SYNTHESIS AND ANALYSIS OF PLASTIC CURVED FACEWIDTH SPUR GEARS

Transcription

1 8 SYNTHESIS AND ANALYSIS OF PLASTIC CURVED FACEWIDTH SPUR GEARS Laurentia ANDREI, Gabriel ANDREI, Alexandru EPUREANU, Iulian Gabriel BÎRSAN The University Dunarea de Jos of Galati, Romania ABSTRACT The paper gives an overview of a non-standard spur gear whose geometry is defined by both tooth curvature and variable height along the gear face-width. The gears are mainly designed for plastic gears, in order to enhance their transmissible power. Virtual models are used to investigate gear mesh, tooth deflections and strength, and a comparison to standard spur gears is drawn. Due to the complex gear geometry, a cutting process is appropriate for the gear manufacture. KEYWORDS: non-standard curved face width spur gear, plastic gear, gear performances, gear manufacture 1. INTRODUCTION The conventional plastic spur gears are continuously being investigated in order to increase their transmissible power level, either by developing performing materials, such as nanostructured composites, or by modifying the gear tooth geometry. The curved facewidth spur gears are not popular for gear industry and not much has been published on their performances and manufacture, to our knowledge. The advantages of these gears, compared to standard spur gears are [1]: - high contact ratio, - higher bending resistance and reduced contact stresses, - better meshing in plane misalignment conditions, and - there are no axial forces as are inherent in helical gears. Against these advantages there are limitations: - the difficulty in gear design due to the complex tooth geometry compared to conventional designs, - the difficulty in gear train mounting and - the sensitivity to center distance variations. The gears described in this paper have a modified geometry, controlled by two parameters: the radius of the tooth curvature along the gear face width and the reduction on gear tooth height towards the gear face width ends. This non-standard gear is specially designed for plastic gear, less sensitive to variations in tooth profile geometry than their metal counterparts. This modified tooth geometry increases the gear transmissible power level; it also enhances lubrication under operating conditions, but reduces the gear manufacture possibilities. The non-standard gear geometry and mesh are investigated by both numerical method and solid modeling technique. They enable the tooth flanks, along the gear face width, to be drawn and compared to the standard involute shape. Studies on virtual models enable further analysis to be developed such as tooth bending resistance and deflection, modifycations in sliding velocity, tooth generation errors. The simulation of the gear generation, based on a simple kinematics, enables the first attempt in gear manufacture. Virtual gears are used to manufacture the gear prototype, by the selective laser sintering as a material incress manufacturing technique. Injection moulding is the traditional technique for plastic gear manufacture. Unfortunately, the complexity of the shape of the curved face width gear, with modified geometry, makes the design of a die almost impossible. Therefore, it was decided to cut the gear on a traditional milling machine, with specially designed equipment and tools. The tooth geometry precision and quality are investigated as they are essential in gear operating conditions. Experimental tests carried out by the authors on the running curved face width spur gears, with modified geometry, are focused on the gear thermal behaviour and noise. This paper resumes the significant theoretical and experimental investigations on the non-standard gear geometry and mesh.

2 9. THEORETICAL INVESTIGATIONS ON GEAR PERFORMANCES.1. Non-Standard Gear Generation Process Gear performances are developed on virtual gears, using the traditional conjugate surface generation theory as well as the solid modeling techniques. Simulations of gear generation consider the following kinematics: - the tool performs the rotational motion about its inclined axis; - the gear blank is rotated about its axis and is provided with translational motion, tangential to the base circle of the gear. A special device provides the rolling motion, required for the involute tooth flank profile generation; - a simple indexing motion is used to generate all the concave teeth flanks and then the convex flanks, using a different cutter. A numerical computation [], based on the conjugate surface generation theory, enables to produce the gear tooth forms in several sections along the gear face width (fig. 1), as well as the line of action (fig. ); h indicates the distance from gear centre. The following design specification is considered: mm modulus, 3 the number of teeth, 4 mm gear face width, 5 tool axis inclination and mm is the radius of the generating circle. Figure 1 shows that the generated tooth profile is of involute form except at the extreme ends of the gear face. The straight lines of action, shown in figure, indicate the gear conjugate motions. y [mm] Simulations of gear generation are also developed using the solid modeling technique [3]. Several tools are designed, with similar shape but different geometrical parameters, shown in figure 3: R g - the radius of the generating circle varies from 16 mm to 8 mm, for a 4 mm gear face width, and β - the tool axis inclination is considered at - 5. AutoLISP routines are used for the curved spur gear generation process simulation and a representation of the entire gear is produced (fig. 4). a) b) Fig. 3. Illustrating the virtual tools for the tooth convex (a) and concave (b) flanks. Fig. 4. Illustrating the virtual gear blank with two generated teeth [3]... Gear Tooth Geometry y [mm] h=1mm h=1mm h=6mm h=11mm Fig. 1. Tooth profiles along the gear face width []. h=6mm h=11mm Fig.. Lines of action along the gear face width []. x [mm] x [mm] Several non-standard gears are generated, with the above design specification, in order to investigate the variation of some tooth geometrical parameters such as the maximum base circle radius R b, the maximum dedendum circle radius R f and the maximum pressure angle α, recorded at the end of the gear face width, as they influence the gear meshing condition (table 1). R g [mm]/β [ ] R b [mm] R f [mm] α[ ] Table 1 16 / / / / / /

3 1 start of mesh 3 gear rotational angle 8 gear rotational angle 13 gear rotational angle 17 gear rotational angle gear rotational angle gear rotational angle 7 gear rotational angle Fig. 5. Illustrating the path of contact during gear mesh. It is noticed that, for the same tool axis inclination, the decrease in the gear generating circle leads to an increase of the dedendum circle radius, implying a decrease in gear backlash for R g = 16 mm, the gear backlash is reduced to.6 mm which definitely implies gear interference. Also, the tooth flanks exhibits the involute profile on 7% of its height compared to 85% in the case of the standard tooth. The pressure angle also decreases leading to a decrease in plastic gear resistance to wear, as it is shown [4]. For the same value of the generating circle radius, the increase in the tool axis inclination increases the dedendum circle radius and decreases the pressure angle, damaging the gear operating conditions. A limited increase in the tool axis inclination should be considered for each tooth curvature, so that the gear mesh avoids interference..3. Gear Mesh The solid modeling technique also enables the gear path of contact to be produced. Figure 5 illustrates the path of contact on the pinion tooth convex flank, in a theoretical mesh of a gear train with the contact ratio equal to 1 and 6 mm center distance. The gears have been generated with a geometry defined by mm generating circle radius and 5 the tool axis inclination. The gears contact starts at the beginning of the theoretical line of action, specific to the gear center section, where the tooth flank exhibits the ideal involute; an interference, produced by an additional gear rotational angle of.1, is used. The analysis of the gear path of contact shows the peculiarities of the modified curved face width gear mesh: - the line of contact is a spatial curve, with a variable curvature during the gear mesh; - there are points along the line of action where three tooth pairs are in contact, showing the high gear contact ratio; - a longer contact is specific to the gear tooth, a consequence of both the tooth curvature and the increase of gear base circle radius. Compared to its ideal value of.3, calculated for the standard spur gear, the gear rotational angle, for a single tooth contact, is increased to Gear Bending Resistance Curved face width spur gears are modeled using COSMOS/M, version.5, set to the following geometrical parameters: modulus mm, 3 teeth and 4 mm face width. Poisson s ratio for the plastic material used in the gear manufacture, ERTALON 66SA, is.3 and the tensile modulus of elasticity is 345 MPa. The gear 3D model consists of SOLID elements - 8 vertex per solid and 3 possible translational motions per vertex [5]. As a first step, the load is applied successively on the concave and convex flanks of the pinion tooth. A gear geometry is considered at mm the generating circle radius and at the tool axis inclination. For an applied load of 4 Nm, the tooth bending resistance analysis shows that: - the maximum von Mises stresses, recorded on the tooth concave flank, is of 14.8 N/mm occurring on limited areas close to the gear face ends, while most of the tooth exhibits a medium of 5-7 N/mm stresses; - the maximum Von Mises stresses, recorded on the tooth convex flank, decrease significantly by 3%

4 11 a) b) c) Fig. 6. Illustrating the Von Mises stresses specific to curved face width gear (a), double helical gear (b) and standard spur gear (c). compared to the previous stresses values, and occur on a large area along the tooth dedendum, on the opposite flank. As regarding the tooth displacements, it is found out that the recorded displacement is reduced by 31.5% when load is applied on the tooth convex flank. The maximum displacement of 35 4 µm occurs on the entire tooth length, while the maximum displacement of 5 6 µm, recorded when load is applied on the tooth concave flank, moves to limited areas near the gear face ends. Generally the pinion is the first one to fail in a plastic gear train [6]; therefore, the previous analysis is important in the attempt of improving the pinion behaviour through a convenient mounting, i.e. the load should be applied on the pinion tooth concave flanks. Both the tooth stresses and displacements are lower than those specific to convex flanks, ignoring the highest values recorded in limited areas at the end of gear face width analysing the gear path of contact, the load acting on this areas is much reduced than the theoretical one, considered in finite element analysis. Secondly, a comparison of the standard and non-standard gear bending resistance is developed. Loads are applied on the tooth concave flanks, related to a torque of 4 Nm, in the assumption of having two teeth in contact. Analysing the illustrations from figure 6, concerning the maximum Von Mises stresses as well as the stresses distribution, it is noticed that: - on the standard spur gear tooth (fig. 6c), the maximum stresses are constant, at about 8 N/mm. The maximum stresses are developed on a large area on the tooth dedendum, along the gear face width; - on the curved face width spur gear tooth (fig. 6a), the maximum stresses increase to 11 N/mm, but they are distributed on reduced areas close to the gear ends; - on the double helical gear tooth (fig. 6b), the maximum stresses are concentrated in gear center sections (35 N/mm ) and towards the gear face width end sections (4 N/mm ), on extremely reduced areas. In conclusion, the non-standard gear, with a constant tooth height, has a higher bending resistance than its standard counterparts. Even the recorded maximum stresses are by 7% higher than in case of standard spur gear, the stresses are distributed on limited areas where the load is not as high as it has theoretically been considered. The double helical gear is also convenient regarding the bending resistance, but the recorded maximum stresses exceed by more than 1% the specific values for the curved spur gear. A third analysis on non-standard gear bending resistance is focused on the influence of the gear tooth modified geometry on the Von Mises stresses and tooth deflections. Figure 7 shows the variation of the tooth bending stresses versus the tool axis inclination the tooth height respectively, for a chosen mm radius of the generating circle. It is noticed that the maximum stresses increase with the reduction in tooth height along the gear face width, but it is recorded on extremely reduced areas close to the tooth tip; moreover, the areas are reduced by the increase in tool axis inclination. A similar distribution of the bending stresses is exhibited by each tooth geometry the most affected is the unload tooth flank, at the tooth dedendum, and the limited sections of the loaded tooth flanks. The tooth maximum displacement is recorded for β = and decreases with the increase in the tool axis inclination (the reduction in tooth height). The maximum tooth displacement is specific to the gear center, where the tooth has the standard involute profile. Figure 8 shows the variation of the tooth bending stresses versus the generating circle radius the tooth curvature on the gear face width respectively, for a chosen 5 the tool axis inclination. It is noticed that the increase in the radius of the tooth curvature decreases the maximum Von Mises stresses and extends the area they occur along the gear face width a distribution specific to the spur gear. For reduced values of the generating circle radius, the bending stresses are not induced in the unload tooth flank. The maximum tooth displacement is recorded for R g = 16 mm and occurs at the gear face ends, where the applied load is not as high as theoretically considered. Increasing the radius of the tooth curvature, the maximum tooth displacement rapidly decreases, but it extends along the entire tooth length.

5 1 Von Mises stresses [N/mm ] helical gear curved face width gear standard spur gear The tool axis inclination [ ] Fig. 7. The influence of the tool axis inclination on the tooth bending resistance. 1 laser beam Fig. 9. Gear manufacture by SLS [8]. Von Mises stresses [N/mm ] helical gear curved face width gear standard spur gear The radius of the generating circle [mm] Fig. 8. The influence of the generating circle radius on the tooth bending resistance. 3. NON-STANDARD GEAR MANUFACTURE 3.1. Gear Manufacture by DCM The authors used the selective laser sintering (SLS), as a material incress manufacturing technique, to form the solid three-dimensional non-standard gear [7]. Figure 9 shows a schematic representation of the process. The unsintered nylon powder, from the powder feed 1, is preheated to a temperature close to its melting point and flatted by a leveling drum 4 into the part cylinder 3. Here, the selective solidification happens by further heating, up to the sintering temperature, by means of the XY controlled pulsed laser beam 5. The powder grains being melt and stocked together, the base plate moves down slowly and a new layer of powder is spread across the surface. The powder that is not scanned by the laser is unaffected and remains in place to support the next layer of powder. The gear shape is built in a discrete way, with a point-to-point D layer technique. At the end of the building process, the entire cake of sintered and unsintered powder is allowed to cool down and is lifted out of the machine. Then the loose powder is shaken off and the sintered gear is free. Fig. 1. Curved face width gear with modified geometry [8]. Figure 1 illustrates a pair of the designed nonstandard gears, manufactured by SLS. The relatively poor accuracy of the tooth flank is caused by the particular simulation of the gear generation the increment of the rolling motion was limited by the extremely large size of the file as well as by the relatively low precision of the manufacture technique. 3.. Gear Cutting Process The curved face width gear, with modified geometry, is designed to be manufactured on FUS 5 milling machine. Based on the kinematics presented in.1, the cutting process requires special equipment, in order to provide both the rolling and the dividing motions, and special tools [8] (fig. 11). The plastic gear blank 1 and a metal spur gear, with the same module and number of teeth, form an opposite set. The blank is fitted on the shaft 9 by two specific bushing 14 and pin 11, which enable the gear being generated to be further tested on a special rig [5]. The screw 1 overcomes an additional blank rotational motion due to the exerted cutting forces. The gear being generated is connected to both the machine-tool table, by the shaft 9 and the bearing block 13, and the spindle support, by the following elements: shaft 9 - metal spur gear - barrel 4 - band 5 wedge slide 8.

6 D b Fig. 11. A schematic presentation of the specific equipment [8] Rg R b Rg Fig. 1. The tool heading configuration and position related to the machine milling spindle [8]. The translational motion of the milling table 1 is transmitted to the entire equipment, including the barrel 4 which simultaneously induces the rotational motion for the gear set, thus the required rolling motion necessary for the gear tooth involute profile generation. A proper rolling motion is assured if the outside diameter of the barrel (D b ) is equal to the gear base circle diameter, calculated in the gear centre section. At one stage, one gear tooth flank is generated. The blank and the standard gear are then simply dividing; the pin 7 initially disengages the system rolling motion. Variations of tooth geometry are generated by different tool-headings. Figure 1 shows a schematic illustration of the cutter heading 1 with its inclined axis (β) related to the milling spindle direction. The cutter 3 is located and accurately positioned into the cutter heading 1 by special screws 4. The translation of the cutter, along its axis, is controlled by the guide pin and enables the variation of the radius of the generating circle (R g ). The limitation of the designed cutter heading is that the tool axis inclination cannot be varied and the generation of new gear geometries, relative to the variation of the tooth height, requires new cutter headings. Two different cutters are used for the tooth convex and concave sides. The normal sections of the cutters show straight lines for the imaginary rackcutters flanks, with zero pressure angle, necessary to generate an involute tooth profile in all sections along the gear face width. 4. EXPERIMENTAL TESTS A number of plastic curved face width gear, with modified geometry, were cut to the following specifications: gear modulus = mm, number of teeth = 3, gear face with = 4 mm, the radius of the generating circle = mm, the tool axis inclination = 5. Investigations on gear flanks are focused on the tooth profile precision and on the flank surface quality. In concern to gear running behaviour, with no lubrication, measurements of the gear temperature and noise are developed Tooth Profile Precision The tooth profile is investigated using a Coordinate Measuring Machine. A comparison to the standard involute shows that [, 9]: - at gear center, the tooth has the standard involute profile, as expected due to the gear kinematics;

7 14 - in sections away from the gear center, the tooth profile is slightly changed; it gets close to a different involute, specific to a base circle whose radius is R bi R b a consequence of both the tool axis inclination and tool cutting edge geometrical configuration. 4.. Tooth Surface Quality measurements showed that the run-out errors of the curved face width spur gears were quite large and they would lead to an effective reduction of the center distance and non-uniform wear on the tooth rear face. 98 The tooth flank surface roughness is measured on a Talysurf machine and the data is exported to the 3D mapping program Toposurf which produces a topographical map of the investigated area (fig. 13) and is able to calculate the physical and statistic roughness parameters in sections along the gear face width and along the tooth height. It is found out that, along the gear face width, the flanks roughness is appropriate, considering the manufacturing conditions: a single point cutter is generating the tooth flank; there is no cooling liquid during the gear manufacture as the cutting forces, specific to plastics, were considered relatively low. The manufacturing conditions influence mainly the tooth surface roughness along the gear height. Temperature [ C] N = 15 revs/min N = 1 revs/min N = 5 revs/min = the predicted surface temperature Time [min] Fig. 14. Temperature of CFW spur gears as a function of speed [1] Fig. 13. Showing a 3D map of the tooth concave flank surface [8] Gear Temperature The curved face width spur gear temperature is measured [1] using the test rig designed by Walton [11], to investigate the wear of polymer and composite gears. The test rig also enables the gear temperature to be measured using an infrared thermocouple placed 5 mm away from the test gear, after the gear comes out of mesh. Suitable computer data-based monitoring and data logging systems allow continuous measurement of the average surface temperature of the gear in operating conditions. Figure 14 shows the significant influence of speed on gear temperature - a torque of 6 Nm is applied to the non-standard pinion. There are several reasons for the rapid increase in gear temperature: the tooth flank surface finish and a definite evidence of contact on the rear face the transmission error Temperature [ C] T = 15 Nm T = 1 Nm T = 9 Nm = the predicted surface temperature Time [min] Fig. 15. Temperature of CFW spur gears as a function of torque [1]. Figure 15 shows that the increase in torque does not influence the increase in gear temperature significantly and the recorded maximum surface temperature is lower than its predicted temperature a speed of 5 rev/min is considered. The extremely high rate of the temperature rise is also explained by 3

8 15 the tooth flank surface finish and running errors; in addition, the mechanical hysteresys losses, caused by the viscoelastic nature of polymers, should be considered. Other tests were carried out, at higher torques, but the gears were not allowed to run for too long in order to avoid the overheating. The gears failed at 5 Nm while a similar standard spur gear train would fail at about 15 Nm. The failure was caused by the high temperatures induced that affected the gear mesh rather than by excessive wear Gear Noise The non-standard gear noise is recorded using the above mentioned test rig; a small microphone is positioned in front of the gear train at exactly 5 mm away from the meshing point. The microphone is held at the end of a long aluminum tube, clad in pipe insulation, which helps to isolate it from mechanically transmitted vibrations. The microphone is connected to a standard PC computer that is running a program called Creative Wave Studio. This samples the microphone signals and gives a plot of noise pressure level against time. Figure 16 shows the influence of gear speed on noise (sound pressure in kn/m ), at a low torque of 5 Nm for speeds of 5 and 15 rev/min. It can be seen that the noise amplitude increases with speed, due to the increase in tooth collisions. A similar standard spur gear is quieter than the CFW gear, as recorded, and reaches lower surface temperature. Sound pressure [db] Sound pressure [db] spur gear CFW gear 35 C 33 C 47 C 5 C 6 C 37 C 43 C 5 C Speed [revs/min] Fig. 16. Sound pressure level versus speed, for a 5 Nm torque [8]. 47 C 68 C 37 C 44 C 7 C 8 C 75 C 9 C 85 C spur gear CFW gear 8 C Torque [Nm] Fig. 17. Sound pressure level versus torque, at 1 rev/min [8]. Figure 17 shows the influence of the applied torque on gear noise at 1 rev/min speed. It can be seen that the noise amplitude decreases with torque. This is explained by the reduction in friction forces, as for plastics the coefficient of friction decreases with load, so there is less excitation for gear teeth. Also, the increase in temperature modifies the flexibility of the teeth and the transmission error is changed. The main reason for the higher value of the non-standard spur gear noise is the manufacturing process leading to important transmission errors and rough tooth flank surface that induces high friction forces. It is noticed that the temperature for curved face width gears are extremely sensitive with respect to gears speed. The higher the speed, the higher the rate of gear temperature increase; consequently, more deflection occurs. 5. CONCLUSIONS A non-standard geometry is specially designed for the plastic spur gears in order to increase their transmissible power level. The gear tooth has a circular shape along the gear face width and a variable tooth height, decreasing to the gear end sections. Based on a simple specific kinematics, the gear generation process is simulated both with numerical and solid modeling methods. Investigations on gear tooth geometry show that the tooth flank is an involute or near involute in all locations around the face width. Simulation of the gear mesh enables the path of contact to be produced and the increased contact ratio, compared to the standard spur gears, is proven. The finite element analysis was used for the non-standard gear bending resistance study. The double curvature of the gear tooth as well as the reduced tooth height decrease the recorded Von Mises stresses and locate the maximum stresses on limited areas towards the end sections of the gear face width. Based on the virtual gear, produced by the solid modeling method, the curved face width spur gear prototype is manufactured by selective laser sintering. Due to the difficulty of a moulding die design, the gears are then cut on a conventional milling machine. To provide the required kinematics of the gear generation process, special tools and equipment are designed. Measurements of the gear tooth profile, developed on a CMM, showed the involute or near involute tooth profile. The tooth flank surface quality, investigated by a Talysurf machine, is reasonable for a cutting process with a single point generation process and with no lubrication during manufacturing. The temperature of the running non-standard gear was measured on a specially designed test rig; it

9 16 was found that, due to the gear generation errors and due to the rough surface of the tooth flank (compared to the standard moulding injected spur gears), the temperature of the CFW gears increases rapidly and it is more influenced by an increase in speed rather than in the applied load. Measurements of sound pressure level, with variations in torque and speeds, showed that noise level is very much dependent on speed. The curved gears noise level decreases for high torques, mainly as a consequence of the high temperature induced that modifies the tooth stiffness. There are reasons to encourage the curved face width spur gear development and its application in the gear industry, but the manufacture process should be improved. Also, lubrication is recommended for gear operating conditions. REFERENCES 1. Sidorenko A.K., 1984, 7-NKMZ, Mashinostroenie, Moscow.. Andrei L., Andrei G., Epureanu Al., Oancea N., Walton D.,, Numerical simulation and generation of curved face width gears, Int. J. Machine Tools Manufact., 4, pp Andrei L., Andrei G., Mereuta E.,, Simulation of curved-face-width spur gear generation and mesh using the solid modelling method, Proc. 1 th Int. Conf. on Geometry and Graphics, Kiev, Ucraine, Vol. 1, pp Parsons B.N.V., Walton D., Andrei L., Andrei G., 4, Non- Standard Cylindrical Gears, Gear Technology, The Journal of Gear Manufacturing, Randall Publishing, Inc. USA, Nov./Dec, pp Gafiţeanu M., Poteraşu V. F., Mihalache N., 1987, Elemente finite şi de frontieră cu aplicaţii în calculul organelor de maşini, Ed. Tehnică, Bucureşti. 6. Breeds A.R. et al., 1993, Wear behaviour of acetal gear pairs, Wear, 166, pp Kruth J.P., 1991, Material incress manufacturing by rapid prototyping technique, Annals of the CIRP, vol.4/, pp Andrei L.,, Study of plastic curved face-width spur gear generation and behaviour, PhD Thesis, University Dunarea de Jos of Galati. 9. Andrei L., Walton D., Epureanu A., Andrei G., 3, Experimental assessment of plastic curved face width spur gears behaviour, The Annals of University Dunărea de Jos of Galaţi, Fascicle VIII, Tribology, pp Andrei L., Epureanu A., Andrei G., Walton D., 4, Investigation of the thermal behaviour of non-metallic curved face width spur gears, Tribotest, Leaf Coppin, UK, pp Walton D., Hooke C.J. et. al., 199, A new look at testing and rating non-metallic gears, 3 rd World Congress on Gearing and Power transmission, Paris.