MECHANICAL ENGINEERING DESIGN 1 MEC 531

Part B:

Design of Machine Elements

By:

NURZAKI IKHSAN

Chapter Outline

6.1 Gears 6.2 Parallel & non-Parallel Shaft

Chapter 6:

Design of Power Transmission

Introduction

• Gear is a mechanical element/component by which a power can be

transmitted from one shaft to another.

• Types of gear:

1. Spur gears

2. Helical gears

3. Bevel gears

4. Worm gears

• Parallel shaft → spur gears & helical gears.

• Perpendicular two intersecting shaft → bevel gears & worm gears.

Spur Gears

• Teeth parallel to the axis of rotation.

• Used to transmit motion from one

shaft to another, parallel, shaft.

• Spur gear is the simplest and, for this

reason, will be used to develop the

primary kinematic relationships of

the tooth form.

Helical Gears

• Teeth inclined to the axis of rotation.

• Can be used for the same applications as spur gears.

• Not noisy, because of the more gradual engagement of the teeth during meshing.

• The inclined tooth also develops thrust loads and bending couples, which are not present with spur gearing.

• Helical gears are used to transmit motion between parallel and sometimes are used to transmit motion between nonparallel shaft.

Bevel Gears

• Teeth formed on conical surfaces and

are used mostly for transmitting

motion between intersecting shafts.

• Three types:

1. Straight-tooth bevel gears

2. Spiral bevel gears

3. Hypoid gears

Worm Gears

• Worm gear sets are used to transmit rotary motion between nonparallel and nonintersecting shafts.

• Direction of rotation of the worm gear depends upon the direction of rotation of the worm and upon whether the worm teeth are cut right-hand or lefthand.

• Made so that the teeth of one or both wrap partly around the other.

• Two types:

1. Single-enveloping worm-gear sets.

2. Double-enveloping worm-gear sets.

• Mostly used when the speed ratios of the two shafts are quite high → 3 or more.

Nomencluture

Pitch circle – a theoretical circle upon which all calculations are usually based.

Pitch diameter – diameter of pitch circle.

Module, m – ratio of the pitch diameter to the number of teeth.

Circular pitch, p – distance, measured on the pitch circle, from a point on one tooth to a corresponding point on the next tooth. (tooth thickness + width of space)

Nomencluture

Addendum, a – radial distance between the top land and the pitch circle.

Nomencluture

Dedendum, b – radial distance from the bottom land to the pitch circle.

Whole depth, ht = Addendum, a + Dedendum, b

Nomencluture

Clearance circle – circle that is tangent to the addendum circle of the mating gear.

Nomencluture

Clearance, c – the amount by which the dedendum in a given gear exceeds the addendum of its mating gear.

Nomencluture

Gears Relationship

Relations between the previous nomenclatures: Where; P = diametral pitch, teeth per inch N = number of teeth d = pitch diameter, in m = module, mm d = pitch diameter, mm p = circular pitch

Gears Relationship

Example

An 18-tooth pinion is to mesh a 30-tooth gear. Given that the module of

the gear set is 12 mm/teeth. Determine the pitch diameter for the

pinion and gear.

CONSTRUCTING GEARS

• Learning how to construct and draw the teeth on a pair of meshing gears is not for manufacturing or shop purpose

• More to obtain an understanding of the problems involved in the meshing of the mating teeth.

• Take previous example’s pitch diameters to start constructing our gear.

Constructing Involute Curve

1. Divide the base circle into a number of equal parts and construct radial lines.OA0, OA1, OA3, etc.

2. At OA1, construct perpendicular lines A1B1, A2B2, A3B3, etc.

3. Mark the distance between A1A0 along A1B1, twice along A2B2 and so on producing points.

4. Connect the points to construct the involute curve.

Steps to Construct a Gear

1. Known that the pinion’s pitch diameter is 216 mm and gear’s diameter is 360 mm. Center distance;

2. Locate pinion and gear centers O1 and O2 at 288 mm apart.

3. Construct pitch circles of pinion, r1=108 mm and pitch circles of gear, r2=180 mm radii.

4. Locate pitch point (contact of the circles) that are tangent at P.

5. Draw a line, ab through the pitch point, P tangent to the circles. Gear 1 will act as the pinion (driver rotating counter-clockwise) and gear 2 as the gear.

6. Draw a line, cd (pressure line or generating line or line of action) through the pitch point, P at an angle Φ to the common tangent. Angle Φ (pressure angle) values of 20o or 25o.

Static Loading Example

7. On each gear, draw circle tangents (base circles) to the pressure line, cd. The pressure angle, Φ determines their size.

8. Radius of the base circle;

9. Generate an involute on each base circles. Draw the addendum and dedendum;

10. To draw a tooth, the tooth thickness, t must be known. Tooth thickness is given as;

11. Determine the clearance, c = b – a, The fillets of the teeth is taken from the above clearance, c.

* To draw the actual shape of the gear teeth, a template for drawing gear teeth is needed and may not be the same depending how the profile is generated.

Example

A gearset consists of a 16-tooth pinion driving a 40-tooth gear. The diametral

pitch is 2 and the addendum and dedendum are 1/P and 1.25/P respectively.

The gears are cut using a pressure angle of 20o.

a) Compute the circular pitch, the center distance and the radii of the base

circles.

b) In mounting these gears, the center distance was incorrectly made . in

larger. Compute the new values of the pressure angle and the pitch-circle

diameters

Contact Ratio

• Tooth contact begins and ends at the intersections of the two addendum circles with the pressure line.

• When a tooth is just beginning contact at a, the previous tooth is simultaneously ending its contact at b.

• Contact ratio can be defined as;

• Where;

qt = arc of action

qr = arc of recess

qa= arc of approach

• This ratio is equal to the length of the path of contact between the teeth divided by the base pitch.

• Contact ratio for gears should not be less than 1.20 → inaccuracies in mounting might reduce the contact ratio, therefore may increase impact of teeth and generate noise.

Forming Of Gear Teeth

• Ways of forming teeth of gears may vary from:

– Sand casting – Permanent-mold casting

– Die casting – Shell molding

– Centrifugal casting – Investment casting

– Powder metallurgy process – Extrusion process

– Cold forming or cold rolling

• Gear teeth may be machined by milling, shaping, or hobbing.

• They may be finished by shaving, burnishing, grinding, or lapping.

• Gears made of thermoplastics such as nylon,n polycarbonate, acetal are quite popular and are easily manufactured by injection molding.

• Are low to moderate precision, low in cost for high production quantities, and capable of light loads, an run without lubrication.

Milling

• Gear teeth may be cut with a form milling cutter shaped to conform to

the tooth space.

• Need to use a different cutter for each gear because a gear having 25

teeth, for example, will have a differentshaped tooth from one having,

say, 24 teeth.

• A separate set of cutters is required for each pitch

Forming Of Gear Teeth

Shaping

• Teeth may be generated with either a pinion cutter or a rack cutter.

• The pinion reciprocates along the vertical axis and is slowly fed into

the gear blank to the required depth.

• When the pitch circles are tangent, both the cutter and the blank

rotate slightly after each cutting stroke.

• Since each tooth of the cutter is a cutting tool, the teeth are all cut

after the blank has completed one rotation.

Forming Of Gear Teeth

Hobbing

• The hob is simply a cutting tool that is shaped like a worm.

• The teeth have straight sides, as in a rack, but the hob axis must be turned

through the lead angle in order to cut spur-gear teeth.

• For this reason, the teeth generated by a hob have a slightly different shape

from those generated by a rack cutter.

• Both the hob and the blank must be rotated at the proper angular-velocity ratio.

• The hob is then fed slowly across the face of the blank until all the teeth have

been cut.

Forming Of Gear Teeth

Gear Trains

• Speed of a pinion 2 driving a gear 3 is given by:

Where n = revolutions or rev/min

N = number of teeth

d = pitch diameter

• Above equation applies to any gearset → spur, bevel, helical or worm.

• Absolute value → user can define positive or negative gear rotation direction.

• But for spur and parallel helical gears → positive for counterclockwise

• rotation.

Driven Gear’s Speed

• The speed of the above driven gear 6 is given as:

• No ratio between N4 and N5 because the gears attached to each other.

• For spur gears, e is positive if the last gear rotates in the same way as the first gear.

• •Train value, e is defined as:

• Product of either driving or driven tooth numbers:

– Number of teeth, N

– Pitch diameters, d

• Relation between the speed of the last gear and the first gear in

the train:

Driven Gear’s Speed

Example

A 17-tooth spur gear (pinion) with a diametral pitch, P of 8, n2 = 1120 rpm,

n3 = 544 rpm. Determine the number of teeth in the gear and center to

center distance between the gears.

FBD of Forces and Moments

Force Analysis-spur Gearing

Force Analysis-spur Gearing

• Designation of gear will start with gear 2 as the input and then number the gears successively 3, 4, etc until the last gear in the train.

• Shafts will be designated as lowercase letters → a, b, c, etc.

• Force exerted by gear 2 against gear 3 → F23

• Force of gear 2 against a shaft a → F2a

• Radial component of force → Fr

• Tangential component of force → Ft

• Torque exerted by shaft a against pinion 2 → Ta2

• Transmitted load: or [kN]

Where H = power, kW

d = gear diameter, mm

n = speed, rev/s

• Torque:

• Power transmitted:

• Gear data is often tabulated using pitch-line velocity, V

Where V = pitch-line velocity, mm/s

d = gear diameter, mm

n = gear speed, rev/s

Force Analysis-spur Gearing

Force Analysis-bevel Gearing

• Transmitted load:

• Other forces acting at the center of the tooth:

Force Analysis-helical Gearing

• Three (3) components of the total

(normal) tooth force W:

• If Wt is given:

Force Analysis-worm Gearing

Considering coefficient of friction, f :

Three (3) orthogonal components:

Efficiency of worm gearset: Coefficient of friction is dependent on the relative or sliding velocity, VS