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Precision Engineering 38 (2014) 379–390

Contents lists available at ScienceDirect

Precision Engineering

jo ur nal ho me p age: www.elsev ier .com/ locate /prec is ion

n assessment of “variation conscious” precision fixturingethodologies for the control of circularity within largeulti-segment annular assemblies

tewart Lowth, Dragos A. Axinte ∗

olls-Royce UTC in Manufacturing Technology, University of Nottingham, UK

r t i c l e i n f o

rticle history:eceived 4 February 2013eceived in revised form0 September 2013ccepted 4 December 2013vailable online 25 December 2013

eywords:ixturingssemblyircularity

a b s t r a c t

The fixturing of large segmented-ring assemblies is of importance to a number of key high value indus-tries such as the aerospace and power generation sectors. This study examines methods of optimisingthe circularity of segmented-ring assemblies, and how the manufacturing variation within each element(i.e. segment wedge) contributes to overall assembly variability. This has lead to the definition of twooriginal assembly methodologies that aim to optimise an assembly, so that circularity errors are minimi-sed for a given set of components. The assembly methods considered during this study include a radialTranslation Build (TB) and a Circumscribed Geometric (CG) approach, both of which are compared to atraditional Fixed Datum (FD) build method. The effects of angular, radial, parallelism/flatness and chordlength variability within the component geometry, and their effect on the circularity of the final annularassembly are examined mathematically and experimentally. Furthermore, the inherent loss of assembly

egmented annular assemblyariation consciousuild methodologies

circularity due to differences between component and assembly sagitta is also considered, along with thestepping caused by dissimilar adjacent component radii as a result of manufacturing variation. Exper-imental results show that the CG build method offers a significant improvement in circularity in mostsituations over the benchmark FD build method. This contrasts the TB results that proved to be the leastconsistent in terms of circularity, but better in the control of angular breaking errors within the assembly.

201

©

. Introduction

The creation of large geometrically accurate annular compo-ents is of importance to a number of manufacturing industriesuch as aerospace and power generation. These high precisionings are frequently manufactured using a stock material or aear-net-shape blank, on to which various conventional or non-onventional machining processes are employed until the finaleometry is achieved. However, as the ring size or geometricomplexity increases, it may become economically or technicallyeneficial to assemble the ring from smaller segment components.

he down-side to this is that the additional assembly stages willnevitably introduce new sources of possible variation into the pro-uction process.

∗ Corresponding author. Tel.: +44 (0) 115 951 4117.E-mail address: dragos.axinte@nottingham.ac.uk (D.A. Axinte).

ttp://dx.doi.org/10.1016/j.precisioneng.2013.12.004141-6359/© 2014 The Authors. Published by Elsevier Inc.

Open access under CC BY-NC-ND l

4 The Authors. Published by Elsevier Inc.

It is common practice, where the production of repeatable com-ponents is required, that jigs and fixtures are used to aid the processof manufacture. These manufacturing aides deterministically locateand securely clamp single or multiple workpieces, permitting oneor more stages of manufacture to be performed. They offer a benefi-cial impact on the manufacturing process in terms of productivity,cost-per-part and quality [1]. However, designing and manufactur-ing a fixturing system for a specific task can be both time consumingand expensive, with their creation constituting 10–20% of the totalset-up cost of a manufacturing process [2]. Moreover, a badlyrealised fixture concept can lead to manufacturing and machin-ing errors through poor location, flexing of the fixture or geometricinaccuracy [3].

It is scientifically well established that a 3-2-1 locationdatum methodology offers reliable deterministic positioning ofa workpiece relative to its fixture. Nevertheless, even using thisconventional approach positional variability will occur [4]. Thisis especially the case when fixturing an assembly (e.g. Fig. 1), as

Open access under CC BY-NC-ND license.

the location of components is defined by the other componentswithin the assembly as well as directly by the fixture. Camelio et al.[5] expressed these location errors as the deviation between theworld coordinate system (WCS) or workpiece nominal position and

icense.

380 S. Lowth, D.A. Axinte / Precision Engineering 38 (2014) 379–390

Nomenclature

EICi effective inner chord length for component i,adjusted for out-of-plane error (mm)

Ia identifies inner face aIb identifies inner face bICi inner chords of component i, generated via IRi a and

I�i (mm)IC’i adjusted inner chord for component i, based on IRi

and � i (mm)IRi inner radius of component i (mm)I�i inner angle of component i, between faces Ia and Ibn number of components in annular assemblyOa identifies outer face aOb identifies outer face bOCi outer chord of component i, generated via ORi and

O�i (mm)ORi outer radius of component i (mm)O�i outer angle of component i, between faces 0a and 0bPa out-of-plain measurement (parallelism) of compo-

nent, between face Ia and Oa (mm)Pb out-of-plain measurement (parallelism) of compo-

nent, between face Ib and Ob (mm)RAi radial assembly position of component i, for the

translational build assembly method (mm)Rb nominal assembly inner radius diameter (mm)Rcir theoretical build radius of circumscribed geometric

(GC) assembly method (mm)SAi sagitta for chord EICi and theoretical assembly build

radius Rcir (mm)SCi sagitta for chord EICi and inner radius IRCi of com-

ponent i (mm)˛i angular position of fixture datum locator i, for cir-

cumscribed geometric (CG) build methodˇi angle of component inner chord relative to assembly

centre� i calculated true angle of across componentεi inherent loss of circularity due to sagitta differences

between assembly and component (mm)εmax the largest value of εi within the assembly’s compo-

nents (mm)�i deviation (stepping) of component i from assembly

build radius (mm)

LhCtgZthpcftcpttc

i

ϕi breaking angle between assembled components

ocal Coordinate System (LCS) or workpiece measured position; aomogeneous transformation matrix, within Euclidean space usingartesian coordinates express this deviation. This matrix describeshe Six Degrees of Freedom (6-DOF), by combining the three ortho-onal XYZ translations and the angular rotation about the X, Y and

axes, as required for the positioning of the workpiece withinhree dimensional Euclidean space. Song and Rong [6] built on theomogeneous matrix method and the 3-2-1 methodology, to pro-ose a technique for establishing if a component is over or underonstrained by its fixture. More recent scientific interest has beenocused upon variation propagation within multi-stage manufac-uring processes using a mathematical state-space model [7–10],ommonly referred to as the Stream-of-Variation (SoV). The SoVaradigm combines the error contributions in matrix form fromhe current and preceding manufacturing operations, manifesting

he result as a state vector representation of the total error in theurrent process.

Predicting the quality of workpiece-fixture location compliances also an important consideration in the understanding of fixturing

Fig. 1. Face–face location of assembly components.

error, on which topic a number of scientific reports have beenpublished [11–13] using a Finite Element Analysis (FEA) approach.These studies examine the clamping of components misshapen bymanufacturing errors, with respect to component deflection andfixture deformation as a result of the changes in fixture reactionforces. The idea allowed for the adjustment of successive CNCmachining operations to compensate for the errors, Abellan-Nebot et al. [14] proposed fixture embedded sensors to measureworkpiece error. Furthermore, a number of active fixture designapproaches have also been proposed [4,15–17], where location andclamping conditions are adjusted to balance errors transmittedfrom previous manufacturing stages. The fixture compliance andthe active clamping studies published to date have generallyconcentrated on how a single component interacts with its fixture,but there has been little consideration of multi-part assemblies.When considering an assembly it is usual that some of the degreesof freedom for the individual components are constrained bytheir interface with other assembly components, rather than thefixture itself. These component–component interactions will playa role in deformation modes and conformity when considering theclamping forces in a fixture.

In the main, the interest of the science within the field offixturing has been levelled at deterministically holding a singlecomponent. However, Huang et al. [18] extended the SoV method-ology to propose a model for the assembly of rigid-bodies withina single fixture. Various types of inter-component joint types weremathematically defined with two matrices. The first, called the“twist matrix” which defined the DOF in which the componentwas kinematically permitted to move, while the second “wrenchmatrix” contained the remaining constrained DOF. Validation ofthis SOV method was carried out using a 10,000 item Monte Carlostudy, which was compared to a 3DCS Analyst a commercial Com-puter Aided Tolerancing (CAT) system, using a 5000 item study. TheMonte Carlo SOV and CAT results were found to be <1.5% differentin their results.

Two related studies [19,20] attempting to minimise axial errorbuild-up within the linear stacking of cylindrical gas turbine com-ponents have been conducted. The authors used various buildprotocols in an attempt to evaluate and improve the co-axialitybetween segments and thus build a straighter final assembly. Bothstudies showed the optimum build algorithm to be one of minimis-ing the distance between the component axis and the axis of the ref-erence “table”, rather than minimising the component–componentaxial distance. Although the face–face stacking contact is analo-

gous to the inter-component contact seen within a segmented-ringassembly, these studies are based very specifically on linear assem-blies where the use of cylindrical segments allows for the rotation ofa component to improve its fit. This advantageous situation cannot

S. Lowth, D.A. Axinte / Precision Eng

MZC

(0,0,0)

Diametric mean

Measurand

OMZCI

IMZCI

bm

ADbpDdstaateptaihiisroc

btaocSommsdbOdrobaIcmtv

assembly using the chord length variation in the components. These

Fig. 2. The measurement of circularity.

e extended to the assembly of segmented rings, where the seg-ent components are axially asymmetric.A number of researchers [21–24] have investigated Computer

ided Tolerancing (CAT) as a method of assessing the Geometricimensioning and Tolerancing (GD&T) of assemblies, as definedy ASME Y14.5-2009 or ISO 1101:2004 standards. These softwareackages typically interface with commercial 3D Computer Aidedesign (CAD) packages, and allow the user to specify geometriceviation and assembly variance though the use of assembly con-traints or directly as parametric dimensions embedded withinhe geometry. Once specified the described manufacturing vari-tion is then shown as a transparent envelope surrounding thessembly’s components. The component’s assembled position ishen altered based on this boundary, thus showing the effect of therror/variance within the assembly system. Byungwoo et al. [25]roposed and demonstrated the use of CAT combined with FEAo predict the assembly deviation of non-rigid component whenssembled. CAT could easily be levelled at the problem of predict-ng circularity within annular assemblies and it would demonstrateow segment manufacturing errors and build conditions resulted

n differing amounts of circularity in the final assembly. However,t would be more beneficial to achieve more fundamental under-tanding of the cause of deviation from circularity in segmented-ing assemblies. This fundamental approach will allow the designf a “variation conscious” fixture to pre-determine the best possibleircular arrangement from any given set of ring segments.

To fully understand how to create a high quality annular assem-ly through fixturing, it is important to understand how to evaluatehe assembly’s circularity. There is a body of metrological literaturevailable concerned with the challenge of assessing the circularityf a dataset made entirely from discrete Cartesian points, asollected from a Coordinate Measuring Machine (CMM). Leastquares Method (LSM) is a technique used to gauge the amountf deviational error using curve fitting within a dataset of CMMeasurands relative to a true circle [26]. Despite LSM being mathe-atically robust it is not fully compliant with geometrical tolerance

tandards such as ISO 1101:2004. As Fig. 2 shows, this standardefines the circularity as the deviation of the measured geometryetween an Inner Minimum Zone reference CIrcle (IMZCI) and anuter Minimum Zone reference CIrcle (OMZCI), referring to theistance as the Minimum Zone Circle (MZC). Due to this, moreecent research has adopted a geometric approach through the usef simultaneous equations derived from the Euclidean distancesetween the measurands [27,28]. These are then used to calculate

best fit MZC for the measured component, offering a moreSO standard compliant approach than the LSM measurement of

ircularity. Continuing the geometric approach, Samuel and Shun-ugam [29] created circumscribed and inscribed hull boundary

hrough the measured points, these boundaries are then subdi-ided using the equi-distant (Voronoi) and equi-angle techniques

ineering 38 (2014) 379–390 381

to work back to circle centres. This method was later expandedto improve the accuracy of a roundness form measuring machinedata, as previous studies had shown that errors in the measuringmachine were creating non-circular limacons as they were sweptaround the test subject [30]. Although these studies are useful tothe understanding of what constitutes circularity and the qualityof the finished assembly, they convey little about the componentsused or the impact of manufacturing variance within them.

On review of current literature, there appears only to be a limitedscientific understanding of how best to build annular assemblies,especially with regards to how circularity is affected by the varianceof individual components. There is a great need for the creation ofassemblies with high degrees of circularity, as expensive following-on machining process can be reduced if a “near finished part”condition can be achieved during assembly. Furthermore, subse-quent circular components (a hub for example) may be added inlater assembly stages, therefore a more predictable ring assemblywill make it easier to accommodate these items.

With this in mind, the scope of this study is to assess theuse of various assembly fixture design methodologies for the cre-ation of geometrically optimal annular assemblies. This will beaccomplished via a mathematical model of each method, whereeach assembly methodology will be expressed in terms of OuterMinimum Zone reference CIrcle (OMZCI), Inner Minimum Zonereference CIrcle (IMZCI) and Minimum Zone Circle (MZC). Experi-mental testing enabled by an innovative flexible fixture and a setof generic components, will then be used to substantiate the valid-ity of these build approaches. The expressed methodologies willbe “variation conscious”, as the assembly method will be optimisein response to manufacturing variability within the given set ofcomponents. During this study a number of assumptions (with rel-evance to a given industrial application) about the assembly weremade, namely:

• The holistic circularity of the assembly is of greater functionalimportance than the absolute position of the individual compo-nents or size of the segmented-ring assembly.

• This study is only interested in the internal diameter of the assem-bly.

• Each component is unique and cannot be exchanged for anotherin the assembly.

• The assembly consists of I-shaped wedge segments.• The components comprising the assembly are considered per-

fectly rigid non-deformable bodies.• To create a closed segmented-ring assembly each wedge compo-

nent must contact the components on either side.

2. Definition of assembly methodologies for annularassemblies

This study will examine three build methods used for the con-struction of annular assemblies. The assembly’s will be constructedfrom I-shaped wedge components, intended to simulate the geo-metric characteristics of common aerospace components. The buildmethods to be examined in this study are:

• Fixed Datums (FD)• Translational Build (TB)• Circumscribed Geometric (CG)

Each method (with the exception of FD) calculates a best fit

methodologies will be discussed in greater detail below, beforebeing tested experimentally. This will determine the receptivenessof each build method to component manufacturing variation, whiletrying to control the circularity of final assembly.

382 S. Lowth, D.A. Axinte / Precision Engineering 38 (2014) 379–390

Face Oa Face Ob

Face Ia Face Ib

IC

OCi

γi

Oθi

Iθi

ORi

IRi

2

tIr

ppAImfioba

�n ((1

ORi

a

I

pbm

eIa

E

wao

i

Fig. 3. Calculating the true angle across an I-shaped wedge component.

.1. The effect of manufacturing deviation on chord length

Before the build methods are explained, it is useful to analysehe manufacturing and geometric variability implicit to each of the-shaped wedge components, as these will ultimately influence theesulting circularity of the annular assembly.

As with any manufactured component, I-shaped wedge com-onents will be subject to implicit variation caused/induced byrocessing methods (e.g. mechanical/thermal induced distortions).mong which, there is likely a degree misalignment between faces

a and Oa, and similarly between Ib and Ob, as shown in Fig. 3. Thisakes measuring the true angle across the entire component dif-

cult. Nevertheless, it can be calculated using the inner (ICi) anduter OCi) chord lengths. Looking at Fig. 3 again, the true angle (� i)e found using Eq. (1), where I�i is ∠IaIb, O�i is ∠OaOb, IRi and ORire the inner and outer radii respectively.

i = 2 · tan−1

⎛⎜⎜⎝ IRi sin ((1/2)I�i) − ORi si

ORi − IRi +(

IRi −√

IR2i

− (IRi sin ((1/2)I�i))2

)−(

Using the value � i, it is possible to estimate the componentsdjusted inner chord (IC′

i) using Eq. (2).

C ′i = 2 · IRi sin

(12

�i

)(2)

However, as faces Ia and Oa, and Ib and Ob, are likely out oflane from each other, so the parallelism must also be consideredefore the final effective inner chord length (EICi) can be deter-ined (Fig. 4).Using Pa and Pb as the out-of-plane variation (parallelism) of

ach side of the I-shaped wedge component, and considering theSO 1101:2004 standard, the resultant effect of the parallelism devi-tion on the chord length can be calculated using Eq. (3).

ICi = IC ′i +(

(1/2)Pa

cos ((1/2)�i)

)+(

(1/2)Pb

cos ((1/2)�i)

)(3)

It is worth noting that solid non-I-shaped wedge segmentsould obviously only have, when assembled, a single mating face

nd in these instances the surface flatness should be used insteadf parallelism.

/2)O�i)

−√

OR2i

− (ORi sin ((1/2)IOi))2

)⎞⎟⎟⎠ (1)

Fig. 4. Extending the inner chord in accommodate out-of-plane errors.

The quantity EICi therefore, indicates the adjusted chord lengthwith consideration of the manufacturing variation. Furthermore,this value offers a basis on which to base the proposed assemblymethods.

2.2. Method 1 – The Fixed Datum (FD) build approach

The simplest scheme for datuming an annular assembly is aseries of fixed datums pins that are positioned at the intendednominal internal or external diameter. While a simplistic methodto instigate fixed datums are not “variation conscious”, i.e. paylittle attention to the manufacturing variations within the assem-bly’s components. This is the “traditional” fixturing method for asegmented-ring assembly, and will be included in this study as abenchmark by which to compare the other methodologies. For thepurpose of this study the angular position of the pins will be equi-spaced, along the nominal centre line of the assembly components.

The obvious issue with this assembly method is that, shouldthe manufacturing variability (previously discussed) within the I-shaped wedge segments result in an undersized ring, it will nolong fit properly to the fixed datum locators. Similarly, if the resul-tant assembly is oversized then the assembly’s inner diameter will

be too large to touch the locators at all points around the ring. Bothof these scenarios lead to the assembly’s components no longerbeing deterministically positioned by the fixture, allowing circu-larity errors to be induced in the assembly. Furthermore, a humanoperator is likely to try and force fit the components to the loca-tors exacerbating the deviation in the assembly, as these are theonly real visual assembly references available. Therefore, this is asituation that needs to be avoided as it induces further geometricaldistortion to the assembly.

A point of note with the FD or indeed any build method thatplaces a single datum block at the centre of the wedge component’sinner circumference edge, is its propensity to produce steppingwithin the finished assembly. These steps occur as each segmentcomponent will have a slightly different inner radius from that ofits neighbouring components, and as all the assembled components

are positioned at the same build diameter, so stepping in the assem-bly is to be expected. This stepping will inevitably have an adverseaffect on the assembly’s circularity. Eq. (1) can be used to calculatethe step deviation (�i) of a given segment (i) from the nominal build

S. Lowth, D.A. Axinte / Precision Engineering 38 (2014) 379–390 383

I - S h a p e d w e d g e c o m p o n e n t

Rb

Fixeddatum βi

ρi

Rb

Iθi

βiIRi

dprrttt

�

n ((1

tativdI

2

mcb

t

sn

∑

aˇecat

Ri si

Fig. 5. Traditional fixed datum build with stepping example.

iameter, where IRi as the inner radius of component i (as shownreviously in Fig. 3), I�i the Inner angle of component i, Rb the buildadius of the fixture, and ˇi the wedge angle of the componentelative to the fixture centre (see Fig. 5). The equation subtractshe perpendicular sagitta distance of the fixture’s build radius fromhat of the component, and projects the result on to the angle ofhe wedge.

i = Rb −

(IRi −

√IR2

i− IRi sin ((1/2)I�i)

2

)−(

Rb −√

R2b

− Rb si

cos ((1/2)ˇi)

Eq. (4) offers a quantification of the non-circularity generated viahe FD build method, due to the described stepping. The final ringssembly’s IMZCI can be considered the lowest value of �i withinhe components that populate the assembly or the Rb value shouldt be smaller. While, the OMZCI of the assembly will be the largestalue of �i or the value of Rb should be it be greater. The MZC oregree of non-circularity can then be derived by subtracting the

MZCI from the OMZCI.

.3. Method 2 – The Translational Build (TB) approach

The fundamental premise of the TB approach (Fig. 6) is that theanufacturing variance in each component’s chord length can be

ompensated by altering its radial position relative to the assem-ly’s centre.

Fig. 6 shows ˇi as the angle of the component’s chord relative tohe assembly centre (marked 0, 0, 0). Therefore, to create a closed

egmented-ring assembly Eq. (5) must be satisfied, where n is theumber of segments used to construct the assembly.

n

i=1

ˇi = 2� (5)

Obviously, in an ideal assembly with no manufacturing vari-tion, both � i (the I-shaped wedge components true angle) andi (assembly relative chord angle) would both be identical. How-

�i = RAi −

(IRi −

√IR2

i− I

ver, variation is inevitable, so it is unlikely that the sum of theomponent � i angles will equal exactly 2� radians. Even so, asngle ˇi is relative to the chord’s distance from the assembly cen-re, so translating the components radial position will alter the

/2)ˇi)2

)(4)

Fig. 6. The translational build method.

angle ˇi. Consequently, adjusting the radial position of each assem-bly component independently based on angle ˇi, it is possible to

compensate manufacturing variations in the components innerchord length (and thus as they are interrelated the IRi and � i dimen-sions). By adjusting all the components that populate an assemblyin this manner, it is possible to fulfil the requirements of Eq. (1).Fig. 6 demonstrates how this would work in practise, when appliedto a theoretical assembly. The radial position RAi, required to correctˇi for a given component can be calculated using Eq. (6).

RAi = 12

(EICi

sin (�/n)

)(6)

It is worth noting that this method of error compensation gen-erates stepping at the inner and outer radius of the assembly (asshown in Fig. 6), and as with the FD method, this is a source of non-circularity within the final assembly. The degree of the stepping inthe TB method (�i) can be quantified using Eq. (7); this is a variationof Eq. (4) (employed for calculating the stepping in the FD method),that uses the translated distance (RAi) of the segment as shown inEq. (6), as opposed to the Rb build radius.

n ((1/2)I�i)2

)−(

RAi −√

RA2i

− RAi sin (�/2)2

)cos ((1/2)ˇi)

(7)

Therefore, calculating �i with Eq. (7) the adverse effect of step-ping on the circularity of a TB assembly can be calculated in asimilar manner to the previous FD method. Furthermore, like theFD method, the assembly’s IMZCI can be realised as the lowest valueof �i within the assembly’s components or the RAi value should itbe smaller. While, the OMZCI of the assembly will be the largest ofthese �i values or the RAi value should it be larger. Again, the MZCor degree of non-circularity can then be derived by subtracting theIMZCI from the OMZCI.

2.4. Method 3 – Circumscribed Geometric (CG) polygon approach

A segmented-ring assembly can be geometrically simplified byconsidering it a cyclic n-sided polygonal chain, which is constructed

384 S. Lowth, D.A. Axinte / Precision Eng

fRtcpt

tatbdtftttt

ceai

2

Fig. 7. The circumscribed build method.

rom the chord lengths of the wedge components effective inneradii (EICi). However, to maintain the circularity of the assemblyhis polygonal chain must be constrained to a theoretical datumircle. The circumscribed approach (Fig. 7) places the intersectionoints of the polygon’s sides directly onto the circumference of thisheoretical base circle.

As discussed previously, the component’s chord length is subjecto manufacturing variations originating within the angle, radiusnd out-of-plane deviations of the component. However, unlikehe TB method where the manufacturing variation of each assem-led component is corrected independently, the use of a theoreticalatum circle creates a dependency between all of the segments inhe assembly system. The implication of which, is that deviationrom nominal in any of the chord lengths will cause a change in theheoretical datum circle, causing the entire assembly build diame-er to shrink or grow. Nonetheless, the benefit of this approach ishat it eliminates the stepping of the inner and outer diameters ofhe assembly as seen on FD and TB approaches.

To build an assembly using a CG approach the theoreticalircumscribed datum circle must be calculated. Eq. (5), can bexpanded to express the angle of each component relative to thessembly, in terms of the chord length, to create Eq. (8) where Rcir

s the radius of the theoretical base circle.

� =n∑

i=1

2 sin−1(

EICi

2Rcir

)(8)

Undersize Σγi < 360°

φi

γi

Fig. 8. Effect of angular breaking error on overs

ineering 38 (2014) 379–390

Within Eq. (8), all the variables can be found (by measuringthe physical components), with the exception for the value of Rcir.Therefore, the equation can be solved to find the value of Rcir, gener-ating the required build diameter for the given set of components.

As Fig. 7 demonstrates, the circumscribed build method alsorequires that the fixture’s locator blocks to be placed at the inter-face between the assembly wedge segments. Again, the position ofthese interfaces is dependent on each component’s effective chordlength (which are subject to manufacturing variation, see Section2.1). Finding the angular position of a given fixture datum locator isfurther complicated as the build radius of the assembly shrinks orgrows due to segment chord length variation. This in effect changesthe ˇi angle of every component relative to the assembly. Eq. (9)shows how to calculate the angular position of the datum locatorbased on effective chord length. Where ˛i is the angular position ofthe current fixture locator, which is based on the angular positionof preceding fixture locators.

˛i = 2 sin−1

((1/2) EICi

Rcir

)+

n−1∑i=1

˛i−1 (9)

To summarise the CG build methodology; firstly, the theoret-ical build diameter Rcir must be derived from Eq. (8). This valuewill determine the radial position of all the datum locators basedon the chord lengths of the I-shaped wedge components that willcomprise the assembly. The angular position of each datum loca-tor should then be found with Eq. (9). Setting the fixture’s datumlocators at these angular and radial positions will enable the assem-bly to be constructed using the circumscribed method as shown inFig. 7.

2.5. Angular breaking error

As stated in Eq. (5), to create a closed ring the sum of the angleswithin the assembly’s constituent components must equal 2� radi-ans. The CG build method adjusts the position of all the componentsrelative to the assembly, modifying the ˇi angles of the compo-nents so that the assembly is in accordance with Eq. (5). However,as the physical geometry of the I-shaped wedge components areun-altered, so the sum of angular deviation still exists physicallywithin the assembly. This deviation manifests itself as a series of

angular breaks at either the inner or outer diameter of the assem-bly. If the sum of the � i angles is less than 2� radians, the angularbreaking will occur in ring assembly’s outer diameter. Whereas, ifthe sum of the � i angles is greater than 2� radians, then the angular

Oversize Σγi > 360°

φi

γi

ized and undersized annular assemblies.

S. Lowth, D.A. Axinte / Precision Eng

EICi

RCIR

IRi

SAi

SCi

εi

Fs

ba

bo(

∑

se�Fmte

2

iitpr

ig. 9. Inherent loss of circularity due to difference in component and assemblyagitta.

reaking errors will occur in the inner ring, both of these situationsre demonstrated in Fig. 8.

The amount of the angular breaking error present in the assem-ly can be determined using Eq. (10), where ˙ϕi is the total amountf angular breaking error at either the outer (if negative) or innerif positive) diameter of the ring.

n

i=1

ϕi = 2� −n∑

i=1

�i (10)

This known, a conclusion that oversized ring assemblies whereum of the � i angles is larger than 2�, it would be preferential to usexternal datum locators. While, assemblies where the sum of thei angles is less than 2�, would perform better if internally located.urthermore, in a practical production or manufacturing environ-ent, it would stand to reason to avoid transitional builds, where

he component’s tolerances allow for a plus or minus situation toxist within the same fixture.

.6. Inherent loss of circularity

A point of note with the circumscribed build method is theres an inherent loss of circularity, due to variability in the theoret-

cal base circle. As Fig. 9 demonstrates, a diametric change in theheoretical build circle (Rcir) results in a mismatch with the com-onent’s inner radius (IRi). It is this difference between these twoadii that creates the inherent loss of circularity in addition to any

ALL DIMENSIONS IN MM

4 4 0

2 0 0

X °

= =

2 5

Fig. 10. Experimen

ineering 38 (2014) 379–390 385

manufacturing deviation in the components radius. The loss of cir-cularity can be considered as the difference between the sagitta ofIRi and Rcir as they share a common chord length. This differencecan be calculated with Eq. (11), where εi is the loss of circularity atsegment i, the SCi is the sagitta of component i, and SAi the sagittafor the theoretical base circle using chord EICi.

εi = SCi − SAi (11)

Eq. (11) can be further expanded into Eq. (12), using classicalgeometric equations.

εi =(

IRi −√

IR2i

−(

EICi

2

)2)

−(

Rcir −√

R2cir

−(

EICi

2

)2)

(12)

Because of the manner in which the circumscribed method con-structs an annular assembly, the greatest value for εi within theassembly (εmax) is in effect the MZC of the assembly, as describedin ISO 1101. Furthermore, the theoretical base circle (Rcir) is theIMZCI, and from this information the OMZCI can be calculated byadding the MZC to the IMZCI. In contrast to the traditional FD andproposed TB methods, where �i (the stepping) is the final enablerfor the calculation of non-circularity, the CG method uses the quan-tity εi instead, and as such allows a comparison of non-circularitywith the previous build methods.

3. Experimental validation of build approaches

3.1. Validation approach: test-bench

The premise of the experimentation is that, by buildingsegmented-ring assemblies which contain the same known com-ponent deviation and measuring the resultant circularity, the buildmethod’s sensitivity to component variability can be quantified.To validate the FD, TB, and GC build methods experimentally, a 10segment assembly (shown in Fig. 10) with I-shaped wedge com-

ponents was used as a demonstrator to evaluate the build qualitywithin a generic annular assembly system. The wedge componentswere designed so that their angle (shown as X in Fig. 10) could becontrolled, permitting the intentional creation of known explicit

3RD ANGLE

6 H 7 ( 2 x P E R C O M P O N E N T )

TYPICAL "I"-COMPONENT(10x PER RING)

tal assembly.

386 S. Lowth, D.A. Axinte / Precision Engineering 38 (2014) 379–390

comp

awne1ttp

3

c

Fig. 11. Plot of the combined implicit and explicit errors of test assembly

ngular (and thus chord) deviation. For this study a total of 18edge segment were manufactured, 10 were manufactured with aominal 36◦ wedge angle (labelled as 36-1 to 36-10). The remainingight segments were spilt, with four having a +2◦ (labelled as 38-

to 38-4), and the remaining four a −2◦ explicit error added intohe component (labelled as 34-1 to 34-4). The components werehen CMM measured to quantify any implicit angular, radial, orarallelism deviation (Fig. 11).

.2. Fixturing test-bench: as validation enabler

Fig. 12 shows the make-up of the fixture and its design, theore feature of which is its flexibility. As Fig. 13 shows each of the

Fig. 12. General assembly of fix

onents in terms of the angle (�i), radius (IRi) and chord (EICi) dimensions.

10 clamping units is constrained between two circular runningedges milled into the base unit. Each clamping unit runs on a plainbearing (Fig. 12 item 008), connected to the underside of the lowerradial carriage (Fig. 12 item 002). This arrangement allows eachunit and thus component to be positioned angularly. The I-shapedwedge component is clamped between a threaded clampingblock (Fig. 12 item 007) and the datum locator block (Fig. 12 item005). This assembly sits atop a linear slide (Fig. 12 item 009),allowing the component unit to be slid along the radial line. Once

positioned, the unit is then locked in place with the angular lockingplate (Fig. 12 item 003) and split collar rod clamp (Fig. 12 item011). This flexibility permits a total inner diametric constructionrange of 180–225 mm, and a ±10◦ angular envelope per unit. This

ture and clamping units.

S. Lowth, D.A. Axinte / Precision Engineering 38 (2014) 379–390 387

Table 1Results of CMM fixture accuracy test.

Angle between adjacent unit datum dowels (inner Ø6) Radial distance from fixture centre to unit datum dowel (inner Ø6)

◦ ◦ ◦ CAD

82.0

fldu1ts

ilttruawt

3

ftc

CAD nominal ( ) CMM mean ( ) Max/min ( ) Standard dev.

36.000 36.024 +0.628/−0.353 0.280

exibility enables the test fixture (Fig. 14) to build with: fixedatum, radial translated, or the circumscribed build method. Eachnit’s upper carriage (Fig. 12 item 001) can also be rotated though80◦, allowing for the fixture to also build using datum locators onhe external radius, although this feature will not be used for thistudy.

Each of the 10 clamping units contains two datum dowels (Ø6nner most and Ø3 outermost), these are used to reference the angu-ar and radial position of each unit. A datum plate was also createdo align these unit dowels relative to the fixture centre, so thathe fixture is set-up for the nominal build with a good degree ofepeatable consistency. The nominal unit position being with eachnit equi-spaced at 36◦ and the datum locator blocks positioned to

build diameter of 200 mm. To check the accuracy of the fixture, itas measured in this nominal state (after using the datum plate),

he result being shown in Table 1.

.3. Validation approach: experimental evaluation of build errors

The study considered the FD, TB, and GC build methods against

our basic scenarios, each with differing amounts of angular (andhus chord) error present within the assembly’s I-shaped wedgeomponents, these were:

Linear slide toadjust radialposi�on

Adjust angle

Fixture datum(0,0,0)

Timing datum (angular zero)

Outer unit running edge

Inner unit running edge

Locking collar

Angular Lock plate

Fig. 13. Diagram demonstrating fixture adjustment with single unit.

Fig. 14. Photograph of experimental flexible fixture.

nominal (mm) CMM mean (mm) Max/min (mm) Standard dev.

00 82.1301 +0.192/−0.140 0.120

(A) Nominal build with no intentional error, only implicit manu-facturing variation (�� i ≈ 360◦).

(B) Balanced, the explicit errors present but they broadly cancel-out (�� i ≈ 360◦).

(C) Oversized, intentional explicit oversized error (�� i > 360◦).(D) Undersized, intentional explicit undersized error (�� i < 360◦).

For each of these scenarios four randomised builds were cre-ated from the manufactured components. As an example, to complywith scenario A, just the components labelled 36-1 to 36-10 wereused in the fixture. While the scenario D, used primarily the 36-1to 36-10 components, but also had included one or two from the34-1 to 34-2 (undersized) components. To ensure that the samebuild was used in each fixture set-up, the fixture units were num-bered 1–10 clockwise, with unit one being the unit in-line withthe angular timing datum (see Fig. 13). The full list of builds, withcomponent and unit numbers can be seen in Table 2.

Each of these builds, were then constructed using the FD, TBand GC methods. The fixture datum plate was used to set-up thefixture units for the FD build: being a 200 mm build diameter and36◦ between the locators. For the TB method, Eq. (6) was used tocalculate the translated radial position of each unit, based on theI-shaped wedge component that was loaded into it. To set-up thefixture for the TB method build, it was initially set to the FD “home”position using the fixture datum plate. Each unit was then slid intoits individually calculated radial position. The unit’s radial posi-tion was the checked between the fixture datum dowel (Fig. 13)and the unit primary datum dowel (Fig. 12 item 012) with a digitalcalliper, before the unit was locked in place. The 36◦ angular unitposition was maintained for the TB builds. For the GC build methodthe build diameter was derived using Eq. (8), and the angular posi-tion of each unit set using Eq. (9). Again, for the CG build the fixturewas primed using the fixture datum plate, before all the units wheremoved to the same calculated build diameter. The unit positionswere checked using a digital calliper before being radially lockedinto place. The fixture datum plate also contains an angular scale(with 0.5◦ accuracy), this was used to set the angular position ofeach unit as calculated. After each assembly was constructed theinternal diameter of the assembly was measured (at 20 points, 2per segment), in terms of mean inner diametric distance and cir-cularity using a Mitutoyo Euro-C-A-121210 coordinate measuringmachine. The results of these measurements can be seen in Table 3and Fig. 15.

Looking at the raw data the results show the GC approach out-performed both the FD and TB methods in terms of circularityby some considerable margin, when each build method is con-fronted with the same amount of assembly error. Whereas, theworst performing build method is the TB method, which consis-tently produced the worst build circularity, due to the presence ofsubstantial stepping at the inner diameter. However, this is not theentire story, as during the experimental measurements the pres-ence of any angular breaking error was noted, along with its generalposition (see Table 4). Looking at the angular breaking errors anumber of the build methods created breaks at the inner radius

in response to the given errors. These builds should be consideredas failed, as inner radius is not completely closed, and as stated inthe assumptions it is the inner radius that is the focus of this study.Looking at the builds purely in terms of closure the TB method is the

388 S. Lowth, D.A. Axinte / Precision Engineering 38 (2014) 379–390

Table 2Experimental build combinations.

Component number at uni t Pos i�on

Buil d ID Expli cit

error

Unit

1

Unit

2

Unit

3

Unit

4

Unit

5

Unit

6

Unit

7

Unit

8

Unit

9

Unit

10

A1 0° 36-4 36-8 36-10 36-5 36-1 36-6 36-3 36-9 36-7 36-2 Nom

inal

A2 0° 36-5 36-4 36-2 36-8 36-9 36-3 36-10 36-7 36-6 36-1

A3 0° 36-1 36-4 36-3 36-6 36-8 36-9 36-10 36-2 36-7 36-5

A4 0° 36-2 36-7 36-5 36-8 36-4 36-1 36-9 36-10 36-6 36-3

B1 ±2 ° 36-3 36-1 34-2 36-8 36-4 36-7 38-3 36-9 36-2 36-5 Balanced B2 ±2 ° 36-1 34-3 36-6 36-2 36-8 36-10 36-5 36-7 36-4 38-4

B3 ±4 ° 36-10 36-5 34-1 36-2 38-4 34-4 36-3 36-1 38-2 36-4

B4 ±4 ° 34-2 36-2 36-8 38-3 36-7 38-1 36-1 36-6 34-3 36-9

C1 +2 ° 36-8 38-4 36-5 36-4 36-2 36-10 36-6 36-9 36-1 36-3 Oversize

C2 +2 ° 36-10 36-9 36-3 36-1 36-5 36-4 38-2 36-8 36-6 36-7

C3 +4 ° 38-3 36-10 36-2 36-8 36-3 36-4 36-7 38-1 36-5 36-9

C4 +4 ° 36-1 36-6 38-2 36-3 38-3 36-9 36-5 36-7 36-10 36-4

D1 -2° 36-7 36-6 34-2 36-9 36-4 36-2 36-8 36-3 36-5 36-10 Undersize

D2 -2° 36-6 36-9 36-10 36-5 36-1 34-4 36-7 36-4 36-2 36-3

D3 -4° 36-1 36-4 36-7 34-3 36-6 36-2 36-9 36-8 36-3 34-1

D4 -4° 36-2 34-3 36-3 36-8 36-6 36-10 36-5 34-4 36-4 36-7

38-X = 34° Ove rsized segment 34-X = 34° Und ersized segment 36-X = Nominal (n o expli cit err or)

198.0

199.0

200.0

201.0

202.0

203.0

204.0

205.0

206.0

207.0

195.0

196.0

197.0

194.0

A1 A2 A3 A4 B1 B2 B3 B4

Nominal

Balanced BuildNominal Build Oversize Build Undersize Build

C1 C2 C3 C4 D1 D2 D3 D4

Mean MZC (A1-4)

FD = 1.141 mm

TB = 2.003 mm

CG = 0.242 mm

Mean MZC (C1-2)

FD = 1.552 mm

TB = Unable to build

CG = 0.900 mm

Mean MZC (C3-4)

FD = 2.817 mm

TB = Unable to build

CG = 1.0775 mm

Mean MZC (D1-2)

FD = 2.029 mm

TB = 5.412 mm

CG = 0.391mm

Mean MZC (D3-4)

FD = 2.105 mm

TB = 7.405 mm

CG = 0.228 mm

Mean MZC (B1-4)

FD = 2.292 mm

TB = 7.235 mm

CG = 0.456 mm

- FD assembly diameter (mm)

- TB assembly diameter (mm)

- CG assembly diameter (mm)

- OMZCI (mm)

- IMZCI (mm)

Build identifier

Ass

embl

y di

amet

er a

nd IM

ZC

I / O

MZ

CI /

MC

Z c

ircul

arity

dev

iatio

n (m

m)

Fig. 15. Plot of experimental results for diameter, OMZCI, IMZCI and MZC, against build type.

S. Lowth, D.A. Axinte / Precision Engineering 38 (2014) 379–390 389Ta

ble

3B

uil

d

resu

lts

for

TD, T

B

and

CG

buil

ds.

Bu

ild

ID

n ∑ i=1

�i

(◦)

Fixe

d

dat

um

s

(FD

)

Tran

slat

ion

al

buil

d

(TB

)

Cir

cum

scri

bed

geom

etri

c

(CG

)

Mea

nd

iam

eter

(n

=

20)

(mm

)

MZC

(mm

)

+OM

ZCI

−IM

ZCI

(mm

)

Mea

nd

iam

eter

(n

=

20)

(mm

)

MZC

(mm

)

+OM

ZCI

−IM

ZCI

(mm

)

Cal

cula

ted

dia

met

er2R

ab

(mm

)

Mea

nd

iam

eter

(n

=

20)

(mm

)

MZC

(mm

)

+OM

ZCI

−IM

ZCI

(mm

)

Idea

l

360.

000

200.

00

0.00

0

+0.0

00−0

.000

200.

000

0.00

0

+0.0

00−0

.000

200.

000

200.

00

0.00

0

+0.0

00−0

.000

A1

359.

496

200.

706

0.84

1

+0.5

11−0

.330

200.

661

2.28

7

+1.2

39−1

.048

200.

324

200.

740

0.22

5

+0.1

24−0

.101

Nom

inal

buil

dA

2

359.

496

200.

755

1.25

8

+0.7

11−0

.547

200.

667

1.57

3

+0.7

15−0

.858

200.

324

200.

702

0.18

8

+0.0

69−0

.119

A3

359.

496

200.

709

1.59

5

+1.0

11−0

.585

200.

661

2.38

9

+1.3

17−1

.072

200.

324

200.

747

0.28

0

+0.1

56−0

.124

A4

359.

496

200.

766

0.88

8

+0.4

67−0

.401

200.

668

1.76

3

+1.0

30−0

.763

200.

324

200.

822

0.27

4

+0.1

44−0

.130

B1

359.

427

203.

013

2.18

6

+0.9

31−1

.235

200.

587

5.83

0

+2.4

69−3

.361

201.

049

200.

967

0.32

1

+0.1

48−0

.173

Bal

ance

d

erro

rB

2

359.

588

203.

309

2.08

5

+0.9

23−1

.162

200.

619

6.37

3

+2.7

28−3

.645

200.

274

200.

817

0.56

3

+0.2

67−0

.295

B3

359.

426

202.

648

2.75

3

+1.2

49−1

.504

200.

086

12.8

51

+5.2

76−5

.296

200.

009

200.

647

0.67

8

+0.3

66−0

.311

B4

359.

388

203.

939

2.14

4

+1.0

41−1

.103

200.

606

6.16

4

+3.0

79−3

.086

200.

781

200.

786

0.26

0 +0

.172

−0.0

88C

1

361.

600

203.

032

1.68

8

+0.7

03−0

.995

aB

uil

d

not

pos

sibl

e

201.

218

203.

170

1.01

5 +0

.427

−0.5

88O

vers

ize

C2

361.

549

203.

032

1.41

5

+0.7

04−0

.711

aB

uil

d

not

pos

sibl

e

201.

269

203.

103

0.78

5 +0

.299

−0.4

86C

3

363.

215

205.

364

2.47

1

+1.0

67−1

.404

aB

uil

d

not

pos

sibl

e

202.

219

205.

417

1.07

9

+0.5

15−0

.564

C4

363.

366

205.

259

3.16

4

+1.5

60−1

.604

aB

uil

d

not

pos

sibl

e

202.

138

205.

407

1.43

1

+0.6

62−0

.769

D1

357.

475

200.

276

2.68

1

+1.4

48−1

.233

199.

555

4.64

8

+1.4

80−3

.168

200.

108

200.

047

0.33

7

+0.2

52−0

.086

Un

der

size

D2

357.

428

200.

332

1.37

6

+0.7

86−0

.590

199.

430

6.17

5

+2.0

04−4

.171

199.

232

199.

482

0.44

4

+0.2

51−0

.194

D3

355.

559

200.

183

0.88

3

+0.5

25−0

.359

198.

281

9.38

9

+2.7

86−4

.393

198.

180

198.

363

0.54

6

+0.3

46−0

.201

D4

355.

396

200.

294

1.29

7

+0.7

71−0

.526

198.

329

7.63

0

+2.5

20−5

.110

198.

189

198.

457

0.34

9

+0.1

82−0

.167

aB

uil

d

not

pos

sibl

e

–

fin

al

com

pon

ent

cou

ld

not

be

fitt

ed

into

asse

mbl

y,

as

rem

ain

ing

angu

lar

gap

in

buil

d

was

too

smal

l.

MZC

=

Min

imu

m

Zon

e of

Cir

cula

rity

(or

MC

C-M

IC)

OM

ZCI =

Ou

ter

Min

imu

m

Zon

e

refe

ren

ce

CIr

cle.

IMZC

I =

Inn

er

Min

imu

m

Zon

e

refe

ren

ce

CIr

cle.

Table 4The presence and position of angular breaking errors in the experimental builds.

Build ID Error FD TB CG

A1–4 Implicit only None None NoneB1–4 Balanced (±) Both None Outer radiusC1–4 Oversized (+) Inner radius Build not

possibleInner radius

D1–4 Undersized (−) Inner radius None at◦

Outer radius

−2 /outerradius at−4◦

best performing method, offering the most complete builds. Whilethe FD build method performed poorly against this measure.

The data shows that all the build methods performed at theirleast reliable when loaded with a scenario D oversized assembly(˙� i > 360◦). When building an oversized assembly all the buildmethods failed to locate on the inner diameter, as the resultantinner assembly diameter was too large to contact the fixturesdatum locators, or the inner diameter was subject to angularbreaking error. This is due to the calculations being based aroundthe component’s inner chord length, when the build quality ofan oversized ring is determined by the component’s outer chordlength. The CG method performed marginally better than the otherapproaches, thanks to the clamps pushing at the joints of the outerdiameter. An observation made during the experiment was, thatthe CG method controlled an extra degree of freedom in compo-nent location which the other build methods did not. As a singlelocator block sits at the centre of the wedge component’s innerradius, for the FD and TB methods, so the components were able topivot about this point. Any angular breaking errors in the assem-bly gave the components room to move, until they interacted withthe adjacent wedge components. This may explain why the cir-cularity deteriorates as the angular breaking error increases. Thelinear distance from the locator to the component’s edge amplifiesthis rotational movement, exaggerating the stepping present in theFD and TB methods. In contrast, the CG method contacts the com-ponents inner radius in two positions so this rotational motion isblocked, leading to a fundamentally more consistent build, espe-cially as the amount of geometric deviation increases.

4. Conclusions

Annular ring assemblies are of critical importance to high valueindustry such as aero-engines and power generation. Furthermore,there are situations where such assemblies are made of wedgeshaped parts and the use of traditional fixturing methods mightnot address the challenge of generating assemblies at low errorlevels. This paper has presented for the first time a critical analysisof the manufacturing variations that might influence the geom-etry and thus accuracy of such assemblies. This research bringsto the attention of the wider manufacturing community that thetraditional fixture design using the FD method is highly sensitiveto the manufacturing variability inherent to each wedge part. Amean circularity error of 1.11% (2.292 mm) of the inner diameterwas observed for the builds with roughly balanced errors (B1–4).Furthermore, the FD method was prone to produce assemblies thatwere not fully closed. In response to this drawback two original“variation conscious” build methodologies have been presented.

• Translational Build (TB): A method of compensating chord lengthvariability in the assembled components, by translating the com-

ponent relative to the assembly centre along the radial line.

• Circumscribed Geometric (CG): A geometrically derived buildmethod that fits the assembly’s build diameter to match the mea-sured variations in the components to be assembled.

3 on Eng

bkabstt0tbf3tcb

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[form data using computational geometric techniques. Precision Engineering

90 S. Lowth, D.A. Axinte / Precisi

To validate these concepts an original flexible test fixture haseen designed and constructed to build assemblies of parts withnown geometric deviations. The unique fixture offers radial andngular adjustment to the components positions, while enablingoth internally and externally datum locator options. Using thiset-up the efficiency of the traditional FD build has been substan-ially improved upon. Looking at the mean circularity error forhe builds with roughly balanced errors (B1–4), a mean error of.23% (0.456 mm) relative to the build diameter was experimen-ally observed using the CG method. This is a 483% improvementetween CG and FD methods. While it is true that TB method per-ormed the least consistently in terms of circularity, exhibiting a.6% (7.235 mm) error in circularity error relative to build diame-er. It did produce the most complete assemblies, and created fullylosed rings with component deviation levels that caused the otheruild methods to become unreliable.

This metrological analysis of a generic annular ring assemblyas proven the overarching effect of component variability on theeometric quality of such structures. Furthermore, it proved theivotal importance of in-depth geometric analysis for the under-tanding of the sources of variability in real aerospace components.his facilitates the future ability to manufacture larger ring assem-lies at higher precisions, as demanded by a number of modernanufacturing industries.

cknowledgements

This work is supported by Rolls-Royce Plc and the Engineeringnd the Physical Sciences Research Council (EPSRC). The authorsish to thank both Rolls-Royce and ESPC for their financial and

echnical support in this research. The authors would also like tohank Mr. Peter Winton at Rolls-Royce for his continuing supportn this research area. Finally, the authors would like to acknowl-dge the work of Mr. Ifeanyi Emmanuel Ogueri for his assistance inbtaining some of the experimental results within this study.

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