An Algorithm for Robust Gear Modifications Design

Algorithms present all possible solutions to help designers decide what is the best gear modification.


The design of gear modifications is one of the key tasks for high performance gears, where high performance may be defined as, a maximized torque capacity, minimized vibration level, highest scuffing resistance or lowest wear risk. Gear modifications, or the gear micro geometry, are assessed in the tooth contact analysis TCA where the true gear geometry, considering both gear macro and micro geometry is combined with the true gear misalignment which again is a function of many parameters as shown below:

– Line load distribution to calculate KHβ e.g. along ISO6336-1, Annex E
– Contact stress distribution to check e.g. against stress peaks
– Transmission error, to assess the gear vibration excitation
– Local sliding speeds and pressure, to assess the risk of wear in low speed gears
– Local lubricant film thickness to assess the risk of micropitting

The TCA allows for the assessment of the effectiveness of a gear modification Figure 1. Yet, the obvious question remains “what is the best gear modification”. A range of papers has been published on the subject of gear modifications, where the different types available and their effect e.g. on the transmission error of the loaded mesh are described. These papers have in common that they usually look at one particular correction at operating condition only. However, if the profile and lead modifications in a gear pair is to be optimized, different amounts of each modification should be combined and checked for suitability for a range of operating torque levels. This means that we have to find a combination of modifications that:

– Lead to a good level in a particular parameter (e.g. a low variation in the transmission error or a low KHβ value) at the nominal torque level
– Result in little variation in the level of this particular parameter if the torque level changes, meaning, it is a robust design

Furthermore, the different modifications in lead and profile direction may be combined in any manner and it may not always be intuitive which combination is the best. A suitable solution to the above problem is to calculate the TCA for different combinations of modifications, for different sizes of the respective modification and for different load levels automatically. Then, for each combination, parameters of interest (e.g. PPTE or KHβ) are found and may be assessed by the gear designer. This also means that the final selection of the “best” solution is up to the gear designer and is not left for the calculation algorithm to decide, as the importance of each parameter is, remains and has to be a subjective choice based on experience and design philosophy. While this approach is not highly refined, it shows to be highly effective as the below examples will illustrated. Figure 2

Profile Modifications
Profile modifications are either limited to the root or tip area or cover the whole tooth height. The former are called tip and root relief, and different types are possible. The later may be a pressure angle modification or profile crowning. The definition of profile corrections may be found in ISO21771:2007.

Profile modifications have an effect on the gear strength rating along ISO6336:2006 in the sense that they affect the theoretical contact ratio and hence the contact stress under load. However, it is recommended not to consider the profile modifications in the gear rating for pitting and bending. Furthermore, they have a considerable effect on the scuffing rating e.g. along ISO/TR13989: 2000 where they much affect the flash temperature at the start and end of mesh. They also have an effect on micropitting rating, e.g. along ISO15144: 2010, method B where the profile modifications are considered. If method A (where the local contact pressure from a tooth contact analysis is used to calculate the EHD film thickness) is used, then, obviously, the profile modifications have a direct influence as they affect the contact stresses calculated.

In case of poorly lubricated gears such as dry running plastic gears or slow running, highly loaded girth gears they strongly affect the local pressure, in particular at the start and end of mesh and hence the wear rate. Furthermore, profile corrections are applied to avoid point-surface-origin macropitting in the root of a driving pinion. Figure 3

Another focus of the profile modifications typically is to design them such that the vibration excitation during the gear meshing is minimized even if load levels center distance and gear quality vary. This excitation is typically assessed by means of the variation of the transmission error, the peak to peak transmission error PPTE and it’s Fourier analysis or the variation of the resulting meshing / bearing reaction forces (as the bearing forces ultimately are responsible for housing excitation).

Lead Modifications
Typically, three types of lead modifications are combined in a mesh as listed below. Each of them serves a distinctive, different purpose.

Helix angle modification: account for the shaft deflection at design load level.
Crowning: account for variations in shaft deflection e.g. if the load varies or if machining errors in the housing are present.

End relief: ensure that at extreme load levels (when gears are severely misaligned for a short time), no stress concentrations occur at the end of the face width.

Variable corrections: to compensate uneven thermal expansion of the gears in case of e.g. turbo gears.

Typically, the end relief is applied on the gear (assuming the pinion has the lower face width) only. The helix angle modification and the crowning may be distributed between the two gears in the mesh. Below, the resulting flank modification without and with superimposed profile directions are shown. The lead modifications are then assessed through the calculation of KHβ along ISO6336-1: 2006, Annex E.

Application Example, Automotive Transmission
Let us consider a gear mesh in an automotive transmission. The gear has a high theoretical contact ratio of εα=1.72. The gear modifications should be optimized such that the PPTE is minimized for different torque levels, starting at 50% nominal torque up to 110% nominal torque. In parallel, there is the interest to achieve a low contact stress, allowing for a higher power density in the mesh resulting in a lower gear mass. To maximize the performance, the best combination of tip/root relief, profile crowning and pressure angle modification has to be found. Just by guessing it is unlikely to find a good solution. A systematic search by varying all this parameters stepwise has to be performed. Figure 4 shows the user window, where 3 groups of modifications can be defined including the stepwise variation. All possible combinations are then analyzed, giving results in a table as shown in the Figure 4.

The display of the different results such as PPTE, KHb, ea, Micropitting safety, for the different modification variants shows clearly the tendencies. The gear designer can choose carefully his optimum solution, having a low PPTE combined with modest Hertzian pressure. Figure 5
Application Example, Plastic Gears
One of the key design problems with plastic gears, especially those running in dry condition, is wear. Wear is the most common failure mode in plastic gear and the wear of plastic gears may greatly be improved through optimized modifications. Furthermore, plastic gears are often used in medical devices, vehicles (as actuators), kitchen appliances or consumer electronics where a low noise and vibration level is desirable. On the other hand, applying profile modifications on plastic gears is simple and has no impact on the manufacturing costs. Applying lead modifications however is a major challenge and may be limited to helix angle modifications. Figure 6

In this example, we seek an optimal modification to reduce the wear of the gear while achieving also a low PPTE. The gears are made of thermoplastic POM and they are running without lubrication, having a specific wear rate of kW=1.03 mm^3/Nm/10e6. The gear data used (reference profile) is for a high contact ratio gear, which has superior wear performance to start with, see e.g Figure 7 . The wear calculation follows e.g. and may be described as follows:

Wear depth in [mm] on a point on the gear flank, basic formula (product of specific wear rate, times pressure, times sliding distance):
dw (mm) = KLNPdmetric (mm2/N) *
P(N/mm2) * V(mm/s) * T(s)

dw (mm) = 0.001*KLNPd (mm3/Nm) * P(N/mm2) * V(mm/s) * T(s)

From the above basic formula, we find the wear depth per gear:
dw_i = 0. 001*KLNPd  i * F * vg / b / vp_i;  i = 1,2

Or, using the specific sliding z =vg/vp_i :
dw_i =  0. 001*KLNPd  i * F * z_i / b;  i = 1,2
And finally, the wear after n load cycles
dw_i =  n * 0. 001*Kfactord_i * F / b * z_i;  i = 1,2

F (N)    Load
b (mm)    Face width
vp1, vp2 (mm/s)    Velocity tangential to the flank of gear1, gear2
vp1,2*Dt (mm)    Moving distance of a point on the flank (Gear1, 2)
vg (mm/s) = vp1 – vp2     Sliding velocity
Dt (s)    Time
vg*Dt (mm)    Sliding distance
z    Specific sliding
A = b*(vpi*Dt)    Surface
P = F / b/(vpi*Dt)    Pressure
dw (mm)    Wear depth (mm)
i                     Index for gear No, i=1, 2

In the optimization, we will combine different amount of tip and root relief with different correction diameters to find the wear rate for each case as shown in Figure 8.

The resulting wear on the flank of one of the gears in the gear mesh before and after optimization is shown in the next figure. It can be seen that by applying a combined correction, the total wear can be reduced to about half of the original value, effectively doubling the gear lifetime at no additional manufacturing cost. Figure 9

Application Example, High Speed Stage of a Wind Gearbox
Consider a high-speed stage (HSS) in a wind turbine gearbox with a rated power of 3MW. The larger gear – the wheel, drives the smaller gear – the pinion. The shafts are both supported by a non-locating bearing on one side (the side facing the rotor of the turbine) and a locating, paired taper roller bearing in X arrangement on the other side (the side facing the generator). The wind turbine operates in a wide range of wind speed, resulting in a wide range of torque levels acting in the HSS. Typical load show that most of the time is spent at a torque level in a range of 80% to 120% of the nominal torque Tnom. However, even in operation, peak loads may reach 140% or 150% of the rated torque. Furthermore, extreme load cases may reach 200% or more percent of the rated torque, however, this torque levels are rare. Figure 10

Here, the key objective is to find lead modifications:

1) That result in a low KHβ value in the operating range of the torque
2) Allows for the change in shaft and bearing deflection due to the load changes
3) Which ensures that at the peak loads, no stress concentrations at the edges of the face width occur.

We will split the above three requirements as follows:
– Requirement 1) and 2) will be resolved through a combination of the helix angle modification and crowning. To optimize the modification, we will use the optimization tool
– Requirement 3) will be resolved through an end relief, which will be applied based on experience

Furthermore, we will check the effectiveness of the modifications in two distinctive torque domains:

– Torque domain A: operating load range, from 80% to 150% nominal torque. Here, we want to achieve a low KHβ value and a good stress distribution.
– Torque domain B: extreme load range, at 200% nominal torque. Here, we want to ensure that there are no stress concentrations at the end of the face width.

For the purpose of this study, we will put all modifications on the pinion and none on the wheel and we will assume that the face width of both gears is equal. The deformation of the bearings due to the gear forces is based on a non-linear bearing stiffness, which is calculated from the bearing inner geometry. The bearing stiffness is non-linear due to two effects:

1) There is a finite bearing clearance, which results in a horizontal line in the displacement-force curve. This finite bearing clearance is calculated from the assembly clearance (here, for the CRB, C3), the fit between the races and the shaft and housing, the temperature of the bearing races and the centrifugal forces on the races.
2) With increasing load, more rolling elements get into contact, effectively increasing the stiffness with increasing load

The shaft deflection considers these bearing stiffness effects. The results are calculated using a non linear FEM solver (meaning that the equilibrium of all external forces is solved in the deformed state of the shaft), considering the shaft by means of Timoshenko type beam elements and considering the bearings by means of a non linear spring / gap element.
Figure 11

Considering nominal torque and without any modifications, we find a contact stress distribution as shown below, left side. Using the shaft calculation, we find an approximate value for the helix angle modification Cβ=80μm and from experience, we estimate a suitable crowning of Cb=20 μm and find a contact stress distribution as shown below, right side. We can see that the contact pattern has improved but is not optimized. Furthermore, only nominal load condition (100% of nominal torque) has been considered till now whereas in wind gearboxes, a load level slightly higher than nominal torque should be considered in the design of the gears. Figure 12

Figure 13 Finally, let us consider the effect of an end relief, which is of particular importance in case of overloads. From the TCA for a peak load of 250% nominal torque, we find stress concentrations at the end of the face width (Figure 14 left). After we apply a curved end relief (here, over 10% of the face width on each side) at both sides of the face width, we can reduce this stress concentration. Obviously, the KHβ increases with this modification as now; the effective face width and now the face width experiencing load under nominal load condition is reduced.

Application Example, Sun-Planet Mesh in a Mill Gearbox
Horizontal (e.g. for sugar mills) or vertical (e.g. for cement mills) output planetary stages transmit very high power levels at very low speed, resulting in large planetary stages having with ratios d/b>1.00 – in particular on the sun gear. Due to the torsional wind up of the planet carrier and the sun shaft, corrections in the sun-planet mesh are applied, the basic considerations are explained e.g. Care has to be taken not to select to high levels of crowning, as the resulting loss in load carrying face width is severe. The design objective is hence to find a combination of helix angle correction and crowning resulting in a low KHβ value for different planet pin deflections and sun gear torsion, an objective that has been reportedly achieved in the industry. Figure 15

In this example, a gear face width of b=430mm was applicable and the helix angle and the crowning on the sun gear were selected as variables (while an end relief on the planet, having somewhat lower face width, was applied). Figure 16

The load level was varied in five steps between 90% and 130% of nominal load.

The crowning was varied between 10μm and 50μm and the helix angle modification between -150μm and -190μm, both in five steps, giving a total of 125 calculations (25 different combinations of modifications, each for five different load levels). The result of these are shown in the below graphics:

It can be seen that for modification 3:2:-, the lowest KHβ values result and that there, the variation of the KHβ value is also small (KHβ is not much affected by the load level). For this correction, the crowning is 30μm and the helix angle correction is -170μm. Figure 17

If we now look at the tooth contact pattern, the contact stress and KHβ for different load levels (nominal load, lowest load and highest load) with the above corrections applied, we find that:

– KHβ (not considering manufacturing errors fma and fHβ) is low for all load cases
– KHβ varies only very little with the variation of the external loads
– For no load conditions, stress concentrations at the edge of the face width exists (also due to the end relief)

We may therefore conclude that the design is robust, see the below images for the results at lowest, nominal and highest load. Figure 18

For 100% load and 240μm tilt
For the face load distribution factor KHβ we find:
For the contact stress in the plane of contact and the contact pattern on the sun see Figure 19 (pg. 61).

For 90% Load and Scaled Tilt, as Lowest Load
For the face load distribution factor KHβ we find: KHβ=wmax/wm=1920N/mm/1782N/mm=1.08

For the contact stress in the plane of contact and the contact pattern on the sun see Figure 20 (pg. 61).

For 130% Load and Scaled Tilt, as Highest Load
For the face load distribution factor KHβ we find:

For the contact stress in the plane of contact and the contact pattern on the sun see Figure 21 (pg. 61).

In the above four examples, general guidelines on the use and purpose of profile and lead modifications have been presented. It is pointed out that the key to a successful design is not only to optimize the modifications for a certain load level, but also to take into account a range of operating loads and selected overloads. While for overload cases, experience based application of gear modifications makes sense, a search algorithm to find a robust modification design for the whole range of operating loads is proposed.

We consider it critical that such an algorithm does not propose a single solution claiming it is the best, but that the algorithm presents all solutions found for the designer to assess, e.g. using graphics as shown above. While such an assessment could be automatic based on the numerical results and user defined weights for each parameter, we strongly believe that gear design is, remains and should be also intuitive. Above, a search algorithm, suitable graphics for this intuitive assessment of the results and the application of the method in various industries for different optimization targets are presented.  

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