TY - JOUR
AU - Zorko, Damijan
AB - Abstract A reliable method of optimization of polymer gears remains, to date, an open challenge, due to the lack of specific material characterization of polymers and to the complex nonlinear relations between different geometric and operating parameters. For spur and helical gears, the authors herein have developed the optimization algorithm, which primarily enables variation of geometry according to various criteria: the number of teeth (z1, z2), face width (b), helix angle (β), and normal module (mn). The method enables a better insight into how design parameters influence the target criteria. The main paper contribution is a newly developed multicriteria function that enables a simultaneous consideration of different criteria such as root/flank stress, gear bulk/flank temperature, wear, deformation, quality, cost, and volume. Graphical Abstract Open in new tabDownload slide Graphical Abstract Open in new tabDownload slide polymer gears, failure modes, multicriteria function, optimization, material characteristics Highlights The study introduces multicriteria polymer gear optimization function. Load level and operation conditions determine polymer gears’ failure modes. The function simultaneously considers different criteria: root/flank stress, gear bulk/flank temperature, wear, deformation, quality, cost, and volume. Rules and guidelines for polymer gear optimization were established. 1. Introduction Polymer gears dampen vibrations well, operate without lubrication, and can be manufactured at low costs in a multicavity tool by means of injection molding. A weak property of polymer gears is their sensitivity to higher temperatures, as well as a variety of failure modes that depend on the load level and many other gear drive parameters (Senthilvelan & Gnanamoorthy, 2004; Singh et al., 2018; Černe et al., 2020). The mechanical properties of a meshing polymer gear can be improved by using reinforcements such as glass fibers. Gear temperatures must be controlled for them to work properly. The durability of the gear pair can be increased by selecting a tribologically compatible polymer material pair (Mao et al., 2009). Metal gears have valid standards for calculating load capacity, such as DIN 3990: 1987 and ISO 6336: 2006. Such well-defined standards do not exist for plastic gears. The method for calculating the strength of plastic gears was for several years the German guideline VDI 2545: 1981, which was introduced in 1981 and withdrawn in 1996. In 2014, a new design guideline for plastic gears, the VDI 2736: 2014, was released. It is a simplified version of the metal gears standard DIN 3990:1987, which is supplemented with gear temperature and a wear calculation model. However, the guideline still has only a very limited number of material data available. Designers therefore face a lack of material data and material characterization guidelines for gear calculation and design. Polymer gears fail due to different failure modes. The main types of failure include temperature-induced failure, wear, and fatigue (Singh et al., 2018; Jain et al., 2019). The types of polymer-gear failures can vary depending on the operating conditions – which in turn define the acting loads, the gear geometry, the material pairing, and the chosen type of lubrication. For example, an identical material pair at a relatively high load will fail due to an excessive temperature load, while at a lower load it will fail due to wear, and in the event of lubrication it might fail due to fatigue, i.e. tooth fracture (Singh et al., 2018). Due to the large variation of materials, mechanical characteristics, tribological conditions, and thermal conductivity, the failure behavior of polymer gears is very diverse. Das et al. proposed simultaneous selection of material and geometry to ensure a rational design of gears (Das et al., 2016). The gear designer’s additional knowledge and expertise over interrelations and specific properties of the materials and design criteria are important (Milani et al., 2013). Bravo et al. (2015) provided a review of the multiple damage modes for plastic gears. By applying a range of loads on a plastic gear, it was verified that the damage mode depends highly on the applied load (Bravo et al., 2015). Bravo et al. pointed out that the prediction accuracy of fatigue is limited due to the lack of operating data, which restricts further use of polymer gears in heavy-duty conditions (Bravo et al., 2015). It often happens that a combination of several types of failures occurs, e.g. the material melts and the tooth fractures or the tooth wears out and breaks at the root as a result of reduced tooth thickness. Various authors deal with the several types of defects and the conditions in which they occur in different ways: material melting and tooth deflection (Pogačnik & Tavčar, 2015; Singh et al., 2018; Trobentar et al., 2020), fatigue tooth root fracture (Zorko et al., 2017; Tavčar et al., 2018), wear (Zorko et al., 2019), and pitting on the active flank (Hasl et al., 2018). Experimental testing of gears provides an insight into what type of failure should be expected in a real-life application. Gears should be designed according to the expected type of failure. Lupinetti et al. developed a framework for the retrieval of similar models based on geometry and design details (Lupinetti et al., 2018). There is an increasing number of studies devoted to polymer gears. However, each such study is focused on a specific technical issue and it is difficult to apply results generally to different industrial applications and operation conditions. Currently, the German guideline VDI 2736: 2014 for plastic gears offers the most comprehensive approach – it considers various failure modes and proposes models for calculation. However, a weak point of the VDI guideline is, apart from the lack of available material data, that it fails to provide the user with a holistic picture, and it fails to consider all design criteria and interconnections between failure modes. The aim of this study was to provide a more comprehensive approach in the design and calculation of polymer gears. The study starts with a review of failure modes and sensitivity on design parameters. The proposed optimization procedure is based on the multicriteria function (MCF) that concurrently considers all key design criteria. The geometry variation algorithm and a novel MCF are demonstrated using software that enables simultaneous optimization of the root/flank stress, gear bulk/flank temperature, wear, deformation, quality, cost, and volume. The optimization procedure is based on iterative modifications of gear geometry, requests, and criteria checking; it still requires a design engineer in the loop. The accuracy of the calculation of root stress, flank pressure, contact ratio, and deformation was compared and critically assessed by using the finite element method (FEM). 2. The Review of Failure Modes, Design Parameters, and Sensitivity on Polymer Gear Acceptance Criteria Polymer gears have, due to material specifics, several constraints that must be checked during the design process: root and flank strength, temperature, wear, deformation, gear quality, and final quality of installation. The polymer gears’ design must consider all given criteria. The most critical criteria and failure mode are determined with the load level, used material pair, and operating conditions (speed of rotation, environment temperature, and lubrication). Polymer gear design is not a straightforward procedure, but it is rather iterative and must consider and balance several criteria, which are in nonlinear interrelation (Singh et al., 2018). The designer needs to understand the basic approaches how to improve specific criteria of a polymer gear pair (Tavčar et al., 2019). For a more illustrative presentation, the influence of the design parameters was investigated on a specific gear pair case (refer to Section 5). The parameters were selected in specific ranges and they were applied to different material pairs. In the following sections, the models for each of the criteria according to VDI 2736: 2014 are summarized. The rules on how to influence design criteria are presented in Fig. 1. The corrective measures are based on the physical model of each criterion. The proposed optimization procedure is result of comprehensive literature review and research experiences of the authors. The identified sensitivity of different design parameters was evaluated with the OptiTooth software developed by the authors, and it is presented in several diagrams below (Figs 3, 6, and 7). The models for gear quality, quality of installation, and costs are the authors’ new contributions. A detailed presentation of the OptiTooth is given in Section 4.2. Figure 1: Open in new tabDownload slide Gear pair optimization procedure and criteria influencing the gear pair design. The input data and preliminary calculation determine the initial material selection and gear geometry. The optimization procedure is carried out as an iterative search for equilibrium between several criteria with support of the OptiTooth. Figure 1: Open in new tabDownload slide Gear pair optimization procedure and criteria influencing the gear pair design. The input data and preliminary calculation determine the initial material selection and gear geometry. The optimization procedure is carried out as an iterative search for equilibrium between several criteria with support of the OptiTooth. 2.1 Tooth root stress When tooth root stress is high, an increase of the normal module or gear face width linearly reduces root stress (equation 1). Such measures also increase the size of the transmission drive, which is not always acceptable. The model for root stress calculation in equation (1) was defined in VDI 2736. An alternative solution is to use a reinforced polymer material that allows higher stress levels (Tavčar et al., 2018). Tooth root stress can be reduced by modifying the tooth shape – such as a positive profile shift (x1). The tooth root fracture failure mode is a consequence of bending fatigue or root stress overload (Fig. 2). The multicriteria optimization algorithm does not directly consider the various techniques employed to enhance polymer gear performance, such as asymmetric tooth profile (Karthik Pandian et al., 2020), variable face width (Düzcükoğlu, 2009; İmrek, 2009), eccentric gears (Lin et al., 2020), or cooling inserts (Singh et al., 2017). Different enhancements can be applied additionally for improved robustness. Figure 2: Open in new tabDownload slide Root fatigue failure mode was identified on a PPS + CF (carbon fiber) polymer gear. PPS + CF has outstanding quasistatic mechanical characteristics, but it is due to low root fatigue durability not being a good option for gears (Tavčar et al., 2018). Figure 2: Open in new tabDownload slide Root fatigue failure mode was identified on a PPS + CF (carbon fiber) polymer gear. PPS + CF has outstanding quasistatic mechanical characteristics, but it is due to low root fatigue durability not being a good option for gears (Tavčar et al., 2018). Figure 3 shows how the normal module influences the root stress. If the center distance is constant, the tooth root stress significantly increases as the normal module gets smaller (the number of teeth increases). However, root stress is not the only parameter. At the same time, the module size has a negative influence on gear temperature, especially on the pinion. A higher number of teeth (smaller module) improves transmission efficiency and therefore reduces the gear bulk temperature (Fig. 3). That means, if the high temperature is critical for the gear pair failure mode, an increase of module size can make the situation even worse. The tooth root stress level and temperature were calculated according to the VDI 2736 guideline. The root stress variation in Fig. 3 is a consequence of different profile shifts (x1,2) as a mechanism to adapt to the fixed center distance. Figure 3: Open in new tabDownload slide Tooth root stress and gear bulk temperature in relation to the normal module. Gear pair parameters: center distance a = 50 mm, pressure angle α = 20o, helical angle β = 0o, profile: involute (ISO 53 A), face width b = 10 mm, load torque Td = 0.5 Nm, n = 2400 rpm, transmission ratio i = 4, and µ = 0.18 (POM/PA66 material pair). Diagram data are calculated in OptiTooth. Figure 3: Open in new tabDownload slide Tooth root stress and gear bulk temperature in relation to the normal module. Gear pair parameters: center distance a = 50 mm, pressure angle α = 20o, helical angle β = 0o, profile: involute (ISO 53 A), face width b = 10 mm, load torque Td = 0.5 Nm, n = 2400 rpm, transmission ratio i = 4, and µ = 0.18 (POM/PA66 material pair). Diagram data are calculated in OptiTooth. Tooth root stress: $$\begin{eqnarray} {\sigma _{\rm{F}}} = {K_{\rm{F}}}\,\, \cdot {Y_{{\rm{Fa}}}} \cdot {Y_{{\rm{Sa}}}} \cdot {Y_{\rm{\varepsilon }}}\,\, \cdot {Y_{\rm{\beta }}} \cdot \frac{{{F_{\rm{t}}}}}{{b \cdot {m_{\rm{n}}}}} \end{eqnarray}$$(1) KF tooth root load factor YFa shape of tooth factor YSa tension concentration factor Yε contact ratio factor Yβ helix angle factor Ft nominal tangential force b face width mn normal module 2.2 Flank pressure Flank pressure is not very critical for tribologically compatible materials (Singh et al., 2018). According to VDI 2736, flank pressure does not need to be checked for a limited number of high load peaks (<1000) or for dry runs. However, the flank pressure can be reduced by carefully selecting the gear geometry (contact ratio and tooth contact curvature) (Zorko et al., 2017), and by adding suitable profile modifications (Černe et al., 2019/1). Increasing the gear face width and gear diameter reduces the flank pressure. Equation (2) presents the VDI 2736: 2014 model to calculate tooth flank pressure. It is important to understand the nature of each failure mechanism; different load levels and different operating periods generate different failure modes (Singh et al., 2018). Sudden load peaks can cause tooth root fracture (Fig. 2). Short-term overload generates high temperature, especially if the speed of rotation is high, which causes teeth deformation (Fig. 5) or even melting of the polymer gear. Lubricated polymer gears usually suffer from pitting and tooth root fatigue fractures (Hasl et al., 2018; Lu et al., 2019) beyond merely the excessive wear of nonlubricated polymer gears (Zorko et al., 2019). The failure mode prediction model is on steel/POM gears modeled with FEM analysis and proved with experiments on a test rig (Lu et al., 2020), and its results are summarized in Fig. 4. With a decreasing load level, the tooth flank fatigue life falls much faster than the root bending fatigue life. The tooth flank fatigue damage, such as pitting or spalling, is prone to occur under light loading conditions, while tooth root bending fatigue failure may appear under heavy-duty conditions, as observed for PEEK (Illenberger et al., 2019), and POM (Lu et al., 2020). When load level is reduced, failure mode is changed from overload root fracture into fatigue root fracture. Additional reductions in load level increase lifespan and change the failure mode into wear or flank fatigue (Fig. 4). However, flank fatigue can only be achieved in most cases with oil lubrication; otherwise, wear failure mode occurs first (Hasl et al., 2018). Figure 4: Open in new tabDownload slide Correlation between root and flank fatigue failure modes in relation to load level and number of cycles for POM gears. Tooth root fatigue failure appears under heavy-duty conditions, while flank fatigue occurs under lighter loading conditions (Lu et al., n.d.). Figure 4: Open in new tabDownload slide Correlation between root and flank fatigue failure modes in relation to load level and number of cycles for POM gears. Tooth root fatigue failure appears under heavy-duty conditions, while flank fatigue occurs under lighter loading conditions (Lu et al., n.d.). Tooth flank pressure: $$\begin{eqnarray} {\sigma _{\rm{H}}} = \left( {Z_{\rm{Hx}}} \right){\rm{\,\,}}\cdot\sqrt {\frac{{F_{\rm{t}} \cdot K_{\rm{H}} \cdot \left( {i + 1} \right)}}{{\left( {b\cdot d_{\rm{1\,\,}}\cdot i} \right)}}} \end{eqnarray}$$(2) ZHx set of factors KH factor for flank load i gear ratio d1 pinion diameter Similar results were gained by Lu et al. on a steel/PEEK material pair (Lu et al., 2019). The wear measurements and the tooth surface microtopography help us to identify that the micropits near the pitch line lead to the final tooth breakage under moderate loading conditions (Lu et al., 2019). The gear’s material also determines failure modes, as Sarita and Senthilvelan observed pitting near the pitch region under dry conditions when the tested gear pairs of steel/PA66 were subjected to a low load (Sarita & Senthilvelan, 2019). Polymer material and fiber reinforcement have a significant influence on durability. Glass fiber reinforcement significantly increases maximum load level up to few millions of load cycles (Tavčar et al., 2018). However, unreinforced materials are less sensitive to fatigue failure in long running times (Tavčar et al., 2018). 2.3 Gear temperature Gear bulk temperature has a significant influence on polymer gear durability, while a difference of 10°C to 15°C is already significant for the strength of polymer materials. The ambient temperature must be reduced and heat dissipation improved. The gear bulk temperature can be reduced by minimizing the coefficient of friction (COF) or by decreasing the power transmitted over gears (equation 3) through lowering of the torque or running speed (Miler et al., 2019). The model for calculating gear bulk temperature in equation (3) is based on VDI 2736. Using grease or oil can significantly reduce heat generation (Sarita & Senthilvelan, 2019). For dry meshing, an internal lubricant can be helpful. The COF is not a property of a single material, but a pair of materials, wherefore the material pair must be selected carefully (Tavčar et al., 2018; Černe et al., 2019/1). The larger size of a polymer gear improves heat dissipation, and so reduces the gear temperature; the pinion is therefore critical. In several applications, the gear bulk and flash temperatures determine failure mode. Therefore, accurate temperature modeling (Černe et al., 2020) is of key importance. Černe et al. considered the influence of the gears’ quality level on heat generation (Černe et al., 2019/2). A detailed design of the gear tooth profile can additionally reduce gear temperature (Zorko et al., 2019). Fernandes et al. predicted the bulk and flash temperature of polymer gears by FEM, which allows us to predicted bulk temperature under oil lubrication and nonlubricated conditions (Fernandes et al., 2018). Tooth deformation as seen in Fig. 5 or even gear melting is a typical failure mode caused by the gears’ temperature overload. Figure 5: Open in new tabDownload slide Tooth deformation due to thermal and root stress overload. The degradation of tooth geometry increases heat generation and gear’s temperature. Figure 5: Open in new tabDownload slide Tooth deformation due to thermal and root stress overload. The degradation of tooth geometry increases heat generation and gear’s temperature. The pinion is loaded significantly more than the driven gear; the pinion bulk temperature is higher for small center distances (Fig. 6), and the number of load cycles is for a transmission ratio higher on pinion than on the driven gear. The pinion made from a much stronger and temperature-resistant material, such as steel, is an often applied solution. In Fig. 6, it is shown how gear bulk temperature decreases if the center distance between gears and therefore number of teeth increase. The larger gears enable the dissipation of friction-generated heat at lower temperatures due to their larger surface. The pinion, small in size, must have a significantly higher temperature to enable the same heat flow as gear 2. The larger gears also reduce tooth root stress (Fig. 6). Flash temperature model according to VDI 2736 is not realistic [36], [39], while the calculated temperature is often over polymer melting temperature. The flash temperature is also difficult to measure; therefore, it is excluded from the presented examples of multicriteria optimization. Figure 6: Open in new tabDownload slide Gears’ bulk temperature and tooth root stress in relation to the center distance between gears. Gear pair parameters: module mn = 1.0 mm, pressure angle α = 20o, helical angle β = 0o, profile: involute (ISO 53 A), face width b = 10 mm, load torque Td = 0.5 Nm, speed n = 2400 rpm, transmission ratio i = 4, ambient temperature ϑo = 30°C, and µ = 0.18 (POM/PA66). Diagram data are calculated in OptiTooth. Figure 6: Open in new tabDownload slide Gears’ bulk temperature and tooth root stress in relation to the center distance between gears. Gear pair parameters: module mn = 1.0 mm, pressure angle α = 20o, helical angle β = 0o, profile: involute (ISO 53 A), face width b = 10 mm, load torque Td = 0.5 Nm, speed n = 2400 rpm, transmission ratio i = 4, ambient temperature ϑo = 30°C, and µ = 0.18 (POM/PA66). Diagram data are calculated in OptiTooth. Gear bulk temperature: $$\begin{eqnarray} {\vartheta _{{\rm{bulk}}}} = {\vartheta _o}\,\, + P\cdot\mu \cdot{H_{\rm{v}}} \cdot \frac{{{k_{\rm{f}}}}}{{b\cdot\,\,z \cdot {{\left( {v\cdot{m_{\rm{n}}}} \right)}^{0.75}}}} \end{eqnarray}$$(3) ϑo ambient temperature P power μ COF Hv loss degree level kf heat transfer coefficient z the number of teeth v tangential speed 2.4 Gear wear An increased wear rate is typical for a pair of a metal pinion and polymer-driven gear, due to the great difference between the hardness of the surfaces in contact. Proper metal gear processing and tooth tip relief on the pinion can significantly reduce the wear rate (Zorko et al., 2019). In the event of increased wear, the geometry of the tooth profile is damaged. In the first phase, the noise level and temperature increase due to incorrect meshing. Later, the teeth cross-section is reduced significantly, leading to a tooth fracture (Zorko et al., 2019). There are three known wear mechanisms when a polymer comes into contact with a metal: adhesive, abrasive, and thermal wear (Stachowiak & Batchelor, 2014). It would be expected that the lower roughness of steel gears reduces the wear; however, studies have shown that wear for extremely smooth surfaces is comparable with that of relatively rough surfaces (Moder et al., 2018). When a surface is too rough, it accelerates the polymer wear, as the rough surface of a hard metal abrasively wears the polymer. Polymer wear depends on how deep the asperities of the metal surface penetrate, on the shear angle and on the sliding distance. In practice, the wear rate changes, as the valleys gradually become filled with the removed polymer material forming a transfer layer, which reduces the penetration and consequently slows the material erosion (Moder et al., 2018). Conclusion of several researchers is that tribological tests such as pin-on-disc test do not give a good prediction for gear applications and that gear tests are the best option for characterizing the wear factor (Wright & Kukureka, 2001; Cathelin, 2019; Matkovič et al., n.d.). Cathelin proposed improvements on strength, fatigue, and wear measurement techniques (Moder et al., 2018). An alternative for determining the wear coefficient and COF is a twin-disc test (Moder et al., 2018). Moder et al. presented an advanced disc-on-disc machine with sophisticated control technology for both dry and lubricated setups (Moder et al., 2017). Wear is, for a tribologically compatible polymer material pair such as POM/PA66, not a critical failure mode (Pogačnik & Tavčar, 2015). However, the pinion has, under higher load levels, a significantly higher wear rate than driven gears do (Fig. 7). One design solution is to use a wear-resistant material for the pinion; an often applied solution is to use a metal pinion to enable a more compact drive design (Zorko et al., 2019). However, metal pinions in general increase the wear rate on polymer gears, which means that polymer gears must be examined (Fig. 7). $$\begin{eqnarray} {W_{\rm{m}}} = {\rm{\,\,}}\frac{{{T_{\rm{d}}}\cdot 2 \cdot \pi \cdot{N_{\rm{L}}} \cdot {H_{\rm{V}}} \cdot {k_{\rm{w}}}}}{{\left( {b\cdot z\cdot{l_{{\rm{Fl}}}}} \right)}} \end{eqnarray}$$(4)Wear: Td torque load NL the number of load cycles Hv loss degree level kw wear coefficient b gear face width z the number of teeth lFl the length of active flank Figure 7: Open in new tabDownload slide Wear in relation to gear width for POM/PA66 and for Steel/POM material pair. Gear pair’s parameters: z1 = 20, z2 = 80, mn = 1 mm, and NL = 107; Td = 0.5 Nm. Diagram data are calculated in OptiTooth. Figure 7: Open in new tabDownload slide Wear in relation to gear width for POM/PA66 and for Steel/POM material pair. Gear pair’s parameters: z1 = 20, z2 = 80, mn = 1 mm, and NL = 107; Td = 0.5 Nm. Diagram data are calculated in OptiTooth. The model for wear (equation 4) from VDI 2736 has a linear relationship with the load level, number of load cycles (NL), and wear coefficient (kw). In most cases, it is not critical till up to a few million load cycles. However, a high number of load cycles makes wear significant even for a small wear coefficient (kw). Above 106 load cycles, the progress of wear is often more critical than bending fatigue, especially for metal/polymer material combinations in dry running as seen in Fig. 8. Different operation conditions determine the failure mode for the steel/PEEK material pair. In the event of dry running or grease lubrication, wear is a dominant failure mode (Zorko et al., 2019). PEEK gears break at the root due to reduced tooth thickness and not due to fatigue or temperature overload. If steel/PEEK gears operate in oil, durability is improved and pitting on the active flank is failure mode (Illenberger et al., 2019). Figure 8: Open in new tabDownload slide Significant wear of PEEK gear that was running in pair with a steel gear is typical failure mode (Zorko et al., 2019). Figure 8: Open in new tabDownload slide Significant wear of PEEK gear that was running in pair with a steel gear is typical failure mode (Zorko et al., 2019). 2.5 Tooth deformation The deformation of polymer gears is significantly greater than that of metal gears. If the noise level must be reduced or if the transmission error tolerance is small, especially, the tooth deformation must be checked (Hiltcher et al., 2006; Cathelin et al., 2013; Banodiya & Karma, 2017). Designing gears made from identical or similar materials reduce transmission error with an increased contact ratio. When different materials are used, one method of reducing the transmission error variation is balancing teeth stiffness between the pinion and driven gear (Meuleman et al., 2007). If other design criteria are fulfilled, the tooth deformation is not critical. While the tangential force increases deformation, the gear’s face width and elastic modulus decrease tooth deformation. The model for calculating tooth deformation according to VDI 2736 is presented in equation (5). Tooth deformation: $$\begin{eqnarray} \lambda \,\, = {\rm{\,\,}}\frac{{7.5\cdot{F_{\rm{t}}}}}{{\left( {b\cdot\cos (\beta )} \right)}}{\rm{\,\,}}\cdot{\rm{\,\,}}\left(\frac{1}{{{E_1}}}{\rm{\,\,}} + {\rm{\,\,}}\frac{1}{{{E_2}}}\right) \end{eqnarray}$$(5) Ft tangential force b gear face width β helix angle E1,2 elastic modulus (1 pinion, 2 gear) Table 1 compares tooth tip deformation according to VDI 2736 model and FEM analysis. For larger numbers of teeth (mn = 0.6 mm), the FEM deformation is significantly smaller than that with VDI 2736 model due to increased contact ratio. Deformation is presented as movement of a tooth tip, wherefore the deformation of larger tooth (mn = 1 mm) is bigger even if the angle of deformed tooth is smaller than that at mn = 0.6 mm (Table 1). Figure 13: Open in new tabDownload slide Comparison of root stresses calculated using FEM, considering a module variation between 0.6 and 1.0 mm and different load levels: (a) for drive gear (POM) and (b) for driven gear (PA66). Due to contact ratio >2 at module mn = 0.6 mm, typical mesh points B and D do not exist. Figure 13: Open in new tabDownload slide Comparison of root stresses calculated using FEM, considering a module variation between 0.6 and 1.0 mm and different load levels: (a) for drive gear (POM) and (b) for driven gear (PA66). Due to contact ratio >2 at module mn = 0.6 mm, typical mesh points B and D do not exist. Table 1: Comparison of tooth deformation for different load levels between VDI 2736 model and FEM analysis (POM/PA66 pair, face width b = 6 mm, d1 = d2 = 20 mm). . Module = 1.0 mm z1 = z2 = 20 . Module = 0.6 mm z1 = z2 = 34 . Load level (Nm) . Deform. VDI 2736 (mm) . Deform. FEM (mm) . Deform. VDI 2736 (mm) . Deform. FEM (mm) . 0.4 0.027 0.031 0.031 0.020 0.5 0.039 0.034 0.039 0.025 0.6 0.046 0.040 0.046 0.030 . Module = 1.0 mm z1 = z2 = 20 . Module = 0.6 mm z1 = z2 = 34 . Load level (Nm) . Deform. VDI 2736 (mm) . Deform. FEM (mm) . Deform. VDI 2736 (mm) . Deform. FEM (mm) . 0.4 0.027 0.031 0.031 0.020 0.5 0.039 0.034 0.039 0.025 0.6 0.046 0.040 0.046 0.030 Open in new tab Table 1: Comparison of tooth deformation for different load levels between VDI 2736 model and FEM analysis (POM/PA66 pair, face width b = 6 mm, d1 = d2 = 20 mm). . Module = 1.0 mm z1 = z2 = 20 . Module = 0.6 mm z1 = z2 = 34 . Load level (Nm) . Deform. VDI 2736 (mm) . Deform. FEM (mm) . Deform. VDI 2736 (mm) . Deform. FEM (mm) . 0.4 0.027 0.031 0.031 0.020 0.5 0.039 0.034 0.039 0.025 0.6 0.046 0.040 0.046 0.030 . Module = 1.0 mm z1 = z2 = 20 . Module = 0.6 mm z1 = z2 = 34 . Load level (Nm) . Deform. VDI 2736 (mm) . Deform. FEM (mm) . Deform. VDI 2736 (mm) . Deform. FEM (mm) . 0.4 0.027 0.031 0.031 0.020 0.5 0.039 0.034 0.039 0.025 0.6 0.046 0.040 0.046 0.030 Open in new tab 2.6 Quality of gear geometry and installation Gear geometry quality and the quality of installation have significant impacts on durability, heat generation (Li et al., 2018), and noise level. However, a quantitative model that would facilitate numerical optimization according to a quality level is not available yet, and it therefore requires additional research (Černe et al., 2019/2). Molding parameters such as packing time, cooling, molding and melting temperatures, packing and injection pressures, and fiberglass percentages are the most important factors affecting warpage and shrinkage – quality of gear geometry (Hakimian & Sulong, 2012). The model according to equation (6) considers the worst of the discussed quality grades, according to ISO 1328 (ISO 1328-2:1997: Cylindrical Gears—ISO System of Accuracy—Part 2: Definitions & Allowable Values of Deviations Relevant to Radial Composite Deviations & Runout Information, 1997). The authors would like to raise awareness about quality. Quality level: $$\begin{eqnarray} Q\,\, = {\rm{\,\,max}}\left( {{Q_{{\rm{geom}}}},{\rm{\,\,}}{Q_{{\rm{inst}}}}} \right) \end{eqnarray}$$(6) Qgeom quality level of gear geometry Qinst quality level of gear installation 2.7 Cost and volume In most cases, there is a request to have high-performance gears, at low cost, and in a small design volume. Gear characteristics such as root stress, temperature, wear, and deformation improve as gear size increases. However, larger gears increase the cost and design volume. Selecting high-performance polymers with internal lubricants and reinforcements can reduce volume, but can also significantly affect gear cost. Better gear and installation quality can also contribute to better gear performance (Černe et al., 2019/2). Pairing with a metal pinion is a common solution for a compact design. The current model for gear cost is a product of gear and additional processing volume for filling channels, specific mass, cost of gear material, and processing cost (equation 7a). If manufacturing process with cutting is applied, the cost of cut-off volume is additionally considered in the equation (equation 7b). Gear designers must find a balance between functional requests, material characteristics, volume, and costs. The authors propose to determine processing and material costs according to the context of application and gear supplier capability. The cost awareness in the conceptual design phase and checking of key design options as soon as possible is essential for commercial polymer gear applications. The simultaneous considering of the technical and financial aspects of polymer gears is a unique approach in the scientific publications according to the authors’ knowledge. Gear costs: $$\begin{eqnarray} C = \left\{ \begin{array}{ll} \left( {{V_{1,2}} + {V_{{\rm{p}}1,2}}} \right)\cdot{\rho _{1,2}}\cdot\left( {{C_{{\rm{kg}}}} + {C_{{\rm{pi}}}}} \right), & \quad \left( a \right)\,\,\rm{if\,\,injection\,\,molding\,\,is\,\,used} \\ ({V_{{\rm{c}}1,2}}\cdot{\rho _{1,2}}\cdot{C_{{\rm{kg}}}} + {V_{{\rm{m}}1,2}}\cdot{C_{{\rm{pm}}}}), & \quad \left( b \right)\rm{if\,\,machining\,\,\left( {{\rm{gear\,\,cutting}}} \right)\,\,is\,\,used} \end{array} \right. \end{eqnarray}$$(7) V1,2 volume of gear 1,2 Vp1,2 additional processing volume (cold filling channels of gear 1,2) Vc1,2 volume of cylinder 1,2 for gear cutting Vm1,2 cutoff volume from cylinder 1,2 ρ1,2 specific mass of gear material 1,2 Ckg cost of material per kg Cpi processing cost per kg Cpm machining cost per dm3 2.8 Summary of failure modes and optimization criteria A summary of polymer-gear failure modes and references to researchers is presented in Table 2. Failure modes are determined with load level, type of lubrication, operation speed, and material pair. There are some differences between different polymer materials, though general design rules can be set. Temperature overload with gear deformation is critical for pairs of polymer gears in the event of dry running, high load level, and high rotational speed. Running in oil reduces friction and dissipates heat better, and therefore prevents thermal damages. Lifespan is significantly extended; pitting is a typical failure mode for running in oil. A root or pitch fatigue fracture is the expected failure mode for highly loaded polymer gears with grease lubrication. The wear failure mode is critical for polymer gears in pairs with a steel pinion, moderate load level, and dry running or even with grease lubrication. Table 2: A comparison between operation condition and failure modes with references to researchers. Operation conditions . Failure mode, material pair, and references . Very high load level dry run if not specified Melting due to high temperature: steel/PA66 (Senthilvelan & Gnanamoorthy, 2004); PA66-GF30/PA66-GF30 (Yakut & Düzcükoglu, 2014); steel/PA66 (Sarita & Senthilvelan, 2019); steel/PEEK (Lu et al., 2019) Root fracture due to peak overload: steel/POM—oil lubrication (Hasl et al., 2017/1, 2018); different materials (Bravo et al., 2015); POM/PA6 (Pogačnik & Tavčar, 2015); POM/PA6 (Trobentar et al., 2020); POM/POM (Duhovnik et al., 2015); POM/PA6 (Duhovnik et al., 2016); POM/POM (Miler et al., 2019) Thermal teeth deformation: POM/PA6 (Pogačnik & Tavčar, 2015); POM/PA6 (Trobentar et al., 2020); steel/POM—oil lub. (Hasl et al., 2018) Fracture at the pitch line: PA66/PA66 (Mao et al., 2009); steel/PA66 (Sarita & Senthilvelan, 2019) Wear of teeth: PA66/POM, POM/PA66 [4] High load level dry run if not specified Plastic deformation of teeth: steel/PA66 (Senthilvelan & Gnanamoorthy, 2004); steel/PA66 (Sarita & Senthilvelan, 2019); POM/PA6 (Pogačnik & Tavčar, 2015) Root fracture due to root stress overload: steel/ABS, steel/HDPE, steel/POM [2]; POM/PA6 (Pogačnik & Tavčar, 2015); POM/PA6 (Duhovnik et al., 2016); steel/PA66 (Sarita & Senthilvelan, 2019); PA66-GFa/POM, PPS-CFa/POM [16]; steel/POM—oil lub. [18], (Hasl et al., 2017/1); steel/PEEK—oil lub. (Lu et al., 2019); steel/POM—oil lub. (Lu et al., n.d.) Wear of teeth: steel/ABS, steel/HDPE, steel/POM (Singh et al., 2018); POM/POM (Mao et al., 2015) Low load level, long time of operation dry run Root fracture due to fatigue: steel/PEEK (Zorko et al., 2019); steel/PEEK; steel/PA66-GF, steel/PA66-CF (Senthilvelan & Gnanamoorthy, 2004); PA66-GF/POM (Tavčar et al., 2018), PPS-CF/POM (Tavčar et al., 2018); POM/PA6 (Pogačnik & Tavčar, 2015); POM/PA6 (Trobentar et al., 2020) Wear of teeth: steel/PEEK (Zorko et al., 2019; Kurokawa et al., 2000); steel/PA66-GF(Senthilvelan & Gnanamoorthy, 2004); steel/PA66-CF (Senthilvelan & Gnanamoorthy, 2004); POM/POM (Jain et al., 2019); different materials (Bravo et al., 2015) Pitch cracks: steel/PA66 (Senthilvelan & Gnanamoorthy, 2004) Pitting in pitch region: steel/PA66 (Sarita & Senthilvelan, 2019) Low load level, long time of operation oil lubrication Pitting on flanks (flank fatigue): steel/PEEK (Lu et al., 2019); steel/PEEK (Illenberger et al., 2019) Fatigue fracture between the pitch and the tooth tip: steel/POM (Lu et al., n.d.) Scuffing in region close to tooth root: steel/PA66 (Sarita & Senthilvelan, 2019) Operation conditions . Failure mode, material pair, and references . Very high load level dry run if not specified Melting due to high temperature: steel/PA66 (Senthilvelan & Gnanamoorthy, 2004); PA66-GF30/PA66-GF30 (Yakut & Düzcükoglu, 2014); steel/PA66 (Sarita & Senthilvelan, 2019); steel/PEEK (Lu et al., 2019) Root fracture due to peak overload: steel/POM—oil lubrication (Hasl et al., 2017/1, 2018); different materials (Bravo et al., 2015); POM/PA6 (Pogačnik & Tavčar, 2015); POM/PA6 (Trobentar et al., 2020); POM/POM (Duhovnik et al., 2015); POM/PA6 (Duhovnik et al., 2016); POM/POM (Miler et al., 2019) Thermal teeth deformation: POM/PA6 (Pogačnik & Tavčar, 2015); POM/PA6 (Trobentar et al., 2020); steel/POM—oil lub. (Hasl et al., 2018) Fracture at the pitch line: PA66/PA66 (Mao et al., 2009); steel/PA66 (Sarita & Senthilvelan, 2019) Wear of teeth: PA66/POM, POM/PA66 [4] High load level dry run if not specified Plastic deformation of teeth: steel/PA66 (Senthilvelan & Gnanamoorthy, 2004); steel/PA66 (Sarita & Senthilvelan, 2019); POM/PA6 (Pogačnik & Tavčar, 2015) Root fracture due to root stress overload: steel/ABS, steel/HDPE, steel/POM [2]; POM/PA6 (Pogačnik & Tavčar, 2015); POM/PA6 (Duhovnik et al., 2016); steel/PA66 (Sarita & Senthilvelan, 2019); PA66-GFa/POM, PPS-CFa/POM [16]; steel/POM—oil lub. [18], (Hasl et al., 2017/1); steel/PEEK—oil lub. (Lu et al., 2019); steel/POM—oil lub. (Lu et al., n.d.) Wear of teeth: steel/ABS, steel/HDPE, steel/POM (Singh et al., 2018); POM/POM (Mao et al., 2015) Low load level, long time of operation dry run Root fracture due to fatigue: steel/PEEK (Zorko et al., 2019); steel/PEEK; steel/PA66-GF, steel/PA66-CF (Senthilvelan & Gnanamoorthy, 2004); PA66-GF/POM (Tavčar et al., 2018), PPS-CF/POM (Tavčar et al., 2018); POM/PA6 (Pogačnik & Tavčar, 2015); POM/PA6 (Trobentar et al., 2020) Wear of teeth: steel/PEEK (Zorko et al., 2019; Kurokawa et al., 2000); steel/PA66-GF(Senthilvelan & Gnanamoorthy, 2004); steel/PA66-CF (Senthilvelan & Gnanamoorthy, 2004); POM/POM (Jain et al., 2019); different materials (Bravo et al., 2015) Pitch cracks: steel/PA66 (Senthilvelan & Gnanamoorthy, 2004) Pitting in pitch region: steel/PA66 (Sarita & Senthilvelan, 2019) Low load level, long time of operation oil lubrication Pitting on flanks (flank fatigue): steel/PEEK (Lu et al., 2019); steel/PEEK (Illenberger et al., 2019) Fatigue fracture between the pitch and the tooth tip: steel/POM (Lu et al., n.d.) Scuffing in region close to tooth root: steel/PA66 (Sarita & Senthilvelan, 2019) a GF – glass fibers; CF – carbon fibers. Open in new tab Table 2: A comparison between operation condition and failure modes with references to researchers. Operation conditions . Failure mode, material pair, and references . Very high load level dry run if not specified Melting due to high temperature: steel/PA66 (Senthilvelan & Gnanamoorthy, 2004); PA66-GF30/PA66-GF30 (Yakut & Düzcükoglu, 2014); steel/PA66 (Sarita & Senthilvelan, 2019); steel/PEEK (Lu et al., 2019) Root fracture due to peak overload: steel/POM—oil lubrication (Hasl et al., 2017/1, 2018); different materials (Bravo et al., 2015); POM/PA6 (Pogačnik & Tavčar, 2015); POM/PA6 (Trobentar et al., 2020); POM/POM (Duhovnik et al., 2015); POM/PA6 (Duhovnik et al., 2016); POM/POM (Miler et al., 2019) Thermal teeth deformation: POM/PA6 (Pogačnik & Tavčar, 2015); POM/PA6 (Trobentar et al., 2020); steel/POM—oil lub. (Hasl et al., 2018) Fracture at the pitch line: PA66/PA66 (Mao et al., 2009); steel/PA66 (Sarita & Senthilvelan, 2019) Wear of teeth: PA66/POM, POM/PA66 [4] High load level dry run if not specified Plastic deformation of teeth: steel/PA66 (Senthilvelan & Gnanamoorthy, 2004); steel/PA66 (Sarita & Senthilvelan, 2019); POM/PA6 (Pogačnik & Tavčar, 2015) Root fracture due to root stress overload: steel/ABS, steel/HDPE, steel/POM [2]; POM/PA6 (Pogačnik & Tavčar, 2015); POM/PA6 (Duhovnik et al., 2016); steel/PA66 (Sarita & Senthilvelan, 2019); PA66-GFa/POM, PPS-CFa/POM [16]; steel/POM—oil lub. [18], (Hasl et al., 2017/1); steel/PEEK—oil lub. (Lu et al., 2019); steel/POM—oil lub. (Lu et al., n.d.) Wear of teeth: steel/ABS, steel/HDPE, steel/POM (Singh et al., 2018); POM/POM (Mao et al., 2015) Low load level, long time of operation dry run Root fracture due to fatigue: steel/PEEK (Zorko et al., 2019); steel/PEEK; steel/PA66-GF, steel/PA66-CF (Senthilvelan & Gnanamoorthy, 2004); PA66-GF/POM (Tavčar et al., 2018), PPS-CF/POM (Tavčar et al., 2018); POM/PA6 (Pogačnik & Tavčar, 2015); POM/PA6 (Trobentar et al., 2020) Wear of teeth: steel/PEEK (Zorko et al., 2019; Kurokawa et al., 2000); steel/PA66-GF(Senthilvelan & Gnanamoorthy, 2004); steel/PA66-CF (Senthilvelan & Gnanamoorthy, 2004); POM/POM (Jain et al., 2019); different materials (Bravo et al., 2015) Pitch cracks: steel/PA66 (Senthilvelan & Gnanamoorthy, 2004) Pitting in pitch region: steel/PA66 (Sarita & Senthilvelan, 2019) Low load level, long time of operation oil lubrication Pitting on flanks (flank fatigue): steel/PEEK (Lu et al., 2019); steel/PEEK (Illenberger et al., 2019) Fatigue fracture between the pitch and the tooth tip: steel/POM (Lu et al., n.d.) Scuffing in region close to tooth root: steel/PA66 (Sarita & Senthilvelan, 2019) Operation conditions . Failure mode, material pair, and references . Very high load level dry run if not specified Melting due to high temperature: steel/PA66 (Senthilvelan & Gnanamoorthy, 2004); PA66-GF30/PA66-GF30 (Yakut & Düzcükoglu, 2014); steel/PA66 (Sarita & Senthilvelan, 2019); steel/PEEK (Lu et al., 2019) Root fracture due to peak overload: steel/POM—oil lubrication (Hasl et al., 2017/1, 2018); different materials (Bravo et al., 2015); POM/PA6 (Pogačnik & Tavčar, 2015); POM/PA6 (Trobentar et al., 2020); POM/POM (Duhovnik et al., 2015); POM/PA6 (Duhovnik et al., 2016); POM/POM (Miler et al., 2019) Thermal teeth deformation: POM/PA6 (Pogačnik & Tavčar, 2015); POM/PA6 (Trobentar et al., 2020); steel/POM—oil lub. (Hasl et al., 2018) Fracture at the pitch line: PA66/PA66 (Mao et al., 2009); steel/PA66 (Sarita & Senthilvelan, 2019) Wear of teeth: PA66/POM, POM/PA66 [4] High load level dry run if not specified Plastic deformation of teeth: steel/PA66 (Senthilvelan & Gnanamoorthy, 2004); steel/PA66 (Sarita & Senthilvelan, 2019); POM/PA6 (Pogačnik & Tavčar, 2015) Root fracture due to root stress overload: steel/ABS, steel/HDPE, steel/POM [2]; POM/PA6 (Pogačnik & Tavčar, 2015); POM/PA6 (Duhovnik et al., 2016); steel/PA66 (Sarita & Senthilvelan, 2019); PA66-GFa/POM, PPS-CFa/POM [16]; steel/POM—oil lub. [18], (Hasl et al., 2017/1); steel/PEEK—oil lub. (Lu et al., 2019); steel/POM—oil lub. (Lu et al., n.d.) Wear of teeth: steel/ABS, steel/HDPE, steel/POM (Singh et al., 2018); POM/POM (Mao et al., 2015) Low load level, long time of operation dry run Root fracture due to fatigue: steel/PEEK (Zorko et al., 2019); steel/PEEK; steel/PA66-GF, steel/PA66-CF (Senthilvelan & Gnanamoorthy, 2004); PA66-GF/POM (Tavčar et al., 2018), PPS-CF/POM (Tavčar et al., 2018); POM/PA6 (Pogačnik & Tavčar, 2015); POM/PA6 (Trobentar et al., 2020) Wear of teeth: steel/PEEK (Zorko et al., 2019; Kurokawa et al., 2000); steel/PA66-GF(Senthilvelan & Gnanamoorthy, 2004); steel/PA66-CF (Senthilvelan & Gnanamoorthy, 2004); POM/POM (Jain et al., 2019); different materials (Bravo et al., 2015) Pitch cracks: steel/PA66 (Senthilvelan & Gnanamoorthy, 2004) Pitting in pitch region: steel/PA66 (Sarita & Senthilvelan, 2019) Low load level, long time of operation oil lubrication Pitting on flanks (flank fatigue): steel/PEEK (Lu et al., 2019); steel/PEEK (Illenberger et al., 2019) Fatigue fracture between the pitch and the tooth tip: steel/POM (Lu et al., n.d.) Scuffing in region close to tooth root: steel/PA66 (Sarita & Senthilvelan, 2019) a GF – glass fibers; CF – carbon fibers. Open in new tab 3. Research Method Research on the design and optimization of polymer gears started with a comprehensive review of polymer gear design parameters, failure modes, and calculation models. The review summarizes the results of other researchers, design guidelines VDI 2736, and the authors’ prior research, and builds upon existing knowledge with several sensitivity diagrams and correlations between design parameters and gear acceptance criteria. The review finishes with a comparison between operation conditions and failure modes. Based on systematic research, the authors designed a gear pair optimization procedure and elaborated criteria influencing the gear pair design as presented in the next section and in Fig. 1. The generally presented model was supplemented in the next step with a mathematical model, namely a multicriteria polymer gear function (MCF; equation 8). The MCF is presented in one step; nevertheless, it is the result of several years of experimental and analytical research on polymer gears by the authors (Pogačnik & Tavčar, 2015; Zorko et al., 2017; Tavčar et al., 2018; Černe et al., 2019/1, 2020; Zorko et al., 2019). The MCF represents an integrated view on how design parameters determine acceptance criteria. MCF’s applicability was demonstrated in the OptiTooth software developed by the authors. OptiTooth facilitates variations in gear pair geometry and calculates the MCF for each variant, which is the basis for quickly and reliably assessing a high number of geometric variants. However, OptiTooth and the proposed gear optimization model still require a design engineer in the optimization loop. The optimization process goes through several iterations. The role of design engineer is first the active setting of design constraints and material selection. The case study proves the applicability of MCF and demonstrates how variation in design requirements influences costs and gear pair design. 4. Results 4.1 Gear pair optimization procedure and MCF Gear durability is measured as a product of the rotational speed and operating time, which determines the number of load cycles. The durability, or lifespan, is influenced by various operation conditions: gear geometry and quality, load level, ambient temperature, lubrication, and material. It is important to know how to select gear material, with reinforced fibers or without, and how to influence tooth root stress or gear temperature. The rules on how to influence design criteria are summarized in Fig. 1, as a result of literature review and research experiences of the authors. Table 3 supplements corrective measures with more general rules or guidelines for gear/drive design improvements. The rules are based on the present authors’ experience acquired over several years of testing polymer gears, conducting applicable projects, and by studying publications (Pogačnik & Tavčar, 2015; Singh et al., 2018; Tavčar et al., 2018; Zorko et al., 2019; Černe et al., 2020). The MCF (equation 8) represents a model that goes toward mathematical optimization of a complex technical problem as is polymer gear design. Table 3: Rules for polymer gear optimization. Gear drive application request . Optimization rules . Compact drive • Apply reinforced/high-performance materials with improved mechanical characteristics – especially for pinion • Apply steel (metal) pinion • Improve the geometric quality of gears (Černe et al., 2019/1) • Control overload peaks and do volume optimization • Apply planetary gears Normal lifespan (NL < 5 million load cycles) • Use reinforced polymers (Tavčar et al., 2018) • Conduct precise calculation Long lifespan (NL > 10 million load cycles) Reduced wear • Check fatigue durability before applying reinforced polymers (basic polymers can be better; Tavčar et al., 2018) • Apply lubrication (reduce temperature) • Check wear coefficient and tribological compatibility of the gear pair • High-quality gears • Apply lubrication with oil Low cost • Apply basic polymer materials • Optimize gear size/volume Low noise • Apply helical gears (contact ratio >2) • Use at least one polymer gear in gear chain (to damp vibration) • Improve the quality of gears (quality grade Q9 or better) • Reduce transmission error (Meuleman et al., 2007) • Apply proper tooth tip (Zorko et al., 2019) Robust drive • Control load peaks and gear temperature • Use polymers with low friction coefficient (internal lubricant; Tavčar et al., 2018) • Apply lubricant to reduce friction heat generation • Apply materials with good heat dissipation (glass fibers) • Apply polymers that resist higher temperature • Lubrication with oil; oil reduces friction and dissipates heat • Increase gear size • Combine polymer gear with a metal pinion • Apply high-performance polymers Robust drive specific approaches • Width-modified gear teeth (Düzcükoğlu, 2009) • Asymmetric polymer gears (Karthik Pandian et al., 2020) • Application of S-tooth profile (Zorko et al., 2019; Trobentar et al., 2020) • Steel-pin inserts into gear tooth (Kim, 2006) • Applications of dry film lubricants (Dearn et al., 2013) Gear drive application request . Optimization rules . Compact drive • Apply reinforced/high-performance materials with improved mechanical characteristics – especially for pinion • Apply steel (metal) pinion • Improve the geometric quality of gears (Černe et al., 2019/1) • Control overload peaks and do volume optimization • Apply planetary gears Normal lifespan (NL < 5 million load cycles) • Use reinforced polymers (Tavčar et al., 2018) • Conduct precise calculation Long lifespan (NL > 10 million load cycles) Reduced wear • Check fatigue durability before applying reinforced polymers (basic polymers can be better; Tavčar et al., 2018) • Apply lubrication (reduce temperature) • Check wear coefficient and tribological compatibility of the gear pair • High-quality gears • Apply lubrication with oil Low cost • Apply basic polymer materials • Optimize gear size/volume Low noise • Apply helical gears (contact ratio >2) • Use at least one polymer gear in gear chain (to damp vibration) • Improve the quality of gears (quality grade Q9 or better) • Reduce transmission error (Meuleman et al., 2007) • Apply proper tooth tip (Zorko et al., 2019) Robust drive • Control load peaks and gear temperature • Use polymers with low friction coefficient (internal lubricant; Tavčar et al., 2018) • Apply lubricant to reduce friction heat generation • Apply materials with good heat dissipation (glass fibers) • Apply polymers that resist higher temperature • Lubrication with oil; oil reduces friction and dissipates heat • Increase gear size • Combine polymer gear with a metal pinion • Apply high-performance polymers Robust drive specific approaches • Width-modified gear teeth (Düzcükoğlu, 2009) • Asymmetric polymer gears (Karthik Pandian et al., 2020) • Application of S-tooth profile (Zorko et al., 2019; Trobentar et al., 2020) • Steel-pin inserts into gear tooth (Kim, 2006) • Applications of dry film lubricants (Dearn et al., 2013) Open in new tab Table 3: Rules for polymer gear optimization. Gear drive application request . Optimization rules . Compact drive • Apply reinforced/high-performance materials with improved mechanical characteristics – especially for pinion • Apply steel (metal) pinion • Improve the geometric quality of gears (Černe et al., 2019/1) • Control overload peaks and do volume optimization • Apply planetary gears Normal lifespan (NL < 5 million load cycles) • Use reinforced polymers (Tavčar et al., 2018) • Conduct precise calculation Long lifespan (NL > 10 million load cycles) Reduced wear • Check fatigue durability before applying reinforced polymers (basic polymers can be better; Tavčar et al., 2018) • Apply lubrication (reduce temperature) • Check wear coefficient and tribological compatibility of the gear pair • High-quality gears • Apply lubrication with oil Low cost • Apply basic polymer materials • Optimize gear size/volume Low noise • Apply helical gears (contact ratio >2) • Use at least one polymer gear in gear chain (to damp vibration) • Improve the quality of gears (quality grade Q9 or better) • Reduce transmission error (Meuleman et al., 2007) • Apply proper tooth tip (Zorko et al., 2019) Robust drive • Control load peaks and gear temperature • Use polymers with low friction coefficient (internal lubricant; Tavčar et al., 2018) • Apply lubricant to reduce friction heat generation • Apply materials with good heat dissipation (glass fibers) • Apply polymers that resist higher temperature • Lubrication with oil; oil reduces friction and dissipates heat • Increase gear size • Combine polymer gear with a metal pinion • Apply high-performance polymers Robust drive specific approaches • Width-modified gear teeth (Düzcükoğlu, 2009) • Asymmetric polymer gears (Karthik Pandian et al., 2020) • Application of S-tooth profile (Zorko et al., 2019; Trobentar et al., 2020) • Steel-pin inserts into gear tooth (Kim, 2006) • Applications of dry film lubricants (Dearn et al., 2013) Gear drive application request . Optimization rules . Compact drive • Apply reinforced/high-performance materials with improved mechanical characteristics – especially for pinion • Apply steel (metal) pinion • Improve the geometric quality of gears (Černe et al., 2019/1) • Control overload peaks and do volume optimization • Apply planetary gears Normal lifespan (NL < 5 million load cycles) • Use reinforced polymers (Tavčar et al., 2018) • Conduct precise calculation Long lifespan (NL > 10 million load cycles) Reduced wear • Check fatigue durability before applying reinforced polymers (basic polymers can be better; Tavčar et al., 2018) • Apply lubrication (reduce temperature) • Check wear coefficient and tribological compatibility of the gear pair • High-quality gears • Apply lubrication with oil Low cost • Apply basic polymer materials • Optimize gear size/volume Low noise • Apply helical gears (contact ratio >2) • Use at least one polymer gear in gear chain (to damp vibration) • Improve the quality of gears (quality grade Q9 or better) • Reduce transmission error (Meuleman et al., 2007) • Apply proper tooth tip (Zorko et al., 2019) Robust drive • Control load peaks and gear temperature • Use polymers with low friction coefficient (internal lubricant; Tavčar et al., 2018) • Apply lubricant to reduce friction heat generation • Apply materials with good heat dissipation (glass fibers) • Apply polymers that resist higher temperature • Lubrication with oil; oil reduces friction and dissipates heat • Increase gear size • Combine polymer gear with a metal pinion • Apply high-performance polymers Robust drive specific approaches • Width-modified gear teeth (Düzcükoğlu, 2009) • Asymmetric polymer gears (Karthik Pandian et al., 2020) • Application of S-tooth profile (Zorko et al., 2019; Trobentar et al., 2020) • Steel-pin inserts into gear tooth (Kim, 2006) • Applications of dry film lubricants (Dearn et al., 2013) Open in new tab The optimal design and calculation of different variants can be supported with an MCF (equation 8). A multicriteria approach considers the right balance between all of the noted criteria according to operating conditions and design constraints. It is a challenge to model multiple design criteria with a single function, while the complexity of the interconnected criteria is high. The model for the MCF in equation (8) is based on the target maximum value for each of the criteria, which is set in the denominator. If any of the criteria exceed the target maximum, the MCF increases rapidly with a square function. The maximum values are different for each gear if the material characteristics differentiate. The proposed model facilitates comparison and consideration of all criteria at the same time. The MCF and each other criterion are calculated separately for each of the gears in the pair. The MCF of the gear pair is the larger MCF value between the one calculated for the pinion and the one for the driven gear. The MCF model can also be used with a limited set of selected criteria, if not all the specific material data are available, or if some criteria are not applicable or have but minor importance for the specific application. Application of the MCF model is demonstrated in a case study, as presented in the next section. Several tests were conducted primarily for dry running, due to the prevalent requirements of such type of operation in small polymer drives. It is not necessary to check the flank pressure during dry running according to VDI guideline. The flash temperatures are not considered in the example, while the VDI 2736 model predicts very high, even unrealistic temperatures, and it is technically difficult to measure temperature in the contact (Černe et al., 2019/1). MCF: $$\begin{eqnarray} MCF = {{\left( {\frac{{{{\rm{T}}_{{\rm{bulk1,2}}}}}}{{{{\rm{T}}_{{\rm{bmax}}}}}}{\rm{\,\,}}} \right)}^2} + {{\left( {\frac{{{{\rm{T}}_{{\rm{flash1,2}}}}}}{{{{\rm{T}}_{{\rm{flmax}}}}}}{\rm{\,\,}}} \right)}^2} + {{\left( {\frac{{{{\rm{\sigma }}_{{\rm{\,\,F1,2}}}}}}{{{{\rm{\sigma }}_{{\rm{Flim}}}}}}{\rm{\,\,}}} \right)}^2} + {{\left( {\frac{{{{\rm{\sigma }}_{{\rm{H1,2}}}}}}{{{{\rm{\sigma }}_{{\rm{Hlim}}}}}}{\rm{\,\,}}} \right)}^2} + {{\left( {\frac{{{{\rm{W}}_{{\rm{1,2}}}}}}{{{{\rm{W}}_{{\rm{max}}}}}}{\rm{\,\,}}} \right)}^2} + {{\left( {\frac{{{{\rm{\lambda }}_{{\rm{1,2}}}}}}{{{{\rm{\lambda }}_{{\rm{max}}}}}}{\rm{\,\,}}} \right)}^2} + + {{\left( {\frac{{{{\rm{Q}}_{{\rm{1,2}}}}}}{{{{\rm{Q}}_{{\rm{min}}}}}}\,\,} \right)}^2} + {{\left( {\frac{{{{\rm{C}}_{{\rm{1,2}}}}}}{{{{\rm{C}}_{{\rm{max}}}}}}\,\,} \right)}^2} + {{\left( {\frac{{{{\rm{V}}_{{\rm{1,2}}}}}}{{{{\rm{V}}_{{\rm{max}}}}}}\,\,} \right)}^2} \end{eqnarray}$$(8) Tbmax maximum bulk temperature Tflmax maximum flash temperature σFmax maximum root stress σHmax maximum flank pressure Wmax maximum wear λmax maximum deformation Qmin minimum quality grade Cmax maximum gear cost Vmax maximum gear volume 4.2 MCF software application for polymer gear optimization The developed software for polymer gear optimization is a target application of the MCF. It enables the automatic variation of geometry according to input data and predefined constraints. A user has an overview of all possible options and of the way design parameters influence key criteria during gear calculation. An algorithm for gear pair geometry variation is presented in Fig. 9; this geometry variation is the basis for the optimization procedure in the first level of calculation. The gear geometry is created inside several loops according to the input data and variation range (the number of teeth, helix angle, and face width). Figure 9: Open in new tabDownload slide Flowchart showing the algorithm for gear geometry variation. The gear geometry is created and optimized through several loops according to the input data and variable ranges of teeth numbers, helix angles, and face widths. Figure 9: Open in new tabDownload slide Flowchart showing the algorithm for gear geometry variation. The gear geometry is created and optimized through several loops according to the input data and variable ranges of teeth numbers, helix angles, and face widths. OptiTooth software was developed using Visual Basic for Applications (MS Excel). The platform offers a user-friendly interface, debugging tools, and sufficient performance. In calculating gears and for research purposes, flexibility is more important than computational power. Commercially available software such as KISSsoft also has an option to generate geometric variants of polymer gears. However, KISSsoft does not have any function for the holistic assessment of gear variants such as an MCF. Therefore, the design engineer must select the optimal gear design manually on the basis of partial calculation results for each criterion. The normal module is calculated in relation to the load level, material characteristics, proposed number of teeth, and relative face width as defined in VDI 2736. The next-largest standard normal module is used in the variation procedure. Checking for geometry consistency includes calculating whether the proposed geometry fits into the requested center distance (Fig. 9). The gear calculation results are presented even if some of the criteria are not fulfilled. Searching for an optimal design requires the gear designer to see the whole picture, including gear design variants that, according to the specified safety specifications, fail to meet requirements. By applying different kinds of improvements, such gear designs can lead to the optimal solution. The results of the first level calculation are presented in the main window in the center of the user interface (marked with a red-dashed rectangle; Fig. 10). In each line, one possible gear pair design variant is presented with the basic parameters: the number of teeth (z1, z2), gear ratio (i), normal module (mn), face width (b), helix angle (β), gear diameter (d1, d2), profile shift (x1, x2), root stress safety factor (sfn1, sfn2), total contact ratio factor (εtotal), and MCF value (multicriteria), as defined in Section 4.1. Figure 10: Open in new tabDownload slide Main user interface for OptiTooth with input parameters on the upper part of the window. The results of the first level calculation are presented in the center. The lower part of the window presents results according to different criteria. Critical values are marked in red. For the selected gear pair design option, a detailed calculation is made and exported as a report. Figure 10: Open in new tabDownload slide Main user interface for OptiTooth with input parameters on the upper part of the window. The results of the first level calculation are presented in the center. The lower part of the window presents results according to different criteria. Critical values are marked in red. For the selected gear pair design option, a detailed calculation is made and exported as a report. For each selected gear pair, the user can modify the face width (b) and run calculations on the second level. This option for face width setting gives the designer additional possibility to check different options and sensitivity to design criteria. The results are presented in the bottom of the window for Gear 1 and Gear 2 separately (marked with the blue dashed–dot rectangle) as shown in Fig. 10. If any of the criteria [bulk or flank temperature (Tfu, Tfl), root stress (σf), flank stress (σh), wear (Wm), or deformation (λ)] exceeds the allowed limits, it is marked in red. Users have the option to switch each criterion on or off with a tick. If the selected criterion is switched on, it is considered in the MCF according to equation (7). The user also has the option to set a maximum value for the criteria according to the known data and specific requests of the application (in the middle row: Tfumax, Tflmax, σfmax, σhmax, …, etc.). An overview on how gear design parameters affect different design criteria is the main advantage of the presented computer program. A gear designer has a tool to optimize gear design in a balanced way within the set constraints. The important part of the optimization procedure is material parameters such as COF, wear coefficient, material strength in relation to temperature, and the number of load cycles. 4.3 Application of multicriteria optimization on e-bike drive The application design requirements are the input data for gear pair optimization. It is assumed that the center distance is a part of boundary conditions for the drive, as well as power, rotational speed, transmission ratio, and target cost. The material of both gears and the detailed gear geometry must be specified in the optimization process. An example of the geometry and transmission ratio is taken from an e-bike drive. Gear pair parameters are as follows: center distance a = 50 mm, input torque Td = 0.5 Nm, speed n = 2400 rpm, transmission ratio i = 4, the number of load cycles NL = 107, and cost (€0.30 for the pinion). Optimization was started with a tribologically compatible pair of basic materials POM/PA66. In the next steps, it was checked what can be achieved with high-performance polymers and steel pinion. According to the geometry variation algorithm presented in Fig. 9, face width (b) was in the range between 4 and 30 mm, the number of teeth (z1) was between 11 and 50, and helix angle β was between 0 and 40o. Result of MCF in relation to gear face width is presented in Fig. 11. It can be seen that the optimal face width is between 8 and 10 mm. Module size and tooth profile shift (x1, x2) are selected according to the number of teeth and design constraints (center distance). Figure 11: Open in new tabDownload slide MCF in relation to gear face width and contribution of face width to gear cost. Gear pair parameters are as follows: mn = 1 mm; Td = 0.5 Nm; n = 2400 rpm; i = 4; POM/PA66 pair. Data are calculated in OptiTooth. Figure 11: Open in new tabDownload slide MCF in relation to gear face width and contribution of face width to gear cost. Gear pair parameters are as follows: mn = 1 mm; Td = 0.5 Nm; n = 2400 rpm; i = 4; POM/PA66 pair. Data are calculated in OptiTooth. The MCF was checked additionally in relation to the center distance for the gear parameters (b = 10 mm; mn = 1 mm). The optimal MCF for the POM/PA66 material pair is at a center distance a = 45 mm. Figure 7 shows how the wear decreases if the gear face width is expanding; critical, here, is the pinion. On the other side, gear volume and cost are increasing linearly with the face width; the driven gear has a predominant contribution here (Fig. 12). A logical conclusion is that a high-performance material should be used for pinion to have a balanced load between pinion and driven gear. Figure 12: Open in new tabDownload slide MCF and root stress in relation to the face width. Gear parameters are as follows: a = 40 mm; i = 4; Td = 0.5 Nm; mn = 1 mm; NL = 107; steel/PEEK pair. Diagram data are calculated in OptiTooth. Figure 12: Open in new tabDownload slide MCF and root stress in relation to the face width. Gear parameters are as follows: a = 40 mm; i = 4; Td = 0.5 Nm; mn = 1 mm; NL = 107; steel/PEEK pair. Diagram data are calculated in OptiTooth. The gear drive design constraints can already set the center distance. Another target is to make the drive design as compact as possible. If the pinion is made from steel, the driven gear becomes a weak point. The center distance can be reduced from 50 to 40 mm, but it is recommended to keep face width b = 10 mm due to root stress (Fig. 12). The steel pinion is, due to processing costs, in such a configuration, more expensive than a driven gear made from POM and manufactured by injection molding. If a high-performance material such as PEEK is used for a driven gear, the volume can be further reduced; at center distance a = 40 mm, the face width can be reduced from 10 to 5 or even 4 mm. However, due to the high cost of PEEK material, the MCF in Fig. 12 does not have a minimum any more. If the target cost remains at Cmax = €0.30, as it is in the POM/PA66 combination, then the cost criterion is so dominant that the MCF grows indefinitely along with the gear face width (and volume). In such cases, the target cost shall be set to a realistic value or each of the criteria must be checked one by one. 5. Discussion The paper’s main contribution is a holistic approach to polymer gear optimization. A good understanding of the expected failure modes and how modifying design parameters can influence different criteria is essential for an effective optimization. Models for different criteria are presented in the introduction section, while a summary of how to improve each individual design criterion is given in Fig. 1. The corrective measures are based on the physical model of each criterion. Table 3 supplements corrective measures with more general rules or guidelines for gear/drive design improvements. The MCF (equation 8) represents a model that goes toward mathematical optimization of a complex technical problem as is polymer gear design. The case study in Section 5 demonstrates how geometry parameters and material selection influence the gear design. The idea of the MCF is fast feedback to the polymer gear design engineer. The current version of the program enables the design engineer to see the sensitivity of MCF and recognize the weak point of design parameters in relation to the constraints. The authors have been experimenting with gear geometry variations on metal gears for several years already. MCF has made optimization with geometric variation applicable also for polymer gears with a large number of criteria. Several applications have shown the advantage of using MCF in combination with geometry variation. MCF enables fast assessment of high number of geometric variants that cannot be done manually. The result of checking the whole design space is better design – smaller volume or cost for the same gear performance (durability). The design engineer is expected to make decisions and set constraints, and in this way actively determine the solution space. The optimization procedure, therefore, assumes the active role of the design engineer in the iterative optimization loop. Some optimization steps could be done automatically. For the specific constraints, the optimum of MCF could be calculated with numerical tools, e.g. a search for the minimum. However, for replacing design engineers and their broad knowledge, an advanced, expert system would be needed to combine different materials and design constraints. This is a new topic that poses a potential challenge for future research. 5.1 Characterization of material data The material characterization is conducted at specific gear geometry, temperature, and speed of rotation. If the geometry and operating conditions in the application differ significantly from the test gears, the reliability of lifespan predictions and the accuracy of calculations are lower. The optimization model is based on specific material characteristics. Without reliable material data, optimization cannot be conducted precisely, especially for different pairs of high-performance materials. Table 5 specifies the polymer material characteristics that are needed for accurate gear optimization. Testing a single polymer can take several months. Producing a realistic comparison between different polymer materials requires standardized procedures. In recent decades, several polymer materials have been tested and analysed, but very often in a specific way as defined by researchers. Results therefore cannot be compared and used in an optimization procedure. Most scientific durability tests take only up to a few million load cycles and have a specific research focus; such results cannot be used for applications where more than 10 million load cycles are needed. A step forward was made in the fourth part of the VDI 2736-4:2016 guideline, where gear samples and test conditions are specified. VDI 2736-4 proposes tests with a steel pinion and polymer gear. Such tests are necessary, while a combination of a small metal pinion and polymer gear is very often a solution in applications. However, for the optimization of a polymer/polymer gear pair, the obtained information from such test is insufficient. COF and wear factor are properties of a material pair and not of a single material (Tavčar et al., 2018). Besides the material characteristics from Table 5, there are some additional characteristics needed for gear optimization, such as the heat transfer coefficient, which in most cases is available on a material data sheet. Milani and Shanian propose six material performance indices for material selection based on design criteria (Milani & Shanian, 2006). 5.2 Limitation of the conducted research The optimization algorithm is derived by calculating the criteria based on models defined in VDI 2736. Due to the fact that the VDI guidelines originate from the metal gear standard, there are some shortcomings for polymer gears (Zorko et al., 2019). The root stress calculated by the VDI 2736 model is prominently higher than that when calculated with the FEM. Details of the conducted FEM analyses are presented at the end of this section. The bending strength of polymer gears with a higher number of teeth and lower Young’s modulus withstands higher tooth root stresses according to VDI 2736 due to neglecting load-induced deflections (Hasl et al., 2017/2, 2018). The additional safety in the VDI 2736 model reduces the accuracy of the optimized gear geometry. There is room for upgrading the VDI 2736 gear calculation model, so that it will result in a better match between the calculated and experimental results. However, the authors argue that it does not change the proposed model for multicriteria optimization. If gear design is based on gear tests results (material characteristics) with similar gear geometry and operational parameters as in the application, the influence of the calculation model is mitigated. An optional approach is to apply the VDI 2736 calculation model for conceptual design and FEM analysis for final validation. Motivation for the multicriteria polymer gear optimization approach was the results of gear tests with reduced module size. The used gear testing rig had a limited maximum torque (Tavčar et al., 2018). Because of the plan for testing high-performance polymers, the test gear module was reduced from 1.0 to 0.6 mm (Tavčar et al., 2018). For applied loads of 0.6 Nm, root stress increased by 54.22%, up from 30.45 to 46.96 MPa according to VDI 2736 (Table 4). The tested gear pairs of POM/PA66 had the same diameter for driving and driven gears (d1 = d2 = 20 mm), face width b = 6 mm, and involute profile with a pressure angle of α = 20o, without profile shift, z1 = z2 = 20 when mn = 1 mm and z1 = z2 = 34 when mn = 0.6 mm. The results of durability tests without temperature control on the open-loop test rig have shown that lifespan of gears with module 0.6 mm is on average similar to gears with mn = 1 mm (Table 4). For a moderate load level (Td = 0.4 Nm), the lifespan of gears with mn = 1 mm was 5.8 million load cycles vs. 8.6 million load cycles with the module 0.6 mm. For load level Td = 0.5 Nm, the durability of gears with the module 1 mm was better (Table 4). In all cases, the teeth root fatigue of POM gear was a failure mode. However, for the higher load level Td = 0.6 Nm, the thermal failure mode dominated at gears with larger module. The gears’ bulk temperature measurements have shown that gears with the module 0.6 mm operate on average at 7°C lower than those with a larger module (Table 4). Based on the multicriteria approach and FEM analyses, these initially unexpected results could be explained. Additional FEM analysis has shown smaller difference in root stress (30.9% at drive gear and 20% at driven gear) due to increased contact ratio in comparison with VDI 2736: (2014) (Table 4). In Fig. 13, difference in the duration of higher root stress level due to difference in the number of teeth can be seen. Higher number of teeth improve efficiency in the torque transmission and therefore reduce the gears’ temperature. 7°C is already a significant difference, especially at the higher temperatures associated with polymer materials. Table 4: Comparison of root stress, lifespan, total contact ratio, and gear’s bulk temperature between gear with the module 1.0 and 0.6 mm. The temperature was measured during the test using a thermal camera (Flir A320, Flir, USA), ϑambient = 23°C. Load level (Nm) . 0.4 . 0.5 . 0.6 . . Root stress VDI 2736 (MPa) 20.3 25.4 30.5 mn = 1.0 mm; z1,2 = 20 Root stress FEM (MPa) 21.0 24.0 27.5 Life span (106) 5.5 2.3 0.04 Total contact ration 1.814 1.831 1.868 Gear’s bulk temp. VDI 2736 (°C) 60 67 75 Measured gear’s bulk temp. (°C) 49.5 68 89 Root stress VDI 2736 (MPa) 31.3 39.1 47.0 mn = 0.6 mm; z1,2 = 34 Root stress FEM (MPa) 25.0 30.0 36.0 Life span (106) 8.6 1.6 0.71 Total contact ratio 2.078 2.145 2.171 Gear’s bulk temp. VDI 2736 (°C) 49 54 58 Measured gear's bulk temp. (°C) 43 62.5 82 Load level (Nm) . 0.4 . 0.5 . 0.6 . . Root stress VDI 2736 (MPa) 20.3 25.4 30.5 mn = 1.0 mm; z1,2 = 20 Root stress FEM (MPa) 21.0 24.0 27.5 Life span (106) 5.5 2.3 0.04 Total contact ration 1.814 1.831 1.868 Gear’s bulk temp. VDI 2736 (°C) 60 67 75 Measured gear’s bulk temp. (°C) 49.5 68 89 Root stress VDI 2736 (MPa) 31.3 39.1 47.0 mn = 0.6 mm; z1,2 = 34 Root stress FEM (MPa) 25.0 30.0 36.0 Life span (106) 8.6 1.6 0.71 Total contact ratio 2.078 2.145 2.171 Gear’s bulk temp. VDI 2736 (°C) 49 54 58 Measured gear's bulk temp. (°C) 43 62.5 82 Open in new tab Table 4: Comparison of root stress, lifespan, total contact ratio, and gear’s bulk temperature between gear with the module 1.0 and 0.6 mm. The temperature was measured during the test using a thermal camera (Flir A320, Flir, USA), ϑambient = 23°C. Load level (Nm) . 0.4 . 0.5 . 0.6 . . Root stress VDI 2736 (MPa) 20.3 25.4 30.5 mn = 1.0 mm; z1,2 = 20 Root stress FEM (MPa) 21.0 24.0 27.5 Life span (106) 5.5 2.3 0.04 Total contact ration 1.814 1.831 1.868 Gear’s bulk temp. VDI 2736 (°C) 60 67 75 Measured gear’s bulk temp. (°C) 49.5 68 89 Root stress VDI 2736 (MPa) 31.3 39.1 47.0 mn = 0.6 mm; z1,2 = 34 Root stress FEM (MPa) 25.0 30.0 36.0 Life span (106) 8.6 1.6 0.71 Total contact ratio 2.078 2.145 2.171 Gear’s bulk temp. VDI 2736 (°C) 49 54 58 Measured gear's bulk temp. (°C) 43 62.5 82 Load level (Nm) . 0.4 . 0.5 . 0.6 . . Root stress VDI 2736 (MPa) 20.3 25.4 30.5 mn = 1.0 mm; z1,2 = 20 Root stress FEM (MPa) 21.0 24.0 27.5 Life span (106) 5.5 2.3 0.04 Total contact ration 1.814 1.831 1.868 Gear’s bulk temp. VDI 2736 (°C) 60 67 75 Measured gear’s bulk temp. (°C) 49.5 68 89 Root stress VDI 2736 (MPa) 31.3 39.1 47.0 mn = 0.6 mm; z1,2 = 34 Root stress FEM (MPa) 25.0 30.0 36.0 Life span (106) 8.6 1.6 0.71 Total contact ratio 2.078 2.145 2.171 Gear’s bulk temp. VDI 2736 (°C) 49 54 58 Measured gear's bulk temp. (°C) 43 62.5 82 Open in new tab Table 5: Required polymer material characteristics for accurate gear optimization. Material characteristics . Type of testing . Comment . Elastic modulus E(ϑ) (MPa) Universal testing machine with temperature chamber It is preferred to have E for different temperatures or at least for the operation temperature of gears Bending fatigue strength σF (ϑ, NL) (MPa) Gear test on test rig in pair with steel gear or specific dynamic test that enables application of cyclic load In relation to temperature and number of load cycles Flank strength σH (ϑ, NL) (MPa) Gear testing on test rig, running in oil Test in oil is necessary; otherwise, other failure modes occur first COFa µ (/) Gear testing on test rig + temperature measurement Material pair specific characteristic; COF is not a property of a single material, but of a pair of materials Twin-disc test – secondary option Wear coefficienta kw (10−6 mm3/Nm) Gear testing on test rig + wear measurement Material pair specific characteristic Twin-disc test – secondary option Material characteristics . Type of testing . Comment . Elastic modulus E(ϑ) (MPa) Universal testing machine with temperature chamber It is preferred to have E for different temperatures or at least for the operation temperature of gears Bending fatigue strength σF (ϑ, NL) (MPa) Gear test on test rig in pair with steel gear or specific dynamic test that enables application of cyclic load In relation to temperature and number of load cycles Flank strength σH (ϑ, NL) (MPa) Gear testing on test rig, running in oil Test in oil is necessary; otherwise, other failure modes occur first COFa µ (/) Gear testing on test rig + temperature measurement Material pair specific characteristic; COF is not a property of a single material, but of a pair of materials Twin-disc test – secondary option Wear coefficienta kw (10−6 mm3/Nm) Gear testing on test rig + wear measurement Material pair specific characteristic Twin-disc test – secondary option a If application operates in extreme conditions, COF and kw must be measured at such extreme temperature and other conditions. Open in new tab Table 5: Required polymer material characteristics for accurate gear optimization. Material characteristics . Type of testing . Comment . Elastic modulus E(ϑ) (MPa) Universal testing machine with temperature chamber It is preferred to have E for different temperatures or at least for the operation temperature of gears Bending fatigue strength σF (ϑ, NL) (MPa) Gear test on test rig in pair with steel gear or specific dynamic test that enables application of cyclic load In relation to temperature and number of load cycles Flank strength σH (ϑ, NL) (MPa) Gear testing on test rig, running in oil Test in oil is necessary; otherwise, other failure modes occur first COFa µ (/) Gear testing on test rig + temperature measurement Material pair specific characteristic; COF is not a property of a single material, but of a pair of materials Twin-disc test – secondary option Wear coefficienta kw (10−6 mm3/Nm) Gear testing on test rig + wear measurement Material pair specific characteristic Twin-disc test – secondary option Material characteristics . Type of testing . Comment . Elastic modulus E(ϑ) (MPa) Universal testing machine with temperature chamber It is preferred to have E for different temperatures or at least for the operation temperature of gears Bending fatigue strength σF (ϑ, NL) (MPa) Gear test on test rig in pair with steel gear or specific dynamic test that enables application of cyclic load In relation to temperature and number of load cycles Flank strength σH (ϑ, NL) (MPa) Gear testing on test rig, running in oil Test in oil is necessary; otherwise, other failure modes occur first COFa µ (/) Gear testing on test rig + temperature measurement Material pair specific characteristic; COF is not a property of a single material, but of a pair of materials Twin-disc test – secondary option Wear coefficienta kw (10−6 mm3/Nm) Gear testing on test rig + wear measurement Material pair specific characteristic Twin-disc test – secondary option a If application operates in extreme conditions, COF and kw must be measured at such extreme temperature and other conditions. Open in new tab The FEM analysis in Fig. 14 shows a significant difference in flank pressure at the beginning and end of meshing between gear pairs with module 1.0 and 0.6 mm. The gear pair with the module 0.6 mm and a larger number of teeth (34) has, due to the contact ratio of over 2, a lower flank pressure, which additionally influences the gears’ temperature and lifespan. Figure 14: Open in new tabDownload slide Comparison of flank pressure between gear with module 1.0 and 0.6 mm for different load levels POM/PA66 gear pair. Due to the contact ratio >2 at mn = 0.6 mm, the typical mesh points B and D do not exist. Figure 14: Open in new tabDownload slide Comparison of flank pressure between gear with module 1.0 and 0.6 mm for different load levels POM/PA66 gear pair. Due to the contact ratio >2 at mn = 0.6 mm, the typical mesh points B and D do not exist. The numerical FEM models of meshing gear pairs were set up in ANSYS/19.2 (Ansys, Inc., USA) software. All simulations were run in 2D, taking into account a planar stress state. Quadratic-order PLANE183 elements were used to discretize gear geometry. CONTA172 (drive gear) and TARGE169 (driven gear) elements were used to model the contact between the meshing flanks and a frictional contact was considered. The used value of the COF was µ = 0.29 as determined for this material pair in the work of Pogačnik & Tavčar (2015). Mesh convergence was conducted with an h-refinement method, confirming the accuracy of the calculated stress. The accuracy of the calculated stress also depends on the element quality, wherefore a high-quality mesh with an average composite quality of 0.95 was acquired. The entire gear body was modeled, but only five teeth, where the postprocessing was done on the third tooth, which meshed through all characteristic meshing points. The hole of drive gear 1 was constrained to the fixed point A, located at the origin of the coordinate system X1Y1 (Fig. 15). Translations in the directions X1 and Y1 were constrained and only rotation around point A was allowed. In the same manner, the drive gear was constrained to fixed point B, located at the origin of coordinate system X2Y2. Rotation was prescribed around point A on the drive gear, and torque was prescribed on the driven gear in the direction opposite to rotation. Material properties were modeled as linearly elastic, since it was confirmed that such an assumption is appropriate for the numerical modeling of polymer gear applications (Černe et al., 2020). Elastic modulus EPOM = 3100 MPa and Poisson’s ratio νPOM = 0.35 were considered for POM and EPA6 = 3400 MPa and νPA6 = 0.4 for PA6 gears. Figure 15: Open in new tabDownload slide Numerical model of the gears and mesh refinement in the contact region. Figure 15: Open in new tabDownload slide Numerical model of the gears and mesh refinement in the contact region. 5.3 Possibilities for further upgrades The developed multicriteria optimization method fails to consider a wide variety of factors that can influence the polymer gear’s performance, e.g. tooth profile modifications, chosen gear rim/web geometry, influence of fillers on the tribological properties, and the gear’s thermal response. These factors can be studied using experimental and numerical analysis methods and the result data extrapolated for an application inside the multicriteria method. The latter could, hence, be upgraded so as to consider the influence of these additional factors on the main parameters defining the MCF function. This would presumably enable an even more refined and optimal gear design. 6. Conclusions This paper’s main contribution is the mathematical algorithm for the MCF (equation 8) and optimization procedure. The polymer gear optimization procedure is based on a variation of polymer gear design parameters and the simultaneous consideration of criteria for root and flank stress, gear temperature, wear, tooth deformation, quality, cost, and volume of each variant. The authors developed the OptiTooth software, which demonstrates the applicability of the MCF in polymer gear optimization. Additionally, polymer gear design guidelines and rules were established to accelerate the initial selection of the parameters and for better overview over possible improvement options during the optimization process. The sensitivity of each design criterion on gear geometrical parameters is presented together with typical failure modes. Failure modes depend on the load level, number of load cycles, lubrications, speed of rotation, material pair, and method of interrelation. Relationships between design criteria such as gear face width were investigated. A better understanding of sensitivity on design criteria and failure modes is crucial for a successful optimization. An important conclusion is that a larger normal module reduces root stress linearly, but at the same time bulk temperature increases. The bending strength of polymer gears increases with a higher number of teeth due to load-induced deflection. Therefore, if gear diameter is unchanged, a higher number of teeth with a proportionately smaller module can, in some cases, result in better performance than larger module with a lower number of teeth. Sensitivity analyses have shown that the pinion is critical for the temperature, and therefore using a metal pinion is often a good design solution. The model for multicriteria polymer gear optimization is additionally presented with a case study. The demonstration includes initial gear pair design constraints and a variant with a steel pinion and driven gear made from a high-performance polymer PEEK. A prerequisite for the presented gear optimization model is the availability of material data. Required polymer material characteristics and type of testing are specified for accurate gear optimization and good matching between calculated and experimental results. If the COF or the wear coefficient for a specific pair of polymer materials is not available, then durability cannot be predicted in any detail. Therefore, the authors’ research is now focused on the systematic testing of promising material pairs and collecting specific material data. The authors see an additional challenge in gear profile optimization, with an aim for reduced contact pressure outside the kinematic point and for reduced frictional heat generation. ACKNOWLEDGEMENTS The research was financed partly by the MAP gears project (project is cofinanced by the Republic of Slovenia and the European Union under the European Regional Development Fund, contract number C3330-18-952014) and partly by the Slovenian Research Agency (contract number 630-33/2019-1). Conflict of interest statement None declared. References Banodiya B. , Karma V. K. ( 2017 ). Measurement of transmission error in spur gears . International Research Journal of Engineering and Technology , 4 ( 8 ), 2369 – 2375 . Google Scholar OpenURL Placeholder Text WorldCat Bravo A. , Koffi D., Toubal L., Erchiqui F. ( 2015 ). Life and damage mode modeling applied to plastic gears . Engineering Failure Analysis , 58 , 113 – 133 . https://doi.org/10.1016/j.engfailanal.2015.08.040 . Google Scholar Crossref Search ADS WorldCat Cathelin J. ( 2019 ). Material data for advanced plastic gear simulation . In International Conference on Gears 2019 (Vol. 2355 , pp. 1379 – 1390 .). Google Scholar OpenURL Placeholder Text WorldCat Cathelin J. , Letzelter E., Guingand M., de Vaujany J.-P., Chazeau L. ( 2013 ). Experimental and numerical study of a loaded cylindrical PA66 gear . Journal of Mechanical Design , 135 ( 041007 ). https://doi.org/10.1115/1.4023634 . Google Scholar OpenURL Placeholder Text WorldCat Černe B. , Duhovnik J., Tavčar J. ( 2019/1 ). Semi-analytical flash temperature model for thermoplastic polymer spur gears with consideration of linear thermo-mechanical material characteristics . Journal of Computational Design and Engineering , 6 ( 4 ), 617 – 628 . https://doi.org/10.1016/j.jcde.2019.03.001 . Google Scholar Crossref Search ADS WorldCat Černe B. , Zorko D., Duhovnik J., Tavčar J., Žavbi R. ( 2019/2 ). Flash temperature analysis method for polymer gears with consideration of deviations in meshing kinematics . In IDETC-CIE2019 . https://doi.org/10.1115/DETC2019-97824 . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Černe B. , Petkovšek M., Duhovnik J., Tavčar J. ( 2020 ). Thermo-mechanical modeling of polymer spur gears with experimental validation using high-speed infrared thermography . Mechanism and Machine Theory , 146 , 103734 . https://doi.org/10.1016/j.mechmachtheory.2019.103734 . Google Scholar Crossref Search ADS WorldCat Das D. , Bhattacharya S., Sarkar B. ( 2016 ). Decision-based design-driven material selection: A normative-prescriptive approach for simultaneous selection of material and geometric variables in gear design . Materials & Design , 92 , 787 – 793 . https://doi.org/10.1016/j.matdes.2015.12.064 . Google Scholar Crossref Search ADS WorldCat Dearn K. D. , Hoskins T. J., Andrei L., Walton D. ( 2013 ). Lubrication regimes in high-performance polymer spur gears . Advances in Tribology , 2013 , 1 – 9 . https://doi.org/10.1155/2013/987251 . Google Scholar Crossref Search ADS WorldCat DIN 3990 . ( 1987 ). Calculation of load capacity of cylindrical gears . German National Standard . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Duhovnik J. , Zorko D., Sedej L. ( 2015 ). The effect of tooth flank geometry on the lifetime of injection moulded polymer gears . In Proceedings of International Conference on High Performance Plastic Gears 2015 (pp. 1203 – 1218 .). Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Duhovnik J. , Zorko D., Sedej L. ( 2016 ). The effect of the teeth profile shape on polymer gear pair properties . Tehnicki Vjesnik - Technical Gazette , 23 ( 1 ), 199 – 207 . https://doi.org/10.17559/TV-20151028072528 . Google Scholar OpenURL Placeholder Text WorldCat Düzcükoğlu H. ( 2009 ). PA 66 spur gear durability improvement with tooth width modification . Materials & Design , 30 ( 4 ), 1060 – 1067 . https://doi.org/10.1016/j.matdes.2008.06.037 . Google Scholar Crossref Search ADS WorldCat Fernandes C. M. C. G. , Rocha D. M. P., Martins R. C., Magalhães L., Seabra J. H. O. ( 2018 ). Finite element method model to predict bulk and flash temperatures on polymer gears . Tribology International , 120 , 255 – 268 . https://doi.org/10.1016/j.triboint.2017.12.027 . Google Scholar Crossref Search ADS WorldCat Hakimian E. , Sulong A. B. ( 2012 ). Analysis of warpage and shrinkage properties of injection-molded micro gears polymer composites using numerical simulations assisted by the Taguchi method . Materials & Design , 42 , 62 – 71 . https://doi.org/10.1016/j.matdes.2012.04.058 . Google Scholar Crossref Search ADS WorldCat Hasl C. , Oster P., Tobie T., Stahl K. ( 2017/1 ). Bending strength of oil-lubricated cylindrical plastic gears . Forschung Im Ingenieurwesen , 81 ( 2 ), 349 – 355 . https://doi.org/10.1007/s10010-017-0224-2 . Google Scholar Crossref Search ADS WorldCat Hasl C. , Liu H., Oster P., Tobie T., Stahl K., & Forschungsstelle fuer Zahnraeder und Getriebebau (Gear Research Centre) . ( 2017/2 ). Method for calculating the tooth root stress of plastic spur gears meshing with steel gears under consideration of deflection-induced load sharing . Mechanism and Machine Theory , 111 , 152 – 163 . https://doi.org/10.1016/j.mechmachtheory.2017.01.015 . Google Scholar OpenURL Placeholder Text WorldCat Hasl C. , Illenberger C., Oster P., Tobie T., Stahl K. ( 2018 ). Potential of oil-lubricated cylindrical plastic gears . Journal of Advanced Mechanical Design, Systems, and Manufacturing , 12 ( 1 ), JAMDSM0016 – JAMDSM0016 . https://doi.org/10.1299/jamdsm.2018jamdsm0016 . Google Scholar Crossref Search ADS WorldCat Hiltcher Y. , Guingand M., de Vaujany J.-P. ( 2006 ). Load sharing of worm gear with a plastic wheel . Journal of Mechanical Design , 129 ( 1 ), 23 – 30 . https://doi.org/10.1115/1.2359469 . Google Scholar Crossref Search ADS WorldCat Illenberger C. M. , Tobie T., Stahl K. ( 2019 ). Flank load carrying capacity of oil-lubricated high performance plastic gears . Forschung Im Ingenieurwesen , 83 ( 3 ), 545 – 552 . https://doi.org/10.1007/s10010-019-00332-x . Google Scholar Crossref Search ADS WorldCat İmrek H. ( 2009 ). Performance improvement method for Nylon 6 spur gears . Tribology International , 42 ( 3 ), 503 – 510 . https://doi.org/10.1016/j.triboint.2008.08.011 . Google Scholar Crossref Search ADS WorldCat ISO 1328-2:1997 . ( 1997 ). Cylindrical gears—ISO system of accuracy—Part 2: Definitions and allowable values of deviations relevant to radial composite deviations and runout information . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC ISO 6336 . ( 2006 ). Calculation of load capacity of spur and helical gears, Parts 1–6, International standard . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Jain M. , Patil S., Ghosh S. S. ( 2019 ). A review on failure characteristics of polymeric gears . AIP Conference Proceedings , 2148 ( 1 ), 030057 . https://doi.org/10.1063/1.5123979 . Google Scholar Crossref Search ADS WorldCat Karthik Pandian A. , Gautam S. S., Senthilvelan S. ( 2020 ). Experimental and numerical investigation of the bending fatigue performance of symmetric and asymmetric polymer gears . Proceedings of the Institution of Mechanical Engineers, Part L: Journal of Materials: Design and Applications , 234 ( 6 ), 819 – 834 ., https://doi.org/10.1177/1464420720909486 . Google Scholar Crossref Search ADS WorldCat Kim C. H. ( 2006 ). Durability improvement method for plastic spur gears . Tribology International , 39 ( 11 ), 1454 – 1461 . https://doi.org/10.1016/j.triboint.2006.01.020 . Google Scholar Crossref Search ADS WorldCat Kurokawa M. , Uchiyama Y., Nagai S. ( 2000 ). Performance of plastic gear made of carbon fiber reinforced poly-ether-ether-ketone: Part 2 . Tribology International , 33 ( 10 ), 715 – 721 . https://doi.org/10.1016/S0301-679X(00)00111-0 . Google Scholar Crossref Search ADS WorldCat Li W. , Zhai P., Tian J., Luo B. ( 2018 ). Thermal analysis of helical gear transmission system considering machining and installation error . International Journal of Mechanical Sciences , 149 , 1 – 17 . https://doi.org/10.1016/j.ijmecsci.2018.09.036 . Google Scholar Crossref Search ADS WorldCat Lin C. , Wei W., Wang S., Xia X., Xin Q. ( 2020 ). Bending stress analysis of eccentric straight and helical curve-face gear pair . International Journal of Mechanics and Materials in Design , 16 ( 2 ), 401 – 414 . https://doi.org/10.1007/s10999-019-09475-9 . Google Scholar Crossref Search ADS WorldCat Lu Z. , Zhu C., Song H., Yu G. ( 2020 ). The effect of injection molding lunker defect on the durability performance of polymer gears . International Journal of Mechanical Sciences , 180 , https://doi.org/10.1016/j.ijmecsci.2020.105665 . Google Scholar OpenURL Placeholder Text WorldCat Crossref Lu Z. , Liu H., Zhu C., Song H., Yu G. ( 2019 ). Identification of failure modes of a PEEK-steel gear pair under lubrication . International Journal of Fatigue , 125 , 342 – 348 . https://doi.org/10.1016/j.ijfatigue.2019.04.004 . Google Scholar Crossref Search ADS WorldCat Lu Z. , Liu H., Wei P., Zhu C., Xin D., Shen Y. ( 2020 ). The effect of injection molding lunker defect on the durability performance of polymer gears . International Journal of Mechanical Sciences , 180 , 105665 . https://doi.org/10.1016/j.ijmecsci.2020.105665 . Google Scholar Crossref Search ADS WorldCat Lupinetti K. , Giannini F., Monti M., Pernot J.-P. ( 2018 ). Multi-criteria retrieval of CAD assembly models . Journal of Computational Design and Engineering , 5 ( 1 ), 41 – 53 . https://doi.org/10.1016/j.jcde.2017.11.003 . Google Scholar Crossref Search ADS WorldCat Mao K. , Li W., Hooke C. J., Walton D. ( 2009 ). Friction and wear behaviour of acetal and nylon gears . Wear , 267 ( 1–4 ), 639 – 645 . https://doi.org/10.1016/j.wear.2008.10.005 . Google Scholar Crossref Search ADS WorldCat Mao K. , Langlois P., Hu Z., Alharbi K., Xu X., Milson M., Li W., Hooke C. J., Chetwynd D. ( 2015 ). The wear and thermal mechanical contact behaviour of machine cut polymer gears . Wear , 332–333 , 822 – 826 . https://doi.org/10.1016/j.wear.2015.01.084 . Google Scholar Crossref Search ADS WorldCat Matkovič S. , Pogačnik A., Kalin M. ( 2019 ). Comparison between VDI 2736 wear calculation and experimentally obtained results . In International Conference on Gears 2019 (Vol. 2355 , pp. 1311 – 1321 .). Google Scholar Crossref Search ADS WorldCat Meuleman P. K. , Walton D., Dearn K. D., Weale D. J., Driessen I. ( 2007 ). Minimization of transmission errors in highly loaded plastic gear trains . Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science , 221 ( 9 ), 1117 – 1129 . https://doi.org/10.1243/09544062JMES439 . Google Scholar Crossref Search ADS WorldCat Milani A. S. , Shanian A. ( 2006 ). Gear material selection with uncertain and incomplete data. Material performance indices and decision aid model . International Journal of Mechanics and Materials in Design , 3 ( 3 ), 209 – 222 . https://doi.org/10.1007/s10999-007-9024-4 . Google Scholar Crossref Search ADS WorldCat Milani A. S. , Shanian A., Lynam C., Scarinci T. ( 2013 ). An application of the analytic network process in multiple criteria material selection . Materials & Design , 44 , 622 – 632 . https://doi.org/10.1016/j.matdes.2012.07.057 . Google Scholar Crossref Search ADS WorldCat Miler D. , Hoić M., Domitran Z., Žeželj D. ( 2019 ). Prediction of friction coefficient in dry-lubricated polyoxymethylene spur gear pairs . Mechanism and Machine Theory , 138 , 205 – 222 . https://doi.org/10.1016/j.mechmachtheory.2019.03.040 . Google Scholar Crossref Search ADS WorldCat Moder J. , Grün F., Stoschka M., Gódor I. ( 2017 ). A novel Two-Disc machine for high precision friction assessment . Advances in Tribology , 2017 , 1 – 16 . https://doi.org/10.1155/2017/8901907 . Google Scholar Crossref Search ADS WorldCat Moder J. , Grün F., Summer F., Kohlhauser M., Wohlfahrt M. ( 2018 ). Application of high performance composite polymers with steel counterparts in dry rolling/sliding contacts . Polymer Testing , 66 , 371 – 382 . https://doi.org/10.1016/j.polymertesting.2018.01.009 . Google Scholar Crossref Search ADS WorldCat Pogačnik A. , Tavčar J. ( 2015 ). An accelerated multilevel test and design procedure for polymer gears . Materials & Design (1980-2015) , 65 , 961 – 973 . https://doi.org/10.1016/j.matdes.2014.10.016 . Google Scholar Crossref Search ADS WorldCat Sarita B. , Senthilvelan S. ( 2019 ). Effects of lubricant on the surface durability of an injection molded polyamide 66 spur gear paired with a steel gear . Tribology International , 137 , 193 – 211 . https://doi.org/10.1016/j.triboint.2019.02.050 . Google Scholar Crossref Search ADS WorldCat Senthilvelan S. , Gnanamoorthy R. ( 2004 ). Damage mechanisms in injection molded unreinforced, glass and carbon reinforced nylon 66 spur gears . Applied Composite Materials , 11 ( 6 ), 377 – 397 . https://doi.org/10.1023/B:ACMA.0000045313.47841.4e . Google Scholar Crossref Search ADS WorldCat Singh P. K. , Siddhartha, Singh A. K. ( 2017 ). An investigation on the effects of the various techniques over the performance and durability of polymer gears . In Fifth International Conference of Materials Processing and Characterization (ICMPC 2016) (Vol. 4 ( 2, Part A ), pp. 1606 – 1614 .). https://doi.org/10.1016/j.matpr.2017.01.184 . Google Scholar OpenURL Placeholder Text WorldCat Singh P. K. , Siddhartha, Singh A. K. ( 2018 ). An investigation on the thermal and wear behavior of polymer based spur gears . Tribology International , 118 , 264 – 272 . https://doi.org/10.1016/j.triboint.2017.10.007 . Google Scholar Crossref Search ADS WorldCat Stachowiak G. W. , Batchelor A. W. ( 2014 ). Engineering tribology . (4th ed.) [Computer software] . BH, Butterworth-Heinemann/Elsevier . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Tavčar J. , Grkman G., Duhovnik J. ( 2018 ). Accelerated lifetime testing of reinforced polymer gears . Journal of Advanced Mechanical Design, Systems, and Manufacturing , 12 ( 1 ), JAMDSM0006 – JAMDSM0006 . https://doi.org/10.1299/jamdsm.2018jamdsm0006 . Google Scholar Crossref Search ADS WorldCat Tavčar J. , Kos J., Zorko D., Duhovnik J. ( 2019 ). A multi-criteria polymer gears design optimisation procedure . In Proceedings of the third International Conference on High Performance Plastic Gears 2019 (pp. 1473 – 1484 .). Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Trobentar B. , Kulovec S., Hlebanja G., Glodež S. ( 2020 ). Experimental failure analysis of S-polymer gears . Engineering Failure Analysis , 111 , 104496 . https://doi.org/10.1016/j.engfailanal.2020.104496 . Google Scholar Crossref Search ADS WorldCat VDI 2545 . ( 1981 ). Zahnräder aus thermoplastischen Kunststoffen . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC VDI 2736 . ( 2014 ). Blatt 2, Thermoplastische Zahnräder, Stirngetriebe, Tragfähigkeitsberechnung. VDI Richtlinien . VDI-Verlag GmbH, Duesseldorf . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC VDI 2736-4: 2016 .( 2016 ). Thermoplastic gear wheels—Determination of strength parameters on gears . Verlag des Vereins Deutscher Ingenieure . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Wright N. A. , Kukureka S. N. ( 2001 ). Wear testing and measurement techniques for polymer composite gears . Wear , 251 ( 1–12 ), 1567 – 1578 . https://doi.org/10.1016/S0043-1648(01)00793-1 . Google Scholar Crossref Search ADS WorldCat Yakut R. , Düzcükoglu H. ( 2014 ). Examining the abrasion behaviour of PA 66 gears in different cycles . Advances in Materials Science and Engineering , 2014 , 721731 . https://doi.org/10.1155/2014/721731 . Google Scholar Crossref Search ADS WorldCat Zorko D. , Kulovec S., Tavčar J., Duhovnik J. ( 2017 ). Different teeth profile shapes of polymer gears and comparison of their performance . Journal of Advanced Mechanical Design, Systems, and Manufacturing , 11 ( 6 ), JAMDSM0083 – JAMDSM0083 . https://doi.org/10.1299/jamdsm.2017jamdsm0083 . Google Scholar Crossref Search ADS WorldCat Zorko D. , Kulovec S., Duhovnik J., Tavčar J. ( 2019 ). Durability and design parameters of a Steel/PEEK gear pair . Mechanism and Machine Theory , 140 , 825 – 846 . https://doi.org/10.1016/j.mechmachtheory.2019.07.001 . Google Scholar Crossref Search ADS WorldCat © The Author(s) 2021. Published by Oxford University Press on behalf of the Society for Computational Design and Engineering. This is an Open Access article distributed under the terms of the Creative Commons Attribution-NonCommercial License (http://creativecommons.org/licenses/by-nc/4.0/), which permits non-commercial re-use, distribution, and reproduction in any medium, provided the original work is properly cited. For commercial re-use, please contact journals.permissions@oup.com
TI - A multicriteria function for polymer gear design optimization
JF - Journal of Computational Design and Engineering
DO - 10.1093/jcde/qwaa097
DA - 2021-01-14
UR - https://www.deepdyve.com/lp/oxford-university-press/a-multicriteria-function-for-polymer-gear-design-optimization-Ks2PL1urFQ
SP - 1
EP - 1
VL - Advance Article
IS - 
DP - DeepDyve
ER -