Kinematic Accuracy Method Of Mechanisms Based On Tolerance . - Hindawi
HindawiMathematical Problems in EngineeringVolume 2020, Article ID 5023092, 18 pageshttps://doi.org/10.1155/2020/5023092Research ArticleKinematic Accuracy Method of Mechanisms Based onTolerance TheoriesLi Zhang,1,2 Hong Nie ,1,2 and Xiaohui Wei1,21Key Laboratory of Fundamental Science for National Defense Advanced Design Technology of Flight Vehicle,Nanjing University of Aeronautics and Astronautics, Nanjing 210001, China2State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics,Nanjing 210001, ChinaCorrespondence should be addressed to Hong Nie; hnie@nuaa.edu.cn and Xiaohui Wei; wei xiaohui@nuaa.edu.cnReceived 5 May 2020; Revised 3 August 2020; Accepted 19 August 2020; Published 14 October 2020Academic Editor: Gordon HuangCopyright 2020 Li Zhang et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.Traditional tolerance analysis is mostly restricted to static analysis. However, tolerances of different components also affect themovement accuracy in a mechanism. In this paper, the idea of kinematic tolerance analysis is advanced. In the interest of achievingmovement precision considering tolerance, a kinematic Jacobian model is established on the basis of a traditional dimensional chain andan original Jacobian model. The tolerances of functional element (FE) pairs are expressed as small-displacement screws. In addition, jointclearances resulting from tolerance design also influence the kinematic accuracy, and they are modeled by FE pairs. Two examples arepresented to illustrate the rationality and the validity of the kinematic tolerance model. The results of the two examples are shown, andthe discussion is presented. A physical model of the 2D example is also built up in 3DCS software. Based on the discussion, a comparisonbetween the statistical and physical models is carried out, and the merits and demerits of both are listed.1. IntroductionFor an assembly, component tolerance reflects the actualrelationships between mating parts. Conventionally, tolerance analysis is used to estimate the accumulation of theassembly dimensions. As a consequence, tolerance analysisis mostly restricted to static tolerance analysis in an assembly. However, the tolerances of different parts also affectthe accuracy of movement in an assembly. With this need fortolerance analysis, the effects of component tolerances onthe kinematic accuracy of mechanisms require study.Tolerance analysis includes three steps: the tolerancerepresentation, propagation model, and analysis method[1–13]. In the early years, the propagation model and analysismethod were often discussed together. The work of this papermainly focuses on the topic of tolerance propagation.In terms of tolerance propagation and analysis, thetraditional approach is a dimensional chain. The mainmethods that have been developed in recent years are thematrix model [14, 15], vector loop model [16–19], SOVmodel [20, 21], and Jacobian model [22, 23]. Concerning thetraditional method, a tolerance stack-up function is established on the basis of the relevant dimensional chain. Thefirst-order differential value of the corresponding dimensionis taken as the deviation. This method is mainly used in statictolerance analysis and has been applied in many cases[24, 25]. Utilizing homogeneous matrix transformation, amatrix model transfers the displacements from local reference frames to a global reference frame. The matrixpropagation and representation models are used together toperform the tolerance analysis. A vector loop model usesvectors to represent the dimensions in an assembly [17, 19].The vectors are arranged in chains or loops representingthose dimensions that stack together to determine the resultant assembly dimensions [14]. In an SOV model, thedimensional deviations that represent the influences oftooling and part errors (including part-to-fixture, part-topart, and interstation interactions) are considered as productand process factors; the whole assembly process is modeledas a state-space model. The accumulations of dimensional
2deviations are represented by state transfer functions. TheSOV model is suitable for complex assemblies. In recentdecades, increasingly more attention has been directed to thedevelopment of three-dimensional (3D) tolerance analysis,and a Jacobian model is advanced with the tide. Laperrièreand Lafond adopted virtual joints in robotics to explore atolerance transformation process, and the Jacobian modelwas obtained by first associating a coordinate frame to everyvirtual joint [23, 26]. In this model, functional element (FE)pairs and functional requirement (FR) are used to representthe relationship between parts. Based on this innovation,many other studies ensued [27–33]. In addition to themainstream tolerance analysis methods, new ideas are alsoburgeoning [18, 34, 35]. Wang et al. [36] invented an assembly dimensional model based on the shortest path. Thetolerance propagation function can be acquired through theshortest path, as described in their theory. Based onequivalence of the deviation source, Zhao et al. [18] proposed a combined deviation accumulation method. Although the angles of view are new, the core propagationapproaches are mentioned above. Among the propagationmodels introduced in recent years, the Jacobian model is themost suitable one for the kinematic tolerance analysis.Regarding tolerance representation, different modelsexist in the relevant literature. Requicha introduced themathematical definition of tolerance semantics and proposed a solid offset approach initially [37]. Salomons et al.specified tolerances based on technologically and topologically related surfaces (TTRS) [38, 40]. Desrochers andRiviere presented a homogeneous matrix approach coupledwith the notion of constraints for the representation oftolerance zones [15]. T-Maps were set up as well as a hypothetical volume of points that correspond to all possiblelocations and variations of a segment of a plane that can arisefrom tolerances on size, form, and orientation [41]. The skinmodel is a basic concept within GeoSpelling and the ISOstandards. Skin model shapes [42–47] are skin model representatives that comprise various kinds of geometric deviations. With this method, geometric deviations arerepresented by discretizing features into points and measuring the distances between the points of the nominalmodel and the skin model shape. The main advantage of thismethod is to model geometric tolerance more practically. Toapply geometric tolerances in the Jacobian model, Polini andCorrado [48–50] presented an approach and, whereafter,also integrated manufacturing signature and operationconditions into the Jacobian model based on the skin modelshapes. In recent years, there has been an increasing amountof literature on small-displacement screw models [51–55]. Asmall-displacement screw model was first discussed byBourdet and Clement [56]. As variations of a surface and itsfeatures from the nominal position can be represented by ascrew, a small-displacement model was gradually developedfor tolerance analysis in the following years.Most of the tolerance studies are limited to static tolerance analysis. However, for a mechanism, the tolerance ofevery part in an assembly also affects the kinematic performance, e.g., the accuracy of movement. Walter et al. [57]used “the integrated tolerance analysis in motion” approachMathematical Problems in Engineeringto obtain the effects of manufacturing-caused and operationdepending deviations on a system’s FKCs. The propagationfunction is derived from a dimensional chain, while in thethree-dimensional space, the ideal propagation method forkinematic tolerance analysis is the Jacobian matrix; meanwhile, the study concerns more about the operationdepending effect on the tolerance, not the tolerance influence on the kinematic accuracy. Utilizing the skin modelshapes, Schleich et al. [58, 59] constituted a framework forthe deviation analysis of contact and mobility in an assembly. The skin model shapes are a tolerance representationmethod, and their study mainly focuses on the geometricmodeling of the contact in motion. The mobility is involved;however, the propagation method is also a dimensionalchain, and the application is limited to some extent. Zhouet al. [60] proposed a kinematic accuracy method based onDP-SDT theory. Although the influences of motional displacements, force direction, and vibration can be calculatedusing the method, the propagation function is complicated.The purpose of this paper is to propose a novel kinematicaccuracy method for mechanisms based on toleranceanalysis. First, joint clearances resulting from the fit tolerances are represented as tolerances by small-displacementscrews. Then, the kinematic Jacobian model is established bycombining an original tolerance Jacobian model and atraditional dimensional chain. Finally, the kinematicmovement error can be obtained through the analysis of thekinematic tolerance Jacobian model. These ideas will generate important insights into tolerance design and kinematicaccuracy; a small constituent part of the establishmentmethod can also be used independently and integrated withother theories (e.g., robust theory) so that several kinds ofpractical problems in engineering can be solved [61].The outline of this work is as follows: Section 2 reviewsrelated theory. Section 3 discusses the details of the proposedmethod. Section 4 presents a case study to verify the feasibility and efficiency of the new method, and a physicalmodel is also given to prove the legitimacy of the results inthis section. Meanwhile, a practical mechanism in engineering is also chosen as an example in Section 5, and thediscussion of the results is given. Section 6 summarizes thework and points out the future study.2. Related Theory2.1. Jacobian Model. A Jacobian model adopting kinematictheory has been advanced in robotics. Several basic conceptsof an original Jacobian model are listed in Table 1 [61, 62].In an assembly, it is essential to establish the tolerance chainfirst and then define FE pairs in the established tolerance chain.2.2. Deviation of a Functional Element Pair from Its NominalValue (Based on Dimensional Chain Theory). A tolerancechain is a representation of how a functional requirement xdepends on a known set of functional dimensions di [61, 63].Each dimension has a deviation from its nominal value:Δdi di di , and, as a consequence, a deviation will occuron the functional requirement: Δx x x.
Mathematical Problems in Engineering3Table 1: Basic concepts of a Jacobian model.ParameterDefinitionPoints, curves, or surfaces that belong to parts in the assembly. An FE can be real, e.g., the plane surface of aFunctional element (FE)block, or constructed, e.g., the axis of a cylinder.Functional requirement (FR)An important condition to be satisfied between two FEs on different parts, e.g., a fitting condition.If two FEs are on different parts and there is physical contact between them, then they constitute a kinematicKinematic pairpair.If two FEs are on the same part and both of them participate in a contact relation with some other parts, thenInternal pairthey form an internal pair.Functional element pair (FETwo FEs on different parts or on the same part, including kinematic pairs and internal pairs.pair)For the purpose of tolerancing, some coordinate frames are associated with the toleranced FEs in an FE pair,Virtual jointsassuming a set of virtual joints exist in each FE pair and can make the toleranced FEs “move” relative to theother FEs, in order to simulate manufacturing inaccuracies.Global reference frameThe frame in which the FR and global dimensional chain are established.(GRF)Local reference frame (LRF)The frames in which FE pairs are established.In the general case, x f(d1 , d2 , . . . , dn ) is nonlinearand possibly unknown, and the above assumption Δx isdifficult to calculate. However, the equation can be linearizedby a first-order Taylor approximation (as the deviations onthe dimensions are small compared to nominal values,higher-order terms of a Taylor series can be omitted):nzfxΔdi ,zdii 1Δx nzfyΔdi ,zdii 1Δy nzfzfzfx Δx f d1 , di , . . . , dn Δd · · ·Δd · · ·Δd .zd1 1zdi izdn n(1)Then,zfzΔdi ,zdii 1Δz n(6)zfΔθx θx Δdi ,zdii 1nzfΔdi .zdii 1Δx (2)FR and FE pairs are both established between two FEs.Even the length of an FE pair can be seen as a FR of its elements,letting x f(d1 , d2 , . . . , dn ) be the length of an FE pair, then.Then,nzfΔdi ,zdii 1Δx (3)where x is the small deviation of the length of the functional element pair and di is its deviation from the nominalvalue of di (in tolerancing, it can be seen as a designedtolerance of dimension d).For an FE pair in Cartesian coordinates, the projectionsof lengths along three axes arex fx d1 , d2 , . . . , dn , y fy d1 , d2 , . . . , dn , z(4) fz d1 , d2 , . . . , dn ,and the respective angles about three axes areθx fθx d1 , d2 , . . . , dn , θy fθy d1 , d2 , . . . , dn , θz fθz d1 , d2 , . . . , dn .(5)Then,nzfθyΔdi ,zdii 1Δθy nzfθzΔdi ,zdii 1Δθz respectively, where Δx Δy, Δz, Δθx , Δθy , and Δθz are, respectively, small deviations along and about the threeCartesian axes.3. Jacobian Model for Kinematic ToleranceAnalysis (Kinematic Jacobian Model)In this article, the generic forms of small-displacementscrews are used to model the tolerances associated withfeatures and gaps in an assembly.3.1. Tolerance Modeling Using the Small-Displacement Screw.Let Δ [δ1 δ2 δ3 ; θ1 θ2 θ3 ] be a small-displacement screw model,where [δ1 δ2 δ3 ] is a small translational displacement and [θ1 θ2θ3] is a small rotational displacement.Concerning the existence of tolerances, when mechanisms move, FE pairs move, the lengths of FE pairs change,and the small-displacement screw models change as well.We let η be the input variable of a mechanism; then, theprojections of lengths and angles, respectively, along andabout three axes are
4Mathematical Problems in Engineeringx fx (d, η), y fy (d, η), z fz (d, η),θx fθx (d, η), θy fθy (d, η), θz fθz (d, η),z6(7)y6x6z5and then,nzfx (d, η)Δdi ,zdii 1Δx z4nzfy (d, η)Δy Δdi ,zdii 1z3z2nzfz (d, η)Δdi ,zdii 1Δz (8)nzf (d, η)Δθx θxΔdi ,zdii 1z1nzfθy (d, η)Δdi ,zdii 1z0 y0Δθy x0nzfθz (d, η)Δdi ,zdii 1Figure 1: Virtual joints and coordinate frames to FE pairs.Δθz respectively, where Δx Δy, Δz, Δθx , Δθy , and Δθz are, respectively, small deviations that are functions of the inputvariables η and Δdi . As a consequence, the small-displacement screw can be expressed asnzf (d, η) xΔdi zdi i 1 n zfy (d, η) Δd i Δxδ1 zd i i 1 Δyδ 2 n zf(d,η) z Δdi zdi i 1 Δz δ3 . Δ nΔθx θ1 zf(d,η) θx Δd i zd i i 1 θΔθ 2y n zfθy (d, η) θ3Δθz Δd i zdi i 1 n zfθz (d, η) Δd izdii 1(9)In the LRF, For the ith FE pair of a mechanism whichcontains n FE pairs, the small-displacement screw model isnzf (d, η)Δdi xi i 1zdi n zf (d, η) yi Δdi zdi i 1Δxiδi1 δi2 Δyi n zfzi (d, η) Δdi i 1 zdi δi3 Δzi .ΔFEi n θΔθ i1 xi zf (d, η) θxi Δd i zdi θ Δθ i 1 i2 yi n zfθyi (d, η)θi3Δθzi Δdi zdi i 1 n zfθzi (d, η) Δdi zdii 1(10)Here, the clearance in an assembly is seen as an FE pair ofthe tolerance chain.3.2. Jacobian Matrix. For the Jacobian model, FE pairs areused to represent the dimensions and variations in an assembly. The representations of virtual joints and coordinateframes in an FE pair are shown in Figure 1.The transformation matrix can be deduced to be
Mathematical Problems in Engineering5S ω43 C ω54 C ω65 C ω43 S ω65 S ω43 S ω54 S ω43 C ω54 S ω65 C ω43 C ω65 D32 4645646456462 Sωω Cωωω Cωω Cωωω Sωω D S C C S C S C 3534535345351 . T60 565561 ωCω SωωD Sω C S 454450 0001The Jacobian matrix introduced previously is written asJ J1 J2 J3 J4 J5 J6 FE1 , . . . , J6n 5 J6n 4 J6n 3 J6n 2 J6n 1 J6n FEn .(12)[J1J2. . .J6] is the 6 6 Jacobian matrix associated with theFE of the ith FE pair (internal or kinematic) to which thetolerances are applied, with i 1 n.For small rotational virtual joints, the ith column of theJacobian matrix Ji is computed as i 1 n i 1Z 0 d 0 d 0 ,(13)Ji i 1Z0 i 1 i 1where Z 0 is the third column of Ti 10 and d 0 is the lastcolumn of Ti 10 . In a mechanism, a kinematic chain movesconsistently. If a kinematic chain is constrained to a plane, n i 1 i 1Z 0 does not change, and ( d 0 d 0 ) changes with themovement of the kinematic chain; therefore, Ji changes with n i 1the movement of the kinematic chain. Here, ( d 0 d 0 ) isδ1 δ2 δ3 J1 J2 J3 J4 J5 J6 FE1 θ1 θ2 θ3 FE··· n i 1a function of variable η. We let ( d 0 d 0 ) fi (η), whereη is the input variable of the mechanism.For small rotational virtual joints, the ith column of theJacobian matrix Ji can be changed to i 1 n i 1 i 1 Z 0 d 0 d 0 Z 0 fi (η) Ji , (14) i 1 i 1ZZ00while regarding small translational virtual joints, they do notcontribute to the small rotational displacements of the finalFR, and the ith column of the Jacobian matrix Ji is writtensimply as i 1Z(15)Ji 0 .0For a mechanism containing n FE pairs, the Jacobianmodel for kinematic tolerance analysis in the GRF can beexpressed as J1 J2 J3 J4 J5 J6 FEiThis model is called the kinematic Jacobian model, whereδ1 δ 2 δ3 is the small displacement of the FR in the GRF. The θ1 θ2 θ3 FRJacobian matrix (J) and small-displacement screws of FEpairs ( FEi) change with input variable η in the mechanism.The clearance between parts is seen as an FE pair andmodeled as a small-displacement screw.3.3. Steps to Establish a Kinematic Jacobian Model in Kinematic Tolerance Analysis. The basic steps of the kinematicJacobian model for tolerance analysis are as follows:(1) Identify FE pairs and define the LRF for each FE andthe virtual joints(11)···ΔFE1 J1 J2 J3 J4 J5 J6 FEn ΔFEi . ΔFEn(16)(2) Create the tolerance chain in the mechanism, and obtainsmall-displacement screws for FE pairs; then, establishthe Jacobian model for kinematic tolerance analysis(3) Compute the Jacobian model using a statisticalmethod, worst-case method, or Monte Carlo method(4) Analyze the results4. Case Study 1 for the KinematicTolerance AnalysisIn Section 4, a crank-slider mechanism is taken as an example to analyze the tolerance effect. In addition to theresults obtained using the kinematic Jacobian model introduced above, a software simulation is also performed andis discussed in Section 4. Figure 2 shows the vector representation of a classic crank-slider mechanism with aclearance re in the revolving joint resulting from the tolerance design. B and C are centers of the hole and shaft,
6Mathematical Problems in Engineeringrespectively. The distance between A and D is seen as afunctional requirement. The frame established at point A isseen as the GRF, while the other frames are taken as LRFs.The effect of small angle change is omitted in this case.As re is the clearance resulting from the fit tolerancespredesigned, considering the contact theories stated by Johnson,the following two assumptions are provided before the analysis:(1) There is no deformation caused by the contacts injoints(2) Contacts between the holes and shafts are supposedto be line contactsBased on the above two basic assumptions, the anglebetween re and the horizontal line can be concluded to beequal to the angle between r2 and the horizontal line. Theparameters of the crank-slider are listed in Table 2.In this case, the Monte Carlo method is used in the calculation; therefore, the tolerance of the connecting rod obeysnorm distribution, as shown in Table 2. The functional pairs ofthe crank-slider are listed in Table 3. It is worth noting that the zdirections of small-displacement screws are different from theoriginal virtual joints; therefore, for functional pairs, smalldisplacements in the z directions are negative. n i 1In Table 4, ( d 0 d 0 ) fi (d, η), which changes withthe nmovement i 1 of the kinematic chain. In this case, η θ, and( d 0 d 0 )s are actually the coordinates of each functionalpair in the GRF.4.1. Kinematic Tolerance Analysis Using the Jacobian Model.To establish small-displacement screws for the three FEpairs, we haveδ11000010000010000010The overall Jacobian model isδ1 δ2 ΔFE1 δ3 J · ΔFE2 . θ1 ΔFE3 θ2 θ3 FR0000010000010000010 0 0 1 0 0 0 0 0 .1 0 0 0 0 0 1 0 0(18)(19)In this case, δ3 is the kinematic tolerance change in the xdirection of the FR; calculating the Jacobian model, thestack-up function of δ3 is formalized asδ3 1 · δ13 z1 · θ13 1 · δ23 z2 · θ23 1 · δ33 1 · Δr1 cos θ z1 · 0 Δre cos c z2 · 0 1 Δr2 cos c Δr1 cos θ Δre cos c Δr2 cos c. Δr1 · sin θ δ12 0 δ13 Δr1 · cos θ , ΔFE1 θ11 0 θ0 12 θ130 Δre · sin cδ21 δ22 0 δ23 Δre · cos c ,ΔFE2 θ0 21 θ0 22 θ230 Δr2 · sin cδ11 δ12 0 δ13 Δr2 · cos c . ΔFE3 θ11 0 θ0 12 θ130001 0 10 100 0 0 x2 x3 0z01 z0 x2 x3 1 001 0 10 100 T J 0x03 0z20 0 x3 z2 001 0 10 100 000 000 000(20)(17)To establish the Jacobian matrix for tolerance analysis,we haveIn the GRF, (r2 re ) · sin c r1 · sin θ 0; therefore,sin c ( r1 · sin θ/(r2 re )).Letting θ to change from 0 to 2π, calculations are taken every15 . For the tolerances of the crank-slider, the Monte Carlomethod is used here, and the test number is chosen as 100,000.The histograms for each angle are shown in Figures 3–8.PD represents the probability density of deviation withevery δ, while the small possible deviations of the functionalrequirements are denoted as offset values in Figures 3–8.Meanwhile, mean offset values are plotted in Figure 9.4.2. Kinematic Tolerance Analysis Using the Physical Model.As tolerance analysis can be processed with 3DCS software,here a physical model has been established, and numericalsimulation has also been completed (Figure 10). In 3DCS, thefits are endowed with tolerances, and the analysis is carried out.Basic parameters are the same as analysis using the kinematicJacobian model. The mean values are listed in Figure 11.
Mathematical Problems in Engineering7z2B reC x2x1r1z1γz3x3z1Global framez0Br1Cr2DAr2(a)z2θADx0x1x2x3(b)Figure 2: (a) Classic crank-slider with clearance. (b) Vector presentation.Table 2: The parameters of the crank-slider.Parametersre (mm)re (mm) re (mm) r1 (mm) r2 (mm)θγDescriptionLength (AB)Length (CD)Clearance (BC)Tolerance (AB)Tolerance (CD)Input variableThe angle between re and the horizontal lineValue50160 re N (0.165, 0.0362) r1 N (0, 0.12) r2 N (0, 0.162)0 2πTable 3: Screw models of FE pairs.Functional pairsParameters of each functional pairx1 r1 · cos θ y1 0 z1 r1 · sin θδ11 Δr1 · sin θδ12 0 δ11 Δr1 · cos θx2 re · cos c y2 0 z2 re · sin cδ21 Δre · sin c δ22 0 δ23 Δre · cos cx3 r2 · cos c y2 0 z1 r2 · sin cδ33 Δr2 · sin c δ32 0 δ33 Δr2 · cos cFE pair 1 (A and B)FE pair 2 (B and C)FE pair 3 (C and D)FR (A and D)Unknown and must be calculated n i 1Table 4: ( d 0 d 0 ) of FE pairs in the GRF.Functional pairsFE pair 1 (A and B)FE pair 2 (B and C)FE pair 3 (C and D) n i 1( d 0 d 0 ) of each functional pairx1′ x2 x3 , y1′ 0, z1′ z1x2′ x3 , y2′ 0, z2′ z2x3′ 0, y3′ 0, z3′ 0For comparative analysis, results of both the statisticalmodel and the physical model are represented in Figure 12.4.3. Discussion and Comparison. When the kinematic Jacobian model is used as the stack-up function, Figures 3–8 showthe probability densities of deviations for functional requirements at every angle. Meanwhile, the mean values of all offsetvalues are drawn in Figure 9, from which it can be seen that theoffset values of the functional requirements are strongly dependent on the input variable of the crank-slider; that is, theychange as the input variable changes. The figure reaches thepeak when θ is near π i/2(i 2k 1, k 1, 2,. . ., n).Figure 10 illustrates the results obtained when the crankslider is analyzed using 3DCS software. In general, the resultsare influenced by the input variable of the crank-slider aswell, and the figure also reaches the peak when θ is near π·(i/2 1/12) (i 2k 1, k 1, 2,. . ., n).In Figure 12, the results obtained from the kinematicJacobian model and the results obtained by 3DCS softwareare both plotted. From the comparison in Figure 12, it can beconcluded that the trends of two curves are almost the same.However, for the curve obtained by 3DCS, the peak of thecurve is brought forward or postponed compared with thecurve of the kinematic Jacobian model results. The reason isthat the tiny unalterable tolerance setting in software willinevitably affect the tolerance results.It can be seen from the above results that the kinematicJacobian model obtains comparatively stable results andclearly shows the trend of error change. The results obtainedwith the kinematic Jacobian model are conservative, however, and the basic error-change tendency is the same as thatfound via the software analysis.
8Mathematical Problems in 0–0.50.400.5Offset valueOffset 50–11–0.50Offset valueOffset value(c)(d)0.51Figure 3: Probability density of deviation (θ π/12, π/6, π/4, π/3). (a) θ π/12. (b) θ 2 π/12. (c) θ 3 π/12. (d) θ 4 fset value12Offset et value(c)20–1–0.500.51Offset value(d)Figure 4: Probability density of deviation (θ 5π/12, π/2, 7π/12, 2π/3). (a) θ 5 π/12. (b) θ 6 π/12. (c) θ 7 π/12. (d) θ 8 π/12.
Mathematical Problems in 5Offset value0–0.5100.51Offset 0–10.401Offset value2 10–16Offset value(c)(d)Figure 5: Probability density of deviation (θ 9π/12, 5π/6, 11π/12, π). (a) θ 9 π/12. (b) θ 10 π/12. (c) θ 11 π/12. (d) θ 12 ��10.2–0.5Offset value00.5Offset 00.50–1–0.50Offset valueOffset value(c)(d)0.51Figure 6: Probability density of deviation (θ 13π/12, 7π/6, 5π/4, 4π/3). (a) θ 13 π/12. (b) θ 14 π/12. (c) θ 15 π/12. (d) θ 16 π/12.
10Mathematical Problems in –21–10Offset valueOffset 0–11–0.50Offset valueOffset value(c)(d)0.51Figure 7: Probability density of deviation (θ 17π/12, 3π/2, 19π/12, 10π/6). (a) θ 17 π/12. (b) θ 18 π/12. (c) θ 19 π/12.(d) θ 20 10.5Offset value–0.500.5Offset .200.20–4–2Offset valueOffset value(c)(d)02 10–16Figure 8: Probability density of deviation (θ 7π/4, 11π/6, 23π/12, 2π). (a) θ 21 π/12. (b) θ 22 π/12. (c) θ 23 π/12.(d) θ 24 π/12.
Mathematical Problems in Engineering11Mean values of every angle0.250.20.15Offset 2π3π/22πAngleFigure 9: Mean values of all offset values (Jacobian).(a)(b)Figure 10: (a) Physical model of the classic crank-slider. (b) Details of endowed tolerances.Mean values of every angle (3DCS)0.30.2Offset ure 11: Mean values of all angles (from 3DCS).The kinematic Jacobian model is comparatively simple inthe example, provided in this paper. This coincides withexpectation. Major trends are clearly observed using thekinematic Jacobian model, implying that it can show thetrends of a mechanism more distinctly. As the basic trends ofthe statistical model and physical model are almost identical,
12Mathematical Problems in EngineeringMean values of every angle0.3δ11ΔAB · cos θx1 δΔ 12 AB · cos θy1 δΔ·cosθ 13ABz1 , ΔFE1 ϕ0 11 0ϕ 12 0.2Offset value0.1ϕ13δ21ΔrB · cos θx1
Traditional tolerance analysis is mostly restricted to static analysis. However, tolerances of different components also affect the . traditional method, a tolerance stack-up function is estab-lished on the basis of the relevant dimensional chain. e . tolerance zones [15]. T-Maps were set up as well as a hy-
UNIT 1 SIMPLE MECHANISMS Simple Mechanisms Structure 1.1 Introduction Objectives 1.2 Kinematics of Machines 1.3 Kinematic Link or an Element 1.4 Classification of Links 1.5 Degree of Freedom 1.6 Kinematic Pairs 1.7 Different Pairs 1.7.1 Types of Lower Pair 1.7.2 Higher Pair 1.7.3 Wrapping Pair 1.8 Kinematic Chains
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