TUTORIAL SHEET 1: - EME 602

(Submitted by Achintya Mehrotra, 0802740006)

Q1 Give an account of the classification of gears?

Ans: - Gears are classified into four groups:

a) Spurb) Helicalc) Beveld) Worm

SPUR GEARS

1) The teeth are cut parallel to the axis of the shaft.2) The profile of the tooth is in the shape of an involute curve and remains identical along the

entire width of the gear wheel.3) These gears impose radial loads on the shaft.

HELICAL GEARS

1) The teeth are cut at an angle with the axis of the shaft.2) The involute profile is similar to that of the spur gear.3) The magnitude of the helix angle of pinion and gear is same however the hand of helix is

opposite, i.e. a right hand pinion meshes with a left hand gear and vice versa.4) These gears impose radial and tangential loads on the shaft.5) A special type of helical gear known as Herringbone gear consists of two helical gears with

opposite hand of helix resulting in equal and opposite thrust reactions.

BEVEL GEARS

1) These gears are used for shafts which are at right angles to each other.2) The tooth can be spiral or straight.3) It imposes radial and thrust load on the shaft.

WORM GEAR

1) It consists of a worm and worm wheel. The worm is in the form of a threaded screw which meshes with the matching wheel.

2) The threads on the worm can be single start or multi start and usually have a small lead.3) Worm gears are used for shafts which are perpendicular but do not intersect.4) The worm imposes high thrust load, while worm wheel imposes high radial loads on the shaft.

5) Worm gear drives are characterized by high speed reduction ratio.

Q2 What are the assumptions made in the force analysis of gears?

Ans: - The following assumptions are made in the force analysis of gears:-

1) As the point of contact moves, the magnitude of resultant forces changes. However this effect is neglected.

2) It is assumed that only one pair of teeth takes the entire load.3) The analysis is valid under static conditions, i.e. when the gears are running at very low

speeds.4) The effect of dynamic forces is neglected.

Q3 What are the assumptions made in the Lewis Beam equation?

Ans: - The following assumptions are made in the Lewis beam equation:-

1) The effect of radial component which induces compressive stresses is neglected.2) The tangential component is uniformly distributed over the face width of the gear. This is

possible when the gears are rigid and accurately machined.3) Effect of stress concentration is neglected.4) At any time only one pair of teeth is in contact and takes the total load.

Q4 Compare involute and cycloidal profile of gear tooth.

Ans:-

Advantages of involute gears Advantages of cycloidal gears

The centre distance for a pair of involute gears can be varied within limits without changing the velocity ratio. However in cycloidal gears the exact centre distance needs to be maintained.

Since cycloidal teeth have wider flanks, thus they are stronger than the involute gears for the same pitch.

The face and flank of the involute teeth are generated by a single curve whereas in cycloidal gears, double curves are required for face and flank respectively.

The contact takes place between a convex flank and concave surface, whereas in involute gears, convex surfaces are in contact. This results in less wear in cycloidal gears.

The pressure angle from the start of the engagement of the teeth to the end of the engagement remains constant. This result in smooth running and less wear of gears. But in cycloidal gears, the pressure angle is maximum at the beginning of the engagement, reduces to zero at the pitch point, starts increasing and again becomes maximum at the end of engagement.

Interference does not occur at all.

Q5 Discuss briefly the various systems of gear tooth used in gear drives.

Ans:- The following four systems of gear teeth are used:-

14.5 degree composite system

1) It is used for general purpose gears

2) Stronger but no interchangeability.

3) This profile of the tooth system has cycloidal curves at the top and bottom and involute curve at the middle portion.

14.5 degree full depth involute system

1) The basic rack for this system is composed of straight sides except for the fillet arcs.

2) Interference occurs when number of teeth on pinion is less than 23.

3) This system is satisfactory when the number of teeth on gears is large.

20 degree full depth involute system

1) The basic rack for this system is composed of straight sides except for fillet arcs.2) Interference occurs when number of teeth on pinion is less than 17.3) This system is the most widely used system of gear standards.4) It reduces the risk of undercutting and interference.5) Due to increased pressure angle, the tooth becomes slightly broader at the root. This

increases the load carrying capacity and makes the tooth stronger.

20 degree stub involute system

1) The gears in this system have shorter addendum and shorter dedendum.2) The interfering portion of the tooth, i.e. a part of addendum is thus removed. This reduces

undercutting.3) The minimum number of teeth on pinion to avoid interference is 14.4) Stub teeth are stronger than full depth teeth because of smaller moment arm of bending

force.5) Stub system transmits very high load.6) It also results in low cost.

S.No Particulars 14.5 degree composite or full depth involute system

20 degree full depth involute system

20 degree stub involute system

1. Addendum 1m 1m 0.8m2. Dedendum 1.25m 1.25m 1m3. Working depth 2m 2m 1.6m4. Min total depth 2.25m 2.25m 1.8m5. Tooth thickness 1.5708m 1.5708m 1.5708m6. Min clearance 0.25m 0.25m 0.2m7. Fillet radius 0.4m 0.4m 0.4m

Q6 Define the following:1. Circular pitch

2. Diametral pitch 3. Module

4. Backlash in gears5. Tooth thickness6. Width of space7. Addendum8. Dedendum

Ans:-

a) Circular pitch:- The circular pitch(p) is the distance measured along the pitch circle between two similar points on adjacent teeth

b) Diametral pitch: - The diametral pitch (P) is the ratio of the number of teeth to the pitch circle diameter.

c) Module: - It is defined as the inverse of the diametral pitch.d) Backlash: - It is defined as the amount by which the width of the tooth space exceeds the

thickness of the engaging tooth measured along the pitch circle.e) Tooth thickness: - The length of the arc on the pitch circle subtending a single gear tooth is

called circular tooth thickness.f) Width of space: - The width of the space between two adjacent teeth measured along the

pitch circle is called tooth space.g) Addendum: - It is the radial distance between pitch and the addendum circle. It indicates the

height of tooth above the pitch circle.h) Dedendum: - It is the radial distance between the pitch and the dedendum circle. It indicates

the depth of tooth below the pitch circle.

Q7 Prove the fundamental law of gearing.

Ans: -  Fundamental Law of Gear-Tooth Action

The figure shows two mating gear teeth, in which

Tooth profile 1 drives tooth profile 2 by acting at the instantaneous contact point K. N1N2 is the common normal of the two profiles. N1 is the foot of the perpendicular from O1 to N1N2

N2 is the foot of the perpendicular from O2 to N1N2.

Although the two profiles have different velocities V1 and V2 at point K, their velocities along N1N2 are equal in both magnitude and direction. Otherwise the two tooth profiles would separate from each other. Therefore, we have

We notice that the intersection of the tangency N1N2 and the line of centre O1O2 is point P, and

(7-3)

Thus, the relationship between the angular velocities of the driving gear to the driven gear, or velocity ratio, of a pair of mating teeth is

(7-4)

Point P is very important to the velocity ratio, and it is called the pitch point. Pitch point divides the line between the line of centres and its position decides the velocity ratio of the two teeth. The above expression is the fundamental law of gear-tooth action.

7.2.2 Constant Velocity Ratio

For a constant velocity ratio, the position of P should remain unchanged. In this case, the motion transmission between two gears is equivalent to the motion transmission between two imagined slipless cylinders with radius R1 and R2 or diameter D1 and D2. We can get two circles whose centre are at O1 and O2, and through pitch point P. These two circles are termed pitch circles. The velocity ratio is equal to the inverse ratio of the diameters of pitch circles. This is the fundamental law of gear-tooth action.

The fundamental law of gear-tooth action may now also be stated as follow (for gears with fixed centre distance) The common normal to the tooth profiles at the point of contact must always pass through a fixed point (the pitch point) on the line of centres (to get a constant velocity ratio).

Q8 Discuss in brief the involute profile and its important characteristics.

Ans:-

The curve most commonly used for gear-tooth profiles is the involute of a circle. This involute curve is the path traced by a point on a line as the line rolls without slipping on the circumference of a circle. It may also be defined as a path traced by the end of a string which is originally wrapped on a circle when the string is unwrapped from the circle. The circle from which the involute is derived is called the base circle.

In the figure, let line MN roll in the counter clockwise direction on the circumference of a circle without slipping. When the line has reached the position M'N', its original point of tangent A has reached the position K, having traced the involute curve AK during the motion. As the motion continues, the point A will trace the involute curve AKC.

Properties of Involute Curves

1. The distance BK is equal to the arc AB, because link MN rolls without slipping on the circle.2. For any instant, the instantaneous centre of the motion of the line is its point of tangent with

the circle.Note: We have not defined the term instantaneous centre previously. The instantaneous centre or instant centre is defined in two ways

1. When two bodies have planar relative motion, the instant centre is a point on one body about which the other rotates at the instant considered.

2. When two bodies have planar relative motion, the instant centre is the point at which the bodies are relatively at rest at the instant considered.

3. The normal at any point of an involute is tangent to the base circle. Because of the property (2) of the involute curve, the motion of the point that is tracing the involute is perpendicular to the line at any instant, and hence the curve traced will also be perpendicular to the line at any instant.

4. There is no involute curve within the base circle.

Q9 What is interference, explain with a neat diagram. And discuss briefly the various methods to prevent interference.

Ans:-

1) The gear tooth has involute profile only outside the base circle. In some cases the dedendum is so large that it extends below this base circle. In such situations, the portion of the tooth below the base circle is not involute.

2) The tip of the tooth on the mating gear, which is involute, interferes with the non-involute portion of dedendum. This phenomenon of tooth profiles overlapping and cutting into each other is called interference.

3) Interference is non conjugate action and results in excessive wear, vibrations and jamming.4) It is maximum when the smallest pinion is in mesh with the largest gear.

Methods to prevent interference

1) Undercutting: - When the gears are generated by involute rack cutters, this interference is automatically eliminated because the cutting tool removes the interfering portion of the flank. This is called undercutting. However, undercut tooth is considerably weaker. Also, the loss of involute profile may cause a serious reduction in the length of contact.

2) Increase in pressure angle: - This results in smaller base circle so that more portion of the tooth profile becomes involute.

3) Increase number of teeth on pinion: - However, this increases the size of gearbox and also the pitch line velocity.

4) Use long and short addendum gearing: - The addendum of pinion is made longer than the standard addendum. Also, the addendum of mating gear is made shorter than the standard addendum. However, this results in non-standard and non-interchangeable gears.

Q10 What do you understand by conjugate action of teeth. Justify that involute teeth do not fulfil the conjugate action.

Ans: -

Conjugate tooth action

The shape of teeth necessary for the speed ratio to remain constant during an increment of rotation; this behaviour of the contacting surfaces (i.e. the teeth flanks) is known as conjugate action.

Consider the two rigid bodies 1 and 2 which rotate about fixed centres, O, with angular velocities ω. The bodies touch at the contact point, C, through which the common tangent and normal are drawn. The absolute velocity v of the contact point reckoned as a point on either body, is perpendicular to the radius from that body's centre O to the contact point. For the bodies to remain in contact, there must be no component of relative motion along the common normal, so that from the velocity triangles :- v2 cosθ2 = v1 cosθ1 where v1 = ω1 . O1C ; v2 = ω2 . O2C Note that the tangential components of velocity are generally different, so sliding must occur. For the speed ratio to be constant therefore, from the above and similar triangles :-

( 3 ) ω2 /ω1 = v2 . O1C/v1 . O2C = O1C.cosθ1 /O2C.cosθ2

= O1 C1 /O2 C2 = O1 P / O2 P ie. this ratio also must be constant. This indicates that, since the centres are fixed, the point P is fixed too.

In general therefore, whatever the shapes of the bodies, the contact point C will move along some locus as rotation proceeds; but if the action is to be conjugate then the body geometry must be such that the common normal at the contact point passes always through one unique point lying on the line of centres - this point is the pitch point referred to above, and the pitch circles' radii are O1 P and O2 P.

There exists a host of shapes which ensure conjugacy - indeed it is possible, within certain restrictions, to arbitrarily choose the shape of one body then determine the shape of the second necessary for conjugacy. But by far the most common gear geometry which satisfies conjugacy is based on the involute, in which case both gears are similar in form, and the contact point's locus is a simple straight line - the line of action.

Q11 Write short notes on

a) Modes of gear tooth failureb) Gear materialsc) Gear manufacturing methods

Ans: - Modes of gear tooth failure

1) Bending failure: - Every gear acts as a cantilever. If the total repetitive dynamic load acting on the gear tooth is greater than the beam strength of the gear tooth, then the gear tooth will fail in bending.

2) Pitting: - It is the surface fatigue failure which occurs due to many repetitions of hertz contact stresses. The failure occurs when the contact stresses are higher than the endurance limit of the material. The failure starts with the formation of pits which continue to grow resulting in the rupture of tooth surface.In order to avoid pitting, the dynamic load between the gear teeth should be less than the wear strength of the gear tooth.

3) Scoring: - The excessive heat is generated when there is an excessive surface pressure, high speed or supply of lubricant fails. It is a stick-slip phenomenon in which alternate shearing and welding takes place rapidly at high spots.This type of failure can be avoided by properly designing the parameters such as speed, pressure and proper flow of lubricant.

4) Abrasive wear: - Foreign particles in the lubricant such as dirt, dust or burr enter between the tooth and damage the form of tooth. It can be avoided by providing filters for the lubricating oil or by using high viscosity lubricant oil which enables the formation of thicker oil film and hence permits easy passage of such particles without damaging the gear surface.

5) Corrosive wear: - The corrosion of the tooth surfaces is mainly caused due to the presence of corrosive elements such as additives present in the lubricating oil.

Gear materials

The desirable properties of gear materials are as follows:-

1) The gear material should have sufficient strength to resist failure due to breakage of tooth. When the gear is subjected to fluctuating forces, then the endurance strength of the tooth is the deciding factor.

2) The gear material should have sufficient surface endurance strength to avoid failure due to destructive pitting.

3) For high speed transmissions, the sliding velocities are high and the material should have a low coefficient of friction to avoid failure due to scoring.

The various materials used for gear manufacture are cast iron, steel, bronze, wood, rawhide and synthetic resins.

1) Large size gears are made of cast iron of grades FG 200, FG 260 or FG 350. They are cheap and generate less noise compared with steel gears. However, their main drawback is poor strength.

2) Case hardened steel gears offer the best combination of a wear-resisting hard surface together with a ductile and shock absorbing core.

3) Plain carbon steels are used for medium duty applications such as 50C8, 45C8, 50C4.4) For heavy duty applications, alloy steels 40Cr1, 30Ni4Cr1 and 40Ni3Cr65mo55 are used.5) Bronze is mainly used for worm wheels due to its low coefficient of friction and excellent

conformability.