DESIGN PROCEDURE FOR INPUT SHAFT

FME 461: ENGINEERING DESIGN IIEXAMPLE-DESIGN OF INPUT GEAR-SHAFTIDENTIFICATION 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 SHAFTTorque 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 TRANSMITTEDTANGENTIAL 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

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

9880 Newtons

(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 SHAFTLOADING DIAGRAM OF THE INPUT SHAFT-CONSIDERED AS A SIMPLY SUPPORTED BEAM SUBJECT TO TRANSVERSE POINT LOADSThe 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 DIAGRAMThe 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 SHAFTThe 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 SHAFTThe 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 SHAFTIn 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 SITUATIONThe 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 ELEMENTSThe 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) MATERIALThe 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

EMBED Equation.3Where

,

The design equation then becomes

EMBED Equation.3

(15)SOLVING DESIGN EQUATIONSubstituting the TORQUE and BENDING MOMENT loads into design equation

(16)

(17)

EMBED Equation.3Substituting for yield strength of chosen material and the factor of safetyFactor of safety =2.5 and Tensile yield strength

EMBED Equation.3

(18)SELECT SHAFT SIZE FROM PREFERRED METRIC RANGESelect the shaft size to be used form the nearest size in the range of preferred metric sizes (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 DESIGNDetermine 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 =3.5APPENDIX A: LOADING, SHEAR FORCE, AND BENDING MOMENT DIAGRAMS

LOADING DIAGRAM

SHEAR FORCE DIAGRAM

BENDING MOMENT DIAGRAM

APPENDIX B: MECHANICAL PROPERTIES OF SOME STEELSMaterialBritish Standard

Production processMaximum section size, mm.Yield Strength MpaTensile Strength, MpaElongation %Hardness Number, HB

0.20C070M20HR

15221543022126-179

25420040020116-170

CD

1338553012154

7634043014125

0.30C080M30HR15224549020143-192

25423046019134-183

CD1347060010174

6338553012154

H&T

63385550-70013152-207

0.40C080M40HR15028055016152-207

CD6343057010165

H&T63385625-77516179-229

0.50C080M50HR15031062014179-229

CD6351065010188

H&T150430625-77511179-229

1Cr530M40H&T100525700-85017202-255

29680850-100013248-302

1.5MnMo605M36H&T150525700-85017202-255

29755925-107512269-331

1.25NiCr640M40H&T152525700-85017202-255

102585770-93015223-277

64680850-100013248-302

29755930-108012269-331

3NiCr653M31H&T64755930-108012269-331

680850-100012248-302

1CrMo708M40H&T150525700-85017201-255

139401075-122512311-375

3CrMo722M24H&T152680850-100013269-331

755930-108012269-331

2.5NiCrMo826M40H&T150755925-107512269-331

8501000-115012293-352

10201150-130010341-401

3NiCrMo830M31H&T254650850-100013248-302

152680850-100012248-302

649401080-124011311-375

1.5MnNiCrMo945M38H&T152525690-85017201-255

64680850-100013248-302

298501000-116012293-352

APPENDIX C: STEEL APPLICATION AND HEAT-TREATING GUIDE

USE

OR

PARTLow-CarbonMedium-CarbonHigh-Carbon

Plain

Carbon

Or

Lean

AlloyAlloyPlain

Carbon

Or

Lean

AlloyMedium

AlloyRich

Alloy

C 1020

C 1117A2315-20

3115-20

4615-20

5120

8620C1040-50A3140-50

4140-50

5145

8640-50

8740-50

6145A 4340

3250Oil

Hard-ening

Tool

Steel Water

Hard-ening

Tool

Steel

ArborsN,TTT

Armature shaftsTTT

AxlesCCN,T,A,S,T,TT

Ball racesCSTTT

Bolts and studsT,ATT

BushingsCCT

CamsCTT

CamshaftCCTT

Cant dogsT

Chain LinksT

Chain PinsCC

Chuck JawsCTT

Chuck screwsN,AT

ClutchesTT

ColletsTT

Connecting RodsTT

CrankshaftsN,S,AS,TS,T

Drift PinsNT

Engine boltsCCN,TT

GearsCCN,S,T,AS,TS,TT

Guide PinsTT

MandrelsCCT

PinionsCCN,S,TS,TS,TT

PinsCTT

PistonsCT

Pump ShaftsN,T,AT

RollersCC

RollsCCSS,TS,TTT

Lead ScrewsN,AT

Set ScrewsTT

SpindlesCCS,T,A,S,T,S,TTT

Stay BoltsNA

Thrust washersCT

Turbine ShaftsN,T,AT

TurnbucklesTT

U boltsTT

Universal Joint PinsCC

Universal joint bodiesN,T,A,TT

Worm GearsCCS,TS,T

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

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

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

British Standards Institution, BS 970: Part 1: 1983

HR-Hot rolled and normalised

CD-Cold drawn

H&T-Hardened and tempered

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

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