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Concerning the motion of the center of mass of a crank mechanism
Jnl. Mechanisms Volume 3, pp.335-338/Pergamon Press 1968/Printed in Great Britain
Concerning the Motion of the Center of Mass of a Crank Mechanism R. Kreutzinger Abstract After explaining the general background, it is shown by the author how one may generate, as a coupler curve the trajectory, of the total center of mass of the movable links of a four-bar linkage. This is done by means of basic configurations, which have been supplemented by appropriate parallelogram mechanisms, and by reference to the Roberts-Chebyshev theorem. The work also gives directions for the construction of a crank mechanism (four-bar linkage) whose center of mass trajectory has been generated as a coupler curve.
1. General Fundamentals FIGURE 1 shows a crank mechanism F1ABF z. Let it be required to examine the trajectory of its center of mass. A
z
S3
Figure 1. Four-bar linkage with positions of centers of mass S~, $2, 3'3 of the movable links.
0
K
".
A.
', ,, /
~H2
.'i /
S2
.....l / ~ c
/
Figure 2. Graphical determination of the principal points Ar t and H2 of a two-bar configuration determined from centers of mass St and S z of both links. * Getriebetechnilc
10 (9), 397-398 (Sept. 1942). 335
336
Since it is assumed that the ground F1Fz is fixed in space, one need only consider the kinematic chain which consists of the links FtA, AB and BF, when examining the motion of the center of mass. The centers of mass S t, S z and S 3 are identical to the centerpoints of the links connecting the joints, if one assumes that all links are equally thick and homogeneous. If one considers with the above assumptions an open two-link kinematic chain (two-bar configuration of Fig. 2), then, as is well known, its center of mass St2 lies on the line connecting St and S_,, and divides the distance S t S 2 in the inverse proportion to the forces (weights) which act at S t and S 2. In the case at hand these forces might be represented possibly by means of half the respective lengths of the rods, i.e. the distances St A and S2A. It has proven to be advantageous, for all questions relating to centers of mass, to introduce certain invariable points which lie on the link axes and which are called "principal points".* Thus, for example, one finds the principal point Ht of link FtA as the center of mass of the points St and A, with weights of the links FtA and A B assumed to act at these points respectively. As stated above, the magnitudes of these weights are expressed by the lengths StA and SzA. The points A, H t, St2, H 2 form the corners of a parallelogram. This parallelogram may be thought of as weightless and articulated, thereby serving to constrain the center of mass to the motion of the two-bar configuration. Figure 2 shows a simple construction of the points Ht, H2 and St2 by means of the easily obtained auxiliary points C, D, E, K, J. 2. T h e C e n t e r o f M a s s S o f a C r a n k M e c h a n i s m
If one now goes to the open three-linked kinematic chain which is similar to the mechanism FtABFx (see Fig. 3), and if one determines the principal points H i , / / 2 , H3 on the individual links of this chain,t then it is also in this case always possible to connect the center of mass S of the mechanism, by means of articulated parallelograms which have H t, H2, H 3 as vertices, to the moving links of the chain. The position of S is given, as is well known, by the intersection of lines S t $23 and S3St 2. The parallelogram chain, which results from the above, may be considered weightless and, for whose determination it was only necessary to find the position of Ht, H3, and S, also contains S as one of its vertices. Since this parallelogram partakes of all motions of the mechanism, it is possible to trace out the trajectory of the total center of mass by attaching a pencil at point S. Similar procedures are valid for kinematic chains of greater numbers of links. If the open chain contains n links, it is necessary for the creation of a center of mass chain to construct (n/2)(n-1) articulated parallelograms.§ * See also the fundamental work of O. FISCHER,Concerning the reduced system and the principal points of the links of a mechanism. Z. Math. Phys. 47, 429 et seq. The same author treats this topic more broadly in his book: Theoretische Grundlagen fiir eine Mechanik der lebenden K6rper. Leipzig (1906). Short synopses of the above can be found in F. Wn'TErCSAtmR,Graphische Dynamik. Berlin (1923), pp. 371 et seq.; R. BEYER,Teehnische Kinematik, Leipzig (1931), pp. 471 et seq. and K. FEDERrlOFER,Graphische Kinernatik und Kinetostatik. Berlin (1932), p. 33. l" Hi and//3 are found according to the idea given in Fig. 2 where, for example, for the determination of Hi, a certain point S~. appears (in place of ,72) which has the distance of AS~=AS2+BS3 from A. In the determination of H3, one must analogously assume S~ in such a manner that BS~ = BS2 + A $1. H2 is the center of mass of the weights of the links FtA, AB and BF2 which act at points A, $3 and B respectively. ~, Also look at the above mentioned book by FISCHERon p. 20. Figure 449 in Wrr'rENBAUERhas some errors.
§ See FEDERHO~, Ibid., or WITTENBAUER,Ibid., p. 378.
337
,,
A
-/jr
,,'
Figure 3. Four-bar linkage with principal points H~, H 2, H 3 and the center of mass position S given by articulated parallelograms.
/\
Figure 4. Total center of mass S describing the coupler curve of four-bar linkages F,,C1DtF3, FaGIEIFs, F4KLFs, with couplers C1D1, GIE 1, and KL. 3. Locus of all S of a Coupler Curve In order to determine more closely the character of the trajectory of the center of mass as the locus of all S, one may proceed as follows: Consider the articulated parallelogram F~AH2C or F2BH2E which is added to the given mechanism (in Fig. 4, the positions of H l , / / 2 , / / 3 and S are transferred from Fig. 3). One finds on the links H2C or H2E, besides C and E, certain points D and G respectively, which trace out circles during the course of motion. They are the points of intersection of lines FtB and F2A respectively, with the above mentioned rods. In order to show, for example, that D goes through a circle, it is sufficient to look at the pantograph BH2AFICDH2, whose fixed turning point is F~, and where B and D, lying on a ray through F~ are points which must go through similar curves. Since B 2 describes a circle about F 2, point D must also describe a circle with the eenterpoint F3 on F1F2 so that DF3 is parallel to BF2. The same goes for G, where GF3 must be parallel to AFt. One recognizes that F3 must be that point which, according to the Roberts theorem concerning the three-fold generation of each coupler curve,* represents the third turning * See R. BEYI~, Ibid., pp. 306 et seq., or F. EBr~R, Leitfaden der technisch wichtigenKurven. Leipzig (1906), pp. 81 et seq.
338
point which lies, in this case, on FtF 2 of the coupler curve which is generated by H_,, where
Ft F3 : F3F2= AH2 : H2 B. All other points of the link HzC will describe coupler curves, since CDH 2 may be considered the coupler of the linkage F t CDF 3. These coupler curves are, because of the above mentioned pantograph, similar to the curves which are generated by points on AB. The same is true for the points of HzE. If one now defines all points of H2C or HzE as points which experience " p r i m a r y " coupling by means of a two-bar configuration to mechanism FtABF,., then the center of mass S itself experiences "secondary" coupling by the addition of the articulated parallelogram H2PS Q to the given mechanism. According to the similarity principle, C, F3, and E lie on a straight line. If one now intersects this line with an elongation of QS or PS and draws parallels to AB through the thus obtained points, i.e. C t and E t, respectively, then one obtains the points F 4 and F5 on a ground line FtF 2. The latter are fixed turning points for three further mechanisms F.~CtDtF3, F3GtEtF5 and F4KLFs, which are again connected by means of the Roberts theorem.* Since the center of mass S may be interpreted as the coupler point of the above mentioned mechanism, the following theorem is proven: The center of mass of the moving links of a crank mechanism describes a coupler curve.~f One recognizes that the same theorem may also be applied to linkages which stem from slider crank chains, as well as for the case where the mass distribution of the individual links of the kinematic chain and therefore the positions of points St, S 2, $3 are different than the ones assumed here. With the help of Fig. 4, one can also solve the inverse problem: to find that mechanism whose center of mass trajectory has been kinematically generated as a coupler curve of another linkage. * It is to be observed that F4DIL and FsGtK must lie on a straight line. Further note that F4 divides the distance F1F3in the same proportion as Q divides the distance HzG'. Similar statements are true for F5 and P. t F3, F4, F5 are focal points of this coupler curve.
Concerning the Motion of the Center of Mass of a Crank Mechanism R. Kreutzinger Abstract After explaining the general background, it is shown by the author how one may generate, as a coupler curve the trajectory, of the total center of mass of the movable links of a four-bar linkage. This is done by means of basic configurations, which have been supplemented by appropriate parallelogram mechanisms, and by reference to the Roberts-Chebyshev theorem. The work also gives directions for the construction of a crank mechanism (four-bar linkage) whose center of mass trajectory has been generated as a coupler curve.
1. General Fundamentals FIGURE 1 shows a crank mechanism F1ABF z. Let it be required to examine the trajectory of its center of mass. A
z
S3
Figure 1. Four-bar linkage with positions of centers of mass S~, $2, 3'3 of the movable links.
0
K
".
A.
', ,, /
~H2
.'i /
S2
.....l / ~ c
/
Figure 2. Graphical determination of the principal points Ar t and H2 of a two-bar configuration determined from centers of mass St and S z of both links. * Getriebetechnilc
10 (9), 397-398 (Sept. 1942). 335
336
Since it is assumed that the ground F1Fz is fixed in space, one need only consider the kinematic chain which consists of the links FtA, AB and BF, when examining the motion of the center of mass. The centers of mass S t, S z and S 3 are identical to the centerpoints of the links connecting the joints, if one assumes that all links are equally thick and homogeneous. If one considers with the above assumptions an open two-link kinematic chain (two-bar configuration of Fig. 2), then, as is well known, its center of mass St2 lies on the line connecting St and S_,, and divides the distance S t S 2 in the inverse proportion to the forces (weights) which act at S t and S 2. In the case at hand these forces might be represented possibly by means of half the respective lengths of the rods, i.e. the distances St A and S2A. It has proven to be advantageous, for all questions relating to centers of mass, to introduce certain invariable points which lie on the link axes and which are called "principal points".* Thus, for example, one finds the principal point Ht of link FtA as the center of mass of the points St and A, with weights of the links FtA and A B assumed to act at these points respectively. As stated above, the magnitudes of these weights are expressed by the lengths StA and SzA. The points A, H t, St2, H 2 form the corners of a parallelogram. This parallelogram may be thought of as weightless and articulated, thereby serving to constrain the center of mass to the motion of the two-bar configuration. Figure 2 shows a simple construction of the points Ht, H2 and St2 by means of the easily obtained auxiliary points C, D, E, K, J. 2. T h e C e n t e r o f M a s s S o f a C r a n k M e c h a n i s m
If one now goes to the open three-linked kinematic chain which is similar to the mechanism FtABFx (see Fig. 3), and if one determines the principal points H i , / / 2 , H3 on the individual links of this chain,t then it is also in this case always possible to connect the center of mass S of the mechanism, by means of articulated parallelograms which have H t, H2, H 3 as vertices, to the moving links of the chain. The position of S is given, as is well known, by the intersection of lines S t $23 and S3St 2. The parallelogram chain, which results from the above, may be considered weightless and, for whose determination it was only necessary to find the position of Ht, H3, and S, also contains S as one of its vertices. Since this parallelogram partakes of all motions of the mechanism, it is possible to trace out the trajectory of the total center of mass by attaching a pencil at point S. Similar procedures are valid for kinematic chains of greater numbers of links. If the open chain contains n links, it is necessary for the creation of a center of mass chain to construct (n/2)(n-1) articulated parallelograms.§ * See also the fundamental work of O. FISCHER,Concerning the reduced system and the principal points of the links of a mechanism. Z. Math. Phys. 47, 429 et seq. The same author treats this topic more broadly in his book: Theoretische Grundlagen fiir eine Mechanik der lebenden K6rper. Leipzig (1906). Short synopses of the above can be found in F. Wn'TErCSAtmR,Graphische Dynamik. Berlin (1923), pp. 371 et seq.; R. BEYER,Teehnische Kinematik, Leipzig (1931), pp. 471 et seq. and K. FEDERrlOFER,Graphische Kinernatik und Kinetostatik. Berlin (1932), p. 33. l" Hi and//3 are found according to the idea given in Fig. 2 where, for example, for the determination of Hi, a certain point S~. appears (in place of ,72) which has the distance of AS~=AS2+BS3 from A. In the determination of H3, one must analogously assume S~ in such a manner that BS~ = BS2 + A $1. H2 is the center of mass of the weights of the links FtA, AB and BF2 which act at points A, $3 and B respectively. ~, Also look at the above mentioned book by FISCHERon p. 20. Figure 449 in Wrr'rENBAUERhas some errors.
§ See FEDERHO~, Ibid., or WITTENBAUER,Ibid., p. 378.
337
,,
A
-/jr
,,'
Figure 3. Four-bar linkage with principal points H~, H 2, H 3 and the center of mass position S given by articulated parallelograms.
/\
Figure 4. Total center of mass S describing the coupler curve of four-bar linkages F,,C1DtF3, FaGIEIFs, F4KLFs, with couplers C1D1, GIE 1, and KL. 3. Locus of all S of a Coupler Curve In order to determine more closely the character of the trajectory of the center of mass as the locus of all S, one may proceed as follows: Consider the articulated parallelogram F~AH2C or F2BH2E which is added to the given mechanism (in Fig. 4, the positions of H l , / / 2 , / / 3 and S are transferred from Fig. 3). One finds on the links H2C or H2E, besides C and E, certain points D and G respectively, which trace out circles during the course of motion. They are the points of intersection of lines FtB and F2A respectively, with the above mentioned rods. In order to show, for example, that D goes through a circle, it is sufficient to look at the pantograph BH2AFICDH2, whose fixed turning point is F~, and where B and D, lying on a ray through F~ are points which must go through similar curves. Since B 2 describes a circle about F 2, point D must also describe a circle with the eenterpoint F3 on F1F2 so that DF3 is parallel to BF2. The same goes for G, where GF3 must be parallel to AFt. One recognizes that F3 must be that point which, according to the Roberts theorem concerning the three-fold generation of each coupler curve,* represents the third turning * See R. BEYI~, Ibid., pp. 306 et seq., or F. EBr~R, Leitfaden der technisch wichtigenKurven. Leipzig (1906), pp. 81 et seq.
338
point which lies, in this case, on FtF 2 of the coupler curve which is generated by H_,, where
Ft F3 : F3F2= AH2 : H2 B. All other points of the link HzC will describe coupler curves, since CDH 2 may be considered the coupler of the linkage F t CDF 3. These coupler curves are, because of the above mentioned pantograph, similar to the curves which are generated by points on AB. The same is true for the points of HzE. If one now defines all points of H2C or HzE as points which experience " p r i m a r y " coupling by means of a two-bar configuration to mechanism FtABF,., then the center of mass S itself experiences "secondary" coupling by the addition of the articulated parallelogram H2PS Q to the given mechanism. According to the similarity principle, C, F3, and E lie on a straight line. If one now intersects this line with an elongation of QS or PS and draws parallels to AB through the thus obtained points, i.e. C t and E t, respectively, then one obtains the points F 4 and F5 on a ground line FtF 2. The latter are fixed turning points for three further mechanisms F.~CtDtF3, F3GtEtF5 and F4KLFs, which are again connected by means of the Roberts theorem.* Since the center of mass S may be interpreted as the coupler point of the above mentioned mechanism, the following theorem is proven: The center of mass of the moving links of a crank mechanism describes a coupler curve.~f One recognizes that the same theorem may also be applied to linkages which stem from slider crank chains, as well as for the case where the mass distribution of the individual links of the kinematic chain and therefore the positions of points St, S 2, $3 are different than the ones assumed here. With the help of Fig. 4, one can also solve the inverse problem: to find that mechanism whose center of mass trajectory has been kinematically generated as a coupler curve of another linkage. * It is to be observed that F4DIL and FsGtK must lie on a straight line. Further note that F4 divides the distance F1F3in the same proportion as Q divides the distance HzG'. Similar statements are true for F5 and P. t F3, F4, F5 are focal points of this coupler curve.