The number of plastic gears used in different applications is increasing every year, mostly due to their cost effectiveness for large series production and lubrication-free running. With a rapid development of plastic materials in terms of strength and allowable temperature range, plastic gears are finding their way also into more demanding applications, where high transmittable torque at elevated temperatures is required. Unfortunately for gear designers, “gear fatigue data” (S-N curves) are rarely measured for new materials.

If the decision is made to measure gear-fatigue data, it is worth doing it correctly so that the measured data can be used. This paper provides an overview of the testing procedure for plastic gears according to the VDI 2736-4, which was published in 2016. The procedure to measure temperature and other challenges that arise are discussed. Furthermore, the statistical procedure to evaluate test results is explained.

Measured data according to the VDI 2736-4 must be converted correctly to become usable for strength calculation according to the VDI 2736-2. To help test engineers, a software tool was developed. The measured data is introduced, then the software performs the evaluation of the test results automatically and calculates the permissible root/flank stress data for different temperatures.

In the conclusion, some remaining problems are discussed. For the calculation of achievable lifetime or for loads based on duty cycles, the S-N curves (Wöhler lines) should be defined up to 10^{30} cycles, which is far beyond the obtainable test results. Therefore, an extrapolation of the measured data to higher cycle numbers should be considered.

**Introduction**

Due to the well-known advantages of plastic gears, their use is increasing every year, especially in the automotive industry where lubrication-free running, low noise, and high serial production are required [1,2]. With a rapid development of plastic materials in terms of strength and allowable temperature range, plastic gears are finding their way also into more demanding applications, where high transmittable torque at elevated temperatures is required [3,4]. But this is a problem for gear engineers as gear data (permissible stresses) are not being measured at the same rate as new plastics are developed. In fact, there are just a few new materials measured every year, for which permissible root/flank stresses are publicly available. The initiative is mainly coming from the major plastic material companies, which want to promote their materials also for gear applications.

As an engineer, how can you design plastic gears if no reliable fatigue data is available for the material that is used in the application? It is possible to calculate root/flank safety factors with fatigue data from a material, which has gear data available. But due to uncertainties between the two materials in question, the safety factors should be increased/decreased. For high temperatures, there is very little gear data available. Often, extrapolation is used to project permissible stresses at high temperatures, but this can be very inaccurate. At the end, all these uncertainties lead to gear designs that are not optimized in terms of strength and can be far from an optimal solution in terms of price.

An alternative to the above-mentioned procedure is to generate gear data for a material that is being used in the application. Generating root/flank fatigue data on actual gears (on a gear test rig) is a huge effort in terms of time and money spent. Generating permissible root stresses for three different temperatures and for five different numbers of cycles (between 0.1·10^{6} and 5·10^{6}) can easily take between 3-4 months and can cost up to 50,000 euros.

If decision is taken to measure gear data, it is worth doing it correctly so that the measured data can be used at the end. In this paper, an overview of the testing procedure for plastic gears according to the VDI 2736-4 [5], which was published in 2016, is provided. The procedure to measure temperature and other challenges that arise are discussed. The statistical procedure to evaluate test results is also explained. A comparison of the permissible tooth root strength, measured on different test gears, will be discussed, indicating a large scatter between the measured permissible stresses.

Measured data according to the VDI 2736-4 must be converted correctly to become usable for strength calculation according to the VDI 2736-2 [6]. The calculation of tooth root and flank safety factors requires S-N curves, which are temperature dependent. The wear calculation requires wear factors.

To help test engineers, a tool was developed based on the VDI 2736-4. The measured data is introduced, then the tool performs the evaluation of the test results automatically and calculates the permissible root/flank stresses. This information is documented in a text file, which can directly be used by a calculation tool for the calculation of plastic gears according to the VDI 2736-2.

For the safety factor calculation with duty cycles, the S-N curves should be defined up to the 10^{30} cycles, which is far beyond the obtainable test results. Therefore, an extrapolation of the measured data to higher cycle numbers should be considered.

**Converting Cycles to Failure**

According to the VDI 2736-4 guideline, each test condition (T(torque), ϑ(temperature)) must be measured at least three times. To measure a S-N curve for one temperature, cycles to failure should be measured at four different loads, each being repeated at least three times. In total, at least 12 measurements are necessary to calculate a S-N curve for 1 temperature. Cycles to failure for every (T, ϑ) condition and every repetition can be written as

Since the S-N curve has a logarithmic scaling for the number of load cycles, a variable for the logarithm of the number of load cycles is introduced for simplifying the equation.

For every test condition (Ti, ϑi), the mean value of cycles to failure is calculated as

The mean value is related to 50 percent damage probability. The standard deviation is estimated through

To improve the quality of estimates, standard deviations of all (T,ϑ) pairs are normalized and averaged, as shown in

The logarithm of the number of cycles with 10 percent failure probability (assuming normal distribution) are calculated as

and the statistically determined cycles to failure at 10 percent damage probability as

The VDI 2736-4 always calculates with the 10 percent damage probability. If necessary, cycles to failure can easily be calculated also for lower damage probabilities.

In Table 1, a statistical calculation of cycles to failure at different damage probabilities for POM gear is shown. Measured cycles to failure were 143300, 100780, and 94020 cycles (calculated standard deviation of cycles to failure: 26715). It can be seen that, at 10 percent damage probability, the statistically calculated cycles to failure are 30 percent lower than the calculated average value of cycles to failure. Without recalculating cycles to failure to lower damage probabilities, 50 percent of the gears would fail even before the desired lifetime (112700 cycles) is achieved. It is evident that statistical evaluation of the results is necessary to have a reliable lifetime calculation of plastic gears.

To have a valid gear test, the test must run until failure, otherwise it should not be used in the statistical evaluation.

**Geometry of the Test Gears**

The VDI 2736-4 guideline defines three sets of possible test gears, ranging from normal module 1 mm to 4.5 mm. For testing small, injection molded plastic gears, “size 1” gears is preferred. Parameters of the size 1 gears are shown in Table 2. The material for the pinion (DIN quality 6) is steel and for the gear (DIN quality 10), the plastic material that we want to test. A possible material for the pinion is 100Cr6, tempered and hardened to 55 HRC. However, other materials can also be used.

Nevertheless, it is not always possible to use the test gear geometries defined in the VDI 2736-4. The reason could be that the gears are too “strong” for the given test rig (would not be able to get any failure in reasonable testing time) or that the center distance does not fit. In such cases, it is recommended to design your own test gears with similar properties to the VDI defined gears (similar theoretical contact ratio and contact ratio under load, similar specific sliding). This should keep the measured data comparable with the measurements on the VDI defined gear geometries.

It is also very important to check the operating backlash at expected maximum temperatures to avoid gear jamming as this can lead to “wrong” cycles to failure. If jamming occurs, instead of having contact just on the working flanks, additional contacts occur also on the non-working flanks, which results in additional forces, and heating. In cases where contact occurs on both flanks, the gears usually fail after a short number of cycles.

If root or flank fatigue is investigated, it is necessary to prevent excessive wear on the plastic gear as it can affect the outcome of the tests significantly. The surface roughness R_{a} of the steel pinion should be around 0.3 µm or lower (R_{z} < 1.5 µm) [7]. However, smaller roughness does not necessarily lead to lower wear; there might be an optimum with minimal wear. For plastic/plastic material combinations, the presence of fiber reinforcements (glass, carbon) has bigger influence on the wear of the gears than surface roughness [8].

**Measuring and Controlling Temperature**

When conducting gear testing, it is important to have the correct temperature measurements. Here it is crucial that not the ambient temperature is measured, but the relevant material temperature. The most convenient way to measure gear temperatures is using a thermal camera; however, other options are possible (thermal couples). But measuring with a thermal camera is only possible for dry running gears. If oil or grease is used for lubrication, other methods must be used for temperature measurements (measuring oil temperature for instance).

Based on the design of the test rig, there are usually two options to measure gear temperatures: from the top (Figure 1a) and from the side (Figure 1b). If the temperatures are measured from the top, then the flank temperature is measured on the working flank, and the root temperature on the non-working flank. When measuring temperatures from the side, the root temperature can be measured just below the root diameter.

Depending on the expected failure mode (root, flank, wear), the corresponding root and/or flank temperature (see Table 6) must be controlled.

To get reliable fatigue data, a climate chamber is necessary to control the temperatures. If the goal is to measure fatigue data at room temperature (20° C), then it might be necessary to set the ambient temperature in the chamber to e.g. -10° C (or lower, depending on the testing conditions), so the temperature control system must also enable negative temperatures.

**Comparison of Calculated Tooth Root Stresses**

The calculated permissible root/flank stresses are calculation-method dependent. An example below will show how much the results can differ if different calculation methods are used for calculating the permissible stresses and for the calculation of the lifetime. The comparison is done for the following calculation methods: VDI 2736 (YF-C), VDI 2545 (YF-C), and VDI 2545 (YF-B) [11]. The difference between methods YF-B and YF-C is in the location of the applied bending force. For method C, the force is applied at the tip of the tooth, whereas for method B the force is applied at the point of single tooth contact. For a manual calculation, method C is much easier to apply, but method B is potentially more accurate. Both VDI guidelines originally follow method C, so applying method B is a modification of the VDI method.

Table 3 shows the calculated safety factors for size 1 test gears made from POM. If permissible stresses and safety factors are calculated with the same calculation method, then the safety factors are 1 (results on the diagonal from top left to bottom right). But if methods are different, then the results deviate from 1. Safety factor below < 1 indicates that the calculated lifetime is smaller than the actual achievable lifetime (254 h), so the results are on the safe side. In contrary, if the safety factors are > 1, then the calculated lifetime is higher than the achievable lifetime. In worst-case conditions for this given example, the calculated lifetime is by a factor 2.1 higher than the actual achievable lifetime.

**Comparison of Measured Permissible Tooth Root Stresses**

The VDI guidelines contain permissible stress data for various plastic materials. Compared to measured, values this data fits more or less OK to the measurements.

Table 4 shows the comparison of the tooth root stresses for different POM materials at given temperature and cycles to failure. It can be seen that the values for Delrin 100 deviate significantly from the VDI and Delrin 100P measurements. The values for VDI and Delrin 100P are very similar.

Table 5 shows the permissible tooth root stresses for material PA66. At 20° C, the difference between the materials is factor 2, but at 80° C, the difference is much smaller. Still, however, the VDI material has higher permissible stresses.

The results from Table 4 and Table 5 show that there can be big differences between the measurements and the values in the VDI. The problem is that there is hardly any description available on how the permissible stresses in the VDI were measured. But it seems that, with proper testing procedures, it is possible to achieve the VDI measured values for POM material.

With dry running steel/plastic gears, there are a lot of factors that influence the test results (quality of the gears, temperature of the gears, injection molding process, humidity, different test rig layouts, etc.). Consequently, there will always be differences in measured permissible stresses, however they should be within the reasonable limits.

The data contained in the guidelines should only be used if no other data is available. The closer the measured data fits to the real material used, the more reliable the results will be.

**Data Extrapolation**

The permissible stresses for plastic materials are usually measured until 2·10^{6} or 5·10^{6} cycles. However, if the S-N curves are not measured until 10^{30} cycles [12], it is not possible (in KISSsoft) to calculate lifetime safety factors with defined load spectrum or if the number of cycles exceed the measured number of cycles. To make load spectrum calculation possible, measured data should be extrapolated according to some principles. To our knowledge, there are no rules available for extrapolating data from plastic gear tests.

Several proposals were evaluated at KISSsoft. The one shown above was selected as the best. We are proposing a combination of Corten/Dolan approach and Haibach approach.

From the last measured point, we extrapolate the curve with an average slope until 10^{10} cycles. Between 10^{10} and 10^{30} cycles, extrapolation is performed with half the calculated average slope. An example is shown in Figure 2.

**Analyzing Test Results**

Table 6 shows an overview of the parameters that must be measured (or monitored) during gear testing. The measured parameters depend on the desired outcome of the tests (root fatigue, flank fatigue, wear, etc.).

A software program was developed that calculates the corresponding S-N curves based on the results from the gear fatigue measurements. Test results can be imported from a txt file. Figure 3 shows the input fields for the measured data, calculation settings, and test gear geometry. Torque and cycles to failure must be defined on the failed gear.

The extrapolation of the measured points is also possible. As an additional option, the measured S-N curve at lowest temperature can be extended to even lower temperatures using a factor to scale the permissible stresses. Extrapolation can be independently carried out for different lubrication regimes (dry, grease, oil).

Figure 4 shows the S-N curves that were calculated from the measured points. The values at 20° C were extended from 32° C with a factor of 1.2.

**Conclusions**

The number of materials for which gear fatigue data is available is limited. To measure fatigue data for new materials, gear testing is necessary. However, gear testing can be expensive and time consuming, so it should be done correctly, so the measured data can be used for gear calculations. This paper gives a brief overview of the measuring and calculations procedures for plastic gear testing described in the VDI 2736-4.

When measuring cycles to failure, it is important to scale the number of cycles to 10 percent (or lower) damage probability. As shown in Table 1, this results in lowering the average number of cycles for 30 percent, depending on the deviation of the data. When comparing gear-fatigue data, it must be compared at the same damage probability; otherwise this can cause significant inconsistencies.

If designing test gears for the specific test rig, a VDI 2736-4 defined gear geometry should be used. If this is not possible (due to the load capacity or center distances), gears with comparable properties (similar theoretical contact ratio and contact ratio under load, similar specific sliding) should be used. This should keep the measured data comparable with the measurements on the VDI defined gear geometries.

The same calculation method should be used for the calculation of the permissible fatigue stresses from the measured data and for the calculation of safety factors. If this is not the case, then significant differences in lifetime can occur (resulting in under- or over- dimensioned gears), as shown in %%1217-KS-T3%.

Fatigue data for plastic gears is usually measured until 2·10^{6} or 5·10^{6} cycles. The reason is to obtain results in reasonable time. But to calculate lifetime safety factor with load spectrum, it is necessary to have fatigue data available until 10^{30}. To make load spectrum calculation possible, measured data should be extrapolated. A proposed extrapolation method for plastic gears is briefly discussed (Figure 2).

To help test engineers, a program was developed based on the VDI 2736-4. The measured data is introduced, and then the software performs the evaluation of the test results automatically and calculates the permissible root/flank stresses. This information is documented in a text file, which can directly be used by a calculation tool for the calculation of plastic gears according to the VDI 2736-2.

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