GLOBAL COORDINATE METHOD FOR DETERMINING SENSITIVITY IN ASSEMBLY TOLERANCE ANALYSIS

Transcription

1 GOBA COORDINATE METOD FOR DETERMINING SENSITIVIT IN ASSEMB TOERANCE ANASIS Jinsong Gao ewlett-packard Corp. InkJet Business Unit San Diego, CA Kenneth W. Chase Spencer P. Magleb Mechanical Engineering Department Brigham oung Universit Provo, UT ABSTRACT Tolerance sensitivit indicates the influence of individual component tolerances in an assembl on the variation of a critical assembl feature or dimension. Important applications include assembl tolerance analsis and tolerance allocation. This paper presents a new method for determining tolerance sensitivit using vector loop assembl tolerance models. With a vector loop model, the assembl kinematic constraints or assembl functions can be established automaticall as implicit functions. This method evaluates the derivatives of the kinematic constraint equations with respect to both manufactured dimensions and assembl variables. The derivative matrices are then used to calculate the tolerance sensitivit matri. This is a closed form method which relates the derivatives of the assembl functions to the coordinates of the joints or nodes and the orientations of the vectors and the local joint aes in an assembl. It is accurate, simple and ver suitable for design iterations.. INTRODUCTION Manufactured parts are seldom used as single parts. The are used in assemblies of parts. The dimensional variations which occur in each component part of an assembl accumulate statisticall and propagate kinematicall, causing the overall assembl dimensions to var according to the number of contributing sources of variation. The resultant critical clearances and fits which affect performance are thus subject to variation due to the tolerance stackup of the component part variations. Tolerance analsis is a quantitative tool for estimating the effect of the accumulation of component variation in assemblies. Assembl variations accumulate or stackup statisticall b rootsum-squares: δui Σ((S ij δ j ) ) () where δui is an assembl feature variation, δ j is the set of component variations, and Sij is the sensitivit δu δ To perform an analsis, assembl functions must be derived which describe the nominal geometr of each assembl feature U i in terms of the component dimensions j. Current analsis methods use both implicit and eplicit assembl functions. The relationship between critical assembl features and the component dimensions which govern them are epressed algebraicall and analzed to determine the effects of variation. Assigned tolerances are introduced and the resulting variation of critical assembl features is evaluated to see if design limits will be eceeded. Tolerance sensitivit is an essential aspect of tolerance analsis of mechanical assemblies in -D and -D space. It indicates the influence of the individual component tolerance in an assembl on the variation of a critical assembl feature or dimension. B eamining the sensitivities, a designer can decide how to control the component tolerances to meet the design specifications. For eample, tolerances ma be loosened on epensive processes and tightened on others to reduce cost, while assuring that the design specs will be met. A new method, called the Global Coordinate Method, has been developed which can accuratel and effectivel determine the sensitivit matri of both eplicit and implicit assembl functions. This paper presents the new method and includes: ) research review, ) a procedure for obtaining closed form epressions for the derivative matrices with respect to manufactured dimensions and assembl or kinematic dimensions, ) the geometric interpretation of the derivatives in -D and -D assemblies, ) two eamples to demonstrate the procedures of appling the new method to mechanical assemblies, and ) conclusions. j i.

2 . RESEARC REVIEW The sensitivit evaluation method is related to the tpe of assembl function. If an eplicit function can be established to epress the assembl variable in terms of the manufactured dimensions, the sensitivit can be obtained b straightforward procedures [Knappe 96, Fortini 967, Co 986]. owever, finding such an eplicit function for all but the simplest assemblies is ver difficult. The authors previous papers [Chase, Gao & Magleb 99, Gao, Chase & Magleb 99] developed vector loop models for representing mechanical assemblies in -D and -D space, with which the assembl kinematic constraint equations can be obtained automaticall in the form of implicit functions. This is ver useful for computer-aided tolerance analsis and allocation, since no user intervention is needed to establish assembl functions. With implicit functions, the tolerance sensitivities must be derived from the derivatives of the implicit assembl equations b algebraic or numerical operations. In the previous papers, the derivatives with respect to each manufactured dimension and the assembl or kinematic variables, are arranged in matrices, which can be used to calculate the tolerance sensitivit matri of the assembl. So, a method for efficientl and accuratel evaluating derivatives is an essential part of evaluating tolerance sensitivit of an assembl described b implicit functions. Methods have been developed to calculate the derivative matrices from vector loop-based models of mechanical assemblies. In such models the vectors represent chains of dimensions which contribute to tolerance stackup in the assembl. Closed vector loops describe the kinematic constraints between mating parts. Open loops describe assembl dimensions and features resulting from the dimension chains. Marler [988] used simple -D vector assembl functions for evaluating the derivatives. Figure shows a closed vector loop. The vectors are joined tip-to-tail. Each vector length i represents either a component dimension, with its corresponding tolerance, or a kinematic variable, describing an adjustable assembl parameter. The angles i describe the relative rotation from one vector to the net Figure. A sample vector loop-based assembl model. The assembl constraints can be resolved into three scalar equations in the global coordinate directions, representing the sum of vector components in the and directions and sum of rotations about Z: n i Σ i cos Σ j i j () n i Σ i sin Σ j i j () n+ Σ j i () From equations (), () and (), the derivative with respect to i and i can be easil obtained in closed form. For translation: For rotation: n k δ δ i cos δ δ i sin δ. i Σ j i Σ j j j () δ - k sin δi Σki n δ Σ k δi ki δ. δi cos j Σj k Σ j δi This closed form for evaluating derivatives of the vector loopbased assemblies in -D is straightforward and accurate. owever, it is ver difficult to etend to -D assemblies. A closed -D loop can be epressed in terms of a concatenation of coordinate transformation matrices: [R][T][R][T]...[Ri][Ti]...[Rn][Tn][Rf] [I] (7) where [Ri] and [Ti] represent the rotational and translational operations at Joint i, [Rf] is the final rotation required to bring the loop to closure, and [I] is the identit matri. This implies that if a kinematic loop describing a mechanism is closed, the coordinate sstem at the end of the loop must be parallel to and located at the same point as the coordinate sstem at the beginning. Robison [989] used a small perturbation method for evaluating the derivatives on the assembl constraint equation (7). For a -D case, if the rotation at Joint i is the variable with respect to which the derivatives are desired, a small angle perturbation δ is added to the original value, then the matri multiplication of equation (7) is performed: {,,} t [R][T][R][T]...[Ri( +δ)][ti]... j [Rn][Tn]{ } t { } t (8) The derivatives can then be approimated numericall b: () δ (9) δ (6)

3 where and are the scalar constraints in global and directions, and are the resultants. A similar procedure is followed for derivatives with each of the other variables. This method can easil be etended to -D assembl models. The disadvantage of this approach is that it is computationall intensive and its accurac depends on the size of the perturbation and the size of the matri equation. Sandor [98] introduced a method for accuratel evaluating the derivatives of the vector loop-based mechanisms in -D and -D. e emploed a derivative operational matri and inserted it into the constraint equation (7). Then the matri multiplications are performed and the derivatives are obtained. A similar procedure was used b Whitne, et al [99]. This method reduces the derivative to a matri operation which is equivalent to a small perturbation. Although it is more suitable for automated computation, it still requires substantial matri multiplications for each sensitivit. uo [996] introduced the variation polgon, a new method for obtaining tolerance sensitivities. It is based on vector polgons, in which dimensional variations are added vectoriall, similar to velocit polgons for mechanism analsis. The provide closed form relationships between the dimensional variations and resultant assembl variations. The sensitivities ma be derived from the vector polgons. This method has not been generalized for -D and -D assemblies. The Global Coordinate Method for computing sensitivities offers some advantages over the current methods described above. It works with both eplicit and implicit assembl functions, is simple to use, readil automated, and offers good accurac and reduced computation. The basics are described in the following sections and demonstrated with eamples.. GOBA COORDINATE METOD FOR DETERMINING SENSITIVIT The -D model described in figure represented an assembl b vector chains and relative rotations between adjacent vectors. -D assemblies ma be represented b similar vector chains. The derivative formula for the new approach can be derived b the differentiation of the D vector epression. V i + j + Zk () et, and Z as well as i, j and k be the functions of variable u. Then differentiating equation () gives [Chisholm, 978]: dv d du du i + d du j + dz du k + ω V () where ω V (ω Z - ω )i + (ω - ω Z)j + (ω - ω )k () and ω (ω,ω,ω ) are the direction cosines of the ais of rotation. The first three terms in equation () result from the change in length of V. The cross product is from it's rotation.. Derivative with Respect to a ength Variable If vector V represents the sum of a vector chain, the derivative with respect to a length variable can be obtained b letting u i. From equation (), it can be seen that the last term drops out due to no local rotation. Then V i lim δ i δi + δ j + δz k δ i δ i δ i () Since the rates of δ/δi, δ/δi and δz/δi do not change for an variation δi along the vector i, the derivatives of the assembl function to such a variable are constants, which are equal to the direction cosines of the vector measured in the global coordinate sstem. For translational equations: cosα i cosβ i z cosγ i () (6) For rotational equations: i i z i (8) (9) (7) () where α, β and γ are the direction cosine angles of vector i and variation vector δi;, and z are the scalar sum of vectors in the global, and Z directions; and, and z are the sum of, and Z rotations.. Derivative with Respect to a Rotation Variable The derivative with respect to a rotation variable can also be obtained b letting ui in equation (), which is a rotation about one of the aes of the local joint coordinate sstem. Since this rotation variable is not related to the first three terms in equation (), the are zeros. Onl the last term of the equation is undecided. Equation () gives the derivatives of the assembl functions with respect to the angular variable. In that equation, the rotation is applied at the joint and the variation is measured at the origin of the global coordinate sstem. V ( ω - ωz)i + (ωz - ω) j + (ω - ω)k i () So, it is eas to write the derivatives for the translational constraint equations in terms of the global coordinates, and Z of joint i and the direction cosines of the rotation: For translational equations: For rotational equations: ω i i ω - ωz ωz - ω () z ω - ω i i i ω z ω i () If the rotation variable is considered as a vector, equations () have the same meanings as equations () to (7), that is, the each describe the direction cosines of the rotation variation vector.

4 If the global coordinates of the joints and the direction cosines of the vectors and local joint aes in an assembl are known, the derivatives with respect to both translational and rotational variables can be obtained ver easil. For a -D assembl, simpl let Z, ω, ω and ω ±, and the derivatives are: For translational variable: For rotational variable: i cosα i cosβ sinα () i i i i ω -ω ω () where ω could be either or -, depending upon the direction of the relative rotation at joint or node i.. GEOMETRIC INTERPRETATION OF TE DERIVATIVES The geometrical interpretation of the derivatives with respect to a variable will help one to understand the relationship between the derivatives and variations in geometr. An small perturbation at a joint or node will cause the vector loop to fail to close. The variation propagates around the loop and results in a gap at the starting point. If a closure vector is added to close the loop, the ratio of the closure vector to the disturbance at the joint is related to the derivative.. -D Mechanical Assemblies First consider the derivative of the -D assembl in Figure with respect to length variable. The procedure includes: ) perturbing the variable b δ, ) finding the resultant variations of and at the origin and ) dividing these variations b δ. In this case, the variation and are the projections of δ on and aes, so the ratios cos α δ cos β sin α δ are all constants, where α and β are the direction cosine angles of vector. δ 7 Figure. Figure with perturbation δ along vector. Now consider the derivative with respect to a rotational variable. Figure illustrates a loop with an angle perturbation δ at Joint. Since each vector direction is measured relative to the preceding vector, this is equivalent to rotating the remaining vectors, from Joint to the origin, as a rigid bod through the angle δ. Such a rigid bod rotation around Joint will ield the resultant variations of the end point of the last vector with the magnitudes of and. From Figure, it is eas to see that for small angle δ ω δ -ω δ so ω δ -ω δ ω 6

5 δ 7 Figure. Figure with angle perturbation δ.. -D Mechanical Assemblies The geometrical description of the derivatives of a vector assembl in -D is more complicated as compared with the -D case. Figure shows a -D vector loop with a perturbation δ along the vector,, and Z are the resultant variations at the end point of the last vector, and these variations divided b δ give the derivatives with respect to. Just as the case in -D, the derivatives with respect to a translational variable can be written as: δ δ z Z δ cos α cos β cos γ 6 z where α, β and γ are the direction cosine angles of vector, or the angles between and, and Z ais respectivel. Z Z Figure. A D vector loop with perturbation δ. The most difficult part of the geometrical illustration is to visualize the derivatives with respect to a rotation in -D space. δ Figure describes a -D vector loop with an angle variation δz around local z ais. This rotation is equivalent to rotating the line connecting Joint and the origin of the global coordinate sstem b an angle variation δz about the local z ais. Z δ z Joint Figure. A -D vector loop with angle perturbation δz. This angle variation at Joint will produce translational variations, and Z as well as rotational variations Θ, Θ and Θz at the global coordinate origin. It is eas to obtain the resultant angle variations at the global coordinate sstem, if we place a unit vector ω representing the rotational variation, whose components in the global, and Z directions are ω, ω and ω respectivel. Then:. Θ ωδz Θ ωδz Θz ωδz If the angle variable δz is considered as a vector, ω, ω and ω have the same meanings as direction cosines as have been discussed in translational variable case. So the remaining work is to find out, and Z caused b δz. Figure 6 illustrates how is calculated. From the drawing, it can be seen that onl rotations around and Z affect. Z Z ( Z) Θ ω δ z Θ ω δ z δ z ω δ Θ ω δ z z Figure 6. Components of caused b δz. ωδz - Zωδz z z z Zω δ z

6 In the same wa, and Z can be found. Zωδz - ωδz Z ωδz - ωδz So the derivatives can be epressed as: z z z z δz δz Z δz ω - Zω Zω - ω ω - ω z z z z Θ δz Θ δz Θz δz ω ω ω This is called the Global Coordinate Method for determining the scalar derivatives of the kinematic constraint equations. These derivatives can be used to form matrices from which the sensitivities ma be derived, as demonstrated in the net section.. EAMPES The -D one-wa clutch assembl [Chase, Gao & Magleb 99] and the -D crank slider mechanism [Gao, Chase & Magleb 99] will be re-eamined in this section to show how to appl the Global Coordinate Method to determine the sensitivit matri for tolerance analsis of mechanical assemblies.. -D One-Wa Clutch Assembl Figure 7 shows the vector loop model of the one-wa clutch assembl, as described b Fortini [967]. This is a common device used to transmit rotar motion in onl one direction. When the outer ring of the clutch is rotated clockwise, the rollers wedge between the ring and hub, locking the two so the rotate together. In the reverse direction, the rollers just slip, so the hub does not turn. Table lists all the information necessar for evaluating the derivatives with respect to both manufactured and assembl or kinematic dimensions. Joint Φ c Joint ub Joint a b c Roller Joint e Joint Ring Φ Figure 7. Vector loop model of one-wa clutch assembl. Table. Dimensions of one-wa clutch vector loop Part-Joint Name Orientation Joint Coordinates a Joint α 9.. b Joint α c Joint α 9 c Joint α e Joint α In this assembl, dimensions a, c and e are the manufactured variables, while b, and are the assembl or kinematic dimensions. From equations () and (), the derivatives ma be calculated and grouped into matrices [A] and [B].

7 [ A ] e c a e c a e c a [ B ] b b b where [A] are derivatives with respect to the manufactured variations and [B] are derivatives with respect to the assembl variables. Finall, the sensitivit matri [S] can be calculated as shown b Chase, et al [99]: [S] -[B] - [A] D Crank Slider Mechanism The vector loop model of the -D crank slider mechanism is illustrated in Figure 8 with all dimensions marked. In this assembl, dimensions A, B, C, D and E are the manufactured variables, while,,, and U are the assembl or kinematic dimensions. Table lists the joint coordinates and vector orientations as well as the orientations of the local joint aes around which the rotations will be the variables with respect to which the derivatives are desired. Those data can be easil obtained if the assembl model has been established using assembl modeling software, such as Pro/E, CATIA and etc. A B C D E U Crank Base ink Slider D z 6 Figure 8. -D slider crank assembl Table. Dimensions of crank slider vector loop Part-Joint Name Orientations ( ω ω ω) Joint Coordinates ( Z) A Joint (,, ) B Joint (,, ) C Joint (,.77,.77) D Joint (.99,.6,.6) Joint () (,.77,.77) (-.,.666, 9.9) Joint (z) (.66,.698,.698) (-.,.666, 9.9)

8 E Joint (,, ) Joint () (.9,.6,.76) (-9.76,.,.) Joint (z) (.87,.89, ) (-9.76,.,.) U Joint 6 (,, ) From equations () and (), the derivative matri with respect to the manufactured dimensions [A] and with respect to the assembl or kinematic variables [B] can be obtained. A B A A z ŽA A A z A Ž z B B z B B B z B Ž z C C z C C C z C z D D z D D D z D Ž Ž z Ž Ž z Ž z z z E E z E E E z E U Ž U z U U U z U Since [B] is not a square matri, it can not be inverted directl. In such cases, the sensitivit matri ma be calculated using a least square fit b inverting the product [B] T [B] and multipling with the product [B] T [A]. This procedure is described b Gao, et al [998]. [S] - ([ B ] T [ B ]) - [ B ] T [ A ] The statistical variation in displacement U can be estimated b root sum squares from equation () using elements of [S]: δu [ (.86 δa) + ( δb) + (.7 δc) + (.6677 δd) + (.86 δe) ] / where δa, δb, δc, δd and δe are the tolerances on dimensions A, B, C, D and E, respectivel. In a similar manner, the calculated tolerance sensitivities [S] ma be used to calculate predicted tolerance stackup in an assembl feature b worst case or statistical sums of the component tolerances times their respective sensitivities. Design decisions about which tolerances to tighten or loosen to avoid assembl problems or to reduce cost ma also be made based on the sensitivities, as described in man design publications [Fortini 967, Chase 99]. 6. CONCUSIONS This research work presented the Global Coordinate Method for evaluating the sensitivit matri for assembl tolerance analsis. The Global Coordinate Method relates the sensitivit of assembl functions to the geometric information of an assembl, such as the coordinates of the joints, the orientations of the vectors and the local coordinate reference sstems in the global coordinate frame. Such data can be obtained easil after the assembl has been generated using modeling software. The

9 outstanding features of this method include its simplicit, accurac and efficienc. REFERENCES Chase, K. W., Gao, J. and Magleb, S. P., 99, General -D Tolerance Analsis of Mechanical Assemblies with Small Kinematic Adjustments, Journal of Design and Manufacturing, v, 6-7. Chase, K. W. and A. R. Parkinson, 99, A Surve of Research in the Application of Tolerance analsis to the Design of Mechanical Assemblies. Research in Engineering Design, (99): --7. Chisholm, J. S. R., 978, Vectors in Three-Dimensional Space, Cambridge Universit Press. Co, N. D., 986, "Volume : ow to Perform Statistical Tolerance Analsis," American Societ for Statistical Qualit Control. Fortini, E. T., 967, " Dimensioning for Interchangeable Manufacture," Industrial Press. Gao, J., Chase, K. W. and Magleb, S. P., 998, Generalized -D Tolerance Analsis of Mechanical Assemblies with Small Kinematic Adjustments, IIE Trans, v, uo, anqi, 996, "Variation Polgon-A New Method for Determining Tolerance Sensitivit in Assemblies," Proc. CSME Forum 996, Ontario, Canada, Ma 7-9. Knappe,. F., 96, " A Technique for Analzing Mechanism Tolerances," Mechine Design, April, pp. -7. Marler, Jaren D., 988, Nonlinear Tolerance Analsis Using the Direct inearization Method, M.S. Thesis, Mechanical Engineering Department, Brigham oung Universit. Robison, R.., 989, A Practical Method for Three- Dimensional Tolerance Analsis Using a Solid Modeler, M.S. Thesis, Mechanical Engineering Department, Brigham oung Universit. Sandor, G. N., Erdman, A. G., 98, Advanced Mechanism Design: Analsis and Snthesis, Volume, Prentice-all, INC. Whitne, D. E., Gilbert, O.. and Jastrzebski, M., 99 Representation of Geometric Variations Using Matri Transforms for Statistical Tolerance Analsis in Assemblies, Research in Engineering Design, vol. 6, pp. 9-.