start of contact pitch point(s) end of contact

Legend
The flank pressure decreases at the end of contact and in the area around the pitch points, whereas it increases at the start of contact.For the sliding path on the plastic wheel, the values decrease at each position.Accordingly, the optimum geometry requires a suitable combination between the contact ratio and the load characteristics to correspond to the requirements of the application.
In order to investigate the influence of the polynomial coefficients on the gear behavior and the load parameters, a simulation was carried out with around 300 geometry variations.Figure 4 illustrates the influence of q2 using a ZC example and a pressure angle αn of 5°.
As the value increases, the intensity of the changes in the curve decrease.For the parameter range shown, higher values also result in a lower contact ratio and flank pressures at the start of contact, as well as in an increasing efficiency.At the end of the contact, the pressure and the sliding paths on the plastic wheel increase.
Due to a curvature optimization with an S-modification, the coefficient q3 is negative.It influences the curvature especially at the beginning and end of contact.The influence increases with increasing distance of the reference diameter position from the mentioned locations.The pitch points are located in the middle area and, due to the parameter definition, do not experience any deviations from a change in q3.In the example, the reference diameter is in the direction of the worm tooth tip [1].The results of the parameter study are shown in Fig. 5.The contact ratio increases as the coefficient gets closer to zero.In the range considered the change in contact ratio looks linear.The flank pressure Fig. 7 Path of contact from a crossed helical gear compared to the plane of contact from a worm gear [1] X Y X Y Plane of contact of a worm gear path of contact of a crossed helical gear high contact ratio low contact ratio at the start of contact slightly increases for small q3 values, whereas it decreases at the end of contact.The biggest change occurs between -0.008 and -0.006.The efficiency shows a decrease for small q3 at the start of contact.For small values also the sliding path at the wheel becomes significantly smaller at the beginning and at the end.
Figure 6 presents the results of the coefficient q4 which, in the example, only shows an influence on the parameter at the start of contact.While the efficiency remains nearly constant, with increasing values the contact ratio decreases and the sliding path at the start of contact increases.With regard to the flank pressure, the values around the pitch points and at the end of contact are not influenced, whereas there is a clear drop in the curve at the start of contact.Similar to q3 the changes and effects in q4 depends a bit on the position of the reference diameter.
The results of the parameter study show that some of the parameters for the load-carrying capacity influence each other.A reduction in pressure can lead to an increase in the sliding path, especially with the larger contact area of a ZC geometry.This parameter study can help to find the value range and optimal compromise of the polynomial coefficients for the best possible geometry design.

Transferability of the ZC geometry from worm gears
The ZC geometry has already been successfully used in worm gears due to its many advantages.When transferring the flank contour to crossed helical gears the question arises, how far the aspects from worm gears are transferrable to crossed helical gears.Predki [9] described the geometry derivation and some design recommendations for ZC worm gears in detail.These calculations are integrated in the programs ZSB/ZSP [6], always involving a cylindrical worm with a globoidal wheel.For crossed helical gears, a cylindrical worm is paired with a helical spur gear.The helical gear to an identical ZC worm in worm gears is mentally within the globoidal wheel.Parameter calculations with the previously mentioned program show that the existing criteria from worm gears provide a good approximation to estimate the basic operating behavior.However, there are a number of other aspects to be considered which, if ignored, will result in unsatisfactory gearsets [1].In addition to the different material behavior of bronze and plastic wheels and the deviation of an oil and grease lubrication, the dif- ferences in contact behavior of the worm to a globoid or helical wheel should not be neglected.The plane of contact results from line contact, a path of contact in crossed helical gears is created as a result of point contact.When the geometry is transferred directly between the gear types, the path of contact of the crossed helical gear can result in low or high contact ratio, depending on how it is located in the contact plane of the worm gear.This has an effect on the contact ratio behavior of the gears and becomes obvious in the concrete calculation of the crossed helical gear example.Figure 7 shows the plane of contact, calculated with ZSB/ZSP [6] and the path of contact from the new approach in [1], which is summarized in Chap. 2.
Existing worm gear programs and guidelines are suitable for an initial estimate, but not all influencing parameters are sufficiently considered for a detailed calculation, so ZC crossed helical gears are insufficiently described and require the new approach.

Geometry design for the ZC test gears
Based on the ZI reference gear of Pech [8] and considering the above-mentioned parameter studies, the new ZC profile is derived [1].For an optimized geometry, the normal pressure angle αn was reduces in favor of the contact ratio.In addition, the tooth thickness factor smx* on the worm were selected smaller, so the tooth thickness shifts in favor of bigger plastic teeth.The optimization process taking into account the state of research with regard to the smallest possible flank pressure in highly stressed areas.To address the issue of non-involute geometries in general, a ZC-S geometry is added for a theoretical comparison.Table 1 shows the basic gear data of the geometries.
The deviations of the ZC and ZC-S contour from the involute are small and amount to a maximum difference of 1 mm at the worm and 0.15 mm at the wheel at the ZC geometry.A reference profile on ZC and ZC-S that differs from ZI causes smaller diameter on the worm and equivalently larger diameter on the helical gear.In general, smaller diameter on the worm could have a positive effect on the efficiency.The torque at the worm results from the applied contact force and the radius of the force application point on the worm.So theoretically, the local geometrical efficiency increases with smaller diameter at the worm.Due to the larger contact area of the ZC geometry, the flank pressure is reduced.Combined with the S-modification, this results in a complete optimized gear.Table 2 compares the optimized load parameter based on the calculation results of the new approach to the ZI geometry.
In a first step, the ZC geometry was considered.FEM calculations confirm the positive effects of a ZC geometry with regard to the calculated pressure advantages.The geometry model is available as output of the calculation and can be imported into the program area of MSC Marc/Mentat.During the simulation, all degrees of freedom were locked except for the axial displacement of the worm.A nonlinear material model takes into account the plastic material behavior in detail.A complete description is given in [1].In addition to the lower contact stresses, which are evaluated as van Mises stresses, there is also a reduction in tooth deformation due to the larger tooth thickness of the plastic gear.The FEM simulation verify the smooth contact behavior of the ZC geometry which leads to a beneficial effect on the load-carrying capacity. Figure 8 shows the normal section of the ZI and ZC gears with the results from the FEM simulation.

Manufacturing and testing
To introduce the new crossed helical gear calculation for general flank geometries, the ZC geometry in Table 1 was tested in test rigs.The geometry definition of the ZC geometry is done considering the grinding disc parameters and a so-called three-roll size for the production control of worms.Such a calculation method for general flank geometries is not available so it has to be adapted within the new calculation approach.

Manufacturing of the ZI and ZC geometry
The ZI and ZC worms are manufactured in a hobbing process.Except the flank geometry, the requirements of both worms are identical.For the ZC geometry, there is a special grinding disc with a circular arc profile which gives the ZC geometry its characteristic contour.It is arranged on the reference diameter dm1 of the worm at the center pitch angle perpendicular to the worm shaft axis.The grinding disc width smS0 influences the gap of the worm, and the grinding disc diameter dmS0 can influence the flank contours in the width direction for large numbers of teeth and helix angles.Studies [1] have shown that the influence of the grinding disc diameter is of minor importance for the geometries considered.The radius ρmS0 of the grinding disc determines the arc of the ZC geometry.Like the pressure angle αmS0, it is defined at the reference diameter dm1 of the worm.Figure 9 shows a grinding disc with the descriptive parameters.Increasing the radius on the grinding disc results in a larger contact ratio and, at the end of contact, a reduction in flank pressure and the sliding path on the wheel.The efficiency shows a slight reduction.The three-roll size M0 is used to control the manufacturing of worms.Three cylindrical rollers are located in the tooth gaps of the worm as shown in Fig. 10.The three-roll size in the geometry-independent calculation results from the distance of the cylinder centers to the axis of rotation of the worm and the roller diameter dkug according to Eq. 3.
For the calculation, the cylinders are simplified as spheres with identical diameters.A sphere touches the worm flank at any point S as a function of the parameter u.The worm contour is available from the geometry calculation in the transverse section (Chap.2; [1]), so the point S results from Eq. 4.
S.u/ = .xs1 .u/;y s1 .u/;z s1 .u//(4) The coordinates of the center of the sphere are located in the direction of the normal vector on the tooth flank at K point S. Two tangent vectors are required to calculate the normal vector TN.The first vector TV1 is tangential to the flank in the transverse section of the worm at point S. The second vector TV2 is spatially tangential to a helical line through point S.
The normalized normal vector TNE at the point S together with S and the radius of the sphere leads to the coordinates of the center of the sphere according to Eq. 5.
M kug .u/= S.u/ + d kug TNE.u/ 2 (5) The calculation scheme is shown in Fig. 11.The sphere moves along the helix until the center has the values x and y equal to zero (Eqs.6 and 7).For this purpose the angle vkug is introduced.Using Eq. 8, the z-coordinate provides the desired three-roll size from Eq. 3.
The plastic gears are manufactured additively by laser sintering which is part of the Powder-Bed-Fusion category in [5].Due to the manufacturing process, the accuracy is limited by the thickness of the laser beam and the diameter of the powder grains.The quality inspection shows that the working flank area meets the requirements.Only the tooth root area has a larger diameter than required.With regard to the selected tip/root clearance, there will be no influence on the operating behavior.Figure 12 compares the calculated geometry (black) with the manufactured contours (colored).Due to the manufacturing process, the quality is 11.

ZC geometry in experimental tests
The practical tests are carried out on crossed helical gear test rigs at the Chair of Industrial and Automotive Drivetrains at the Ruhr-University Bochum.The speed-controlled drive takes place via a servo motor.The output via a hysteresis brake is torque-controlled.Measuring shafts are located in front of and behind the test gearbox to check the speeds and torques.This enables the efficiency of the test gearing to be determined.The temperature in the free-flowing grease sump is measured directly below the tooth contact.Figure 13 shows the test rig on the left and the tested ZI worm (top) and ZC worm (bottom) on the right.
From further research work [1,3], a load of 8 Nm is obtained for the plastic wheel and a speed of 1500 rpm -1 on the worm.Both geometries show a uniform operating behavior with constant speed and torque curves for a service life of 300 h.The new ZC geometry not show any deviations in behavior.
During the test period of 300 h, the temperature of the ZC geometry is constantly 4-5 °C lower than that of the ZI geometry.At the same time, there is an average efficiency increase of 4%, which corresponds to the calculation results.Both curves are shown in Fig. 14.
The advantage in efficiency as well as the engagement location on the tooth flanks agree with the practical results and verify the new calculation approach.The improved flank contact and the larger contact area of the concave-convex ZC geometry leads to a lower local load and a beneficial effect on the service life of the crossed helical gears.

Discussion and conclusion
In this paper, influencing factors in the design of non-involute crossed helical gear geometries were described and illustrated with the aid of two examples.The geometry-independent manufacturing parameters enable production and control of geometries that result in long-lasting and highperformance crossed helical gears, as shown in the example of the ZC gear.
Based on the extended crossed helical gear calculation, suitable value ranges and influences of the polynomial coefficients for the profile description are obtained with the aid of gear studies.In addition to this, the normal pressure angle and the radius of curvature on the grinding disc also have a decisive influence on the operating behavior.The parameters contact ratio, flank pressure, efficiency and sliding K path partly behave in opposite directions when the corresponding values are varied, so the best compromise from the parameters mentioned must be achieved depending on the application.Design recommendations and programs for calculating ZC worm gears are only suitable to a limited extent due to the differences between line and point contact.Additionally, there are deviating properties as a result of the bronze and the plastic material, which, in combination with the geometry differences of the globoidal and helical gears, lead to different operating properties.Between the theoretical calculation and the practical tests is the manufacturing of non-involute crossed helical gears.With plastic gears and modern manufacturing processes, there are almost no restrictions due to the manufacturing processes.Conventional manufacturing processes worm gears can be used to manufacture the worm.The tool profile is available with the counterpart rack.For manufacturing control, the three-roll size was derived for a geometry-independent calculation.
Based on these results, an optimized ZC and ZC-S geometry was designed using the basic gear data of a ZI reference gear.The practical tests on the ZC gear show temperature advantages of up to 5 °C and up to 4% higher efficiencies, which are also reflected in the calculation results of the new crossed helical gear calculation.The transmission behavior of the ZC geometry is comparable to the ZI geometry and shows no deviations.
Crossed helical gears with non-involute flank geometries offers advantages over the ZI geometry without showing any non-uniformity in transmission behavior.Based on these results, there is a great potential for the optimization of crossed helical gears.In addition to the design of the ZC-S geometry, this will also be tested in test rigs in further research work.With the help of the extended crossed helical gear calculation, all basic parameters for the consideration of the EHD theory as well as the wear behavior are available.All of these topics lead to a better understanding and a more efficient design of crossed helical gears.

Fig. 3 Fig. 4
Fig. 3 Dependence of the flank pressure and sliding path on the normal pressure angle αn

Fig. 5
Fig.5 Dependence of the contact parameter on the coefficient q3

Fig. 6
Fig.6 Dependence of the contact parameter on the coefficient q4

2 Fig. 8 FEM
Fig.8FEM simulation to compare the ZI and ZC geometry in the normal section; right: van Mises stress, left: tooth deformation[1]

Table 2
Benefits of load characteristics with non-involute geometry ZI compared to ZC | ZC-S