Analytic-Statistical Analysis of Measuring Errors in Technological Systems

Analytic-Statistical Analysis of Measuring Errors in Technological Systems

Cop\Tigh t © I F.-\. C St oc ha sti c CO llt rol \ ·illli lls. l .i til ll'lIliall SS R . L·SS R. I " Kt; ANALYTIC-STATISTICAL ANALYSIS OF MEASURING ...

609KB Sizes 0 Downloads 58 Views

Cop\Tigh t © I F.-\. C St oc ha sti c CO llt rol \ ·illli lls. l .i til ll'lIliall SS R . L·SS R. I " Kt;

ANALYTIC-STATISTICAL ANALYSIS OF MEASURING ERRORS IN TECHNOLOGICAL SYSTEMS G. Lazenas, M. Gricius, V. Pukas and

J.

Skuchas , 3.

h a ll ll{/.\ :'< lI ltl//(J SlIiwhlwlI.\ P,,/ill'Chllir / 1/\1/11111'. [) (I/I/' /ai,hill

h all/w."

[ 'SSR

Abstract. Simulation algorithm and analytic-statistical analysis results of measuring errors of angle-positioning measuring equipments that are employed in precision and Dleasuring machines are presented. Ke words. Error; estimation ; simulation; algorithm; probability distribut on function; trapezoidal tooth.

1

a mean number of contacting teeth. The peripherial displacement error X dispersion ~Jl.2. is defined in such a way

INTRODUCTION Modern systems of robotized production must ensure a high accuracy process of technological operations. The errors of measuring equipment blocks call work surface positional inaccuracy, which leads to reduction of product quality. Estimation of errors becomes more complicated because positional equipment errors cannot be directly calculated to the inaccuracies of the performance of the technological operation itself. The error analysis becomes more complicated when there are a variety of errors components distribution laws.

6'2. .. 62. + 6:1. JI.

where

1

Nv

t

2. ) /

2.N y

Here :

622. "6r:1. I sint d..

(5 )

oj2. a hlg4clC-../Sin2.c( (tgd..-tgAcC_x)

(6)

The notations are following : X~i (i-1,1. I .. ' I N v I - the random independeni values of the normal distribution law ; the teeth contact sides thickness dispersion ; Ah. a mean deviation of disk contacting depth : d - a coefficient depending on geometrical parameters of teeth ; D -dispersion operator symbol.

6; -

In practice we take an assumption the random values Xsi (i· i .2. .... . N.. ) are normal with the prObability distribution density (Fig. 2) f(Xal 2e.tp(-(x,-dI2 6':) IVR OS (HO 1- F (~s-11 I Ss)) ,

Recent work by Lazenas, Mikuckas, Petrauskas and Skuchas (1984) shows the peripherial displacement error X of the upper disk in relation with the lower one depends on pressureforce F , disk surfaces contact angle , teeth geometrical parameters and may be represented in the following way

f.) ( Ny,c;(, ,x41) .

(

~~ .. 2~ + 0,)

ALGORITHM FOR DETERMING OF PERIPHERIAL DISPLACEMENT ERROR DISPERSION

+

2.

2.

6'1 ,. 8( FI d.·sin2~) D(1f.[ Xsi ) I t.1

The work deals with the estimation of measuring errors of angle-positioning measuring equipments (APE) used in precision and measuring machines by statistical simulation. The angle-positioning is realized by a flat disks with trapezoidal toothed surfaces in those machines. The contact geometry fragment of two teeth of the upper disk in relation with the lower disk is mentioned in Fig. 1. The notations in this figure are following : QC - tooth slope angle ; 4X - tooth deformations; x - tooth productions errors; h tooth height.

X.. ft (F, "',ilX,)... f,. ( Nw~' Xp)

(2)

2~'

i

when x.' x s ' i

and

fb:) "0 , when X$ <. ::C. I :x.~ > t • The probability distribution function canonical normal law. Let us sign of random values +- X.. , by '( then mathematical expectation and sion of this value

(1)

Here : f .+,. f ~ - functions of random values .x:;Xf"X~ Namely, Xs - a random value, that characterizes tooth contact area scattering ; Xp - a random value, that characterizes plane deviation : XA~ - a random value , that characterizes tooth slope angle deviation , and Nv -

F( ,) -

of the the sum • and disper(7 )



The probability distribution density of the value Y 2

t

H';:Il=~p(-(~tm)!261)/12i b1 (F!tl-F(-tl

-19,)

)



-t!lli

my-t 6y0; !:j ~

when

1'\"1.'1+

t 6'1

was confirmed by other computer experiments with variety of technological data.

and

f(y)~O )

when Y<'l'YI.l " t by y ., my" t fiy . There t~ ", 2., ~ . In the fact the probability distribution density of the value '( is known, we may obtain the mathematical expectation M (11 '(l and dispersion D(1/ y) of the random value (1/ Y1 • The pressureforce F , the deformation Ax and teeth geometrical parameters are connected by the following relation

F=A"id·NX s NvSln2oC,

(8)

where 4 X - the mean deformation value, being determed from

SUMMARY ERROR DISPLACEMENT DISPERSION The summary displacement error dispersion 6'i:. may be expressed by the sum of the peripherial displacement error dispersion 6~ and disk teeth contacting depth dispersion 5~ (10) 1 1 the expressions 642. 11. = 6" .9 d.. and formulas (5), ) (6) , the expression (10) may be reduced to the form

Accordin~ly

6'=

o"(k +1/ZN v

or As is seen from (2) - (6), the peripherial displacement error dispersion depends on pressureforce and disk teeth parameters, so we present here an algorithm for calculation of the relative dispersions 5,,1/61. 6,,1./ S2. Eir!/6'1. wi th respect to the generalized value k~F/6'MX~d ·slI\2~

0; ,

o)

Algorithm. 1. Give the numerical values of;c N) t and the series of relation N,,/ N .) from the segment [0.1 i j 1 ; here N the disk teeth quantity; 2. Determine the value ( from the equation

A

~I

6

-

t

N /ND(F('~;c /6'-tl-FH))/F(t}-FH); 3. Calculate the value formula (9) ; 4. Determine the value tion (7) in such a way

from the

4:t;

k

from the equa-

k =AX N.,l6'j

~

1

5. Calculate the value Ou 1 fi a from the equation (4) in such a way 5:~/ 0" '" 1/2 N,,;

6. Calculate the mathem~tical expectation m y and dispersion 0"( of the random value Y from the formulas (7) ; 7. Calculate the mathematical expectation M (if Y} and dispersion D (" 1 y) of the random value (11'( 1 by special subroutine ; 8. In accordance with the results of the step 5 calculate the value Ii; / fia in such a way

9. Obtain the peripherial displacement error relative dispersion from the formula

0 1 /62. = 6 1 /6 1 %

Let us determine the tooth slope angle ~, that minimizes the summary dispersion when the values A. B. Ei~ are fixed. According to the expression (11) let us write a neoessary condition for extremal value
"

...

b." / b" u

The Fig. 3 shows dependences of the relative dispersions 0.'/0' 6.2./6 1 6'l~/6a on value k • As is seen from the charts, the peripherial displacement error dependence has an expressed minimum. This fact

<1:.'*: d.6;/ dat.. .. 0 <: d-. <: T/t. This condition results in the following equation

(A/~ink .. ~/c.os~1"(2.~/~2 .. 1/sin1oe)(A" tg \d =0. (12) Solution of this equation must be fulfilled numerically. CONCLUSIONS 1. In the process of designing APE the fact of extremal dependance of peripherial displacement dispersion on pressureforce F , disk teeth quantity N ,tooth s~ope at.. (there exists a zone of optimal parameters choice) must be taken into consideration. 2. The minimal value of summary displacement error dispersion may be achieved, when teeth slopes are chosen from the interval [36° ; 42°J • However this question requires futher research. 3. In the process of constructing APE the main attention should be drown to reducing dispersion of equivalent roughness in teeth contact places,as the part of peripherial displacement error is caused by the above mentioned dispersion when the pressureforce is being increased. 4. The part of the summary dispersion depending on teeth geometrical parameters may be reduced to a minimal value if to choose the pressureforce F from the condition F~2.6'd.·NMX~.5in2~. REFERENCES 1. Lazenas, G.B., Mikuckas, J.J., Petrauskas, K. V• ,Skuchas, I. J. ( 1 984). The examination of linked toothing disk errors in precision angle-positioning measuring equipments. Vibration, 3(51).(In Russian).

.-\na h tic·sta ti st ica l .-\nah·sis of \leasu rin g Erru rs

2. lom.G •• Kom, T. (1976). Mathematical handbook for scientists and engineers. Moscow. (In Russian).

Fig. 1. Contact geometry fragment of two teeth of the upper and lower disks.

o

:Xo

Fig. 2. Random value 'X.. probability distribution density chart • •2 ·10 0 .6

0.4 0.2.

ol-----==:1=::==~=~2.=~::::~~- k. .10 Fig. ). Relative dispersions ~t~/.. 6" 6'.2/""t 0 6z~/ 6'2 dependence on the generalized value k.

SCp·h' "

3

-197