Design of fully rotatable, roller-crank-driven, cam mechanisms for arbitrary motion speci cations

Transcription

1 Mechanism and Machine Theory 36 (2001) 445±467 Design of fully rotatable, roller-crank-driven, cam mechanisms for arbitrary motion speci cations G.K. Ananthasuresh * Mechanical Engineering and Applied Mechanics, 297 Towne, Bldg. 220 S., 33rd Street, University of Pennsylvania, Philadelphia, PA , USA Received 21 June 1999; received in revised form 9 October 2000 Abstract The design of a non-traditional cam and roller-follower mechanism is described here. In this mechanism, the roller-crank rather than the cam is used as the continuous input member, while both complete a full rotation in each revolution and remain in contact throughout. It is noted that in order to have the cam fully rotate for every full rotation of the roller-crank, the cam cannot be a closed pro le, rather the roller traverses the open cam pro le twice in each cycle. Using kinematic analysis, the angular velocity of the cam when the roller traverses the cam pro le in one direction, is related to the angular velocity of the cam when the roller retraces its path on the cam in the other direction. Thus, one can specify any arbitrary function relating the motion of the cam to the motion of the roller-crank for only 180 of rotation in the angular velocity space. The motion of the cam in the remaining portion is then automatically determined. In specifying the arbitrary motion, many desirable characteristics such as multiple dwells, low acceleration and jerk, etc., can be obtained. Useful design equations are derived for this purpose. Using the kinematic inversion technique, the cam pro le is readily obtained once the motion is speci ed in the angular velocity space. The only limitation to the arbitrary motion speci cation is making sure that the transmission angle never gets too low, so that the force will be transmitted e ciently from roller to cam. This is addressed by incorporating a transmission index into the motion speci cation in the synthesis process. Consequently, in this method we can specify any arbitrary motion within a permissible zone, such that the transmission index is higher than the speci ed minimum value. Single-dwell, double-dwell and a long hesitation motion are used as examples to demonstrate the e ectiveness of the design method. Force closure using an optimally located spring and quasi-kinetostatic analysis are also discussed. Ó 2001 Elsevier Science Ltd. All rights reserved. * Tel.: ; fax: address: gksuresh@seas.upenn.edu (G.K. Ananthasuresh) X/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S X(00)

2 446 G.K. Ananthasuresh / Mechanism and Machine Theory 36 (2001) 445±467 Nomenclature a length of the roller-crank d distance between the roller-crank and cam pivots k spring constant of the force-closure spring n angular velocity ratio of the cam and the roller-crank in the rst half of the cycle n angular velocity ratio of the cam and the roller-crank in the second half of the cycle r transmission index T torque a angular acceleration b angle between the common normal and the line joining the cam and roller-crank pivots / rotation of the cam g angle between the line joining the cam and the roller-crank pivots and the line joining the cam pivot and the center of the roller k pressure angle l transmission angle h rotation of the roller-crank x angular velocity w angular position of the force-closure spring location on the cam 1. Introduction The functionality of a machine comes from the coordinated relative motion among its moving and xed members. Despite the advances in electronic and electrical hardware and software, mechanically coordinated motion cannot be dispensed with in many practical applications (for example, see [11]). Consolidation of parts, improved reliability, low cost, and reduction in weight are some of the advantages of mechanically controlled machines [2]. Guiding a point on a member along a desired path (path generation), guiding the entire rigid body in a desired manner (motion generation), and achieving a desired relationship between a type of motion on one member and another type of motion on a second member (function generation) are three classes of coordinated motion studied extensively in the kinematics literature. In this paper, the function generation problem between two rotating planar members is considered. Linkages, cam-follower systems, and gears are widely used for this purpose. For example, a double-crank four-bar linkage can give a wide variety of functional relationships between its two cranks while both rotate through 360 in every revolution. However, intermittent motion that includes nite dwells is not possible with a four-bar linkage. Six and more links are needed to obtain dwells with linkages [12]. Furthermore, designing a linkage to achieve a desired function relationship exactly over a long range of motion is very di cult even with the use of optimal synthesis and other types of methods [15]. Cam-follower systems, on the other hand, are more versatile and can be designed easily to achieve wider variety of functional relationships between the rotating input cam and oscillating follower. Full rotation of both the cam and the follower is not possible in the conventional single camfollower systems where the cam drives the follower. Geneva mechanisms, star-wheels, conjugate

3 G.K. Ananthasuresh / Mechanism and Machine Theory 36 (2001) 445± cams, ratchets, escapements, etc. are some of the means of achieving intermittent motion with nite dwells [3,10]. Indexing is one type of intermittent motion where a machine member moves with alternate periods of rest and motion with or without reversal of motion in a cycle. The Geneva mechanism, one of the most popular indexing devices, is a special type of cam-follower system in which intermittent motion with three or more dwells is possible. However, that involves intermittent contact between the driving and driven members. This results in high acceleration and jerk (time derivative of the acceleration) hindering its performance in high-speed applications. Using two Geneva wheels in series [18], Geneva wheels with curved slots [6,13], Geneva wheels with a nonlinear spring [4], double-crank Geneva wheels [1], Geneva wheels with curved slots [17] and a four-bar mechanism in conjunction with the Geneva wheel [9] are some of the ways to improve the high-speed performance of the Geneva mechanism. Gonzalez±Palacios and Angeles [7] present a general method for generating the contact surfaces for indexing devices, but the planar cams in this work had unacceptable pressure angle of 90. In contrast to the aforementioned mechanisms, the cam-follower device presented in this paper provides intermittent motion while the cam and roller-crank are always in contact and the transmission index is within prescribed acceptable limits. It can be designed to have better performance even at high speeds, as there is no intermittent impact unlike most other intermittent motion devices. Single or multiple dwells of non-uniform dwell periods can be obtained. By specifying the motion appropriately, maximum acceleration and jerk can be minimized. Furthermore, an appropriate transmission index can also be incorporated in the design procedure. The construction of this device is also simple because it consists of only two members each rotating about separate xed pivots, and a roller that keeps them together. It should be noted that dwell motion is one of the many possible function generation motions that can be achieved with this mechanism. The remainder of the paper is organized as follows. In Section 2, the general description of the device and its kinematic principles are discussed. This is followed by the kinematic analysis of the output motion of the cam in Section 3. In Section 4, the design procedure for specifying the desired motion and generating the cam pro le is described. Three examples are presented in Section 5. Some issues related to force closure, quasi-kinetostatic analysis and optimization are discussed in Sections 6 and 7. Concluding remarks are provided in Section Kinematic principles The physical arrangement of the planar cam-follower system considered in this paper is shown schematically in Figs. 1(a) and (b). The roller-crank and the cam and are pivoted about two xed points, A 0 and B 0, respectively. A roller mounted at the free end of the roller-crank is in contact with the cam. The rotations of the roller-crank and cam are denoted by h and /, respectively. For the purpose of kinematic analysis that follows, the radius of the roller is reduced to zero as shown in Fig. 1(b). However, the radius of the roller will be duly accounted for when generating the cam pro le later in the paper. The design objectives for this compact and simple cam-follower mechanism are: 1. both cam and the roller-crank must rotate through 360 in each cycle; 2. they should always be in contact with each other;

4 448 G.K. Ananthasuresh / Mechanism and Machine Theory 36 (2001) 445±467 Fig. 1. (a) Basic arrangement of the cam-follower mechanism; (b) schematic of the cam-follower mechanism with a roller of zero radius. 3. arbitrary speci cation of motion relating h and / should be possible including nite dwells and reversal of motion; 4. transmission criteria should be adequately satis ed to be of practical use Transmission criteria In mechanism synthesis, it is necessary to have a suitable transmission criterion in order to be able to judge the e ciency of transmission of motion and force. Chapter 9 of Hain's book [8] gives an excellent account of classical approaches to quantifying the quality of motion transmission. Transmission angle (or its complementary angle, pressure angle) is often used as the transmission criterion. Ref. [5] summarize various interpretations of this angle in the modern kinematics literature. One de nition of pressure angle, by Shigely and Uicker [16], is the acute angle between the direction of the output force and the direction of the velocity of the point where the output force is applied. Since the transmission angle is the complement of pressure angle, the former can be de ned as the acute angle between the common normal and the output point path normal. In camfollower systems, the direction of force is along the common normal at the point of contact. The pressure angle and transmission angle are indicated in Fig. 2 when the roller-crank drives the cam. The smaller the pressure angle the better the transmission and vice versa. Likewise, ideal transmission angle being 90, the closer it is to 90 the better the transmission. In this paper, the above de nition of transmission angle will be used as the criterion for the e ciency of transmission in the following discussions. A more suitable transmission index based on the transmission angle will be de ned in Section 3.2 for use in the design procedure in Section 4.

5 G.K. Ananthasuresh / Mechanism and Machine Theory 36 (2001) 445± Fig. 2. Pressure angle k, and transmission angle l for the mechanism when the cam is the output member Open vs closed pro le for the cam It can be inferred from a description of cam mechanisms by [8] that for full rotation of the cam and the roller-crank, the cam cannot have a closed pro le. This can be explained as follows. Consider a kinematic inversion of the basic arrangement of Fig. 1(b) after xing the cam instead of the frame consisting of the two pivots A 0 and B 0. As shown in Fig. 3, this corresponds to the guided motion of the end point of a two-link open serial chain along the pro le of the cam. A Fig. 3. A kinematic inversion of the basic arrangement of Fig. 1(b). Here, the cam is the xed link and the end point of the two-link open serial chain follows the cam pro le.

6 450 G.K. Ananthasuresh / Mechanism and Machine Theory 36 (2001) 445±467 Fig. 4. A con guration in the kinematic inversion of the cam-follower system when the output member, the rollercrank, momentarily comes to rest. possible cam pro le is shown in the gure for a certain type of functional relationship between h and /. B 0 A 0 A is a general con guration of the open chain. To ensure a full 360 range of motion for h and /, the cam pro le must contain two points corresponding to the fully extended and fully folded con guration of the two-link open chain. B 0 A 0 0 A0 is the fully extended con guration and B 0 A 00 0 A00 is the fully folded con guration. At these two con gurations, the cam pro le should be tangential to the boundaries of the annular work space of the two-link open chain. Therefore, the common normal is aligned to the line of centers and the roller-crank, resulting in zero transmission force between the cam and the roller-crank. Thus, a closed pro le for the cam is not desirable. This implies that the cam must be an open pro le as shown in Fig. 4. Consequently, the roller must traverse the open pro le of the cam twice in each revolution. That is, in Fig. 3 as B 0 A 0 0 rotates to B 0 A 00 0 following counterclockwise direction, point A0 moves to A 00 via A, and then when B 0 A 00 0 continues its counterclockwise rotation to B 0A 0 0, A00 moves to A 0 again via A. This is an important feature of the mechanism as this stipulates that we can prescribe the functional relationship between h and /, for only half the cycle. Barring this limitation and transmission criteria, there is no other restriction to specify the motion Selection of the input member It can be reasoned that with the cam as the input member, even instantaneous dwell is not possible if the roller-crank has to rotate completely with a good transmission angle. As shown in Fig. 4, at the con guration where the roller-crank momentarily comes to rest, if A is the point of contact, then the common normal at that point must pass through the cam center B 0. The acute angle between the common normal B 0 A, and output point path normal A 0 A, is the transmission angle, l. A 0 is the position of the roller-crank pivot when the roller comes to the same point on the pro le during its reverse travel along the pro le. Therefore, A coincides with A. The transmission angle remains as l. As can be seen in Fig. 4, the output link A 0 A shifts its position from one side of the common normal to the other side as A goes along the pro le and comes back to the same point as A. This means that somewhere in between, the output link will be aligned with the normal to the pro le resulting in zero transmission angle. We therefore conclude that the cam cannot be the input member. With the roller-crank as the input member, a suitable transmission criterion will be de ned and it will be adequately met in the design procedure described in Section 4.

7 G.K. Ananthasuresh / Mechanism and Machine Theory 36 (2001) 445± Analysis of the output motion of the cam As discussed above, if the input member roller-crank completes one revolution, the roller must traverse the open cam pro le twice. Consequently, the output motion of the cam while the roller retraces its path is governed by the pro le that prescribes the motion of the cam in the rst part of the cycle. It is now clear that we can specify the motion of the cam in only a half the cycle and the remaining half is determined automatically, thus limiting control over the complete motion. However, by studying the motion in the second half we can design the cam pro le to have better control over its motion Output rotation of the cam in the second half of the cycle Fig. 5 shows a kinematic inversion of the basic arrangement of Fig. 1(b) where B 0 A 0 A is the con guration during the initial traverse of the roller on the cam. B 0 A 0 A is the con guration when the roller returns to the same point during the reverse traverse on the cam. The angle between A 0 B 0 extended beyond B 0 and a reference (horizontal) line on the xed link (i.e., the cam in this inversion) /, indicates the rotation of the cam in the basic arrangement. Fig. 5 shows the same angle / for the reverse traverse. The following relationships can be observed from the gure: h h ˆ 2p; / / ˆ 2g; 1a 1b where g is the angle between the line joining the cam-pivot and the roller center, and the line joining the cam pivot and roller-crank pivot. By di erentiating Eq. 1b with respect to input h, we get d/ dh d/ dh ˆ 2 dg dh : 2 Fig. 5. Kinematic inversion of the mechanism to determine The motion in the second part of the cycle.

8 452 G.K. Ananthasuresh / Mechanism and Machine Theory 36 (2001) 445±467 Eq. (2) can be re-written as d/ dh d/ dh dh dh ˆ 2 dg dh : 3 Denoting d/=dh as n and d/ =dh as n, and using the fact that dh =dh ˆ 1, Eq. (3) becomes n n ˆ 2 dg dh : 4 It should be noted that n refers to the part of the motion that we specify, and n refers to the motion in remaining part. The integration of n and n gives the cam rotation /. Therefore, the area under the )2dg=dh vs h gives the sum of the motion of cam in both halves of the cycle. Thus, using Eq. 4, we can predict and control the motion in the second half of the cycle. Denoting the distance between the cam and the roller-crank pivots by d, and the length of the roller crank as a, applying the cosine rule to the triangle A 0 B 0 A in Fig. 6, g can be written as g ˆ cos 1 d 2 x 2 a 2 ; 5 2dx where p x ˆ a 2 d 2 2ad cos h: Di erentiating g in Eq. 5 with respect to h, we get dg dh ˆ 1 sin g sin h 1 d=a cos h : 6 1 d=a 2 3=2 2 d=a cos h The above expression is useful in specifying the desired motion as well as generating the cam pro le. This will be demonstrated in Section Transmission index Although the transmission angle is a useful measure to quantify the performance of a given mechanism, it is not always the most convenient measure to use while designing a new mechanism. Here, we use a more appropriately de ned index of transmission as shown in Fig. 7. The torque Fig. 6. Schematic to derive an expression for g and its derivative.

9 G.K. Ananthasuresh / Mechanism and Machine Theory 36 (2001) 445± Fig. 7. Schematic sketch of the mechanism for the purposes of writing the design equations. exerted on the output member, the cam, is the moment due to the contact force between the roller and the cam, which acts along the common normal. The moment arm is given by r, which is the perpendicular distance to the common normal from the cam pivot. The higher the value of r is, the better the transmission and vice versa. As shown in Section 3.3, the non-dimensionalized r is a convenient measure of transmission for the mechanism considered in this paper. The ratio r=d is the non-dimensional transmission index for this mechanism Design equations As shown in Fig. 7, point C is the instantaneous center of roller-crank and the cam. Therefore, this point has the same velocity on both the cam and the roller crank. This leads to the following relationship: n ˆ d/ dh ˆ x y : 7 Further, since x ˆ y d, Eq. 7 can be re-written as 1 y ˆ n 1 : 8 d Now, applying the sine rule to the triangle A 0 AC of Fig. 7, we get sin 180 h b ˆ sin b x a ; 9 which, after eliminating x, can be simpli ed to sin h cot b cos h ˆ y d : 10 a From the right angled triangle B 0 PC, p r y cot b ˆ 2 r 2 y 2 ˆ 1 : 11 r r

10 454 G.K. Ananthasuresh / Mechanism and Machine Theory 36 (2001) 445±467 Combining Eqs. 10 and 11 yields r y 2 sin h 1 ˆ y r a d cos h: a 12 Squaring both sides of Eq. (12) and re-arranging, we get sin 2 h sin 2 h r 2 ˆ r r d cos h 2r2 y a y a ay d a cos h : 13 Substituting for 1=y from Eq. 8 into the above equation gives a quadratic equation in n: ( r ) 2 2 d cos h sin 2 h n 1 2 2r2 d cos h n 1 r 2 sin 2 h ˆ 0: d a ad a a 14 Several observations can be made about the above equation. (1) Since Eq. 14 remains the same if we replace h with 2p h, the two solutions of n for a given value of h correspond to the angular velocity ratios of the cam and the roller-crank during the initial and reverse traversal of the roller on the cam. That is, n solution1 ˆ n ˆ d/ ; dh at h 15 n solution2 ˆ n ˆ d/ dh : at h ˆ 2p h (2) When h ˆ 0orp, Eq. (14) becomes independent of r. 2 d 1 n 1 2 2d d 1 n 1 d 2 ˆ 0: 16 a a a a This can be seen in Fig. 8 which shows the solution curves for n for di erent r=d ratios. It can also be observed in this gure that region bounded by the two solution curves for on value of r=d ratio contains similar curves of larger ratio. As mentioned before a larger value of r=d implies better transmission. Therefore, the region between two solutions curves can be used to specify the angular velocity ratio n so that the transmission index r=d is larger than a minimum speci ed value. This is a useful feature in designing the mechanism. (3) Fig. 9 shows the solutions curves for n for a xed ratio of r=d and varying ratios of d=a. It can be seen that n ˆ 0 occurs at the same two points. This is also evident from Eq. (14) if we substitute n ˆ 0 and solve for h. n ˆ 0 ) h ˆ sin 1 r=d or p sin 1 r=d ; 17 which is clearly independent of the ratio d=a. This feature of the design (Eq. 14) has an important signi cance if a long dwell or reversal of motion of the cam is desired in this mechanism. That is, if we specify a lower bound on the transmission index r=d, then the permissible region for n has zero or a negative value in a limited range of h. For example, the maximum possible single-dwell in the initial traversal of the roller on the cam is p 2 sin 1 r=d. Thus, the ratio is not only useful to achieve a prescribed minimum transmission index, but also in selecting the

11 G.K. Ananthasuresh / Mechanism and Machine Theory 36 (2001) 445± Fig. 8. Solutions of Eq. (14) for a constant value of d=a ˆ0:6, and varying values of r=d ˆ0:3; 0:5; and 0.6. Fig. 9. Solutions of Eq. (14) for a constant value of r=d ˆ0:5, and varying values of d=a ˆ0:3; 0:5; and 0.6.

12 456 G.K. Ananthasuresh / Mechanism and Machine Theory 36 (2001) 445±467 Fig. 10. Design curves for specifying the desired motion of the mechanism ( d=a ˆ0:6; r=d ˆ0:5). longest possible dwell or reversal of motion. It is also worth noting that unless the transmission index zero, a single-dwell of 180 or more is not possible. Furthermore, shorter dwell or no dwell at all imply that the transmission index is greatly improved. This is a useful feature in designing a mechanism for prescribed function generation without a dwell. (4) In Fig. 10, a dg=dh vs h curve is also drawn along with the solution curves for n for sample values of r=d and a=d. Recalling Eq. 4, it can be seen that the dg=dh vs h curve is the average of two segments of the solution curves for n. Therefore, by re ecting the n curve for the range 0 6 h 6 p about dg=dh vs h curve, we get n curve for the range p 6 h 6 2p. This makes the prediction and control of the output motion of the cam in the second half of the cycle possible, as Eq. (4) is true for any value of n, not just the solutions of Eq. (14). It should be noted that dg=dh is independent of the transmission index r=d. It is also worth noting that by choosing n along the 2dg=dh curve, n can be made zero to get a dwell in the second half of the cycle. (5) The area under the n vs h curve gives /, the rotation of the cam which can be used to generate the cam pro le by using the kinematic inversion shown in Fig Design procedure Using the results of the foregoing analysis, a systematic procedure for designing fully rotatable, roller-crank driven cam mechanism for arbitrary motion speci cations is described in this section. An important design speci cation is the minimum value the transmission index can take

13 G.K. Ananthasuresh / Mechanism and Machine Theory 36 (2001) 445± throughout the motion of the mechanism. The overall size of the mechanism is not important as this is a function generation problem. Consequently, the non-dimensional ratios of r=d and d=a are used. The value of d determines the size of the mechanism. The procedure consists of the following steps. Step 1: For the speci ed value of r=d and a chosen value of d=a, using Eq. (14), the two segments of n vs h curve are drawn. This determines the permissible zone as the region enclosed between the two curves for specifying the motion in the rst half of the cycle. Drawing the dg=dh vs h curve in the same graph aids the visualization of the motion in the remaining half of the cycle. Step 2: Depending on the nature of the desired function / h, the n vs h can be speci ed within the permissible region. The d=a ratio can be changed if necessary. Second order (acceleration) and third order (jerk or shock) derivatives of the function / h can be controlled in this process. Continuity of the functions / h and its derivatives can also be ensured. Harmonic, cycloidal, polynomial, spline, and other types of curves can be used to specify the n vs h curve to satisfy all of the design objectives in the design of the cams. The n, the motion in the second half of the cycle is easily computed using Eq. 4. The dg=dh vs h curve should be appropriately used to control the motion in the second half of the cycle by specifying a curve in the rst half. Step 3: The rotation of the cam in the entire 360 range can be readily obtained by computing the cumulative area under the n vs h curve from 0 to 360. This can either be done analytically or numerically depending on how the n vs h curve is speci ed. Note that the analytical expression for dg=dh is readily available from Eq. 6. Step 4: Using the kinematic inversion technique the end point of the two-link open serial chain can be made to trace the pitch curve of the cam pro le as per the function / h obtained in Step 3. The nite radius of the roller is then used to obtain the actual pro le of the cam as an envelope of all the positions of the rollers. This can be done by computing the equi-distant o set curve of the pitch pro le of the cam using the numerically determined normal at the point of contact. In practice, by choosing a milling cutter of the same radius as the roller, the pitch pro le is su cient. All of the above steps are implemented in [14] and the entire design procedure automated. The Matlab scripts generate the G-code for the computer numerically controlled Fadal vertical machining center to manufacture the cam. Examples below illustrate the procedure for three di erent types of design speci cations. 5. Numerical examples 5.1. Example 1: single-dwell motion In this design example, the objective is to obtain a mechanism with the longest possible dwell in the rst half of the cycle while satisfying the requirements of good transmission and smoothness of motion. A minimum transmission index r=d of 0.5 is desired. The design procedure begins with choosing d=a equal to 0.6. Fig. 11 shows the design curves and the speci ed curve for n in the range 0 6 h Since n is zero or negative from 30 to 150, the maximum possible dwell period is 120. It should be noted that the speci ed n must lie in the area enclosed by the n and n curves to meet the minimum value speci cation for r=d. To meet the smoothness of motion

14 458 G.K. Ananthasuresh / Mechanism and Machine Theory 36 (2001) 445±467 Fig. 11. Design curves and the speci cation of single-dwell motion using half-cycloidal curves to meet the smoothness requirements and the transmission criterion. requirements, a half-cycloidal motion [16] is speci ed in the range 0 6 h 6 30 and h This ensures that the second derivative of n (i.e., the jerk) is also smooth. PQ! half-cycloidal motion, QR! zero (corresponds to dwell in the rst half of the cycle), RS! half-cycloidal motion. The motion in the second half of the cycle is obtained by re ecting n about the dg=dh curve. Therefore, it retains the smoothness at the transition points. The segment SR Q P corresponds to this part of the motion as seen in Fig. 11. The n curve is then used to obtain the cam rotation, /. Fig. 12 shows the resulting cam motion along with its three derivatives. By knowing the function / h, kinematic inversion is used to obtain the cam pro le, which is shown in Fig. 13. A fabricated laboratory prototype is shown in Fig. 14 in two positions. Fig. 14(a) shows the roller-crank and the cam in the starting con guration and Fig. 14(b) shows the dwell con guration. It can be seen that in the dwell con guration, the center of curvature of the cam coincides with the roller-crank pivot Example 2: double-dwell motion This example illustrates double-dwell motion with each dwell occurring in the rst and second halves of the cycle. Through this example, it can be seen how motion in the second half of the cycle can be controlled in the design stage. The same values of r=d and d=a are considered in this example as in the previous example. Fig. 15 shows the design curves. n is speci ed using half-

15 G.K. Ananthasuresh / Mechanism and Machine Theory 36 (2001) 445± Fig. 12. Single-dwell motion of the cam and its derivatives. Fig. 13. Cam pro le obtained using the inversion technique for the single-dwell motion.

16 460 G.K. Ananthasuresh / Mechanism and Machine Theory 36 (2001) 445±467 Fig. 14. Single-dwell prototype (a) starting con guration (b) dwell con guration. Fig. 15. Speci cation of the double-dwell motion. cycloidal curves and the design curves as described below. The portion PQRSTUV is the speci ed portion in the range 0 6 h 6 180, while the portion VU T S R Q P is automatically determined by re ecting the rst half about the ) dg=dh curve. PQ! half-cycloidal motion, QR! zero (corresponds to dwell motion in the rst half of the cycle), RS! half-cycloidal motion, ST! half-cycloidal motion, TU! )2 dg=dh curve (corresponds to dwell in the second half of the cycle), UV! half-cycloidal motion,

17 G.K. Ananthasuresh / Mechanism and Machine Theory 36 (2001) 445± In Fig. 15, it can be seen that the rst dwell occurs in the range 30 6 h 6 80 between Q and R, and the second occurs in the range h between U and T. Any other period for each dwell and the spacing between them can be chosen. Fig. 16 shows the resulting cam motion and its derivatives. The cam pro le obtained using the kinematic inversion technique is shown in Fig. 17. Fig. 18 shows the prototype in di erent positions Example 3: hesitation Fig. 16. Double-dwell motion of the cam and its derivatives. In this third example, a long hesitation motion will be demonstrated. Hesitation is a type of intermittent motion where the motion reverses brie y and then continues to complete the full rotation. As shown in Fig. 19, n is speci ed along the curve which is a solution for Eq. 14 for r=d ˆ0:3 and d=a ˆ0:6. As this curve is less than zero in the range 17:5 6 h 6 162:5, reversal of motion occurs in this range. After 162.5, the cam accelerates and catches up with the rollercrank to complete the 360 in time. The resulting cam pro le is shown in Fig. 20. It should be noted that this pro le is an involute of the circle of radius equal to r centered around the cam pivot. A slotted cam is required in this mechanism. 6. Force closure Form-closure and force-closure are two ways in which the cam and roller can be kept in constant contact even at high speeds. Force-closure with a helical extension spring is used here.

18 462 G.K. Ananthasuresh / Mechanism and Machine Theory 36 (2001) 445±467 Fig. 17. Cam pro le for the double-dwell motion. Fig. 21 shows the arrangement of the spring. Three parameters s 1, s 2, and w (see Fig. 21) are identi ed to locate the spring on the roller crank and the cam. These three parameters are determined to meet the following objectives. As the mechanism is in motion, the length of the spring should not change excessively as this will impose additional loads on the mechanism. The direction of the spring force should be such that it counteracts the separative forces acting on the cam and roller. An optimization problem was solved for the single-dwell cam to minimize the maximum variation of the spring length with s 1, s 2, and w as the design variables. Nelder±Mead simplex algorithm implemented in fmins ( ) function in [14] was used for this purpose. The optimum values obtained were: s 1 ˆ 2:04, s 2 ˆ 3:37, and w ˆ 140:17. As shown in Fig. 14, a spring is attached between the cam and the roller-crank. The prototype con rmed that the contact is maintained even when the roller-crank is turned at higher speeds. 7. Quasi-kinetostatic analysis In this section, the expressions for computing the input torque required at the roller-crank are presented. If the mechanism is operated at high speeds, inertia forces should also be taken into account. The following quasi-kinetostatic analysis is also useful in selecting the correct value of the spring constant for the force-closure spring by way of optimization. Fig. 22 shows the forces acting on the roller-crank and the cam. F c is the contact force between the roller and the cam and

19 G.K. Ananthasuresh / Mechanism and Machine Theory 36 (2001) 445± Fig. 18. Prototype of the double-dwell mechanism in various positions: (a) rst-dwell con guration; (b) after rstdwell; (c) second-dwell; (d) after second-dwell. acts in the direction of the common normal. F s is the spring force whose direction is also known at every instant during the motion. No load other than the inertia forces is shown on the cam. Radial, tangential, and angular acceleration components of the inertial forces are shown both on

20 464 G.K. Ananthasuresh / Mechanism and Machine Theory 36 (2001) 445±467 Fig. 19. (a) Speci cation of the motion with a long hesitation; (b) resulting motion of the cam. Fig. 20. Cam pro le for the hesitation motion.

21 G.K. Ananthasuresh / Mechanism and Machine Theory 36 (2001) 445± Fig. 21. Force-closure for the mechanism using a spring. Fig. 22. Contact, spring, and inertial forces acting on the roller-crank and the cam. the cam and the roller-crank. From the results of the kinematic analysis and the mass properties of the roller-crank and the cam, all inertia forces can be computed at every instant. The unknown in the force analysis then are the four ground reaction forces (two at each xed pivot), F c, and T in. Since it is a planar problem, three Newtonian force/moment balance equations arise for each body, here the roller-crank and the cam. Thus, the six equations in six unknowns are easily solved. The contact force and the required input torque are given by: I c a c m c R 2 c a c F c R Fc F c R s ˆ 0; 18 T in I r a r m r R 2 r a r F c r Fc F c r s ˆ 0; where I c is the moment of inertia of the cam about its center of mass, a c is the angular acceleration of the cam, m c is the mass of the cam, R 2 c is the distance from the pivot to the center of mass of the cam, R Fc is the moment arm of the contact force about the cam pivot, R s is the moment arm of the spring force about the cam pivot, I r is the moment of inertia of the roller-crank about its center of mass, m r is the mass of the roller-crank, R 2 r is the distance from the pivot to the center of mass of the roller-crank, r Fc is the moment arm of the contact force about the roller-crank pivot and r s is the moment arm of the spring force about the roller-crank pivot. Fig. 23 shows T in as a function of the roller-crank position for the single-dwell mechanism. In this calculation, constant angular velocity of 30 rpm T in was assumed for the roller-crank. Other properties estimated for the single-dwell prototype were: density of the cam material = 19

22 466 G.K. Ananthasuresh / Mechanism and Machine Theory 36 (2001) 445±467 Fig. 23. Inertia torque of the cam T c, and the required input torque on the roller-crank with and without the spring. 0:04 lb=in 3 ; m c ˆ mass of the cam ˆ 0:32 lb; I c ˆ moment of inertia of the cam about its center of mass ˆ 1:7264 lb in 2 ; R c ˆ distance from the cam pivot to its center of mass ˆ 2:2 in. Inertia forces on the roller-crank were neglected. Fig. 23 shows the input torque requirement with and without the force-closure spring. Optimizing the location of the spring and the spring constant to decrease the peak input torque to improve the performance of the device at high speeds based on the quasi-static analysis is part of the ongoing work. 8. Conclusions The design of a cam-roller mechanism in which the roller-crank drives the cam is presented here. Using the kinematic design equations described in this paper, it is possible to obtain the cam pro le and other dimensions to specify any arbitrary motion of the cam. The unique feature of this mechanism is the complete rotation of the cam for every full rotation of the input rollercrank. Intermittent motion is one of the many types of motions possible with this mechanism. Unlike the Geneva mechanism, star wheels, etc., here the roller and the cam always remain in contact resulting in better overall performance of the mechanism. It is shown that this mechanism can only have an open cam pro le and that cam cannot be the input member if we desire a dwell. An appropriate transmission index is de ned and used in the design equations. Three examples are presented to illustrate that long, non-uniformly spaced multiple dwells and hesitations (backward rotations) are possible with this mechanism. Force-closure with an optimally located

23 G.K. Ananthasuresh / Mechanism and Machine Theory 36 (2001) 445± spring and quasi-kinetostatic analysis are also brie y presented. Optimization to achieve the best dynamic performance is currently in progress. Acknowledgements The author is indebted to Professor K. Lakshminarayana (Indian Institute of Technology, Madras, India) for introducing him to this topic and for his valuable guidance in this research. The help received from Mr. Wade Bennett and Mr. Charles Nappen (both at the University of Pennsylvania) in fabricating the prototypes and running some computer simulations, is also gratefully acknowledged. References [1] A.K. Al-Sabeeh, Double-crank external geneva mechanism, J. Mech. Design 115 (3) (1993) 666±670. [2] S. Ashley, Liftgate device uses geneva mechanism, news and notes, Mech. Eng. 119 (7) (1997) 10. [3] J.H. Bickford, Mechanisms for Intermittent Motion, Industrial Press, New York, [4] C.Y. Cheng, Y.Y. Lin, Improving dynamic performance of the geneva mechanism using nonlinear spring, Mech. Machine Theory 30 (1) (1995) 119±129. [5] T.L. Dresner, K.W. Bu ntgon, De nition of pressure and transmission angles applicable to multi-input mechanisms, J. Mech. Design 113 (4) (1991) 495±499. [6] R.G. Fenton, Y. Zhang, J. Xu, Development of a new geneva mechanism with improved kinematic characteristics, J. Mech. Design 113 (1) (1991) 40±45. [7] M.A. Gonzalez-Palacios, J. Angeles, The generation of contact surfaces of indexing cam mechanisms ± a uni ed approach, J. Mech. Design 116 (2) (1994) 369±374. [8] K. Hain, Applied Kinematics, McGraw-Hill, New York, [9] H.K. Hunt, N. Fink, J. Nayar, Proc. Inst. Mech. Eng. London 174 (1960) 643±656. [10] P.W. Jensen, Classical and Modern Mechanisms for Engineers and Inventors, Marcel Dekker, New York, [11] S. Kota, A.G. Erdman, Motion control in product design, Mech. Eng. 119 (8) (1997) 74±77. [12] S. Kota, Generic models for designing dwell mechanisms ± a novel kinematic design of stirling engines as an example, J. Mech. Design 113 (4) (1991) 446±450. [13] H.P. Lee, Design of a geneva mechanism with curved slots using parametric polynomials, Mech. Machine Theory 33 (3) (1998) 321±329. [14] Matlab, 2000, Numerical analysis software from Math Works, Inc., Natick, MA. [15] D. Rosen, D. Riley, A. Erdman, A knowledge based dwell mechanism assistant designer, J. Mech. Design 113 (3) (1991) 205±212. [16] J.E. Shigley, J.J. Uicker, Theory of Machines and Mechanisms, McGraw-Hill, New York, [17] Y. Zhang, R.G. Fenton, J. Xu, Two station geneva mechanisms, J. Mech. Design 116 (2) (1994) 647±653. [18] R.G. Fenton, Geneva mechanism connected in series, J. Eng. Industry 97 (1975) 603±608.