Program 60-554—Cylindrical Worm Gear Analysis Introduction

Program 60-554—Cylindrical Worm Gear Analysis

Introduction This model has been prepared to help you with the design of new worm drives or to analyze existing worm sets. The model has been configured to help you design a good worm drive and design the cutting tools required at the same time. If you wish to start with a worm gear hob the model will allow you to do this quickly and efficiently. There are a number of problems associated with worm gears. A gear set of a given size may have a load capacity and efficiency that vary through a wide range due to relatively minor differences in the geometry used. A recurring problem is the actual form used on the thread of the worm. The profile on the worm thread can vary from straight sides in the axial section through an infinity of forms to an involute helicoid. The form of the profile depends on the manufacturing method used to produce it. Straight sides in the axial plane will be produced by a V-shaped tool set in a plane containing the axis of rotation of the worm. This is the form used for screws and is called a “screw helicoid”. If the tool is set to the lead angle of the worm then a “chased helicoid” is produced. Many worms are milled or ground with a double conical cutter or wheel. If the milling cutter or grinding wheel has straight sides in the axial section and is set to the worm lead angle it will produce an approximation of a “convolute helicoid” on the worm. (The convolute helicoid has straight line generators tangent to a base cylinder that is concentric to the helicoid. The limiting forms of the convolute helicoid are the screw helicoid and the involute helicoid.) The actual shape of the worm thread will vary with the diameter of the milling cutter or grinding wheel and is called a “milled helicoid”. If the worm is hobbed or rolled the form will be an “involute helicoid”. (This is the form of a helical involute gear.) The actual form used does not matter greatly, but it is ESSENTIAL that the hob used to machine the worm gear have the same form as the worm. The method of manufacture used for the hob should also be used for the worm. If the worm must be milled or ground (a milled helicoid) and the method of hob manufacture can not be duplicated then it may be necessary to introduce curvature into the milling cutter or grinding wheel to approximate as closely as possible the form on the hob. Most of the “standards” and conventions used in worm drive design have been

UTS Integrated Gear Software

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established to ease manufacture, not to optimize the operation of the drive in the field and this should be kept in mind by the designer who wishes to get the most out of a his design. Program 60-554 is based on an involute helicoid worm. This form has the largest base cylinder of all the convolute helicoids. Conjugate (smooth) gear tooth action can not take place below the points where the pressure angle becomes zero. The size of the field of conjugate action of the involute helicoid is the limit for all convolute helicoids. If we keep the contact of the worm gear above a plane tangent to the base cylinder of the involute helicoid we will be safe whatever helicoid we actually use. In extreme cases, where the worm must be kept to the smallest possible size, it may be necessary to use a screw or chased helicoid to utilize the field of conjugate action closer to the center of the worm. In this case an analysis must be made of the conjugate action field for the type of worm used. TK Solver models can be solved in almost any manner we please because of TK’s “backsolving” capability. Note that, for your convenience, the usual inputs have been marked with “E” or “H” for starting with an existing design or with an existing worm gear hob. It is not necessary to follow this procedure exactly if some of the information you have is not the same as the data marked. Examples Example 1 For our first example we will analyze an existing design (or a design we wish to complete) and design a worm gear hob and worm cutting tool as we proceed. We will assume that the speed of the worm is 3600 RPM and obtain an estimate of the starting and running efficiency. Also we will want a size over measuring pins for the worm. Figures 1-1A and 1-1B show the input data before solving.

60-554–Cylindrical Worm Gear Analysis

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Fig. 1-1A


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Fig. 1-1B

We entered the worm lead but could have input the axial pitch instead. We could have entered the ratio instead of the number of gear teeth. The solved model is shown in Report 1-1.


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Report 1-1

Model Title : Program 60-554 Unit System: US Description Value Unit Comment

m1 RATIO

m2 PA_LA

m3 GEAR_OD

m4 OK

m11 FACES

m12 GEOMETRY

m13 WORM_TT

m14 OK

m21

m22

m23

m24

COMMON DATA

Gear Ratio 14.0000

Recommended Minimum Gear Teeth 25.0000

Center Distance E 4.375 in

Normal Pressure Angle EH 25.0000 deg

Minimum Backlash (Axial) EOHO 0.004 in

Maximum Backlash (Axial) EOHO 0.006 in

Worm Root Clearance E1H 0.036 in


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Recommended Root Clearance 0.025 in CONTACT FACTORS

Gear OD Factor (Rec 1.0 to 1.5) H 1.45589

Pitch Line Offset Factor H 0.67068 WORM DATA

Number of Threads EH 3

Axial Pitch 0.500 in

Lead of Thread E 1.500 in

Axial Pressure Angle 25.4962 deg

OUTSIDE DIAMETER E 2.471 in

Max Allow OD (Set by hob OD & clear) in

Normal Thread Thickness at OD 0.131 in

WORM LENGTH (Default=WFrec+) 2.850 in *EOHO

Recommended Minimum Worm Length 2.825 in

EFF PITCH DIAMETER (Thread=Space) 2.228 in E

Normal Pitch 0.489 in

Lead Angle 12.0946 deg

Transverse Pressure Angle 65.8042 deg

Axial Thread Thickness 0.250 in

Normal Space Width 0.245 in

WHOLE DEPTH 0.339 in

Working Depth 0.303 in

Addendum (from Eff PD) 0.121 in


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Dedendum (from Eff PD) 0.218 in

PITCH DIAMETER 2.065 in

Lead Angle at Pitch Diameter 13.0160 deg

ROOT DIAMETER (H Default=dr1max) 1.792 in E2

Max Allow Worm Root (Set by hob) in

BASE DIAMETER (Involute Helicoid) 0.913 in WORM CUTTING TOOL

Setting Angle 12.0946 deg

Tip to Reference Line 0.218 in

Thickness at Ref Line 0.245 in

Tip Radius (Default=rtl_max) 0.032 in

Max Allowable Tip Radius 0.032 in

WORM MEASUREMENT

Measuring Wire Diameter EOHO 0.270 in

0 Backlash: Size Over 3 Wires 2.611 in

Projection of Wire Above Worm OD 0.070 in

Min Backlash: Size Over 3 Wires 2.603 in

Max Backlash: Size Over 3 Wires 2.599 in GEAR DATA

Number of Teeth EH 42

Outside Diameter E 6.875 in

Pitch Diameter 6.685 in

Throat Diameter EO 6.885 in

Whole Depth (from Throat Dia) 0.339 in


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Throat Form Radius EOHO in

Actual Face Width (Default=Frec-)EOHO 1.270 in

Recommended Face Width 1.280 in GEAR HOB DATA

Outside Diameter H 2.557 in

Lead H 1.500 in


Lead Angle at Ref Line deg

Normal Pitch at Ref Line in

Normal Diametral Pitch at Ref Line 1/in

Normal Module mm`

Tooth Tip to Ref Line H in

Normal Tooth Thickness (Ref line) H in

Axial Tooth Thickness (Ref line) in

Tooth Tip Radius (Default=rhmax) H in

Max Allowable Tooth Tip Radius in

Root Diameter H in

Ref Diameter in

EFF HOB DIA (Hob Tooth = Hob Space): 2.242 in

Clearance at Leaving Edge of Gear 0.003 in


Hob Addendum (Tooth Tip to Eff Dia) 0.158 in

HOBBING CENTER DISTANCE 4.382 in


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Model Title : Program 60-554 Unit System: US Description Value Unit Comment EFFICIENCY

WORM SPEED EOHO 3600.0 rpm

Sliding Velocity at Pitch Diameter 1998.02 ft/min

Coefficient of Friction 0.0205

Running Efficiency 91 %

Starting Efficiency 57 % AUXILIARY DATA Worm

Eff Pitch Radius (Thread=Space) 1.114 in

Thread Thickness at Eff Pitch Radius In 1.167 in Plane of Rotation

Outside Radius 1.235 in

Lead Angle at Outside Radius 10.9372 deg

Transverse PA at Outside Radius 68.3082 deg

Pitch Radius 1.033 in

Pressure Angle at Pitch Radius 63.7593 deg

Root Radius 0.896 in

Base Radius (Involute Helicoid) 0.457 in

Lead Angle on Base Cylinder 27.6016 deg AUXILIARY DATA Gear



Throat Radius from Gear Center 3.443 in AUXILIARY DATA Hob

Axial Pressure Angle at Hob Ref Dia deg

Trans Pressure Angle at Hob Ref Dia deg


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Notice that the model is not completely solved. It will give you as much information as it can with the input data it has to work with. This allows you to solve a model progressively if desired. All the geometrical data for the worm and worm gear has been determined. If desired, the inputs can be changed or outputs can be changed to inputs and the model solved again (using the Power User form) until you are satisfied with the results. There are three message areas at the top of the report or TK Solver Variable Sheet to inform you if there is any problem with the solution. It is always best to solve enough of the model so that the messages are not blank. (If you wish to see which items are checked see the functions lim0, lim1 and lim2 on the TK Function Sheet.) In our example the items concerning the geometry of the worm and worm gear have been checked but not the cutting tools, as we have not yet defined the hob to be used to machine the worm gear. Next we will move to the section titled “Worm Measurement” on the wizard input form and enter the wire diameter we wish to use under Measuring Wire Diameter. The default value is about the size of wire to contact at the effective PD of the worm. We will assume the closest wire on hand is .28 inch. To complete the design of the hob, we must decide where we want the reference diameter on the hob. Since this will be a new hob, we will make the reference diameter the same as the effective diameter. Copy the effective hob diameter, deh, to the input column of the reference diameter, dhref, using the copy and paste commands, as shown in Figure 1-2. Notice that the effective hob diameter was calculated enough larger than the worm effective diameter to provide about .003 inch clearance between the worm teeth and worm gear teeth at the leaving side of the worm gear face. If we wish a different amount of clearance we need only to enter a value for the clearance we desire and solve again. We might wish to start with a larger clearance to obtain more hob oversize and, as the hob is sharpened back, and the hobbing center distance reduced, let the clearance diminish. In no event will we let the clearance become negative. These calculations are based on the hob being mounted at right angles to the worm gear axis. As the lead angle increases the clearance due to hob oversize will produce more deviation between the hobbed tooth form and the theoretical form. This can be partly corrected by mounting the hob with the hob swivel set “off angle”. (The amount of “off angle” is lam_eh minus lam_e. The edge clearance will no longer be equal to the value calculated by the model.)


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If the hob is set at right angles, the conditions obtained during the sharpening procedure can be easily checked by solving the model with the various hob configurations produced as the hob is sharpened back. We also copy the worm root diameter, dr1, to max allowable diameter, dr1max, to obtain a root diameter for the hob. This is also shown in Figure 1-2.

To finish our input specifications we will enter 3600 for WORM SPEED to obtain the efficiency.

The complete solved model is shown in Report 1-2.

Fig. 1-2

Report 1-2


m1 RATIO

m2 PA_LA

m3 GEAR_OD

m4 OK


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m11 FACES

m12 GEOMETRY

m13 WORM_TT

m14 OK

m21 TOOL

m22 GEOMETRY

m23 CHECKED

m24

COMMON DATA

Gear Ratio 14.0000









Pitch Line Offset Factor H 0.67068 WORM DATA




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Max Allow OD (Set by hob OD & clear) 2.519 in

















Max Allow Worm Root (Set by hob) 1.792 in

BASE DIAMETER (Involute Helicoid) 0.913 in


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Model Title : Program 60-554 Unit System: US Description Value Unit Comment WORM CUTTING TOOL






WORM MEASUREMENT











Throat Form Radius EOHO 0.939 in





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Lead H 1.500 in


Lead Angle at Ref Line 12.0208 deg

Normal Pitch at Ref Line 0.489 in

Normal Diametral Pitch at Ref Line 6.4241 1/in

Normal Module 3.9539 mm`

Tooth Tip to Ref Line H 0.158 in

Normal Tooth Thickness (Ref line) H 0.245 in

Axial Tooth Thickness (Ref line) 0.250 in

Tooth Tip Radius (Default=rhmax) H 0.021 in

Max Allowable Tooth Tip Radius 0.021 in

Root Diameter H 1.879 in

Ref Diameter 2.242 in





HOBBING CENTER DISTANCE 4.382 in EFFICIENCY





Starting Efficiency 57 %


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Model Title : Program 60-554 Unit System: US Description Value Unit Comment AUXILIARY DATA Worm














Axial Pressure Angle at Hob Ref Dia 25.5243 deg

Trans Pressure Angle at Hob Ref Dia 65.9658 deg

Trigger variable for Iterative Solver 1.0064

This design has more recess action than a “standard” design where the worm PD = the worm Eff PD if the worm is the driver. (The Pitch Line Offset Factor is greater than “0”.) In general more recess action will give longer life and better efficiency. Care must be taken with low pressure angles, coarse pitch, etc. that the worm gear is not undercut. If a problem is suspected see the reference book listed at the top of the TK Variable Sheet for further information.


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Example 2 For our next example we will start with a hob and design a worm gear drive to utilize our existing tooling. Figures 2-1A and 2-1B show our model with only the input data. Note that we have decided to make a full recess action gear set (with worm driving) by setting the Pitch Line Offset Factor equal to the Gear OD Factor. The data in the GEAR HOB DATA section would be obtained from the hob manufacturer’s hob specification drawing. Fig. 2-1A


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Fig. 2-1B

After solving we have Report 2-1. This time TK Solver used the Iterative Solver to obtain a solution. This was triggered by the trigger variable, gC, at the bottom of the report and the TK Variable Sheet. The Direct Solver could not evaluate the variable “gC” with the information we entered. When this occurs the variable is checked for a “First Guess” value and, if one is present, makes the variable a guess. The TK Iterative Solver will then converge on the solution.


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Report 2-1


m1 RATIO

m2 PA_LA

m3 GEAR_OD

m4 OK

m11 FACES

m12 GEOMETRY

m13 WORM_TT

m14 OK

m21 TOOL

m22 GEOMETRY

m23 CHECKED

m24

COMMON DATA

Gear Ratio 14.0000








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Pitch Line Offset Factor H 1.50000

WORM DATA



















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WORM MEASUREMENT










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Lead H 1.500 in


















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WORM SPEED EOHO rpm

Sliding Velocity at Pitch Diameter ft/min

Coefficient of Friction

Running Efficiency %

Starting Efficiency % AUXILIARY DATA Worm













Throat Radius from Gear Center 3.189 in


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Model Title : Program 60-554 Unit System: US Description Value Unit Comment AUXILIARY DATA Hob



Trigger variable for Iterative Solver 0.9554 Notice that, in this case, the hob reference diameter was about the same as the effective diameter. This will usually, but not always, be the case. The worm has been designed by the model so that the worm effective diameter is enough smaller than the hob effective diameter to give us about .003 inch clearance at the leaving side of the worm gear. To complete the design we will enter a .2800 inch diameter measuring pin and 3600 RPM worm speed and solve one more time. The final model is on Report 2-2. Report 2-2


m1 RATIO

m2 PA_LA

m3 GEAR_OD

m4 OK

m11 FACES

m12 GEOMETRY

m13 WORM_TT


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m14 OK

m21 TOOL

m22 GEOMETRY

m23 CHECKED

m24

COMMON DATA

Gear Ratio 14.0000









Pitch Line Offset Factor H 1.50000

WORM DATA







26
























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WORM MEASUREMENT















Lead H 1.500 in




28





















Starting Efficiency 61 % AUXILIARY DATA Worm


Thread Thickness at Eff Pitch Radius In 1.167 in


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Model Title : Program 60-554 Unit System: US Description Value Unit Comment Plane of Rotation














Trigger variable for Iterative Solver 0.9556 TK Solver used the Iterative Solver again because the trigger variable still could not be evaluated by Direct Solver. If we had changed the values marked “E” to input and blanked (or changed to output) the values marked “H” then Direct Solver would have been used. The model has a number of default values which are used if you do not put an input in the input column:

The OD of the worm will be set to the maximum allowable OD set by the hob OD and the clearance.


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The worm length will be set slightly larger than the recommended minimum worm length. The worm root diameter will be set to the maximum allowable root diameter set by the hob. The tip radius on the worm cutting tool and on the gear hob will be set to the maximum radius that will avoid interference. The measuring wire diameter will be set to the PD wire. The gear throat form radius will be set to the root radius of the hob. The actual gear face width will be set slightly smaller then the recommended face width. The clearance at the edge of the gear will be set to the smaller of .012 inch times the axial pitch and .003 inch. The coefficient of friction will be calculated to correspond to empirical data gathered from worm set tests.

The default values have been set for the suggested sequence of solving. If solving in a different manner is employed the default values may not always be supplied. UTS Model 60-554 was designed to help you design a good worm gear drive and give you the opportunity to consider alternative specifications quickly and accurately. The model, however, is not a substitute for careful analysis of the drive by you. Care must be taken to insure that everything has been considered before putting the design into the field. See UTS Gear Programs 60-542, 60-544, and 60-546 if you wish to use American Gear Manufacturers Standards or start with a design close to these standards. Also see UTS Gear Programs 60-548 and 60-552 if you are working with double enveloping wormgears.

Program 60-554—Cylindrical Worm Gear Analysis Introduction

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Transcript of Program 60-554—Cylindrical Worm Gear Analysis Introduction