An accelerated multilevel test and design procedure for polymer gears

Materials and Design 65 (2015) 961–973

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An accelerated multilevel test and design procedure for polymer gears Aljazˇ Pogacˇnik a, Jozˇe Tavcˇar b,⇑ a b

Iskra Mehanizmi d.o.o., Lipnica 8, 4245 Kropa, Slovenia Faculty of Mechanical Engineering, Aškercˇeva 6, 1000 Ljubljana, Slovenia

a r t i c l e

i n f o

Article history: Received 15 July 2014 Accepted 3 October 2014 Available online 13 October 2014 Keywords: Plastic gear Accelerated testing Lifetime Failure modes Temperature

a b s t r a c t This paper presents a new accelerated testing procedure for plastic gears that is based on several different levels of testing. The iterative testing procedure fulfils requests from the product development process. The following criteria are considered for testing: reduced number of tests, shorter test time and reliable results for different applications. The proposed method was applied over the full range on a gear pair made from polyacetal (POM) and polyamide 6 (PA6). Different rotational speeds and torque loads, and therefore different transferred powers, were used for testing. During testing, gear temperature and cycles to failure were monitored. The paper also includes a comparison between the measured and theoretically calculated gear temperatures. A prediction of the life span on the basis of statistical methods is a part of the proposed test procedure. The presented procedure enables testing within acceptable cost and time consumption limits. The testing method can be reproduced and applied to plastic gears from different materials. Testing has shown that polymer gears fail in two typical ways: by fatigue and by sudden melting. The wear fail mode can be avoided by using an appropriate material pair. Fatigue can be measured by life span tests and is predictable. However, the melting of gears, which is a consequence of high gear temperatures, is not easily predictable. In most cases, melting failure mode occurs during the first few hours of gear testing. For reliable and optimal gear design, gear testing cannot be avoided because the tribological interaction between gears is specific for each combination of materials. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction Plastic gears have been in use since the 1950s, and their popularity has increased significantly in the last decade. The mass production of plastic gears using injection moulding and new plastic materials with improved properties have propelled the application of plastic gears in several different areas, e.g., automotive and medicine. The primary advantages of plastic gears are low manufacturing costs for serial production, no need for external lubrication and good noise damping properties [1,2]. There are also some disadvantages that exclude plastic gears from certain fields of use: plastic materials have inferior mechanical and thermal properties compared to typical gear materials (steel or brass), lower operating temperatures, lower manufacturing tolerances and water absorption [3–5]. A wide variety of different types of polymer materials (PA, POM, etc.), different reinforcements (carbon fibres, glass fibres, nanoparticles, etc.) and internal lubricants (PTFE, MoS2, etc.) can be used to ⇑ Corresponding author. Tel.: +386 40 527 578. E-mail addresses: [email protected] (A. Pogacˇnik), joze.tavcar@ lecad.fs.uni-lj.si (J. Tavcˇar). http://dx.doi.org/10.1016/j.matdes.2014.10.016 0261-3069/Ó 2014 Elsevier Ltd. All rights reserved.

tailor a polymer for a specific application. However, due to the large number of different materials available, it is very difficult to determine the optimal material combination for a specific gear drive, especially when also considering noise and vibration properties [6]. The most widely used standard for the strength calculation of plastic gears is German guideline VDI 2545 [7], which was introduced in 1981 and withdrawn in 1996. It is a simplified version of the metal gears standard DIN 3990 [8]. One of the major restrictions of the VDI 2545 is that fatigue data are only available for three different materials (PA12, PA66, and POM). In 2013, a draft version of a new VDI 2736 guideline for plastic gears was released [2]. However, the guideline still has only a limited number of material data available. AGMA 909-A06, ANSI/AGMA 1006-A97 and ANSI/AGMA 1106A97 address only the geometry of the plastic gears [9], while the British standard BS 6168 [10] includes some details from standard VDI 2545 [7] and contains pure empirical modification of Hachmann–Strickle gears’ temperature model [11]. In the literature and guidelines, the allowable gear endurance limits (for bending and contact stresses) are mainly given for polyamides (PA) and polyacetals (POM). However, because of the

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large number of suitable polymer gear materials available, the standards offer little support for the lifetime calculations of polymer gears from other materials. Only a few attempts have been made to compare the allowable endurance limits of the standards with the results obtained from gear testing [5,12]. The common conclusions of these comparisons show that wide discrepancies exist between standards and tests. The polymer composite gears’ potential use in power transmission is therefore limited due to the lack of understanding of their behaviour at working conditions [12,13]. The fact that polymer materials have much lower mechanical strengths at elevated temperatures, lower thermal conductivities and higher coefficients of thermal expansion makes them highly sensitive to temperature changes [12]. Because several key parameters (torque, speed, material and temperature) have a significant impact on polymer gear performance [5], testing gear geometry and materials combinations cannot be avoided in the gear drive design. Gear testing is very time consuming and expensive, especially when testing several different material combinations at different testing conditions. However, by using accelerated tests, gear testing times and costs can decrease significantly. 1.1. Modes of polymer gear failures Several damage mechanisms can occur on polymer gears according to the literature. The main ones are excessive tooth wear, fatigue (tooth cracking) and tooth deformation [14,15]. The failure mechanism depends on the testing conditions (load, speed and temperature) and material combination [15]. Surface cracks were dominant in unreinforced gears subjected to low stresses. Severe deformation was observed at higher stress levels. Reinforced gears exhibited longer life compared with unreinforced gears due to their superior mechanical strength and thermal resistance [15]. An increase in tooth temperature decreases polymer mechanical properties and also decreases the performance of the gear pair. The enhancement in mechanical properties of polyamide nanocomposite gears results in higher power transmission efficiency compared to unreinforced polyamide gears [16]. Mao performed extensive testing of different gear geometries, loads and with different material combinations [14]. Wear rate is the predominate form of failure for polyacetal (POM) gears, especially above critical loads [14]. Other specific research was performed with the goal of better understanding gear rolling– sliding contact [17] and gear durability [18]. 1.2. Calculation of polymer gear temperature One of the most critical parameters that effects the lifetime of polymer gears is the contact and bulk temperature of the engaging gears. There are several different temperature calculation methods available for polymer gears [5,11,19,20]. The most popular temperature model for plastic gears is the Hachmann–Strickle temperature model [11]. This model is based on thermal equilibrium between the generated heat (at meshing) and the dissipated heat (to the gears, housing and the environment). The drawback of this model is that temperature calculations depend on the coefficient of friction of the materials used and on the coefficient k2, which determines the material pair compatibility. Both coefficients are material related and, therefore, must be determined via testing. In the literature, the values are only known for a limited number of plastic materials and material combinations (like PA/POM and PA/PA), which makes this calculation method very difficult to use on new material calculations. The Hachmann– Strickle model is also used for temperature calculation in the VDI 2545 guideline [7].

The coefficient of friction has a dominant influence on the temperature model. In the literature and standards [2,21], there are different values for the coefficient of friction for the same pairs of materials. Mao suggests designing gears based on the surface temperature of the gears at different operating conditions. For a given gear geometry, a critical operating torque can be calculated from its surface temperature calculation [5]. However, it has limited value for new gear designs using new pairs of plastic materials. Takanashi developed gear temperature equations that contain not only friction generated heat, but also heat generated from gear deformation [1]. The model according to Takanashi and Shoji [20] proved to be precise enough for practical application. However, several of the parameters needed for the calculation of the deformation part are difficult to determine. The Takanashi model was developed using a pair of plastic and metal gears; therefore, it has limited value for pairs of plastic gears. Determination of the gear temperature during meshing is an indispensable step for the precise calculation of plastic gears. The existing models for temperature calculation can give a rough estimate, but they do not provide reliable results. Particularly for the meshing of new material pairs, there is a lack of specific material data. The presented accelerated test model contains temperature calculations using the Hachmann–Strickle temperature model, however with corrected and improved input parameters, which were determined from the test results. 1.3. Accelerated testing Testing of appropriate gear geometries for different material combinations under different operating conditions can be very time consuming. To decrease the necessary test times, accelerated testing can be used. Accelerated testing is an indispensable part of time-effective product development and product validation. At the end of the design process, the design must meet the specified performance, safety, reliability and durability requirements [22]. Reliability and durability testing requires a large number of tests because of uncertainty. Researchers have investigated different methods that correlate the lifetime under accelerated test conditions to the lifetime under normal conditions [23,24]. To reduce the duration of the tests, a highly accelerated life testing method (known as HALT) can be used. It is an effective approach because it is conducted at relatively high stresses for short periods of time. It is very important to prepare a test plan that can effectively assess the robustness of the product (system) for a short duration tests [23]. An alternative to HALT testing is step-stress accelerated life testing (SSALT), a subsection of the accelerated life test (ALT) family [22]. SSALT is a design strategy, where stress is increased or decreased over time [24]. Step increases in the torque are used in first level testing of the proposed accelerated testing model. In the literature, no papers related to the accelerated testing of polymer gears were found. 1.4. Problem definition There is a question of how to accelerate testing and improve the calculation reliability of plastic gears. There is a large variety of different plastic materials, but there is limited information on the mechanical and tribological properties of these materials. One option is to use only well-known materials, for which data is available. However, with a large number of different polymer materials available, gear testing is necessary to determine the performance of modern polymer materials in gears. To determine the polymer gear endurance limits with acceptable testing times and costs, a new multilevel accelerated testing

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procedure is proposed in this paper. The method was tested on polymer gears made of polyamide 6 (PA6) and polyacetal (POM) materials under different testing conditions (up to 2500 rpm and 0.82 N m torque load). A comparison was performed also with other material pairs (using reinforced polyamide 6) using step load increased tests. The test results of the life span and gear temperature measurements (using a thermal camera) were compared with the calculation procedure according to the VDI 2736 guideline. The method enables the determination of the coefficient of friction and an accurate calculation of the bulk temperature of the gears. A prediction of the life span using statistical methods is also a part of the proposed testing procedure. Different rotational speeds and torque values, and therefore different transferred power, were used, and the number of cycles to failure were measured for different material pairs. An additional contribution of the paper is increased accuracy and reliability of the gear calculation method by integrating the existing calculation models and the results of accelerated multilevel testing. Finally, an example is presented that demonstrates the use of the accelerated test results data in gear design.

Preliminary product design: Load level, Speed of rotaon

1. Preliminary Gear design Preliminary gear Design: material selecon

2.1 Accelerated tesng with step increased load (First level tesng) Coefficient of fricon, Esmaon of load level for lifeme tesng Is material pair approved?

(1) Preliminary gear design (simple calculation of gears, material selection). (2) Accelerated multilevel gear testing. 2.1. Preliminary step tests (increasing of test torque). 2.2. Lifetime tests (at selected speed and torque). Full matrix lifetime tests (at different speeds and torque) – optional. (3) Detailed gear design. (4) Gear testing in final application (product validation).

New gear material pair or new tesng parameters are needed

2.2 Lifeme tesng at speed in applicaon (Second level tesng) Life span / load diagram

2. Accelerated multilevel polymer gear testing To have an efficient testing procedure, a balance needs to be found between reliability of the results and the testing time. A proposed accelerated method for polymer gears has several levels of testing, which leads to a reduced number of tests (shorter test times) and also provides reliable test results for different applications. In this paper, a SSALT testing strategy is used to speed-up the gear testing procedure. However, to perform accelerated gear tests, an endurance testing rig for polymer gears is needed. A detailed description of the test rig and measuring equipment is presented in Section 3. A basic scheme of the accelerated gear test/design procedure is shown in Fig. 1. The procedure consists of several steps:

963

Full matrix Lifeme tesng (oponal) Life span / load diagram for all operaon condions 3. Accurate gears calculaon, Detailed gears design

Are Are product product requirements requirements fulfilled? fulfilled?

Detailed Detailed gear gear design, design, Temperature Temperature check check in in applicaon applicaon New gear material pair is needed

4. Validaon tesng on final product (Third level tesng) Fig. 1. Accelerated gear test/design procedure for polymer gears.

2.1. Preliminary gear design

2.2. Accelerated multilevel gear testing

When designing new gear drives for certain application, it is necessary to know the basic requirements for the gears (expected load and tangential sliding speed on gears, expected cycles in life span and environmental conditions). For the preliminary gear design, it is presumed that the loads and speeds of the gear drive are already defined. Based on the standardised gear design procedures [2], approximate gear sizing (module, number of teeth and tooth shape) must be performed first. At the same time, appropriate materials have to be selected for the gears. When selecting materials, special attention must be paid to material pairing (good tribological properties), allowable root and flank stresses, allowable operating temperature of the materials and also to teeth deformations. According to sound level and life span criteria, teeth deformations have to be smaller than 0.4 mm and also lower than 0.1 module [1]. The materials can be selected based on the manufacturer material data or based on the results from previous gear designs.

2.2.1. Preliminary step tests SSALT testing strategy is proposed for the preliminary step tests. Preliminary step tests are made on a polymer gear test rig and on the standard gear geometry (Table 1). The tangential speed in the test must be similar to the tangential speeds expected in the

Table 1 Specification of the test gear parameters. Module Number of teeth Diameter of tip circle Profile shift coefficient x Pressure angle Face width Contact ratio Quality

1 mm 20 22 mm 0 20° 6 mm 1.48 10

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final application. The load during the test is increased in steps, as shown in Fig. 2. The starting load for the step test must be at least 20% lower than the expected load in the application. The duration of the test at 1 load level is determined based on the temperature of the tested gears. The duration must ensure that the gear temperature is stable before moving on to higher loads. The criterion for stable temperature at a single load level: The increase or decrease of average gear temperature after 90% of step test cycle duration has to be lower than 5 °C when compared with average gear temperature after 10% of the step test cycle duration. Additionally, difference between the maximum and minimum temperature during a single load level has to be less than 10 °C. Criteria for stable temperature at a single load level have to be fulfilled in at least 50% of arbitrary load levels of step load test. It is suggested testing for at least 2  105 cycles before increasing the load level. The load level is increased until one of the gears fails. In the case that the first result does not fulfil the requirements, the step load test is repeated using a pair of gears made from different materials or with some other modification (load step, centre distance or number of cycles). Guidelines for preliminary step testing: – Total testing time for one material combination at one speed is limited to 24 h. – Number of cycles at constant load must be chosen so that the gear bulk temperature is stable at more than 50% of load levels. – Fatigue failure mode is analysed using additional tests at constant load levels. – Gear tangential speed during step testing has to be similar to the tangential speed expected in the final application. Information obtained from the preliminary step testing: – – – – – – –

Approval/rejection of selected materials for gears. Coefficient of friction for further gear temperature calculation. Operating bulk temperature of the gear at different load levels. Maximum operating temperature of the gears material. Maximum load for short term gear operation. Expected failure mode (fatigue, wear, . . .). Approval/rejection of gear geometry and centre distances.

the second-level test is to obtain the tribological properties, failure modes and gear pair temperature behaviour. These data are used for detailed gear calculations and, in the next phase, for final product design. For frequently used pairs of materials, full matrix testing at different speeds and loads is recommended (Table 2). Tests at each testing condition are run until one of the gears fails. These tests are very time consuming; however, in-depth knowledge of the selected material combination is collected at different operating conditions, which means that the mechanical and tribological properties of tested materials become well known. An example of full matrix lifetime testing is presented in Section 4. 2.4. Detailed gear design When designing a new gear drive, the material data (root fatigue, coefficient of friction, etc.) are needed for accurate calculations. The result of the step test is the coefficient of friction of the selected material pair, which enables gear temperature calculation. The result of the lifetime test is an acceptable load level (tooth root stress) in relation to the number of cycles. During detailed gear design, several iterations of the stress level calculations are performed for specific gear geometry, according to VDI 2736 [2], as shown in Section 3.4. The calculation model enables the use of test data on standard gears in application specific geometries. At the same time, the gear temperature is checked using the test data updated temperature model [2]. At the end of Section 5, the gear design procedure is demonstrated using an example. 2.5. Gear testing in the final application A gear drive has to be verified in the final application. It is the only way to determine if the gear drive also fulfils all other requirements (noise, shocks, vibrations, etc.), which are difficult to take into consideration during the gear design process. The presented multilevel test and design procedures ensure that gear tooling is performed only once and that the design time and test costs are kept to a minimum. 3. Experimental details

2.3. Lifetime tests

3.1. Gear test rig

The lifetime tests are performed using the same polymer gear test rig and with standard gears; however, this time the tests are performed at constant load and constant speed. It is performed at speeds that are close to the tangential speed in the application. This test is repeated at a few different load levels (Table 2). The goal is to define a diagram that correlates the lifetime (number of loads) and load level (Table 6 and Fig. 7). An additional goal of 0.9 0.8

Torque load, [Nm]

0.7 0.6 0.5 0.4

test speed = constant

0.3 0.2 0.1 0 0

200

400

600

800

1000 1200 1400 1600 1800 2000 2200 2400

Number of cycles [103] Fig. 2. Step test with varying load.

The tests were conducted on an open-loop gear testing machine, which is shown in Fig. 3. The power is provided by a servomotor with a maximum output of 1 N m at 4000 rpm. The motor is connected (via coupling) to the shaft (Fig. 3) on which the driver gear is mounted. The driven gear is mounted on the other shaft, which is connected (via coupling) to the hysteresis magnetic brake. The bearing units, which enable precise rotation of the two shafts, consist of two roller bearings, which are pre-tensioned to eliminate the effect of the bearing backlash. The design of the test rig allows precise positioning of the test gears in x and y directions; the centre distance can be adjusted between 5 mm and 100 mm with a precision of 2 lm. The measuring system consists of a force sensor, which is used to indirectly measure the torque of the test. The software of the test rig allows the test to be run at constant loads and speeds, but it also allows step tests to be performed without the need of stopping the test and changing the torque or speed. The temperature of the gears was measured using a thermal camera (Flir A320, Flir, USA). The bulk gear temperature was measured from the side of the gear, as shown in Fig. 4. The region of temperature measurement was 2 mm  1 mm and was set in the thermal camera software. The emissivity of the test gear materials

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Table 2 Summary of the test conditions and transmitted power. Rotational speed (rpm)

Tang. speed (m/s)

600 840 1176 1646 2305

0.63 0.88 1.23 1.72 2.41

Transmitted power at different tooth root stress (test torque level) 17.0 MPa (0.30 N m)

37 W 52 W 72 W

Table 3 Torque load level and corresponding tooth root stress according to VDI 2736 [2]. Torque (N m) rF (MPa)

0.21 11.9

0.30 17.0

0.42 23.8

0.59 33.4

0.82 45.9

was measured prior to the tests and was set to a constant value of 0.95 throughout the duration of the tests. Digital microscope (Keyence VHX-200) was used to analyse the failure modes of the polymer gears. 3.2. Test gear geometry and selected materials An involute gear geometry was selected for the test gears due to its common use in gear drives [1]. A standard pressure angle of a = 20° was used with no profile shift applied to the geometry (x = 0). A summary of the test gear parameters is presented in Table 1. Because of the thermal expansion of the polymer materials, the operating centre distance was increased from 20.00 mm to 20.05 mm [25]. Three different polymer materials were used for the injection moulding of the test gears. Injection moulding was selected because it is commonly used for serial production of plastic gears. The materials were selected based on their tribological properties [21,26] and also on their common use for polymer gears [27]. Based on tribological test [21] a preferred combination POM/PA6 material was selected. The first abbreviation always refers to the driver gear (POM) and the second to the driven gear (PA6). Selected materials:  Polyamide 6 (PA6, UltramidÒ B3S, BASF).  Polyamide 6 + 30% glass fibres (PA6-30, ZytelÒ 73G30 HSLNC, DuPont).  Polyacetal (POM, DelrinÒ 500P, DuPont). The raw materials were produced in granular form and then injection moulded to the desired gear geometry using a BOY 35 M machine (BOY limited, UK). However, because the selected materials have different mould shrinkages (between 0.6% and 2.0%), different gear moulding inserts were used for production of the test gears. The tool matrix was linearly scaled in all axes. Additional improvements to the gear surface roughness were achieved using tool matrix polishing [28]. After injection moulding, the test gears were measured using a 3D measuring machine (Zeiss, UMC 850) with a standard measuring protocol. According to DIN 58405, the quality of the test gears is 10, which is in the expected range of quality for injection moulded gears in mass production. 3.3. Gear testing 3.3.1. Preliminary step test The rotational speed of 1176 rpm was selected based on the results from our previous tests [25]. It also represents an

23.8 MPa (0.42 N m)

33.4 MPa (0.59 N m)

45.9 MPa (0.82 N m)

37 W 52 W 72 W 101 W

37 W 52 W 72 W 101 W 142 W

52 W 72 W 101 W 142 W

approximate rotational speed of the final gear application. The starting load level was 0.30 N m and increased every 2  105 cycles by 0.05 N m. These values were selected based on the 24 h testing time limitation and also based on the fact, that the temperature must be stable in at least 50% of arbitrary load levels. The criterion was set on the base of several step tests with different material pairs. If the gear temperature is not stable according to the definition in Section 2, the number of cycles at each load level has to be increased. Several preliminary measurements were also conducted to confirm parameter adequacy. The starting load level of 0.30 N m represents the low load level for the selected materials and gear geometry. Different material combinations were tested (PA6/PA6, POM/POM, PA6-30/POM) at room temperature (23 °C) and without lubrication. Based on the results from the gear temperature measurements, the coefficients of friction (COF) for different material combinations and load conditions were calculated. The calculation was based on the VDI 2736 [2]. 3.3.2. Lifetime testing Life span at testing is related to the distribution. According to our experiments, it is better to reduce the number of samples at specified test condition and to introduce so-called matrix test, which increases the number of operating parameters (speed, load, power). The idea on multi-factor experiment originates from the Taguchi design of experiments. At full factorial experiment, each factor or parameter is tested at every level and in every possible combination with the other factors. Systematic approach enables study of paired interactions in economic way. This results in deeper insight into the problem comparing only single factor experiments. The torque and speed for the lifetime test are shown in Table 2. The torque and speed of rotation increase by a factor of 1.4, which is a compromise between the number of tests and the raster of the results. The operating parameters are interrelated, and in this way, it is possible to compare gears that transfer the same quantity of power at different rotation speeds. For some applications, it is better to vary the speed of rotation at the same load level. Testing under different loads conditions typically takes 4 weeks on one testing rig. The accelerated testing procedure is focused on the working conditions close to the application. The determination of the life span over the entire range of load and speed levels (Table 2) or so-called full matrix testing would be much more time consuming. If there is no requirement for extremely long test times, such testing can be finished on a single testing rig in 4–6 months. The paper presents the results of the matrix life span tests for the POM/PA6 material combination. 3.4. Gear calculation methods Tooth root stresses for different torque levels were calculated according to Eq. (1), which is also implemented in the new guideline VDI 2736 [2]. Different coefficients were chosen according to the test gear geometry.

rF ¼ K F  Y Fa  Y Sa  Y   Y b  ðF t =ðb  mn Þ 6 rFP

ð1Þ

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966 Table 4 Calculation of the COF from preliminary step tests. Material combination

PA6/PA6 POM/PA6 POM/POM PA6/PA6-30 POM/PA6-30 PA6-30/PA6-30 a

rF at failure, MPa (torque at failure, N m)

25.5 42.5 31.1 31.1 42.5 36.8

a

Gear bulk temperature at different gear stress/torque

rF = 17.0 MPa (0.3 N m)

rF = 22.6 MPa (0.4 N m)

rF = 28.3 MPa (0.5 N m)

rF = 34.0 MPa

rF = 39.6 MPa

(0.6 N m)

(0.7 N m)

71 °C 46 °C 47 °C 66 °C 52 °C 63 °C

83 °C 64 °C 61 °C 77 °C 55 °C 78 °C

/ 75 °C / 85 °C 65 °C 85 °C

/ 84 °C / / 74 °C /

/ 92 °C / / 83 °C /

(0.45) (0.75) (0.55) (0.55) (0.75) (0.65)

COF

0.48 0.29 0.29 0.40 0.27 0.40

COF – calculated coefficient of friction.

KF – tooth root use factor (based on dynamic analysis) (1), YFa – shape of tooth factor (number of teeth, correction coefficient) (2.9), YSa – tension concentration factor (number of teeth, correction coefficient) (1.6), Ye – coverage ratio factor (0.732), Yb – factor of angled teeth (1), Ft – tangential force (depending on the test torque and gear diameter (20 mm)), b – test gear width (6 mm), mn – test gear module (1 mm). Tooth root stress at different load levels is shown in Table 3. For the calculation of the tooth root stress, the rotational speed is not considered (VDI 2736). According to VDI 2736, root stress is approximately 30% higher compared to the old VDI 2545. The difference originates from tension concentration factor. The temperatures of the gears must be calculated to define the acceptable load level (rFN or rFH) at the specified number of load cycles. The calculated gear temperature according to different models [2,7] depends mostly on an estimation of the coefficient of friction (l) and factor k2. According to tribological tests, the coefficient of friction between PA6 and POM is 0.36 [21]. According to the data in VDI 2736 [2], the coefficient of friction is for the same pair of material 0.18, which has a significant influence on the calculated gear temperature and the allowed load level for a specified life span. The major problem is limited data for the polymer materials. There are some specific calculation methods that also consider the viscoelastic behaviour of the material and load sharing [29,30]. 4. Results 4.1. Preliminary step test results (first-level test) Preliminary step tests were conducted at 1176 rpm. Typical gear bulk temperatures for different torque levels and material combinations are shown in Figs. 5 and 6. Temperature of the gears increases with increasing load level. At smaller number of load cycles, a running in process is not always finished – the gear temperature can increase throughout the whole load level as at material pair POM/PA6 at load levels 1, 2, 3 and 4 (Fig. 6). There is a combination of fatigue and melting failure mode in each step test. From the temperature plot (Fig. 5), temperature scattering can be observed (it is greatest for the PA6/PA6 pair). The temperature data scatter is higher if the materials are not tribologically compatible. The material pairs PA6/PA6, POM/POM and PA6-30/PA6-30 have at several testing steps difference between the maximum and minimum temperature more than 10 °C therefore use of this material pairs has to be avoided (Figs. 5 and 6). Material pair POM/PA6-30 has at load levels 2, 3, 4, 5, 6, 7, 8 and

9 fulfilled criterion for stable temperature (Fig. 5). Material pair POM/PA6 has fulfilled criterion for stable temperature at load levels 4, 5, 6, 7, 8 and 9 (Fig. 6). Material pair PA6-30/PA6 has stable temperature at load levels 1, 2, 3, 4 and 5 (Fig. 6). An important characteristic of plastic gears is also the centre distance of gears in engagement. Sudden temperature increases in the gears can be a consequence of thermal expansion of the gears; the gears can jam. The running in process can take from 15 min to the entire life span. There are several typical modes of running in the temperature increase of the gears: (A) After a gradual increase in the temperature at a specific level, the temperature stabilises (typical for gear made from glass-reinforced materials and gear pair with good matching POM/PA6-30). (B) In the first phase of the running in process, the gear temperature increases to a high level. The temperature decreases back to a constant level, which occurred for the PA6/PA6 gear pair. (C) The running in process is not completed during the entire load step. The temperature increases till the end of the life span (at some load levels of POM/PA6, Fig. 6). (D) There is a combination of the first three modes of the running in process. The maximum gear temperatures and load levels at the failure point of the gears differ for the different pairs of materials (Table 4). For example, a PA6/PA6 gear pair fails at a tooth root stress of 25.5 MPa. The coefficient of friction for the PA6/PA6 pair of material is high (0.48); therefore, it generates a high temperature at a lower load level. The gear pairs made from POM/PA6 or POM/ PA6-30 fails at a much higher load level of 42.5 MPa. A lower coefficient of friction and a lower temperature are key factors for better performance. The results of the step test are used as entry data for the second-level life span test. Using the bulk temperature of the gears measured during the step test, the coefficient of friction was calculated. In the VDI 2736 formula for the calculation of the bulk temperature of the gears, the coefficient of friction has a dominant impact. The coefficient of friction was determined in a way that the measured bulk temperature of the gears matches the calculated temperature. The measured gears temperature is the bulk temperature in the ‘‘thermal view’’, as shown in Fig. 4. The coefficients of friction for different pairs of materials are presented in Table 4. 4.2. Life span test experimental data 4.2.1. Temperatures The bulk temperature of the gears calculated according to VDI 2736 is presented in Table 5. Table 5 also contains the bulk temperature of the gears that was measured using the thermal camera

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Table 5 Calculated (according to VDI 2736) and measured gear bulk temperatures for the POM/PA6 gear pair at different load levels and speeds during the life span test; COF l = 0.29 used in temperature calculation (see Table 4). rpm

Speed (m/s)

Gear bulk temperature at different load levels

rF = 17.0 MPa (0.3 N m) 600

0.63

840

0.88

1176

1.23

1646

1.72

2305

2.41

rF = 23.8 MPa (0.42 N m)

rF = 33.4 MPa (0.59 N m)

rF = 45.9 MPa (0.82 N m)

72 °C 61 °C

91 °C 80 °C

Calculated Measured

61 °C 59 °C

76 °C 65 °C

97 °C 87 °C

Calculated Measured

64 °C 60 °C

81 °C 78 °C

104 °C X

Calculated Measured

55 °C 61 °C

68 °C 73 °C

86 °C 89 °C

132 °C X

Calculated Measured

57 °C 76 °C

71 °C 87 °C

92 °C X

Calculated Measured

Table 6 Summary of the life span test of the POM/PA6 gear pairs. rpm

600 840 1176 1646 2305

Cycles to failure (106, 90% survival rate) at different load levels

Speed (m/s)

0.63 0.88 1.23 1.72 2.41

rF = 17.0 MPa (0.3 N m)

rF = 23.8 MPa (0.42 N m)

rF = 33.4 MPa (0.59 N m)

rF = 45.9 MPa (0.82 N m)

5.43 °C 3.42 °C 1.59 °C 0.76 °C 0.01 °C

0.45 °C 0.30 °C 0.01 °C

15.56 °C 11.75 °C

10.98 °C 7.30 °C 3.62 °C 2.35 °C

4.2.2. Cycles to failure The load level during life span testing is defined using the step test results and the target life span. Eq. (2) can be used for load determination during the life span test and was derived from the POM/PA6 test results. The coefficients ks1 and ks2 were calculated from the life span data in Table 6 with regression between load level and logarithm of life span at step test speed 1176 rpm (median speed at testing). The maximum load level achieved during step testing (rFmax) was integrated into Eq. (2) to make direct connection with step test results.

rF ¼ ks1  rFmax  ks2  log10 ðLife spanÞ

ð2Þ

rF – load level for life span (MPa), rFmax – maximum load level achieved during step testing (MPa), ks1, ks2 – coefficient for life span reduction; value for POM/PA6: ks1 = 2.5; ks2 = 12, Life_span – target life span (number of cycles).

Fig. 3. (a) Schematic and (b) picture of the polymer gear test rig.

during the life span tests. The measured temperatures best match the calculated values at a tangential speed of 1.23 m/s because the coefficient of friction was determined at that speed. At higher tangential speeds, the calculated temperatures are in general lower than the measured values (Table 5), however at lower tangential speeds, the calculated temperatures are higher compared to the measured ones. It should be stressed once again that our coefficient of friction was l = 0.29, which was determined from the step test. The calculated temperatures would be much lower using the coefficient of friction from the POM/PA6 material pair (0.18), which is defined in the VDI 2736 guideline [2].

During the life span test, the load level (torque) is fixed and the speed of rotation is close to the speed during the gear application. A comprehensive picture of the characteristics of the gear and its life span can be determined using tests at different loads and speeds during the so-called matrix test. Performing at least 6 tests from Table 2 is recommended: three speeds of rotation at two load levels. With three different speeds from Table 2, a wide range of working conditions can be evaluated. In most cases, three speeds are enough to get reliable data on material pair for detailed gear calculation. It is recommended to conduct life span tests first with one load level. According to the test results, the load level is then increased or decreased (Table 2) in a way that the testing conditions and expected life span are closer to the working conditions in application. If the reliability of the time span prediction needs to be better, then the number of samples must be increased. The life span matrix test for the POM/PA6 gear pair was conducted

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Fig. 4. Temperature measurement of the driver gear bulk temperature (polymer emissivity: e = 0.95).

Step increased torque test PA6 / PA6

POM / POM

POM / PA6-30

120 PA6/PA6

Gears´ bulk temperature [°C]

110

POM/POM

POM/PA6-30

100 90 80 70 60 50 40 30 20 10 0 0

5

10

15

20

25

30

Test duraon [h] Fig. 5. Step test for PA6/PA6 POM/POM and POM/PA6-30 at 1176 rpm. The starting load level was 0.30 N m (17.0 MPa), which was increased by 0.05 N m (2.85 MPa) after every 2  105 load cycles.

Step increased torque test PA6-30 / PA6

120

PA6-30 / PA6-30 PA6-30/PA6

POM / PA6

PA6-30/PA6-30

POM/PA6

Gears' bulk temperature [°C]

110 100 90 80 70 60 50 40 30 20 10 0 0

5

10

15

20

25

30

Test duraon [h] Fig. 6. Step test for PA6-30/PA6-30, POM/PA6 and PA6-30/PA6 at 1176 rpm. The starting load level was 0.30 N m (17.0 MPa), which was increased by 0.05 N m (2.85 MPa) after every 2  105 load cycles.

according to the test conditions presented in Table 2. For each combination of load levels (torque) and rotation speeds, at least two tests were executed. If the life span of the two tests differentiates by more than 30%, an additional test was performed. The Weibull model was used for the life span prediction. Its main advantage is that it can operate using a small number of samples [31]. The first step is the determination of the distribution or

Fig. 7. Relationship between the root stress rF and the expected life span (number of cycles in millions at different rotation speeds (rpm) for the POM/PA6 gear pair) at 90% survival rate.

shape parameter b in the Weibull model. According to the life span tests, the POM/PA6 gear parameter b was determined to be 5. The shape parameter b depends on the failure mode: for the gear melting failure mode, the parameter b is approximately 3, and for fatigue fracture, it can increase to values higher than 15. The results of three life span tests of POM/PA6 gears at load 45.9 MPa and speed 600 rpm were 0.85  106, 0.55  106 and 0.74  106 cycles. The calculated shape parameter b for these values is 7.1. The scatter of life span tests at load 23.8 MPa and speed 2304 rpm was bigger. The calculated shape parameter b for values 3.55  106, 1.51  106 and 4.30  106 is 3.12. In life span test analysis, the calculation is done with an average b parameter of 5. Calculation of the parameter b with regression was conducted as a standard function in a statistical software MiniTAB. If the results of life span tests are plotted as points into Weibull probability form, an estimation of the parameter b can be seen as slope of the line through the points [31]. The distribution of the result of the life span test parameter b depends also on the material pair combination: for example, the PA6/PA6 gear pair distribution is wider. An improved estimate of the shape parameter b can be determined using additional tests and a calculation for a particular plastic pair and failure mode. The second step was a calculation of the characteristic life (g) according to Eq. (3) [31]:

g¼

" !#1=b b N X ti r i¼1

t – number of load cycles, r – number of failed gears (in our case r = N),

ð3Þ

A. Pogacˇnik, J. Tavcˇar / Materials and Design 65 (2015) 961–973

N – total number of failures plus suspensions, b – distribution or shape parameter (b = 5 from historic data for POM/PA6 testing), g – estimation of the characteristic life (63.2% of gears fail at that value). The characteristic life (g) was calculated for each load combination. At least two life span tests results are needed for this calculation. For example, at a torque of 0.59  N m and a speed of 840 rpm, failure occurred at 5.87  106 and 4.53  106 loads. By inserting these values into Eq. (3), the characteristic life g = 5.36  106 loads was calculated. Engineers are typically interested in the life span with at least a 90% survival rate, which is often denoted B10 and indicates the number of cycles at which 10% of population will fail. The B10 life span for the gears was calculated according to Eq. (4). It was derived from the two parameter Weibull cumulative distribution function [31]. The calculated life span values for POM/PA6 gears with a 90% survival rate are presented in Table 6. 1=b

t B10 ¼ g  ½lnð1  FðtÞÞ

ð4Þ

tB10 – number of load cycles, F(t) – probability of failure (up to time t, in our case number of load cycles) (F(t) = 0.1), b – distribution or shape parameter (b = 5 from historic data for POM/PA6), g – estimate of the characteristic life (63.2% of gears fail at that value). By applying the regression data to the results from Table 6, the life span (in relation to the load and speed) can be calculated, as shown in Eq. (5). In some cases, it is more practical to predict the allowable load level in relation to the life span and the speed of rotation (Eq. (6)). Eq. (6) is derived from Eq. (5). ð1=3Þ

LifeSpan ð1=2Þ

Load

ð1=2Þ

¼ 740  92:0  Load

 67:7  Speed

ð1=3Þ

¼ 8:04  0:0109  LifeSpan

 0:736  Speed

ð5Þ ð6Þ

LifeSpan – expected number of load cycles, Load – load level on gears as root stress – rF (MPa), Speed – tangential speed of gears in kinematic point (m/s). The regression correlation coefficients are R-Sq = 96.0% for Eq. (5) and R-Sq = 97.1% for Eq. (6). Calculated results from Eq. (6) in a linear scale of cycles to failure are graphically presented in Fig. 7. Eqs. (5) and (6) are the result of several iterations in Minitab software. When forming equation for lifespan in relation to load and speed, the main criteria was good regression correlation and also simple form. For the first iteration, logarithmic model was used, however it did not pass the correlation criteria. Better results were achieved by using the third square root of life span, the second square root of load and the first square root of speed. With these factors, the correlation of 96% was achieved. The accuracy of the life span prediction can be further improved by applying the regression separately to each rotational speed. A validation of the life span prediction model (Eq. (5)) was performed using an independent set of POM/PA6 tests at different speeds and load levels. Matching the life span in the prediction model to the test results is reliable, especially at speeds below 1.2 m/s. Based on these results, eight of the nine predicted life span values were shorter than the life span test values. A larger number of tests at higher speeds would increase the accuracy of the life span prediction model.

969

4.3. Gear failure mechanisms Polymer gears typically fail in two ways: by fatigue or by sudden melting. Fatigue can be measured using life time tests, and it is more or less predictable (Fig. 10). The melting of gears is the consequence of overload and temperature rise. In most cases, the melting failure happened during the first few hours of the gear test (Fig. 11). If the gear pair survives the first phase and the gear body temperature is stabilised or decreasing, fatigue will most often be the failure mode. Typical gear failure modes are presented in Table 7. Wear was recognised as a damage mechanism only for some gear pairs, e.g., POM/POM (Fig. 8). In this case, the wear rate was high even at acceptable load levels from the root stress and temperature point of view. The wear was determined with visual inspection on measuring microscope after the life span test. For other combinations of materials (PA6/PA6, POM/PA6), wear was not determined to be the failure mechanism (Fig. 9), even at a high number of loads (16 million). For the PA6-30/POM material pair, wear was detected on the POM gear at a large number of loads. Our conclusion is that the tribological characteristics of the polymer materials pair can be detected using preliminary tribological tests or standard gear test [21]. By avoiding combinations of problematic materials, the wear failure mode can be eliminated in most cases. The sudden increases in the wear rate caused by high temperatures are not classified as a wear failure mode. The fatigue and sudden fractured failure mode is typical for lower loads levels and large numbers of cycles (more than 5  106) (Fig. 10). Our test results using PA6 and PA6-30 show that glass fibre reinforcement improves the robustness of the gears towards strength over time. A lower gear bulk temperature is a consequence of the lower heat generation of the sliding surfaces and also better heat dissipation. At higher numbers of loads, the advantages due to reinforcement are lost if it is in a pair with tribologically incompatible reinforced material (PA6-30/PA6-30) (Fig. 12). A high coefficient of friction generates higher temperatures, which reduces the life span. From a chemical point of view, PA6 and POM polymer have polar chemical bonds. The structures of these materials are different; therefore, the materials do not stick together during contact. 5. Discussion and application Advanced gear drive design requires data on the mechanical and tribological properties of the materials. It is important to have specific data for a pair of gears and not on a single material only. Life span testing has shown that the PA6/PA6 gear pair (module = 1 mm) can transfer 36 W of power. The same PA gear enables the transfer of 72 W of power in a pair with a POM gear at the same speed. The relative difference in power or torque transfer is proportional to the coefficient of friction. Tribological testing of all material pairs would require great effort. There are 45 material pair combinations for 10 materials and 4950 pairs for 100 materials. The PA6-30/PA6-30 gear pair is sensitive to high temperatures and the fatigue failure mode (0.7 million loads at rF = 24 MPa, speed = 1.23 m/s). The same PA6-30 gear in combination with a POM gear has a 17 times longer life span (12 million loads) under the same testing conditions. The PA6-30/POM pair has a failure mode due to wear on the POM gear, which is additional proof that gear material matching is important. The presented accelerated testing procedure demonstrates a pragmatic approach of how to obtain the needed material data for optimised gear design. A standardised plastic gear testing procedure is needed for wide use and the exchange of polymer material data. The FZG test for metal gears can be used as a

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970 Table 7 Typical gear failure modes and mechanisms. Failure mode

Failure mechanism with typical examples

Recommended action to avoid failure

Wear

Occurs at tribologically incompatible material combination (POM/POM)

Melting of gears

Occurs at high load level – high transmitted power Occurs for material combinations with high COF (PA6/PA6) High load level Reinforced material in combination with reinforced material have problems achieving a high load number; the failure mode is combined with the melting failure mode (PA6-30/PA6-30)

Select tribologically compatible materials using preliminary testing Material pair selection with low COF, Lower transmitted power, Gear test Load calculations, Gear testing Material data on endurance limits

Fatigue - tooth root fracture

reference [32]. The test of plastic gears has some specific requirements. According to our experiments, the following parameters need to be defined: – Testing procedure, load level, running in process. – Gear geometry (module, number of teeth on pinion and gear, correction coefficient, and centre distance). – Gear quality level and surface roughness. – Specification of the polymer and injection moulding parameters. – Method of comparable gear temperature measurement (bulk or flash). – Environment parameters (temperature, moisture, cooling, and lubrication). Standard pairs of polymer gears are necessary for the calibration of testing rigs. It is expected that a standard pair of gears with known geometry and material generate temperatures and life spans with a predictable distribution. The technical rule for VDI 2736-4 [33] proposes a specification for this polymer gear geometry and test. The geometry and speed of the test gears is often different from the gears in the application. A larger difference reduces the reliability of the test results. A size of test gears (module = 1 mm) was

selected similar to the gears in applications with a module in the range of 0.5–1.5 mm. There is a good match between the test gears and the test in application (between 20% and 30% of the life span). The results from the test are the material data (rF and rH) for a specific number of loads and bulk gear temperatures. The results are then transferred to the new application geometry based on the VDI 2736 root stress calculation and the bulk temperature model. The accuracies of the models in VDI 2736 are improved using the test data. The friction coefficient is determined based on the temperature measurement, which is then used in the bulk gear temperature model. There is an open question about the reliability of the test results if the gear modules and number of teeth in the applications are 100% higher or smaller than the test gears. The model used for the assessment of the gear temperature during application is based on the standard gear test at equal tangential speeds and equal specific root tooth stress (rF). The model was approved for the selected examples. Additional systematic research is needed to establish a generalised temperature model that will determine reliability limits. 5.1. Application of the test data The example demonstrates how to apply the test data result to a new gear pair design. There is a request to design a gear pair for a

Fig. 8. Wear damage on a POM/POM gear pair (load level 0.42 N m, rotation speed 1176 rpm, wear level after 1.2  106 cycles);

Fig. 9. Undamaged POM gear (from POM/PA6 pair) after 16  106 of cycles (load level 0.3 N m, speed of rotation 2305 rpm).

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Fatigue –POM/PA6

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Table 4). By comparing the measured gear bulk temperature and the temperature calculated according to VDI 2736, the coefficient of friction was calculated (l = 0.29). The matrix life span test results for POM/PA6 gear pair (Fig. 7) were used. After several calculation iterations, the following gear geometry was selected: Module = 0.6 mm; x1 = 0.3, x2 = 0.3. Number of teeth: z1 = 15, z2 = 33. Gear width: b1 = 3.5 mm, b2 = 2.5 mm. Material for smaller gear (z1): POM. Material for bigger gear (z2): PA.

Fig. 10. Sudden tooth fracture failure of PA gear is caused by fatigue on a POM/PA6 gear pair (load level 0.3 N m, speed of rotation 2305 rpm, fractured failure after 16  106 cycles).

Based on results from Eq. (6), the allowable gear load was calculated. Load(1/2) = 8.04  0.0109  LifeSpan(1/3)  0.737  Speed (see Eq. (6)) Load = (8.04  0.0109  LifeSpan(1/3)  0.736  Speed)2 Load = (8.04  0.0109  (20  106)(1/3)  0.736  1.41)2 Load = 16.35 MPa (Allowed load level for the selected material pair at 23 °C and without housing) The tooth root stress on gear 1 and gear 2 according to VDI 2736: rF1 = 6.25 MPa, rF2 = 9.78 MPa (Checking the proposed gear geometry; the temperature needs to be checked separately). According to the calculated gear geometry, speed in the application is 1.41 m/s. It is different from preliminary step test (v = 1.23 m/s) therefore the step test is repeated. The temperature model for the standard gears determined from additional step tests for POM/PA6 at v = 1.41 m/s is presented in Eq. (7). It is valid for an environmental temperature of 23 °C. Eq. (7) is determined from measured temperatures at the step test at different load levels with regression.

Temperature ðstandard gearsÞ Fig. 11. Gear tooth deformation caused by melting and overload of a PA6/PA6 material pair (load level 0.42 N m, rotation speed – 1176 rpm, deflection after 40  103 cycles).

small appliance. The preferred pair of materials is POM/PA6 according to our step tests. 5.2. Design constraints Power = 4.5 W. Reduction of speed = 2.2. Speed = 3000 rpm (smaller gear). Number of loads: 20  106. Already done step test at a speed of 1.23 m/s has indicated the tribological compatibility of the POM/PA6 gear pair (Fig. 6 and

¼ 8:46 þ 16:7  sqrtðrF Þ ð CÞ

ð7Þ

Temperature (standard gears) = 59.07 °C is calculated according to Eq. (7) at the load level rF = 16.35 MPa. At the application conditions (speed = 1.41 m/s), there is temperature matching between the calculated temperature model and the measured data during the step test if the coefficient of friction is set to l = 0.35. Therefore, this coefficient of friction is used to calculate the gear temperature according to VDI 2736. The considered variables are the ambient temperature (40 °C) and the close appliance housing with area 0.0036 m2 further influences the gear temperature. Temperature (application) = 54.0 °C is the calculated gear temperature in the appliance according to VDI 2736 with our improved test data. This temperature is smaller than the temperature of the standard gears according to Eq. (7) (59.07 °C). The final validation

Fig. 12. (a) Damaged PA6-30 gear (from PA6-30/PA6-30 pair) after 0.6  106 of cycles at a load level of 0.42 N m, 1646 rpm. (b) Undamaged PA6-30 gear (from PA6-30/POM pair) after 14.6  106 of cycles at load level 0.42 N m, 1646 rpm.

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speed of rotation. The generated gear temperature is integrated in the model. This paper presents detailed results of the life span matrix test for the POM/PA6 material pair. A comparison with some other typical pairs of materials (PA6-30/POM, PA6/PA6, POM/POM, and PA630/PA6-30) is included. Polymer gears typically fail in two ways: fatigue or sudden melting. Fatigue can be measured using the life span tests and is more or less predictable. The melting of gears is a consequence of overload and an increase in temperature. Wear was recognised as a damage mechanism only for some gear pairs (POM/POM, PA6-30/POM). By avoiding combinations of problematic materials, the wear failure mode can be eliminated in most cases. The existing models for temperature calculations can give a rough estimate, but they do not provide reliable calculations. The presented accelerated test model is an upgrade to existing models using data gathered with testing. The result is increased gear life span and temperature calculation accuracy and reliability. The suggested testing and calculation procedure is presented using an example for a small appliance. Acknowledgements The authors would like to thank European Social Fund for partial financial support. References

Fig. 13. Example of the gear test and calculation procedure.

of the application satisfies the life span and temperature. The entire procedure is presented once again in Fig. 13. 6. Conclusions The main advantage of the multilevel accelerated testing procedure is that it follows the gear development process. During the first level of increased step load, the results have limited precision, but they can be performed quickly and enable decision making for the material and design variants on time. During detailed design, there is additional time for the second level and more precise life span tests. The final validation is performed on the gears built into the final assemblies. The main goal is that the gear design is optimised and tooling is performed only once. The step load test has been demonstrated to be very informative. It tells the characteristics of the material pair, such as the temperature generated at different loads, and it enables the calculation of the coefficient of friction. The information gained from the step load test for tested materials is comparable to tribological tests. The coefficients of friction of the material pairs enable more precise determination of the bulk gear temperatures and, subsequently, the acceptable load level. The guidelines for plastic gear step load tests are defined. A novel contribution of the paper is pragmatic life span data handling. The presented procedure shows how to establish a life span prediction model based on a full matrix life span test. The influencing factors are the load level and the

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