1.1 Classification | unit 1 spur gear

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TOM2

Unit -1

Spur Gear

1.1 Classification

The gears or toothed wheels may be classified as follows:

According to the position of axes of the shafts.

Parallel

The two parallel and coplanar shafts connected by the gears is shown in Fig.

These gears are called spur gears and the arrangement is known as spur gearing.

These gears have teeth parallel to the axis of the wheel as shown in figure.

Another name given to the sperm gearing is helical gearing, in which the teeth are inclined to the axis. the single and double helical gears connecting parallel shafts as shown in figure (a) and (b) respectively.

b. Intersecting

The two non-parallel or intersecting, but coplanar shafts connected by gears is shown in figure c.

These gears are called bevel gears and the arrangement is known as bevel gearing.

c. Non intersecting and non-parallel

The two non-intersecting and nonparallel shaft connected by gears is shown in figure d these gears are called skew bevel gears or spiral gears and the arrangement is known as skew bevel gearing or spiral gearing.

2. According to the type of gearing.

External gearing

In external gearing, the gears of the two shafts mesh externally with each other as shown in figure a.

The larger of these two wheels is called spur wheel and the smaller wheel is called the pinion.

b. Internal gearing

In internal gearing, the gears of the two shafts mesh internally with each other as shown in figure b.

The larger of these two wheels is called annular wheel and the smaller wheel is called the pinion.

c. Rack and pinion

Sometimes, the gear of a shaft meshes externally and internally with the gears in a straight line, as shown in the figure such type of gear is called rack and pinion.

The straight-line gear is called rack and the circular wheel is called the pinion.

3. According to the position of teeth on the gear surface

Straight

Inclined

Curved.

1.2 Spur gear: definition

When the teeth of the gear are parallel to the axis of the wheel the gears are called spur gears and the arrangement is known as spur gearing.

They consist of a cylinder or disk with teeth projecting radially.

Spur gears or straight-cut gears are the simplest types of gear.

1.3 Terminology

Pitch Circle:- It is an imaginary circle which by the pure rolling action, would give the same motion as the actual gear.

Pitch circle diameter:-It is the diameter of the pitch Circle. The size of the gear is usually specified by the pitch circle diameter. It is also known as pitch diameter.

Pitch point:- It is a common point of contact between two pitch circles.

Pitch surface:- It is the surface of the rolling discs which the machine gears have replaced at the Pitch Circle.

Pressure angle or angle of obliquity. 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. It is usually denoted by

. The standard pressure angles are 14.5° and 20°

Addendum:- It is the radial distance of a tooth from the pitch circle to the top of the tooth.

Dedendum:- It is the radial distance of a tooth from the pitch circle to the bottom of the tooth.

Addendum circle:- It is the circle drawn through the top of the teeth and is concentric with the pitch Circle.

Dedendum circle:- It is the circle drawn through the bottom of the teeth. It is also called the root circle.

Circular pitch:- It is the distance measured on the circumference of the pitch circle from a point of one tooth to the corresponding point on the next tooth. It is usually denoted by

Mathematically,

Circular pitch.

D=diameter of the pitch Circle, and

T= number of teeth on the wheel

Diametral pitch:- It is the ratio of the number of teeth to the pitch circle diameter in millimetres.

It is denoted by

Mathematically,

Diametral pitch

T=number of teeth

D=pitch circle diameter

Module:- It is the ratio of the pitch circle diameter in millimetres to the number of teeth. It is usually denoted by m. Mathematically,

Module m=D/T

Clearance:- It is the radial distance from the top of the tooth to the bottom of the tooth, in a meshing gear. A circle passing through the top of the machine gear is known as clearance circle.

Total depth:- It is the radial distance between the addendum and the dedendum circles of a gear. It is equal to the sum of the addendum and dedendum.

Working depth:- It is the radial distance from the addendum circle to the clearance circle. It is equal to the sum of the addendum of the two meshing gears.

Tooth thickness:- It is the width of the tooth measured along the pitch Circle.

Tooth space:- It is the width of the space between the two adjacent teeth measured along with the pitch.

Backlash:- It is the difference between the tooth space and tooth thickness, as measured along the pitch Circle. Theoretically, the backlash should be zero, but in actual practice, some backlash must be allowed to prevent jamming of the teeth due to tooth errors and thermal expansion.

The face of tooth:- It is the surface of the gear tooth above the pitch surface.

The flank of tooth:- It is the surface of the gear tooth below the pitch surface.

Top land:- It is the surface of the top of the tooth.

Face width:- It is the width of the gear tooth measured parallel to its axis.

Profile:- It is the curve formed by the face and flank of the tooth.

Fillet radius:- It is the radius that connects the root Circle to the profile of the tooth.

Path of contact:- It is the path traced by the point of contact of two teeth from the beginning to the end of the engagement.

Length of the path of contact:- It is the length of the common normal cut off by the addendum circles of the wheel and pinion.

Arc of contact:- It is the path traced by a point on the pitch circle from the beginning to the end of engagement of a given pair of teeth. The arc of contact consists of two parts i.

Arc of approach:- It is the portion of the path of contact from the beginning of the engagement to the pitch point.

Arc of recess:- It is the portion of the path of contact from the pitch point to the end of the engagement of a pair of teeth.

1.4 Fundamental law of toothed gearing

Consider the portions of the two teeth, one on the wheel 1 and the other on the wheel 2 as shown by the thick line curves in the figure.

Let the two teeth come in contact at point Q, and the wheels rotate in the directions as shown in the figure.

Let T T be the common tangent and MN be the normal curves at the point of contact Q.

From the centres

draw

perpendicular to MN.

A little consideration will show that the point Q moves in the direction QC when considered as a point on wheel 1 and in the direction QD when considered as a point on wheel 2.

Let

be the velocities of the point Q on the wheels 1 and 2 respectively.

If the teeth are to remain in contact, then the components of these velocities along the common normal MN must be equal.

………. I

Also, from similar triangles

……….. II

Combining equations, I and II we have

From above, we see that the angular velocity ratio is inversely proportional to the ratio of the distances of the point P from the centres

or the common normal to the two surfaces at the point of contact Q intersects the line of centres at point P which divides the centre distance inversely as the ratio of angular velocities.

Therefore, to have a constant angular velocity ratio for all positions of the wheels, the point P must be the fixed point for the two wheels.

In other words, the common normal at the point of contact between a pair of teeth must always pass through the pitch point.

This is the fundamental condition which must be satisfied while designing the profiles for the teeth of gear wheels.

It is also known as the law of gearing.

1.5 Involute and cycloidal profile

Involute teeth:-

An involute of a circle is a plane curve generated by a point on a tangent, which rolls on the circle without sleeping.

In connection with toothed wheels, the circle is known as a base circle.

The involute is traced as follows:

Let A be the starting point of the involute.

The base circle is divided into an equal number of parts e.g.

etc.

The tangents at

etc. are drawn and the length

equal to the arcs

are set off.

Joining points A

etc. we obtain the involute curve AR.

A little consideration will show that any instant

the tangent

to the involute is perpendicular to

is normal to the involute.

In other words, normal at any point of an involute is a tangent to the circle.

Now, let

with the fixed centres of the two base circles as shown in the figure.

Let the corresponding involutes AB and

be in contact at point Q.

MQ and NQ are normal to the involutes at Q and are tangents to base circles.

Since the normal of an involute at a given point is the tangent drawn from that point to the base circles, therefore the common normal MN at Q is also the common tangent to the base circles.

The common normal MN intersects the line of centres

at the fixed-point P.

Therefore, the involute teeth satisfy the fundamental condition of a constant velocity ratio.

From similar triangles

Which determines the ratio of the radii of the two base circles. The radii of the base circles are given by

Also, the centre distance between the base circles,

Where

is the pressure angle or the angle of obliquity.

When the power is being transmitted, the maximum tooth pressure is exerted along the common normal through the pitch point.

This force may be resolved into tangential and radial or normal components.

These components act along and at right angles to the common tangent to the pitch circles.

If F is the maximum tooth pressure as shown in figure then

Tangential force

Radial or normal force

The torque exerted on the gear shaft

= where r is the pitch circle radius of the gear.

Cycloidal teeth

Cycloidal is the curve traced by a point on the circumference of a circle which rolls without slipping on a fixed straight line.

When a circle rolls without slipping on the outside of a fixed circle, the curve traced by a point on the circumference of a circle is known as epicycloid.

On the other hand, if a circle rolls without slipping on the inside of a fixed circle, then the curve traced by a point on the circumference of a circle is called hypocycloid.

In the figure, the fixed-line or pitch line of a rack is shown.

When the circle C rolls without slipping above the pitch line in the direction as indicated in figure then the point P on the circle traces epicycloid PA.

This represents the face of the cycloidal tooth profile.

When the circle Drolls without slipping below the pitch line, then the point P on the circle D traces hypocycloid PB, which represents the flank of the cycloidal tooth.

The profile BPA is one side of the cycloidal rack tooth.

Similarly, the two curves P'A' and P'B' forming the opposite side of the tooth profile are traced by the point P' when the circles C and D roll in the opposite directions.

The cycloidal teeth of gear may be constructed as shown in the figure.

The circle C is rolled without slipping on the outside of the pitch Circle and the point P on the circle C traces epicycloid PA, which represents the face of the cycloidal tooth.

The circle D is rolled on the inside of pitch Circle and the point P on the circle D traces hypocycloid PB, which represents the flank of the tooth profile.

The profile BPA is one side of the cycloidal tooth.

The opposite side of the tooth is traced as explained above.

The construction of the two matting cycloidal teeth is shown in the figure.

A point on the circle D will trace the flank of the tooth

when circle D rolls without slipping on the inside of pitch Circle of wheel 1 and face of tooth

when the circle D rolls without slipping on the outside of pitch Circle of wheel 2.

Similarly, a point on the circle C will trace the face of tooth

and flank of tooth

The rolling circle C and D may have unequal diameters, but if several wheels are to be interchangeable they must have rolling circles of equal diameters.

A little consideration will show, that the common normal X X at the point of contact between two cycloidal teeth always passes through the pitch point, which is the fundamental condition for a constant velocity ratio.

Sr No.	Cycloidal teeth	Involute teeth
1	Pressure angle varies from maximum at the beginning of engagement, reduces to zero at the pitch and again increases to maximum at the end of engagement resulting in less smooth running of the gears.	Pressure angle is constant throughout the engagement of teeth. This results in smooth running of the gears.
2	It involves double curve for the teeth, epicycloid and hypocycloid. This complicates the manufacture.	It involves single curve for the teeth resulting simplicity of manufacturing and tools.
3	Owing to difficulty of manufacture these are costlier	These are simple to manufacture and thus cheaper.
4	Exact centre distance is required to transmit a constant velocity ratio	A little variation in the centre distance does not affect the velocity ratio
5	Phenomenon of interference does not occur at all	Interference can occur if the condition of minimum number of teeth on a gear is not followed
6	The teeth have spreading flanks and thus are stronger	The teeth have radial flanks and thus are weaker as compared to the cycloidal form for the same pitch.
7	In this, convex flank always has contact with a concave face resulting in less wear	Two convex surfaces are in contact and do there is more wear.

1.6 Path of contact

Consider a pinion driving the wheel as shown in figure.

When the pinion rotates in clockwise direction, the contact between a pair of involute teeth begins at K and ends at L.

MN is the common normal at the point of contact and the common tangent to the base circles.

The point K is the intersection of the addendum circle of wheel and the common tangent.

The point L is the intersection of the addendum circle of pinion and common tangent.

We know that the length of path of contact is the length of common normal cut off by the addendum circles of the wheel and the pinion.

Thus, the length of path of contact is KL which is the sum of the parts of the path of contacts KP and PL.

The part of the path of contact KP is known as path of approach and the part of the path of contact PL is known as path of recess.

Let. radius of addendum circle of pinion.

radius of addendum circle of wheel.

= radius of pitch Circle of pinion, and

radius of pitch Circle of wheel.

From figure, we find that radius of the base circle of pinion

And radius of the base circle of wheel,

Now from right angle triangle

Length of the part of the path of contact, or the path of approach,

Similarly, from right-angled triangle

And

Length of the part of the path of contact, or path of recess

Length of the path of contact,

1.7 Arc of contact

The arc of contact is the path traced by a point on the pitch circle from the beginning to the end of engagement of a given pair of teeth.

In figure arc of contact is EPH or GPH.

Considering the arc of contact GPH, it is divided into two parts arc GP and arc PH.

The arc GP is known as arc of approach and the arc PH is called arc of recess.

The angles subtended by these arcs at

are called angle of approach and angle of recess respectively.

We know that the angle of the arc of approach

And the length of the arc of recess

Since the length of the arc of contact GPH is equal to the sum of the length of the arc of approach and arc of recess, therefore,

Length of the arc of contact

1.8 Conjugate action

The gears must be designed such that the ratio of rotational speeds of driver and driver gear is always constant.

When the tooth profiles of two meshing gear produce a constant angular velocity during meshing, they are said to be executed and conjugate action point that is

Where =Angular velocity of the driver.

=Angular velocity of the driven

Gears are mostly designed to produce conjugate action.

Theoretically, it is possible to select an arbitrary profile for one tooth and then to find a profile for the meshing tooth, which will give conjugate action.

One of these solutions is involute profile.

The involute profile is universally used for constructing gear teeth with few expectations.

Figure illustrates a conjugate action.

There are two arms A and B with curved surfaces.

When A pushes against B the point of contact between them occurs at C, where the two contacting surfaces are tangent to each other.

The forces at any instant are directed along the common normal ‘ab’ to the two curved surfaces.

The line ab is called the line of action.

The line of action intersects the line of centres of the two curved surfaces at a point P, referred as pitch point.

The angular velocity ratio between the two arms is equal to the ratio of their radii to the point P.

Circles drawn through the pitch point P with centres at

are called pitch circles.

The radii

are pitch radii.

To transmit constant angular velocity to B, for a given angular velocity of arms A, the pitch point P, must remain fixed.

It means that all the line of action must pass through the same point P at all instants.

In the case of involute profile, all points of contact occur on the same straight-line ab.

That is normal to the contacting surfaces.

Thus, these profiles transmit constant angular motion.

Pitch line velocity

1.9 Contact ratio

The contact ratio of the number of pairs of teeth in contact is defined as the ratio of the length of the arc of contact to the circular pitch.

Mathematically,

Contact ratio of number of pairs of teeth in contact

Where.

m = module

1.10 Interference and undercutting

Figure shows a pinion with centre

in mesh with wheel or gear with centre

MN is the common tangent to the base circles & KL is the path of contact between the two mating teeth.

That is the radius of the addendum circle of pinion is increased to

, the point of contact L will move from L to N.

When this radius is further increased, the point of contact L will be on the inside of base circle of wheel and not on the involute profile of tooth on wheel.

The tip of tooth on the pinion will then undercut the tooth on the wheel at the root and remove part of the involute profile of tooth on the wheel.

This effect is known as interference and occurs when the teeth are being cut.

The phenomenon when the tip of tooth undercuts the root on its matting gear is known as interference.

Similarly, if the radius of the addendum circle of the wheel increases beyond

then the tip of tooth on wheel will cause interference with the tooth on pinion.

The points M and N are called interference points.

Obviously, interference may be avoided if the path of contact does not extend beyond interference points.

The limiting value of the radius of the addendum circle of the pinion is

and of the wheel is

So, we can conclude that the interference may only be avoided, if the point of contact between the two teeth is always on the involute profiles of both the teeth.

In other words, interference may only be prevented, if addendum circles of the two mating gears cut the common tangent to the base circles between the points of tangency.

When interference is just avoided, the maximum length of path of contact is MN when the maximum addendum circles for pinion and wheel pass through the points of tangency N and M respectively as shown in figure in such a case,

Maximum length of path of approach,

And maximum length of path of recess,

Maximum length of path of contact

Maximum length of Arc of contact

Undercutting

Figure shows a pinion.

A portion of its dedendum falls inside the base circle.

The profile of the tooth inside the base circle is radial.

If the addendum of the mating gear is more than the limiting value, it interferes with the dedendum of the pinion and the two gears are locked.

However, if a cutting rack having similar teeth is used to cut the teeth in the pinion, it will remove that portion of the pinion tooth which would have interfered with the gear as shown in figure.

A gear having its material removed in this manner is set to undercut and the process undercutting.

In pinion with small number of teeth, this can seriously weaken the tooth, however, when the actual gear meshes with the undercut pinion no interference occurs.

Undercutting will not take place if the teeth are designed to avoid interference.

1.11 Methods to avoid interference

In general, there are three different ways of reducing or eliminating interference and subsequent undercutting at the flank region, when a pinion has less than minimum number of teeth to avoid interference.

By using modified involute or composite system

When is standard addendum is used for involute pinion with the pressure angle of

° the smallest pinion that will gear with a rack without interference has 32 teeth.

In many situations, it becomes obligatory to use pinion which has 12 teeth only.

In such cases, the shape of basic rack tooth may be modified.

The flank portion of pinion tooth lying inside the base circle and the matting portion of gear tooth face may be made cycloidal in place of involute shape.

Remaining portion of the pinion tooth maybe of involute profile.

2. By modifying addendum of gear tooth

To avoid interference, it is required to chop off interfering portion of the face of the gear tooth. This is illustrated in figure.

The resulting tooth is called stub tooth instead of full depth tooth.

As can be seen in figure, portion A'B' of gear tooth measures with non-involute portion in the flank of pinion tooth. If portion AA' B'B of gear tooth is chopped off, interference will be eliminated.

However, this step amounts to reducing radius of addendum circle of gear, which now cuts line of action at point K' instead of K.

In other words, length of path and approach and hence, the contact ratio is reduced.

3. Increased centre distance

When centres

of two mating gears are slightly moved apart, the common tangent to the base circles cut the line of centres at point P'.

then become the pitch Circle radii of the mating gears.

The centre distance between two involute gears may be increased within limits, without disturbing correctness of gearing, and the step will prevent the tip of gear tooth from mating with non-involute flank portion of the pinion, at least to some extent.

There are however two adverse consequences:

Actual pitch circle diameter is increased, this tends to reduce effective addenda and hence the arc of contact too.

As the difference in tooth space width and tooth thickness at new pitch Circle is further increased, considerable backlash will be introduced between the teeth.

1.12 Minimum number of teeth on gear and pinion only

Minimum number of teeth on the pinion in order to avoid interference

In order to avoid interference, the addendum circles for the two mating gears must cut the common tangent to the base circles between the points of tangency.

The limiting condition reaches, when the addendum circles of pinion and wheel pass through points N and M respectively.

Let. t= number of teeth on the pinion.

T= number of teeth on the wheel

m = module of the teeth

r= pitch Circle radius of pinion =m.t/2

G= Gear ratio =T/t =R/r

= Pressure angle or angle of obliquity

From triangle

Where

Limiting radius of the pinion addendum circle,

Let = addendum of the pinion, where is a fraction by which the standard addendum of one module for the pinion should be multiplied in order to avoid interference.

We know that the addendum of the pinion.

This equation gives the minimum number of teeth required on the pinion in order to avoid interference.

Minimum number of teeth on the Wheel in order to avoid interference

Let. T= minimum number of teeth required on the wheel in order to interference,

.m= addendum of the wheel, where is a fraction by which the standard addendum for the wheel should be multiplied.

Using the same notations as in above article we have from triangle

Where

Limiting radius of wheel addendum circle

We know that addendum of the wheel

1.13 Force analysis

In gears, power is transmitted by means of force exerted by the tooth of the driving Gear on the meshing tooth of the driven gear.

Figure shows the tooth of the driving pinion exerting force

on the tooth of driven gear.

According to the fundamental law of gearing, this resultant force

always acts along the pressure line.

The resultant force

can be resolved into components - tangential component

and radial component

at the pitch point as shown in figure.

The tangential component

useful load because it determines the magnitude of the torque and consequently the power, which is transmitted.

The radial component

is a separating force, which is always directed towards the centre of the gear. The torque transmitted by the gears is given by

Where T = torque transmitted by gears

kW = power transmitted by gears

n= speed of rotation

The tangential component acts at the pitch Circle radius. Therefore,

From figure

The resultant force is given by

The above analysis of gear tooth force is based on the following assumptions

As the point of contact moves, the magnitude of resultant force

changes. This effect is neglected in above analysis.

It is assume that only one pair of teeth takes the entire load. At times there are two pairs, which are simultaneously in contact and share the load. This aspect is neglected in the analysis.

The analysis is valid under static conditions, i.e. when the gears are running at very low velocities. In practice, there is dynamic force in addition to force due to power transmission. The effect of this dynamic force is neglected in the analysis.

1.14 Friction in gears

Friction refers to the resistance of one substance moving against the other.

By definition, gears move against each other, and therefore experience friction to various degrees.

Gear friction occurs at the interface of lubricated tooth contacts that are subject to combined sliding and rolling motions.

Engineered gear tooth surfaces that are not smooth experience diverse lubrication conditions, ranging from full film to mixed elastohydrodynamic lubrication (EHL) or boundary lubrication conditions, depending on surface and operating conditions and lubricant characteristics.

In cases where asperity contacts occur, gear friction is defined by a combination of fluid viscous shear, rolling resistance, and dry friction at the contact interfaces.

A fundamental understanding of gear friction is essential as it impacts

(1) gear scuffing failures due to excessive heat generation and gear contact fatigue lives and associated failure modes of spalling and micro-pitting

(2) load-dependent (mechanical) gear mesh power losses, and

(3) a class of gear vibrations along the direction of the relative sliding and the damping effects along the line of action.

Following methods are used to reduce gear friction.

Different gear structures.

We can reduce the amount of friction generated by a gear set by introducing different shapes.

Helical gears, for example, experience friction gradually and distributed along the length of their teeth, making them capable of handling higher levels of friction with no problem.

2. Lower speeds and loads.

Most friction problems only arise in certain conditions—usually, when gears are operating at high speeds, or with high loads that increase the amount of friction present.

Working to lower those speeds and lows can make the friction problem disappear.

3. Lubrication.

Lubrication is one of the simplest ways to address friction.

It instantly reduces the friction between gears but must be reapplied occasionally to keep parts moving.

4. Material selection.

We can also produce gears out of special materials that don’t experience as much friction as their counterparts, such as certain metal alloys.

Numerical :

The number of teeth on each of the two equal spur gears in mesh are 40. The teeth have 20° involute profile and the module is 6 mm. If the arc of contact is 1.75 times the circular pitch, find the addendum.

Solution Given. T=t=40; = 20° ; m=6mm

Circular pitch,

Length of Arc of contact

And length of path of contact

= Length of Arc of contact × cos = 33 cos 20° = 31mm

Let. radius of the addendum circle of each wheel.

Pitch Circle radii of each wheel,

R =r= m.T/2=6×40/2=120mm

And length of path of contact

The addendum of the wheel

= mm

2. The following data relate to a pair of 20-degree involute gears in mesh:

Module=6mm, number of teeth on pinion =17, number of teeth on gear=49 addendum on pinion and gear wheel=1module

Find: 1. The number of pairs of teeth in contact 2. The angle turned through by the pinion and the gear wheel when one pair of teeth is in contact, and 3. The ratio of sliding to rolling motion when the tip of a tooth on the larger wheel a. Is just making contact, b. Is just leaving contact with its matting tooth, c. Is at the pitch point.

Solution

Given =20° m=6mm; t=17 ; T=49 ; Addenda on pinion and gear wheel =1 module =6mm

Pitch Circle radius of pinion

r = m.t/2=6×17/2=51mm

And pitch Circle radius of gear

R = m.T/2=6×49/2=147mm

Radius of addendum circle of pinion

addendum=51+6=57mm

And radius of addendum circle of gear

addendum=147+6=153mm

The length of path of approach

The length of path of recess

= 13.41mm

Length of path of contact

KL = KP + PL =15.5+13.41=28.91mm

Length of Arc of contact =

Circular pitch,

Number of pairs of teeth in contact

=Length of Arc of contact/circular pitch=30.8/18.852=1.6 say 2

Angle turned through by the pinion

And angle turned through by the gear wheel

Ratio of sliding to rolling motion

Let =Angular velocity of pinion

=Angular velocity of gear wheel

Rolling velocity

At the instant when the tip of the tooth on the larger wheel is just making contact with its mating teeth, the sliding velocity

Ratio of sliding velocity to rolling velocity

b. At instant when the tip of a tooth on the larger wheel is just leaving contact with its mating teeth, the sliding velocity.

Ratio of sliding velocity to rolling velocity

b. Since at the pitch point, the sliding velocity is zero, therefore the ratio of sliding velocity to rolling velocity is zero.

3. Two mating gears have 20 and 40 involute teeth of module 10 mm and 20° pressure angle. The addendum on each wheel is to be made of such length that the line of contact on each side of the pitch point has half the maximum possible length. Determine the addendum height for each gear wheel, length of the path of contact, arc of contact and contact ratio.

Solution Given t=20; T=40; m=10mm; =20°

Addendum height for each gear wheel

The pitch Circle radius of the smaller gear wheel,

R = m.t/2 =10×20/2=100mm

And pitch circle radius of the larger gear wheel,

R = m.T/2 =10×40/2=200mm

= radius of addendum circle for the larger gear wheel, and

=radius of addendum circle for the smaller gear wheel.

Since the addendum on each wheel is to be made of such a length that the line of contact on each side of the pitch point has half the maximum possible length, therefore

Path of approach,

Addendum height for larger gear wheel

Now the path of recess

Addendum height for smaller gear wheel

The length of the path of contact

The length of the arc of contact

Circular pitch

Contact ratio

4. Two 20-degree involute spur gear mesh externally and give a velocity ratio of 3. The module is 3 mm and the addendum is equal to 1.1 module. If the pinion rotates at 120 RPM, determine the

Minimum number of teeth on each wheel to avoid interference

Contact ratio

Solution VR=3 Addendum=1.1m m=3mm

Minimum number of teeth on each wheel to avoid interference

Taking the higher whole number divisible by the velocity ratio,

T=51 and t=51/3=17

Pitch Circle radius of pinion

r = m.t/2= 3×17/2=25.5mm

And pitch Circle radius of gear

R = m.T/2= 3×51/2= 76.5 mm

Radius of addendum circle of pinion

addendum = 25.5 + 1.1 x 3 = 28.8mm

And radius of addendum circle of gear

addendum= 76.5 + 1.1 x 3 = 79.8 mm

The length of path of approach

= 8.48 mm

The length of path of recess

= 7.25 mm

Length of path of contact

KL = KP + PL =8.48 + 7.25 = 15.73 mm

Length of Arc of contact =

Circular pitch,

Number of pairs of teeth in contact

=Length of Arc of contact/circular pitch= 16.74/9.42=1.78 say 2

5. Two gears wheels Mahesh externally and are to give a velocity ratio of 3 to 1. The teeth are of involute form module =6 mm, addendum = one module, pressure angle = 20° . The pinion rotates at 90 r.p.m. determine 1. The number of teeth on the pinion to avoid interference on it and the corresponding number of teeth on the wheel. 2. The length of path and arc of contact,. 3. The number of pairs of teeth in contact. 4. The maximum velocity of sliding

Solution :- Given G=T/t =3, m=6mm, x module = 6mm; =20°