Minimization of transmission errors in highly loaded

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Minimization of transmission errors in highly loaded plastic gear trains P Klein Meuleman1 , D Walton2∗ , K D Dearn2 , D J Weale2† , and I Driessen1 1 R&D Department, Océ Technologies BV, Venlo, The Netherlands 2 School of Manufacturing and Mechanical Engineering, The University of Birmingham, Birmingham, UK The manuscript was received on 14 July 2006 and was accepted after revision for publication on 30 April 2007. DOI: 10.1243/09544062JMES439

Abstract: Transmission errors (TEs) are an important source of unwanted noise and vibration in gear drives. Errors can result from geometrical inaccuracies and from elastic deformations. Plastic drives are often loaded in a way that produces high deflections relative to steel gears, and the elastic component of TE is relatively more important. Furthermore, plastic gears are often run in mesh with gears made from steel or other metals. In this case there is a large difference in tooth stiffness, which leads to unusual TE problems. The current paper discusses the origins of elastic TEs and means of their calculation. A simple beam model is used to demonstrate the stiffness of a pair of meshing gear teeth. A finite-element analysis is used to refine this model and to run iterative tooth meshing enabling TEs to be accurately characterized. A number of TE traces from gear pairs running under high loads are included and compared with the theoretical predictions. Several different scenarios are proposed including balancing gear tooth stiffness for dissimilar materials and the adjustment of pressure angle to account for tooth deflection. A set of design guidelines are presented in the conclusions. A case study of a precision printer drive is used to illustrate some of the techniques for the minimization of TEs. Keywords: gears, spur, polymers, plastics, steels, transmission error, finite element, stiffness, involute, pressure angle, contact ratio, tooth accuracy

1

INTRODUCTION

UK. email: [email protected]

As load increases, the effect of tooth stiffness becomes more important. TE is the prime source of vibration in geared systems and is thus important in any noise or vibration study and is particularly important when attempting to minimize noise. In some motion control mechanisms TE can lead to positional inaccuracy – for example it can lead to image distortion in high quality printing transports. This paper considers the effects of material stiffness and tooth geometry on the TEs of polymer gears running in polymer/polymer and polymer/steel combinations. In the former case, the arguments in fact apply to any pairing of materials with similar stiffnesses, whereas the latter applies to any material pairing with large differences (greater than one order of magnitude) in stiffness. In each case, the stiffness of a gear pair is considered, design modifications are proposed and performance improvements demonstrated.

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Transmission error (TE) is defined [1] ‘as the difference between the position that the output shaft of a gear drive would be if the gearbox were perfect, without errors or deflections and the actual position of the output shaft’. TEs may be expressed as an angular displacement from the correct position or as a linear displacement along a line of action, often as a linear displacement at pitch circle radius. Ideally, the TE would be zero, but in practice errors are caused by a number of effects that may be broadly divided into the categories of accuracy and stiffness. At relatively low loads the effects of geometrical accuracy dominate.

∗ Corresponding author: School of Manufacturing and Mechanical

Engineering, The University of Birmingham, Birmingham B15 2TT,

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A case study is presented in which a gear pair for a precision printing process is optimized.

3

GEAR DEFORMATIONS

Fig. 1 Typical TE trace showing a once per revolution waveform with superimposed tooth-to-tooth error (after Smith [1])

Gear teeth will deflect elastically under load, the deflections comprising bending and shear deflection as well as local Hertzian contact deflection. In the case of polymer gears of relatively low modulus, these deflections can be large compared to those experienced by metal gears. The flexibility of non-metallic gears allows gear tooth contact to take place outside the theoretical line of action, and this affects their performance in two opposing ways. In one respect it helps to increase the operating or real contact ratio by bringing more teeth into mesh. On the other hand, it can lead to severe wear because contact outside the theoretical line of action represents interference, i.e. the digging in of the tip of the driver into the driven gear. This has been observed in polyoxymethane (POM/POM) gear pairs running under dry, unlubricated conditions [2]. The earliest study on gear tooth deflection was conducted by Timoshenko and Baud [3]. They measured tooth deflection and put forward an analytical method for its approximation. The effects of elastic deflections on gear performance were studied in the UK in an early work also by Baud and Peterson [4]. In this they modelled gears as two pairs of simple rectangular beams of different lengths, in contact at their ends, to account for the effects of load sharing and tooth elasticity. Using this model the gear stiffness could be determined although the load fraction was unknown at a particular meshing point. This was overcome by using an iteration procedure which changed the load fractions from an initial guess of say 50 per cent each (for two pairs of teeth in contact) until the two load fractions were found which caused the mesh deflections to be equal. An extensive experimental investigation of tooth stiffness was conducted by Walker [5]. From this study he was able to draw a number of important conclusions for example that tooth deflection is proportional to load at any point of the tooth profile, and that the deflection under a given load is independent of the module. Harris [6] and Gregory et al. [7] analysed deflections, load sharing and predicted the effects of elasticity on the vibrations of drive systems. Similar work was carried out by Niemann and Baethge [8]. Weber [9], using strain energy methods, provided a rigorous analytical study of tooth deflection modelling gear teeth using their actual profile. Attia [10] studied the deflection of spur gears cut with thin rims and derived an equation which took into account the effect of neighbouring teeth and rim stiffness, based on the work by Weber. A number of researchers [11–13] have used conformal mapping to determine gear teeth deflection. In this method, the tooth profile is transformed into a semi-infinite boundary so

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2 TRANSMISSION ERRORS Figure 1 shows a typical TE trace for a single pair of spur gears. The trace comprises a low frequency once-per-revolution signal with a higher frequency signal superimposed upon it. The low frequency signal is the run-out error due to the gear’s eccentricity relative to its centre of rotation and, particularly for moulded polymer gears, to a lack of roundness in the gear body. The higher frequency signal is the toothto-tooth error due to manufacturing inaccuracies and tooth deflections. Polymer gears are often specified in motion control systems largely because of their low cost and low noise output. However, they are usually less accurately manufactured than steel gears – high quality injection moulded polymer gears may achieve a DIN accuracy class of around 7 or 6, whereas high quality steel gears may be around 5 to 4. Plastics also have a lower material stiffness by about two orders of magnitude. Young’s modulus for a typical polymer is around 3 GPa compared with 210 GPa for steel. The strength of an engineering thermoplastic polymer compound does not scale in the same way and differs by a factor of about one order of magnitude (for example glass reinforced nylons have a tensile strength of about 80 MPa compared with about 800 MPa for a steel alloy). If a gear pair is primarily designed for strength, then like for like a polymer gear pair will have around one tenth the stiffness of a steel gear pair. Consequently, the difference in manufacturing accuracy on TE for plastic gears tends to be outweighed by the much larger influence of tooth deflection, and stiffness must be considered if high transmission accuracy is required. Furthermore, plastic gears are often meshed with steel gears which can pose specific TE problems related to the imbalance of stiffness between the materials.

Minimization of TEs in highly loaded plastic gear trains

that equations developed for a semi-infinite body can be used to evaluate its deflection. More recently, numerous researchers have used the finite-element (FE) method to evaluate gear deflections. Two important works devoted to accurate gear deflection are those of Coy and Chao [14] and Steward [15]. In reference [14] the specific objective was the selection of a suitable grid size to account for Hertzian deformation. The authors applied a concentrated load at a single point for different grid sizes to Hertzian contact width ratios until they reached an optimum grid size to give the Hertzian deformation accurately. In reference [15] the author included the gear body in his analysis, thereby reducing gear stiffness and compared the pitch point stiffness of gear pairs as determined by different authors. FE methods have been used for the analyses of polymer gears by Walton et al. [16], who examined load sharing. As a result the real contact ratio as distinct from the ideal or theoretical contact ratio for spur gears was determined and plotted using a non-dimensional approach in which the real contact ratio was uniquely defined in terms of the gear geometry, torque and elastic modulus, termed the non-dimensional gear elasticity parameter. This enabled a further study to be made [17] of the tip relief and backlash allowances needed for flexible polymer gears where the tipping and backlash were expressed in terms of the same non-dimensional gear elasticity parameter. Subsequent tests on a number of gears moulded with varying degrees of tipping demonstrated the beneficial effects of tipping polymer gears [18].

Fig. 2

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4 TE FOR GEARS OF LIKE MATERIALS 4.1 Theoretical modelling of gear mesh stiffnesses The variation of gear mesh stiffness through a complete cycle results in TE. Mesh stiffness is affected by the material stiffnesses, the shape of the gear teeth, the position of the contact point, and the number of tooth pairs in contact at any given moment. Before considering a detailed numerical analysis, it is instructive to consider a simplified theoretical system (similar to that used by Baud and Peterson [4]). The mechanism behind stiffness variation can best be understood by reducing the gear teeth to simple rectangular beams, Fig. 2, with a width of m · π/2, a length of 2.25 m, and a thickness of w, where m is the module and w is the facewidth. One beam is rigidly fixed at one end, the other beam can move vertically at its support but it cannot rotate. The load is applied on the beam with the vertical degree of freedom. The point where the upper beam is supported by the lower is represented by a ball, the position of which can be varied between root and tip of each beam. The horizontal position of the contact is measured from the tip of the upper beam, and is normalized with respect to the tooth module. This normalized distance is denoted by x as shown in the figure. The position x = 0 represents contact at the tip of the upper or driving tooth, position x = 1 represents contact at the pitch point and the position x = 2 represents contact at the tip of the lower or driven tooth.

Simple beam model representing the bending stiffness of a pair of spur gear teeth in contact at one instant in the meshing cycle

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The vertical displacement of the point where the load is applied has a shear and a bending component. It can be shown that the deflection due to both shear and bending is independent of the module 2F (x + 0.25) 2F · m(x + 0.25) = wmπG wπG F {m(x + 0.25)}3 δbending (x) = 3EI

δshear (x) =

(1) (2)

where I=

w((m/2)π)3 12

(3)

Substitution results in the module independent relationship δbending (x) =

32F {x + 0.25}3 π3 wE

(4)

Using these relationships, two deflection scenarios are presented: one where the beams have equal stiffness (same materials, in this case POM) and the second where the beams are of different materials (in this case POM against steel). Figure 3(a) shows the total deflection of the system of two idealized teeth against distance x. It also shows the different bending and shear components that add together to give the final composite deflection. The shear deflection of each tooth varies linearly with distance from its support. When added together, the composite of these deflections is a constant. The composite bending deflection varies parabolically and is a minimum when the contact point is half way between the two extremes (x = 1) at a position analogous to the pitch point of a pair of meshing teeth. The magnitude of the composite shear deflection is about the same as the minimum and about a quarter of the maximum composite bending deflection. The total deflection is the sum of all these components and also varies parabolically with distance and is a minimum when x = 1. Figure 3(b) shows the equivalent stiffness of the tooth pair. In a typical gear mesh there may be more than one pair of teeth in contact at any time. For example, the authors have for many years [2, 18] employed a standard spur gear for all their tests on plastic gears, which is used in a 1:1 ratio benchmark gear train. The gears are of module 2 mm, 30 teeth of standard proportions and nominal pressure angle 20◦ . These gears when meshed at the ideal centre distance (60 mm) give a contact ratio of 1.65. This implies that for part of the period of meshing, load is shared between two pairs of teeth. In this case the total stiffness of the mesh is the sum of the stiffnesses of successive tooth pairs with contact at the appropriate positions. This combined mesh stiffness is shown in Fig. 3(c). Figure 3(d) Proc. IMechE Vol. 221 Part C: J. Mechanical Engineering Science

shows the TE determined from the total mesh stiffness during several meshing cycles. Note that the error is a minimum when two pairs of teeth are in contact and increases abruptly as the load is shifted to single pair contact. Changing the nominal pressure angle of the benchmark gear alters the contact ratio. For example changing the pressure angle of the benchmark gear to 13.7◦ gives an integer contact ratio of 2. The geometry (corresponding to AGMA ASK 2348/17) tip relief is applied in order to avoid premature contact. The sharp tip of the incoming tooth would damage the involute flank of the mating tooth at first contact. The stiffness of such a gear pair is represented in Fig. 3(e) and the resultant TE in Fig. 3(f ). Figure 3(g) shows the meshing stiffness for a contact ratio of 2.35 (where for 35 per cent of the meshing cycle, three tooth pairs are active), and Fig. 3(h) shows the resultant TE. Two important characteristics can be seen from the three TE figures. First, the TE decreases as the contact ratio increases. Second, for an integer contact ratio the TE ripple is a minimum. This has important implications for designing gears for minimum vibration and noise.

4.2

Numerical modelling of gear mesh stiffnesses

FE analysis (FEA) is an established tool for predicting stresses and deflections in structures and has often been applied to the analysis of gears [14, 15]. Conventional static FE simulation is not ideal for the prediction of TEs because the system stiffness changes during a mesh and conditions at only one position of the gears can be calculated at a time. The effects of the moving contact such as the stepwise change in stiffness due to the changing number of engaged teeth are not taken into account. However, some FEA codes such as ‘mechanical event simulation’ a feature of the Algor FEA code [19] allow a dynamic system to be modelled, and was employed in the work described in this paper. This gives the possibility to progress iteratively through a gear contact using a series of pseudo-static analyses in which the results of a previous step are used to set up the new contact condition for the subsequent step. The FEA model features gap elements that account for the changing contact patch between engaged gear tooth pairs; an input torque applied to the driving gear that is determined by a ‘time curve’; and an output rotation on the driven gear, also determined by a time curve (Fig. 4). At the initial condition the two gears were represented as almost touching at the first point of contact. The time curves were employed first to apply a torque to the driving gear (the upper gear in Fig. 4, positions 1–3) to bring the two gears into contact – this torque was applied steadily over ten time steps. The second gear was then rotated through 120 equal angular JMES439 © IMechE 2007

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Fig. 3

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Showing the stiffness and deflection of pairs of beams of POM/POM, (a) total deflection of the system, (b) the equivalent stiffness of the tooth pair, (c) the combined mesh stiffness of successive tooth pairs (contact ratio 1.65), (d) calculated TE (contact ratio 1.65), (e) the combined mesh stiffness of successive tooth pairs (contact ratio 2.0), (f) calculated TE (contact ratio 2.0), (g) the combined mesh stiffness of successive tooth pairs (contact ratio 2.35), (h) calculated TE (contact ratio 2.35)

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Fig. 4 TEs due to tooth deflections at various loads as predicted by FEA for benchmark gear geometry

displacements over 120 subsequent time steps such that the gears were taken through one complete cycle of engagement. During rotation the torque on the pinion was kept constant, and friction was ignored. Figure 4 also shows the TE predicted by the FE model for various loads over two tooth meshing periods for the benchmark gear geometry. For each load, the pattern is similar to that predicted by the simple beam model, Fig. 3(c), with an abrupt transition from single to multiple tooth contact. The figure also shows stress plots for a number of discreet mesh positions.

In position 1, there is single tooth contact and this corresponds to the higher part of the deflection curve. In position 2 there are two pairs of teeth in contact. Position 3 shows the first point of contact of the engaging tooth. Note that as the load is increased, the period of single tooth contact decreases because the tooth deflection permits premature tooth contact. This may be thought of as an increase in the true contact ratio of the loaded gear pair, an effect not incorporated into the beam model. Thus the FE model differs from the beam model in that it includes true contact ratio, accounts

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Minimization of TEs in highly loaded plastic gear trains

for the true shape of the gear tooth and also takes into account surface deformation at the contact. Figure 5 shows the TE predicted by the FE model for various loads for a gear pair where the teeth have been modified to give a contact ratio of exactly 2. The variation of error is much smaller than for the benchmark gear geometry as expected, although instead of the peaks that appeared in Fig. 4, small dips

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appear. The depth and width of these dips increase with load. The explanation for this phenomenon is the increase in real contact ratio due to the deflection of the teeth under load. The premature contact of incoming teeth increases the effective stiffness. This can be clearly seen in the stress plot at position 1 where there are three pairs of teeth in contact. Parametric studies not reported here were carried out and it can be

Fig. 5 TEs due to tooth deflections at various loads as predicted by FEA for gears modified to give a theoretical contact ratio of exactly 2 JMES439 © IMechE 2007

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shown that changes to the amount of tip relief or to the centre distance, for example, could return the true contact ratio to exactly 2 for a given load to remove these dips.

4.3

Measured TE

A test rig specifically designed for TE measurement of gear drives was employed to characterize gears moulded to the geometries described in the previous section. This test rig is described in reference [20], and permits measurements of torque, speed, and position for both the driver and the driven gear,

Fig. 6

thus allowing the calculation of TE and efficiency, among other performance indicators. The torques and speeds are computer controlled which enables these parameters to be changed without stopping the machine. Rotary encoders are employed to measure gear angle providing a resolution of 5 × 10−3 degrees. The results presented here were measured at low speed to minimize dynamic effects. Figure 6 shows the measured TE of a pair of POM benchmark geometry gears over a range of applied loads. Figure 6(a) shows a whole revolution of the gear pair where the run-out and the toothto-tooth errors are clearly visible. As the load is increased, the magnitude of the tooth-to-tooth error

Showing the measured TE of a pair of benchmark geometry POM gears for a range of loads over (a) a whole revolution and (b) six tooth meshing intervals

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Minimization of TEs in highly loaded plastic gear trains

also increases, although the run-out is a constant, as would be expected. The total error also increases with load as the gear body deflects. This latter effect is not seen so clearly in the FE results because the gear body was largely modelled as a rigid structure. Figure 6(b) shows the same data over a portion of a revolution showing more clearly the TE pattern. The characteristic peaks predicted by both the simple beam model and by the FE model are clearly visible. Note that, as predicted, the peaks become sharper and narrower with load as the real contact ratio increases. The amplitude of the TE traces predicted by the beam model, the FE model and those measured in practice, are approximately 20, 30, and 50 μm, respectively for an applied torque of 10 Nm. The difference between the two theoretical models has already been discussed. The relatively larger measured error may be due to manufacturing inaccuracies and because the theoretical models do not consider gear body deflections, or movement at the hub attachment.

Fig. 7

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5 TE FOR GEARS OF UNLIKE MATERIALS 5.1 Theoretical modelling of gear mesh stiffnesses When the beams of the simple gear model differ largely in stiffness, e.g. due to a high difference in the Young’s moduli of the materials, the deflection profile is dominated by the most flexible beam. The point of highest stiffness is shifted from x = 1 to the tip of the stiffest tooth, x = 0 (Figs 7(a) and (b)). Figure 7(b) shows the combined stiffness of a steel/POM system under the same conditions as used in Fig. 3. The consequence of this asymmetry applied over a contact ratio of 1.65 is illustrated in Figs 7(c) and (d). The TE trace exhibits a saw tooth shape. Figures 7(e) and (f ) show the effect of increasing the contact ratio to exactly 2. As distinct from the plastic/plastic case there are still abrupt changes in TE and stiffness, and the saw tooth pattern is still evident. However, the amplitude of TE is seen to be lower than for a contact ratio of 1.65. Thus the design guide lines proposed in the previous section do not apply wholly to the case where material stiffness

Showing the stiffness and deflection of pairs of beams of steel/POM, (a) total deflection of the system, (b) the equivalent stiffness of the tooth pair, (c) the combined mesh stiffness of successive tooth pairs (contact ratio 1.65), (d) calculated TE (contact ratio 1.65), (e) the combined mesh stiffness of successive tooth pairs (contact ratio 2.0), (f) calculated TE (contact ratio 2.0)

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Fig. 8

Showing (a) an FE model of an attempt to balance tooth stiffness by modifying pressure angle and tooth thickness, and (b) a comparison of predicted TE for attempted balanced and unbalanced tooth pairs at two torque levels

differences are large. TE may be reduced by increasing contact ratio but abrupt changes cannot be eliminated by the use of integer contact ratios. One method to address this problem might be to attempt to balance stiffnesses of the teeth. This can be achieved by reducing the thickness of teeth of stiffer material and increasing the thickness of the teeth of less stiff material. Tooth bending stiffness is proportional to the cube of the thickness and for a 70 : 1 ratio of material stiffness between steel and POM, a thickness ratio of four would be required. The extent to which this can be done depends on the particular gear configuration in question. Clearly, the thickness of a tooth cannot be reduced by more than the width of the top land. Figure 8(a) shows a potential design based on a 13.7◦ pressure angle in which a steel pinion tooth has been reduced in thickness by the maximum possible amount and the POM wheel tooth increased by the same amount. The chosen pressure angle permitted an integer contact ratio and a wider top land than for a 20◦ pressure angle to maximize the potential modification. Figure 8(b) shows the predicted TE curves for the adjusted gears where it can be seen that a smooth curve such as that seen in Fig. 5 is not attainable because the thickness ratio of the teeth is still insufficient to fully compensate for the material stiffness differences. The figure also shows the TE for an unadjusted pair. The thickness adjustments improve the performance but fail to eliminate the sudden changes in error. It should be noted that where material stiffness differences are smaller, for example if the pinion were made from a glass fibre reinforced nylon and the wheel from POM, a smooth TE trace would be possible.

TEs in the main drive of a colour printer caused unwanted accelerations in the image transfer process leading to distortions of the final image. The drive consisted of a steel pinion and a precision moulded plastic gear, and had a gear ratio of 11 : 1. The design was previously optimized for high contact ratios and wide tolerance to centre distance variation, which resulted in low pressure angle (12◦ ) and long addendum teeth. The TEs were largely due to tooth deflections caused by the relatively high loads. The loads, although high were constant, which helped with the design optimization. The error was diagnosed by measurement and explained by FE analysis. One potential solution was to balance pinion and wheel tooth stiffnesses as proposed in the previous section. However, the geometrical constraints of the system combined with other design considerations did not allow sufficient material to be removed from the steel pinion to achieve a possible design. An alternative approach was to consider the cause of the shape of the TE curve. The deflected plastic

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Figure 9 shows the measured TEs for a benchmark geometry steel pinion running against a POM gear for a torques ranging from 1 to 9 Nm. The saw tooth shape as predicted by the theoretical calculations can be clearly distinguished. The effect of increasing torque on the shape of the peaks is apparent, the amplitude increasing with torque. The height of the peaks corresponds very well with FE simulations.

6

CASE STUDY

Minimization of TEs in highly loaded plastic gear trains

Fig. 9

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Showing the measured TE of a steel pinion of benchmark geometry running against a POM wheel for a range of loads over a whole revolution

tooth has a higher apparent pressure angle to that of the mating steel gear. A possible solution would be to decrease the pressure angle of the plastic gear such that the deflected shape is that of an ideal rigid gear. In this case study, it was considered more practical to change the steel pinion rather than the plastic wheel because the latter was required to mesh with other, relatively low loaded gears. To achieve the same effect it was necessary to increase the pressure angle of the

pinion to match it with the deformed shape of the wheel. To find a pragmatic solution to the problem a number of iterative FE analyses were carried out in which the steel gear pressure angle was increased in steps of 0.5◦ from the nominal 12◦ up to 14◦ . The results are shown in Fig. 10. For the nominal load the figure shows that TE amplitude was minimum for a pressure angle of 13◦ .

Fig. 10 TE predicted by FE analysis at an input torque of 3 Nm and with pinion pressure angle varying from 12◦ to 14◦ JMES439 © IMechE 2007

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Final design of drive train for the Océ CPS-700 colour printer

However, this is only the case for this specific load. This measure was enough to reduce image distortions to an acceptable level – the production gear train is shown in Fig. 11. 7

CONCLUSIONS

This paper has examined the effects of tooth stiffness on the TE of polymer gear trains. The main conclusions are as follows. 1. A simple beam model has been used to demonstrate the effect of tooth stiffness on TE. FEA can be used to refine this approach and to enable the adjustment of tooth designs to minimize TE. 2. When designing gears made from identical or similar materials TE reduces with increasing contact ratio and the amplitude of TE may be greatly reduced by the use of integer contact ratios. 3. When designing gears made from materials with widely differing moduli, TE also reduces with increasing contact ratio but there is no advantage in seeking an integer contact ratio. 4. When different materials are used, one method of reducing TE variation is to attempt to balance tooth stiffnesses. This is usually achieved by thinning the stiffer gear teeth and widening the less stiff teeth proportionally. 5. An alternative method of reducing TE variation when different materials are used is to modify the pressure angle such that under load both sets of teeth have approximately the same operating pressure angle. Either pinion or wheel or both gear teeth can be modified in this way to achieve the same result. This can only be achieved for a specific load. 6. Although it is known that true contact ratio and true pressure angle differ from their kinematic Proc. IMechE Vol. 221 Part C: J. Mechanical Engineering Science

nominal values under load, the effect of these differences is magnified in highly loaded polymer gear combinations. 7. Plastic gears when run dry (unlubricated) and at high loads and speeds will usually reach high bulk body temperatures. This can lead to a significant reduction in Young’s modulus and subsequent gear stiffness. The operating elastic modulus should be used in all calculations when minimizing TE. 8. The method of changing nominal pressure angle has been used successfully to reduce inaccuracies caused by variation in TE in a high precision printer drive. 9. Future developments should include modelling of the gear body and shaft interface. Coefficients of friction for many polymer combinations are known to be high especially under dry running conditions [20]. The inclusion of friction forces would further refine the modelling, but a study on the influence of frictional forces on TE. Finally, the design considerations presented in the above conclusions presuppose that the stiffness component equals or exceeds TEs caused by gear manufacturing inaccuracies. For a well manufactured polymer gear under high load this is usually the case. ACKNOWLEDGEMENTS The authors would like to thank the European Commission for supporting this work under the Industrial and Materials Technologies Programme (Brite-EuRam III) (Contract BRPR-CT98-0703), High Performance Polymer Gears-PROGEAR. REFERENCES 1 Smith, J. D. Gears and their vibration, 1983 (Marcel Dekker, The Macmillan Press Ltd, London). 2 Breeds, A., Kukureka, S. N., Mao, K., Walton, D., and Hooke, C. J. Wear behaviour of acetal gear pairs. Wear, 1993, 166, 85–91. 3 Timoshenko, S. and Baud, R. V. Strength of gear teeth is greatly affected by fillet radius. Autom. Ind., 1926, 5(4), 138–142. 4 Baud, R. V. and Peterson, R. E. Load and stress cycle in gear teeth. Mech. Eng., 1929, 51(9), 653–662. 5 Walker, H. Gear tooth deflection and profile modification. Engineer, 1938, 166, 319–412, 434–436. 6 Harris, S. L. Dynamic loads on the teeth of spur gears. Proc. Inst. Mech. Eng., 1958, 172, 87–112. 7 Gregory, R. W., Harris, S. L., and Munro, R. G. Dynamic behaviour of spur gears. Proc. Instn Mech. Engrs., 1963, 178(Pt 1), 207–226. 8 Niemann, G. and Baethge, J. Drehwegfehler, Zahnfederharte und Geravsch bei Stirnradern. VDI- Z Band 112, Nr 4 and 8, pp. 205–214, 495–499. JMES439 © IMechE 2007

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9 Weber, C. The deformation of loaded gears and the effect of their load carrying capacity. Report no 3, British Scientific and Industrial Research, Sponsored Research (Germany), 1949. 10 Attia, A. Y. Deflection of spur gear teeth cut in thin rims. Trans. ASME, J. Eng. Ind., 1964, 86(4), 333–342. 11 Aida, T. and Terauchi, Y. On the bending stress of a spur gear. Bull. JSME, 1962, 5(17), 161–170. 12 Premilhat, A., Tordion, G. V., and Baronet, C. N. An improved determination of the elastic compliance of a spur gear tooth acted on by a concentrated load. Trans. ASME, J. Eng. Ind., 1974, 96(2), Ser B, 382–384. 13 Terauchi, Y. and Nagamura, K. Study on deflection of spur gear teeth. Bull. JSME, 1980, 23(184), 1682–1688. 14 Coy, J. J. and Chao, H. C. A. A method of selecting grid size to account for Hertz deformation in finite element analysis of spur gears. Trans. ASME, J. Mech. Des., 1982, 104, 759–766. 15 Steward, J. H. The compliance of solid, wide faced spur gears. In Proceedings of the International Power Transmission and Gearing Conference, 1989, pp. 181–189. 16 Walton, D., Tessema, A. A., Hooke, C. J., and Shippen, J. M. Load sharing in metallic and non-metallic gears. Proc. Instn Mech. Engrs, Part C: J. Mechanical Engineering Science, 1994, 208, 81–87. 17 Walton, D., Tessema, A. A., Hooke, C. J., and Shippen, J. M. A note on tip relief and backlash allowances in non-metallic gears. Proc. Instn Mech. Engrs, Part C: J. Mechanical Engineering Science, 1995, 209, 383–388. 18 White, J., Walton, D., and Weale, D. J. The beneficial effect of tip relief on plastic spur gears. Conference Proceedings at ANTEC ’98, Society of Plastics Engineers,

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Atlanta, USA, April 1998, vol. III, pp. 3013–3017 (ISBN 1-56676-669-9). 19 Algor Inc. 150 Beta Drive, Pittsburgh, PA, 15238-2932 USA www.algor.com, 1998. 20 Walton, D., Cropper, A. B.,Weale, D. J., and Klein Meuleman, P. The efficiency and friction of plastic cylindrical gears, part 1: influence of materials. Proc. Instn Mech. Engrs, Part J: J. Engineering Tribology, 2002, 216, 75–92.

BIBLIOGRAPHY Wilson, A. R. Transmission errors in polymer gear trains. Undergraduate Thesis, Department of Mechanical Engineering, The University of Birmingham, 2004.

APPENDIX Notation E F G I m w x

Young’s modulus force shear modulus second moment of area module face width ratio of distance of contact point from tip of driving tooth to tooth module

δ ν

deflection Poisson’s ratio

Proc. IMechE Vol. 221 Part C: J. Mechanical Engineering Science