Shaft LoadingShaft Loading

Lecture 17Lecture 17

Engineering 473Engineering 473Machine DesignMachine Design

Shaft Design IssuesShaft Design Issues

Shafts are one of the most commonly Shafts are one of the most commonly encountered machine components.encountered machine components.

MaterialMaterial

C

IC

e

RKS

q

SS

yt

ut

EnvironmentEnvironmentTemperature

CorrosionMagnetic

LoadsLoadsStationaryRotating

Press FitsKeywaysSplines

Bearings

InterfacesInterfaces

AssemblyAssembly

StiffnessStiffnessTolerancesTolerances

Mott, Fig. 5-1

ShaftShaft: Rotating machine elementthat transmits power.

Parallel Shaft Gear BoxParallel Shaft Gear Box

Shaft design spans most topics taught in a Machine Design Course.

Mott, Fig. 15-7

Design Detail Needed to Design Detail Needed to Specify a ShaftSpecify a Shaft

Mott, Fig. 15-5

Significant detail is required to completely specify the geometry needed to fabricate a shaft.

Common Shaft Common Shaft Loading MechanismsLoading Mechanisms

Spur GearsSpur Gears Unbalanced MassUnbalanced Mass

Helical GearsHelical GearsSpiral Bevel GearsSpiral Bevel Gears Belt DrivesBelt Drives

Chain DrivesChain Drives

Spur Gear LoadsSpur Gear Loads

tanφWW2

DTW

nP63,000T

tr

t

⋅=

=

⋅=

[ ][ ]

[ ][ ]

angle pressureφindiameter pitch Dlbin ueshaft torqTrpm speed rotationaln

hppower dtransmitteP

≡≡

⋅≡≡≡

Mott, Fig. 12-3

Helical Gear LoadsHelical Gear Loads

Mott Fig’s 10-3 & 10-4(a)

Helical Gear LoadsHelical Gear Loads(Continued)(Continued)

Mott Fig’s 10-4(a)

force AxialWforce RadialW

force dTransmitteWforce normalResultant W

angleHelix ψangle pressure Transverseφ

angle pressure Normalφ

x

r

t

n

t

n

≡≡≡≡

≡≡≡

cosψtanφtanφ tn ⋅=

Helical Gear LoadsHelical Gear Loads(Continued)(Continued)

Mott Fig 10-4

Helical Gear LoadsHelical Gear Loads(Continued)(Continued)

ψtanWW

ψ /cosφtan WW

2DTW

nP63,000T

tx

ntr

t

⋅=

⋅=

=

⋅=

Mott Fig 10-4

Chain Drive LoadsChain Drive Loads

Mott Fig 12-4

θ

Belt Drive LoadsBelt Drive Loads

Mott, Fig. 12-5

Net Driving ForceNet Driving Force

2DTF

FFF

n

21n

=

−=

21b FFF +=Total Bending ForceTotal Bending Force

Belt Drive LoadsBelt Drive Loads(Bending Force)(Bending Force)

Net Driving ForceNet Driving Force

2DTF

FFF

n

21n

=

−=

21b FFF +=Total Bending ForceTotal Bending Force

3.0FF

5.0FF

2

1

2

1

=

=

Tension RatioTension Ratio

(V-belts)

(Flat-belts)nB

nB

22

22

22

22

21

21

n

B

F 0.2FF 5.1F

2.0F3FF3FC

5.1F5FF5FC

FFFF

FFC

==

=−+=

=−+=

−+==

(V-belts)

(Flat-belts)

(V-belts)(Flat-belts)

Stationary LoadsStationary Loads

1F

1F

1F

1F

2F

2F

2F

2F

Bending Stresses Due to Bending Stresses Due to Stationary LoadsStationary Loads

3

2

32b

32b

33

23

22

32b

MM-θ tan

0Iθsin rM

Iθ cos rM

θσ

Iθ cosr M

Iθsin r Mσ

IcM

IcMσ

=

=/

/+/

/=∂∂

−=

−=

2M

3M

θ

3

2

θsin r cθ cosr cIII

3

2

3322

==

==

2c3c Eq. 1

Eq. 2

Bending Stresses Due to Bending Stresses Due to Stationary LoadsStationary Loads

2M

3M

θ

3

22c

3c

MMθ cos

MMθsin

θ cosθsin

MM

MM

θtan

2, Eq. with Combining

MMM

3

2

3

2

23

22

=

−=

=−

=

+=

3

2

32b

MM-θtan

Iθ cosr M

Iθsin r Mσ

=

−= Eq. 1

Eq. 3

Eq. 4Eq. 2

Bending Stresses Due to Bending Stresses Due to Stationary LoadsStationary Loads

2M

3M

θ

3

22c

3c

Iθ cosr M

Iθsin r Mσ 32

b −= Eq. 1

MMθ cos

MMθsin

MMM

3

2

23

22

=

−=

+= Eq. 3

Eq�s 4

Combining Eq�3 1,3, and 4

IrMM

σ

IrM

IrMσ

Iθ cosr M

Iθsin r Mσ

23

22

b

23

22

b

32b

⋅+−=

−−=

−=

Bending Stresses Due to Bending Stresses Due to Stationary LoadsStationary Loads

IrMM

σ23

22

maxb,

⋅+=

IrMM

σ23

22

minb,

⋅+−=

Mott, Fig. 5-3(e)

Torsional Torsional Stresses Due to Stresses Due to Stationary LoadsStationary Loads

2M

3M

3

2

1M

JrMτ 1=

r

time

τ

The torsional stress at a point will be constant under steady state conditions.

Axial Stresses Due to Axial Stresses Due to Stationary LoadsStationary Loads

Helical, worm, and spiral gears will generate axial loads in the shaft. Under steady state conditions, the axial stress from these loads will be constant.

AWσ x

x =

Mott Fig 10-4

Unbalanced Mass LoadsUnbalanced Mass Loads

Bending stresses in a shaft due to in-balance loads are complicated by whether the rotational speed is lower or higher than the critical speeds of the shaft. In practice, the in-balance loads are minimized by balancing the shaft and attached components as a system. Rotordynamics theory is required if the magnitudes of the stresses at a particular operating speed is required.

Synchronous WhirlSynchronous Whirl(Due to Unbalanced Mass)(Due to Unbalanced Mass)

Thomson, Fig. 3.4-2

( )( ) ( )222

2

scωmωk

φ-ωtcosmeωx+−

=( )

( ) ( )222

2

scωmωk

φ-ωtsinmeωy+−

=

( ) ( )222

22s

2s

cωmωk

meωyxOS+−

=+= 2mωkcωφtan

−=

φe

m=unbalanced mass

AssignmentAssignment(Problem 1)(Problem 1)

The shaft rotating at 550 rpm carries a spur gear B having 96 teeth and a diametral pitch of 6. The teeth are of the 20o, full-depth, involute form. The gear receives 30 hp from a pinion directly above it.

Compute the torque delivered to the shaft and the tangential and radial forces exerted on the shaft by the gear.

Mott, Fig. 12-20

AssignmentAssignment(Problem 2)(Problem 2)

Mott, Fig. 12-21

The shaft rotating at 200 rpm carries a 20-in-diameter flat-belt pulley at A that receives 10 hp from below.

Compute the torque delivered by the pulley to the shaft and the force exerted on the shaft by the pulley.

AssignmentAssignment(Problem 3)(Problem 3)

The shaft is rotating at 650 rpm and receives 7.5 hp through a flexible coupling. The power is delivered to an adjacent shaft through a single helical gear B having a normal pressure angle of 20o and a helix angle of 15o.

(a) draw free-body diagrams for the shaft in both the vertical and horizontal planes, (b) find the magnitude of the forces shown, (c) draw the shearing force and bending moment diagrams for the shaft in both planes.

DB=4.141 in

Mott, Fig. 12-29

AssignmentAssignment(Problem 4)(Problem 4)

The shaft rotating at 480 rpm carries a 10-in-diameter chain sprocket at C that receives 11 hp from a mating sprocket below and to the left as shown.

Compute the torque delivered to the shaft by the sprocket and the total force exerted on the shaft by the sprocket. Resolve the force into its horizontal and vertical components, and show the net forces acting on the shaft at C in the vertical and horizontal directions.

Mott, Fig. 12-22