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Developing the Epitrochoid Curve in Gear Teeth Below the Base Circle

Developing the Epitrochoid Curve in Gear Teeth Below the Base Circle Plotting from tooth centre line. Developing the Epitrochoid Curve in Gear Teeth Below the Base Circle By H. C. Pepper* Plotting from space centre line The angle Φ is common to both formulae. The term Inv θ is found from Involute Function Tables N text books on gears and gear teeth the form dealt with so far a series of four or five points has and the term Invr. from Polar Involute Function of the tooth below the base circle is usually been found sufficient and if each step is worked Tables. Having calculated the four positions, dismissed with a few words of advice on the out for all four points the calculation can soon be changing the angle β each time, multiply to give evils of undercutting, and while formulae for made. The positions are taken as angular rotations the required magnification. Mark off length S calculating the form of the involute curve abound, of the mating gear at, say, 2, 4, 6, 8 deg. and are below pitch line on the chosen centre line, tooth no one worries about the epitrochoid which is the designated in the formula as angle β. The curve or space, and at right angles from this point form of the curve below the base circle. This may can be drawn from the tooth or space centre line measure off R. This gives the desired point on the be because the involute is a self-respecting curve by using the appropriate formula. curve. that never varies except in size while the epitro­ The data required is : To obtain the point where the curve blends choid varies as the distance below the base circle A Centre distance between gears. with the root radius the following formulae will and the ratio of the rolling circles vary. Of course B Root radius of gear to be drawn. give the required dimensions of S and R, in the case of generated gears, provided that the C Pitch radius of gear to be drawn. root diameter gives sufficient clearance the cutter D Pitch radius of mating gear. Plotting from space centre line will sweep out its own correct path, but with gears N Number of teeth in gear to be drawn ground with formed wheels the question of the N Number of teeth in mating gear. form below the base circle does arise, especially θ Pressure angle. with small pump gears of about nine to sixteen T Thickness of tooth of mating gear at pitch teeth. The easiest way to examine the form of the line. root is by projection and here comes the problem β Angle of rotation of mating gear. (As chosen) of drawing the correct form. Plotting from tooth centre line Then: It used to be standard practice in gear drawing to finish off the flanks by drawing a radial line from where the involute meets the base circle down to the root diameter blending with a small fillet in the corner. If the gear is meshing with another with a Any point of a cutter or gear at the same radius small number of teeth this construction will un­ will describe the same curve which could be necessarily weaken the root of the tooth, and, if obtained without angle a. The use of angle α is the gear has been roughed out on a generating necessary to fix the position of the curve relative type of machine previous to case-hardening, it to the involute. In a further development the will be found out that an excessive amount is writer has simplified the calculations by using ground away in the fillets even to the extent of engraved glass plates which speed up the process locally removing the case; thus leaving a change of laying out the drawing. More actual experience of structure favourable to the commencement of will be obtained before the details of this idea are fractures. Undue wear on the corners of the published. grinding wheel will also ensue. The form to be aimed at is that which would be generated by the tip of the mating gear if it were carried through to the opposing root diameter. A method in which the mating gear is drawn on tracing paper and the successive positions pricked through while rolling the gears round on their respective pitch circles is impracticable owing to the large radii involved at 25 or 50 magnifications. A simple method devised by the writer is by calculating the position of a series of points representing successive positions of the tip of the mating gear. These are laid off on the drawing and connected by a smooth curve. In the gears * Superintendent, Metrology Department, The de Havilland Aircraft Co. Ltd. P. 284, Col. 3, line 58. Read The Estimation of Aerofoil Lifting P. 285, Col. 1, line 44. In (vi), for 'figure' read 'factor'. P. 285, Col. 1, line 49. For 'xn' read ' x n' Characteristics P. 285, Col. 1, line 60. For CHART III, read CHART IV. P. 285, Col. 2, line 8. For CHART III, read CHART IV. It is regretted that, owing to the non-receipt of P. 282, Column 3, equation 7. In denominator, P. 285, Col. 2, line 11. For 'xnxf, read ' x n x f.' his corrected proof from the author till after the read P. 285, Col. 2, line 17. For 'xnx,'' read ' xnx ' issue was published, the following errors appeared P. 285, Col. 2, line 26. For 'xnxfx', read ' x n xfx' P. 283, Column 1, lines 19 and 35. For CHART V, in the article by P. I. Bisgood on pp. 278-286 of P. 285, Col. 2, line 37. Read, (a /a ) the September issue of AIRCRAFT ENGINEERING : read CHART VI. 2 1 P. 285, Col. 3, line 15. Read, Trailing edge P. 284, Column 1, line 44. For 3·92, read 3·29. angle=S. P. 284, Column 1, line 65. Read (0·98-0·60). P. 282, Column 2, equation 6 =value of P. 286, Col. 1, line 25. For CHART III, read CHART IV, in both instances. a /a for aerofoil with central cut-out. P. 284, Col. 2. 3.2(2)(v). Read á= xa 2 1 1 1 P. 286, Col. 3, line 14. For CHART II, read P. 282, Column 3, line 7. For CHART IV (ii), read CHART IV, in both instances. CHART IV (iii). P. 284, Col. 3, line 38. Read 3·125 x 0·89 =2·7 8 320 Aircraft Engineering http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Aircraft Engineering and Aerospace Technology Emerald Publishing

Developing the Epitrochoid Curve in Gear Teeth Below the Base Circle

Aircraft Engineering and Aerospace Technology , Volume 19 (10): 1 – Oct 1, 1947

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Publisher
Emerald Publishing
Copyright
Copyright © Emerald Group Publishing Limited
ISSN
0002-2667
DOI
10.1108/eb031559
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Abstract

Plotting from tooth centre line. Developing the Epitrochoid Curve in Gear Teeth Below the Base Circle By H. C. Pepper* Plotting from space centre line The angle Φ is common to both formulae. The term Inv θ is found from Involute Function Tables N text books on gears and gear teeth the form dealt with so far a series of four or five points has and the term Invr. from Polar Involute Function of the tooth below the base circle is usually been found sufficient and if each step is worked Tables. Having calculated the four positions, dismissed with a few words of advice on the out for all four points the calculation can soon be changing the angle β each time, multiply to give evils of undercutting, and while formulae for made. The positions are taken as angular rotations the required magnification. Mark off length S calculating the form of the involute curve abound, of the mating gear at, say, 2, 4, 6, 8 deg. and are below pitch line on the chosen centre line, tooth no one worries about the epitrochoid which is the designated in the formula as angle β. The curve or space, and at right angles from this point form of the curve below the base circle. This may can be drawn from the tooth or space centre line measure off R. This gives the desired point on the be because the involute is a self-respecting curve by using the appropriate formula. curve. that never varies except in size while the epitro­ The data required is : To obtain the point where the curve blends choid varies as the distance below the base circle A Centre distance between gears. with the root radius the following formulae will and the ratio of the rolling circles vary. Of course B Root radius of gear to be drawn. give the required dimensions of S and R, in the case of generated gears, provided that the C Pitch radius of gear to be drawn. root diameter gives sufficient clearance the cutter D Pitch radius of mating gear. Plotting from space centre line will sweep out its own correct path, but with gears N Number of teeth in gear to be drawn ground with formed wheels the question of the N Number of teeth in mating gear. form below the base circle does arise, especially θ Pressure angle. with small pump gears of about nine to sixteen T Thickness of tooth of mating gear at pitch teeth. The easiest way to examine the form of the line. root is by projection and here comes the problem β Angle of rotation of mating gear. (As chosen) of drawing the correct form. Plotting from tooth centre line Then: It used to be standard practice in gear drawing to finish off the flanks by drawing a radial line from where the involute meets the base circle down to the root diameter blending with a small fillet in the corner. If the gear is meshing with another with a Any point of a cutter or gear at the same radius small number of teeth this construction will un­ will describe the same curve which could be necessarily weaken the root of the tooth, and, if obtained without angle a. The use of angle α is the gear has been roughed out on a generating necessary to fix the position of the curve relative type of machine previous to case-hardening, it to the involute. In a further development the will be found out that an excessive amount is writer has simplified the calculations by using ground away in the fillets even to the extent of engraved glass plates which speed up the process locally removing the case; thus leaving a change of laying out the drawing. More actual experience of structure favourable to the commencement of will be obtained before the details of this idea are fractures. Undue wear on the corners of the published. grinding wheel will also ensue. The form to be aimed at is that which would be generated by the tip of the mating gear if it were carried through to the opposing root diameter. A method in which the mating gear is drawn on tracing paper and the successive positions pricked through while rolling the gears round on their respective pitch circles is impracticable owing to the large radii involved at 25 or 50 magnifications. A simple method devised by the writer is by calculating the position of a series of points representing successive positions of the tip of the mating gear. These are laid off on the drawing and connected by a smooth curve. In the gears * Superintendent, Metrology Department, The de Havilland Aircraft Co. Ltd. P. 284, Col. 3, line 58. Read The Estimation of Aerofoil Lifting P. 285, Col. 1, line 44. In (vi), for 'figure' read 'factor'. P. 285, Col. 1, line 49. For 'xn' read ' x n' Characteristics P. 285, Col. 1, line 60. For CHART III, read CHART IV. P. 285, Col. 2, line 8. For CHART III, read CHART IV. It is regretted that, owing to the non-receipt of P. 282, Column 3, equation 7. In denominator, P. 285, Col. 2, line 11. For 'xnxf, read ' x n x f.' his corrected proof from the author till after the read P. 285, Col. 2, line 17. For 'xnx,'' read ' xnx ' issue was published, the following errors appeared P. 285, Col. 2, line 26. For 'xnxfx', read ' x n xfx' P. 283, Column 1, lines 19 and 35. For CHART V, in the article by P. I. Bisgood on pp. 278-286 of P. 285, Col. 2, line 37. Read, (a /a ) the September issue of AIRCRAFT ENGINEERING : read CHART VI. 2 1 P. 285, Col. 3, line 15. Read, Trailing edge P. 284, Column 1, line 44. For 3·92, read 3·29. angle=S. P. 284, Column 1, line 65. Read (0·98-0·60). P. 282, Column 2, equation 6 =value of P. 286, Col. 1, line 25. For CHART III, read CHART IV, in both instances. a /a for aerofoil with central cut-out. P. 284, Col. 2. 3.2(2)(v). Read á= xa 2 1 1 1 P. 286, Col. 3, line 14. For CHART II, read P. 282, Column 3, line 7. For CHART IV (ii), read CHART IV, in both instances. CHART IV (iii). P. 284, Col. 3, line 38. Read 3·125 x 0·89 =2·7 8 320 Aircraft Engineering

Journal

Aircraft Engineering and Aerospace TechnologyEmerald Publishing

Published: Oct 1, 1947

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