A Simple Approach for Evaluation of Cylindrical Surfaces

A Simple Approach for Evaluation of Cylindrical Surfaces T. S. R. M u r t h y and S. Y. Rao; C e n t r a l Machine Tool Institute, Bangladore/lndia

- Submitted by J. Peters (1)

UMFVrRY: Precise evaluation o f form errors of three dimensional surfaces l i k e c y l i n d r i c i t y . s p h e r i c i t y requires the development o f a matnematical model and an appropriate measuring technique. Though form errors are defined by ISO. t h e i r measurement and evaluation techniques are n o t available. An attempt has therefore been made t o devise methods f o r t h e i r reasurentent and evaluation. CMTI i s developing algorithms f o r the evaluation o f form errors l i k e sphericlty, c y l i n d r i c i t y . c o n i c i t y etc., w i t n the ultimate objective o f evolving user-oriented software f o r computerised d i g i t a l metrology applications. Metnods o f measurement and evaluation procedures o f spherical surfaces have been developed a t CMTI and reported elsewhere. I n t h i s paper some methods by d i f f e r e n t authors f o r evaluation o f c y l i n d r i c a l surfaces are b r i e f l y o u t l i n e d and a new approach developed and practiced a t CMTI has been described. For t h i s method c i r c u l a r i t y traces a t d i f f e r e n t sections o f the c y l i n d e r are obtained and d i g i t i z e d f o r c y l i n d r i c i t y evaluation i n a computer. The p r i n c i p l e o f t h i s method i s t o develop the inlagindry surface o f the wean c y l i n d e r mathematically and locate appropriately the measured data and t r e a t i t as a plane surface w i t h sine wave boundaries f o r evaluation o f flatness. I n t h i s rliethod the chosen g r i d on the developed surface w i l l n o t be rectangular but w i l l be c u r v i l i n e a r o f sine wave type. This method o f analysis i s d e a l t w i t h i n t h i s paper. This p r i n c i p l e can be u s e f u l l y extended t o other developable surfaces.

METHODS OF EVALUATION

c o e f f i c i e n t o f cosine term i n the Fourier series containing orthogonal polynomials coefficient o f sine term i n the Fourier series containi n g orthogonal polynomials deviation o f a data p o i n t f i , j ) w i t h respect t o the l e a s t squares mean surface sum o f the squares o f the deviation = XEZ . s u f f i x i n the Fourier c o e f f i c i e n t , d=0,TzJJ S u f f i x i n the Fourier c o e f f i c i e n t and i n the orthogonal polynomial 1- (2) o f i t h order, k 0 , U number o f angular & v i s i o n s number o f horizontal sections orthogonal polynomial o f Z t h order mean radius o f the c y l i n d e r maximuv order o f the Fourier c o e f f i c i e n t maximum order o f the Fourier c o e f f i c i e n t height o f the j t h horizontal section f r o m some datum surface centre o f the mean c i r c l e obtained by the i n t e r s e c t i o n o f the mean c y l i n d e r w i t h the datum plane &+a constants i n the equation o f the plane, p&+ e c c e n t r i c i t y given as e = (u'+ bL)!a amplitude o f the sine wave boundary o f the developed surface ir = n tan 0 s u f f i x f o r angular co-ordinate s u f f i x f o r height co-ordinate d i r e c t i o n cosines o f the axis of the c y l i n d e r dimensionless d i r e c t i o n cosine r a t i o Z=Zr/nl diniessionless d i r e c t i o n cosine r a t i o m ' / n l n cos .$ 2 sin 0

(a) General Method Tor the measurement o f c y l i n d r i c i t y an instrument w i t h a f e e l e r nloving p a r a l l e l t o the r o t a t i n g a x i s i s required. I n t h i s method c i r c u l a r i t y traces a t d i f f e r e n t horizontal sections o f the c y l i n d e r as shown i n Fig. 1 are taken. With the system o f co-ordinates shown i n Fig. 2 and assuming Z1,ml,nl t o be the d i r e c t i o n cosines o f the axis o f the cylinder, the equation o f the axis passing through a p o i n t fa,L,O) i s given by Ix/ I~ '=f;.-bl /ni'=~/e'

(1)

The points o f i n t e r s e c t i o n o f the axis o f the c y l i n d e r w i t h d i f f e r e n t horizontal sections i n which the c i r c u l a r i t y traces are obtained are faj,bj,Z:) and are given by

a 4

measured r a d i i w l t h some reference c y l i n d e r l o c a l co-ordinate on the developed surface,a=h cosfe-4 I angular co-ordinate f o r e semicone angle angular co-ordinate phase angle angular tilt o f the axis o f the c y l i n d e r w i t h z-axis

114TrlUOUCTI ON The form errors o f machined components are important frori the p o i n t o f the function o f the components i n any assembly. Since most o f the functional components are c y l i n d r i c a l , circul a r i t y and c y l i n d r i c i t y are required t o be established. Evaluat i o n o f c y l i n d r i c i t y i s not simple because i t requires a number o f c i r c u l a r i t y traces t o be taken a t d i f f e r e n t horizontal sections and r e l a t i n g these w i t h the straightness o f the generators. The importance o f these nmsurements i s increasing Decause o f the development o f high precision spindles which are required to give high accuracy o f a x i s o f r o t a t i o n E1.2.31 which g r e a t l y depends on the c y l i n d r i c a l form errors. The c y l i n d r i c i t y o f the spindles and hydrostatic bearing bores are WJ be frequently measured during lapping t o achieve the desired accuracy. I n t h i s paper a simpler method f o r measurement and evaluation o f c y l i n d r i c i t y has been proposed. Some o f the e x i s t i n g methods published are b r i e f l y o u t l i n e d i n the paper. Goto e t al.. [5.&3.9] determined the polygonal coniponents on the c y l i n d r i c a l surface by using orthogonal polynomials. Tsukada e t al., C63 increased the speed o f measurement by s p i r a l tracing. I n the method by Kakino e t al.. [ 7 j inprocess measurement o f c y l i n d r i c i t y i s achieved. The present paper deals w i t h a d i f f e r e n t approach. nanlely. the development o f the imaginary surface o f the m a n c y l i n d e r t o a plane surface w i t h sine wave boundaries. The method can be employed t o evaluate c y l i n d r i c i t y and c o n i c i t y o f surfaces which can be developed. U i t h t h i s method the e l l i p t i c component i n the c i r c u l a r i t y traces due t o the i n c l i n a t i o n o f the c y l i n d e r i s suppressed.

Annals of the ClRP Vol. 30/1/1981

spacing of t r a c e s ( h o r i z o n t a l section) sampled p o i n t s s p a c i n g in a n g u l a r direction

Fiq.1.Dat a

p o i n t s for c y l i n d r i c i t y evaluation

Assuming Y t o be the semi taper angle, R t o be the mean radius, $, t o be the measured data, the deviation from the mean c y l i n d e r i s Ei,j = ri,j

- TZj-R-e.cosfei-aj)

= ri,.j- Y2.j-R-a

cos ei-b s i n

ei-

mZj sinei-

and the sum o f the squares o f the deviation i s

Zs = i f E i , j I 2

EZ. c o s e i J

(3) 14)

Single subscript i n the above equation indicates t h a t i t s value i s independent o f the other subscript.

441

c y l i n d r i c i t y = n t f E i , j max where n' = 11-1t2 -

.

- zi,:mi

1/(1+Z2 t

I71

n)

m2)'2

This nlethod i s applied a t CMTI f o r surfaces o f revolutions

L cos2 C sin

ei

z s i n S i cos ei

e i cos Bi

L sin'

Z Z j COS2 6 i

I: Z;

cos

L

eisin

.8,

ei

i Zj cOS

I cos ei

c

2. J

I

z . sin2 ei

sinei

cosei

J

i: 22

3 L 22. J

eicosei

sin

sin*ei

Z? 3

sinei

z zj

sinei

ei

z . s i n eicosei J

E z;,

sin2ei

I

z . cos2ei

E

z;

z: z?. J

cos2ei

z sinei

E

zj

2

i

(b)

z z j s i n e i cosei L z j cosei

I:

also.

s i n e i cosei

u

I 2. sin e J

\

z

=

rnv2)'2

zj

!SJ

#thod of orthogonal polynomials by I i z u k a e t al..

I n t h i s method the e r r o r o f c y l i n d r i c i t y i s represented by orthogonal functions consisting o f Fourier series and orthogonal polynomials. F i r s t the p r o f i l e o f the c y l i n d e r i s expressed as

cowi

cosei

I cossi

zj

sinei

z sinei

zj

cosei

i

z,

i

zj

sinei

I:

z . sinei

1

z;

where z i s the co-ordinate along the r o t a t i n g axis and i s orthogonal polynomial o f i t h order s a t i s f y i n g J

cosei

P . (2.) b

? L l ( z ; )=

0;lyL'

The c o e f f i c i e n t s Ai

X

and dL,L I

?Lfz) (91

are obtained as

,

i 2. J i l

z zj

I

cosei

L ri,j Z r. Z,

2

--

. J

i*i,j

= 2 I i r.

t,j

sinej where F: orientat&

Zi cosei

z, sine* r~. , j j '

C

r.

.

t,J

.

I f the horizontal sections are equispaced, by s h i f t i n g the o r i g i n t o the centroid o f the c y l i n d e r the above equations o i r e c t l y give the following solutions.

reference

1

denotes the observation o f the p r o f i l e a t the i t h on the j t h section.

I n the above equation the i n d i v i d u a l terms represent term i s a constant red i f f e r e n t geometric meaning. I0,U) presenting the mean radius. S i m i l a r l y (1,0) term i s the e c c e n t r i c i t y o f the average axis o f work and I 1 , l ) term represents the t i l t o f the axis. The r e s t o f the terms represent the form e r r o r .

J

L

.

PL ( Z j l s i n f K e l / [ M i Pi I Z j )

(c)

Spiral t r a c i n g nethod by Tsukada e t al..

L6!

I n t h i s method the speed o f measurenlent has been increased by having a continuous movement o f the s t y l u s on the c y l i n d r i c a l

Fa----

axis

Tracing d i r e c t ion

Fig.3 . s p i r a l t r a c i n g method Fig.2. Co-ordinate cylinder

s y s t e m for least squares

a)-Co-ordinat e system

surface. Tnis i s obtained oy r o t a t i n g the work and moving the s t y l u s a x i a l l y along a d i r e c t i o n p a r a l l e l t o the axis o f the cylinder. Tne matnematical treatment i s on s i m i l a r l i n e s as explained i n (a) except t h a t the height Z v a r i e s continuously and e can also be treated as a continuous variable. The data points f o r t h i s method are shown i n Fig. 3.

for cylinder

b).Details o f data o n a t r a c e

(d)

h u l t i s t y l u s method by Kakino e t al..

[71

This i s developed f o r inprocess measurement o f c y l i n d r i c i t y by using the p r i n c i p l e o f 3 p o i n t roundness instrunent. Pickups 2 and 3 are kept a t an angle 6 and 63 w i t h respect t o pickup I and t h e i r outputs are multip?ied oy calculated amounts and aaded t o r e a l i s e the form errors. (61

PROPOSED METHOD BASED ON THE DEVELOPtEHT OF SURFACE

This method i s based on the p r i n c i p l e o f the development o f the imaginary surface o f the mean cylinder. The d e t a i l s o f t h i s new approach are d e a l t w i t h i n t h i s paper. As t h i s illethod i s based on the flatness evaluation which i s known t o many i n the snop f l o o r , i t can be e a s i l y applied and understood by them i n order t o introduce the importance o f c y l i n d r i c i t y .

442

MATHEMTICAL ANALYSIS F i g . 4 shows a cylinder, whose base i s inclined a t an angle il With the horizontal plane i n which the stylus rotates around the cylinder. The particular orientation of the cylinder is chosen f o r convenience i n showing the developnent of the surface. tlecause of the s l i g h t inclination of the cylinder the section of the cylinder as seen by the stylus will be e l l i p t i c a l . when the surface i s developed by rolling the cylinder t h e Doundary will be a sine wave type as shown in Fig. 4. The

1113

Measuring stvlus

6

i e;

re.

z ei zJ.

z z?

c e

cos ei

z z j cos

sin a;

i

E

i

ai sin e i

i e.

i

z : s i n ai

i

.

developed amplitude

)t

h =A

of the sine wave will be given by

tan 4

(111

cos

18-01

(121

where e i s the angular co-ordinate i n the developed surface. Q i s the phase t o indicate any arbitrary generator o f the cylinder f o r development. Fig. 5 shows the complete grid along w i t h data points ideasured on different sections of the cylinder located appropriately on the sine wave.

R.

cos

9;

2.

J

ai

i cos2

9;

z cos e; sin oi

Z. sin o i

i cos

ei

SS + 22 + c*

i cos ai i sin2

i sin

sin e.i

ei ei

i cos

ei

i sin

ei

$

x

,.

X I

P,.

. ei

La J E ri _. z . J E ri,j COS Bi

B

z:I

>

q

i ri

a

E ri

.

,J

. sin

(811

ei

,j

By shifting the origin t o the centre of the developed surface the above equations get simplified t o

* where

A

1131

i

c

4 .

i:

With this data, the flatness of the developed cylinder with deviations can be obtained from the following mean plane equation. With the system of co-ordinates shown i n Fig. 5. the equation of mean plane i s given by r =

c ei cos i

J i 2. 2

. I

the development can be made from any generator of the cylinder the equation of the wave can be written as =h

J

surface

AS

2

2.

3

ei ei

E

F i g . 4 . L o c a t i o n o f d a t a p o i n t s on t h e

Z

I: y,j sin

z a i sin

z e; i e i sin e i

=

i

$

=

i

i

=

zj ri,j / cos

eisin

ei

ei sin ei ei

i sin2

=I

2

3

ei

i sin2 i Z;,~

ei

ei

ei

I f 221

/ i cos2 Bi

2 pi

.3

. s i n ei

I:i.;,j/MN

where I: denotes double sumnation over i and j and the cylindricity =(Ei,j m(Lz - Zi,

)& t

?tz

+

;2)

APPLICATION

l -

d a t a points

Fig. 5 . G r i d

and

data

points

on d e v e l o p e d s u r f a c e .

In order t o compare the solution given by equation (51 w i t h the solution given by equation 121), different horizontal sections (11 and 12) on cylinders are selected a t equal spacing and the cylindricity evaluated by both the methods. This has been done f o r a nuher of cylinders and also for ,analytically generated cylinders w i t h random data points. A few cases are shown i n Table.1.

443

Table 1 comparison o f c y l i n d r i c i t y by general method and the new method General Method pm

6.759 6.945 tl.937 6 .826 6.871 5.441

New Method

m 6.679 6.555 6.603 6.639 6.604 5.969

The lower value o f c y l i n d r i c i t y as found i n many cases i n the proposed method i s due the suppression o f the e l l i p t i c i t y o f the c y l i n d e r as seen by the s t y l u s due t o the i n c l i n a t i o n o f the cylinder. This indicates the inherent advantage o f t h i s method t h a t the c y l i n d e r need n o t be p e r f e c t l y aligned a t the time o f measurement and t h i s saves considerable time i n measurement. Some s t r a y cases where a higher value i s seen as i n the l a s t s e t i n Table 1 could not be f u l l y explained a t the present juncture. However as the c y l i n d r i c i t y obtained by l e a s t squares method i s n o t the minimum zone deviation, methods o f obtaining minimum zone deviation a r e developed a t CMTI [lo2 i n which the simplex search tecnnique has been found t o be powerful and quick. Hence these techniques are applied f o r minimum zone aeviation. The above methods help as the s t a r t i n g p o i n t f o r search techniques t o determine minimum zone deviation. CONCLUSIONS D i f f e r e n t methods available for c y l i n d r i c i t y evaluation are b r i e f l y o u t l i n e d and a new simple method based on the development o f the surface o f the c y l i n d e r f o r evaluation i s explained. I n t h i s method the inherent e l l i p t i c i t y due t o the i n c l i n a t i o n o f the c y l i n d e r i s suppressed and i s as e f f i c i e n t as other methods. Further, because o f the above advantage much time need not be spent i n a l i g n i n g the c y l i n d e r t o reduce the i n c l i n a t i o n t o minimize e l l i p t i c i t y and hence measurement w i l l be faster. It also helps as a s t a r t i n g s o l u t i o n f o r minimuni zone evaluation. I n t h i s method the taper on the c y l i n d e r can also be included i n the analysis i n which case the developed surfaee w i l l be the p a r t o f a sector o f a c i r c l e and polar co-ordinates w i l l be useful i n f i n d i n g flatness E l l ] . The present e f f o r t s a t C M T I for the evaluation o f form errors o f surfaces, l i k e spheres, cylinders, cones etc. are d i r e c t e d towards the u l t i m a t e o b j e c t i v e o f developing e f f i c i e n t a1 g o r i thms t o enable development o f software f o r computer-aided metrology. Software packages f o r evaluation o f s p h e r i c i t y , c y l i n d r i c i t y etc.. using minimum zone evaluation incorporating simplex search and other techniques are avallable a t CMTI. ACKNOWLEDGEMENTS This work was c a r r i e d o u t i n the Department o f Research and Testing, Central Machine Tool I n s t i t u t e , Bangalore. India. The authors are thankful t o t h e i r colleague Mr. B.R.Satyan f o r h i s valuable suggestions t o the contents o f t h i s paper. The authors are thankful t o the Director f o r permission t o publish t h i s work.

444

RE FEKENCES U n i f i c a t i o n document Me. Axis o f r o t a t i o n . o f C I R P 25/2/1976.

Annals

T S R MURTHY e t al., New methods o f evaluating axis o f r o t a t i o n error; Annals o f CIRP 27/1/1978 p 365-369. C MKLLANNA. T S i( MURTHY and M E VISVESUARAk4. I n fluence o f errors i n wheel and work axes o f r o t a t i o n on work piece i n c y l i n d r i c a l grinding; Annals o f CIRP 29/1/1980 p 373-375. T S R MURTHY e t al., Evaluation o f spherical surfaces; I n t e r n a t i o n a l conference on metrology and properties o f engineering surfaces. Leicester (1979). Wear. 57(1979) p 167-184. M GOTO and K IIZUKA (1975). A m t h o d f o r evaluation o f form e r r o r s o f c y l i n d r i c a l parts; J.JSPE 41(5). p 477.

T TSUKADA e t al., An evaluation o f form e r r o r s o f c y l i n d r i c a l p a r t s by s p i r a l t r a c i n g method. 19th I. hTDR p 529-535.

Y KAKINO e t al., I n s i t u measurement o f c y l i n d r i c i t y ; Annals o f CIRP 27/1/1978 p 371-375 K IIZUKA and M GOTO. Application o f m u l t i p l e regression analysis i n measurement and c a l i b r a t i o n ; ACTk IMEKO (1973) Vol. 1 p 275-286.

M GOTO and K IIZUKA. An analysis o f r e l a t i o n between minimum zone deviation and l e a s t squares deviation i n c y l i n d r i c i t y and c i r c u l a r i t y ; ICPE New Delhi (1978) p x61-70.

T S R MURTHY and S 2 AIDIN. Minimum zone evaluation of surfaces; I n t . J MTDR (1980) Vol. 20 p 123-136. T S R MURTHY. Polar g r i d f o r flatness evaluation. 9 t h AIMTDR I I T Kanpur (1980) p 427-434.