MECHANICAL MEASUREMENTS AND METROLOGY

GUIDED BY:- D.K.PATEL

NAME ENROLLMENT NO

PATEL JAIMIN 150280119080

PARMAR ASHISH 150280119068

4TH SEMESTERB1 BATCH

Gear is an mechanical device used for transmission of power and motion

Gears can also be used to achieve variable speeds by using different drives

They transmit power by meshing with other gear

Manufactured precisely and accurately

There are mainly two types of profiles are used for manufacturing of the gear teeth. They are

1) Involute profile

2) Cycloidal profile

1) Involute Profile:

Involute gear design was designed Leonhard Euler it is defined as the locus of a point on a straight line which rolls around the cylinder without slipping

2) Cycloidal Profile:

It is defined as the curve traced by a point on the circumference of the circle which rolls without slipping on a fixed straight line

TERMINOLOGY OF

GEAR TEETH

1)PITCH CIRCLE:

It is an imaginary circle by which pure rolling action. Would give the same motion as the actual gear

2)PITCH CIRCLE DIAMETER:

It is the diameter of the circle which by pure rolling action would produce the same motion as the toothed gear.

3)PRESSURE ANGLE:

It is the angle between the common normal to two gear teeth at the point of contact and the common tangent at the pitch point

4)ADDENDUM:

It is the radial distance from the pitch circle to the tip of the tooth

5)DEDENDUM:

It is the radial distance from the pitch circle to the bottom of the tooth

Dedendum=Addendum + Clearance

6) CLEARANCE:

it is defined as the radial distance from tip of a tooth to the bottom of the mating tooth space

7) FACE OF TOOTH:

It is that part of the tooth surface which is above the pitch surface

8) FLANK OF THE TOOTH:

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

9) CIRCULAR PITCH:

it is the distance measured on the circumference of the pitch circle from a point on one tooth to the corresponding point on the adjacent tooth

10) DIAMETRICAL PITCH:

It is the ratio of number of teeth on the pitch circle to the diameter of the pitch circle

Pd=T/D

11) MODULE:

It is the ratio of pitch circle diameter in millimeters to the number of teeth

m=D/T

12) TOTAL DEPTH:

It is the radial distance between the addendum and dedendum circle

13) TOOTH THICKNESS:

it is the width of the tooth measured along the pitch circle from the intercept with one flank to its intercept with the other flank of the same tooth

14) FACE WIDTH:

It is the width of the gear tooth measured parallel to its axis

15) WORKING DEPTH:

it is the radial distance from addendum circle to the clearance circle

16) BACKLASH:

It is the difference between the tooth space and the mating tooth thickness

GEAR MEASUREMENT

Gears are the devices meant for power transmission.For proper working the gears should be measured and inspected in each steps as:

Raw materials.Machining the blanks. Heat treatment.The cutting and finishing operations.

The accuracy depends on:

The measuring equipment available. Errors in the tooth surface finish

GEAR INSPECTION

Inspecting the dimensions and the surface for getting a designed thread.

Two types Analytical: slow process which check all individual elements. Less

preferred

Functional: carrying out running test with master and deciding. Much preferred

MEASUREMENT OF TOOTH THICKNESS

Defined as the length of an arc, which is difficult to measure directly.

This is the most important measurement a gear should undergo.

The tooth thickness is generally measured at pitch circle and is therefore, the pitch line thickness of tooth.

Gear tooth thickness varies from the tip of the base circle of the tooth,

The instrument must be capable of measuring the tooth thickness at a specified position on the tooth.

Mounting the gear between the bench centers, placing a standard roller in each tooth space and measuring the deviation using a dial indicator

Using a projector in which case the teeth are brought against a stop and each image of the tooth on the screen should coincide with a line on the screen

Using a gear testing fixture fitted with a spring loaded slide and dial indicator, in which the spring exerts a constant pressure on the mating teeth and the movements of the dial indicator gives the measure of the eccentricity of the teeth

Tooth thickness is generally measured along the pitch circle and is therefore the pitch line thickness of the tooth

Following are various methods of measuring the gear tooth thickness

a) Measurement of tooth thickness by gear tooth Vernier caliper

b) Constant chord method

c) Base tangent method

d) Measurement by dimension over pins

MEASUREMENT BY GEAR TOOTH VERNIER CALLIPER

It’s very conveniently measured by a gear tooth vernier caliper.

The gear tooth vernier has two vernier scales and they are set for the width (w) of the tooth and the depth (d) from the top.

Each of which is adjusted independently by adjusting screw on graduated bars.

It measures the tooth thickness at the pitch line.

It can also measure chordal addendum, the distance from top to chord.

Only used to verify the theoretical measurements.

GEAR TOOTH VERNIER CALLIPER

In fig.

w=AB=2AD

<AOD=x= 360°4N , Where N is the no.of teeth

w=2AD=2*AO.sin x=2R.sin360

4𝑁

Module m=P.C.D

no.of teeth=2R/N

Therefore

R=N.m/2

w=2N.m

2sin

360

4𝑁=N.m.sin

90

𝑁

Also d=OC-OD

But OC=OE+addendum=R+m=(Nm/2)+m

And OD=Rcos 𝑥=𝑁𝑚

2cos

90

𝑁

Therefore

d=𝑁𝑚

2+ 𝑚 −

𝑁𝑚

2cos

90

𝑁=

𝑁𝑚

21 +

2

𝑁− cos

90

𝑁

• This method is simple and inexpensive.

• However it needs different setting for a variation in number of teeth for a given pitch.

• Accuracy is limited by the least count of instrument.

• The wear during use is concentrated on the two jaws.

• The calliper has to be calibrated at regular intervals to maintain the accuracy of measurement.

ADVANTAGES AND LIMITATIONS

CONSTANT CHORD METHOD

Defined as “The chord joining those points, on opposite faces of tooth, which make contact with the mating teeth when the centerline of the tooth lies on the line of the gear centers”.

Constant chord measured where the tooth flank touches the flank of the basic rack.

The teeth of the rack are straight and inclined to their Centre line at the pressure angle.

Tooth thickness of rack along pitch line is equal to the arc tooth thickness of gear round

its pitch circle.

Property used :

“The gear tooth and rack space are in contact in the symmetrical position at the

points of contact of the flanks, the chord is constant at this point irrespective of the

system of gear in mesh”.

In fig.

PD=PF=arc PF=1/4*circular pitch

=1

4∗

π∗PCD

N= 1

4 ∗ π ∗ m

AP is tangential to the base circle, <CAP=x

∴ in ∆APD, AC = PD cos x = π

4 m. cos2 x

C= constant chord =2AC= π 2 mcos2x …….................→ 1

Where x→pressure angle

For helical gear , constant chord= π

2 mncos2xn

Where mn=NORMAL MODULE xn=NORMAL PRESSURE ANGLE

Now PC=APsin x= π

4 m cos x sin x

∴ d = addendum − PC = m −π

4 m cos x sin x

=m 1 −π

4cos x sin x ……………………………………………..….→2

for helical gear , d = mn 1 −π

4cos xn sin xn

Also PC=πm

4sin x cos x =

πm

8sin 2x ……………………….....→3

ADVANTAGES

For all number of teeth value of constant chord is same.

It readily lends itself to a form of comparator which more sensitive than the gear tooth vernier

Base Tangent Method

The limitation of gear tooth Vernier calliper can be overcome by measuring the span of a convenient number of teeth with a reviewed calliper .

In this method the span of a convenient number of teeth is measured with the help of a David brown tangent comparator or a micro meter with flanged anvils .

The anvils in the comparator are first set of the base tangent length with the help of slip gauges .

The distance is adjusted by setting the fixed anvils at a desired place with the help of locking ring and setting tubes .

Then the slip gauges are replaced by the gear which is to be measured and again the reading are taken by the comparator as shown in figure .

Advantages It depends only on one Vernier reading unlike gear tooth Vernier calliper where we required too readings .

This measured values of base tangent length is less than compared with the calculated value , the difference between the calculated value and measured value gives the error in the tooth thickness .

The number of teeth over which the measurement is to be made for particular gear is selected from the gear Handbook .

Base tangent length=Arc AB +Arc BC

Number of base pitches = S * π m cosΦ

Where Φ is pressure angle.’. Base tangent length = Arc AB + S *π m cosΦ

The arc length AB can be calculated as follows:Arc AB = 2 * Arc AF

=2 X(Arc AD + Arc DF)

As Arc AD = involute function of Φ in radiansRbase= tan Φ – Φ

.’. (tan Φ - Φ) = Arc AD /Rbase

.’. Arc AD = Rbase X (tan Φ - Φ)

cos Φ = Rbase/Rpitch = Dbase/Dpitch.’. Rbase = Rpitch X cos Φ = Nm/2 cos Φ

Arc AD = Nm/2 cos Φ(tan Φ - Φ)

CUIET '13

Now ,as Arc CG = ¼ *circular pitch=1/4 X π m

Θ radians = Arc CG /R pitch = ¼ πm/Nm/2= ¼ πm X 2/Nm

Θ = π/2N radians

Also from figure Θ radians = Arc DF / R base Arc DF = R base X θ in radians

=Nm/2 X cos Φ X π/2N

Arc AB = 2 X[ (Arc AD +Arc DF )]=2 X [(Nm/2 cos Φ)(tan Φ - Φ)+ (Nm/2 cos Φ X π/2N)]

Arc AB = Nm cos Φ [(tan Φ - Φ)+ π/2N]

Base tangent length = Nm cos Φ{(tan Φ - Φ)+π/2N}+S X π m cos Φ

=N m cos Φ [(tan Φ - Φ)+ π/2N + Sπ/N]

By Dimension Over PinsA convenient method of checking tooth thickness and obtaining some indication of some accuracy of involute profile is to measure a gear over roller placed in opposite tooth spaces .

From figure OD = circular pitch /4 =π/4 .mAngle OBD = 90 , Angle BOD = Φ = pressure anglecos Φ = OB/OD , OB = OD cos Φ = π/4 m cos Φ

Diameter of roller = 2 X OB = π/2 m cos Φ

Gauging diameter over rollers M= P.C.D + diameter of roller M= m N + π/2mcos Φ = m[N+π/2 cos Φ]

If the gear has an odd number of teeth a radian measurement with the gear between centres be carried out using a comparator with the gear .

WORKING

Two carriages one fixed and other movable are mounted on the base

The movable carriage is spring loaded towards the fixed carriage

Two spindles are mounted in a parallel plane on each carriage and are made to suit the bore of the gear wheels

A dial gauge is made to rest against the movable carriage

The two gears in mesh are then rotated by hand and variations in the dial gauge readings are observed

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