5. measurement of gears

IIT Gandhinagar, Discipline of Mechanical Engineering ME 352 Mechanical Engineering Laboratory II HPM Jan/Feb 2014

Experiment 5 - Measurement of Gears

1. AIM

To measure various parameters of a spur gear

2. THEORY

Gears are used in a wide variety of machines (automobiles, machine tools, processing

machine for cement, sugar, etc., material handling equipment like cranes). The material and

the geometric features of the gears will influence the performance of those machines.

There are many types of gears. Spur gears are the simplest one very commonly used. Here

the teeth will be parallel to the axis of the gear. Both the driving and the driven gears are

mounted on the parallel shafts.

To measure its various features one has to be familiar with the gear tooth terminology. All

gears have generally involute profile for teeth. The generation of an involute curve is

shown in (Fig-1). (Fig-2) shows the terminology.

Figure 1


Figure 2

The various errors that occur during manufacture are

1. Profile error of the teeth

2. Error in tooth thickness

3. Error in circular pitch

4. Eccentricity between axis of rotation and pitch circle diameter etc.

For practical applications, an assessment of total composite error will be adequate (Fig-3a)

and (Fig-3b) show the composite error in one revolution of gear when meshed with the

master gear (Fig-3a) shows the error as displacement along pitch circle, whereas (Fig-3b)

shows the errors as the displacement along centre line.

Figure 3(a): Typical single flank error chart


Figure 3(b): Errors are shown as the displacement along center line

However, this cannot be separate out individual errors. It can be used as a quality test, but

not for analysis of the errors.

For analyzing, the individual errors are to be measured. The error on tooth can be

measured with a Vernier Caliper. Gear tooth Vernier Calipers (Fig-4), has a horizontal and

vertical scale so that one can set as reference and measure the variation in the other. Since

the tooth thickness varies throughout its height, it is measured at the pitch circle diameter.

Hence, the vertical scale can be set for addendum and horizontal scale can measure the

error in tooth thickness.


Figure 4: Gear tooth Vernier Caliper

For measuring the error along pitch circle, tangent micrometer is used (Fig-5b). The

opposing involute on one or more teeth will have the same dimension if it is measured

across root of one and tip of the other and vice versa (Fig-5a). Hence if the dimension

across 2-3 teeth is measured, the corresponding theoretical value can be evaluated and the

difference fives the error. By repeating this process along the entire pitch circle, it is

possible to measure circular pitch errors at the different parts of the gear.

Figure 5(a)


Figure 5(b): ‘David Brown’ Base Tangent Comparator

Where, S: No. of teeth covered during measurement

m: Module

N: No. of teeth

3. PROCEDURE

1. Measure the outer diameter of the gear and the number of teeth

2. Calculate the module and adjust it to standard value

3. Calculate all dimensions of the tooth, circular pitch and tooth clearance

4. Using gear tooth caliper, measure thickness of all teeth and tabulate

5. Using Flange Micrometer, measure circular pitch of the hear at different points of

the pitch circle, tabulate the results

(Please refer to Table-1 in the Appendix)


APPENDIX

Involute Curve

An involute curve is defined as the locus of a point on a straight line which rolls around a

circle without slipping. It could also be defined in another way as the locus on a piece of

string which is unwound from a stationary cylinder.

Thus it is obvious that in an involute curve the length of the generator (G1R1) will always be

equal to the arc length (GR1) of the base circle from the point of tangency to the origin of

involute at G. (Fig.15.1)

Similarly generator G2R2=arc GR2

It is also clear that the tangent to the involute at any point will be perpendicular to the

generator at that point. This condition fulfills the requirements of laws of gearing.

Further, will also be noticed that the shape of the involute curve is entirely dependent upon

the diameter of the base circle from which the involute is generated. The curvature of the

involute goes on decreasing as the base circle diameter goes on increasing and finally

involute become straight line when the circle diameter is infinity.

Terminology of Gear Tooth

A gear tooth is formed by portions of a pair of opposed involutes. Most of the terms used in

connection with gear teeth are explained in Fig. 15.2.


Base Circle: It is the circle from which involute form is generated. Only the base circle on a

gear is fixed and unalterable.

Pitch Circle: It is an imaginary circle most useful in calculations. It may be noted that an

infinite number of pitch circles can be chosen, each associated with its own pressure angle.

Pitch Circle Diameter (P.C.D.): It is the diameter of a circle which by pure rolling action

would produce the same motion as the toothed gear wheel. This is the most important

diameter in gears.

Module: It is defined as the length of the pitch circle diameter per tooth. Thus if P.C.D. of

gear be D and number of teeth N, then module (m) = D∕N. It is generally expressed in mm.

Diametral Pitch: It is expressed as the number of teeth per inch of the P.C.D.

Circular Pitch (C.P.): It is the arc distance measured around the pitch circle from the flank

of one tooth to a similar flank in the next tooth.

.’. C.P. =πD∕N=πm

Addendum: This is the radial distance from the pitch circle to the tip of the tooth. Its value

is equal to one module.

Clearance: This is the radial distance from the tip of a tooth to the bottom of a mating

tooth space when the teeth are symmetrically engaged. Its standard value is 0.157 m.

Dedendum: This is the radial distance from the pitch circle to the bottom of the tooth

space.

Dedendum=Addendum + Clearance

=m+0.157 m=l.157 m.

Blank Diameter: This is the diameter of the blank from which gear is a t. It is equal to

P.C.D. plus twice the addenda.


Blank diameter =P.C.D + 2m.

=mN+2m = m (N+2).

Tooth Thickness: This is the arc distance measured along the pitch circle from its

intercept with one flank to its intercept with t le other flank of the same tooth.

Normally tooth thickness=½ C.P. =πm∕2

But thickness is usually reduced by certain amount to allow for some amount of backlash

and also owing to addendum correction.

Face of Tooth: It is that part of the tooth surface which is above the pitch surface.

Flank of the Tooth: It is that part of the tooth surface which is lying below the pitch

surface.

Line of Action and Pressure Angle: The teeth of a pair of gears in mesh, contact each

other along the common tangent to their base circles as shown in Fig. 15.3. This path is

referred to as line of action. As this is the common generator to both the involutes, the load

or pressure between the gears is transmitted along this line. The angle between the line of

action and the common tangent to the pitch circles is therefore known as pressure angle ø.

The standard values of ø are 14½ 0 and 20°.

In Fig. 15.3

𝑂𝐴

𝑂𝑃= 𝑐𝑜𝑠𝜙 =

𝑅𝑏𝑅𝑝

=𝐷𝑏𝐷


.’. Dia. Of base circle 𝐷𝑏 = 𝑃.𝐶.𝐷.𝑋 𝑐𝑜𝑠𝜙

Base Pitch: It is the distance measured around the base circle from the origin of the

involute on the tooth to the origin of a similar involute on the next tooth.

Base Pitch=Base Circumference/ No. of teeth= π × Dia. of base circle/N

=π × D cosø /N= πmcosø.

Involute Function: It is found from the fundamental principle of the involute, which it is

the locus of the end of a thread (imaginary) unwound from the base circle.

Mathematically its value is Involute function δ=tan ø—ø, where ø is the pressure angle.

The relationship between the involute function and the pressure angle can be derived as

follows:

In Fig. 15.4,

OA=base circle radius=Rb

OP=Pitch circle radius=Rp, and

BP=involute profile of gear tooth.

AP is tangent to base circle at A,

AOC=ø=Pressure angle

Now OA=OP cosø, or Rb=Rp cosø


COB=Involute function of ø.

By definition of involute, length=AP=arc Ab

and tan ø=AP/OA=AP/Rb=arc AB/Rb, Also ø+δ=arc AB/Rb

.’. ø+δ=tan ø or δ=tanø−ø.

Helix Angle: It is the acute angle between the tangent to the helix and axis of the cylinder

on which teeth are cut.

Lead Angle: It is the acute angle between the tangent to the helix and plane perpendicular

to the axis of cylinder (Refer Fig. 15.5).

Back Lash: It is the distance through which a gear can be rotated to bring its non-working

flank in contact with the teeth of mating gear (Ref. Fig 15.6).


Table 1: Basic tooth Proportions for Involute Spur Gears

Pressure Angles

20° 14½° Addendum

Dedendum

Teeth Depth

Circular teeth thickness

Fillet radius

Clearance

m

1.25 m

2.25 m

πm/2

0.3 m

0.25 m

m

1.157 m

2.157 m

πm/2

0.157 m

0.157m

Table 2: Some Important Relationships between Various Elements of Gears

To find Having Formula (a) Spur Gears Module (m) Module Outside diameter (Do) Base circular diameter (Db)

No. of teeth (N) and pitch diameter (D) Circular pitch (p) Pitch diameter and Module Pitch diameter and pressure angle

𝑚 = 𝐷 𝑁 𝑚 = 𝑝 𝜋 𝐷𝑜 = 𝐷 + 2𝑚 𝐷𝑏 = 𝐷𝑐𝑜𝑠𝜙

5. measurement of gears

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