mechanical analysis gear trains - city, university of londonra600/me2105/analysis/me2104-a-4.pdf ·...

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Ahmed Kovacevic, City University London 1 1 Mechanical Analysis Gear trains Prof Ahmed Kovacevic Lecture 4 School of Engineering and Mathematical Sciences Room C130, Phone: 8780, E-Mail: [email protected] www.staff.city.ac.uk/~ra600/intro.htm Mechanical Analysis and Design ME 2104

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Page 1: Mechanical Analysis Gear trains - City, University of Londonra600/ME2105/Analysis/ME2104-A-4.pdf · Mechanical Analysis Gear trains ... Week 4 – Gears and gear trains Week 5 –

Ahmed Kovacevic, City University London 1 1

Mechanical Analysis Gear trains

Prof Ahmed Kovacevic

Lecture 4

School of Engineering and Mathematical Sciences

Room C130, Phone: 8780, E-Mail: [email protected]

www.staff.city.ac.uk/~ra600/intro.htm

Mechanical Analysis and Design ME 2104

Page 2: Mechanical Analysis Gear trains - City, University of Londonra600/ME2105/Analysis/ME2104-A-4.pdf · Mechanical Analysis Gear trains ... Week 4 – Gears and gear trains Week 5 –

Ahmed Kovacevic, City University London 2

Plan for the analysis of mechanical elements

Objective:

Procedures for design and selection of

mechanical elements

Week 1 – Shafts and keyways

Week 2 – Bearings and screws

Week 3 – Belt and chain drives

Week 4 – Gears and gear trains

Week 5 – Design Project Review

Page 3: Mechanical Analysis Gear trains - City, University of Londonra600/ME2105/Analysis/ME2104-A-4.pdf · Mechanical Analysis Gear trains ... Week 4 – Gears and gear trains Week 5 –

Ahmed Kovacevic, City University London 3

Plan for this week

Gears

Gear trains

Examples

Page 4: Mechanical Analysis Gear trains - City, University of Londonra600/ME2105/Analysis/ME2104-A-4.pdf · Mechanical Analysis Gear trains ... Week 4 – Gears and gear trains Week 5 –

Ahmed Kovacevic, City University London 4

Gear types

Page 5: Mechanical Analysis Gear trains - City, University of Londonra600/ME2105/Analysis/ME2104-A-4.pdf · Mechanical Analysis Gear trains ... Week 4 – Gears and gear trains Week 5 –

Ahmed Kovacevic, City University London 5

How gears work

Basic

circle

Pressure

line

Law of Gearing:

A common normal to the tooth profiles at their point of contact must, in all positions of the contacting teeth, pass through a fixed point on the line-of-centres called the pitch point.

Any two curves or profiles engaging each other and satisfying the law of gearing are conjugate curves

Pressure

angle

Pitch point

Pitch

circle

Page 6: Mechanical Analysis Gear trains - City, University of Londonra600/ME2105/Analysis/ME2104-A-4.pdf · Mechanical Analysis Gear trains ... Week 4 – Gears and gear trains Week 5 –

Ahmed Kovacevic, City University London 6

Diametral

pitch

Circular

pitch

Module

Standard modules are 0.5, 0.8, 1, 1.25, 1.5, 2, 2.5, 3, 4, 5, 6

Module and Pitch

m=6.35

m=2.11

m=0.97

1

25.4[ ];

G P

G P

N NP in

D D

Dp m mm

P N

Dm mm m

N P

Page 7: Mechanical Analysis Gear trains - City, University of Londonra600/ME2105/Analysis/ME2104-A-4.pdf · Mechanical Analysis Gear trains ... Week 4 – Gears and gear trains Week 5 –

Ahmed Kovacevic, City University London 7

Relations between gear parameters

Centre distance:

Pressure angle: 20o (14.5o)

Addendum: a = module

Dedendum: b = 1.25 module

Clearance: c = 0.25 module

Backlash: clearance measured on the

pitch circle of a driving gear

Backlash is function

of module

and centre distance

2

P GD DC

Page 8: Mechanical Analysis Gear trains - City, University of Londonra600/ME2105/Analysis/ME2104-A-4.pdf · Mechanical Analysis Gear trains ... Week 4 – Gears and gear trains Week 5 –

Ahmed Kovacevic, City University London 8

Relations between gear parameters

Gear ratio

2 2

;30 30

ow G G P P

G G G GP PG P

P P G P

G G P P

P T T T

D D N TD nv GR

D N n T

n n

Radius of the base circle

Pitch of the

base circle

cos

cos cos

b

b

r r

Dp p

N

Page 9: Mechanical Analysis Gear trains - City, University of Londonra600/ME2105/Analysis/ME2104-A-4.pdf · Mechanical Analysis Gear trains ... Week 4 – Gears and gear trains Week 5 –

Ahmed Kovacevic, City University London 9

Contact ratio:

Interference (the smallest number of teeth for contact without interference):

2 2 2 2

cos

sin

ab abr

b

ab op bp og bg d

L Lc

p p

L r r r r C

Contact ratio and interference

Page 10: Mechanical Analysis Gear trains - City, University of Londonra600/ME2105/Analysis/ME2104-A-4.pdf · Mechanical Analysis Gear trains ... Week 4 – Gears and gear trains Week 5 –

Ahmed Kovacevic, City University London 10

Helical gears

cos

cos

cotsin

cn c

dn d

cna c

p p

p p

pp p

Number of teeth for helical

gear

Normal pressure angle

bw – gear width

Contact ratio for helical gear

Axial contact ratio

3cos

tantan

cos

2cos 2cos

tan

n

n

p g p g

d

rt r ra

wra

c

NN

D D N Nc

C C C

bC

p

Normal circular pitch

Normal diametral pitch

Axial Pitch

tan

tan

cos cos

a t

r t

t

F F

F F

FF

Axial load

Radial load

Normal load

Page 11: Mechanical Analysis Gear trains - City, University of Londonra600/ME2105/Analysis/ME2104-A-4.pdf · Mechanical Analysis Gear trains ... Week 4 – Gears and gear trains Week 5 –

Ahmed Kovacevic, City University London 11

Example 11 –helical gear

An involute gear drives a high-speed centrifuge. The speed of the centrifuge is

18000 rpm. It is driven by a 3000 rpm electric motor through 6:1 speedup

gearbox. The pinion has 21 teeth and the gear has 126 teeth with a diametral

pitch of 14 per inch. The width of gears is 45.72 mm and the pressure angle is

20o. Power of the electric motor is 10kW.

Determine :

a) A contract ratio of a spur gear

b) A contact ratio of a helical gear with helix angle of 30o.

c) A helix angle if the contact ratio is 3

d) Axial, radial and contact (normal) force for both helix angles.

a) Cr=1.712

b) Cr1=6.343

c) 9.122o

d) Fa1=160.8 N

Fa2=44.7 N

Page 12: Mechanical Analysis Gear trains - City, University of Londonra600/ME2105/Analysis/ME2104-A-4.pdf · Mechanical Analysis Gear trains ... Week 4 – Gears and gear trains Week 5 –

Ahmed Kovacevic, City University London 12

Page 13: Mechanical Analysis Gear trains - City, University of Londonra600/ME2105/Analysis/ME2104-A-4.pdf · Mechanical Analysis Gear trains ... Week 4 – Gears and gear trains Week 5 –

Ahmed Kovacevic, City University London 13

Page 14: Mechanical Analysis Gear trains - City, University of Londonra600/ME2105/Analysis/ME2104-A-4.pdf · Mechanical Analysis Gear trains ... Week 4 – Gears and gear trains Week 5 –

Ahmed Kovacevic, City University London 14

Page 15: Mechanical Analysis Gear trains - City, University of Londonra600/ME2105/Analysis/ME2104-A-4.pdf · Mechanical Analysis Gear trains ... Week 4 – Gears and gear trains Week 5 –

Ahmed Kovacevic, City University London 15

Examples of gear trains

Two stage compound

Gear train

Planetary gear train

Compound reverted

gear train

Planetary gear train

on the arm

Page 16: Mechanical Analysis Gear trains - City, University of Londonra600/ME2105/Analysis/ME2104-A-4.pdf · Mechanical Analysis Gear trains ... Week 4 – Gears and gear trains Week 5 –

Ahmed Kovacevic, City University London 16

Example 12 – Compound gear train

A gearbox is needed to provide an exact 30:1 increase in speed, while

minimizing the overall gearbox size. The input and output shafts should be in-

line.

Governing equations are:

N2/N3 = 6

N4/N5 = 5

N2 + N3 = N4 + N5

Specify appropriate teeth numbers.

N2 = 108; N3 = 18; N4 = 105; N5 = 21

Page 17: Mechanical Analysis Gear trains - City, University of Londonra600/ME2105/Analysis/ME2104-A-4.pdf · Mechanical Analysis Gear trains ... Week 4 – Gears and gear trains Week 5 –

Ahmed Kovacevic, City University London 17

Example 13 – Planetary gear train

The sun gear in the figure is the input, and it is driven clockwise at 100 rpm. The

ring gear is held stationary by being fastened to the frame.

Find the rev/min and direction of rotation

of the arm and gear 4.

n3 = - 20 rpm, n4 = 33.3 rpm,

Page 18: Mechanical Analysis Gear trains - City, University of Londonra600/ME2105/Analysis/ME2104-A-4.pdf · Mechanical Analysis Gear trains ... Week 4 – Gears and gear trains Week 5 –

Ahmed Kovacevic, City University London 18

Gear Force Calculation

Stress based on the Force acting

The force is caused by the

transmitted torque. That force

always acts along the pressure line.

The force induces stress

concentration on gear.

We need to answer:

How much power a pair of gears in

question can transfer?

Ft

F Fr

R

a

30[ ]

30

cos cos

t t

t

PPT F R F Nn R

PFF F

n R

a a

Page 19: Mechanical Analysis Gear trains - City, University of Londonra600/ME2105/Analysis/ME2104-A-4.pdf · Mechanical Analysis Gear trains ... Week 4 – Gears and gear trains Week 5 –

Ahmed Kovacevic, City University London 19

Basic gear stress calculation

Basic analysis of gear-tooth is

based on Lewis Equation which has

following assumptions:

» The full load is applied on the tip of a

single tooth (the worse case)

» Radial component is negligible

» The load is distributed uniformly

along the teeth width

» Friction forces are negligible

» Stress concentration is negligible.

Basic stress in

teeth

Power transmitted

tB

y

B

s

F

mbY

SP n D mbY

f

Page 20: Mechanical Analysis Gear trains - City, University of Londonra600/ME2105/Analysis/ME2104-A-4.pdf · Mechanical Analysis Gear trains ... Week 4 – Gears and gear trains Week 5 –

Ahmed Kovacevic, City University London 20

Correction factors

Basic stress must be corrected for: - Shocks, - Size effects, - Uneven load distribution, - Dynamic effects.

a s mB

v

a s mB

v

K K K

K

K K KP P

K

Ka – Application factor

Ks – Size factor

Km – Load distribution factor

Kv - Dynamic factor

Page 21: Mechanical Analysis Gear trains - City, University of Londonra600/ME2105/Analysis/ME2104-A-4.pdf · Mechanical Analysis Gear trains ... Week 4 – Gears and gear trains Week 5 –

Ahmed Kovacevic, City University London 21

Gear materials

Page 22: Mechanical Analysis Gear trains - City, University of Londonra600/ME2105/Analysis/ME2104-A-4.pdf · Mechanical Analysis Gear trains ... Week 4 – Gears and gear trains Week 5 –

Ahmed Kovacevic, City University London 22

Example 14 – Gear stress calculation Pinion A and gear B are shown in figure. Pinion A rotates

at 1750 rpm, driven directly by an electric motor. The driven machine is an industrial saw consuming 20 kW. The following conditions are given:

NP=20 m=3 mm Qv=6

NG=70 b=38 mm fs=1.5

np=1750 rpm Pow=20 kW

What is the centre distance? Compute the stress due to bending in the pinion and gear and find required Brinell hardness for this application.

SOLUTION:

Centre distance:

The pitch diameter of the pinion is: 3 20 60[ ] 0.06P PD mN mm m

The pitch velocity is: P PP

n D 1750 0.06v 5.5[m / s] 1090 ft / min

60 60

Transferred load (Tangential force): 60 60 20000

36381750 0.060

t

P P

PowF N

n D

( ) ( ) 903 1352 2 2

P G P GD D N Nc m mm

Page 23: Mechanical Analysis Gear trains - City, University of Londonra600/ME2105/Analysis/ME2104-A-4.pdf · Mechanical Analysis Gear trains ... Week 4 – Gears and gear trains Week 5 –

Ahmed Kovacevic, City University London 23

Example 14 – cont.

6

6

363894 10

0.003 0.038 0.34

363876 10

0.003 0.038 0.42

tBP

P

tBG

G

FPa

mbY

FPa

mbY

Basic bending stress is: pinion –

gear -

62.64 94 10 248a s mP BP

v

K K KMPa

K

Correction factors are: Application factor Ka=1.5

(from diagrams and tables) Size factor Ks=1.0

Load distribution Km=1.2

Dynamic factor Kv=0.68

Corrected pinion bending stress:

248 1.5 372s PS f MPa Allowable stress required for

this application:

From the diagram, any material with Brinel hardness higher than HB=400 will satisfy application

requirements.

From the diagram: YP=0.34 and YG=0.42

Page 24: Mechanical Analysis Gear trains - City, University of Londonra600/ME2105/Analysis/ME2104-A-4.pdf · Mechanical Analysis Gear trains ... Week 4 – Gears and gear trains Week 5 –

Ahmed Kovacevic, City University London 24

Example 15 – Industrial mixer

A gear train is made up of helical gears with their shafts in a single plane, such as the

arrangement in the vertical mixer shown. The gears have a normal pressure angle of 20°

and a 30° helix angle. The middle shaft is an idler. Gears AB and CD are engaged, the

others in the illustration are not in contact. The module of all gears is 2 mm. The table

gives the number of teeth per gear. Gear A exerts a load of 1200 N onto gear B. The

shaft containing gear A is driven by the motor at a speed of 200 rpm. All gears are Grade

2 and have been

hardened to

HB 350 and have

50 mm face widths.

Gear Number of Teeth

A 20

B 50

C 12

D 75

Find:

The normal load exerted by gear C on gear D

and the speed of gear D and the safety factor

for gear D based on bending and contact

stresses.

F=4066 N; nD=12.8 rpm; fsb=2.76; fsc=1

Coursework