program 60-1441—pin and span measurement: external ... · pdf fileand the gear is...

Program 60-1441—Pin and Span Measurement: External Involute Gears

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Introduction This model calculates either the measurement over pins/balls or the length over a block of gear teeth from the tooth thickness. Measurement over pins or balls is used extensively for gears and splines and is a very accurate method for control of tooth thickness. By using different size pins (or balls) a rough check can also be made of the involute profile. If the number of teeth in the gear is even the pins will be opposite each other and the micrometer will lie in the plane of rotation of the gear (transverse plane) regardless of the helix angle. The measuring contact points will be on a centerline of the gear. If the number of teeth is odd and the gear is a spur gear the micrometer will lie in the plane of rotation but the contact points will not be on a centerline of the gear. If the number of teeth is odd and the gear is helical the micrometer will NOT lie in the plane of rotation and the contact points will not be on a centerline of the gear. The tipping of the micrometer out of the transverse plane makes the manual calculation of the size over pins difficult as the equations can not be solved directly and a loop procedure is necessary. For cylindrical involute gears, spur or helical, the measuring pins or balls will have their centers on the ball helix of the gear. The ball helix is the location of the center of a ball that is free to roll in the tooth space of the gear. The problem of finding the measurement over pins for a given tooth thickness and pin diameter then becomes a problem of finding the distance between the ball helixes. This is quite straight-forward for all gears except odd tooth helicals. In this case it is necessary to find the minimum distance between the ball helixes by some method where the first derivative of the distance can be forced to zero. This model contains such a procedure. The model also calculates the diameter of a pin (or ball) that will contact the tooth flanks at the diameter where the tooth thickness is equal to the space width. The closest “standard” pin is then found and given on the Power User form or TK Solver Variable Sheet. The model may be used when the tooth thickness is known and the measurement over pins or balls is needed or when the measurement is known and the tooth thickness is needed. In the first case we enter the tooth thickness and solve for the measurement. In the second case we enter the pin measurement and solve for the tooth thickness.

UTS Integrated Gear Software

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Measurement over a block of gear teeth (span size) to determine the tooth thickness is also widely used, especially for larger gears where measurement over pins is not practical due to the large distance to be measured. Another advantage in using span size instead of measuring over pins is the ease of handling only the micrometer or caliper. In addition, the right size pins may not be available or the gear may still be mounted on a machine and access to both sides may be difficult. The span measurement can be made with a “disc micrometer” or with a caliper. If the pitch is finer than about 12 or 13 it will be difficult to make the measurement with a micrometer or caliper. When the tooth space is not large enough to provide access, the measurement can be made on a shadowgraph at an enlarged scale in the case of spur gears. For high production work a simple go/no go gage can be used with a large saving of inspection time. When making span measurements using disc micrometers or calipers, the precise location on the tooth profile where the measuring device makes contact is not important. This is due to the fact that the distance being measured is the length of a base tangent line which remains constant no matter where the teeth are contacted. The “feel” will be much like measuring a cylindrical shaft. The model may be used when the tooth thickness is known and the span size is needed. In this case we enter the tooth thickness and solve for the span size. If the tooth thickness is not known we enter the number of teeth spanned and the measurement.

60-1441—Pin and Span Measurement: External Involute Gears

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Examples Example 1 This example was selected to demonstrate the procedure to be followed for a solution for the size over pins given the tooth thickness and also to solve for the tooth thickness given the measured size over pins. The example is an odd tooth helical that does not have a “standard” pitch or tooth thickness. First we will find the measurement over pins when we know the normal tooth thickness at the reference pitch diameter. (The reference pitch diameter is the diameter found by dividing the number of teeth by the transverse pitch.) We have the following data available:

Number of teeth Normal Diametral Pitch Normal Pressure Angle Helix Angle Face Width # Outside Diameter * Root Diameter * True Involute Form Diameter * Normal Tooth Thickness # The face width is only necessary to enable the

model to check the possibility of using 3 pins or using span measuring. The model will solve the problem without the face width.

* These values are not necessary to obtain a size

over pins but are required for the model to check the contact diameter of the pins (or the mean caliper contact diameter for span measuring and to create a plot of the gears).


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Open a new analysis and enter the known data. The data entry sequence and their numeric inputs are as follows: Number of teeth—13 Normal diametral pitch—12.5 Normal pressure angle—20 Helix angle—30 Face width—.75 Outside diameter—1.51 Root diameter—1.12 True involute form diameter—1.185 At this point the application will ask you to choose Tooth Thickness, Measurement Over Pins or Balls, or Span Size. For this example, choose Tooth Thickness and enter .16 for actual diameter. A dialog box will ask whether you want a plot; click Yes. The completed inputs are shown in Figure 1-1. The inputs and outputs of the solved model should look like Report 1-1. Fig. 1-1


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Report 1-1

Model Title : Program 60-1441 Unit System: US

Plot Ball / Teeth? y

Number of teeth 13

Normal diametral pitch 12.500000 1/in `

Normal module 2.032000 mm `

Normal pressure angle 20.000000 deg

Nominal helix angle at Ref pitch dia 30.000000 deg

Face width 0.7500 in

Normal tooth thickness at Ref PD 0.1760 in

Transverse tooth thickness at Ref PD 0.2032 in

Pointed tooth diameter 1.5290 in

Outside diameter 1.5100 in

True involute form dia TIF (Opt) 1.1850 in

Root diameter 1.1200 in

Ref pitch diameter 1.2009 in

Base diameter 1.1071 in

Transverse diametral pitch 10.8250 1/in `

Transverse module 2.3460 mm `

Transverse pressure angle 22.7959 deg

Normal circular pitch at ref PD 0.2513 in

Transverse circular pitch at ref PD 0.2902 in

Normal base pitch 0.2362 in


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Model Title : Program 60-1441 Unit System: US Transverse base pitch 0.2680 in

Axial pitch 0.5030 in

Lead 6.5345 in

Base helix angle 28.0240 deg

PIN OR BALL MEASUREMENT

Pin dia for contact at tooth = space 0.15915 in

Closest standard inch series pin 0.16000 in `

Closest standard metric series pin 4.00000 mm `

Actual pin (or ball) diameter 0.16000 in

Radius over one pin 0.7785 in

Measurement over 2 balls in same plane 1.5469 in

Measurement over 2 pins 1.5389 in


Minimum face width for use of 3 pins 0.5278 in

Diameter where tooth = space 1.31467 in

Radius to center of pin 0.69852 in

Pressure angle at center of pin 0.65598 rad

Involute of PA at center of pin 0.11370 rad

Ball path helix angle 33.8875 deg

Pressure angle at pin contact diameter 0.57080 rad

Actual pin contact diameter 1.3157 in

SPAN MEASUREMENT

Min recommended teeth spanned 3.0000

Recommend number of teeth spanned 3.0000

Max recommended teeth spanned 5.0000


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Notice that the tooth=space pin is output along with the closest “standard” pin—in this case .16000 inches.

Tables of “standard” inch series pins and metric pins are available in the TK Solver model. Pick the table from the Tables List on the Object Bar. Table 1


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Table 2

The model has solved for the radius over one pin, the measurement over two balls , the measurement over two pins and the measurement over three pins. If two balls are used they MUST be constrained to the transverse plane. If three pins are used the two pins on one side should be placed in adjacent axial spaces. The required face width to use three pins is also given. (If the face width of the gear is close to the minimum the use of 3 pins may be difficult.) A plot of the ball in contact with the teeth is available. Figure 1-2 is the plot. The section in the plot is through the plane of rotation (transverse plane) where the ball is in contact with one of the teeth. The dashed circle is the circle on the ball through the contact point. The solid circle is in front of the section and is the actual diameter of the ball. It can be seen that the ball extends above the outside diameter, is well above the root diameter and contacts the teeth above the TIF diameter. For external gears a pin may be

used instead of a ball and the location of the pin will be the same as the location of the ball.


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Fig. 1-2

This gear is a low tooth number, odd tooth helical with a relatively high helix angle. This gear was chosen to demonstrate the difference between the size over two balls and the size over two pins. The size over two balls is the size obtained using the “Cosine(90 deg/number of teeth)” correction used for odd tooth spur gears. Unfortunately, it has been the practice by many in the gear industry to use this method for odd tooth helicals under the assumption that it is “good enough". This was done primarily because of the difficulty of the calculation without the computer and because it is time consuming to move the gear and mount it where a reading can be taken from the center of the arbor over one pin.

For odd tooth helicals of low helix angle and/or large numbers of teeth the difference between 2 ball and 2 pin measurements is not great. However, as the number of teeth comes down and/or the helix angle goes up the measurements using the spur gear correction factor soon become useless for any purpose. It may be of interest to you to try a few gears using this model to get a feel for the problem.

The next example illustrates how to use the model if you have measured a gear and wish to find the tooth thickness. In this case it is only necessary to enter the measurement over one pin, two balls, two pins or three pins along with the pin (or ball) size used and solve the model. We have left the numerical data the same so that you can see that we get the same tooth thickness and measured size any direction we choose to go.


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The inputs and outputs are shown in Figure 1-3 and Report 1-2.

Fig. 1-3

Report 1-2


Plot Ball / Teeth? n

Number of teeth 13










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Model Title : Program 60-1441 Unit System: US Outside diameter 1.5100 in











Transverse base pitch 0.2680 in


Lead 6.5345 in












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Model Title : Program 60-1441 Unit System: US Minimum face width for use of 3 pins 0.5278 in







Actual pin contact diameter 1.3157 in Now we will look at some examples using span size instead of pins. When making span measurements using disc micrometers or calipers, the precise location on the (involute) tooth profile where the measuring device makes contact is not important. This is due to the fact that the distance being measured is the length of a base tangent line which remains constant no matter where the teeth are contacted. The “feel” is much like measuring a cylindrical shaft. The model may be used when the tooth thickness is known and the span size is needed. In this case we enter the tooth thickness and solve for the span size. The same data is required for span size from tooth thickness as is required for pin size from tooth thickness.


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Example 2 Figure 2-1 shows the inputs and Report 2-1 is the solved model for the length over 4 teeth for the gear data shown. Fig. 2-1

Report 2-1



Number of teeth 23







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Model Title : Program 60-1441 Unit System: US Normal tooth thickness at Ref PD 0.1420 in



Outside diameter 2.2500 in













Lead 15.4106 in










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Model Title : Program 60-1441 Unit System: US Measurement over 2 pins 2.3024 in


Minimum face width for use of 3 pins 0.7035 in







Actual pin contact diameter 2.1082 in

SPAN MEASUREMENT




Actual number of teeth spanned 4

Span size 0.9054 in

Mean measuring contact diameter 2.1116 in

Minimum face for span measurement 0.35 in

The minimum and maximum number of teeth to span are calculated along with a recommended number of teeth. We may wish to find the tooth thickness after we have measured a gear. Then we just enter the size measured along with the number of teeth spanned.


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Fig. 2-2

Report 2-2



Number of teeth 23










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Model Title : Program 60-1441 Unit System: US Outside diameter 2.2500 in













Lead 15.4106 in






Actual pin (or ball) diameter ENTER PIN in

Radius over one pin in

Measurement over 2 balls in same plane in

Measurement over 2 pins in



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Model Title : Program 60-1441 Unit System: US Minimum face width for use of 3 pins in


Radius to center of pin in

Pressure angle at center of pin rad

Involute of PA at center of pin rad

Ball path helix angle deg

Pressure angle at pin contact diameter rad

Actual pin contact diameter in

SPAN MEASUREMENT





Span size 0.9054 in


Minimum face for span measurement 0.35 in


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Example 3 It is sometimes necessary to duplicate a gear which is no longer serviceable. If no specifications are available for the gear it is then necessary to determine what cutter might be used to duplicate the working flanks of the teeth. This model can be used to find a suitable diametral pitch and pressure angle to duplicate the gear. Once a cutter has been selected other UTS gear models can be used to check for usable gear outside diameter, root diameter, TIF diameter, etc. As an example, suppose we have a spur gear, or a segment of a gear, which we want to duplicate. Trial and error show that we can measure across 5 teeth without contacting the OD and 4 teeth without contacting the root. The difference between the 4 and 5 tooth readings will be the normal base pitch of the gear. (The normal base pitch is the distance between same handed flanks on the generating rack normal to the flanks. It is also the distance from one tooth flank on the gear to the next same handed flank along a line tangent to the base circle.) Better accuracy will be obtained if a number of readings are taken at 4 and at 5 teeth and then averaged. After taking a number of readings we find that the average measurement over 4 teeth is 1.8280 inches and over 5 teeth is 2.3118 inches. The difference is 0.4838 inch and this should be the normal base pitch of the gear. (CAUTION: If the readings vary by more than 0.003 inch or so, the gear may be too badly damaged to use this procedure.) The first step is to enter the known data—the number of teeth, the helix angle and the normal base pitch—in the Power User form, as shown in Figure 3-1. Any combination of nominal diametral pitch and pressure angle which produces the correct normal base pitch may be used. We will guess at a nominal pressure angle of 20 degrees. Enter 20 degrees and solve. The solution is shown in Report 3-1


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Fig. 3-1

Sheet 3-1



Number of teeth 25





Face width in

Normal tooth thickness at Ref PD in

Transverse tooth thickness at Ref PD in

Pointed tooth diameter in


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Model Title : Program 60-1441 Unit System: US Outside diameter in

True involute form dia TIF (Opt) --- in

Root diameter in










Axial pitch --- in

Lead --- in

Base helix angle 0.0000 deg This results in a nominal diametral pitch of about 6.102. (A cutter of this pitch and pressure angle would indeed duplicate the gear if the cutter had the right tooth thickness but the cutter would have to be made special.) Let’s try again with a guess of 22.5 degrees for the normal pressure angle. Report 3-2 is the solved model.


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Report 3-2



Number of teeth 25





Face width in

Normal tooth thickness at Ref PD in

Transverse tooth thickness at Ref PD in

Pointed tooth diameter in

Outside diameter in


Root diameter in









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Model Title : Program 60-1441 Unit System: US Normal base pitch 0.4838 in


Axial pitch --- in

Lead --- in

Base helix angle 0.0000 deg This time the diametral pitch is very close to 6 and the gear was probably made with a 6 pitch, 22.5 degree pressure angle cutter. Next we will determine the tooth thickness of the gear. (If the gear is badly worn adjustment may have to be made to the calculated tooth thickness.) Change the normal diametral pitch to 6, enter one of the measured span sizes, blank the base pitch and solve. Figure 3-2 shows the completed Power User form, Report 3-3 the solved model.


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Fig. 3-2


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Report 3-3



Number of teeth 25





Face width in




Outside diameter in


Root diameter in










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Model Title : Program 60-1441 Unit System: US Transverse base pitch 0.4840 in

Axial pitch --- in

Lead --- in






Actual pin (or ball) diameter ENTER PIN in

Radius over one pin in

Measurement over 2 balls in same plane in



Minimum face width for use of 3 pins in


Radius to center of pin in

Pressure angle at center of pin rad

Involute of PA at center of pin rad

Ball path helix angle deg

Pressure angle at pin contact diameter rad

Actual pin contact diameter in


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Model Title : Program 60-1441 Unit System: US SPAN MEASUREMENT





Span size 1.8280 in


Minimum face for span measurement 0.00 in The normal tooth thickness is 0.3182 inch. If a 6 pitch, 22.5 degree pressure angle cutter is available the remaining step is to check the root diameter and TIF diameter produced by the cutter. If a cutter must be designed then, of course, the TIF and root diameter can be specified. If we also wanted the size over pins we can easily get it at this point. Change the tooth thickness to an input, enter the pin diameter, blank the span size and solve. Report 3-4



Number of teeth 25 Normal diametral pitch 6.000000 1/in ` Normal module 4.233000 mm `


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Model Title : Program 60-1441 Unit System: US Normal pressure angle 22.500000 deg Nominal helix angle at Ref pitch dia 0.000000 deg Face width in Normal tooth thickness at Ref PD 0.3182 in Transverse tooth thickness at Ref PD 0.3182 in Pointed tooth diameter 4.7560 in Outside diameter in True involute form dia TIF (Opt) --- in Root diameter in Ref pitch diameter 4.1667 in Base diameter 3.8495 in Transverse diametral pitch 6.0000 1/in ` Transverse module 4.2330 mm ` Transverse pressure angle 22.5000 deg Normal circular pitch at ref PD 0.5236 in Transverse circular pitch at ref PD 0.5236 in Normal base pitch 0.4837 in Transverse base pitch 0.4840 in Axial pitch --- in Lead --- in Base helix angle 0.0000 deg

PIN OR BALL MEASUREMENT Pin dia for contact at tooth = space 0.31080 in Closest standard inch series pin 0.32000 in ` Closest standard metric series pin 8.00000 mm ` Actual pin (or ball) diameter 0.32000 in Radius over one pin 2.3886 in Measurement over 2 balls in same plane 4.7685 in


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Model Title : Program 60-1441 Unit System: US Measurement over 2 pins 4.7685 in Measurement over 3 pins in Minimum face width for use of 3 pins in Diameter where tooth = space 4.29261 in Radius to center of pin 2.22862 in Pressure angle at center of pin 0.52833 rad Involute of PA at center of pin 0.05535 rad Ball path helix angle 0.0000 deg Pressure angle at pin contact diameter 0.46409 rad Actual pin contact diameter 4.3048 in

SPAN MEASUREMENT Min recommended teeth spanned 3.0000 Recommend number of teeth spanned 4.0000 Max recommended teeth spanned 5.0000 Actual number of teeth spanned 4 Span size 1.8280 in


Minimum face for span measurement 0.00 in NOTE: The same procedure can be used with helical gears but the axial pitch or lead must be known (or measured).

program 60-1441—pin and span measurement: external ... · pdf fileand the gear is...

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