B. TROBENTAR et al.: THE INFLUENCE OF SURFACE COATINGS ON THE TOOTH TIP DEFLECTION ... 517–522 THE INFLUENCE OF SURFACE COATINGS ON THE TOOTH TIP DEFLECTION OF POLYMER GEARS VPLIV POVR[INSKIH PREVLEK NA POVES VRHA ZOBA POLIMERNIH ZOBNIKOV Bo{tjan Trobentar1, Sre~ko Glode`1, Jo`e Fla{ker1, Bo{tjan Zafo{nik2 1University of Maribor, Faculty of Mechanical Engineering, Smetanova ulica 17, 2000 Maribor, Slovenia 2Prometheus, Bo{tjan Zafo{nik s.p., Tacenska cesta 125 E, 1000 Ljubljana, Slovenia bostjan.trobentar@gmail.com Prejem rokopisa – received: 2015-03-09; sprejem za objavo – accepted for publication: 2015-07-29 doi:10.17222/mit.2015.056 When designing gear drives made of polymer, the tooth tip deflection is a crucial parameter in respect to the proper gear drive operation. Excessive tooth tip deflection can lead to serious disturbances of gear meshing and consequently to increased noise and wear of the teeth flanks. In such cases the tooth tip deflection can be reduced through the use of stiff surface coatings on the tooth flanks. In this paper the influence of different coating materials and thicknesses on the tooth tip deflection of polymer gears is analysed using comprehensive finite element computational analysis. The numerical results obtained are then used to define an approximate equation for the calculation of gear tooth tip deflection for the coating material used and the thickness of the surface coating layer. The results show that the tooth tip deflection decreases with large values of the coating material Young’s modulus and with the coating layer thickness. Keywords: polymer gears, surface coatings, tooth deflection, numerical analysis Pri konstruiranju zobni{kih dvojic s polimernimi zobniki je poves vrha zoba zobnika eden klju~nih dejavnikov, ki vpliva na pravilno delovanje zobni{ke dvojice. Prevelik poves vrha zoba namre~ vodi do resnih motenj pri ubiranju zobni{ke dvojice, kar povzro~i ve~ji hrup in tudi ve~jo obrabo zobnih bokov. V tak{nih primerih lahko poves vrha zoba zobnika zmanj{amo z uporabo togih povr{inskih prevlek. V predlo`enem prispevku je predstavljena obse`na numeri~na analiza po metodi kon~nih elementov, ki zajema vpliv materiala trde prevleke in debeline prevleke na poves vrha zoba zobnika. Numeri~ni rezultati so nato uporabljeni za dolo~itev aproksimativne ena~be za izra~un povesa vrha zoba zobnika v odvisnosti od izbranega materiala prevleke in njene debeline. Kon~ni rezultati ka`ejo, da se poves vrha zoba zmanj{uje z ve~anjem modula elasti~nosti prevleke in debeline utrjenega povr{inskega sloja zobnih bokov. Klju~ne besede: polimerni zobniki, povr{inske prevleke, poves zoba, numeri~ne analize 1 INTRODUCTION The primary function of gears is to transmit rotary or linear motion. The design for the kinematics of a gear set is geometrically controlled when the deflections of gear teeth during the gear drive operation are taken into account. Therefore, the selection of appropriate gear material is crucial in the design process. Recently, the development of new materials and technologies have resulted in increased use of engineering polymers for machine elements (e.g. gears) due to benefits such as low cost for injection moulding, light weight, resilience and their ability to operate under dry, unlubricated condi- tions.1 Polymer gears, when used in moderate power transmission applications without lubrication may have potentially conflicting tribological properties (low fric- tion, high resistance to wear) and mechanical properties (stiffness, fracture strength). This situation is further complicated by the complex loading and contact pheno- mena that change throughout the meshing cycle. As the transmissible power levels increase, problems of surface temperatures arise due to the frictional losses between mating gear teeth.2,3 Most of these coatings are based on zinc and alumi- nium, which are mainly used for radio frequency shield- ing and electrical conductivity where the mechanical properties of the coatings are not of prime importance. The use of polymer with good strength to weight ratio as a base material is an alternative to metallic components. The main disadvantage is the poor wear resistance of contact surfaces. Recent research work regarding these problems has indicated that thermal spray coatings on polymers can be applied in many engineering applica- tions. Careful selection of material combinations and the use of special process parameters can produce relatively thick metal, ceramic or carbide coatings which can be machined or ground.4 It is very important to optimize the gear design before the tools for manufacturing and experimental testing are made to decrease their cost. Traditionally, the gear designer had a limited number of analytical tools as well as his own experience to achieve this goal. Some of the most well known analytical tools for the prediction of gear tooth strength are the procedures according to ISO, VDI and AGMA standards.5–7 As noted earlier, the tooth tip deflection becomes important for gears made of Materiali in tehnologije / Materials and technology 50 (2016) 4, 517–522 517 UDK 669.058:621.833:678.7 ISSN 1580-2949 Original scientific article/Izvirni znanstveni ~lanek MTAEC9, 50(4)517(2016) materials with low stiffness (e.g. polymers) as it can lead to disturbances in gear meshing in the case of excessive tooth tip deflection. The procedure of gear meshing considering the deflection effect is described in the VDI 2736 standard,7 where spur gear tooth tip deflection is influenced by the force, tooth face width and material stiffness, which is represented by the Young’s modulus and the gear geometrical parameters. It is known that the Young’s modulus of polymer materials is much smaller than that of steels, which are often used as gear materials. Furthermore polymer stress-strain behaviour is also affected by changes in the temperature. Lately, thermoplastic polymers are widely used in manufacturing polymer gears. They have different stress-strain behaviour in the tensile than in the com- pression region which leads to different values of the Young’s modulus in both regions. Furthermore, polymers can also show nonlinear behaviour in the elastic region which cannot be described with a Young’s modulus. Therefore, the different stress-strain behaviour in ten- sile/compression regions and the nonlinear behaviour in an elastic region have to be described with an appro- priate constitutive model using an appropriate compu- tational approach (e.g. Finite Element Method). P. Wyluda and V. Wolf1 performed an elastic-plastic finite element analysis of meshing of two acetal copolymer spur gears under quasi-static loading conditions. Their computa- tional results were in a relative good agreement with the experimental data. The effect of PTFE on the tribological behaviour of polymers in rolling sliding contact has been investigated for two of the most widely used polymers – PA 66 and polyacetal. The experimental results showed that the friction and wear performance of the PTFE filled polymers was superior to that of the unfilled polymers. In addition, the surface cracking found in unfilled PA66 and thought to be responsible for the premature fracture of components such as gear teeth was suppressed by the PTFE. It is suggested that a combination of high surface temperature and high surface tensile stress, produced by friction, is required to initiate these cracks and that PTFE inhibits crack formation by reducing friction.8 In general, the gear tooth tip deflection can be de- creased using a stiffer material with higher Young’s modulus. If a gear is made of low stiffness material (e.g. polymer) the gear tooth stiffness can be increased using a stiff surface coating. Because the standard VDI proce- dure for the determination of gear tooth deflection includes only homogenous material properties, it is not possible to determine the deflection of a coated gear tooth using the equations available in the above quoted standards.7 For this reason, an appropriate computational model to analyse the tooth tip deflection of coated polymer spur gears with different coating materials is presented in this paper. 2 MATERIALS AND METHODS Polymer materials have low stiffness that can lead to greater deflection of elements made of such materials. Excessive deflection can lead to serious problems in the meshing of polymer gears and consequently the addi- tional loading of the gear teeth. According to the VDI 27367 the deflection of the tooth tip of a spur gear in the circumferential direction can be approximately calcu- lated from Equation (1):  = ⋅ + F c E E E E E t steel b ' 2 1 2 1 2 (1) where Ft is the tangential force in N, b is the face width in mm, c’ is the maximum tooth stiffness per unit face width (single stiffness) of a tooth pair in N/mm × μm2, Esteel is the modulus of elasticity of steel (Esteel = 210000 MPa) and E1,2 are the elastic modulii of gears 1 and 2 in MPa at room temperature. The maximum tooth stiffness per unit face width can be ex- pressed in the Equations (2) and (3)6: c c C C Cth M R B' ' cos= ⋅ ⋅ ⋅ ⋅  (2) c qth ' ' = 1 (3) Equation (2) uses the theoretical single tooth stiffness c’th as given by Equation (3), where q’ is the minimum value for the flexibility of a gear pair. Individual factors in Equation (2) can be determined as described in the ISO 6336 standard: CM is the correction factor that takes into account the difference between theoretical and actual values for solid disc gears (CM = 0.8 according to the standard ISO 6336); gear blank factor CR takes into account flexibility of gear rims and webs (CR = 1.0 for a solid disc according to the standard ISO 6336); the factor CB reflects the influence of deviation from the gear basic rack profile according to standards ISO 6336 and ISO 53.6,9 The deflection of the tooth tip  determined with Equation (1) must be smaller than the estimated per- missible deformation P = 0.07·mn, where mn is the normal module in mm. Overcoming of the permissible deflection P can lead to serious disturbances of the meshing gear teeth and consequently to higher noise and a shorter service life of the gear pair.7 As mentioned above, Equation (1) cannot be used for calculation of the tooth tip deflection of coated gears. Therefore, an appropriate numerical model should be used in that case. The first step in developing a nume- rical model for the simulation of tooth tip deflection of coated gears was the creation of a numerical model that will yield proper results to be verified by analytical or experimental solutions, or solutions taken from stan- dardised procedures. For this purpose a two dimensional B. TROBENTAR et al.: THE INFLUENCE OF SURFACE COATINGS ON THE TOOTH TIP DEFLECTION ... 518 Materiali in tehnologije / Materials and technology 50 (2016) 4, 517–522 plane strain model with homogeneous material proper- ties (Hooke’s law) and static load was generated. The stress-strain behaviour of the polymer material used in this paper consists of linear and non-linear regions (Fig- ure 1). The assumption of using Hooke’s law for poly- mer gears was verified by comparison of the tooth tip deflections based on material models that consider linear elastic material properties. The static load used in the present model is based on the assumption that the number of loading cycles during the service life of the gear is small and that the gear is used only for partial rotation (e.g. application of the four bar mechanism and for slow motion of the gear rack). This is necessary due to the presence of polymer hyste- resis during loading and unloading. If the material is viscoelastic, the path of loading is not coincident with the path of unloading, which occurs when the material changes significantly in volume and is experiencing large deformations. Furthermore, it is assumed that there is enough time for strain recovery and a permanent set effect of the polymer can be neglected.11 An accurate two-dimensional model of the bench- mark gear was developed and imported into the ABAQUS software. The contact position of engaging gear tooth flanks was chosen to be on the outer point of single engagement, where the bending load is highest. Contact between engaging teeth was prescribed as a standard surface to surface contact, with the master surface set on the driving gear and the slave surface on the driven pinion gear. Assuming good lubrication, con- tact interaction was modelled as tangentially frictionless with finite sliding with no adjustments or smoothing between the surfaces. The influence of coating wear during the service life and consequently its influence on tooth toughness and deflection were not considered. The pinion in Figure 2a was fully constrained in the centre bore, while displacements were only constrained in the gear, and the rotations remained free. The load was applied by torque load in the centre of the gear. The material properties of the pinion were defined as hyper-elastic in the core and as linearly elastic in various coatings. The mesh was generated with CPS4R-type elements with locally controlled element size at tooth flanks (Figure 2b) and with at least ten elements in the coating layer thickness direction (Figure 2c). The nume- rical model was also tested for convergence analysis. Figure 3 shows a significant improvement when the number of elements in contact was increased from 467 to 3456. An increase to 5538 elements had no influence on the results or calculation time. Our workflow was combined from several steps. First, the gear deflection according to VDI 27367 and ISO 6336 6 standards are calculated. The analysed gears (gear and pinion presented in Table 1) have properties of the steel and the polymer material. Young’s modulus and Poisson’s ratio for the steel gear was E = 206000 MPa and  = 0.33, respectively, while the polymer material was described with the properties for POM (Figure 1) and  = 0.35. A reference tooth tip deflection was calcu- lated according to Equation (1). In the next step, a model with a coated polymer gear was generated. The pinion gear was partitioned between a coating layer (Figure 2c) and core material. Different models were created for different types of coatings and their thicknesses. Two different core material properties from DuPont and BASF (POM Delrin 100 NC01012 and PA Ultramid® 820213) and several different materials used for coating B. TROBENTAR et al.: THE INFLUENCE OF SURFACE COATINGS ON THE TOOTH TIP DEFLECTION ... Materiali in tehnologije / Materials and technology 50 (2016) 4, 517–522 519 Figure 2: a) Numerical model, b) locally adjusted mesh on gear teeth and c) coating model on pinion Slika 2: a) Numeri~ni model, b) lokalna zgostitev mre`e na zobu in c) prikaz sloja na pastorku Figure 1: Material properties of pinion core10,11,13 Slika 1: Materialne lastnosti jedra pastorka10,11,13 Figure 3: Mesh converge of pinion gear Slika 3: Konvergenca mre`e pastorka were studied; Polytetrafluoroethylene – PTFE (E=560 MPa), aluminium – Al (E = 70 GPa), Boron nitride – BN (E = 33.85– 85.91 GPa) and Molybdenum disul- phide – MoS2 (E = 210–400 GPa) with various Young’s modulii between 560 MPa and 210 GPa.14 Four different thickness of the coating were modelled on the pinion: t = 25 μm, 50 μm, 75 μm and 100 μm. Table 1: Details of gear pairs Tabela 1: Podatki zobni{kih dvojic Gear pair 1, 2, 3 Gear pair 4, 5 Parameter Pinion Gear Pinion Gear Module m (mm) 4 4 Centre distance a (mm) 80 100 Pressure angle  (°) 25 20 Width b (mm) 25 25 25 25 Number of teeth z (-) 9 31 23 27 Profile shift x (mm) 0.4871 –0.4871 0 0 Addendum height haP (mm) 4.750 2.000 4 4 Dedendum Height hfP (mm) 3.051 6.949 5 5 Root radius profile (mm) 1.24 48 1.83 1.78 Material POM 42CrMo4 POM/PA 42CrMo4 Young’s modulus E (MPa) 560 206000 560/ 3000 206000 Poisson’s ratio v (-) 0.35 0.33 0.35/0.35 0.33 The computational results based on the numerical procedure as described above have then been used to determine the appropriate mathematical formulation where the tooth tip deflection  is expressed according to the Young’s modulus of the coating material E and the coating thickness t. Several forms of polynomials were used to find the best fit, however the best correlation was found using the following Equation (4):  = + ⋅ + ⋅ + ⋅ + ⋅A B E C t D t F Eln ln (4) The computational parameters A, B, C, D and F were calculated using the least squares method. The large value of the coating material’s Young’s modulus and large coating thickness cause a minimum deflection , and vice versa. This fact can lead to verification of the model and the derivation of extreme values. Using the global optimization method, the coating material Young’s modulus E and coating thickness t parameters were determined, yielding the minimum and maximum deflections. Equation (4) is valid for general nonlinear elastic base materials. 3 COMPUTATIONAL RESULTS As described in section 2, the computational analysis of a gear without coating has been carried out first to compare the numerical results with the results of the analytical procedure as described in VDI 2736.15 The first finite element analysis (FE-E model) using the standard Young’s modulus and with torque T = 10 Nm (tangential force Ft = 555.6 N) results in a tooth tip deflection of the uncoated pinion = 0.388 mm, while the standardised VDI calculation gives a value  = 0.361 mm. Using a modified computational model (FE-HE model), the tooth tip deflections are also determined for coated gears with different Young’s modulii of coating materials and constant coating thickness t = 0.1 mm. As shown in Figure 3, the tooth tip deflection of the coated pinon is reduced compared to the tooth tip deflection of an uncoated gear and is for a tangential force, Ft = 555.6 N, inside the permissible limit according to VDI 2736. It is evident from Figure 4 that a reasonably thick coating of t = 0.1 mm can have a significant influence on the deflection values of polymer gears. Combining the computational results with Equation (4), valid parameters for treated Gear pair 1 with re- gression R = 0.991 are determined: A = 0. 5173140, B = 8.42502·10–8, C = –0.236158, D = 0.000644889 and F = –0.0251878. B. TROBENTAR et al.: THE INFLUENCE OF SURFACE COATINGS ON THE TOOTH TIP DEFLECTION ... 520 Materiali in tehnologije / Materials and technology 50 (2016) 4, 517–522 Figure 5: The effect of Young’s modulus and coating thickness on the tooth tip deflection for gear pair 1 and R = 0.991 Slika 5: Vpliv modula elasti~nosti in debeline prevleke na poves vrha zoba za zobni{ko dvojico 1 in R = 0,991 Figure 4: Comparison of gear tooth tip deflection with and without coating (Gear pair 1) Slika 4: Primerjava povesa vrha zoba zobnika z in brez prevleke (zobni{ka dvojica 1) For different gear pairs we obtained the coefficients A, B, C, D, F (Table 3) and with different regression values R which can be found in Table 2. The effect of the Young’s modulus E and coating thickness t on the tooth tip deflection according to Equation (4) is presented in Figure 5. The minimum tooth tip deflection min = 0.2132 mm is observed for Young’s modulus E = 210 GPa and a coating thickness t = 0.1 mm, with the maximum tooth tip deflection max = 0.3717 mm seen for Young’s modulus is E = 560 MPa and coating thickness t = 0.025 mm. The influence of Young’s modulus on the tooth tip deflection for minimum (t = 0.025 mm) and maximum (t = 0.1 mm) coating thicknesses is presented in Figure 6. In region I it can be seen that an increase of the Young’s modulus of the surface coating can significantly reduce the tooth tip deflection of the treated gear (about 26.4 % relative to that of Young’s modulus E = 560 MPa and E = 20000 MPa). A somewhat smaller decrease of the tooth tip deflection with increase of Young’s modulus is ob- served in region II (about 13 % relative to that of at E = 20000 MPa and E = 105000 MPa), while in region III the toot tip deflection drops only about 6.1 %. Similar results to those described above are also found when the tooth tip deflection is studied for two given Young’s modulii (560 MPa and 210 GPa) and different surface coating thickness (Figure 7). For uncoated gears (t = 0) a tooth tip deflection of 0.388 mm is observed. When a surface coating is applied, the significant drop of tooth tip deflection is observed especially for harder surface coatings. In region I (up to coating thickness of 0.025 mm) the decrease of tooth tip deflection is about 40 %, while in region II this value is only 9 %. The deflection behaviour of five gear pairs was examined at the given operating conditions (Table 2). Two distinct geometries were tested with different core materials. Table 2 shows the values for regression of all five cases. The largest deflection has been found in the gear pair 1, with torque 10 Nm and POM as the core material. The smallest deflection was observed in gear pair 5, with torque 10 Nm and PA as a core material. Table 2: Fitting of equation with FE values Tabela 2: Ujemanje ena~be z MKE rezultati Gear Torque (Nm) Material Regression R Gear pair 1 i = 9/31 10 Delrin 100 NC010 0.9910 Gear pair 2 i = 9/31 5 Delrin 100 NC010 0.9806 Gear pair 3 i = 9/31 1 Delrin 100 NC010 0.9790 Gear pair 4 i = 23/27 10 Delrin 100 NC010 0.9781 Gear pair 5 i = 23/27 10 Ultramid ® 8202 0.9650 In Table 3, the A, B, C, D and F coefficients are given for different gear pairs. It is evident that by reducing force or increasing the stiffness of the base material (gear pair 5), these coefficients approach zero. This is a direct consequence of the fact that the tooth tip deflection decreases with stiffer materials and/or reduced B. TROBENTAR et al.: THE INFLUENCE OF SURFACE COATINGS ON THE TOOTH TIP DEFLECTION ... Materiali in tehnologije / Materials and technology 50 (2016) 4, 517–522 521 Figure 7: Comparison of tooth tip deflection for two different coating materials on gear pair 1 Slika 7: Primerjava povesa vrha zoba za dva tipa prevlek na zobni{ki dvojici 1 Figure 6: Comparison of tooth tip deflection for two coating thick- nesses on gear pair 1 Slika 6: Primerjava povesa vrha zoba za dve debelini prevlek na zobni{ki dvojici 1 Table 3: Equation coefficients for equation 4 Tabela 3: Koeficienti za ena~bo 4 Gear A B C D F Gear pair 1 i = 9/31 0.5173140 8.42502E-08 –0.236158000 0.000644889 –0.025187800 Gear pair 2 i = 9/31 0.2407840 3.56586E-08 0.053658500 –0.004475320 –0.012522600 Gear pair 3 i = 9/31 0.0463095 7.10510E-09 0.000250842 –0.001774000 –0.002599930 Gear pair 4 i = 23/27 0.1549290 3.23817E-08 –0.009230920 –0.006828120 –0.010816000 Gear pair 5 i = 23/27 0.0304427 –1.67453E-08 –0.009067370 –0.000983624 –0.001365980 torque. For gear pair 1 (the weakest material and the largest torque) the maximum coefficient deviations and consequently the highest tooth tip deflection was observed. 4 CONCLUSION The influence of different coating materials and coat- ing thicknesses on the tooth tip deflection of polymer gears made of POM and PA is presented in this paper. The tooth tip deflection was determined numerically using a finite element model. Here, the mechanical pro- perties of the core material were defined using the Marlow hyperelastic material model while the coating materials’ mechanical properties were defined using the linearly elastic Hooke’s law. The computational results obtained have then been used to define the approximate analytical equation, where linear and logarithmic terms of Young’s modulus and coating layer thickness were used. The obtained analytical equation form describes the tooth tip deflection. It is valid for coating material Young’s modulii in the range from 560 MPa to 210 GPa, coating thicknesses between 0.025 mm to 0.1 mm, diffe- rent loads, different nonlinear elastic base materials and different geometries. The final results showed that the use of stiff surface coatings on polymer gears signifi- cantly reduce the tooth tip deflection and consequently improve the operation of such gear pairs. Further investigations for coated polymer gears should cover the potential cracking of hard coatings in the case where the core material stiffness is too low. The coating layer wear due to the relative sliding of gear flanks and its influence on the tooth tip deflection should also be studied. To confirm the computational results, the experimental testing of bending behaviour of coated polymer elements should be carried out. Acknowledgement The doctoral study of the first author is partly co- financed by the European Union through the European Social Fund. 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