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A review of two models for tolerance analysis of an assembly: vector loop and matrix

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A review of two models for tolerance analysis of an assembly: vector loop and matrix
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  ORIGINAL ARTICLE A review of two models for tolerance analysisof an assembly: vector loop and matrix Massimiliano Marziale  &  Wilma Polini Received: 3 June 2008 /Accepted: 2 October 2008 /Published online: 31 October 2008 # Springer-Verlag London Limited 2008 Abstract  Mechanical products are usually made by assem- bling many parts. The dimensional and geometricalvariations of each part have to be limited by tolerancesable to ensure both a standardized production and a certainlevel of quality, which is defined by satisfying functionalrequirements. The appropriate allocation of tolerancesamong the different parts of an assembly is the fundamentaltool to ensure assemblies that work rightly at lower costs.Therefore, there is a strong need to develop a toleranceanalysis to satisfy the requirements of the assembly by thetolerances imposed on the single parts. This tool has to be based on a mathematical model able to evaluate thecumulative effect of the single tolerances. Actually, thereare some different models used or proposed by the literatureto make the tolerance analysis of an assembly, but none of them is completely and univocally accepted. Some authorsfocus their attention on the solution of single problemsfound in these models or in their practical application incomputer-aided tolerancing systems. But none of them hasdone an objective and complete comparison among them,analyzing the advantages and the weakness and furnishinga criterion for their choice and application. This paper  briefly introduces two of the main models for toleranceanalysis, the vector loop and the matrix. In this paper, thesemodels are briefly described and then compared showingtheir analogies and differences. Keywords  Toleranceanalysis.Vectorloopmodel.Matrix model.Functionalrequirements 1 Introduction As technology increases and performance requirementscontinually tighten, the cost and required precision of mechanical assemblies increase as well. Then, there is astrong need for industries to produce high-precisionassemblies at lower costs. Therefore, there is a strong needto use tolerance analysis to predict the effects of thetolerances that have been assigned to the components of anassembly on the functional requirements of the assemblyitself. The aim of the tolerance analysis is to study theaccumulation of dimensional and/or geometric variationsresulting from a stack of dimensions and tolerances. Theresults of the analysis are meaningfully conditioned by theadopted mathematical model. Some are the models pro- posed by the literature to carry out a tolerance analysis of an assembly, but they still appear not adequate under manyaspects: the schematization of the form deviations, theschematization of the joints with clearance between the parts, the solution of complex stack-up functions due tothe network joints among the components, and so on.Moreover, there does not exist in the literature a paper that compares the different analytical methods on the basis of acase study that underlines in a clear way all the advantagesand the weakness. In the literature, some studies comparethe models for tolerance analysis by dealing with their general features [1, 2]. Other studies compare the main computer-aided tolerancing softwares that implement someof the models of the tolerance analysis [3, 4]; but these studies focus the attention on the general features. However,a complete comparison of the models proposed to solve the Int J Adv Manuf Technol (2009) 43:1106  –  1123DOI 10.1007/s00170-008-1790-0M. Marziale : W. Polini ( * )Università degli Studi di Cassino,via G. di Biasio 43,03043 Cassino, Italye-mail: polini@unicas.it   tolerance analysis does not exist in the literature and,therefore, no guidelines exist to select the method moreappropriate to the specific aims.The purpose of this work is to analyze two of the most significant models for tolerance analysis: the model called “ vector loop ”  and the model called  “ matrix. ”  The comparisonof the models starts from their application to a case study.Dimensionalandgeometricaltoleranceshavebeenconsideredas part of stack-up functions. The worst and the statisticalapproaches have been taken into account. The application of the envelope principle [5] and of the independence principle[6] has been deeply investigated. Finally, the guidelines for the development of a new and original model able toovercome the limits of the compared models have beenunderlined. “ Section 2 ”  gives an overall explanation of the vector loop and matrix models.  “ Section 3 ”  gives a comprehensivecomparison of the two models by means of a case studythat is characterized by 2-D tolerance stack-up functions.Finally,  “ Section 4 ”  offers some guidelines for those whowill have to make the choice. 2 Tolerance analysis models 2.1 Vector-loop-based methodVector-loop-based model uses vectors to represent thedimensions in an assembly [7, 8]. Each vector represents either a component dimension or assembly dimension. Thevectors are arranged in chains or loops representing thosedimensions that stack together to determine the resultant assembly dimensions.Three types of variations are modeled in the vector loopmodel: dimensional variations, kinematic variations, andgeometric variations. In a vector loop model, dimensionsare represented by vectors, in which the magnitude of thedimension is the length (  Li ) of the vector. Dimensionaltolerances are incorporated as ± variation in the length of the vector. Kinematic variations are small adjustmentsamong mating parts, which occur at assembly time inresponse to the dimensional variations and geometricfeature variations of the component analysis. There are sixcommon joints in 2-D assemblies and 12 common joints in3-D assemblies; at each kinematic joint, a local datumreference frame (DRF) has to be defined for. These jointsare used to describe the relative motions among mating parts. The degrees of freedom ( df   ) that are constrained or not by the mating part are controlled and the tolerances arespecified only for the constrained  df   . Geometric tolerancesare considered by adding micro- df    to the joints just described [9], i.e., a virtual transformation (a 0-lengthvector or a rotation matrix along the directions admitted bythe applied tolerance) is added to the joints.Although geometric tolerances may affect an entiresurface, they introduce a variation at the contact amongmating parts. Appropriate geometric variations may beadded to displacements at the joint. If the variation caused by the geometric tolerance  –  kinematic joint combination is arotation, the geometric variable can be represented by arotational matrix or a combination of rotational matrices inthe assembly kinematic constraints. If the variation caused by the geometric tolerance  –  kinematic joint combination is atranslation, a translation matrix may be inserted at theappropriate node in the assembly kinematic constraints.To better understand this method, the basic steps to builda vector loop scheme and to carry out a tolerance analysisare given below [10  –  12]:1. Create assembly graph  —  the first step is to create anassembly graph. The assembly graph is a simplifieddiagram of the assembly representing the parts, themating conditions, and the measures to perform. Anassembly graph assists in identifying the number of vector loops required for the analysis of the assembly.2. Locate the DRFs for each part   —  the next step islocating the DRFs for each part. These DRFs are usedto locate features on each part. If there is a circular contact surface, its center is considered as a DRF too.3. Locate kinematic joints and create datum paths  —  eachcontact among the parts is translated into a kinematic joint. The kinematic joints of the assembly are locatedat the points of contact and they are oriented in such away that the joint degrees of freedom align with theadjustable assembly dimensions; in this way, each joint introduces kinematic variables into the assembly whichmust be included in the vector model. The datum pathsare created as chains of dimensions which locates the point of contact with respect to the DRF of the parts.4. Create vector loops  —  using the assembly graph and thedatum paths, vector loops are created. Each vector loopis created by connecting the datum paths of the datumtraverse by the loops. Avector loop may be called openor closed if it is related to a measure or not.5. Derive the equations  —  the assembly constraints withinvector-loop-based models may be expressed as aconcatenation of homogeneous rigid body transformationmatrices: R  1    T 1    ...    R  i    T i    ...    R  n    T n    R  f   ¼  H  ð 1 Þ where: R  1  is the rotational transformation matrix between the  x -axis and the first vector;  R  i  is the rotational transformationmatrix between the vectors at node  i ;  T i  is the translationalmatrix of vector   i ;  R  f   is the final closure rotation with the  x -axis; and  H  is the resultant matrix. For example, in the 2-D Int J Adv Manuf Technol (2009) 43:1106  –  1123 1107  case, the rotational and the translational matrices assumethe following shape:  R  i  ¼ cos f i  sin f i  0sin f i  cos f i  00 0 1 2435  and  T i  ¼ 1 0  L i 0 1 00 0 1 2435 where  φ i  is the angle between the vectors at node  i , and  L i  is the length of vector   i.  If the assembly isdescribed by a closed loop of constraints,  H  is equal to theidentity matrix; otherwise,  H  is equal to the  g  vector representing the resultant transformation that will lead tothe final gap or clearance and its orientation when applied toa DRF.6. Tolerance analysis  —  we consider an assembly consti-tuted by p-parts. Each part is characterized by the  x -vector of the dimensions and by the  α  -vector of thegeometrical variables that are known. When these partsare assembled together, the resultant product is charac-terized by the  u -vector of the assembly variables and bythe  g  vector of the measures required on the assembly.It is possible to write  L  ¼  J     P   þ  1 closed loops,where  J   is the number of the ties among the parts that looks like: H  x ; u ; a  ð Þ ¼  0  ð 2 Þ while there is an open loop for each measure to do that looks like: g  ¼  K   x ; u ; a  ð Þ ð 3 Þ Equation 3 allows us to calculate the measure  g  after having solved the equations ’ system (Equation 2). Equation 2 is not linear and it is solved by means of the direct linearization method (DLM):d H  ffi  A    d x  þ  B    d u  þ  F    d a   ¼  0  ð 4 Þ d u  ffi  B  1   A    d x    B  1   F    d a   ð 5 Þ d g  ffi  C    d x  þ  D    d u  þ  G    d a   ð 6 Þ with  A ij  = ∂  H  i /  ∂  x   j  ,  B ij  = ∂  H  i /  ∂ u   j  ,  F  ij  = ∂  H  i /  ∂ α    j  ,  C  ij  = ∂  K  i /  ∂  x   j  ,  D ij  = ∂  K  i /  ∂ u   j  ,  G  ij  = ∂  K  i /  ∂ α    j  .From Eqs. 4 to 6: d g  ffi  C    D    B  1   A     d x  þ  G    D    B  1   F     d a  ¼  S x    d x  þ  S a     d a   ð 7 Þ where  S x  ¼  C    D    B  1   A    and  S a   ¼  G    D    B  1   F   are the sensitivity matrices. When the sensitivity matricesare known, it is possible to calculate the solution in theworst-case scenario as: Δ  g  i  ¼ X k   S x ik     t x k  j j þ X l   S a  il    t a  l j j ð 8 Þ while in the statistical scenario (root sum of squares) as: Δ  g  i  ¼ X k   S x ik     t x k  ð Þ 2 þ X l   S a  il    t a  l ð Þ 2 h i 1 = 2 ð 9 Þ The DLM is a very simple and rapid method, but it isapproximated too. When an approximated solution isunacceptable, it is possible to use a numerical simulation by means of a Monte Carlo technique to improve the exact solution [13  –  15].2.2 Matrix modelMatrix-based model uses displacement matrices  D  whichdescribe the small displacements a feature may have oneinside the tolerance zones to represent the variability of the parts. For each feature to which a tolerance is applied, thereis a local DRF and a displacement matrix which is definedwith respect to the local DRF. Besides, for each clearance between two parts, there is a displacement matrix too andthe maximum value of the clearance is assumed as a “ virtual ”  tolerance. The displacement matrices are arrangedtogether through the principle of effects overlapping inorder to determine the resultant assembly measure. In fact,the displacements are so small that it is possible to assumethat the total displacement is the sum of the displacementsdue to the single cause (tolerances and assembly gaps)[16, 17]. For each assembly measure, it is needed to define the points used to model it. For example, if we have to measurethe clearance between two axes, it results by combining theminimum and maximum distances that separate them.Therefore, this implies the optimization of two distancefunctions corresponding to the worst-case scenario [18].To completely define the problem, the constraints due tothe displacement matrix  D  have to be added. Theseconstraints limit the features to remain inside the tolerancezones that are applied to them. The constraints are appliedto the points characterizing the features with tolerances. For example, if there is a location tolerance applied to a hole,this means that its axis should remain inside a cylindricalzone with an assigned diameter. It is enough that both theextreme points of the axis remain inside the cylindricalzone, once assumed that the axis maintains its nominalshape. This involves two constraints on the displacementsof matrix  D .The matrix model is based on the positional tolerancingand the technologically and topologically related surfaces 1108 Int J Adv Manuf Technol (2009) 43:1106  –  1123  (TTRS) criteria [19]. Moreover, the matrix model considersonly the worst-case solution and the features are assumed asideal, i.e., the form tolerances are considered as null. To better understand this method, the basic steps are given below for conducting a tolerance analysis:1. Transform the tolerances applied on the drawing  —  thefirst step is to transform the tolerances applied on adrawing according to the positional tolerancing and theTTRS criteria.2. Create assembly graph  —  the second step is to create anassembly graph. The assembly graph allows us tolocate the global DRF and the linkages among thefeatures on which the tolerances are assigned.3. Locate the local DRF of each part feature  —  there is aneeded to assign a DRF to each part feature.4. Define the measures  —  there is a need to define the points to use in order to evaluate the displacements. For each point, there is a need to define the path connectingthe point to the global DRF up to the assemblymeasure. The contributions of each point to theassembly displacement have to be identified.5. Define the single displacement contributions and theconstraints  —  it is necessary to define the contribution of each displacement to the total displacements field andits constraints. Each surface can be classified into oneof the seven classes of invariant surfaces and thisallows us to annul some displacements in order toobtain a simplified displacement matrix. Therefore,considering the generic  i th feature, once indicated with R  i  the local DRF, with  R   the global DRF, with  P  1R   > R  i the homogeneous matrix transformation from  R   to  R  i (this matrix depends only from the nominal geometry),with  m R   the vector which components are the coor-dinates of the generic point M of the  i th feature in thelocal DRF, and with  D i  the displacement matrix by thetolerances applied on the  i th feature, the displacement of point   M  in the global DRF is: m Ri  ¼  P  1 R   R  i   D i    I ½    m R   ð 10 Þ Equation 10 gives the range of displacements of the  i thfeature in the global DRF that is allowed by the tolerancezone. Further points are necessary to specify the constraintsassuring the feature remains inside the bounds of thetolerance zone. The additional constraints are defined bylimiting the displacement of a set of points  M  i  belonging tothe feature inside the  t   tolerance range m Ri    y  ¼  D i    I ½    m Ri    y    t  = 2  ð 11 Þ 6. Apply the principle of effects overlapping and performthe optimization  —  if more than one tolerances areapplied on the same part, their effects is calculatedthrough the principle of effects overlapping. For example, if there are  n  tolerances applied to the samefeature that is characterized by the local DRF  R  i , thedisplacement of a generic point   M   belonging to thefeature is simply expressed as the sum of the singlecontributors: m R   ¼ X ni ¼ 1 P  1  R !  R i h i   D i    I ½    m Ri  ð 12 Þ Adding the constraints obtained for each feature withtolerances, a typical problem of optimization under con-straints is gotten. The optimization problem is solvedthrough standard optimization algorithms. 3 Models comparison 3.1 Case studyTo compare the two models previously described, the casestudy shown in Fig. 1 has been used. It is constituted by a box containing two circles. The aim of the toleranceanalysis is the measurement of the variation of the gap  g   between the second circle and the top side of the box ( Δ  g  )as a function of the tolerances applied to the components.The first analysis has considered only the dimensionaltolerances that are shown in Fig. 2. The envelope principlehas been applied, i.e., rule #1 of the American Society of Mechanical Engineers (ASME) standards.Then, the toleranceanalysis has considered the geometrical tolerances too, as Fig. 1  The case studyInt J Adv Manuf Technol (2009) 43:1106  –  1123 1109  shown in Fig. 3. Both the envelope principle (according toASME Y19.4 standard) and the independent principle(according to ISO 8015 standard) have been considered.The case study has been solved through both the worst-case and the statistical approaches. The case study containsall the characteristics and the critical aspects of the problem, but at the same time it is so simple to calculatethe exact geometric worst-case value of the required range Δ  g   in order to compare the results of the models. The exact geometric worst-case results are: ±0.89 mm for the caseconsidering dimensional tolerances only and ±0.91 mm for the case considering both dimensional and geometricaltolerances (see Table 4).3.2 Vector loop model 3.2.1 Dimensional tolerances only Once the dimensions of the box was indicated as  x 1  and  x 2 ,the diameter of the two circles as  x 3  and  x 4 , and theassembly variables as  u 1 ,  u 2 ,  u 3 ,  u 4  (see Fig. 4), theassembly graph of Fig. 5 has been built. It shows two jointsof   “ cylinder slider  ”  kind between the box and the circle 1 at  point A and point B, respectively, one joint of parallelcylinder kind between the circle 1 and the circle 2 at point C, one joint of cylinder slider kind between the circle 2 andthe box at point D, and the measure to perform (  g  ).A DRF has been assigned to each part; it is centered inthe point   Ω  of Fig. 6 for the box and in the centers O 1  andO 2  of the two circles. All the DRFs have the  x -axishorizontal. The DRF of the box is also considered as theglobal DRF of the assembly. Then, the datum paths have been created; they are shown in Fig. 6.The vector loops have been created and placed on theassembly using the datum paths as a guide. There are  L  ¼  J     P   þ  1  ¼  4    3  þ  1  ¼  2 closed loops and one openloop. The first (closed) loop joins the box and the circle 1 by the links passing from points A and B. The second(closed) loop joins the subassembly box  –  circle 1 and the Fig. 2  The case study with only dimensional tolerances Fig. 3  The case study with dimensional and geometrical tolerances(and rule #1) x 1 = 50 ± 0.20u 2  u 3  u 1 u 4     x    2   =    8   0        ±    0 .   5   0 Ω A B C D E FGH O 1  O 2  x 3  x 3  x 3  x 4  x 4  x 4  x 3 = 20 ±  0.05x 4 = 20 ±  0.05g Fig. 4  Assembly variables and tolerances of vector loop model(dimensional tolerances only)1110 Int J Adv Manuf Technol (2009) 43:1106  –  1123

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