Interactive computer program for the selection of interference fits

Interactive computer program for the selection of interference fits A G Lagodimos and A d Scarr*

A procedure and an interactive computer program that allow the computer-aided selection o f standard manufacturing limits for interference fit joints are presented in detail. The mathematical model describing the problem is outlined together with the algorithms developed for its solution. The relative merits o f the computer-aided selection compared with existing manual methods are discussed. A numerical example is provided to demonstrate the capabilities o f the proposed computer program.

of CADCAM techniques suggested that an alternative form that uses digital computer capabilities to perform the selection task should be investigated. The work presented in this paper forms part of a wider investigation undertaken on behalf of the British Standards Institution and the International Standards Organization (ISO) aimed at updating the current standard on limits and fits 1. This paper presents a computer program developed for assisting designers in the selection of interference fits. A case study showing how the program can be used in practice is also provided.

interactive computing, interference f/ts, numerical iteration

NOTATION

The selection of limits and fits is often regarded as a minor design activity. The existence of tables giving values of manufacturing tolerances and deviations I may lead one to think that the designer has merely to refer to them and that the s~lection itself is then just a trivial operation. This is not the case in practice. Engineers are aware that the selection of correct limits and fits demands the consideration of both the functional and operational requirements of the product, in addition to the information contained in the standard tables, if components of the required quality are to be produced. Dealing with the particular case of interference fits, standards have been developed in the past aimed at assisting the designer in selecting the appropriate fit. These standards may be classified according to their form as follows:

da di df ds E

•

recommendation form 2 - recommendations based on experience gained from industrial practice (these are highly customized and this form was soon abandoned) as a way of handling non-trivial applications • equation-flow form 3 - a set of equations to be solved in sequence to calculate the values of the functional parameters (a pre-requisite for this method is the adaptation of linear and simple mathematical models; this form is error prone and time consuming and is little used today) • nomographic form 4 - a set of diagrams to be followed in sequence to calculate the values of the functional parameters (this overcomes some of the limitations of the equation-flow form and has been widely used in European industry, but it remained time consuming and error prone) The increasing rate of computer applications in industry during the past decade together with the recent evolution HAF Technology ResearchCentre (KETA), Palio Faliro, Athens, Greece *Cranfield Institute of Technology, Cranfield, Beds MK43 0AL, UK

272

e Ft Pf Ps (Pf)el-pl (Pf)s

qa qi qs Ra Rz S

outer diameter of hub inner diameter of shaft nominal joint diameter plasticity diameter Young's modulus deformation of part transmissible load interface pressure pressure causing plastic yield initiation pressure at the limit where one part deforms elastically and the other elastic-plastically maximum pressure permitting elastic deformation of both parts hub diameter ratio (df/da) shaft diameter ratio (di/df) plastic ratio (ds/df) centre-line-average value of surface finish mean peak-to-valley height of surface finish surface smoothing (0.8Rza + 0.8Rzi)

6 6el-pl

effective interference effective interference occurring at the limit where one part deforms elastically and the other elastic-plastically measurable interference (6 +S) ~m ~n~ax, ~rnin calculated border (effective) interference limits effective interference that permits elastic 6s deformation of both parts ~ t 6max, 6min effective interference limits of a selected fit radial stress at the border between the elastic Ors and the plastic region of the shaft deformation yield stress Os P Poisson ratio

0010-4485/84/050272-07 $03.00 © 1984 Butterworth & Co (Publishers) Ltd

computer-aided design

izo

are the dimensions and materials of the component parts, the magnitude of the interference and the value of the friction at the interface. For a given assembly configuration the designer is required to determine the appropriate interference. By assigning manufacturing limits to the hole and the shaft a maximum and a minimum value of interference are implicitly specified. For the fit to function in accordance with the design requirements two conditions have to be satisfied:

i zo ~17

I00

~~ 1' i5~

oeo

-~"\ ~ ~~

21

ioo

o~

~--~_ ~ ~-Ib'OSO

g

g

c

_

o.~-I¢

o~

# o~o

~

oao

• The assigned maximum interference must be such that it does not generate unacceptable stresses in either of the mating parts. • The assigned minimum interference must be sufficient for the transmission of the required load.

% ' ............ o~o

o~o

o4o

020

o

o ~o

•;

040

OSO

oeo

i oo

~

Figure I. The plasticity ratio o f hub and shaft as function o f the applied pressure

The use of standard tables does not provide sufficient information to enable interference fits to be selected which satisfy the requirements stated above. It was therefore necessary to develop a logical algorithm for this purpose. This is presented in the following section.

When not specifically otherwise noted, subscripts denote a the hub i the shaft ac a maximum acceptable value r a required value Combinations of the above are often used. THE

SELECTION

PROCEDURE

The selection procedure proposed consists of the following generalized stages:

PROBLEM

•

specification of the ultimate stresses for the mating parts that could be considered as acceptable • determination of the load to be transmitted by the assembly and of the stresses generated by the application of the load

Interference fits are obtained by assembling holes with shafts which are of slightly larger diameter. The initial oversize of the shaft is referred to as interference. After assembly a common interface pressure is produced, generating stresses that force the shaft to contract and the hub to expand in order to accommodate the interference. Frictional forces at the interface allow for loads which are applied to one part to be transferred to the other, making interference fits an attractive alternative to splines and keys in load transmission. The magnitude of the load that a fit can transmit is a function of various factors. The most important of these

Comparison of the results of these two stages shows whether an interference fit is viable for the given configuration. In cases where the stress induced by the required load exceeds the ultimate limits, two alternatives can be considered: either to change one or more design parameter(s) or to look for another assembly method. The next two stages of the selection procedure are: 1.20

1.20 i

,.ool

/0"2

/

J

0.4 ~

/

/~,~

a80

1.00

-

\

,~ 0.60

\

0.~

.~ ~

~ /

,

/ / / /

°:I b° a6o

!

'\

- - 0.5 -

0.40

_

~

~

0.6 / --

/

,

I 0.6 ~ 1

0.60 ~Ylb"

/

/ / /

- 0.40

0.7

- - 0.7 I

0.20 ~ 0.8

S I

"

',

- -

0.~

/

I - - 0.9

0 1.50

,

/

~ I

1.00

0.50

i~l~

0

~ ~ 0.50

~

I /

1.00

-

0.~0

/

, ,

0.9

~

Z ~ 1.50

2.00

t 2.50

t_~-

3.00

o

3.BO

~= ~

df ~o df ~=i Figure 2. The non-dimensional deformation o f hub and shaft as function of the applied pressure

volume 16 number 5 september 1984

273

the graphic facilities of the Prime-550 computer at the HAF Technology Research Centre. To carry out the computer-based interference fit selection procedure, the equations and the system of simultaneous equations obtained from the Lundberg model had to be solved. The Newton Raphson iteration method incorporating penalty functions was chosen for this purpose. Details are available in the literature 6 . The basic equations governing the deformation of parts are given in Appendix 1.

3.10

2.80

2.50 ._o ~ .~

2.20

.~

COMPUTER IMPLEI~IENTATION

~. E ~ ._

1.90

g

:~

1.60

1.30

1.00 0

0.15

030

0.45

0.60

0.75

0.90

Diameter ratio q~

Figure ~. The mex~mum plestic retio o f the hub as function o f the diemeter ratio •

•

calculation of the border interference limits within which the maximum and minimum interference values should lie, corresponding to the selected ISO fit selection of the ISO fit with maximum and minimum interference such as to satisfy the previous stage

If the second of these two steps does not provide a satisfactory fit combination then new modifications to the design parameters are necessary. The final step in the selection procedure is: • determination of the performance of the selected fit, eg by evaluating the load that the fit can transmit for the maximum and the minimum interference value The iterational form of the procedure presented suggests that by using a computer to perform the calculations significant time savings can be achieved.

THEORETICAL BASIS The aim of this section is not to provide a detailed description of the theory underlying this work s'6 but to give a general outline of some important concepts. The theoretical model adopted here was developed by Lundberg 7 to describe the behaviour of thick-walled cylinders under external pressure(s). It assumes an even axial stress situation with oz = 0 along the cylinder length. The model covers the whole range between fully elastic to fully plastic deformation of both parts and has been experimentally verified by Biederstedt8. The model has been produced by using the yon Mises yield hypothesis and for elastically deformed cylinders it provides identical results to those obtained by using the Lam~ 9 approach. In the elastic-plastic and the fully plastic region of deformation the prevailing equations are strongly non-linear and cannot be solved analytically. To overcome this difficulty, Lundberg provided graphs showing the interconnection of the various parameters with each other. Figures I and 2 show Lundberg's diagrams and Figure 3 shows how the maximum plasticity ratio of the hub varies as a function of its diameter ratio. The figures were produced by using

274

An important decision that had to be made at an early stage was to define which part of the selection procedure was to be computerized and which was to remain manual. After the non-linear expressions of the Lundberg model had been solved numerically, it was decided that all but the fourth stage could be computerized. The reason for this decision was the construction of the ISO tables together with our aim to use a desktop microcomputer with medium memory capacity to perform the selection of fits. Although the values in the ISO tables can be calculated by using existing formulae, the tables as they have been constructed present considerable deviations from the calculated values l°. We were interested in the tabular values. Hence, after taking into account the existing memory limitations, we looked first at the possibility of creating an external database. The fact that its use would necessitate the incorporation of an external storage device, with a consequential increase in the system cost, obliged us to abandon this idea. Thus for the time being the use of the ISO tables remains manual. The algorithms developed to calculate the border interference limits and the loads that a selected ISO fit can transmit are shown in Figures 4 and 5 respectively. The computer program developed for the selection of interference fits was written in BASIC 4.0 and runs on a CBM-PET 8032 microcomputer. The program occupies 25k of memory. Minor modifications of the program are necessary in order to run it on a micro using another version of BASIC. As written the program provides all the information related to usual interference fit calculations on the CRT. Only when an elaborate study of a given situation is required can a printed output of all the calculated parameters be provided. In this way the capital expenditure is minimized. The program operates in two modes: • •

mode 1 for the implementation of the complete selection procedure described mode 2 for cases where only an assessment of the performance of a selected fit is required

The mode to be implemented is selected by the user at the start of each run. Mode 2 is of particular value as it allows the designer to assessfits for which no design information is apriori available, as often happens in practice. At every stage of the run, users are provided with all the information necessary for decision making with respect to the design parameter values. Diagnostics are offered throughout the run, irrespective of the mode in use.

RESULTS AND DISCUSSION Manual calculations were performed to verify the results obtained from the computer program. A very good correlation was shown for all the particular cases studied. The fact that the Lundberg model is non-linear made it essential

computer-aided design

Specify do, di , dt , /f , Eo, E~ , % a , ~ , , Vo , H , Rza ' Rz~ , ~ ,

Calculate e, for shaft elastic-plastic

(~t)r, (qsa)ac,(qsi)ac

½~

i

Alter parameters

alculateqa 'qi

@

I

¼ (*)

I

I c°~,u'~ (~o)oo I

(*)

J C°mpute (~'~l)ac

i

I

Interferencefit not possible

i

Assign I P= (Pf)oc,arnin =a

J J l

I Calculate(Pf )r '(Pf )ac

Calculate Psi' Pso

AssignBmax--~

l

i I ~o,c0,o,0~ I i i ~o,co,o,e,~o~o..,~.,~,oI

No

Assign P= (Pf)r

I

J

J Calculate(Pi)ac,(Pa)ac

J

~°'°°'°'°~

]

I ( * ) Newton- Rapheon numerical iteration

(-)1 C°m°°te~'°=~'°(~'~)l I

Calculate ea for hub elastic - plastic

Calculate ea for hub elastic J

No

(.)

P

Compute qsi =qsi (~s--~,qi ) °~s _ ~'rs P, qi °"i ~ i

(~-'~)

I

Calculate e i for shaft elastic I

C

Figure 4. A I~oHt~m for the determinotion o f t~e border inte~erence Hmits

to use the diagrams shown in Figures I and 2 when performing the manual calculations. In addition, consideration was given to the fact that Figure 2 is strictly valid only for materials with a Poisson ratio u = 0.3. For this reason, only materials with such a Poisson ratio were considered for volume 16 number 5 september 1%4

testing the program. It is of interest to note that no such limitation exists for the computer program developed and it provides accurate results independent of the specific material properties. Tables 1-3 present in detail the results of a particular case study. 275

Specify

~

do,~ , d, , I,,Eo,~,%o,

1

~si,V~,Vi,Rzo, Rzi, /~ ,

I ~)oo. amJam)'mo..~aml'~,. i~ I ~,,~r.o.~=~,~ J ~ ] [ . Calculates

Yest

~ '

~

No

~

I~*~1 co~,~= II Co~.~,e I~*~ I I~¢a~.q.,=~.,~a~.~=~a~l I~=~a~.q.~=~.~a~orl

~

J For shaft elastic ~'pla~ic

I I hub e'asflc-pmstic and I

I

I I

.. I ' ~ No

~.~.~,~s,,~

+

I

Resultingconditions I not acceptable I

J

C.lculete~ ~

~ I

Calculate~i'~a

J

I

Calculate(Pf)s

~

J

I

~

.

~.o.~,o..c

I

1 J

a=~.. ~

...,~.

I~

J (~)~ax=~'6=6~in~

~ No

I ~s~,~,'~,n=~ ]

I Calculate (Pf)el- pl

I

T"

Calculate ~el-pl

I

( * ) Newton-Raphsonnumerical iteration

~

for

hubelastic-plastic and shaftelastic .=L. T

J J

I ~s~,~n~=~o. I

I

calculate~el-pl f°r shaftelastic-plastic

@

on0.u.,o~,,c

N~' C~Icul.le ~ =~(~) Yes~

.

~ ~ ~

~

~ ~"

~

B='~l-pl~

J J

~ @

_

( ~ Compute ~=@(B),qso=qsa(B) Yes _ ,~, %s_%s($)fo r

qsi=qsit / ~ - ~ bothportselastic-plastic

Figure 5. AIgorithm for the determination o f the transmissible load limits

Compared with any existing manual selection method, the computer-aided selection of interference fits almost eliminates any calculation errors that would otherwise prevail. Being relieved from tedious activities the design engineer is able to concentrate on a strictly design operation,.the correct assessmentof the values of the various parameters involved. The highly interactive nature of the computer program allows the specified parameters 276

to be modified quickly until an optimum solution is arrived at. Comparison with the most recent manual selection method in use4 shows that the designer is now provided with extended capabilities not available before. The computer program allows the user to assessthe performance of a fit for which no information is available concerning the deformation which prevails at the given interference computer-aided design

CONCLUSIONS

Table 1. Design parameters as specified d a (mm) d i (mm)

160

Osa (N mm -=) Ea (kN mm-:)

6

df (mm)

63

~a

/f (mm) Rza (/~m)

50 6

(~si (N mm -=)

Rzi

(#m)

~i

($m)~nin (g/m) 142" (6m)~nax (#m) ~

191" 0.125

210 0.3

E i (kN mm -~)

6

320

420 210 0.28

(qsa)ac (qsi)ac

1.3

(Ft) r (kN)

0.2 200

*Correspondsto the selectedISO fit H7/z6

The problem of using a desktop microcomputer to select interference fits was investigated in detail. Algorithms were developed and a computer program produced that allows the computer-aided selection of interference fits. This was compared with the existing manual selections and showed considerable advantages. It is believed that industrial implementation of the work presented could enhance the use of interference fits in practice. ACKNOWLEDGEMENT The authors would like to thank George Michael, senior scientist of the HAF Technology Research Centre, Greece, for his assistance in plotting Lundberg's graphs with the Prime computer of the Centre.

Table 2. Obtained output corresponding to distinct 6 limits ~ = ~min ea (/~m) leil (/~m) F t (kN) Pf (kN mm -2) qsi 6 (#m) 6m (/~m)

--

116.7t 126.Mr

137.2 51.3 286.2t 231.38 1.3" 0.1029 188.5~ 198.1-~

Ors/CYsi

--

-0.076

qsa

80.9 35.8 200* 161.68 1.0204

~ = (~max

~ - ~min ~

~ = Smax '

92.3 40.1 223.8) 180.96 1.0875 t 132.4 142"

131.5 49.9 279.9t 226.27 1.2752~0.1011~ 181.4 191"

-

-0.059t

-

*Specified value. ~-Parameterdisplayedon the screen. Table 3. Other parameters calculated Psa (kN mm -2)

155.49

Psi (kN mm -2) (Pa)ac (kN mm -~)

208.10 231.38

(Pi)ac (EN mm -~) (Pf)el-pl (kN mm -=) (Pf)s (kN mm -~) qa qi

(qsa)ma× (qsi)min

2.5397*

qsa7~

0.0952* 1.1938

360.61 208.10 155.49

qsi~"

-

~Ss (/~m)

112.2

0.3938 0.0952

ars/asit

-

6el-pl (/~m) 159.1

*Parameter displayedon the screen. ~-At ael_pl. limits; with the algorithms developed it is possible to identify automatically the deformations of the mating parts and consequently to calculate the load that the fit can transmit. As with all the existing manual methods, the computer program ignores the effects that centrifugal forces, working temperatures which are different from the ambient and temperature gradients could have on the transmissible load. It also relies on a theoretical model strictly valid only for interfering cylinders at plane stress conditions and not immediately applicable for other shapes. Theoretical work is under development to overcome these limitations and to extend the model used. This will eventually require the use of desktop computers with a memory capacity of more than 32k.

volume 16 number 5 september 1984

REFERENCES 1 International .Standards Organization 'Limits and fits' ISO/R 286 rE) (1962) 2 British Standards Institution 'Guide to the selection of fits in BS 1916 : Part 1' B5 1916 : Pert 2 (1953) 3 Deutsches I nstitut f/ir Normung 'Calculation of simple press fits' DIN 7190 (August 1943) (BSI translation, London, UK) (1969) 4 Deutsches Institut fiir Normung 'Berechnung und Anwendung yon Pressverbanden' DIN 7190 (March 1981) 5 Lagodimos, A G and Scarr, A J 'Computer-ai'ded selection of interference fits' ASME Comput. Mech. Eng. Vol 2 No 2 (September 1983) pp 49-55 6 Lagodimos, A G 'An investigation into the computeraided selection of interference fits' MSc Thesis Cranfield Institute of Technology, Cranfield, U K (September 1981) 7 Lundberg, G 'The strength of interference fits' Des Kuge/lager Jahr 19 No 1/2 (1944) pp 1-11 (CIT translation (1981)) 8 Biederstedt, W 'Presspassungenim elastischen, elastischplastischen und plastischen verformungsbereich' Technische Raundschau H.57 Berne, Switzerland (1963) 9 Timoshenko, S 'Strength of materials - part I1' Van Nostrand Reinhold, Toronto, Canada, 3rd edn (1958) 10 Baenninger, E 'The revision of ISO tolerance system - a study of some possibilities' ISO/TC3/WG4 Document 14 International Standards Organization

(1981)

A P P E N D I X 1: E Q U A T I O N S P R E V A I L I N G FROM LUNDBERG'S MODEL When a cylinder is subjected to internal and/or external pressure the resulting stressesforce it to deform elastically. When the part is made of ductile material and the applied pressure exceeds a certain limit (Ps), a ring-shaped region is formed around the paws inner diameter, where the material has deformed plastically. This pressure limit has been calculated for the hub to be 1 -qa ~

- -

(3+qa.),,2

Osa

(1)

277

and for the shaft to be

Psi -

1 -qi 2 %i

2

(2)

When the applied pressure ensures a pure elastic state of the parts (Pr ~
Pfdf

Ea

I + qa~

(

I - qa~

+/~a )

(3)

With the parameters Pf/asa , qsa and qa related through the expression pf pf 2 ~,'2. 1 ( .... + [ 4 - 3 ( - ) l }exp[~/3sin -1 2 C~sa Osa X/3 Pf ) ] = 2qs? (~ asia [3 + (qaqsa)4l '/~

ex~ ( ~ ~i~' ~V~ 2

and the deformation of the shaft (outer diameter) is Pfdf

ei-

(

1 -

dfosa e a -

Ea

3 + (qaqsa) 2 (

(qa qsa) ~ [3 + (qaqsa)4] v~ })

dfOsi [~Pf ( ~+qsi 1 ~ _ 2qsi ~ Ei osi 1 -qsi ~ ~i) + OrsOsi1-qsi 2 ]

ei-

pf

Ors 3 + qsi 2

-

~

Osi

Osi

1 - qs~"~

+ ~

4

Ors ~

[4-3(

)

4

]

~"~

(8)

Osi

2 [3 + (qa qsa) 4 ] ,/2

~/3

qsi : = q i ~ e x p [ - ~ 3 s i n -~ ( ~ 3 2

Osa

I - (qaqsa)~

2 [3;]q~-~q

278

(7)

where Pf/Osi , Ors/Osi , qsi and qi are related through the equations

exp [ x/3 {(sin-' ( ~ - - P f ) - s i n - '

( ~/3

(6)

Accordingly the deformation of the shaft is

1 + qi 2

/~i) (4) qi ~ In cases where the applied pressure causes the elasticplastic state to occur (Pf > Ps), the deformations of hub and shaft are non-linear; they are of parametrical form, the equation for the hub being E i

1

,

I

Pf

''2 ~ ) ] - ( ~ - va ) osa-- ; (s)

~

I

~'+[4

2

Osi

Ors ) ] / Osi 2

_

3(0~') ]

/ I/2

)

(9)

~si

computer-aided design