Path Curvature of the Single Flier Eight-Bar Linkage

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1 Gordon R. Pennock ASME Fellow Associate Professor Edward C. Kinzel Research Assistant School of Mechanical Engineering, Purdue University, West Lafayette, Indiana Path Curvature of the Single Flier Eight-Bar Linkage This paper presents a graphical technique to locate the center of curvature of the path traced by an arbitrary coupler point of the single flier eight-bar linkage. The first step is to locate the pole for the instantaneous motion of the coupler link; i.e., the point in the fixed plane coincident with the absolute instant center of the coupler link. Since the single flier is an indeterminate linkage, comprised of one four-bar and two five-bar chains, then the Aronhold-Kennedy theorem cannot locate this instant center. The paper presents a novel graphical technique which can locate this instant center in a direct manner. Then the paper focuses on a graphical method to locate the center of curvature of the path traced by the coupler point. The method locates six equivalent four-bar linkages for the two five-bar chains, investigates six kinematic inversions and obtains a four-bar linkage from each inversion. This systematic procedure produces a four-bar linkage with a coupler link whose motion is equivalent up to, and including, the second-order properties of motion of the single flier coupler link. The radius of curvature and the center of curvature of the path traced by the coupler point can then be obtained in a straightforward manner from the Euler-Savary equation. DOI: 0.5/ Introduction Contributed by the Mechanisms and Robotics Committee for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received July 2003; revised October Associate Editor: G. Ananthasuresh. The planar four-bar linkage is the simplest single degree of freedom linkage and many graphical techniques exist for investigating the properties of a coupler curve. However, because of the limited number of links this linkage cannot generate certain complex coupler curve shapes 2. In fact, the four-bar cannot satisfy a wide variety of design requirements, for example, a point on the coupler link cannot dwell during a continuous input, and a link cannot move with straight translation. When increased demands are imposed on the design then more links must be employed. The next number of links, for a linkage with a mobility of one, is the six-bar linkage and then the eight-bar linkage. These carefully designed linkages are not only more versatile than the four-bar linkage but can provide the required performance while maintaining economy and reliability without resorting to electronic controls and actuation. The linkage investigated in this paper is a planar, single-degree-of-freedom, eight-bar linkage, comprised of one four-bar and two five-bar chains. The linkage is commonly referred to in the kinematics literature as the single flier eight-bar linkage 3. The focus of the research is a graphical technique to locate the center of curvature of the path traced by an arbitrary coupler point of the single flier linkage. The initial step in the procedure is to locate the pole of the coupler link, which is coincident with the absolute instantaneous center of zero velocity henceforth abbreviated as instant center. However, it is important to note that this instant center cannot be obtained directly from the Aronhold- Kennedy theorem and the linkage is referred to as indeterminate 4. Dijksman 5 presented a geometric approach to determine the instant centers of an indeterminate linkage by applying first-order reduction through joint-joining. Foster and Pennock 6 proposed a more direct approach based on an iterative procedure. The technique was illustrated by Pennock and Sankaranayananan 7 in a study of the path curvature of a geared seven-bar mechanism. The method will also be used in this paper since it is a simple geometric approach, unlike most techniques where velocity information is required 3,8. The second step is to locate the center of curvature of the coupler curve for a specified position of the single flier linkage. Dijksman 9 presented a geometric approach to this problem, which he termed second-order joint-joining, and is an extension of his earlier paper on first-order reduction through joint-joining. The technique presented in this paper uses kinematic inversion to determine ten virtual links which can be used to form a series of six four-bar chains. These chains are then manipulated to create an equivalent four-bar linkage for the motion of the coupler link. For the purposes of this paper, an equivalent linkage is defined as having the same kinematic properties of the original linkage up to, and including, the second-order. This stipulation also applies to the four-bar chains formed by the ten virtual links. The inflection circle and the center of curvature of the coupler path can then be obtained in a straightforward manner from the Euler-Savary equation 0. This equation has proved to be important in the study of curvature relationships for the analysis and synthesis of planar, single-degree-of-freedom mechanisms,. The important contribution of this paper is that the techniques presented here are purely geometric; i.e., the methods do not require the velocity and acceleration of points fixed in links. The paper is arranged as follows. Section 2 presents the graphical technique to locate the velocity pole for the coupler link of the single flier linkage. Then Sec. 3 presents the graphical technique to locate the center of curvature and determine the radius of curvature of the path traced by a coupler point. Section 4 presents a numerical example to illustrate the graphical techniques and compares the result for the radius of curvature of the coupler curve with an analytical method presented by Pennock and Kinzel 2. Finally, Sec. 5 presents some conclusions and suggestions for future research. 2 Graphical Technique to Locate the Pole A schematic drawing of the single flier eight-bar linkage is shown in Fig.. The ground is denoted as link, the input link is denoted as link 2 and the coupler link is denoted as link 8. Link 2 is pinned to the ground at O 2 and link 4 is pinned to the ground at O 4. Links, 5, 6 and 7 are binary links and links 2, 3, 4 and 8 are ternary links. The revolute joints connecting the moving links are denoted as A, B, C, D, E, F, G and H. For purposes of generality, pin A is not coincident with pin F on link 2, pin E is not coincident with pins F or G on link 3, pin G is not coincident with pin H on link 4, and pins B, C and D are separated by finite distances on the coupler link. The coupler point will be denoted as point Q and is an arbitrary point fixed in the coupler link. The instant center for 470 Õ Vol. 26, MAY 2004 Copyright 2004 by ASME Transactions of the ASME Downloaded 02 Feb 203 to Redistribution subject to ASME license or copyright; see

2 Fig. The single flier eight-bar linkage for the instant center I 8-2 then the corresponding instant centers I 8-3 and I 8-4 are coincident with point M. In other words, the locus of I 8-4 for the two-degree-of-freedom seven-bar linkage see Fig. 2 c must pass through point M. Similarly, if link 5 is removed from the single flier linkage to form a two-degree-of-freedom seven-bar, the locus of all possible instant centers I 8-2 can be obtained by guessing locations of I 8-4 along link 7 or the link extended. Then the location of I 8-2 for the single flier linkage will be the point of intersection of link 5 or link 5 extended and the locus of I 8-2, as shown in Fig. 2 e. Note that this locus must pass through point N, which is defined as the point of intersection of links 6 and 7 or the links extended. In addition to this first-order property, points M and N have additional properties which make them important in the graphical technique for path curvature that is presented in Section 3. These properties will be discussed in some detail in that section. Special Configurations. There are some special configurations of the single flier where the Aronhold-Kennedy theorem can be used to directly obtain the unknown instant centers. For example, consider the configuration when the line through pins A and F on link 2 is collinear with the line through pins A and B on link 5. In this configuration, the instant center I 8-3 can be obtained directly from the Aronhold-Kennedy theorem; i.e., it is the point of intersection of link 6 or link 6 extended with link 5 or link 5 extended. Similarly, when link 6 i.e., the line through pins D and E is collinear with the line through pins B and D on link 8. In this configuration, the instant center I 5-3 can be obtained directly from the Aronhold-Kennedy theorem; i.e., it is the point of intersection of the line though pins A and F with the line through pins B, D and E. link i relative to link j is denoted as I i-j and will also be referred to as the velocity pole or pole for link i relative to link j, and denoted as P i-j. For example, the instant center or pole for a study of the instantaneous kinematics of the coupler link 8 with respect to ground link, henceforth referred as 8/, is denoted as I 8- or P 8-. The first step in the study of the curvature of the path traced by the coupler point Q is to locate the instant center of the coupler link. Foster and Pennock 6 developed a graphical technique to locate the unknown instant centers of an indeterminate linkage. For the convenience of the reader, a summary of this technique, as applied to the single flier linkage, is presented here. First, remove a link say link 7 to produce a two-degree-of-freedom seven-bar linkage, see Fig. 2 a. Note that the instant center I 8-2 can lie anywhere on link 5 or link 5 extended. Therefore, choose any point on this line as a first guess for the location of this instant center denoted as I 8-2 ). The Aronhold-Kennedy theorem can then be used to locate the corresponding instant centers I 8-3 and I 8-4,as shown in Fig. 2 a. This procedure is repeated with a second guess 2 for the location of the instant center I 8-2 denoted as I 8-2 ) on link 5 or link 5 extended. The Aronhold-Kennedy theorem will locate 2 2 the corresponding instant centers I 8-3 and I 8-4, as shown in Fig. 2 b. Now draw a line through the two instant centers I 8-4 and 2 I 8-4, see Fig. 2 c. This line represents the locus of all possible instant centers I 8-4 for the two-degree-of-freedom seven-bar linkage because instant centers are a function of velocity and, therefore, a first-order property. Any guess of the location of the instant center I 8-2 on link 5 or link 5 extended will map I 8-4 on to this line for the seven-bar linkage. If link 7 is now replaced, to restore the single flier eight-bar linkage, the Aronhold-Kennedy theorem states that I 8-4 must lie on link 7 or link 7 extended. Therefore, the instant center I 8-4 is the point of intersection of this line with the locus of I 8-4 obtained in the previous step, see Fig. 2 d. Note that if point M defined as the point of intersection of links 5 and 6 or the links extended, see Fig. 2 d is chosen as a guess 3 The Center of Curvature of a Coupler Path This section presents the graphical technique to locate the center of curvature of the path traced by the coupler point Q shown in Fig.. The goal is to obtain a four-bar linkage that is equivalent to the single flier linkage up to the second-order properties of motion i.e., the acceleration properties of the coupler link of the equivalent four-bar linkage will be identical to the acceleration properties of the coupler link of the single flier linkage. To construct this four-bar linkage, two links are required to connect the coupler link 8 to the ground link. These two links will be obtained by investigating kinematic inversions of six intermediate four-bar linkages. The procedure begins by considering the kinematic inversion 8/2. The location of the pole P 8-2 is known from the procedure presented in Sec. 2 and is shown in Fig. 3 a. For this inversion, the center of curvature of point B denoted as O B ) is coincident with point A, and the inflection point for point B denoted as J B ) can be obtained from the Euler-Savary equation; i.e., P 8-2 J B P 8-2 B (a) P 8-2 O B However, there is not sufficient information at this time to draw the inflection circle for 8/2. Therefore, consider a point denoted as M that is fixed in link 8 and lies on link 5 or link 5 extended. This point lies on the ray that passes through point B and the pole P 8-2 as shown in Fig. 3 b. The inflection point for the path traced by M denoted as J M ) is coincident with the inflection point J B. Therefore, the Euler-Savary equation can be written as P 8-2 J M P 8-2 J B P 8-2 M (b) P 8-2 O M where O M denotes the center of curvature of the path traced by M. Substituting Eq. a into Eq. b, and rearranging, the center of curvature can be obtained from the equation Journal of Mechanical Design MAY 2004, Vol. 26 Õ 47 Downloaded 02 Feb 203 to Redistribution subject to ASME license or copyright; see

3 2 Fig. 2 a The instant center I 8-4 of the seven-bar linkage. b The instant center I 8-4 of the seven-bar linkage. c The locus of the instant center I 8-4 of the seven-bar linkage. d The instant center I 8-4 and point M. e The instant center I 8-2 and point N. 472 Õ Vol. 26, MAY 2004 Transactions of the ASME Downloaded 02 Feb 203 to Redistribution subject to ASME license or copyright; see

4 Fig. 3 a The poles P 8-,P 8-2 and P 8-3. b Point M, and the locus of M and M* for inversions 8Õ2 and 8Õ3. c The reduced four-bar linkage 0, 9, 2 and 3. d The reduced four-bar linkage, 2, 4 and 3. P 8-2 O M P 8-2 M (2) P 8-2 O B P 8-2 B Similarly, for the kinematic inversion 8/3, the location of the pole P 8-3 is known from the procedure presented in Section 2 and is shown in Fig. 3 a. The center of curvature of the path traced by point D denoted as O D ) is coincident with point E, and the inflection point for point D denoted as J D ) can be obtained from the Euler-Savary equation; i.e., P 8-3 J D P 8-3 D (3a) P 8-3 O D Now consider a point denoted as M* that is fixed in link 8 and lies on link 6 or link 6 extended. This point lies on the ray that passes through point D and the pole P 8-3 as shown in Fig. 3 b. The inflection point for the path traced by M* denoted as J M* )is coincident with the inflection point J D. Therefore, the Euler- Savary equation can be written as P 8-3 J M * P 8-3 J D P 8-3 M* P 8-3 O M * (3b) where O M * denotes the center of curvature of the path traced by M*. Substituting Eq. 3a into Eq. 3b, and rearranging, the center of curvature can be obtained from the equation P 8-3 O M * P 8-3 M* (4) P 8-3 O D P 8-3 D Recall that point M in link 8, see Sec. 2, is defined as the point of intersection of links 5 and 6 or the links extended. In other words, point M is the point of intersection of the loci of M and M* as shown in Fig. 3 b. Point M is a unique point in link 8 because it is possible to find the centers of curvature of the paths traced by this point for the kinematic inversions 8/2 and 8/3. From this observation, the five-bar loop 2, 5, 8, 6 and 3 can be reduced to the four-bar loop 0, 9, 2 and 3 that is shown in Fig. 3 c. The binary links 9 and 0, defined by MO M and MO M *, respectively, are referred to in this paper as virtual links and are valid up to and including the second-order properties of motion. This technique can also be used to find the centers of curvature of the paths traced by point N in link 8 for the same two inversions. The point is defined as the point of intersection of links 6 and 7 or the links extended ; i.e., the intersection of the loci of N and N*. Again, from this observation the five-bar loop 4, 3, 6, 8 and 7 can be Journal of Mechanical Design MAY 2004, Vol. 26 Õ 473 Downloaded 02 Feb 203 to Redistribution subject to ASME license or copyright; see

5 reduced to the four-bar loop, 2, 4 and 3 that is shown in Fig. 3 d. The binary links and 2, defined by NO N and NO N *, respectively, are virtual links which are valid up to and including the second-order properties of motion. The center of curvature of the path traced by point M, for the study of 8/, can now be obtained in a systematic manner using six kinematic inversions. For convenience, the center of curvature O M * will be denoted as point T note that this point lies in links 3 and 0. The center of curvature of the path traced by T, relative to the ground link, can be obtained from the inflection circle for 3/. This circle will pass through the pole P 3- and the inflection points J A and J G. The location of the inflection points can be obtained from the Euler-Savary equation; i.e., J A A P 3-A 2 and J O A A G G P 3-G 2 (5) O G G The sign convention is that the direction from J A to A is the same as the direction from O A to A and the direction from J G togisthe same as the direction from O G to G. Note that Eqs. 5 are an alternative form of the Euler-Savary equation presented in Eqs. 4. The inflection point J T will be coincident with the point of intersection of the inflection circle for 3/ and the line TP 3-. The center of curvature of the path traced by point T will also lie on the line TP 3- and the distance from O T to T from the Euler- Savary equation is O T T P 3-T 2 (6) J T T where the direction from O T to T is the same as the direction from J T to T. Points O T and T define a virtual link which is denoted as link 3 as shown in Fig. 4 a. The next step is to find the center of curvature of point O 2 for the kinematic inversion 2/0 as shown in Fig. 4 b. The location of the pole P 2-0 is known; i.e., the point of intersection of links 3 and 9 or the links extended and the inflection circle for 2/0 can be drawn from the Euler-Savary equation as shown in Fig. 4 b. The inflection point J O2 is coincident with the point of intersection of the line O 2 P 2-0 with the inflection circle for 2/0. The center of curvature of the path traced by O 2 for this inversion, denoted here as point V, will also lie on the line O 2 P 2-0. From the Euler- Savary equation, the distance from V to O 2 can be written as VO 2 P 2-0O 2 2 (7) J O2 O 2 where the direction from V to O 2 is the same as the direction from J O2 to O 2. Points V and O 2 define a virtual link which is denoted as link 4 as shown in Fig. 4 b. The four-bar linkage that is formed by links, 3, 0 and 4 will be used to find the center of curvature of the path traced by point M for the inversion 0/, see Fig. 4 c. Note that point M is fixed in both links 8 and 0 and the center of curvature of the path traced by point M for inversions 0/ and 8/ is the same point. Since the location of the pole P 0- is known then the inflection circle for 0/ can be drawn. The center of curvature of the path traced by point M, denoted as O M, from the Euler-Savary equation is O M M P 0-M 2 (8) J M M where the direction from O M to M is the same as the direction from J M to M. Since O M does not lie on the page then the location is indicated by a broken line, see Fig. 4 c. Also, note that the inflection point J V is close to the pole P 0-, however, the two points are not coincident. Points M and O M define a virtual link which is denoted as link 5 and connects the coupler link to the ground link. Fig. 4 a The center of curvature of Point T for 3Õ. b The center of curvature of Point O 2 for 2Õ0. c The center of curvature of Point M for 0Õ. 474 Õ Vol. 26, MAY 2004 Transactions of the ASME Downloaded 02 Feb 203 to Redistribution subject to ASME license or copyright; see

6 Fig. 5 a The center of curvature of Point U for 3Õ. b The center of curvature of Point O 4 for 4Õ. c The center of curvature of Point N for Õ. Journal of Mechanical Design MAY 2004, Vol. 26 Õ 475 Downloaded 02 Feb 203 to Redistribution subject to ASME license or copyright; see

7 Fig. 6 a The equivalent four-bar linkage for 8Õ. b The inflection circle and osculating circle for 8Õ. One additional link is required to connect the coupler link to the ground link to form the equivalent four-bar linkage. This virtual link can be obtained by finding the center of curvature of the path traced by point N for 8/ following the procedure that was presented to find O M. First, two additional virtual links denoted as links 6 and 7 are obtained, see Figs. 5 a 5 c. The relevant centers of curvature and inflection circles are also shown on these figures. Then the center of curvature O N can be obtained from the Euler-Savary equation. The points N and O N define a virtual link which is denoted as link 8 and connects the coupler link to the ground link. Therefore, links, 5, 8 and 8 form the equivalent four-bar linkage for a study of the motion 8/, see Fig. 6 a. Note that the pole P 8- is located at the intersection of links 5 and 8 or the links extended which is a check that the four-bar linkage satisfies the first-order properties of motion of the single flier linkage. Using the Euler-Savary equation, the inflection circle can be drawn and is shown in Fig. 6 b. Finally, the center of curvature of the path traced by coupler point Q can be obtained from the Euler- Savary equation; i.e., O Q Q P 8-Q 2 (9) J Q Q where the direction from O Q to Q is the same as the direction from J Q to Q. 4 Numerical Example The figures presented in this paper are drawn to scale using a CAD system with an integrated parametric solver which provides very accurate results. In fact, the results are identical to the results obtained from a vector loop analysis and the method of kinematic coefficients using MATLAB. Angular dimensions are measured counterclockwise from the horizontal i.e., a line parallel to the ground link O 2 O 4 ) and are specified to the nearest 0.0. Linear dimensions can be presented without units because they can be scaled uniformly to any convenient scale 3. For convenience, the metric length of cm will be used in this section. The dimensions see Fig. are the same as presented by Pennock and Kinzel 2 and are tabulated here in Table for the convenience of the reader. The link dimensions and the angular positions of the ten virtual links shown in Figs. 4 a 5 c are tabulated in Table 2. From the Euler-Savary Equation, the location of the inflection points for points M and N of the equivalent four-bar linkage, respectively, are J M M P 8-M cm O M M 476 Õ Vol. 26, MAY 2004 Transactions of the ASME Downloaded 02 Feb 203 to Redistribution subject to ASME license or copyright; see

8 Links Table and J N N P 8-N cm (0) O N N The diameter of the inflection circle for 8/ is measured as cm. The coupler point Q is located on the ray BC, a distance of cm from pin B. The distance from the pole P 8- to point Q is measured as.28 cm and the distance from J Q to point Q is measured as 9.55 cm. Substituting these results into Eq. 9, the radius of curvature of the path traced by the coupler point Q is O Q Q 3.32 cm () This answer agrees with the result obtained from an analytical method, referred to as the method of kinematic coefficients, that is presented by Pennock and Kinzel 2. 5 Conclusions Dimensions and angular positions of the links Link Dimensions cm Angular Position of Each Link degrees O 2 O 4.50 O2 O O 2 A 5.00, O 2 F 2.50 O2 A 80.00, O2 F GE 5.44, FG 6.50 GE 26.85, FG O 4 G 3.50, O 4 H 2.50 O4 G 87.0, O4 H AB 5.00 AB ED 0.00 ED HC 7.50 HC BC 27.50, DB 0.00 BC 6., DB 66. The paper begins with a review of a graphical technique to locate the instant centers of the indeterminate eight-bar linkage commonly referred to as the single flier linkage. The paper then presents a graphical technique to locate the center of curvature of the path traced by an arbitrary coupler point of the single flier Table 2 links Links Dimensions and angular positions of the ten virtual Link Dimensions cm Angular Position of Each Link degrees 9 O M M 2.6 OM M TM 3.58, VM 7.23 TM 7.23, VM 77.9 UN 3.55, WN 0.49 UN 7.23, WN O N N 5.92 ON N O T T 2.73 OT T O 2 V 2.59 O2 V O M M OM M O U U 3.03 OU U O 4 W 7.3 O4 W O N N 5.83 ON N 6.3 linkage. This technique is believed to be an original contribution to the kinematics literature and will prove useful in a study of the curvature theory of an indeterminate linkage. The graphical technique locates the center of curvature of the path traced by a coupler point in a straightforward manner with little geometric construction. The Euler-Savary equation is used in a consistent manner and a series of kinematic inversions are investigated. The direct graphical technique presented in this paper is simple to use and can be applied to the analysis and synthesis of planar mechanisms in general. The technique offers the advantage that it is easy to implement in the design process using CAD software. The paper includes a numerical example, for purposes of illustration, and shows that the results of the graphical technique are in complete agreement with the answers that were obtained previously from an analytical method. A future paper will extend the graphical technique to include the higher-order curvature properties of a coupler curve of the single flier eight-bar linkage. The graphical technique will also be used to investigate the path curvature of other indeterminate mechanisms. The authors believe that the novel concept of virtual links used in this paper represents an important contribution to the kinematics literature on the curvature of a point path. This concept will be investigated further in a future paper on curvature theory. References Hall, A. S., Jr., 986, Kinematics and Linkage Design, Waveland Press, Inc., Prospect Heights, Illinois. Originally published by Prentice-Hall, Inc., Engelwood Cliffs, N.J., Hain, K., 967, Applied Kinematics, Second Edition, McGraw-Hill Book Co., Inc., New York. 3 Klein, A. W., 97, Kinematics of Machinery, McGraw-Hill Book Company, Inc., New York. 4 Bagci, C., 983, Turned Velocity Image and Turned Velocity Superposition Techniques for the Velocity Analysis of Multi-Input Mechanisms Having Kinematic Indeterminacies, Mechanical Engineering News, 20, February, pp Dijksman, E. A., 977, Why Joint-Joining is Applied on Complex Linkages, Proceedings of the Second IFToMM International Symposium on Linkages and Computer Aided Design Methods, Vol. I, paper 7, pp , SY- ROM 77, Bucharest, Romania, June Foster, D. E., and Pennock, G. R., 2003, A Graphical Method to Find the Secondary Instantaneous Centers of Zero Velocity for the Double Butterfly Linkage, ASME J. Mech. Des., 25 2, June, pp Pennock, G. R., and Sankaranarayanan, H., 2003, Path Curvature of a Geared Seven-Bar Mechanism, Mech. Mach. Theory, 38 2, October, pp , Pergamon Press, Ltd., Great Britain. 8 Rosenauer, N., and Willis, A. H., 953, Kinematics of Mechanisms, Associated General Publications Pty. Ltd., Sydney, Australia. 9 Dijksman, E. A., 984, Geometric Determination of Coordinated Centers of Curvature in Network Mechanisms Through Linkage Reduction, Mech. Mach. Theory, 9 3, pp , Pergamon Press, Ltd., Great Britain. 0 Hartenberg, R. S., and Denavit, J., 964, Kinematic Synthesis of Linkages, McGraw-Hill Book Co., Inc., New York. Uicker, J. J., Jr., Pennock, G. R., and Shigley, J. E., 2003, Theory of Machines and Mechanisms, Third Edition, Oxford University Press, Inc., New York. 2 Pennock, G. R., and Kinzel, E. C., 2003, The Radius of Curvature of a Coupler Curve of the Single Flier Eight-Bar Linkage, Proceedings of the 2003 ASME International Design Engineering Technical Conferences and the Computers and Information in Engineering Conference, CD ROM Format, Chicago, Illinois, September Hasan, A., 999, A Kinematic Analysis of an Indeterminate Single-Degreeof-Freedom Eight-Bar Linkage, M.S.M.E. Thesis, School of Mechanical Engineering, Purdue University, West Lafayette, Indiana, December. Journal of Mechanical Design MAY 2004, Vol. 26 Õ 477 Downloaded 02 Feb 203 to Redistribution subject to ASME license or copyright; see

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