G.Ameta School of Mechanical and Materials Engineering
Washington State University,
Pullman,WA99164-2920
e-mail:gameta@wsu.edu
S.Serge
M.Giordano
Universite de Savoie,
BP806,
Annecy,Cedex,74016,France Comparison of Spatial Math Models for Tolerance Analysis: Tolerance-Maps,Deviation Domain,and TTRS
This paper prents a comparison of degree of freedom(DOF)bad math models,viz., tolerance-maps,deviation domain,and TTRS,which have shown most potential for retro-fitting the nuances of th
e ASME/ISO tolerance standards.Tolerances specify allowable uncertainty in dimensions and geometry of manufactured products.Due to the charac-teristics and application of tolerances,it is necessary to create a math model of toleran-ces in order to build a computer application to assist a designer in performing full3D tolerance analysis.Many of the current efforts in modeling tolerances are lacking,as they either do not completely model all the aspects of the ASME/ISO tolerance standards or are lacking the requisite full3-D tolerance analysis.Some tolerance math models were developed to suit CAD applications ud by the designers while others were devel-oped to retrofit the ASME/ISO tolerance standard.Three math models developed to retro-fit the ASME/ISO standard,tolerance-maps,deviation domain,and TTRS are the main focus of this paper.Basic concepts of the math models are summarized in this paper, followed by their advantages and future issues.Although the three math models repre-nt all aspects of the ASME/ISO tolerance standard,they are still lacking in one or two minor aspects.[DOI:10.1115/1.3593413]
1Introduction
1.1Background and Importance of Tolerances.Manu-facturing invariably leads to uncertainty in dimensions and geom-etry of manufactured products.Usually,products with higher precision(dimensional and geometric)cost more than the prod-ucts with lower precision.A designer s
自作多情 英语pecifies the dimensions and geometry of the manufactured product,while satisfying functional and other constraints for the product.Therefore,the designer must also specify the amount of uncertainty in dimensions and geome-try of the manufactured product.The amount of uncertainty in dimensions and geometry of the manufactured product is specified through tolerances.
The modern method of specifying tolerances is through geomet-ric dimensioning and tolerancing(GDT),as indicated in the ASME Y14.5and ISO1101standards[1,2].According to the Standards [1,2],the variations of a feature are bounded within tolerance zones that permit locational,orientational,form,profile,runout,and sym-metry variations of the feature in3-dimensions.Figure1shows dif-ferent class of tolerances and associated symbols.
In the prent era,tolerances affect the cost of production, inspection procedure,asmblability,performance,nsitivity, lection of process,process related tools,andfixtures.The lec-tion of type and value of tolerances for a part or an asmbly is an important issue for any manufacturingfirm as it affects decision-making process at all the echelons of a production cycle.There-fore,a designer should discern the need and the effect of tolerances that he/she lects.
Usually,a designer can arrive at initial/preliminary t of toler-ances by utilizing(a)design history from similar products and(b) trial and error.The initial value of tolerances can be optimized for cost and function of the product using one of two approaches:(a) tolerance analysis or(b)tolerance synthesis[3].With analysis,the designer estimates values for individual tolerances on a target fea-ture for each dimension in a“stackup,”and then us an analysis tool,often automated on a computer,to determine the contribution from each of the tolerances to the accumulation of variations at one or more functional target features of the entire stackup.(Note that a“stackup”is a dependent relationship among dimensions that may all reside on one part or be distributed over veral parts in an asmbly.)With synthesis,often called tolerance allocation,the desired ,a range of clearance to ensure proper lubrica-tion or control of noi)at the target feature is chon,and often ratios among tolerances are also chon,to minimize cost of manu-facture.Then,the tolerances are generated from the math model to meet the choices.Tolerance analysis and allocation can be done using a worst-ca method or a statistical method.With the worst-ca approach,the tolerances chon will ensure100%acceptabil-ity of the asmblies;with the statistical method,the tolerances chon will ensure acceptability of a certain large percentage of asmblies.The statistical method allows a tradeoff between big-ger variations at all the parts,and a correspondingly lowered cost of their manufacture,and a small number of asmblies for which the variations at the target features are not within acceptable limits.
1.2Need for Math Model of Tolerances.Current tools for assisting designers in assigning satisfactory t of tolerances(toler-ance analysis)are neither comprehensive nor accurate.The designer either us manual/automated tolerance charts or simulation bad commercial tolerance analysis tools.Tolerance charting is consist-ent with ASME/ISO standards[1,2],but limited to1-D worst ca analysis only.Simulation bad commercial tolerance packages typically perform both worst ca and statistical analysis,but they are bad on point-to-point constraint solving and,therefore,incom-patible with the tolerance standards that specify variation within tol-erance zones.Comprehensive3D analysis of stack-ups involving all types of dimensional and geometric variations is only possible if a mathematical model of such variations exist.
The current international standards are created by collecting knowledge from years of engineering practice and are,therefore, ca-bad for each individual feature type and tolerance type.
Contributed by the Computer Aided Manufacturing Committee of ASME for pub-
lication in the J OURNAL OF C OMPUTING I NFORMATION S CIENCE IN E NGINEERING.Manu-
script received November8,2010;final manuscript received April9,2011;
七夕快乐英文怎么说published online June15,2011.Assoc.Editor:J.Shah.
新年快乐用英语怎么说
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Journal of Computing and Information Science in Engineering JUNE2011,Vol.11/021004-1
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Emerging methods are attempting to address the challenge to build a math model of geometric variations that are consistent with the already existing tolerance standards and are capable of supporting comprehensive 3D analysis of stack-up conditions.Many different methods for math models of the standard for tol-erance analysis have been approached in the literature,which will be discusd in brief in Sec.2.2.For a detailed and comprehensive review of tolerance analysis methods,plea refer to other review articles [3–8].Three spatial math models for tolerance analysis have en the most development and have been pursued consis-tently,by the rearchers,in the last decade.The only focus of this paper is the three spatial math models for tolerance analy-sis,viz.,tolerance-map,deviation domain,and technologically and topologically related surfaces (TTRSs)model.Section 2prents basics of tolerance analysis and various rearch efforts related to tolerance analysis.Section 3provides an overview of three math models for tolerance analysis,while Sec.4compares the three math models.
2Tolerance Analysis
2.1Basics of Tolerance Analysis.The objective of tolerance
analysis is to check the extent and nature of the variation of an an-alyzed dimension or geometric feature of interest for a given GDT scheme.The variation of the analyzed dimension aris from the accumulation of dimensional and/or geometric variations in the tolerance chain.The analysis include:(1)the ,the dimensions or features that cau variations in the analyzed dimension,(2)the nsitivities with respect to each contributor,(3)the percent contribution to variation from each contributor,and (4)worst ca variations,statistical distribution,and accep-tance rates.
For example,consider the asmbly of three parts shown in Fig.2.Dimensions d 1,d 2,and d 3are known dimensions with asso-ciated dimensional tolerances (6t 1/2,6t 2/2,and 6t 3/2).Dimen-sion d f is the dimension of interest for the asmbly.It is quite evident that
d f ¼d 1Àd 2þd 3ðÞ
(1)
Correspondingly,the worst cast variation of dimension d f can be identified by the tolerance t w
t w ¼t 1þt 2þt 3
(2)
The contributors are dimensions d 1,d 2,and d 3.All the three dimensions have equal nsitivities (1)and equal contribution (0.33).The worst ca variation is d f þt w /2and d f Àt w /2.For sta-tistical tolerance analysis,root sum of squares method from statis-tics is utilized leading to Eq.(3)under the following assumptions:(1)d i parameters are independent random variables
(2)the index capability C p ¼t i /(6SD i )has the same value for
udtoall t i and t s .SD i are the standard deviations of d i and d f .
t 2s ¼t 21þt 22þt 2
3
(3)
报告如何写The above scheme is only suitable for such simple linear chains with only dimensional tolerances.Addition of geometric toleran-ces,with their nuances,and nonlinear chains complicates the tol-erance analysis procedure.Although,some simple linearization or rule bad methods have b
een developed to tackle tolerance anal-ysis,but the methods fall short in achieving full 3D tolerance analysis with geometric
tolerances.
Fig.1Different class of tolerances in the standards [1
]
Fig.2A simple 1-dimensional example showing tolerance analysis with only dimensional tolerances
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2.2Various Rearch Efforts in Tolerance Analysis.Vari-ous rearch efforts in tolerance analysis can be classified into two major categories.Tolerance reprentations retrofitted for com-puter aided design(CAD)and retrofitted to model variations,as specified by the standards.Furthermore,the classification of rearch efforts for developing a math model of standard as given by Davidson et al.[9]can be reclassified into the two categories of rearch efforts in tolerance analysis.Parametric mo
dels,offt zone models,and variational surfaces bad models are repren-tations retrofitted for CAD,while kinematic models and degrees of freedom(DOF)bad models are reprentations retrofitted for variations as specified by the standards.Other recent reviews in tolerance analysis have been conducted for Jacobean and torsor models[10]and matrix and vector models[11].A brief descrip-tion of the rearch efforts is prented below.
2.2.1Tolerance Models Retrofitted for CAD.Initial efforts, during1980’s,utilized parametric CAD to develop models for tol-erance analysis.The models can be called parametric models for tolerance analysis.Parametric CAD utilizes a t of explicit dimensions and constraints to reprent nominal shape and size. The explicit dimensions and constraints can be ud to obtain a t of equations relating the dimension of interest to individual chain of dimensions.Tolerances are incorporated by allowing6variations in the dimensions[12,13].
As is evident,this method is similar to the one-dimensional tol-erance analysis discusd in Sec.2.1.Limitations of attributed to parametric methods include inability(a)to incorporate to all geo-metric tolerances in the standard and their interactions and(b)to conduct full3D tolerance analysis.
About the same time,rearchers attempted to model the con-cept of tolerance zone for tolerance an
alysis,by creating zones for the toleranced features in a CAD model.The main idea was to model tolerance zone as Boolean subtraction of maximal and min-imal object volumes that are obtained by offtting the object by amounts equal to the tolerances on either side[14–16].The models are called offt zone models for tolerance analysis.The construction of such a composite tolerance zone from boundary surfaces of the part(a)does not allow one to model each type of geometric variation parately and(b)to study their interactions as specified in the standard[1].Various issues related to the models are also discusd in Ref.[17].
Extending the same idea of offt zone for simulating the varia-tions of features in CAD models,variational surfaces bad mod-els for tolerance analysis were developed in early1990’s.Each surface is varied independently by changing the values of model variables from which surface coefficients are calculated[18,19]; positions of the vertices and edges are computed from the surface variations.When using this concept in CAD tools,it leads to some topological problems,such as(a)maintenance of tangency and(b)incidence conditions.This model too is incompatible with the ASME Standard[1].A modified version of this method,which us abstracted geometry instead of the CAD model itlf is uti-lized in various simulation bad tolerance analysis(VSA)tools.
2.2.2Tolerance Models Retrofitted to Reprent Variations as Specified in the Standard.A different approach was adopted by Cha et al.[20]that incorporated kinematics to model asm-bly and tolerances for tolerance analysis.Such models can be clas-sified as kinematics bad model for tolerance analysis.Initially, Rivest et al.[21]utilized transformation matrices to analyze toler-ance stack-up in mechanisms.Bad on the idea,Cha and co-workers.[5,20,22,23]developed a kinematic approach to toler-ance analysis.In this approach,three types of variations(dimen-sional,kinematic,and geometric)are modeled in the vector loop. In a vector loop,dimensions are reprented by vectors,in which the magnitude of the dimension is the length(L i)of the vector. Kinematic variations are small adjustments between joints(mat-ing relations),which occur at the asmbly time in respon to the dimensional and geometric variations.Geometric tolerances are considered by adding micro DOFs to particular ones of the joints. Not all interactions of geometric tolerances have been incorpo-rated in the model.
Extending the idea behind kinematic models,degrees of freedom allowed by each tolerance type to each feature was being utilized by veral rearchers.Such models can be classified as Degree of Freedom bad models for tolerance analysis.Kramer[24]ud symbolic reasoning to demonstrate the determination of degrees of freedom of parts in an asmbly and to determine asmbly feasi-bil
ity bad on nominal dimensions.The three math models(toler-ance-maps,deviation domain,and TTRS)discusd in Sec.3,u the concept of DOF to model geometric tolerances and then utilize kinematics or transformations to assist in tolerance analysis.
3Math Models for Tolerance Analysis
Although there are many different math models for tolerance analysis(e Sec.2.2),this paper discuss tolerance-maps,devia-tion domain,and TTRS briefly.All the models u substituted surfaces having no form errors and variations are reprented by real variables.For details of each method,refer to the cited refer-ences in each ction below.
3.1Tolerance-Maps.A Tolerance-Map V R(T-Map V R)is a hy-pothetical Euclidean point-space,the size and shape of which reflects all variational possibilities for a target feature.It is the range of points resulting from a one-to-one mapping from all the variational possibilities of a feature,within its tolerance-zone,to the Euclidean point-space.The variations are determined by the various tolerances that are specified on the feature.
3.1.1Areal Coordinates.The T-Map V R for any combination of tolerances on a feature is constructed from a basis-simplex in a space of dimension n,the value of n corresponding to the freedom of move
ment of the feature within the tolerance-zone;it is described with areal coordinates.A classical description of this subject,a form of affine geometry,is in Coxeter[25].To construct an n-dimensional space and its simplex,nþ1basis points are needed.Therefore,for three-dimensional variations of a feature, the corresponding T-Map is constructed from four basis points that define its basis-tetrahedron.We choo to position the four basis-points r1,r2,r3,and r4as shown in Fig.3.At the four ba-sis-points,we place four mass k1,k2,k3,and k4that may be pos-itive or negative.So long as k1þk2þk3þk4=0,the position of r,the centroid of the mass,is uniquely determined by the lin-ear combination
k1þk2þk3þk4
ðÞr¼k1r1þk2r2þk3r3þk4r4(4)
and we can make r assume any position in the space of r1…r4 by varying k1,k2,k3,and k4.For example,for k1…k4all positive, r identifies any point inside tetrahedron r1r2r3r4.
The four mass k1…k4are the barycentric coordinates of r, yet we note that the position of r depends only on three independ-ent ratios of the magnitudes.Conquently,the four k i’s can be normalized by
tting
Fig.3The basis tetrahedron with its basis points
Journal of Computing and Information Science in Engineering JUNE2011,Vol.11/021004-3
k 1þk 2þk 3þk 4¼1
(5)
then they are areal coordinates and
r ¼k 1r 1þk 2r 2þk 3r 3þk 4r 4
(6)
The shape of the basis-tetrahedron was chon to simplify inter-pretation of T-Maps,particularly to decouple rotational and trans-lational displacements in the tolerance-zone [9].
3.1.2Tolerance-Map for a Face With Size Tolerance.Figure 4shows the end of a rectangular bar of cross-ctional dimensions d x Âd y (d x <d y ).The length of the part shown is ‘with an exag-gerated tolerance t .According to the ASME Standard Y1
4.5[1],all points of the end-face must lie between the limiting planes r 1and r 2and within the rectangular limit of the face.The region (ABCDEFGH)defined by the limiting planes and the rectangular limit of the face is the tolerance-zone for the planar face.The same tolerance zone can be obtained with profile tolerance,t ,specified for the planar face with respect to the opposite end (not shown in the Fig.4)of the bar.
In order to build the T-Map,it is assumed that the variations of the toleranced face in Fig.4are rotations about x and y and trans-lations along z .The shape or form of the face is assumed to be perfectly planar.A coordinate frame is located in the tolerance zone with its origin O at the geometrical center of the tolerance zone as shown in Fig.4(c ).This coordinate frame has its axes par-allel to the edges of the part.Presuming the face at first to be of perfect ,a rectangular gment of a plane,the possible placement of this face is against any one of a three-dimensional t of planes.The planes r 1and r 2are located at maximum dis-tance from the origin of the coordinate frame in the tolerance zone.The planes r 3and r 7are rotated by the greatest allowable amount about
the x -axis in the tolerance zone.Since d x <d y ,the permitted angular variation about the y -axis can be greater than that about the x -axis.The planes labeled r 4and r 8are rotated about the y -axis by the same amount as the planes r 3and r 7are about the x -axis.The planes r 40and r 80are rotated the maximum amount about the y -axis in the tolerance zone.Each of the planes in the tolerance zone is then mapped to a specific point in the T-Map,as shown in Fig.5.Therefore,the construction of a T-Map ensures that each point inside it reprents a single valid con-figuration of the perfect-form feature within its tolerance-zone.The T-Map for a planar face faithfully reprents the 3D varia-tions permitted by the tolerance-zone:translation perpendicular to the plane and rotations about the x -and y -axes (Fig.4(c )).Meas-ures along the s -axis of the T-Map reprent parallel variations of the plane negatively along the z -axis in the tolerance zone,while
the p 0-and q 0-axes reprent the orientational variations of the plane about the y -and x -axes,respectively.
If the toleranced face in Fig.4is not assumed to be perfectly planar,then the shape or form variations of the face are modeled as subts of T-Map as shown in Fig.6.The T-Map for size tol-erance t on the length of the bar remains the same as in Fig.5,but now there are two internal subts,each of the same shape as in Fig.5but of different sizes.The form variation is zero (per-fect form and no warp)for
the large shaded T-Map at the far left in Fig.6.As we move from left to right,the subt for size toler-ance (lower shape)shrinks while the subt for form tolerance (upper shape)enlarges.This tradeoff reprented through the subts basically models Rule#1from the ASME Y14.5standard [1].Further details about the T-Map model for different types of tolerances and features can be found in Refs.[9,26–31].3.1.3Tolerance Analysis.Worst ca tolerance analysis for asmblies with open chain and not consisting of any clearances,can be performed using the following stepshomo是什么意思
ligo
(1)Identify the chain of dimensions and tolerances from the
datum to the target of the asmbly.
(2)Create T-Maps for all the toleranced features in the chain.(3)Using transformation matrices,conform each T-Map to rep-rent the variations at the target feature of the asmbly.(4)Combine the T-Maps using Minkowski Sum to identify the
accumulated T-Map for the target feature.
(5)Create a T-Map for the functional requirement of the
asmbly.This T-Map is called functional T-Map.
(6)Fit the accumulation T-Map within the functional T-Map.
(a)To verify if the assigned tolerances meet the functional
requirement,the accumulation T-Map should be com-pletely inside the functional
T-Map.
Fig.4(a )Rectangular part with size tolerance and (b )rectangular part with profile tolerance on rectangular surface.(c )The tolerance zone on size (specification of (a ))or profile (specification of (b ))for a rectangular bar and a coordinate frame centered within
it.
Fig.5The T-Map for the tolerance zone shown in Fig.4(c )
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(b)To identify the stack up equations,scale the functional
T-Map homogeneously until at least one of the bound-ary points of the accumulation T-Map comes in contact to the boundary of the functional T-Map.Utilizing the geometry of the functional and accumulation T-Map,create the stack up equations.
(c)To optimize the assigned tolerances,change the types
and values of tolerances on each feature such that the accumulation T-Map can fill as much space as possible while remaining confined within the functional T-Map and satisfying other design criteria.3.2Deviation Domain
3.2.1Small Displacement Torsors.Each tolerance zone,allows a small amount of variations of the feature within the toler-ance zone.The small amounts of variations are reprented as small displa
不到长城非好汉的意思cement torsor (SDT).A torsor basically reprents three translations and three rotations of a feature with respect to a coordinate system.For example,a SDT for a planar surface (Fig.4(c ))would be reprented as
SDT planar ¼t x ;t y ;t z ;r x ;r y ;r z
ÈÉ
t x ¼t y ¼r z ¼0(7)
The first three elements of Eq.(4)reprent the translations about x ,y ,and z axis in the tolerance zone (Fig.4(c ))while the last three elements reprent rotations about x ,y ,and z axis in the tolerance zone.Becau of the nature of the feature (planar surface),and the tolerance zone,translations along the x and y axis and the rota-tions about z axis are considered invariant.
3.2.2Deviation and Clearance Domain.In order to reprent the variations of a feature within its tolerance zone,a deviation space is created using the noninvariant components of the SDT.For the example considered in Sec.3.1.1,a deviation domain is created using tz,rx,and ry parameters of the SDT.Since,the devi-ation domain is created for the three parameters of SDT;the do-main is three-dim
ensional.Furthermore,obrving the tolerance zone,toleranced feature and the parameters of the SDT,inequal-ities reprenting the bounds of the tolerance zone are created.The inequalities are then ud to create a bounded deviation do-main.Figure 7shows the deviation domain for the planar surface shown in Fig.
4.
As is evident from Eq.(7),there are six parameters in a torsor.Therefore,the dimensionality of a deviation domain can be six.The clearance in a joint between two parts can also be modeled by SDT called clearance torsor.A coordinate frame is attached to the two parts forming a joint with clearance.The clearance is repre-nted as a SDT of variations of one frame with respect to another.The possible variations in clearance,when reprented in the deviation space (using SDT),is called a clearance domain.Form or shape variations are modeled using vibration modes of the toleranced feature.The vibration modes are then ud to modify the deviation domain in order to reprent form variations.For further details about the deviation domains models plea refer to Refs.[32–38].
3.2.3Tolerance Analysis.Worst ca tolerance analysis,for asmblies
(a)with open chain and not consisting of any clearances,can
out of memorybe performed using the following steps:
(1)Identify the chain of dimensions and tolerances from
the datum to the target of the asmbly.
(2)Create deviation domains for all the tolerance features
in the chain.
(3)Combine the deviation domains using Minkowski Sum
to identify the accumulated deviations for the target feature.
(4)Create a deviation domain for the functional require-ment of the asmbly.This deviation domain is called functional domain.
(5)Align the torsor parameters of the accumulated and
functional domain to obtain the stack up equation for the asmbly (b)with clod chain and consisting of clearances,can be per-formed using the following steps:
(1)Identify the chain of dimensions and tolerances from
the datum to the target of the asmbly.
(2)Create deviation domains for all the tolerance features
in the chain.
(3)Combine the deviation domains using Minkowski Sum
to identify the accumulated deviations for the target feature.
(4)Create a minimal clearance domain (accumulated)for
each joint of the chain.
(5)The asmbly is possible when the accumulated devia-tion domain remains within the accumulated clearance domain.3.3Technologically and Topologically Related Surfaces.The TTRS m
ethod utilizes veral different concepts from con-straints and rigid body motions to model tolerances.In classical kinematics,the constraints between the features of a point,a line,and a plane form six lower pairs of classical kinematics [39].Hunt [40]drew attention to their u in both partial and sufficient constraint of a rigid body.Later,Desrochers and Cle ´ment [41]in-dependently formulated the idea as six TTRSs for the u in appli-cations of dimensioning and tolerancing.The surfaces as derived from kinematic joints are spherical,planar,cylindrical,helical,rotational,and prismatic.Some authors add another surface called “any surface”to create ven TTRSs.
3.3.1MGDE/MGRS.Desrochers [42]and Cle ´ment et al.[43]have integrated TTRS to model the variations in a tolerance-zone with the u of the constraints between a point,line and plane,called “minimum geometric datum elements”(MGDE)or “minimum geo-metric reference surface”(MGRS).Thirteen different constraints have been propod in Ref.[43],as shown in Table 1.
3.3.2Modeling Tolerances.In the TTRS model,various rearchers have ud tensors or torsors or screws combined with some internal parameters to reprent tolerances.Since,the
torsor
Fig.6The tradeoff between the array of sub-ts for form and their companion locations within the T-Map of Fig.
5
Fig.7Deviation domain for the planar surface in Fig.4(c )
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