A probabilistic study relating to tolerancing and path generation error

Mechanism and Machine Theory Vol. 20, No, 1, pp. 71-76, 1985 Printed in the U.S.A.

0094-114X/85 $3.00 + .00 © 1985 Pergamon Press Ltd.

A PROBABILISTIC STUDY RELATING TO TOLERANCING AND PATH GENERATION ERROR J O S E P H R. B A U M G A R T E N ~ " Technische Hogeschool Delft, Delft, The Netherlands; Department of Mechanical Engineering, Iowa State University, Ames, IA 50011, U.S.A. and K. V A N D E R W E R F F Technische Hogeschool Delft, Department of Mechanical Engineering, Delft, The Netherlands (Received June 1982; in revised form 31 January 1984)

AbslractmThe effect of manufacturing tolerances on the performance of constrained kinematic chains is analyzed using probability theory. In particular, the method is appfied to examples of path generating mechanisms to determine the mean and standard deviation of the vector position of a coupler point. The mean and standard deviation of errors in linkage lengths is thus accumulated to produce the mean coupler curve and the deviated curves. Theory is compared with experimental results for a specific coupler curve obtained from microadjustment of the links on a planar coupler path plotter. The probabilistic analysis utilizes a finite-element description of the kinematic chain and partial derivatives of the output variable with each length variable result from easy manipulation of the system coordinate transformation. 1. INTRODUCTION

bilistic model for a four-bar coupler point and compared results with a microadjustable planar coupler The need for analytical tools to predict the effect path plotter. This mechanical model is one of the of tolerances has long been felt. The Babbage commodels used as a basis of comparison in this present puting engine failed from accumulated component study. The present study utilizes the definition of tolerances[I]. While structural error[2] in function the variance of a position vector in a planar mechgeneration has been studied, these techniques apply anism and utilizes virtual displacement to develop to the deviation of desired output position from acthe requisite partial derivatives. Jones and Rootual output position for exact linkage dimensions. ney[11, 12] have also applied the definition of reSadier and Sandor[3] have studied the effect of the sidual (variance) and have evaluated the partial dedynamic deflection of an elastic coupler link on a rivatives of a mechanism constraint function desired coupler curve. The linkage dimensions are (Jacobian) in order to optimize motion generation. exact, and the dynamic coupler point path deviates from the static path by the amount of flexure of the coupler link. Hall and Garret[4, 5] have studied 2. A P R O B A B I L I S T I C M O D E L OF C O U P L E R techniques of randomly combining links of different POSITION dimensions in order to achieve a desired functional relationship between input and output crank angles In Fig. 1, an optimally synthesized four-bar of a planar mechanism, and have applied a statis- mechanism is given whose coupler point traces out tical method for mechanical error analysis of func- a Joukowsky airfoil[13] having a 20.83 cm long tional generators. Dubowsky et al.[6] have com- chord line. Let the input angle be 13, coupler angle pleted a type of sensitivity analysis on the effects T, and linkage dimensions a, b, c, d, u and v as of linkage errors in function generators. The study shown. Now, let the link dimensions a, b, c, etc. utilizes optimization techniques to identify that di- be random variables such that mension whicb contributes greatest to output error. Dhande and Chakraborty[7] have used a stochastic a = ~a "* era (I) model to allocate tolerance and clearance in fourbar function generators. Rao and Ambekar[8] have where ~a is the mean dimension of the link a and extended this probabilistic approach to spherical o., is the population standard deviation. The probfunction generators. lem at hand is to determine the distribution of the Jucha[9] has studied tolerance bands at each random coordinates, x and y, of the coupler point higher and lower pair in an interconnected chain. K, assuming a known distribution of the link diBaumganen and Fixemer[10] developed a proba- mensions. In the case where the deviations or of all linkage dimensions are relatively small and the random length variables are known to have unimodial distributions, an approximation to the variance of

t Guest Professor, Technische Hogeschool Delft; Professor, Iowa State University. 71

72

J. R. BAUMGARTEN and K. VAN DER WURFF a = b = c d =

C~:~,~

/ ~ / ~

=

10.49 36.18 63.68 45.30.

cm cm cm cm

15.09 cm

u

a

~

x

d

Fig. 1. Joukowski airfoil generator. the coordinates x and y is given by[14, 15]

= \Oa/

\Sb/

\'~c/

+ \ad/ ~ + \au/ ~ + kay~ (SY~ 2 , (OY~ z , (OYeZ crec = \ ~ / ';~ + \F~: '~ + \ ~ /

(Oy'~'

(Oy'~2

e,,

elements such as trusses, beams, etc.[16]. When all deformations are kept zero, the mechanism is unable to move. However, when one or more deformations are prescribed to be nonzero, then motion is possible. The deformation parameter ~ of the mechanism (2) can be calculated from the instantaneous coordinate values X by the nonlinear expression e

(Oy)'

then one needs only some knowledge of the tolerance distributions cr of the manufacturing process and an easy ability to evaluate partial derivatives in order to use this estimate of variance. The latter need is fulfilled with finite element analysis of the mechanism. 3. THE FINITE ELF,MF,NT METHOD IN KINEMATIC ANALYSIS In the f'mite element approach one considers the mechanism to consist of deformable interconnected

=

D(X).

(4)

Each element of the mechanism contributes one or more equations to this so-called continuity equation (4). For a two-dimensional truss element this contribution is (see Fig. 2) = [(X~ - Xp): + (Y~ -

yp)2p/2 _ lo,

(5)

in which Xp, Yp, Xq, Yq stand for the end point coordinates of this truss element and lo is the original length. The fundamental problem in the kinematical analysis is the determination of X and e is given. This is called the zero-order kinematics. This solution is obtained by numerical integration of the equations for small deviations. For the mechanism

(Xq ,Yq)

x

Fig. 2. The truss element: coordinate and deformation definitions.

A

probabilistic study relating to tolerancing and path generation error

as a whole, one has for the first-order representation of eqn (4) A¢ = DAX.

(6)

Here D stands for the matrix of derivatives aD/OX which can be calculated for a given position. In the system of linear equations (6) describing the relation between elemental displacements and deformations, not all the displacements AX are unknown. AX is considered to consist of two subvectors, AX ~ and A X °. T h e X ° are the fixed coordinates; they do not vary during the motion. The remaining movable coordinates are called XL As A X ° = 0, we have for (6) A~ = D c ~ X ~,

(7)

where D c is the appropriate submatrix of D. This equation is the basis for the calculation of the partial derivatives because the inverse of D c gives the partial derivatives directly:

be evaluated from eqns (2) and (3). Following this, a finite A¢ is prescribed which advances the mechanism to a new position and a second position is analyzed. When a full rotation is made, the collected results give the variance of the coupler point and plots of the tolerance band can be made. The couple of a normal distribution function is defined as the mean and the standard deviation of a given random variable. In this study, the mean link dimension is interpreted as the nominal dimension with no skew. For the examples considered, the manufacture of links by measuring distances between two centers and boring the holes results in a dimension distribution which has no skew from the nominal center-to-center distance[21]. The link lengths are then assumed to be normally distributed. Sports[21] and Gladman[22] present curves of manufacturing tolerance versus dimension for various machining methods. In the present example of measuring a distance between two points and boring holes, the standard deviation of the dimension is approximately a linear function of length. Thus, for the link a, tro -- 0.00225a.

The relation with the mechanism parameters is established by identifying changes in chain length and the elongations of the corresponding elements. In each position one can thus calculate the position change due to possible changes in deformation. The actual motion of the mechanism is obtained by applying finite coordinate changes AX due to a prescribed nonzero A~. The new position •~new

= Xold "[" D c- I ~k~

(9)

73

(10)

This was the relationship used in establishing the standard deviation of the six lengths in the Joukowski airfoil generator under study. The normal (mean) coordinates of the coupler point and the standard deviations, tr~ and try, were now determined at each 12 degree increment of the input crank. The mean coupler path was computed from the nominal dimensions given in Fig. 1. Computer plots of the mean and deviated paths are shown in Fig. 3. Close examination of this plot of

is then corrected in an iterative scheme until the deformation difference ~,c,~ - (~old + A~) is sufficiently small. The method outlined above has various applications, e.g. analysis of motion limits of any arbitrary kinematic chain[17], second-order kinematics, force analysis, spatial mechanisms, etc.[18, 19]. It is part of a Computer Aided Design of Mechanisms project CADOM[20]. The computer program used is a new development covering arbitrary multidegree-of-freedom spatial mechanisms. 4. COMPUTF~ RESULTS In the problem at hand, the dimensions of all the links of the mechanism of Fig. 1 are taken as random variables. Then each truss has variable length and A lj = ~j, the standard deviation. The mechanism is then considered as a multidegree-of-freedom mechanism with the variable chain lengths and the crank angle 13 as input motion. After the initial position is established, the partiai derivatives are calculated according to eqn (8). The variance of the coupler coordinates can then

Fig. 3. Computer-generated tolerance band for deviated coupler paths.

74

J.R. BAUMGARTENand K. VANDERWURFF

:

)

Fig. 6. Adjustable vernier on coupler link. from Fig. 4 that addition of 3 crx and substraction of 3ay shifts the diamond plot to the right and downward from the mean path; the origin of the coordinate set is at fixed pivot A. It is emphasized, again, that these two variances result from the random effects of six probabilistic couples. Fig. 4. Computer-generated mean and deviated coupler paths. tolerance bands reveals five closed curves. The solid line is the plot of the mean coupler path. The curve indicated by the + symbol is a plot of (x + 3crx), (Y + 3ay) coordinates; that indicated by the x plots the (x - 3~rx), ( Y - 3~y) coordinates; that indicated by the squ411 plots the coordinates (x 3crx), ( Y + 3cry); and that indicated by the diamond plots the (x + 3cry), (Y - 3cry) coordinates. Figure 3 indicates a band of error which would contain 99% of the coupler paths of a population of randomly assembled mechanisms. For the sake of easy comparison with experimental results to be presented later, three of these five curves are replotted in Fig. 4. The computed mean coupler position is the solid line, and the diamond and square plots are defined above. The curves of Figs. 3 and 4 result from probabilistic combinations of the six random variables of the link dimensions defined in Fig. 1. In a given criiask position, the exact coordinates, x and y, of coupler point K are computed from the mean dimensions, and variances of x and y are computed from eqns (2) and (3). One observes

Fig. 5. Universal coupler curve plotter.

5. THE EXPERIMENTAL RESULTS A universal coupler curve plotter was built in the course of this study, and a method for micrometer adjustment of the length of the coupler link was provided. Figure 5 shows the plotter setup with the mean dimensions of Fig. 1. Figure 6 shows the details of the vernier screw length adjustment provided. Close observation of Fig. 6 shows a t-24 right-hand thread to the left of the long nut and a ~-16 right-hand thread to the right. This design provided a differential adjustment of length. Investigation showed that the maximum value of Grx and ~y did not result from the positive adjustment of all six dimensions to the + 3a tolerance. All six dimensions were set to (p. + 3or) lengths on the mechanical plotter, then to the (p. - 3~) lengths and finally to the mean lengths p.. The resulting three plots were disappointingly close together. In-

Fig. 7. Accumulated tolerance causing maximum deviation of the coupler point.

A probabilistic study relating to tolerancing and path generation error

t

75

the mean length, _ l(r, _+2c~, - 3 c 0 , then the total combinations of linkages to be investigated on the mechanical plotter is mind-boggling. Calculations reveal that 117,649 plots would be necessary in order to produce the error band predicted from the probabilistic model plotted in Fig. 3. It is true that the curves of Fig. 8 could have been generated for exact dimensions using the digital computer. However, the plotter was utilized so that a visualization of the effects of change could be readily obtained.

REFERENCES

Fig. 8. Mechanically generated coupler paths. deed, it was found that the maximum deviation from the mean curve was attained when a random collection of tolerances for the six dimensions was constructed. This is reasoned from Fig. 7. With pivot A maintained f'Lxed as it was in computer plots, shoneniag of c and u and elongation of a, b, d and v produces the greatest deviation in point K for the position shown. This combination is shown plotted in Fig. 8 as Case 1. The inverse shortening and lengthening also produces a maximum deviation in the reverse direction. This combination is shown plotted as Case 2 in Fig. 8. Figure 8 also gives a plot of the mean path. Comparison shows that the error bands of Fig. 4 computed from the probabilistic model and that of mechanical generation in Figure 8 agree well. Recall that both figures were produced with a stationary location for pivot A. 6. DISCUSSION AND CONCLUSION It is apparent that the approximation for the variance of the coupler path as given by eqns (2) and (3) gives a useful estimate of the deviated coupler path. The effect of random combinations of linkage tolerances can be predicted with the probabilistic model given. One finds good agreement between the computer-generated and model-generated error band. The maximum width of the error band from the probabilistic analysis plotted in Fig. 4 is found to be 1.47 cm at the 0.73% chord point. The error band produced by the combination of linkage lengths used to plot Fig. 8 on the mechanical plotter was found to be 1.35 cm at this chord point[10]. If one accepted seven dimensions for each link (i.e.

1. P. Morrison and E. Morrison, Charles Babbage and His Calculating Engines. Dover Publications Inc., New York (1961). 2. R. S. Hartenberg and J. Denavit, Kinematic Syntheses of Linkages (esp. pp. 140-230). McGraw-Hill, New York (1964). 3. J. P. Sadler and G. N. Sandor, Kineto-elastodynamic harmonic analysis of four-bar path generating mechanisms. ASME Paper No. 70-Mech-61. 4. R. E. Garret, Optimal Synthesis of Randomly Generated Linkages. Ph.D. Thesis, Purdue University, University Microfilms, Ann Arbor, Michigan. 5. R. E. Garret and A. S. Hall, Jr., Effect of tolerance and clearance in linkage design. Trans. ASME, J. Engng. Ind., 91B, 198-202 (1969). 6. S. Dubowsky, J. Maatuk, and N. D. Perreira, A parameter identification study of kinematic errors in planar mechanisms. ASME Paper No. 74-DET-52. 7. S. (3. Dhande and J. Chakraborty, Analysis and synthesis of mechanical error in linkages--a stochastic approach, Trans. ASME, J. Engng. Ind., 9b'B, 672676 (1973). 8. S. S. Rao and A. G. Ambekar, Mechanical error analysis of spherical function generating mechanisms--a probabilistic approach. ASME Paper No. 74-DET-15. 9. J. Jucha, Bestimmung der ausfilhrungstoleranzen bei vorgegebeuer toleranz mehrerer ergebuisgr6ssen. Feingeratetchnik 23 Jg Heft 11 (1974). 10. J. R. Baumganen and J. V. Fixemer, A note on a probabilistic study concerning linkage tolerances and coupler curves. ASME Paper No. 76-DET-3. ll. J. Rees Jones and G. T. Rooney, Motion analysis of rigid-link mechanisms by gradient optimization on an analogue computer. J. Mechanisms 5, 191-201 (1970). 12. G. T. Rooney and J. Rees Jones, Curve following in kinematic analysis. Proc. 4th World Congress TMM, I. Mech. E., London (1975). 13. A. J. Nechi, A relaxation and gradient combination applied to the computer simulation of a plane four-bar chain. Trans. ASME, J. Engng. Ind. 91B, 113-119 (1971). 14. E. B. Haugen, Probabilistic Approaches to Design. Wiley, New York (esp. pp. 90-93) (1968). 15. P. H. Wirsching and E. B. Haugen, Probabilistic design of helical springs. ASME Paper No. 74-WA/DE21. 16. K. van der Werff, Kinematic and Dynamic Analysis of Mechanisms: A Finite Element Approach. Ph.D. Thesis, Department of Mechanical Engineering, Technische Hogeschool DeLft, Delft University Press, Delft, The Netherlands (1977). 17. A. J. Klein Breteler, Umlanffahigkeit und ubertragungsqualitat bei mehrgliedrigen mechanismen. VDIBerichte Nr. 434 (1982). 18. K. van der Werff, A finite element approach to kinematics and dynamics of mechanisms, Proc. 5th World Congress TMM, ASME, New York (1979). 19. A. J. Klein Breteler, Partial derivatives in kinematic

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J. R. BAUMGARTENand K. VANt)F.aWUt~FF

optimization. Proceedings, 5th World Congress TMM, ASME, New York (1979). 20. H. Rankers and collaborators, Computer aided design of mechanisms: the CADOM project of the Delft University of Technology. Proc, 5th World Congress TMM, ASME, New York (1979).

21. M. F. Spotts, Design of Machine Elements, 4th edition (esp. pp. 530-531). Prentice-Hall Inc., New York (1971). 22. C. A. Gladman, Drawing office practice in relation to interchangeable components, Proc. Institute of Mechanical Engineers 152, London (1945).

WAHRSCHEINLICHTSTHEORETISCHE UNTERSUCHUNG ZU TOLERANZEN UND FEHLERN BEI DER ERZEUGUNG VON BAHNKURVEN Km,zfmsungmDie Auswirkungen der Fertigungstoleranzen auf das Verhalten von Koppelgetrieben mit starren Gliedern werden mit Hilfe der Wahrscheinlichkeitstheorie untersucht. Insbesondere wird diese Vorgehensweise auf Beispiele der Koppelkurven yon Viergelenkgetrieben angewendet, um Mittelwerte und Standardabweichungen des Positionsvektors eine.s Koppelpunktes zu bestimmen. Dutch Sammein der Mittelwerte und Standardabweichungen des Fehlers in den Getriebegliedliingen werden sowohl die Mittelwert- bzw. Hauptkoppelkurve als auch die Abweichkoppelkurven gefunden. Die Ergebnisse der theoretischen Untersuchung werden mit experimentellen Ergebnissen for eine bestimmte Koppelkurve bei Feineinstellung der Gliederlb.ngen an einer ebenen Koppelkurvenzeichanmaschine verglichen. Die Wahrscheinlichkeitsanalyse henutzt eine Finite-Elemente-Beschreibungdes zu untersuchendan Kopl~lgetriebes. In dieser Beschreibong lassen sich die partiellen Ableitungender (Jbertragnngsfunkfionenoder Bahnfunktionen unter EinfluB der Gliedl~ingeniinderungenleicht bestimmen.