Curs 5 Hub & shaft (friction) 2005 - · PDF fileTAPER RINGS - Design solutions Ring nut...

1

nov. 2005 1/32 Traian CICONE

UNTHREADED FASTENERSFriction assemblies - torque transmitted by frictionB

Ff /2

Fa dm 2α

p

SHAFT

HUB

Mt

L

Taper shaft Taper bore

SHAFT

HUB

Tighteningbush

Taper rings(Clamp ring)

Mt

Fs

Fs

FnFf

FnFf

Elastic collar (collet clamp)

dmax

SHAFT

maximuminterference

Dmin

HUB

Interference (Shrink) fit

Internal tightening

External tighteningExternal tightening

External tightening

with intermediary elements


GENERALITIES

1. Friction coefficient, µ = key factor

2. Operating conditions must be taken into account

3. µs ≈2µk

4. Additional safety elements: keys, pins

5. Low roughness; hardened and polished surfaces

Prudence ! Most appropriate value µ∈ µmin÷ µmax

Mf =Mtc=β Mt


TAPER SHAFT

2


TAPER SHAFT

Simple Low stress concentration Hub-to-shaft centering

Technology of tapered hub Useful only on end parts of shafts

Ff /2

Fa dm2α

SHAFT

HUB

Mt

Principle of operationWedge effect

p

Keyed tapered bushing


TAPER SHAFT


TAPER SHAFT - Forces

m

tcfn d

MFF

µµ2

==

( )αµα cossin += na FF

ϕ=atan(µ)( ) ( )

ϕϕα

ϕϕα

sinsin2

cossin +

=+

=m

tcna d

MFF

AssumptionsUniform pressure distribution

Forces act on dm

Ff/2 Fn/2

Mt

Ff/2 Fn/2

p

2αdm

SHAFT

HUB

L

Mt

pFn/2

α

Ff/2

Fa

3


TAPER SHAFT - Cone angle

- α<ϕ is self-locking

- Easy and quick dismounting is required

α > ϕ − 2α > 10°

( )ϕ

αϕsin

sin2 −=

m

tca d

MFDismounting

)(222 mm

tc

m

nk p

LdM

LdF

===πµπ

σ

σak=100-150 MPa.

Bearing pressure

Check the preloading system (threaded assembly)

2αdm

SHAFT

HUB

L

Mt

pFn/2

α

Ff/2

Fa


TAPER RINGS

dSHAFT

HUB

Tighteningbush (collar)

D

Principle of operation Wedge effect

Standardized

Cylindrical hub & shaft

Higher load capacityGreater overall dimensions (esp. radially)

Additional parts


TAPER RINGS – 3D Models

4


TAPER RINGS - Forces I

=−−=−−−

0NFN0NNFF

1n

1n1a

αµαααµµ

sincossincos

α

HUBC

SHAFT

+

−+

= µαµααµα

sincoscossin

11 na FF

( )[ ] ( )µαµϕα 2tantan 111 +≅++= nna FFF

( ) µαϕαϕα +=+≈+ tantantantan

Fn1

Ff1

Fa1Ff2

N

AssumptionsUniform pressure distribution

Coulombian friction force

Zero initial clearance C=0Unique friction coefficient


C

Fa2

α

Ff1

Fn1

Fa1

SHAFT

HUB Fn1

Ff1

d

D

αµ tan2 1112 nnaa FFFF ≅−=

12 aa FF <

TAPER RINGS - Forces II


dSHAFT

HUB

Tighteningbush

Fa2

Fa1

D

dM

F fn µ

11

2=

TAPER RINGS - Forces for z pairs

µµα 2tan2 1

1+

=d

MF f

a

qFFFF nna

n 112

2 2tantan

2tan=

+=

+=

µαα

µα

dM

F fn µ

22

2=

µαα

2tantan

+=q

( )qdFMMMM n1f2f1ftc +=+== 12

µ

( )qqdFqqdFM

z

nnt −−

=+++=11

21

2 12

1 µµ K ( )zt

a qqq

dMF

−−

=112

5


( ) 1123 43.11 === ≈++= zt

zt

zt MMqqM

TAPER RINGS - Design solutions

( ) 112 33.11 === =+= zt

zt

zt MMqM

SHAFT

HUB

d D

HUB

1122 35.12 ==× =⋅= zt

ztt MMM

In practice Mt (z=2)=1.5 Mt (z=1) 5.0=q12.014 ≈°≈ µα


TAPER RINGS - Design solutions


TAPER RINGS - Design solutionsRing nut

Disassembly hole

6


TAPER RINGS - Compensating force

Assumptions

Fn1

Ft

Ftα

HUB

C Fn1

( ) ( ) ( )ϕαϕαϕα +=+=+= tan2tan2tan10m

tn dEACFFF

Clearance is compensated by the tangential stretches (slotted rings) ( )[ ] CdCd mmt πππ =−+=∆ 2

21

l

AEdF mt

tπ

=∆lElastic deformation (Hooke’s law)

LdMp tc

km 22πµ

σ ==Bearing pressure


Mt

Fs

Fs

FnFf

FnFf

CLAMPING (ELASTIC) COLLARS


Mt

Fs

Fs

FnFf

FnFf

Fn

Fs

a d/2

A

Ff

ELASTIC COLLAR Strength calculation

LdMtc

k 22πµ

σ =Bearing pressure

addFF ns 2+

=d

MF tcn µ

2=

( )adMF tc

s 2+=

µTightening force

7


INTERFERENCE FITS

d

SHAFT

interference

D

HUB

Fa

Press-fitted connection

Internal tightening

Without intermediary elements

Assembly

Without temperature effects - press fitting

With temperature effectsHeating the hub - shrink fit

Cooling the shaft - expansion fit

Hydraulically

Shrink - fit Press - fit


INTERFERENCE FITS - Assembly


INTERFERENCE FITS

Low overall dimensions and mass

Relatively low cost

High load capacity, even in the case of alternating loads or shock

Hocate axially the shaft and supports high axial forces

Perfect centering of the hub on shaft

Not recommended for frequent dismounting

Accurate manufacturing (grinding or lapping) and sometimes selective mounting

Special technology for mounting and dismounting

Plus

Minus

8


d

SHAFT

Dd*

p

L

di

De

HUB

INTERFERENCE FITS - schematic


INTERFERENCE FITS - minimum interference

LdFac

k πµσ =1Axial load

LdMtc

k 222πµ

σ =Torque

Ld

FdM

actc

k πµσ

22

3

2+

=Torque + Axial load

mink p=σ

Assumptions

Thick-walled cylindersd*

LFa

Mt

Uniform pressure distributionp


INTERFERENCE FITS - maximum interference

22

22

DDDD

e

ekmaxt −

+=σσTangential stress

kr σσ −=Radial stress

22Yrmaxt

maxτ σσσ<

−=Maximum shear stress theory

−= 21

2 e

Ymaxk D

Dc

σσMaximum permissible stress 3...2.1, =< ccY

maxkσσ

d=DShaft

Hub

σt

di

Deσt max

σr σr

σr

σt

σt

σk

9


INTERFERENCE FITS - deformation

Deformation (Interference) dEK

EK

k

+=

2

2

1

1σδ

Lamé coefficients( )( ) 12

2

1 11 ν−

−+

=ddddK

i

i ( )( ) 22

2

2 11 ν+

−+

=e

e

DDDDK

dEK

EK

k

+=

2

2

1

1minmin σδ d

EK

EK

k

+=

2

2

1

1maxmax σδ

tRcor uu ++= δδCorrected deformation( )212.1 yyR RRu +=

( ) ( )[ ]022011 ttttdut −−−= αα


Smax maximuminterference

Smin minimuminterference

Dd

dmin

dmax

Dmin

Dmax

HUB

SHAFT

TOLERANCES

minmin Scor <δ

maxmax corS δ<


INTERFERENCE FITS - Temperature

ca t

dCStt ∆+

++=

2

max0 α

Only the hub is heated

Clearance for assembly Ca

Heating

• t < 100°C on heating plate

• t < 250°C in oil

• t > 250°C in oven

Assumptions

Cooling during manipulation ∆tc

10


INTERFERENCE FITS - Design recommendations

15÷30°R

•Avoid the press-fitted connections in blind holes (air compression inside the blank hole increases the stresses and the resistance to mounting).

•Additional safety components are recommended to avoid damages due to loosening.

•Use lubricants to easy the engagement.

•Surface coatings are recommended to increase the resistance to wear by fretting.

•Facilitate dismounting.

•Avoid double centering.

•Facilitate the engagement, providing long chamfers or rounded edges (Figure).


NEW, INOVATIVE, SOLUTIONS

(A) Elastic radial rings (A) Fastener with tapered rings (UNI-KEY)


UNI-KEY

11


A

A

A - A

Fax Fax

Fr

p

Fax Fax

Fr p

α

N

Ff

N

α

N

Ff

Ff Ff

Fr/2 Fr/2

α Assumptions

Uniform pressure distribution

Coulombian friction force

Zero initial clearance C=0Unique friction coefficient

( )ϕ

ϕαααcos

cossincos −=+= NFNF fax

( ) ( )ϕαµϕα −=

−=

tantan2 dMFF fr

ax[ ] ( )ϕ

ϕαααcos

sin2cossin2 −=−= NFNF fr 2

dFM rf µ=

UNI-KEY Calculation


YesNoNoNoNoYesYesNoEASYPOSITIONADJUSTING

YesYesYesNoNoNoNoYesACCURATE ANGULARPOSITION

YesNoNoYesYesYesNoNoFATIGUE(CYCLICLOADING)

YesYesYesNoNoYesYesNoEASY HUBREPLACING

YesNoNoYesYesYesYesYesAXIALLOCATIONPROVIDED

YesYesYesYesYesNoNoNoAXIALLYCOMPACT

YesNoNoYesNoYesNoYesLARGEAXIALLOADS

YesYesYesYesNoYesNoNoHIGH TORQUE CAPACITY

TAPERSHAFT/BUSH

KEYSPLINESHRINKFIT

PRESSFIT

TAPER (CLAMP)

RING

GRUBSCREWPIN

METHOD

CRITERIA

OVERVIEW on HUB-ON-SHAFT CONNECTIONS

Curs 5 Hub & shaft (friction) 2005 - · PDF fileTAPER RINGS - Design solutions Ring nut...

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Transcript of Curs 5 Hub & shaft (friction) 2005 - · PDF fileTAPER RINGS - Design solutions Ring nut...