cordinate measuring machine

7/25/2019 Cordinate Measuring Machine

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ABSTRACT

Coordinate measuring machines are relatively recent developments in measurement

technology. Basically, they consist of a platform on which the workpiece being measured is

placed and moved linearly or rotated. A probe attached to a head capable of lateral and vertical

movements records all measurements. Coordinate measuring machines are also called measuring

machines. They are versatile in their capability to record measurement of complex profiles with

high sensitivity (.!" #icro$m% and speed.

&n coordinate measuring machine(C##% research, there is often a need to measure the same

feature repeatedly using multiple settings. The goal of this research is to determine of what

effects the selection of the measurement plane, adaptor style, stylus length, and stylus si'e would

have on the C##s ability to repeatedly measure a single diameter. An analysis of variance

(A)*+A% study was conducted using a C##. Three measurement planes (-, and -% were

selected for the study. /our measurements were taken on a sphere for each variable combination.

The results of this study indicate that if the measurement plane, stylus length, or stylus si'e were

changed, the C## would not repeatedly result in the same measurement reading. 0owever, the

user would be able to alter the adaptor style without affecting the resulting measurement Asfuture research is done on C##s, care will be needed with the assumptions that are made when

researching a specific effect. Based on this study, future researchers will have to determine

whether observed changes are due to the probe head configuration or the changes they are

studying.


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C0A1T23 4

COORDINATE MEASURING MACHINE

INTRODUCTION

3eductions in product life$cycle durations are driving companies to develop and produce

products at an ever$increasing rate. &ndustry experts are predicting the arrival of rapid

manufacturing through the use of flexible manufacturing systems. 2ven a brief examination of

industry periodicals such as #anufacturing 2ngineering, Technometrics, 1roduction, 5uality or

6upply Chain 6ystems, would reveal discussions about highly integrated systems that are

flexible, agile and lean. *ne result of these trends is the incorporation of coordinate measuring

machines (C##s%, which allow companies to perform data collection and process verification

within the manufacturing cell. 3esearch on various coordinate metrology issues have paralleled

the increased usage of C##s in industry as inade7uacies are uncovered and new needs develop.

3esearch topics have covered such areas as the development of new probe compensation

algorithms, sampling strategies, part orientation optimi'ation, and computer generated inspection

paths. As is often the case in research, assumptions have to be made in the interest of ensuring

study feasibility. *ne such assumption is that the part orientation will not affect the

measurements made by the C##.

8ith the advent of numerically controlled machine tools, the demand has grown for some meansto support these e7uipment. There has been growing need to have an apparatus that can do faster

first piece inspection and many times, 49 dimensional inspection. The Coordinate #easuring

#achine (C##% plays a vital role in the mechani'ation of the inspection process. 6ome of the

C##s can even be used as layout machines before machining and for checking feature locations


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after machining.

A coordinate measuring machine is a :; device for measuring the physical geometrical

characteristics of an ob


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MEASUREMENT CAPABI#ITIES

C##s can be designed to perform different types of measurement. These include dimensional, profile,

angularity, depth mapping, digiti'ing>imaging, and shaft measurements.

Di-ensi,nalmeasurements are si'ing measurements made in the x, y, and ' directions.

Pr,ilemeasurements are made to capture information about the form or profile of an

ob


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27uipment design includes a C## systems control mechanism, method of operation, mounting

style, and probe type.

C,ntr,l

C## probes are designed to be controlled either manually or via C)C. 6election is largely a

function of part 7uantity, complexity, and cost.

CNC(Computer )umerical Control% or ;CC (;irect Computer Control% is a control

system built in the C## to control probe movement. C)C$ C##s are best$suited for

production environments re7uiring a higher volume of measurements, and also in

applications re7uiring complex and small measurements with fine features. They tend to

be more expensive than manually controlled machines.

Manualor operator$controlled devices re7uire an operator that physically moves the

probe along the axis to make contact and record measurements. #anual C##s generally

cost less than C)C$C##s of the same si'e, and are better suited for prototype shops

with smaller 7uantities of measurements.

FEATURES

1.Crash Pr,te*ti,n$ #achine has provisions to protect sensitive components in the event of an

unanticipated crash.

%. Oline Pr,gra--ing$ 6oftware supports offline programming using a CA; model.

'. Re4erse Engineering$ #achine software capable of performing reversed engineering. 3everse

engineering captures the geometry of existing physical ob


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The Ma*hine C,,r0inate S+ste-$ There are two types of coordinate systems in the world of

measurement. The first is called the #achineCoordinate 6ystem. 0ere, the , , and - axes

(/igure% refer to the machines motions. 8hen viewed from the front of the machine, the axis

runs from left to right, the axis runs from front to back, and the - axis runs up and down,

vertically perpendicular to the other two.

The Part C,,r0inate S+ste-$ The second coordinate system is called the 1art Coordinate

6ystem where the : axes relate to thedatums or features of the workpiece. Before the

introduction of computer software to coordinate measurement, parts were physically aligned

parallel to the machines axes so that the #achine and 1art Coordinate 6ystems were parallel to

one another. This was very time consuming and not very accurate. 8hen the part was round or

contoured, rather than s7uare or rectangular, the measurement task was nearly impossible.


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O8ER8IE9 OF CMM

:i; Des*ri)ti,n , 9,r


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8hen probe is rotated about $axis it is then called as angle A and when probe is rotated about

-$axis, then it is called as angle B.

The )r,2e *an r,tate in t6, 0ire*ti,ns 4i/ A = B

A>B>$! Angle A?>@ an0 Angle B?>

Angle B$ Pr,2e *an r,tate r,- 1> t, !1> a2,ut (! a"is

Des*ri)ti,n , Re-,te C,ntr,l Unit


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T&PES OF CMM

The machine incorporates the basic concept of three coordinate axes so that precise movement in x, y, and

' directions is possible. 2ach axis is fitted with a linear measurement transducer. The transducers sense

the direction of movement and gives digital display. Accordingly, there may be four types of arrangement


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Cantile4er

The cantilever construction combines easy access and relatively small floor space re7uirements. &t is

typically limited to small and medium si'ed machines. 1arts larger than the machine table can be inserted

into the open side without inhibiting full machine travel. /igure shows a cantilever structure.

C,lu-n T+)e

The column type machine is commonly referred to as a universal measuring machine rather than a C##.

These machines are usually considered gage room instruments rather than production floor machine. The

direction of movements of the arms is as shown in /igure. The constructional difference in column type

with the cantilever type is with x and y$axes movements.


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Gantr+

&n a gantry type arrangement, arms are held by two fixed supports as shown in /igure *ther two arms are

capable of sliding over the supports. #ovements of the x, y and '$axes are also as shown in /igure E.F.

The gantry type construction is particularly suited for very large components and allows the operator to

remain close to the area of inspection.

H,ri/,ntal

/igure shows the construction of a hori'ontal structure. The open structure of this arrangement provides

optimum accessibility for large ob


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.

PROBING S&STEM

There are many probes available for C##s. *ne of the more prevalent C## probes currently

in use is the touch trigger probe (TT1%. TT1s work by sensing the impact of the stylus tip with

the work piece. 6tudies have shown that touch trigger probes, similar to the one used for this

research, have inherent errors (8o'niak D ;obos', !:@ 0ocken, 3a


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purposes of this study, the probe head configuration is comprised of the re7uired probe head

rotational orientation (the selected measurement plane%, whether or not a star adaptor or an

indexable head is used (the selected adaptor style%, the stylus tip si'e and the stylus length.


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In0u*ti4e an0 O)ti*al Trans-issi,n Pr,2e

&nductive and optical transmission probes have been developed for automatic tool changing.

1ower is transmitted using inductive linking between modules fitted to the machine structure and

attached to the probe. /igure E. shows a schematic of the inductive transmission probe. The

hard$wired transmission probe shown is primarily for tool setting and is mounted in a fixed

position on the machine structure.

The optical transmission probe shown in /igure allows probe rotation between gaging moves,

making it particularly useful for datuming the probe. The wide$angle system allows greater axial

movement of the probe and is suitable for the ma


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M,t,ri/e0 Pr,2e

8ith the motori'ed probe, FE positions in the hori'ontal axis, 4" in the vertical axis can be

programmed for a total of I! distinct probe orientations. /igure shows some typical

applications for motori'ed probe. &t shows that with a range of light weight extensions, the head

can reach into deep holes and recesses. The second diagram shows that head of the probe issufficiently compact to be regarded as an extension of the machine 7uill. This enables the

inspection of complex components that would otherwise be impossible or involve complex

setups.


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Multi)le St+luses Pr,2e Hea0s

8ide ranges of styli have been developed to suit many different gaging applications. 6ome of the

different styli available are shown mounted on a multiple gaging head in /igure E.G. The

selection of stylus is done based on the application for which the probe is to be used.


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FACTORS AFFECTING MEASUREMENT

STUS SE#ECTION

8hen assessing how accurate a C## measurement needs to be, it is common practice to use a

ratio of C## uncertainty to feature tolerance of at least 4=" (4=4 is ideal, but is unrealistic in

many cases%. This ratio provides a safety margin that ensures the results have a relatively small

uncertainty compared to the expected range of variation of the component. As long as a 4=" ratiocan be maintained on the tightest tolerance, this should be the end of the accuracy argument.

Jnfortunately something as innocuous as changing the stylus on a probe can have a surprisingly

large influence on the real measuring accuracy that can be achieved and also causing appreciable

variation in the measurement results. &t is not enough to rely on the C##s annual calibration to

check its accuracy as this will only confirm the result with the stylus being used for the test

(usually a very short one%. This is likely to be the best$case accuracy. To get a fuller

understanding of the likely precision of a wider range of measurements an appreciation of howthe stylus contributes to measurement uncertainty is re7uired.


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As)e*ts , st+lus *h,i*e ae*ting CMM a**ura*+

Ball s)heri*it+

The measuring tips of most styli feature a ball, most commonly made of synthetic ruby. Any

error in the sphericity (roundness% of these balls will be a factor in the C##s measurementuncertainty, and it is easy to lose as much as 49 of a C##s accuracy in this way.

3uby balls are manufactured to various levels of precision defined by their Kgrade, which is

related to the maximum deviation of the ball from a perfect sphere. The two most common ball

specifications used are grade " and grade 4 (the lower the grade the better the ball%.

K;owngrading from a grade " to a grade 4 ball saves a little in terms of the cost of the stylus,

but may be enough to threaten the 4=" ratio.

The concern is that the ball grade is impossible to detect visually and is not obviously evident inmeasurement results, making it difficult to calculate if this is significant. *ne solution is to

specify grade " balls as standard= they cost a little more, but this is a minor cost when compared

with the potential of scrapping a good part, or worse, passing a non$conforming one. 1erversely,

the more accurate the C##, the more significant the effect of ball grade is. *n the highest

specification C##s, as much as 49 of accuracy can be lost in this way.

Lets look at an exampleM

A typical probing error according to &6* 4:$! (#121%, established using a stylus with a grade

" ball=

MPEP? 1.> -

This figure is determined by measuring !" discrete points that are each evaluated as !" separate

radii. The range of radii variation is the #121value. 6tylus ball roundness contributes to this

directly, and so swapping from a grade " to a grade 4 ball increases this value by .4! Nm and

degrades the probing error by I9 in this instance=

MPEP? 1.% -

)ote that stylus ball roundness also impacts on #12T01, which uses four scanning paths across a

sphere to evaluate scanning probe performance.

N,tes$


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Hrade " ball sphericity O .4: Pm

Hrade 4 ball sphericity O .!" Pm

/or the most demanding applications a range of styli employing grade : balls, which feature a

sphericity of


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To minimise this effect all styli are calibrated on a reference sphere of known si'e before they are

used. &n an ideal world this process would map the errors at every combination of stylus and

approach angle. &n practice, a sample of angles is often taken to save time, some averaging takes

place, and a small proportion of the error can remain.

&t is difficult to calculate the effect of this onmeasurement uncertainty without carrying out empirical tests. The key fact to note is that any

residual pre$travel variation errors will be magnified by the flexibility of the stylus that is

selected. This emphasises the importance of materials choice in stylus design, weighing up the

flexural rigidity of the stem against other characteristics such as its weight and cost. 8hilst steel

is suitable for many shorter styli, featuring a oungs modulus 2 O !4 k)>mm!, the stiffest

material commonly used in is tungsten carbide (2 O ! k)>mm !%, but this is also dense and is

therefore rarely used on long styli. &n these instances, carbon fibre provides an excellent

combination of stiffness (2 F" k)>mm!% and light weight. #eanwhile, ceramic stems (2 O

: F k)>mm!% are often used in applications where their light weight and thermal stability

are valued.

6tylus stiffness is also affected by


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Ther-al sta2ilit+

/luctuations in temperature can cause serious measurement errors. Choosing the right material

for stylus extensions can provide greater stability under changing conditions, yielding more

consistent measurement results. #aterials with a low coefficient of thermal expansion are

preferable, especially where long styli are being used as thermal growth is length$dependent.

As stated previously, carbon fibre is the material most commonly used for long styli and

extensions as it is stiff, light and does not change its length as temperature varies. 8here metalsare needed Q for


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sphericity. These effects are magnified if one region of the ball is in constant contact with the

part. 2xtensive research into these effects, highlighting two different wear mechanisms=

A2rasi4e 6ear

Abrasive wear is caused when scanning a surface such as cast iron, where tiny particles of

residue cause minute scratches on the stylus and workpiece, resulting in a small Kflat on the

stylus tip. Tough 'irconia stylus tips are the optimum choice for these applications.

A0hesi4e 6ear

Adhesive wear results when the stylus ball and the component material have an chemical affinity

for one another. This may be seen when scanning aluminium parts with a ruby (aluminium

oxide% ball. #aterial passes from the relatively soft component to the stylus, resulting in a

coating of aluminium on the stylus tip, once again affecting its roundness. &n this instance,

silicon nitride is the best choice, as it is shows good wear resistance and is not attracted to

aluminium.

#aterial adhesion is permanent and cannot be removed through normal cleaning techni7ues.

Thus, as the surface material from the workpiece adheres to the ball and contacts with the

surface, like materials attract and build up can occur. 6uch build up will eventually degrade the


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form of the stylus ball and compromise any measuring results. /actors affecting adhesive wear

include=

contact force

distance scanned

hardness of surfaces (if stylus is much harder than surface being measured%

Affinity between ball and surface materials M is it a similar materialR

6ingle point contact.

Other *,nsi0erati,ns

further considerations when selecting a stylus include=

6tylus thread si'e to suit the chosen sensor

6tylus type Q straight, star, swivel, or custom configuration

6tylus tip type Q ball, cylinder, disc, hemisphere

6tylus tip si'e to minimi'e the impact of surface roughness on measurement accuracy.


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CHAPTER %

MEASUREMENT UNCERTAINT& FOR CMM

1. Un*ertaint+ *,ntri2ut,rs ,r CMM -easure-ents

To evaluate the measurement uncertainty of C## measurements, a lot of uncertainty

contributors need to be taken into account. There are several ways to classify all uncertainty

contributors. *ne possible classification is given in fig.

/ollowing five classes can be identified =

Har06are This category contains the errors related to the hardware components of the C##

like probe errors, C## geometric errors including scale errors.

En4ir,n-ent 2nvironmental conditions will have an important influence on the measurement

uncertainty. Temperature is here of extreme importance. )ot only temperature deviations (from

! SC% but also temperature gradients, in time and space, will influence the measurement

uncertainty 4"U. *ther environmental influences like vibrations, non$constant air supply (in case

of air bearings% and lighting conditions (in case of optical probing systems 4, 4IU% can also

influence the measurement uncertainty.

9,r


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Pr,2ing strateg+ The probing strategy determines the number of measurement points and their

location. /urthermore, the measuring se7uence and settings like measurement velocity can be

important. The more complex the probing system that is used, the more parameters will influence

the measurement uncertainty.

/&H= . Jncertainty contributors for C## measurements

E4aluati,n Strateg+ The evaluation strategy covers= the type of fitting criteria (least s7uares,


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minimum 'one, . . . %, the algorithm accuracy, the possible filters used, the alignment strategy, the

selected reference surfaces, the used compensations etc.

%. Tas

The multiplicity of uncertainty contributors and their strong interaction make that measurement

uncertainties will depend a lot on the specific measurement task. That is why uncertainties that

take all (or most of% the uncertainty contributors and their interactions into account, are often

called task-specific uncertainties.

Intera*ti,n , ,r- 0e4iati,ns an0 sa-)ling strateg+$ *ir*ular eatures

The best known example of form deviations interacting with sampling strategy is the

measurement of a :$lobed circular form deviation with six e7ually distributed points. &f the

measurement points are taken in the tops and valleys of the :$lobed contour, the complete out$of$

roundness of the circle can be identified from the measurement, however if the start point is

rotated with : S none of the out$of$roundness of the circle will appear in the measurement.

/igure= #easuring a circle with :$lobed form deviation

Contrary to popular belief, this is not only of importance for the roundness and diameter

uncertainty but also for position uncertainty. This is illustrated in /igure (b% for a four points

measurement of a :$lobed circle (i.e. circle with a :$lobed form deviation%.


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Intera*ti,n , ,r- 0e4iati,ns@ sa-)ling strateg+ an0 e4aluati,n strateg+

#easurement point distribution in combination with feature form deviations can have an

important influence on the measurement uncertainty. /igure 4.4 represents a cut plate with a

large opening that is somewhat curved (exaggerated in the figure%. 6uppose this plate is

measured twice= once with only points (/igure (a%% and once scanned (e.g. with a laser scanner%

resulting in a very large set of measurement points (/igure (b%%.

8hen the orientation of the least s7uares planes is considered, there will be a large difference in

orientation for the two measuring methods (the difference in orientation is somewhat

exaggerated in /igure (b%%. 6ince there are more points on the left side of the plane, the

orientation of the least s7uares plane will be determined mainly by the orientation of this part of

the plane. &f the minimum

'one fitting criterion is used instead of least s7uares, the difference in orientation of the

associated planes will be much smaller. This illustrates how also the evaluation strategy can

influence the measurement uncertainty.


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/igure = #easuring a curved plate

Intera*ti,n , te-)erature 6ith sa-)ling strateg+ an0 *la-)ing

Although most accurate C## measurements are done in a temperature controlled room,

temperature will still have an influence on the measurement uncertainty. Consider the workpiece

of /igure . The most narrow tolerance applies to the distance between the first and last step of the

shaft (F" mm%. This should be kept in mind when measuring the part. &nstead of measuring all

steps se7uentially one should measure the first and last step immediately after each other. This

will reduce the influence of thermal effects (of machine and workpiece% significantly. /or

measurements that take 7uite some time it can be important where the workpiece is clamped. The

error due to thermal expansion (or shrinkage% of the workpiece can be reduced by "9


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/igure = #easuring a stepped axis.

Measure-ents that are ill *,n0iti,ne0$ -easuring a *ir*le seg-ent

A well known example of ill conditioned measurements is the measurement of a circle segment

4EU. The measurement of a small circle segment will always result in very high measurement

uncertainties on both diameter and position. A small measurement error will have a large

influence on the measurement results. This is illustrated in /igure where both circles seem to fit

to the measured points 7uite well.


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Measure-ents that are ill *,n0iti,ne0$ -easuring *,a"ialit+

/igure (a% shows another example of an ill conditioned measurement. The coaxiality tolerance is

given with respect to datum A, which is a very short cylinder. This means that a small

measurement error can result in a large error on orientation of the datum axis. &t is very likely

that the measured coaxiality value is dominated by the measurement error on the datum axis.

This kind of measurements will result in high measurement uncertainties.

/igure (b% shows the same part but now the datum feature and the tolerance feature are

interchanged. 6ince the datum feature is much larger in this case, the errors on the orientation of

the datum axis will be much smaller. As a conse7uence the measurement uncertainty on the

coaxiality value will also be much smaller. 6ome people argue that /igure 4.4:(a% is rather an

example of a badly designed part than it is an example of a part that will result in high

measurement uncertainties.

&t is true that the represented part also suffers from an ill conditioned design, but this does not

necessarily mean that it is a bad design. &t regularly occurs that these types of tolerances are

inevitable. The designs in /igure (a% and (b% are completely different, and should thus also fulfil

another function.


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/igure = #easuring a coaxiality tolerance

'.Inluen*e , ,r- 0e4iati,ns ,n -easure-ent un*ertaint+

/eature form deviations (shortly form deviations% are deviations from the perfect form of the

feature. A circle will never be perfectly round, a line never perfectly straight and a plane never

perfectly flat, whatever manufacturing process is used. The type of form deviation is usually

related to the manufacturing process that is used, therefore it is also called the manufacturing

signature. Chapter : discusses typical form deviations for several types of features. That chapter

also illustrates that

form deviations are often the most important source of measurement uncertainty. 1eople who

tried to determine task$specific measurement uncertainties for C##s focussed until now almost

only on C## hardware uncertainties (see Chapter !%. To obtain reliable uncertainty statements,

all uncertainty contributors should be incorporated, including feature form deviations.


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C0A1T23 :

ANO8A :ANA#&SIS OF 8ARIANCE;

&n statistics,analysis of variance (A)*+A% is a collection of statistical models, and their

associated procedures, in which the observedvariancein a particular variable is partitioned into

components attributable to different sources of variation. &n its simplest form, A)*+A provides

astatistical testof whether or not the meansof several groups are all e7ual, and therefore

generali'es t$testto more than two groups. ;oing multiple two$sample t$tests would result in an

increased chance of committing a type & error./or this reason, A)*+As are useful in comparing

two, three, or more means.

An A)*+A analysis techni7ue was used to evaluate the hypotheses presented. A)*+A was

selected due to its ability to make multiple comparisons without accumulating the effects of

alpha (V%. &n the case of this study, the A)*+A allowed for a simultaneous comparison of each

of the variables, including all interactions. 6pecific attention was paid to the various interactions,

because they would indicate which combinations of variables either encouraged or discouraged

use.
http://en.wikipedia.org/wiki/Statisticshttp://en.wikipedia.org/wiki/Statisticshttp://en.wikipedia.org/wiki/Statistical_modelhttp://en.wikipedia.org/wiki/Variancehttp://en.wikipedia.org/wiki/Variancehttp://en.wikipedia.org/wiki/Variancehttp://en.wikipedia.org/wiki/Statistical_testhttp://en.wikipedia.org/wiki/Statistical_testhttp://en.wikipedia.org/wiki/Statistical_testhttp://en.wikipedia.org/wiki/Meanhttp://en.wikipedia.org/wiki/Student's_t-test#Independent_two-sample_t-testhttp://en.wikipedia.org/wiki/Student's_t-test#Independent_two-sample_t-testhttp://en.wikipedia.org/wiki/Type_I_errorhttp://en.wikipedia.org/wiki/Type_I_errorhttp://en.wikipedia.org/wiki/Statistical_modelhttp://en.wikipedia.org/wiki/Variancehttp://en.wikipedia.org/wiki/Statistical_testhttp://en.wikipedia.org/wiki/Meanhttp://en.wikipedia.org/wiki/Student's_t-test#Independent_two-sample_t-testhttp://en.wikipedia.org/wiki/Student's_t-test#Independent_two-sample_t-testhttp://en.wikipedia.org/wiki/Type_I_errorhttp://en.wikipedia.org/wiki/Statistics


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The tests in an A)*+A are based on the /$ratio= the variation due to an experimental treatment

or effect divided by the variation due to experimental error. The null hypothesis is this ratio

e7uals 4., or the treatment effect is the same as the experimental error. This hypothesis is

re


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compare a list of candidate texts. The random$effects model would determine whether important

differences exist among a list of randomly selected texts. The mixed$effects model would

compare the (fixed% incumbent texts to randomly selected alternatives.

;efining fixed and random effects has proven elusive, with competing definitions arguablyleading toward a linguistic 7uagmire.

The calculations of A)*+A can be characteri'ed as computing a number of means and

variances, dividing two variances and comparing the ratio to a handbook value to determine

statistical significance. Calculating a treatment effect is then trivial, Wthe effect of any treatment

is estimated by taking the difference between the mean of the observations which receive the

treatment and the general mean.

Partitioning of the sum of squares

A)*+A uses traditional standardi'ed terminology. The definitional e7uation of sample variance

is

where the divisor is called the degrees of freedom (;/%, the summation is called the sum of

s7uares (66%, the result is called the mean s7uare (#6% and the s7uared terms are deviations from

the sample mean. A)*+A estimates : sample variances= a total variance based on all the

observation deviations from the grand mean, an error variance based on all the observation

deviations from their appropriate treatment means and a treatment variance. The treatment

variance is based on the deviations of treatment means from the grand mean, the result being

multiplied by the number of observations in each treatment to account for the difference between

the variance of observations and the variance of means. &f the null hypothesis is true, all three

variance estimates are e7ual (within sampling error%.


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The fundamental techni7ue is a partitioning of the totalsum of s7uaresSSinto components

related to the effects used in the model. /or example, the model for a simplified A)*+A with

one type of treatment at different levels.

The number of degrees of freedomDFcan be partitioned in a similar way= one of these

components (that for error% specifies achi$s7uared distributionwhich describes the associated

sum of s7uares, while the same is true for WtreatmentsW if there is no treatment effect.

The F!test

The /$testis used for comparisons of the components of the total deviation. /or example, in one$

way, or single$factor A)*+A, statistical significance is tested for by comparing the / test

statistic

8hereMSis mean s7uare, O number of treatments and O total number of cases

to the /$distributionwith , degrees of freedom. Jsing the /$distributionis a

natural candidate because the test statistic is the ratio of two scaled sums of s7uares each of

which follows a scaled chi$s7uared distribution.

The expected value of / is (where n is the treatment sample si'e%

which is 4 for no treatment effect. As values of / increase above 4 the evidence is increasinglyinconsistent with the null hypothesis. Two apparent experimental methods of increasing / are

increasing the sample si'e and reducing the error variance by tight experimental controls.

The textbook method of concluding the hypothesis test is to compare the observed value of /

with the critical value of / determined from tables. The critical value of / is a function of the
http://en.wikipedia.org/wiki/Sum_of_squares_(statistics)http://en.wikipedia.org/wiki/Sum_of_squares_(statistics)http://en.wikipedia.org/wiki/Degrees_of_freedom_(statistics)http://en.wikipedia.org/wiki/Chi-squared_distributionhttp://en.wikipedia.org/wiki/Chi-squared_distributionhttp://en.wikipedia.org/wiki/F-testhttp://en.wikipedia.org/wiki/F-distributionhttp://en.wikipedia.org/wiki/F-distributionhttp://en.wikipedia.org/wiki/Chi-squared_distributionhttp://en.wikipedia.org/wiki/Sum_of_squares_(statistics)http://en.wikipedia.org/wiki/Degrees_of_freedom_(statistics)http://en.wikipedia.org/wiki/Chi-squared_distributionhttp://en.wikipedia.org/wiki/F-testhttp://en.wikipedia.org/wiki/F-distributionhttp://en.wikipedia.org/wiki/F-distributionhttp://en.wikipedia.org/wiki/Chi-squared_distribution


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numerator degrees of freedom, the denominator degrees of freedom and the significance level

(V%. &f / /Critical()umerator ;/, ;enominator ;/, V% then re


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EPERIMENTA# PROCEDURE

The experimental procedure for calculating C## is as follows=

4.0ere the software used in L?$;#&6 +ersion$"" D 1atch$E

!./irstly,take a probe of diameter 4."mm and now calibrate it.

:. 6tandard sphere of diameter !mm is taken and achieved value is 4G.GG" mm.

F.3ecord the diameter of the sphere.

". 8ith that sphere check the diameter of the probe.

. ;ue to program already present in the software the circle,plane and sphere can be measured

automatically.

I. 3epeat the same above procedure forn different lengths such as " mm, mm, I mm.


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E. #achine si'e specification are=

$A&6$

$A&6$"

-$A&6$F

G.3eading achieved are as shown below

Probe Dia Sample Length Orientation Reading Error

1.5 50 X 50.02 0.02

1.5 60 X 60.0061 0.0061

1.5 70 X 70.0292 0.029

3 50 X 49.9877 0.0123

3 60 X 59.9771 0.0229

3 70 X 70.0032 0.0032

6 50 X 49.9961 0.0039

6 60 X 59.9961 0.0039

6 70 X 70.0021 0.0021

1.5 50 Y 50.266 0.0266

1.5 60 Y 60.007 0.007

1.5 70 Y 70.002 0.002

3 50 Y 49.9671 0.032

3 60 Y 60.003 0.003

3 70 Y 69.99 0.01

6 50 Y 49.9913 0.08

6 60 Y 60.004 0.004

6 70 Y 69.997 0.002


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4. )ow calculate the error for the above readings by A)*+A method.

RESU#TS

1.

Probe Dia

SampleLength

Orientation

Reading Error

1.5 50 Y50.26

60.026

6

1.5 60 Y60.00

7 0.007

1.5 70 Y70.00

2 0.002

3 50 Y49.96

71 0.032

3 60 Y60.00

3 0.003

3 70 Y 69.99 0.01

6 50 Y49.99

13 0.08

6 60 Y60.00

4 0.004

6 70 Y69.99

7 0.002

ANO8A

TAB#E


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SSDO

F MSS F

SS Total5229.6

09 8653.701

1SSProbe

478.8356 2

239.4178

0.736252

SS S.L3450.0

36 21725.01

85.3047

36

SS rror1300.7

38 4325.184

4

%.

ProbeDia

SampleLength

Orientation

Reading Error

1.5 50 X 50.02 0.02

1.5 60 X 60.0061 0.0061

1.5 70 X 70.0292 0.029

3 50 X 49.9877 0.0123

3 60 X 59.9771 0.0229

3 70 X 70.0032 0.0032

6 50 X 49.9961 0.0039

6 60 X 59.9961 0.00396 70 X 70.0021 0.0021

ANO8A TAB#E

SSDO

F MSS F

SS Total

811.028888

9 8

101.37

86

SS Probe348.242222

2 2174.12

111.5109

51SSS.Le!"#t

1.828888889 2

0.914444

0.007935

SS rror460.957777

8 4115.23

94


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'.

ProbeDia SampleLength Orientation Reading Error

1.5 50 X 50.02 0.02

1.5 60 X60.006

10.006

1

1.5 70 X70.029

2 0.029

3 50 X49.987

70.012

3

3 60 X59.977

10.022

9

3 70 X

70.003

2

0.003

2

6 50 X49.996

10.003

9

6 60 X59.996

10.003

9

6 70 X70.002

10.002

1

1.5 50 Y 50.2660.026

6

1.5 60 Y 60.007 0.007

1.5 70 Y 70.002 0.002

3 50 Y 49.9671 0.032

3 60 Y 60.003 0.003

3 70 Y 69.99 0.01

6 50 Y49.991

3 0.08

6 60 Y 60.004 0.004

6 70 Y 69.997 0.002

ANO8A

TAB#E

SSDO

F MSS F

SS Total 6262.54 17368.38

47


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SS Probe13.1433

3 26.5716

670.0376

67

SS S.Le!"#t1797.92

3 2898.96

175.1525

78SSOr$e!tat$o!

2357.853 1

2357.853

13.51451

SS rror 2093.62 12174.46

83

CONC#USION


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REFERENCES

htt)$en.6i

cordinate measuring machine

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