Program External Gear Set Tip Relief Full Depth Teeth

Transcription

1 Program External Gear Set Tip Relief Full Depth Teeth Introduction At the first point of contact (gear tip and pinion root for a reduction gear) the deflection of the teeth already under load causes the incoming driver tooth to seem to be ahead of where it should be. This insures that the load will be picked up very abruptly at the tip of the driven tooth. (This condition is more serious on spur and LACR helicals than on full helicals.) This applies a heavy shock load at the driven tooth tip where the cantilever beam is the longest and produces large root stresses. The same thing happens to the driver tooth at the last point of contact but the effect of dropping the load abruptly is not as severe (mostly noise and vibration along with a rise in compressive stress from the tooth edge breaking the contact foot print"). From the first point of contact to the pitch point the action is approach action. The driven tooth is approaching the driver tooth and the tip acts much like a sharp edged scraper (or in severe cases, a cutting tool). After contact has passed the pitch point the action is recess action. The driven tooth is retreating from the driver tooth. Recess action is much more conducive to building and maintaining a lubricant film than is approach action, especially at start or end of action. Tooth spacing errors, pitch or profile, add to this effect for those teeth that are ahead of where they should be on the driver and behind where they should be on the driven. This means that even lightly loaded gears, with little deflection, suffer the same type of problem due to manufacturing error. One method of reducing the deflection problem (or at least not making it worse) is to make the tolerances on profile such that the base pitch of the driver can never be less than theoretical and the base pitch of the driven never more than theoretical. This will reduce the impact at the first point of contact because it will insure that the teeth on the driver are a little behind and on the driven a little ahead". Of course, this will make things worse at the last point of contact but conditions there are much less severe. The practical application of this method means that the profile tolerance on the driver should always be plus at the tip and on the driven always minus. As long as we have to live with these tolerances we may as well put them where they may help us and, at least, not hurt us. Of course, this must not be overdone as we will lose conjugate action. This depends upon the load level, gear quality class, modulus of elasticity, and tooth pressure angle. The model will split the profile tolerance between plus and minus if the gear is AGMA Q9 or less or if the gear is an idler.

2 UTS Integrated Gear Software In addition to tolerance control the application of tip relief on the gears will reduce the engagement shock and the tendency to scrape the driver root. (This will also help the abrupt dropping of load at the last point of contact.) The load can then be picked up smoothly and the full load (plus shock) on the tooth tips eliminated. Since the tip relief can be built into the production tools the advantages cost us very little except getting the specifications right at the design stage. One of the parameters used in determining the proper amount of tip relief is the allowable tolerance on the elements of the gear (pitch tolerance, profile tolerance and lead tolerance). These tolerances are determined by the AGMA quality class of the gear. The tolerances will be calculated by the model if the quality class is entered. If one or more tolerance is not to be according to the quality class the tolerance can be overridden be entering single values in the input column. Examples Example 1 The first example will establish the tip relief specifications for the following steel spur gear set: 22 tooth driver 55 tooth driven 10 normal diametral pitch 20 degree normal pressure angle 2 inch face width Quality class Q12 ANSI/AGMA 2000-A88 Gear Handbook 100 Horsepower at 3600 RPM driver speed Open a new analysis in and follow the sequence of the input form, which is shown in Figure 1. Use the calculated defaults where they appear. Form diameters will not be entered. Neither gear is an idler. Select power instead of driver torque. Select total lead mismatch between teeth instead of AGMA load distribution factor. If the lead mismatch between the teeth is not known it can be calculated using UTS Model , Total Lead Mismatch for Gear Pair in Housing". We will use a lead mismatch of inch for these gears. The unmodified profile contact ratio is set to 1.05 to avoid a profile contact less than unity at light loads when the modified portions of the teeth are not in contact. Base pitches unmodified from mid-point are set to.525 to split the unmodified portion of 2

3 External Gear Set Tip Relief Full Depth Teeth the line of action about evenly between the driver and driven. (The actual arc length of the modified portions of the teeth can be changed later if desired.) Select tip relief on both gears. Input and output values are shown in Report 1. Fig. 1 3

4 UTS Integrated Gear Software Report 1 MESSAGE m2 m3 m4 Driver & Driven Tip Relief DRIVER, number of teeth 22 DRIVEN, number of teeth 55 NORMAL PLANE Diametral pitch Pressure angle Module Base pitch TRANSVERSE PLANE Diametral pitch Pressure angle Module Base pitch COMMON Helix angle Base helix angle Operating center distance /in ` deg mm ` in /in ` deg mm ` in deg deg in 4

5 External Gear Set Tip Relief Full Depth Teeth Face width DIAMETERS Outside diameter, Driver Base Diameter, Driver Outside diameter, Driven Base Diameter, Driven OPERATING DATA in in in in in Working depth in Operating transverse pressure angle deg Operating PD, Driver in Form diameter (TIF) in Driver is an idler n Operating PD, Driven in Form diameter (TIF) in Driven is an idler n Roll angle at operating PDs deg Length of contact, transverse plane in Profile Helical Total LOADS Power Driver Torque Driver Speed HP lbf-in rpm 5

6 UTS Integrated Gear Software Tangential pitch line load Line of action load Tangential load per unit of face Line of action load per unit of face K factor Unit load MATERIAL lbf lbf lbf/in lbf/in psi psi Modulus of elasticity, Driver Modulus of elasticity, Driven MESH DEFLECTION (FULL DEPTH TEETH) Dudley: Tangent to PD, trans plane Dudley: Normal to tooth AGMA Paper : Normal to tooth AGMA Std 2001: Normal to tooth TIP RELIEF AMOUNT DRIVER TOLERANCE psi psi in in in in AGMA Q Class 12 Pitch error, Driver Profile error, Driver (+ tip) Lead error, Driver TIP RELIEF AMOUNT DRIVEN TOLERANCE in in in AGMA Q Class 12 Pitch error, Driven Profile error, Driven (- tip) Lead error, Driven in in in 6

7 External Gear Set Tip Relief Full Depth Teeth TIP RELIEF AMOUNT DUDLEY (GENERAL DESIGN) AGMA Load Distribution Factor Total lead mismatch between teeth Tooth stiffness constant in psi AGMA Face Load Distribution Factor AGMA Trans Load Distribution Factor Driver tip - last point of contact in Driven tip - first point of contact in TIP RELIEF AMOUNT AGMA PAPER (PRECISION GEARS) Driver tip - last point of contact S_last2 Driven tip - first point of contact in in in S_frst in TIP RELIEF AMOUNT BASED ON DEFLECTION AND ERRORS Trans deflection - AGMA Std 2001 Driver tip - last point of contact Driven tip - first point of contact Driver tip - last point of contact Driven tip - first point of contact TIP RELIEF LOCATION (NO UNDERCUT) in in in in in Tip relief on both gears, 'tb, Unmodified profile contact ratio tb 7

8 UTS Integrated Gear Software TIP RELIEF LOCATION (NO UNDERCUT) DRIVER Start of active profile Roll angle at SAP Roll angle at TIF Mid-point of length of contact in deg deg in Base pitches unmod from mid-point Roll angle at mid-point Start tip relief Roll angle at start of tip relief Arc length of tip relief Roll angle - tip relief to OD Roll angle at OD TIP RELIEF LOCATION (NO UNDERCUT) DRIVEN Start of active profile Roll angle at SAP Roll angle at TIF Mid-point of length of contact deg in deg in deg deg in deg deg in Base pitches unmod from mid-point Roll angle at mid-point Start tip relief Roll angle at start of tip relief Arc length of tip relief Roll angle - tip relief to OD Roll angle at OD deg in deg in deg deg 8

9 External Gear Set Tip Relief Full Depth Teeth TIP AND ROOT RELIEF ON DRIVER Start root relief Roll angle at start root relief TIP AND ROOT RELIEF ON DRIVEN in deg Start root relief Roll angle at start root relief in deg The model offers guidance on tip relief from three different sources. The first is from Dudley's Handbook of Practical Gear Design and suggests that we need about inch at the driver (pinion) tip and about inch at the driven (gear) tip. This is for general design and Mr. Dudley suggests a more detailed analysis for critical drives. The second is from AGMA Paper by Hans Sigg and recommends / (inches) at the driver (pinion) tip and about / at the driven (gear) tip. This data is intended for precision ground gears only (AGMA Q12 and up). The third set of corrections is based on the actual base pitch errors allowed by the tolerances and on the deflections according to the tooth stiffness from AGMA Standard 2001-B88. The values of inch correction for the driver (pinion) tip and inch for the driven (gear) tip are for the mean average error condition with a normal deflection of inch. This correction would leave about 20% load at the first point of contact. The fourth set of corrections is also based on the actual base pitch errors allowed by the tolerances and on the deflections according to the tooth stiffness from AGMA Std 2001-B88. The values of inch at the driver tip and inch at the driven tip would be required to remove all load at the first point of contact with the Root-Mean-Square errors plus the deflection. The RMS errors should cover more than 95% of gears. After the proper amount and location of the tooth modifications have been established it is necessary to transfer this data to the engineering drawings. This is usually done by using K charts on the drawing. These charts usually plot the required and allowed deviation from the true involute form for the tooth. Involute checking machines produce a chart which shows the deviation in form plotted against 9

10 UTS Integrated Gear Software the roll angle for the gear. A straight line indicates no deviation from the involute form. This model includes these plots for driver and driven for all four sets of relief calculations included in the model. Go to the TK Solver Plot Sheet, or the pull-down list of plots on the Toolbar, and select the pair of plots you wish to use. Plot name dr_dudley dn_dudley dr_2001_20 dn_2001_20 dr_2001_0 dn_2001_0 dr_agma_109 dn_agma_109 Tip/root relief method Dudley's Handbook-Driver Dudley's Handbook-Driven AGMA Std % Load-Driver - Ave Errors AGMA Std % Load-Driven - Ave Errors AGMA Std % Load-Driver - RMS Errors AGMA Std % Load-Driven - RMS Errors AGMA Paper 109-Driver AGMA Paper 109-Driven Figures 2 and 3 are the plots from our example for the AGMA Standard 2001 calculations for no load at the first point of contact using the Root-Mean-Square (RMS) values of the allowable tooth errors. (To view plots, toggle to TK Solver and select the plots from the Plots List in the Object Bar.) Fig. 2 10

11 External Gear Set Tip Relief Full Depth Teeth Fig. 3 NOTE: You may change the vertical scale on these plots by changing the Y-Axis range on the plot subsheet. With the pointer on the plot, right-click the mouse and select Display Subsheet. In some cases it may be advantageous to use tip and root relief on a single gear instead of tip relief on both gears. This may occur, for example, with the sun and planet and ring gears in an epicyclic gear set. If tip and root relief is used on the planet gear it will provide relieved contact at the sun and ring gears without any modification to the sun and ring. If root relief is used for any reason it is then necessary to properly locate the diameter at which the root relief would start. Figure 4 and Report 2 show the same model with tip relief on both gears changed to tip and root relief on the driver. The plots for this solution are shown in Figures 5 and 6. 11

12 UTS Integrated Gear Software Fig. 4 12

13 External Gear Set Tip Relief Full Depth Teeth Report 2 MESSAGE m2 m3 m4 Driver Tip & Root Relief DRIVER, number of teeth 22 DRIVEN, number of teeth 55 NORMAL PLANE Diametral pitch Pressure angle Module Base pitch TRANSVERSE PLANE Diametral pitch Pressure angle Module Base pitch COMMON Helix angle /in ` deg mm ` in /in ` deg mm ` in deg 13

14 UTS Integrated Gear Software Base helix angle Operating center distance Face width DIAMETERS Outside diameter, Driver Base Diameter, Driver Outside diameter, Driven Base Diameter, Driven OPERATING DATA deg in in in in in in Working depth in Operating transverse pressure angle deg Operating PD, Driver in Form diameter (TIF) in Driver is an idler n Operating PD, Driven in Form diameter (TIF) in Driven is an idler n Roll angle at operating PDs deg Length of contact, transverse plane in Profile Helical Total LOADS Power HP 14

15 External Gear Set Tip Relief Full Depth Teeth Driver Torque Driver Speed Tangential pitch line load Line of action load Tangential load per unit of face Line of action load per unit of face K factor Unit load MATERIAL lbf-in rpm lbf lbf lbf/in lbf/in psi psi Modulus of elasticity, Driver Modulus of elasticity, Driven MESH DEFLECTION (FULL DEPTH TEETH) Dudley: Tangent to PD, trans plane Dudley: Normal to tooth AGMA Paper : Normal to tooth AGMA Std 2001: Normal to tooth TIP RELIEF AMOUNT DRIVER TOLERANCE psi psi in in in in AGMA Q Class 12 Pitch error, Driver Profile error, Driver (+ tip) Lead error, Driver TIP RELIEF AMOUNT DRIVEN TOLERANCE in in in AGMA Q Class 12 Pitch error, Driven in 15

16 UTS Integrated Gear Software Profile error, Driven (- tip) in Lead error, Driven in TIP RELIEF AMOUNT DUDLEY (GENERAL DESIGN) AGMA Load Distribution Factor Total lead mismatch between teeth Tooth stiffness constant in psi AGMA Face Load Distribution Factor AGMA Trans Load Distribution Factor Driver tip - last point of contact in Driven tip - first point of contact in TIP RELIEF AMOUNT AGMA PAPER (PRECISION GEARS) Driver tip - last point of contact S_last2 Driven tip - first point of contact in in in S_frst in TIP RELIEF AMOUNT BASED ON DEFLECTION AND ERRORS Trans deflection - AGMA Std 2001 Driver tip - last point of contact Driven tip - first point of contact Driver tip - last point of contact Driven tip - first point of contact TIP RELIEF LOCATION (NO UNDERCUT) in in in in in Tip relief on both gears, 'tb, trdr Unmodified profile contact ratio

17 External Gear Set Tip Relief Full Depth Teeth TIP RELIEF LOCATION (NO UNDERCUT) DRIVER Start of active profile Roll angle at SAP Roll angle at TIF Mid-point of length of contact in deg deg in Base pitches unmod from mid-point Roll angle at mid-point Start tip relief Roll angle at start of tip relief Arc length of tip relief Roll angle - tip relief to OD Roll angle at OD TIP RELIEF LOCATION (NO UNDERCUT) DRIVEN Start of active profile Roll angle at SAP Roll angle at TIF Mid-point of length of contact deg in deg in deg deg in deg deg in Base pitches unmod from mid-point Roll angle at mid-point Start tip relief Roll angle at start of tip relief Arc length of tip relief Roll angle - tip relief to OD Roll angle at OD deg in deg in deg deg 17

18 UTS Integrated Gear Software TIP AND ROOT RELIEF ON DRIVER Start root relief Roll angle at start root relief TIP AND ROOT RELIEF ON DRIVEN Start root relief Roll angle at start root relief in 18 deg in deg Fig. 5 18

19 External Gear Set Tip Relief Full Depth Teeth Fig. 6 NOTE: The deflection given by AGMA is for steel only and if the modulus of elasticity for either gear is other than for steel the deflection will not be calculated. (The tip relief amounts also will not be calculated.) 19

20 UTS Integrated Gear Software Example 2 The second example is a 23-degree helical gear set made of powdered metal with a modulus of elasticity of 20,000,000 psi. The pinion has 19 teeth and the gear 45. The normal diametral pitch is 26 with a 20-degree pressure angle. We will establish the tip relief specifications for these gears with average errors and with 20% load left at the first point of contact. Figure 7 and Report 3 are the inputs and outputs for this model. Fig. 7 20

21 External Gear Set Tip Relief Full Depth Teeth Report 3 MESSAGE m2 m3 m4 Driver & Driven Tip Relief DRIVER, number of teeth 19 DRIVEN, number of teeth 45 NORMAL PLANE Diametral pitch Pressure angle Module Base pitch TRANSVERSE PLANE Diametral pitch Pressure angle Module Base pitch COMMON Helix angle Base helix angle Operating center distance /in ` deg mm ` in /in ` deg mm ` in deg deg in 21

22 UTS Integrated Gear Software Face width DIAMETERS Outside diameter, Driver Base Diameter, Driver Outside diameter, Driven Base Diameter, Driven OPERATING DATA in in in in in Working depth in Operating transverse pressure angle deg Operating PD, Driver in Form diameter (TIF) in Driver is an idler n Operating PD, Driven in Form diameter (TIF) in Driven is an idler n Roll angle at operating PDs deg Length of contact, transverse plane in Profile Helical Total LOADS Power Driver Torque Driver Speed 0.62 HP lbf-in rpm 22

23 External Gear Set Tip Relief Full Depth Teeth Tangential pitch line load Line of action load Tangential load per unit of face Line of action load per unit of face K factor Unit load MATERIAL lbf lbf lbf/in lbf/in psi psi Modulus of elasticity, Driver Modulus of elasticity, Driven MESH DEFLECTION (FULL DEPTH TEETH) Dudley: Tangent to PD, trans plane Dudley: Normal to tooth AGMA Paper : Normal to tooth AGMA Std 2001: Normal to tooth TIP RELIEF AMOUNT DRIVER TOLERANCE psi psi in in Q12 Steel in in AGMA Q Class 9 Pitch error, Driver Profile error, Driver (+ tip) Lead error, Driver TIP RELIEF AMOUNT DRIVEN TOLERANCE in in in AGMA Q Class 9 Pitch error, Driven Profile error, Driven (- tip) Lead error, Driven in in in 23

24 UTS Integrated Gear Software TIP RELIEF AMOUNT DUDLEY (GENERAL DESIGN) AGMA Load Distribution Factor Total lead mismatch between teeth Tooth stiffness constant in psi AGMA Face Load Distribution Factor AGMA Trans Load Distribution Factor Driver tip - last point of contact in Driven tip - first point of contact in TIP RELIEF AMOUNT AGMA PAPER (PRECISION GEARS) Driver tip - last point of contact S_last2 Driven tip - first point of contact NA in NA in NA in S_frst2 NA in TIP RELIEF AMOUNT BASED ON DEFLECTION AND ERRORS Trans deflection - AGMA Std 2001 Driver tip - last point of contact Driven tip - first point of contact Driver tip - last point of contact Driven tip - first point of contact TIP RELIEF LOCATION (NO UNDERCUT) in in in in in Tip relief on both gears, 'tb, Unmodified profile contact ratio tb 24

25 External Gear Set Tip Relief Full Depth Teeth TIP RELIEF LOCATION (NO UNDERCUT) DRIVER Start of active profile Roll angle at SAP Roll angle at TIF Mid-point of length of contact in deg deg in Base pitches unmod from mid-point Roll angle at mid-point Start tip relief Roll angle at start of tip relief Arc length of tip relief Roll angle - tip relief to OD Roll angle at OD TIP RELIEF LOCATION (NO UNDERCUT) DRIVEN Start of active profile Roll angle at SAP Roll angle at TIF Mid-point of length of contact deg in deg in deg deg in deg deg in Base pitches unmod from mid-point Roll angle at mid-point Start tip relief Roll angle at start of tip relief Arc length of tip relief Roll angle - tip relief to OD Roll angle at OD deg in deg in deg deg 25

26 UTS Integrated Gear Software TIP AND ROOT RELIEF ON DRIVER Start root relief Roll angle at start root relief TIP AND ROOT RELIEF ON DRIVEN Start root relief Roll angle at start root relief in deg in deg Figures 8 and 9 are the plots from our example for the AGMA Std 2001 deflection for 20% load at the first point of contact using the average values of the allowable tooth errors. Note that at this quality class the profile error is + or - instead of + tip for the driver and - tip for the driven. Fig. 8 26

27 External Gear Set Tip Relief Full Depth Teeth Fig. 9 The model checks various items and, if any problem is found, it will appear in the MESSAGE section of the report, along with advisory messages along the way. All equations and methods used are on the Rule Sheet and in the functions of the TK Solver model. References: Data extracted from AGMA Standard 2001-B88, Fundamental Rating Factors and Calculation Methods for Involute Spur and Helical Gear Teeth, and from AGMA Paper , Profile and Longitudinal Corrections on Involute Gears, with the permission of the Publisher, the American Gear Manufacturers Association, 1500 King Street, Suite 201, Alexandria, VA Handbook of Practical Gear Design, 2nd Edition, by Darle W. Dudley, Dudley Engineering Co., Published by McGraw-Hill, Inc,