FME 461: ENGINEERING DESIGN II

EXAMPLE-DESIGN OF INPUT GEAR-SHAFT

IDENTIFICATION OF EXTERNAL LOAD CARRIED BY GEAR SHAFT

The loading diagram of the gear shaft is shown below:

The specification of performance requirements for shaft is given below in terms of power to be transmitted, and the speed of transmission:

Performance specification required=30 Kw*1450 r.p.m.(1)

SELECTION OF MATERIAL FOR PART

Material selected is medium carbon steel, to British Standard specification BS 970:080M040(H&T), whose mechanical properties are:

(a)Tensile yield strength

(b)Ultimate tensile strength

(c)Elongation %=16%

(d)Hardness number=179-229 HB

The material is a ductile material, having an elongation % of 16 %>5%.

DETERMINATION OF EXTERNAL LOADS CARRIED BY GEAR SHAFT

Torque required to transmit power at the speed specified

The relationship between power and torque transmitted is given by the equation:

Where

Torque transmitted in N-m

Angular velocity in radians/sec

Consequently

and N-m(2)

Where

Angular velocity of shaft in revs/min

Substituting for performance specification required from (1) into (2)

N-m(3)

Torque to be transmitted becomes

(4)

TRANSVERSE LOAD ON GEAR SHAFT ARISING FROM THE TORQUE TRANSMITTED

TANGENTIAL FORCE ON GEAR TEETH REQUIRED TO TRANSMIT TORQUE SPECIFIED

As shown in the gear set diagram, the tangential force on the gear teeth, is the gear tooth force operating at the pitch diameter of the driving gear on the input shaft.

The torque transmitted is then given as function of the tangential tooth force and the pitch diameter of the gear as below:

Therefore Newtons

Substituting for the pitch diameter of the input gear and T=197.6 N-m

9880Newtons(5)

RESULTANT FORCE ON GEAR TOOTH REQUIRED TO TRANSMIT TORQUE SPECIFIED

The tangential force is the component of the resultant gear tooth force that gives rise to the transmitted torque. It acts at the pitch point, along the tangent to the pitch circle diameter of the gear tooth. The resultant gear tooth force is normal to the tooth surface, and therefore inclined to the tangent to pitch circle diameter at an angle equal to the pressure angle of the gear tooth.

The resultant force on the gear tooth is given by the equation

Substituting for =9880 N and =20o

=10154 N

Newtons(6)

DETERMINE BENDING MOMENT LOADS ON THE INPUT SHAFT

LOADING DIAGRAM OF THE INPUT SHAFT-CONSIDERED AS A SIMPLY SUPPORTED BEAM SUBJECT TO TRANSVERSE POINT LOADS

The loading diagram is shown at Appendix A and reproduced below:

Reactions at the simple supports are then given by

Substituting for R=10154 Newtons

(7)

SHEAR FORCE DIAGRAM

The shear force diagram is shown at appendix a. however, the direct shear stress induced by the shear force reaches its maximum value at the centre of the shaft, while the stresses caused by bending and torsion reach their maximum values the surface of the shaft. the effect of the direct shear stress is therefore ignored.

BENDING MOMENT DIAGRAM FOR INPUT SHAFT

The bending moment diagram for a straight beam with intermediate load and simple supports is then as shown below

The maximum bending moment is then given by

(8)

EXTERNAL LOAD ON THE GEAR SHAFT

The external load on the input gear shaft then reduces to combined torsion and bending, where the torsion and bending loads are:

(9)

(10)

Determination of stresses induced by the external loads

STRESSES DUE TO COMBINED TORSION AND BENDING OF SHAFT

In this situation, there is a plane stress at the location of maximum bending moment as shown below

The stress elements are:

Simplifying the loading situation of the input shaft into a static load which remains constant in spite of the rotation of the shaft, determine the significant stress at the location of highest stresses in terms of principal and maximum shear stress arising from the loads on the member

Applying maximum shear stress theory of failure

The MAXIMUM SHEAR STRESS theory of failure states:

When Yielding occurs in any material, the maximum shear stress at the point of failure equals or exceeds the maximum shear stress when yielding occurs in the tension test specimen.

STRESS ELEMENTS IN THE PLANE STRESS SITUATION

The plane stress situation is the stress situation in which the stress elements are , and the stresses on the z-axis are zero,

MAXIMUM SHEAR STRESS IN TERMS OF PLANE STRESS ELEMENTS

The maximum shear stress is the significant stress in this situation and is given by the expression

(11)

Maximum shear stress in the case of plane stress situation with

THE GENERAL CASE OF PLANE STRESS SITUATION WITH

Substituting for into the equation for maximum shear stress yields

==

=(12)

Stresses induced in gear shaft by the external loads

SOLID CIRCULAR SHAFT SUBJECT TO BENDING AND TORSION

In the case of combined torsion and bending, the stress elements in the plane stress situation are:

MAXIMUM SHEAR STRESS IN ELEMENT IN TERMS EXTERNAL LOADS

Substituting for in equation for maximum shear stress yields the expression for maximum shear stress in terms of load and dimension of element as shown below:

(13)

SHEAR STRENGTH OF CHOSEN (DUCTILE) MATERIAL

The yield strength in shear of ductile materials such steel is predicted to be half the tensile yield strength by the maximum shear stress theory of failure, and the shear yield strength of such materials can therefore be derived from the tensile yield strength

Therefore (14)

Where

COMPARE SIGNIFICANT STRESS WITH STRENGTH: DESIGN EQUATION

Design equation then becomes

=OR

Where

,

The design equation then becomes

(15)

SOLVING DESIGN EQUATION

Substituting the TORQUE and BENDING MOMENT loads into design equation

(16)

(17)

Substituting for yield strength of chosen material and the factor of safety

Factor of safety =2.5 and Tensile yield strength

(18)

SELECT SHAFT SIZE FROM PREFERRED METRICRANGE

Select the shaft size to be used form the nearest size in the range of preferred metric sizes[1] (1,1.2,1.6,2,2.5,3,4,5,6,8,10,12,16,20,25,30,35,40,45,50,55,60,65,70,75,80,90,100 mm.)

Nearest shaft size selected is 30 mm.

REVIEW DESIGN

Determine the actual factor of safety resulting from the use of the selected standard shaft size

Rewriting the design equation in terms of the factor of safety

But

Substituting

(16)

(17)

And

Substituting

Factor of safety =3.5

APPENDIX A: LOADING, SHEAR FORCE, AND BENDING MOMENT DIAGRAMS

LOADING DIAGRAM

SHEAR FORCE DIAGRAM

BENDING MOMENT DIAGRAM

APPENDIX B[2]: MECHANICAL PROPERTIES OF SOME STEELS

Material / British Standard[3] / Production process / Maximum section size, mm. / Yield Strength Mpa / Tensile Strength, Mpa / Elongation % / Hardness Number, HB
0.20C / 070M20 / HR[4] / 152 / 215 / 430 / 22 / 126-179
254 / 200 / 400 / 20 / 116-170
CD[5] / 13 / 385 / 530 / 12 / 154
76 / 340 / 430 / 14 / 125
0.30C / 080M30 / HR / 152 / 245 / 490 / 20 / 143-192
254 / 230 / 460 / 19 / 134-183
CD / 13 / 470 / 600 / 10 / 174
63 / 385 / 530 / 12 / 154
H&T[6] / 63 / 385 / 550-700 / 13 / 152-207
0.40C / 080M40 / HR / 150 / 280 / 550 / 16 / 152-207
CD / 63 / 430 / 570 / 10 / 165
H&T / 63 / 385 / 625-775 / 16 / 179-229
0.50C / 080M50 / HR / 150 / 310 / 620 / 14 / 179-229
CD / 63 / 510 / 650 / 10 / 188
H&T / 150 / 430 / 625-775 / 11 / 179-229
1Cr / 530M40 / H&T / 100 / 525 / 700-850 / 17 / 202-255
29 / 680 / 850-1000 / 13 / 248-302
1.5MnMo / 605M36 / H&T / 150 / 525 / 700-850 / 17 / 202-255
29 / 755 / 925-1075 / 12 / 269-331
1.25NiCr / 640M40 / H&T / 152 / 525 / 700-850 / 17 / 202-255
102 / 585 / 770-930 / 15 / 223-277
64 / 680 / 850-1000 / 13 / 248-302
29 / 755 / 930-1080 / 12 / 269-331
3NiCr / 653M31 / H&T / 64 / 755 / 930-1080 / 12 / 269-331
680 / 850-1000 / 12 / 248-302
1CrMo / 708M40 / H&T / 150 / 525 / 700-850 / 17 / 201-255
13 / 940 / 1075-1225 / 12 / 311-375
3CrMo / 722M24 / H&T / 152 / 680 / 850-1000 / 13 / 269-331
755 / 930-1080 / 12 / 269-331
2.5NiCrMo / 826M40 / H&T / 150 / 755 / 925-1075 / 12 / 269-331
850 / 1000-1150 / 12 / 293-352
1020 / 1150-1300 / 10 / 341-401
3NiCrMo / 830M31 / H&T / 254 / 650 / 850-1000 / 13 / 248-302
152 / 680 / 850-1000 / 12 / 248-302
64 / 940 / 1080-1240 / 11 / 311-375
1.5MnNiCrMo / 945M38 / H&T / 152 / 525 / 690-850 / 17 / 201-255
64 / 680 / 850-1000 / 13 / 248-302
29 / 850 / 1000-1160 / 12 / 293-352

APPENDIX C: STEEL APPLICATION AND HEAT-TREATING GUIDE[7]

USE
OR
PART / Low-Carbon / Medium-Carbon / High-Carbon
Plain
Carbon
Or
Lean
Alloy / Alloy / Plain
Carbon
Or
Lean
Alloy / Medium
Alloy / Rich
Alloy
C 1020
C 1117 / A2315-20
3115-20
4615-20
5120
8620 / C1040-50 / A3140-50
4140-50
5145
8640-50
8740-50
6145 / A 4340
3250 / Oil
Hard-ening
Tool
Steel / Water
Hard-ening
Tool
Steel
Arbors / N,T / T / T
Armature shafts / T / T / T
Axles / C / C / N,T,A, / S,T, / T / T
Ball races / C / S / T / T / T
Bolts and studs / T,A / T / T
Bushings / C / C / T
Cams / C / T / T
Camshaft / C / C / T / T
Cant dogs / T
Chain Links / T
Chain Pins / C / C
Chuck Jaws / C / T / T
Chuck screws / N,A / T
Clutches / T / T
Collets / T / T
Connecting Rods / T / T
Crankshafts / N,S,A / S,T / S,T
Drift Pins / N / T
Engine bolts / C / C / N,T / T
Gears / C / C / N,S,T,A / S,T / S,T / T
Guide Pins / T / T
Mandrels / C / C / T
Pinions / C / C / N,S,T / S,T / S,T / T
Pins / C / T / T
Pistons / C / T
Pump Shafts / N,T,A / T
Rollers / C / C
Rolls / C / C / S / S,T / S,T / T / T
Lead Screws / N,A / T
Set Screws / T / T
Spindles / C / C / S,T,A, / S,T, / S,T / T / T
Stay Bolts / N / A
Thrust washers / C / T
Turbine Shafts / N,T,A / T
Turnbuckles / T / T
U bolts / T / T
Universal Joint Pins / C / C
Universal joint bodies / N,T,A, / T / T
Worm Gears / C / C / S,T / S,T

N=Normalised; C= Case-hardened; S= Surface-hardened; T= Through-hardened; A= As-rolled

1

[1] Shigley Joseph, Mechanical Engineering Design, First Metric Edition, , McGraw Hill, 1986,page 660

[2] Shigley, Joseph E., Mechanical Engineering Design, pp. 664, McGraw-Hill Inc., 1986

[3] British Standards Institution, BS 970: Part 1: 1983

[4] HR-Hot rolled and normalised

[5] CD-Cold drawn

[6] H&T-Hardened and tempered

[7] pp. 10, ASME Handbook, Metals Engineering-Processes, McGraw-Hill Book Company, 1958