EES KISSsoft GmbH ++41 41 755 09 54 (Phone) Neudorfstrasse 22 ++41 41 755 09 48 (Fax) 6313 Menzingen P.O. Box 121 Switzerland www.EES-Gear.ch An algorithm for robust gear modifications design Dipl. Ing. ETH Hanspeter Dinner, EES Gear GmbH, Switzerland, [email protected]Dr. Ing. ETH Ulrich Kissling, KISSsoft AG, Switzerland, [email protected]Introduction The design of gear modifications is one of the key tasks for high performance gears, where high performance may be defined e.g. 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) to find - 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 Figure 1: Input parameters and results of the tooth contact analysis The TCA allows for the assessment of the effectiveness of a gear modification. 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 Algorithm for Robust Gear Modifications Design
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EES KISSsoft GmbH ++41 41 755 09 54 (Phone)
Neudorfstrasse 22 ++41 41 755 09 48 (Fax)
6313 Menzingen P.O. Box 121
Switzerland www.EES-Gear.ch
An algorithm for robust gear modifications design Dipl. Ing. ETH Hanspeter Dinner, EES Gear GmbH, Switzerland, [email protected]
Dr. Ing. ETH Ulrich Kissling, KISSsoft AG, Switzerland, [email protected]
Introduction
The design of gear modifications is one of the key tasks for high performance gears, where
high performance may be defined e.g. 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) to find
- 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
Figure 1: Input parameters and results of the tooth contact analysis
The TCA allows for the assessment of the effectiveness of a gear modification. 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
Algorithm for Robust Gear Modifications Design
- 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.
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 [8].
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 centre 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), see e.g. [1], [3], [4], [5].
Figure 2: Left: Profile modification in K-chart (blue) and permissible error (red). A barreling is
superimposed with a progressive tip relief. Right: Linear tip and root relief definition along
ISO21771:2007.
Lead modifications
Typically, three types of lead modifications are combined in a mesh as listed below. Each of
them serves a distinctive, different purpose
1) Helix angle modification: account for the shaft deflection at design load level
2) Crowning: account for variations in shaft deflection e.g. if the load varies or if
machining errors in the housing are present
3) 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
4) 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 e.g. through the
calculation of KHβ along ISO6336-1:2006, Annex E.
Figure 3: Gear modifications. Left: profile and lead modifications. Right: only lead
modifications, where a helix angle modification is superimposed to a crowning and a
progressive end relief on both sides of the flank.
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.
Figure 4: Left: Set up of the definition of profile modification groups using software [7].
Right: Resulting PPTE for the different combinations of modifications for different load
levels.
The display of the different results such as PPTE, KHβ, εa, 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: PPTE for each modification and for different load levels (differently colored curves).
Areas with low PPTE for all torque levels are indicated.
Figure 6: Tooth contact stress for the different modifications combinations. The variations
are considerable. For the highest load level, the stress range is 1350MPa to 1915MPa, a
variation of +42% with respect to the lowest value.
From the graphics above, it can be seen which modification shows both low absolute values
and a low variation in value. This design is then ideal, meaning favorable (low absolute PPTE)
and robust (not susceptible to variations in load ).
Application example, plastic gears
One of the key design problems with plastic gears, especially those running in dry condition,
is wear [9]. Wear is the most common failure mode in plastic gear and the wear of plastic
gears may greatly be improved through optimized modifications [10]. 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.
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. [11]. 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