A STUDY ON A NOVEL QUICK RETURN MECHANISM

Transcription

1 A STUDY ON A NOVEL QUICK RETURN MECHANISM Wen-Hsiang Hsieh and Chia-Heng Tsai Department of Automation Engineering, National Formosa University, allen@nfu.edu.tw Received April 008, Accepted September 009 No. 08-CSME-13, E.I.C. Accession 3051 ABSTRACT This work aims to propose a novel design for quick return mechanisms, and the new mechanism is composed by a generalized Oldham coupling and a slider-crank mechanism. First, the kinematic dimensions that affect the time ratio are found by investigating the geometry of the proposed design. By transforming into its kinematically equivalent mechanism, and then the design equations of time ratio are derived. Furthermore, a design example is given for illustration. Moreover, the design is validated by kinematic simulation using ADAMS software. Finally, a prototype and an experimental setup are established, and the experiment is conducted. The results show that proposed new mechanism is feasible and with reasonable accuracy. In addition, it is more compact and easier to be balanced dynamically than a conventional quick return linkage. ÉTUDE D UN MÉCANISME À RETOUR RAPIDE INNOVATEUR RÉSUMÉ Nous proposons une conception innovatrice d un mécanisme à retour rapide composé d un accouplement de style Oldham et d un système bielle-manivelle. Pour commencer, en étudiant la géométrie du concept proposé, on établit les dimensions cinématiques qui affectent le rapport temps. En les transformant en un mécanisme cinématique équivalent, nous trouvons la dérivée des équations de rapport temps, et pour illustrer le concept on donne un exemple. De plus, le concept est validé par une simulation cinématique à l aide du logiciel ADAMS. Finalement, un prototype est crée, et un cadre expérimental est établi pour mener l expérience. Les résultats révèlent que le mécanisme proposé est réalisable et sa précision est acceptable. En outre, il est plus compact et il est plus facile d effectuer l équilibrage dynamique que pour un mécanisme à retour rapide conventionnel. Transactions of the Canadian Society for Mechanical Engineering, Vol. 33, No. 3,

2 1. INTRODUCTION A quick return mechanism is a mechanism that converts rotary motion into reciprocating motion at different rate for its two strokes. When the time required for the working stroke is greater than that of the return stroke, it is a quick return mechanism. It yields a significant improvement in machining productivity. Currently, it is widely used in machine tools, for instance, shaping machines, power-driven saws, and other applications requiring a working stroke with intensive loading, and a return stroke with non-intensive loading. Several quick return mechanisms can be found in the literatures [1 ], including the offset crank-slider mechanism, the crank-shaper mechanisms, the double crank mechanisms, and the Whitworth mechanism. All of them are linkages. A linkage has its strengths and weaknesses. It is inexpensive to make and easy to lubricate; however, it is bulky and difficult to balance. In situations, if compact space is essential to the design, then a linkage may not be a good choice. Therefore, how to find a new alternative of quick return mechanisms is an open topic that deserves to be examined. There are many scholars devoted to the studies of quick return mechanisms, and many valuable contributions have been made. Dwivedi [3] used the Whitworth mechanism to constructing a high velocity impacting press. Suareo & Gupta [4] performing the design of the spatial RSSR quick return mechanism. Beale [5] employed Galerkin s method to investigate the dynamic and stability of a flexible link used in a quick return mechanism. Fung [6 9] and Lin [10 11] utilized different control approaches to investigate the response of a quick return mechanism with or without a flexible link. Hat et al. [1] proposed a finite difference method with fixed and variable grids to approximate the numerical solutions of a flexible quick-return mechanism. Chang [13] investigated the coupling effect of the geared rotor on the quick-return mechanism undergoing three-dimensional vibration. This work aims to present a new design for the quick return mechanisms, and verify its feasibility by conducting simulation and experimental studies. In this work, Section introduces the composition of the proposed system. The equations for calculating time ratio are investigated in Section 3. Section 4 performs kinematic analysis. A design example is given for illustration in Section 5. Section 6 presents the prototype experiment. Conclusions are summarized in Section 7.. NEW QUICK RETURN MECHANISM An Oldham coupling is used to transmit the motion or the power between two parallel shafts. It can be classified, by its shape of its slots or ribs, as classical Oldham couplings (straight slot), generalized Oldham couplings with circular slots, and generalized Oldham couplings with curvilinear slots [14]. Fig. 1 shows the generalized Oldham coupling which consists of a frame (link 1), an input disk (link ), a floating disk (link 3), and an output disk (link 4). The radii, r 1 and r, of the centerline of the slots need not be equal, and the two centerlines are usually, but not necessary, designed to intersect at the axis of the floating disk. The classical Oldham coupling will transmit uniform motion between the input and output shafts. However, the generalized Oldham coupling with circular slots and curvilinear slots will transmit non-uniform motion and produce quick return motion. And it may lead to many potential applications to devices requiring non-uniform transmission. Hsieh [15 18] applied the generalized Oldham coupling to various mechanical devices, including mechanical presses [15], cam mechanisms [16], ball screw mechanisms [17], and vibrating conveyors. In addition, with proper Transactions of the Canadian Society for Mechanical Engineering, Vol. 33, No. 3,

3 arrangement, Oldham couplings can be used as a balancer to eliminate shaking forces and shaking moment [19]. Figs. (a) and (b) depicts the schematic sketch and the kinematic sketch of the proposed new mechanism, respectively. It consists of a generalized Oldham coupling and a crank-slider mechanism. A motor is connected to the input disc of the generalized Oldham coupling at pivot a 0, and then the coupling drives the crank-slider mechanism to produce quick return motion. The advantages of the new design are easier to be balanced dynamically and more compact in volume (as discussed in Sec. 3) than a conventional quick return linkage. In addition, the building cost is inexpensive if a generalized coupling with circular slots is adopted. 3. MOTION GEOMETRY Fig. 1. Generalized Oldham coupling. Due to the complexity of the geometry for a generalized Oldham coupling with curve slots, the investigation on the geometry is limited to that with circular slots in this work. In Fig., points a 0 and b 0 denote the axes of rotation of the input and the output disks, respectively, and let r 1 5 a 0 b 0 be the distance of centers of two fixed pivots. Furthermore, points a and b denote the arc centers of the slots on the input and the output disks, respectively, and the distance of two centers is set as r 3 5 ab, and point o is the point of intersection of the two arcs. Moreover, let r 5 oa and r 4 5 ob be the radii of the circular slots of input disk and output disk, respectively. Also, angle a 5%aob denotes the intersection angle between two circular arcs, and angle b 5%cb 0 b is that of b 0 b makes with b 0 c, measured in counterclockwise. In addition, let r 4 5 b 0 c and r 5 5 cd be the link lengths of the crank and the connecting rod, respectively. Also, let r 6 be the offset, the perpendicular distance between the line produced by the rectilinear Fig.. New quick return mechanism. Transactions of the Canadian Society for Mechanical Engineering, Vol. 33, No. 3,

4 Fig. 3. Equivalent linkages. motion of joint d and the line passing through two fixed pivots a 0 and b 0. Also, by applying Cosine law to triangle Daob, it leads to r 3 ~r zr 4 {r r 4 cos a (1) Additionally, it is noted that angle b represents the relative position of assembly between the coupling and the slider-crank mechanism. Since the output disk of the coupling will be attached to the crank in the proposed design, b is a constant. Apparently, the output motion will vary with different choices of b. Finally, the kinematic dimensions of the system will be fully defined if r i (i516), r 4, a, and b are specified. The mechanism can be converted into its equivalent mechanism, called the drag link quick return mechanism, as shown in Fig. 3. By comparing Fig. (b) and 3, the proposed new design is more compact than its equivalent linkage. In addition, if a proper design is made as that proposed by Tsai [19]. To perform the time ratio design, the associated design equations have to be derived firstly. The two limit positions of the slider are depicted in Fig. 3, and the subscripts 1 and denote the right limit position and the left limit position of the slider, respectively. Let e be the intersection of perpendicular line of b 0 with the extended line of d 1 d, from Triangles Db 0 d e and Db 0 d 1 e, the stroke S of the slider can be found as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi S~ (r 40 zr 5 ) {r 6 { (r 40 {r 5 ) {r 6 () Also, let c 1 5%b 0 d 1 d and c 5%b 0 d e, we obtain c 1 ~ sin {1 ½r 6 =(r 5 zr 4 0 )Š (3) c ~ sin {1 ½r 6 =(r 5 {r 4 0 )Š (4) and both angles are measured in counterclockwise. Furthermore, let w 1 be the angular displacement of the link when the slider moves from position 1 to (link rotating in clockwise), and w 1 for position to 1. Then, it can be seen from Fig. 3 that w 1 ~p{d 1 zd (5) w 1 ~p{w 1 (6) Transactions of the Canadian Society for Mechanical Engineering, Vol. 33, No. 3,

5 where d 1 5%b 0 a 0 a 1 and d 5%b 0 a 0 a, both measured in counterclockwise. Moreover, let w w and w r denote the angular displacement of the link for the working and the return strokes, respectively. For a quick return mechanism, w w is greater than w r.ifd. d 1, then w 1. w 1. Therefore, w w ~w 1 (7) Otherwise, w w ~w 1 (8) In addition, w w zw r ~p (9) By definition, the time ratio TR is the time of the working stroke t w to that of the return stroke t r. If the input is running at constant speed, then the time ratio t w to t r equals to that of w w to w r. Then, TR~t w =t r ~w w =w r (10) Therefore, the time ratio of the design can be determined if d 1 and d are obtained. Let l 1 5 a 0 b 1, from Da 0 b 0 b 1 in Fig. 4(a), then we have l 1 ~r 1 zr 4 {r 1r 4 cos½p{(b{c 1 )Š (11) Moreover, let g 1 5%b 1 a 0 a 1 and j 1 5%b 0 a 0 b 1, then it yields g 1 ~ cos {1 l 1 zr {r 3 l 1 r (1) from Da 1 a 0 b 1. And j 1 ~ tan {1 r 4 sin (b{c 1 ) r 4 cos (b{c 1 )zr 1 (13) from Db 1 a 0 b 0. Moreover, d 1 is the sum of g 1 and j 1, that is d 1 ~g 1 zj 1 (14) Fig. 4. Limit positions. Transactions of the Canadian Society for Mechanical Engineering, Vol. 33, No. 3,

6 Substituting Eqs. (1) (13) into Eq. (14), we have d 1 ~ cos {1 l 1 zr {r 3 l 1 r z tan {1 r 4 sin (b{c 1 ) r 4 cos (b{c 1 )zr 1 (15) Similarly, let l 5 a 0 b, then from Da 0 b 0 b in Fig. 4(b), it yields l ~r 1 zr 4 {r 1r 4 cos½(b{c )Š (16) Let g 5%b a 0 a and j 5%b 0 a 0 b, where b 0 is a point that b 0 mirrors with respect to the vertical line passing through a 0. Then, we obtain g ~ cos {1 l zr {r 3 l r (17) from Da a 0 b. Furthermore, j ~ tan {1 r 4 sin (b{c ) r 4 cos (b{c ){r 1 (18) from Db a 0 b 0. And d is the sum of g and j, that is d ~g zj (19) Substituting Eqs. (17) (18) into Eq. (19), it yields d ~ cos {1 l zr {r 3 l r z tan {1 r 4 sin (b{c ) r 4 cos (b{c ){r 1 (0) Hence, d 1 and d can be solved from Eqs. (15) and (0). Then, substituting them into Eqs. (5) and (6), w 1 and w 1 can be obtained. Moreover, w w and w r can then be determined through Eqs. (7) (9). Finally, time ratio can be work out by Eq. (10). In addition, the time ratio design for the proposed mechanism can then be conducted through the design equations derived above. 4. KINEMATIC ANALYSIS The main approaches for kinematic analysis of closed loop mechanisms are relative velocity/ acceleration, vector loop, and matrix loop approaches. The vector loop approach can be easily computerized and is suitable for the analysis of planar mechanisms, therefore it is adopted for the kinematic analysis in this study Position Analysis Fig. 5 shows the vector representation of the proposed design. It has two independent vector loops, and their equations can be formulated as: I r z I r 3 { I r 4 { I r 1 ~0 (1) I r 1 z I r 4 z I r 5 { I r 6 { I r 7 ~0 () Transactions of the Canadian Society for Mechanical Engineering, Vol. 33, No. 3,

7 Fig. 5. Vector representation. Eq. (1) can be resolved into its x and y components, respectively, as: r cos h zr 3 cos h 3 {r 4 cos h 4 {r 1 cos h 1 ~0 (3) r sin h zr 3 sin h 3 {r 4 sin h 4 {r 1 sin h 1 ~0 (4) where h Rearranging Eqs. (3) and (4), we have r 3 cos h 3 ~r 4 cos h 4 {r cos h zr 1 (5) r 3 sin h 3 ~r 4 sin h 4 {r sin h (6) The square sum of Eqs. (5) and (6) can be found to be A cos h 4 zb sin h 4 ~C (7) where A~r 4 (r 1 {r cos h ) (8) B~{r r 4 sin h (9) C~(r 3 {r 4 {r 1 {r )zr 1 r cos h 1 cos h (30) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Dividing Eq. (30) by A zb and simplifying it, we obtain where cos w cos h 4 z sin w sin h 4 ~ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C (31) A zb Transactions of the Canadian Society for Mechanical Engineering, Vol. 33, No. 3,

8 cos w~ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A A zb (3) and sin w~ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B A zb (33) Applying the adding formula of trigonometric functions to Eq. (31), then the closed form of h 4 can be solved by h 4 ~w+ cos {1 C pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (34) A zb Dividing Eq. (6) by Eq. (5), and then followed by rearrangement, it yields h 3 ~ tan {1 r 4 sin h 4 {r sin h r 1 zr 4 cos h 4 {r cos h (35) Furthermore, let h 4 5 h 4 b, h u, and h 7 5 0u, and substituting into Eq. (), its x and y components can be solved by r 1 zr 4 0 cos h 4 0 zr 5 cos h 5 {r 7 ~0 (36) r 4 0 sin h 4 0 zr 5 sin h 5 zr 6 ~0 (37) Solving Eq. (37), it yields h 5 ~ sin {1 {(r 0 4 sin h 0 4 zr 6 ) r 5 (38) Substituting Eq. (38) into Eq. (36), then the displacement of the slider is r 7 ~r 1 zr 4 0 cos h 4 0 zr 5 cos h 5 (39) 4.. Velocity Analysis Velocity analysis can be worked out by differentiating the equations deduced in position analysis with respect to time. Due to the limit on the number of pages, the results for velocity and acceleration analysis are presented directly without proofs. For velocity analysis, their deduced equations are: _h 3 ~ r r 3 sin (h 4 {h ) sin (h 3 {h 4 ) _ h (40) _h 4 ~ r r 4 sin (h 3 {h ) sin (h 3 {h 4 ) _ h (41) Transactions of the Canadian Society for Mechanical Engineering, Vol. 33, No. 3,

9 0 _ h4 0 _h 5 ~ {r 4 0 cos h 4 r 5 cos h 5 (4) The velocity of the slider can thus be calculated by _r 7 ~{r 4 0 sin h 4 0 _ h4 0 {r5 sin h 5 _ h5 (43) 4.3. Acceleration Analysis Acceleration analysis can be performed by differentiating the equations derived in velocity analysis with respect to time. The derived equations are: h 3 ~ IK{HL GK{HJ h 4 ~ GL{IJ GK{HJ (44) (45) where G~{r 3 sin h 3 (46) H~r 4 sin h 4 (47) I~r h sin h zr _ h cos h zr 3 _ h3 cos h3 {r 4 _ h4 cos h4 (48) J~r 3 cos h 3 (49) K~{r 4 cos h 4 (50) L~{r h cos h zr _ h sin h zr 3 _ h3 sin h3 {r 4 _ h4 sin h4 (51) Furthermore, h 5 ~ {r 4 0 h 4 0 cos h4 zr 4 0 _ h4 0 sin h 04 zr 5 _ h 5 sin h5 r 5 cos h 5 (5) Finally, the acceleration of the slider can be obtained by r 7 ~{r 4 0 h4 0 sin h4 0 {r 4 0 _ h4 0 cos h4 0 {r 5 h5 sin h 5 {r 5 _ h5 cos h5 (53) Transactions of the Canadian Society for Mechanical Engineering, Vol. 33, No. 3,

10 5. DESIGN EXAMPLE An example is given here for illustrating the design process of the proposed mechanism. It is assumed that the time ratio TR of the design is 1.65, the stroke S is mm, and the working stroke is the one moving from the limit position 1 to position. By specifying r mm, r mm, then substituting them as well as S into Eq. (), we have r mm. Furthermore, r 4, r 5, and r 6 are substituted into Eq. (3) and Eq. (4), then we obtain c u and c u, respectively. Moreover, substituting TR into Eq. (10) and followed by solving simultaneously with Eq. (9), it yields w w u and w r u. Since the design is at working when moves from the limit position 1 to, i.e., w w 5 w 1 and w r 5 w 1, w u and w u. Also, d has to be greater than d 1, based on Eq. (5). Then, assume d u, then d u can be solved from Eq. (5). Moreover, substituting the above known data to Eqs. (1), (11), (15), (16), and (0), we have five equations and eight unknowns (r 1 r 4, l 1, l, a, b). Therefore, three unknowns can be freely specified, and three parameters r mm, r mm, and b 5 10u are assigned here, and then l mm and l mm can be solved by substituting the parameters as well as c 1 and c into Eqs. (11) and (16). Finally, substituting r 1,r 4, l 1, l, and b to Eqs. (1), (15), and (0), we have r 5 60 mm, r mm, and a 5 90u. In addition, the kinematic dimensions of the design are drawn in Fig. 6. To validate the design, its solid model is established by CATIA software, as shown in Fig. 7. Then, the model is introduced into ADAMS software for kinematic simulation, and the output displacement is shown in Fig EXPERIMENT To verify the feasibility of the proposed design, a prototype shown in Fig. 8 is designed and fabricated. In order to have higher precision, an AMT linear guideway with a carriage (MSB 15 TS-BH FC H) and a rail (MSB 15 R /0 H) is used to replace the slider and the guideway, respectively, in the prototype. The slots and ribs are made of S45C Steel, and their profiles are machined and then grinded with a CNC machine. Two 6903ZZ ball bearings are Fig. 6. Kinematic sketch. Transactions of the Canadian Society for Mechanical Engineering, Vol. 33, No. 3,

11 Fig. 7. Kinematic simulation. used for the bearings of the input and the output shafts of the generalized Oldham coupling. Moreover, an experimental system shown in Fig. 9 and Fig. 10 is set up for measuring and testing. The system comprises an AC servo motor (Mitsubishi, HC-KFS73, 750W, 3000 rpm) with a servo driver (MR-JS-70A), a flexible coupling (TSD, MHC-118L48), an linear encoder (Easson, GS10000, 10 mm/m), and a servo amplifier (Elmo, CEL 5/60) for data acquisition. To adjust the input at constant speed more easily, a servo motor is adopted in the system as the power source. In practical application, a motor runs at constant speed can be used instead. By connecting the motor shaft and the input shaft with the flexible coupling, then the servo motor drives the input shaft of the prototype at constant speed with open loop control. The displacement of the slider (carriage) is measured by the linear encoder. The pulses generated by the linear encoder are counted by the servo amplifier, and then transmitted to the Elmo composer software, installed in a PC, for converting it into the output displacement. The comparisons of output motions between the simulation and the experiment at the input speed of 60 rpm are shown in Fig. 11. It can be observed from the figure that both the outputs agree fairly with each other, the experimental output has only a slight lag occurred. Also, the time ratio of the simulation and the experiment can be found by calculating the time for each stroke. For example, the maximum displacement occurred in the simulation is at the time of 0.64 sec in Fig. 11(a), then the time for the working stroke is t w sec, and the time for the return stroke is t r sec. Therefore, by definition, its time ratio is TR Transactions of the Canadian Society for Mechanical Engineering, Vol. 33, No. 3,

12 (a) Assembly drawing (b) Photo Fig. 8. Prototype. Fig. 9. Schematic diagram of the experimental system. Fig. 10. Experimental set up. Transactions of the Canadian Society for Mechanical Engineering, Vol. 33, No. 3,

13 Similarly, in the experiment, t w sec, and t r sec, therefore TR It can be easily found that the error is 0.01 which is 1.7 % to the simulation. Therefore, the proposed design is not only feasible, but also has reasonable accuracy. 7. CONCLUSIONS In this work, a novel design for the quick return mechanism, i.e., a crank-slider driven by a generalized Oldham coupling, has been proposed. The design equations for time ratio have been derived from its geometry. Kinematic analysis has been performed. Then, a design example has been given for illustrating the design process. Moreover, a computer simulation using ADAMS software has been carried out. Finally, to verify the feasibility and accuracy, a prototype has been made, and then an experiment has been conducted. The experimental result shows that the proposed mechanism can produce the designed time ratio with reasonable accuracy. In addition, the new design will consume less space and can be balanced more easily, compared to a conventional quick return linkage. Hence it provides a good alternative for the quick return mechanism that requires more compact volume and better dynamical balance. ACKNOWLEDGEMENT The support of the National Science Council, Republic of China (Taiwan), under Grants NSC 96-1-E and NSC 97-1-E , is gratefully acknowledged. REFERENCES (a) Displacement (b) Velocity (c) Acceleration Fig. 11. Comparisons. 1. Martin, G.H., Kinematics and Dynamics of Machines, nd ed., McGraw-Hill, Norton, R.L., Design of Machinery, nd ed., McGraw-Hill, Dwivedi, S.N., Application of a Whitworth Quick Return Mechanism For High Velocity Impacting Press, Mechanism and Machine Theory, Vol. 19, pp , Suareo, F.O. and Gupta, K.C., Design of Quick-Returning R-S-S-R Mechanisms, Journal of Mechanisms, Transmissions, and Automation in Design, Vol. 110, No. 4, pp , Beale, D.G. and Scott, R.A., Stability and Response of a Flexible Rod in a Quick Return Mechanism, Journal of Sound and Vibration, Vol. 141, No., pp , Fung, R.F. and Chen, K.W., Constant Speed Control Of the Quick Return Mechanism Driven by a DC Motor, JSME International Journal, Series C, Vol. 40, No. 3, pp , Transactions of the Canadian Society for Mechanical Engineering, Vol. 33, No. 3,

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