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Estimation of lifetime for plastic gears

Abstract

The importance of plastic gears for modern industry is growing every year. The engineer sizing plastic gears has a very difficult task. There is no international standard available for the strength analysis. The only method publicly accepted is the German guideline VDI2545. In addition to a lack of calculation methods there is a need for measuring material properties. This paper shall give an overview over the current situation and will provide some guidelines how to rate plastic gears, how to handle the lack of material data available and how to conduct measurements of material properties to make them suitable for the available calculation methods.

Introduction

The number of gears produced out of plastic is getting dramatically larger. This is primarily due to the improvement of plastic materials strength. The properties of plastic can be varied in a large range, especially when compared to steel. It is now possible to select an optimal material for a specific task based on the following properties: strength, wear, stiffness, damping and noise production. In spite of the growing use of plastic gears the scientific research is astonishingly low, especially compared to the resources used for research on metal gears. A mirror of this situation is the availability of standards. The standard used for the strength analysis of cylindrical plastic gears is the German guideline VDI 2545 and is poor compared to the respective standards AGMA 2001 and ISO 6336 for metal gears. AGMA 909-A06, ANSI/AGMA 1006-A97 and ANSI/AGMA 1106-A97 address plastic gears, but only the geometry. ANSI/AGMA 920-A01 offers much general information about the applicability of plastic material for gears and presents the typical test procedures. The VDI 2545 is currently invalid and fits into the overall lack of research into plastic gearing. One of the major restrictions of the VDI 2545 is the availability of data for only three different materials (PA12, PA66, POM). Currently several groups are attempting to produce reliable data for modern materials. These experiments have proven to be expensive and are very time consuming. A design engineer therefore needs to know how he can design new plastic gears successfully without the use of a valid standard and without scientific materials research data. The engineer must rely on knowledge gained from past experience. Usually metal gears are produced in a generating process. Plastic gears most often are injection molded. If the insert for the mold is manufactured with EDM (e.g. wire erosion) the tooth form can be optimized without additional costs. For generated gears this is only possible with special tools, which increases the costs. On the other hand, typical injection molded gears have a relatively low quality (ISO 9-10), a problem, however, which can be handled with special arrangements. Optimized plastic gear tooth forms are also designated “hybrid toothing” in literature.

Pla

stic g

ears

, life rating

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Strength analysis

Material data for plastic gears (Wöhler curves or S-N curves)

For the sizing and optimization of gears, the calculation of root, flank and wear strength for the prescribed lifetime are of large importance. In the same way as with steel gears, for plastic materials the specific parameters (root pulsating strength and flank strength) are dependant on the number of load cycles. For plastic gears these parameters depend strongly on temperature and the type of lubrication (oil, grease or dry running). Where one value for the tooth strength calculation is sufficient for steel, a plastic material requires the necessity of several S-N curves (e.g. for POM fig. 1).

Figure 1: Temperature dependent Wöhler curves for POM. The method according to VDI 2545 [1] for the strength analysis of cylindrical gears made of plastic is the only worldwide known method for the calculation solution. Even though it was cancelled some years ago it is still in common use due to the lack of a replacement. Currently Prof. Werner Krause and Dr. Jürgen Wassermann in Germany along with their associates have plans to develop a replacement for the guideline but it is in the very early stages of development. Today’s materials are much more numerous than the materials mentioned in the VDI 2545. Some of them show a significantly higher strength, e.g. reinforced material. Typically, the producer of the material will only provide values for the tensile strength, the aforementioned data for a gear calculation is not known and can not be derived from the tensile strength. Changing the recipe of a plastic might lead to higher ultimate strength, thus increasing the root strength, while the flank or wear resistance is decreasing at the same time due to tribological effects. A simple solution for the problem is not available. The material properties have to be verified using prototypes or by means of a long term test on a test rig. In most cases it is not necessary to conduct hundreds of measurements, to get enough data points for a diagram like in figure 1 but by determining some data points the diagram can be generated by interpolation with good

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accuracy. These data points can be derived from experience with produced gear boxes or experiments on test rigs. Still the effort is significant.

General layout of the strength calculation of plastic gears

The mechanical properties of plastic parts are strongly dependant on temperature. So for rating plastic gears first the relevant temperature has to be determined. This means that based on the environmental temperature, the local heat production in the meshing of the gear due to frictional and viscoelastic power losses and the dissipation of the heat, the final state of equilibrium must be searched. Figure 2 shows the general layout of the calculation procedure (according to Erhard [2]).

Figure 2: General layout of the strength calculation procedure for plastic parts.

Estimation of the temperature

Several models are available for the calculation of surface and body temperature. For metal gears the flash temperature concept of Blok is used for the calculation of the scoring safety factor. For plastic gears this model was adapted by Siedke [3]. Practical experience however shows that this model may not be appropriate for plastic gears. Takanashi developed equations with an approach containing a friction and a deformation part. The heat production out of the deformation part is based on the model according to Voigt (spring-damping-model). To balance the heat production the dissipation has to be calculated. The difference between heat production and heat dissipation then leads to surface and body temperature. The model according to Takanashi [4] proved to be precise enough for practical application. However, several of the parameters needed are hard to determine. For this reason another model prevailed, based on work of Hachmann and Strickle [5]. The calculation is also based on heat

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balancing. The assumption is, that the heat quantity Q1, which is produced by the power loss in the tooth meshing is equal to the heat quantity Q2 that is dissipated to the inner space of the gearbox housing. This in turn is equal to the quantity Q3 which is dissipated by the housing to the environment. This approach leads to a formula, which is also used in the VDI guideline. The factors in the formula differ slightly between original publication, the formula in the VDI guideline and other publications.

( )

+

××++××+=

A

k

m

k

zbz

uPa

32

2,122,1 3.6

17100

5

1136 κν

µδδ

Where δ1,2 [°C] is the surface or body temperature of gear i, i=1,2, δa [°C] is the ambient temperature, P [kW] is the power transmitted, µ [-] is a coefficient taking friction into account (not the friction coefficient!), zi [-] number of teeth of gear i, i=1,2, u [-] transmission ratio z2/z1, b [mm] face width, ν [m/s] circumferential speed, m [mm] normal module, k2, κ [-] factors described in the VDI2545, see below, k3 [-] takes the influence of the housing into account (none, open, partially open or closed), A [m2] is the surface of the housing. Comparison of the results of the temperature calculation according to Hachmann and Strickle and measured temperatures in tests show slightly lower calculated temperatures in the root area, and significantly higher calculated temperature of the flank surface. Frequently, the calculated temperature exceeds the melting temperature of the material, although the test showed no melting of the flank. Erhard and Weiss [6] proposed a modified calculation of the temperature, taking the ratio of the power-on-time into account. Based on measurements, they defined continuous power-on-time of 75 minutes to be permanent running. For all cases with shorter periods they introduced a factor fED to reduce the calculated temperature. Since this work was done after the publication of the VDI guideline, this correction factor is not included in the calculation formula of the temperature in the VDI 2545. The factors k2 and κ show up during the derivation of the formula. They take the material combinations and lubrication type into account. The factor k2 also determines whether the calculated temperature is the body or the surface temperature. For k2, k3, κ and µ tables are provided in the guideline. The temperature calculation is one of the critical points not only in the VDI 2545, but in the calculation of plastic parts in general. Due to the problems mentioned above it is recommended to use a fixed temperature whenever possible to supersede these problems. For slow running gears (circumferential speed <5m/s) the guideline recommends the use of ambient temperature. Most plastic gears are used for very short running period (less then 1% operating time per hour), so that for all these cases the ambient temperature equals to the operating temperature.

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However, for the ambient temperature a large range might be defined, for example for car applications from –20°C to 80°C.

Basic formulae for the strength analysis

The VDI 2545 is based on the DIN 3990, and adapted to plastic gears. Hence, the two basic formulae are very close to those in the DIN 3990, and, since the main formulae in ISO6336 and DIN3990 are identical, also to those in the ISO6336. Some factors are missing (or in other words constant to 1) since the knowledge about these factors for plastic material was to poor or the factors simply not applicable. The basic formulae are:

minF

FNAF

n

tF S

KYYYmb

F σσ εβ ≤×

=

for the root stress, where Ft [N] is the tangential force at reference diameter, Yβ [-] is the helix factor, taking the influence of the helix angle into account YF [-] is the root form factor, basically the lever arm of the normal force, Yε [-] is the overlap factor to take overlap contact ratio into account, KA [-] is the application factor, to model frequent peak torque,

min

1

H

HNA

tMHH S

Ku

u

db

FZZZ

σσ ε ≤+×

=

for the Hertzian pressure, where ZH [-] is the zone factor, transferring the tangential force from reference diameter to operational pitch diameter, ZM [-] is the material factor, to take the two Youngs moduli into account, Zε [-] is the overlap factor, d [mm] is the reference diameter, u [-] is the transmission ratio, The influence of the temperature is not seen directly in these formulae. It is included in the values of Youngs modulus Ei=Ei(δi), and the permissible stresses σHN(δi), σFN(δi).

Deformation of the teeth

Comparing plastic to steel it is apparent that Youngs modulus for plastic is about two magnitudes smaller than for steel: e.g. about 2800 N/mm2 for POM compared to 206,000 N/mm2 for steel. It should also be noted that the permissible bending stress is about one magnitude lower: for instance 25 N/mm2 compared to some 250-450 N/mm2 for steel. Since most gears are sized according to the strength requirements, the relative deformation of the plastic gear teeth is much higher than for steel gears. This effect raises the need of checking the deformation of the teeth. According to VDI 2545 the shift of the tip is calculated with the formula

Ψ+

Ψ×

=2

2

1

1

cos2

3

EEb

Ff

t

tK ϕ

α,

where ϕ, Ψ1 and Ψ2 are tabulated values.

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The VDI 2545 sets a limit of 0.1 times module for the deformation. This of course is not a firm limit as it is in contrast to the permissible stress. It is important to check the deformation carefully if the teeth remain loaded for a certain time period. If the product of deformation and time period gets to large, permanent deformation might be the result, which can cause severe problems.

Differences between the calculation according to G.Niemann and VDI 2545

In addition to the VDI2545 there is a method for the strength analysis of plastic gears described in Niemann [7]. Although the source of the data and formulae are the same for both methods, the differences are significant. The most important differences are listed in the following table.. Niemann VDI2545 VDI2545-modif. Root calculation: YF DIN 3990, DIN 3990, DIN 3990,

Method C Method B or C Method B or C YS DIN3990 1.0 fixed DIN3990 Yε (only method C) 1.0 fixed = 1 / εα = 1 / εα Yβ 1.0 fixed DIN3990 DIN3990 σFE = 2 × σFlim = σFlim = 2 × σFlim Flank Calculation: Zε 1.0 fixed DIN 3990 DIN 3990 Tooth deformation: Very different calculation methods! One of the major disadvantages of the guideline VDI 2545 is the lack of the stress correction factor YS, or equivalent the constant value of YS =1 in the formula. This factor describes the notch effect in the trochoid part of the root area. It is likely, that the same notch will have a slightly different effect on a steel part than on a plastic part. In either case there will be an increase of the local stress depending on the form of the notch. In Niemann the contact ratio factor Yε is set to one. The argument in the textbook for this is the typically low quality of the plastic gears (ISO 10 or 11), so that only one pair of teeth will have contact. This is not fully correct since plastic teeth will bend much larger than steel teeth (lower Youngs modulus) compromising the pitch errors. In addition the quality of plastic gears has greatly improved since the printing of the textbook mentioned. Based on the available data we propose a method “VDI2545 modified”, which combines the advantages of both methods. With the modified method it is possible, to quantify more accurately the behaviour of variants of gears.

Applying calculation methods

As discussed above the calculation methods for the determination of the estimated lifetime of plastic gears should be applied carefully. Moreover, the necessary minimum safety factors are dependant on the specific application. The question then is how is it possible to receive meaningful results for the sizing of plastic gears? Referencing the calculation method back to existing experiences good results can be achieved. The procedure is illustrated in figure 3. The first step is to define the calculation method with all boundary conditions. Based on this, “known” (existing) gears are analysed and rated. The resulting safety factors are grouped to

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determine the maximum limit and the minimal necessary safety factor. This leads then to the nominal safety factors for the sizing or rating of the new design.

Figure 3: Proposed procedure for the application of calculation methods. Steel material properties can be determined comparably easy (e.g. according to ISO 6336, part 5). One must be very careful in determining the plastic material properties. The nominal safety factors derived by the procedure described are only valid for the material used in the existing designs. When using a material with significantly different properties the safety factors have to be adjusted! The described method has proven to be very successful in practical use. It is important to build up systematically the base of own know how, experience, and to maintain it. Typically, each gearbox with plastic gears is tested before the manufacturing is released. This test data has to be used!

Calculation of gears with high static load

In many applications a gearbox transmits only a relatively low permanent torque, but has to be able to stand a short time peak torque without being damaged. A typical example for this is the actuator for a seat in a car, which is loaded with a high torque when driven into a block. If the number of peak load cycles is low (during the full life time), between 1,000 to maximal 10,000 cycles, the proof of strength can be reduced to a static calculation. This is common use for shafts, where a static and endurance analysis is executed. For the static analysis, the calculation of flank and wear resistance can be omitted. The relevant limit is the check of the root stress versus the ultimate strength. For the calculation the effective root stress is calculated according to VDI 2545 (or ISO 6336) and compared with the ultimate strength. The ultimate strength for non-notched bars is typically found in the documentation from the manufacturer of the plastic material. This calculation can be carried out much easier and can also be applied to material which is not documented in the VDI 2545.

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It is very important to decide when and which method to apply. As a rule by thumb it can be stated: Relation Tp/Td

Frequency of Tp (*) Calculation method

< 1,000..1,0000 Static Strength Check with Tp >= 3.5 >1,000..1,0000 VDI2545 with Tp

< 1,000..1,0000 Static Strength Check with Tp and VDI2545 with Td

2.0….3.5

> 1,000..1,0000 VDI2545 with Tp and VDI2545 with Td

< 1,000..1,0000 VDI2545 with Td <= 2.0 > 1,000..1,0000 VDI2545 with Tp und

VDI2545 with Td Tp : Peak (maximum) torque Td : Nominal (endurance) torque Bold: The most common cases (*) : Number of load cycles with Tp (during full life period)

Strength calculation taking the real tooth form into account

The endurance safety against root failure is highly influenced by an optimized transformation from the involute to the root circle. The manufacturing of a gear with a generating process, even with a well rounded tip of the tool, an optimal rounding often cannot be achieved. With a modification, which of course has to be adapted to the contact behavior with the counter gear, the root strength can be increased significantly. For the strength analysis a reliable algorithm was developed based on literature, hints in standards and comparable calculations by FEA software. With this algorithm the complex effort of a FEA calculation usually can be skipped. All standardized calculation methods determine the root stress based on a simplified model. According to VDI 2545 (in analogy to DIN 3990), the critical cross section is determined by the tangent on the inside of the root curve which intersects the middle line of the tooth at an angle of 30 degrees. Depending on the root rounding, the position of the real critical cross section might deviate more or less. In a paper by B.Obsieger [8] years ago, an approach was proposed for a significant improvement of the calculation method. Depending on the real tooth form for each point in the root area the tooth form YF and the stress correction YS factors are calculated and the point is determined at which the product YF×YS reaches a maximum (see fig. 4). This leads to a much more precise calculation method and can be applied without problems to non-involute tooth forms as well. Applying the formula for YS as defined in the DIN or the ISO standard as proposed by Obsieger exceeds the defined limits for the formula. It is valid only for the point of the 30 degrees tangent, should not be used for a graphical method and is only for involute tooth forms. However, comparing YS in a graphical method and FEA results showed a very good match, so that in most cases the accuracy of both methods should be the same. In addition, the described method is a “worst-case” method, i.e. the calculated safety factors are by definition always smaller than those calculated by the standard: the one point treated in the standard calculation is included in the list of points to be checked according to Obsieger. In contrast to the standard method different root forms can be compared and the benefit or negatives of a root modification can be evaluated.

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Figure 4: Graphical method for the determination of the worst case in the root area. Following the approach of Obsieger it is possible to locate the critical cross section of the tooth. As an option, the force can be applied at the tip as in ISO 6336 Method C, or at the point of single tooth contact – method B. The strength analysis according to VDI 2545 can be carried out later with this specific data. In addition, it is possible to visualize the geometrical course of the stress in the root area and the course of the maximum stress in the root area during the meshing of the gears (figure 5).

Figure 6: Course of stresses in flank and root during meshing of the gears. The calculation of the Hertzian stress can also be conducted along the tooth flank based on the real tooth form. Here for each point of contact the respective radii of both gears are determined and with this the Hertzian stress is calculated. The same data allows the calculation of the sliding velocity, the local heating, the efficiency and the heat production of these tooth forms. With these additional information it is much easier to optimize the tooth form than only with the standard calculation that only gives information about one point of contact.

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Strength analysis of non involute gears

The algorithms described above are derived from the base law of gearing and can be used for modified involute profiles as well as for non involute profiles. Although the large majority of gears used are based on an involute design, sometimes an alternative tooth form can be beneficial. For example for a gear pump in a formula-1-car; the weight could be reduced significantly by replacing the metal gears with an involute profile by plastic gears with a cycloid profile. The Hertzian stress and the sliding velocity were reduced due to the optimal curvature of cycloids (figure 6) by nearly 50%, especially in the critical areas at tip and root circle.

Figure 6: Comparison of an involute design with a cycloid design. Above: stresses during the meshing. Below: sliding velocities.

Optimization of the tooth contour

The geometry of the tooth can be changed in different ways to achieve the optimal situation during meshing. Depending on the targeted design specifications like minimum noise,

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vibration reduction, highest strength, low sliding velocity and even stiffness or smoothness of the rotation, one or the other action may be preferred. The following are typical actions for the optimization of the geometry and give suggestions to which calculation methods should be applied:

1. Change of geometry with a given reference profile

The geometry of the meshing can be influenced strongly be varying module, pressure angle, helix angle and addendum modification while keeping the reference profile. Especially for helical gears in many cases a satisfying solution can be found. This also includes spur gears, which are much more frequent in plastic. In order to find an optimum solution for the before mentioned parameters, a large number of variants have to be checked. Therefore this task should be delegated to a computer program which searches possible solutions, filters them according to diverse criteria, and sorts the remaining solutions to classify each solution with respect to the design specifications. Since the solutions found by this are so called “standard geometries”, usually the standard calculation according to VDI 2545 (preferably the modified version) is sufficient for rating the gears.

2. Variation of the geometry by changing the reference profile

The change of the reference profile (typically an increase of the tooth height) leads to a change of the contact ratio. For gears with low noise production and low vibration level a transverse contact ratio of 2.0 is tried to achieve. With this, the amplitude of the change in tooth stiffness at the point of single tooth contact can be minimized. With a non-standard reference profile, the desired contact ratio can be achieved, however, often contact interference and other problems occur then during meshing. The number of possible solutions is hence limited. To treat this problem the before mentioned computer program should offer the option to seek for all possible solutions with a given contact ratio and present them to the user. To avoid a pointed tip one solution is often to choose a smaller pressure angle than 20°. This generally reduces the calculated strength. The compensation for this reduction is the increased contact ratio. So at least the VDI 2545 should be chosen to have the influence of εα in the method. It can be additionally necessary to apply the graphical method to safely find the real point with the maximum stress in the root area.

3. Elimination of the impact shock by correcting the tooth form

When a new pair of teeth gets into contact at the beginning of the meshing the so called impact shock occurs. This shock is a consequence of the deflection of the teeth due to the load and is greater proportional to the larger the pitch error and the deflection of the teeth usually generating noise. To reduce the impact shock for plastic gears the involute is modified with a tip relief. Typically the tip relief is carried out on both gears. A variation of this can be that the correction is carried out only on one gear, but in tip and root region. The standard methods do not take the real profile modification into account. For this reason analyzing the effect of the profile correction, it is necessary to use the calculation of the real path of contact, and based on this the calculation of the real Hertzian stress, the transmission error and so on.

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4. Optimization of the tooth root

The endurance strength of the root of a gear is mainly determined by the radius in the transition from the involute to the root circle. In a gear produced with a generating process, even with a well rounded tip of the tool, the optimal form is not always produced in the root area. With a suitable modification, which must be done below the used involute area (underneath the SAP) the strength can be increased significantly. Except the engineering effort, which is small when using appropriate software, all these actions do not cause any further costs for the manufacturing of net shaped gears like injection molded gears (at least if the insert for the tool is produced by wire erosion). The graphical method is tailored to rate the effect of the root rounding. So this method should be applied. For the flank, the standard method can be applied, but a careful check of the SAP versus root form diameter (TIF) is necessary to avoid contact interference.

5. Change of tooth thickness for the increase of expected lifetime of

metal/plastic combinations

In worm drives often the worm is made out of steel, while the worm gear is molded with plastic. The life expectancy of this combination can be increased significantly by thinning the teeth of the worm, and on the other side thickening the teeth of the plastic gear. The values for root and tip circle remain more or less the same. The sizing of the geometry of such a pair is not simple since often contact interference, undercut and other problems occur. For the strength analysis, however, the standard method usually is enough. The critical point might be the very different operational pressure angle versus the nominal pressure angle, or the necessity of a modified root rounding due to the reduced space in the root area. Then the calculation of Hertzian stress and the root stress should be carried out based on the real tooth form.

Determination of material data for strength analysis

As mentioned only limited data is available for plastic material concerning strength analysis of gears. The relevant tests where carried out between 1970 and 1980 by Höchst in Germany (now Ticona) and other German companies. Since then a large number of new materials have been developed, especially for higher temperatures. For all of these new materials there is not a complete set of characteristics for the rating of gears. If the nominal safety factors are chosen carefully it is possible to make suitable estimation of lifetime even for these modern plastic materials. A systematic measuring of the characteristics for a set of new plastic materials would be very helpful to make the right choice. The following guidelines are presented for a practical measuring procedure.

Test rig of Scholz GmbH

The published photos show the test rig of Scholz GmbH & Co KG, Kronach, Germany. The test rig was designed together with the University of Erlangen. The original plan was to build a number of these test rigs to make the measurement of several plastic materials possible. The large advantage if several Universities and companies conducting measurements with the same type of test rig is of course the comparability of the results.

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Figure 7: Test rig of company Scholz GmbH & Co. KG, Kronach, Germany

Results from Victrex

One of the producers of plastic material suitable for gear applications, the company Victrex which is producing PEEK, is conducting a measuring project for some years. Although the measurements are not completed, there is some data available. Figure 8 shows the test rig. Figure 9 shows some first results that allow the comparison of PA66 to PEEK. The data obtained from the measurements is already available on request; however, due to the incompleteness it is not yet applicable to all applications. All the measurements where carried out with oil lubrication. One of the final goals of the measurements is to have absolute data available for the sizing of PEEK gears. With the current set of results, however, it is better to use the data for PEEK relative to the measured permissible stress of PA66, and then scale the calculated results. This is an approach which is comparable to the method described before, how to apply calculation methods in general (see figure 3).

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Test gears

X-tables

Output unit

Input unit

Figure 8: Test rig of Victrex Europe GmbH

Gear performance @ 1020 RPM

0

5

10

15

20

25

30

35

40

1,E+04 1,E+05 1,E+06 1,E+07

Cycles

To

rqu

e [N

m]

PA66-RT

PEEK 450G-RT

PA66-120°C

PEEK 450G - 120°C

Figure 9: First results of the Victrex measurements, comparing PEEK with PA66.

Procedure for the measuring

With this document we try to summarize the current state of the art for the treated topic, to derive instructions how to generate the necessary data for strength analysis of plastic gears in the most efficient way.

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General task The properties of plastic material is highly dependant on temperature. Therefore it is not enough – like it is for steel – to determine one Wöhler curve, but several Wöhler curves are needed for different temperatures (e.g. 40, 60, 80, 100°C). In addition the strength of the flank is dependant on the type of lubrication (oil, grease or dry running), which means that the characteristics for the strength of the flank have to be determined for three different types of lubricant! In principal the measurement can be done with some arbitrary pair of gears, for which for instance an injecting mold is available. The module (DP) should be in a typical range. The calculation method can map the measured values quite good to other modules. Nowadays most of the plastic gears are with module 1.0mm or smaller (DP >= 25 in-1). The measurements for the VDI, however, where done with gears with module in the range of 3…5mm. We recommend using gears with module 1.0mm for the tests. All the other gear parameters like number of teeth, width, profile shift can be chosen arbitrarily, e.g. using existing molds. The centre distance of course has to fit to the test rig. The parameters of the gear are taken into account when evaluating the results of the measurement.

Measurement of Youngs modulus

Youngs modulus versus temperature is an important condition for the correct interpretation of the measured results. Typically these values are provided by the material manufacturer. So in most cases a measurement of this parameter can be omitted. However, for plastic material there is not only one Youngs modulus, but the value is dependant on the type of measurement; it usually is different for tension and compression. The same is true for static and dynamic loads.

Measurement of the material parameters for the root strength

For measuring the root strength some assumptions are done: � The strength is independent from the type of lubrication � The measured permissible stress by using a pulsator is a general characteristic of the

material and independent from the material of the mating gear. This is a large advantage and not possible for the flank strength!

The measurement is done on one tooth (typically the other teeth are taken away) by applying a well defined pulsating normal force (going from 0 to Fmax and backwards) at the tip of the tooth in the direction of the pressure angle αn. It is very important to scan the temperature of the tooth (in the root area) during the test, and if necessary control it. This can be done the easiest by heating the gear up.

Measurement of the material parameters for the flank strength

From literature (Ehrenstein, Höchst,...) it is known that the wear of a gear flank has a linear dependency on the number of load cycles. In most cases the failure of the flank of plastic gears is a continuous reduction of the tooth thickness by abrasive wear. This might in the end lead to fracture of the tooth, typically at the operating pitch diameter. For the test carried out in preparation of the VDI 2545 the defined limit for “failure” was 50% loss of tooth thickness. On this basis the permissible stress σHlim was determined for a given number of load cycles.

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To measure the wear, a test rig is necessary that can run gears with given torque and speed at controlled temperature. So the configuration is usually motor -> torque measurement -> pair of gears -> torque measurement ->brake. The flank temperature must be measured continuously and if necessary must be controlled by heating up the test rig. The test rig from Scholz is equipped with a pyrometer for this, scanning the flank. In addition, the tooth thickness is measured continuously, either automatically or manually, by stopping the gears and measuring the backlash of the gear set. For a new material a first test should be done to check if, after a first running in phase, the wear is really linear over the load cycles . So a first gear must be tested until the tooth thickness is reduced to the limit (e.g. 50%). To reduce the test time for this test it is carried out with a relatively high load. After verifying that the wear is really linear, further tests can be carried out on a reduced basis, for instance until the loss of tooth thickness is 5%. As soon as the slope of the wear can be extracted from the measured points, the rest of the associated wear over load cycles curve can be extrapolated. To get a Wöhler curve for a given temperature, several tests are done on different torque levels. Other parameters, like speed for instance, can have an influence. Our experience has shown that in most cases it can be neglected. However, all the parameters should be in the typical range. Varying all parameters that might have an influence would cause the effort to become unmanageable in size. A general problem is that the behavior of the material is strongly dependant on the combination of both gear materials. In principal, all combinations should be treated individually. This is usually not possible. For the VDI 2545, all measurements were done against a steel gear. For the interpretation of the results, the fact that steel was one of the gears has to be taken into account.

Definition of temperature and load for the tests

The temperature for the typical range of application should be determined. Or, if applicable, the planned range of the temperature for a specific application. Since the shape of the Wöhler curves is very similar in the VDI 2545 (and other measurements the author has seen), it can be assumed that after measuring a full set of data points for one temperature, determining one point of the Wöhler curve only and extrapolating the shape of the fully measured curve to this temperature is sufficient. If high quality of the data is necessary, it might be two or three temperatures for which the full curve is measured, and the others are gained by parallel shift to the one measured point. Typically, the temperature range is divided into 15° or 20° degree intervals, e.g. 40°, 60°, 80°, 100°, 120°, 140° for POM. For some materials, for instance PEEK, the assumption that the Wöhler curves for different temperatures are all of the same shape is not true. According to AGMA 920-A01 “a primary thermal transitional common to all thermoplastic resins is the glass transition temperature, or Tg. For crystalline resins, this is the temperature at which the amorphous regions of the polymer begin to soften […]. Mechanically, crystalline polymers begin to lose a major portion of their modulus through this transition. Amorphous resins, which contain no crystalline regions, very quickly lose all modulus at the Tg, becoming unusable for mechanical purposes.” If used for gears at a temperature level below the glass transition temperature PEEK shows pitting. Above this temperature the failure mode of the flank is wear. Due to this, PEEK shows a significantly higher permissible stress for higher temperature than for lower in a wide range of applications. This example show, that for a new plastic material the behavior can be

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different from what is known from other materials, so all assumptions have to be done carefully. For the determination of the load for the tests, a first test should be done with an intermediate value: Based on a first calculation with a data set for a similar material (as similar as possible, if nothing else is available use the data from the VDI 2545) a torque level is determined that leads to a test time of some hours, two days maximum. A very high load often leads to failures which are not well defined, and the time until failure might be hard to determine. One limit that should be avoided in the beginning is the failure mode by melting teeth. On the other hand, a low load would lead to a very long test time (for instance 108 load cycles mean one or two months of measurement). Although these tests have to be carried out anyway to get a full Wöhler curve, it is better to postpone this long running test until the end of the test program. Often during the first few tests some assumptions will be proven to be too far off from reality, and the test procedure has to be modified. Usually all the tests have to be repeated. When the test results are approximately 106 load cycles, the load is increased with a factor of for instance 1.5 and the next test is run. This is done until the gear fails within less then 104 load cycles. Later, tests with a reduced torque, for instance by dividing the first torque value by 1.5, are carried out, reducing the load until the wear is so small, that due to the uncertainties of the measurement, the progress of the wear can not be measured any more (or the budget for the tests is exceeded).

Floating diagram for the evaluation of the test results for wear

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Figure 10: Floating diagram for the measurement wear and the interpretation of the results

Conclusion

Sizing and rating plastic gears is not a simple task, mainly due to two reasons: first the available literature is poor compared to the level for metal gears and second for the only available method, the VDI2545, there is a severe lack of current data for material properties. To supersede the problems the engineer has to apply the calculation method in a careful way. Often material properties have to be measured. Following the guidelines presented in this paper leads to acceptable results.

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Acknowledgements

The author would like to acknowledge the companies Victrex and Scholz for the data and pictures provided. Furthermore the author wants to thank Dr. Ulrich Kissling, KISSsoft AG and Dan Kondritz, KISSsoft, USA, LLC for their contribution to this paper.

References

[1] VDI 2545, Zahnräder aus thermoplastischen Kunststoffen, 1981. [2] Gunter Erhard, Konstruieren mit Kunststoffen, 2nd edition, Carl Hanser, 1999 [3] Siedke, E: Tragfähigkeitsuntersuchung an ungeschmierten Zahnrädern aus thermoplastischen Kunststoffen, Diss. TU Berlin, 1977 [4] Takanashi, S. and A. Shoji: On the Temperature Risc in the Teeth of Plastic Gears, International Power Transmission & Gearing Conference, San Francisco, 1980 [5] Hachmann H. and E. Strickle: Polyamide als Zahnradwerkstoffe, Konstruktion 18 (1966) 3, S. 81-94 [6] Erhard G. and Ch. Weiss: Zur Berechnung der Zahn- und Flankentemperatur von Zahnrädern aus Polymerwerkstoffen, Konstruktion 39 (1987) 11 S. 423-430 [7] Niemann G.: Maschinenelemente, Band 2. Springer Verlag Berlin, 1983 [8] Obsieger: Zahnformfaktoren von Aussen- und Innenverzahnungen. Konstruktion 32 (1980), S. 443-447.