A STUDY ON A NOVEL QUICK RETURN MECHANISM

A STUDY ON A NOVEL QUICK RETURN MECHANISM

Wen-Hsiang Hsieh and Chia-Heng TsaiDepartment of Automation Engineering, National Formosa University,

E-mail: [email protected]

Received April 2008, Accepted September 2009

No. 08-CSME-13, E.I.C. Accession 3051

ABSTRACT

This work aims to propose a novel design for quick return mechanisms, and the newmechanism 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 ofthe proposed design. By transforming into its kinematically equivalent mechanism, and then thedesign equations of time ratio are derived. Furthermore, a design example is given forillustration. Moreover, the design is validated by kinematic simulation using ADAMS software.Finally, a prototype and an experimental setup are established, and the experiment isconducted. The results show that proposed new mechanism is feasible and with reasonableaccuracy. In addition, it is more compact and easier to be balanced dynamically than aconventional quick return linkage.

ETUDE D’UN MECANISME A RETOUR RAPIDE INNOVATEUR

RESUME

Nous proposons une conception innovatrice d’un mecanisme a retour rapide compose d’unaccouplement de style Oldham et d’un systeme bielle-manivelle. Pour commencer, en etudiant lageometrie du concept propose, on etablit les dimensions cinematiques qui affectent le rapporttemps. En les transformant en un mecanisme cinematique equivalent, nous trouvons la deriveedes equations de rapport temps, et pour illustrer le concept on donne un exemple. De plus, leconcept est valide par une simulation cinematique a l’aide du logiciel ADAMS. Finalement, unprototype est cree, et un cadre experimental est etabli pour mener l’experience. Les resultatsrevelent que le mecanisme propose est realisable et sa precision est acceptable. En outre, il estplus compact et il est plus facile d’effectuer l’equilibrage dynamique que pour un mecanisme aretour rapide conventionnel.

Transactions of the Canadian Society for Mechanical Engineering, Vol. 33, No. 3, 2009 487

1. INTRODUCTION

A quick return mechanism is a mechanism that converts rotary motion into reciprocatingmotion at different rate for its two strokes. When the time required for the working stroke isgreater than that of the return stroke, it is a quick return mechanism. It yields a significantimprovement in machining productivity. Currently, it is widely used in machine tools, forinstance, shaping machines, power-driven saws, and other applications requiring a workingstroke with intensive loading, and a return stroke with non-intensive loading.

Several quick return mechanisms can be found in the literatures [1–2], including the offsetcrank-slider mechanism, the crank-shaper mechanisms, the double crank mechanisms, and theWhitworth mechanism. All of them are linkages. A linkage has its strengths and weaknesses. Itis inexpensive to make and easy to lubricate; however, it is bulky and difficult to balance. Insituations, 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 thatdeserves to be examined.

There are many scholars devoted to the studies of quick return mechanisms, and manyvaluable contributions have been made. Dwivedi [3] used the Whitworth mechanism toconstructing a high velocity impacting press. Suareo & Gupta [4] performing the design of thespatial RSSR quick return mechanism. Beale [5] employed Galerkin’s method to investigate thedynamic 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 returnmechanism with or without a flexible link. Hat et al. [12] proposed a finite difference methodwith fixed and variable grids to approximate the numerical solutions of a flexible quick-returnmechanism. Chang [13] investigated the coupling effect of the geared rotor on the quick-returnmechanism undergoing three-dimensional vibration.

This work aims to present a new design for the quick return mechanisms, and verify itsfeasibility by conducting simulation and experimental studies. In this work, Section 2 introducesthe composition of the proposed system. The equations for calculating time ratio areinvestigated in Section 3. Section 4 performs kinematic analysis. A design example is given forillustration in Section 5. Section 6 presents the prototype experiment. Conclusions aresummarized in Section 7.

2. 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 withcurvilinear slots [14]. Fig. 1 shows the generalized Oldham coupling which consists of a frame(link 1), an input disk (link 2), a floating disk (link 3), and an output disk (link 4). The radii, r1

and r2, of the centerline of the slots need not be equal, and the two centerlines are usually, butnot necessary, designed to intersect at the axis of the floating disk. The classical Oldhamcoupling will transmit uniform motion between the input and output shafts. However, thegeneralized Oldham coupling with circular slots and curvilinear slots will transmit non-uniformmotion and produce quick return motion. And it may lead to many potential applications todevices requiring non-uniform transmission. Hsieh [15–18] applied the generalized Oldhamcoupling to various mechanical devices, including mechanical presses [15], cam mechanisms[16], ball screw mechanisms [17], and vibrating conveyors. In addition, with proper


arrangement, Oldham couplings can be used as a balancer to eliminate shaking forces andshaking moment [19].

Figs. 2(a) and 2(b) depicts the schematic sketch and the kinematic sketch of the proposednew mechanism, respectively. It consists of a generalized Oldham coupling and a crank-slidermechanism. A motor is connected to the input disc of the generalized Oldham coupling at pivota0, 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 involume (as discussed in Sec. 3) than a conventional quick return linkage. In addition, thebuilding cost is inexpensive if a generalized coupling with circular slots is adopted.

3. MOTION GEOMETRY

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. 2,points a0 and b0 denote the axes of rotation of the input and the output disks, respectively, andlet r1 5 a0b0 be the distance of centers of two fixed pivots. Furthermore, points a and b denotethe arc centers of the slots on the input and the output disks, respectively, and the distance oftwo centers is set as r3 5 ab, and point o is the point of intersection of the two arcs. Moreover,let r2 5 oa and r4 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, andangle b 5%cb0b is that of b0b makes with b0c, measured in counterclockwise. In addition, letr4’ 5 b0c and r5 5 cd be the link lengths of the crank and the connecting rod, respectively. Also,let r6 be the offset, the perpendicular distance between the line produced by the rectilinear

Fig. 2. New quick return mechanism.

Fig. 1. Generalized Oldham coupling.


motion of joint d and the line passing through two fixed pivots a0 and b0. Also, by applyingCosine law to triangle Daob, it leads to

r23~r2

2zr24{2r2r4 cos a (1)

Additionally, it is noted that angle b represents the relative position of assembly between thecoupling and the slider-crank mechanism. Since the output disk of the coupling will be attachedto the crank in the proposed design, b is a constant. Apparently, the output motion will varywith different choices of b. Finally, the kinematic dimensions of the system will be fully definedif ri (i5126), r4’, a, and b are specified. The mechanism can be converted into its equivalentmechanism, called the drag link quick return mechanism, as shown in Fig. 3. By comparingFig. 2(b) and 3, the proposed new design is more compact than its equivalent linkage. Inaddition, 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 2 denote theright limit position and the left limit position of the slider, respectively. Let e be the intersectionof perpendicular line of b0 with the extended line of d1d2, from Triangles Db0d2e and Db0d1e, thestroke S of the slider can be found as

S~

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi(r40zr5)2{r6

2

q{

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi(r40{r5)2{r6

2

q(2)

Also, let c1 5%b0d1d and c2 5%b0d2e, we obtain

c1~ sin{1½r6=(r5zr40)� (3)

c2~ sin{1½r6=(r5{r40)� (4)

and both angles are measured in counterclockwise. Furthermore, let w12 be the angulardisplacement of the link 2 when the slider moves from position 1 to 2 (link 2 rotating inclockwise), and w21 for position 2 to 1. Then, it can be seen from Fig. 3 that

w21~p{d1zd2 (5)

w12~2p{w21 (6)

Fig. 3. Equivalent linkages.


where d1 5%b0a0a1 and d2 5%b0’a0a2, both measured in counterclockwise. Moreover, let ww

and wr denote the angular displacement of the link 2 for the working and the return strokes,respectively. For a quick return mechanism, ww is greater than wr. If d2 . d1, then w21 . w12.Therefore,

ww~w21 (7)

Otherwise,

ww~w12 (8)

In addition,

wwzwr~2p (9)

By definition, the time ratio TR is the time of the working stroke tw to that of the return stroketr. If the input is running at constant speed, then the time ratio tw to tr equals to that of ww to wr.Then,

TR~tw=tr~ww=wr (10)

Therefore, the time ratio of the design can be determined if d1 and d2 are obtained. Let l15 a0b1,from Da0b0b1 in Fig. 4(a), then we have

l21~r2

1zr24{2r1r4 cos½p{(b{c1)� (11)

Moreover, let g1 5%b1a0a1 and j1 5%b0a0b1, then it yields

g1~ cos{1 l21zr2

2{r23

2l1r2(12)

from Da1a0b1. And

j1~ tan{1 r4 sin (b{c1)

r4 cos (b{c1)zr1

(13)

from Db1a0b0. Moreover, d1 is the sum of g1 and j1, that is

d1~g1zj1 (14)

Fig. 4. Limit positions.


Substituting Eqs. (12)–(13) into Eq. (14), we have

d1~ cos{1 l21zr2

2{r23

2l1r2z tan{1 r4 sin (b{c1)

r4 cos (b{c1)zr1

(15)

Similarly, let l2 5 a0b2, then from Da0b0b2 in Fig. 4(b), it yields

l22~r2

1zr24{2r1r4 cos½(b{c2)� (16)

Let g2 5%b2a0a2 and j2 5%b0’a0b2, where b0’ is a point that b0 mirrors with respect to thevertical line passing through a0. Then, we obtain

g2~ cos{1 l22zr2

2{r23

2l2r2(17)

from Da2a0b2. Furthermore,

j2~ tan{1 r4 sin (b{c2)

r4 cos (b{c2){r1

(18)

from Db2a0b0. And d2 is the sum of g2 and j2, that is

d2~g2zj2 (19)

Substituting Eqs. (17)–(18) into Eq. (19), it yields

d2~ cos{1 l22zr2

2{r23

2l2r2z tan{1 r4 sin (b{c2)

r4 cos (b{c2){r1(20)

Hence, d1 and d2 can be solved from Eqs. (15) and (20). Then, substituting them into Eqs. (5)and (6), w12 and w21 can be obtained. Moreover, ww and wr 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 theproposed 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 easilycomputerized and is suitable for the analysis of planar mechanisms, therefore it is adopted forthe kinematic analysis in this study.

4.1. Position AnalysisFig. 5 shows the vector representation of the proposed design. It has two independent vector

loops, and their equations can be formulated as:

Ir2z

Ir3{

Ir4{

Ir1~0 (21)

Ir1z

Ir4’z

Ir5{

Ir6{

Ir7~0 (22)


Eq. (21) can be resolved into its x and y components, respectively, as:

r2 cos h2zr3 cos h3{r4 cos h4{r1 cos h1~0 (23)

r2 sin h2zr3 sin h3{r4 sin h4{r1 sin h1~0 (24)

where h1 5 0. Rearranging Eqs. (23) and (24), we have

r3 cos h3~r4 cos h4{r2 cos h2zr1 (25)

r3 sin h3~r4 sin h4{r2 sin h2 (26)

The square sum of Eqs. (25) and (26) can be found to be

A cos h4zB sin h4~C (27)

where

A~2r4(r1{r2 cos h2) (28)

B~{2r2r4 sin h2 (29)

C~(r32{r4

2{r12{r2

2)z2r1r2 cos h1 cos h2 (30)

Dividing Eq. (30) byffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA2zB2

pand simplifying it, we obtain

cos w cos h4z sin w sin h4~Cffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

A2zB2p (31)

where

Fig. 5. Vector representation.


cos w~Affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

A2zB2p (32)

and

sin w~Bffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

A2zB2p (33)

Applying the adding formula of trigonometric functions to Eq. (31), then the closed form of h4

can be solved by

h4~w+ cos{1 CffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA2zB2p� �

(34)

Dividing Eq. (26) by Eq. (25), and then followed by rearrangement, it yields

h3~ tan{1 r4 sin h4{r2 sin h2

r1zr4 cos h4{r2 cos h2

� �(35)

Furthermore, let h4’ 5 h42b, h6 5 270u, and h7 5 0u, and substituting into Eq. (22), its x and y

components can be solved by

r1zr40 cos h4

0zr5 cos h5{r7~0 (36)

r40 sin h4

0zr5 sin h5zr6~0 (37)

Solving Eq. (37), it yields

h5~ sin{1 {(r40 sin h4

0zr6)

r5

� �(38)

Substituting Eq. (38) into Eq. (36), then the displacement of the slider is

r7~r1zr40 cos h4

0zr5 cos h5 (39)

4.2. Velocity AnalysisVelocity 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 velocityand acceleration analysis are presented directly without proofs. For velocity analysis, theirdeduced equations are:

_hh3~r2

r3

sin (h4{h2)

sin (h3{h4)_hh2 (40)

_hh4~r2

r4

sin (h3{h2)

sin (h3{h4)_hh2 (41)


_hh5~{r4

0

r5

cos h40 _hh40

cos h5(42)

The velocity of the slider can thus be calculated by

_rr7~{r40 sin h4

0 _hh40{r5 sin h5

_hh5 (43)

4.3. Acceleration AnalysisAcceleration analysis can be performed by differentiating the equations derived in velocity

analysis with respect to time. The derived equations are:

€hh3~IK{HL

GK{HJ(44)

€hh4~GL{IJ

GK{HJ(45)

where

G~{r3 sin h3 (46)

H~r4 sin h4 (47)

I~r2€hh2 sin h2zr2

_hh22

cos h2zr3_hh3

2cos h3{r4

_hh42

cos h4 (48)

J~r3 cos h3 (49)

K~{r4 cos h4 (50)

L~{r2€hh2 cos h2zr2

_hh22

sin h2zr3_hh3

2sin h3{r4

_hh42

sin h4 (51)

Furthermore,

€hh5~{r4

0€hh40cos h4zr4

0 _hh402

sin h0

4zr5_hh5

2sin h5

r5 cos h5(52)

Finally, the acceleration of the slider can be obtained by

€rr7~{r40€hh40sin h4

0{r40 _hh402

cos h40{r5

€hh5 sin h5{r5_hh5

2cos h5 (53)


5. DESIGN EXAMPLE

An example is given here for illustrating the design process of the proposed mechanism. It isassumed that the time ratio TR of the design is 1.65, the stroke S is 102.61 mm, and the workingstroke is the one moving from the limit position 1 to position 2. By specifying r4’ 5 50 mm, r5 5

175 mm, then substituting them as well as S into Eq. (2), we have r6 5 37.5 mm. Furthermore,r4’, r5, and r6 are substituted into Eq. (3) and Eq. (4), then we obtain c1 5 9.59u and c2 5 17.46u,respectively. Moreover, substituting TR into Eq. (10) and followed by solving simultaneouslywith Eq. (9), it yields ww 5 224.15u and wr 5 135.85u. Since the design is at working when movesfrom the limit position 1 to 2, i.e., ww 5 w12 and wr 5 w21, w21 5 224.15u and w12 5 135.85u. Also,d2 has to be greater than d1, based on Eq. (5). Then, assume d1 5 115.99u, then d2 5 159.3u canbe solved from Eq. (5). Moreover, substituting the above known data to Eqs. (1), (11), (15),(16), and (20), we have five equations and eight unknowns (r12r4, l1, l2, a, b). Therefore, threeunknowns can be freely specified, and three parameters r1 5 13.29 mm, r4 5 60 mm, and b 5

120u are assigned here, and then l1 5 69.86 mm and l2 5 53.30 mm can be solved by substitutingthe parameters as well as c1 and c2 into Eqs. (11) and (16). Finally, substituting r1, r4, l1, l2, andb to Eqs. (1), (15), and (20), we have r2 5 60 mm, r3 5 84.85 mm, and a 5 90u. In addition, thekinematic dimensions of the design are drawn in Fig. 6. To validate the design, its solid modelis established by CATIA software, as shown in Fig. 7. Then, the model is introduced intoADAMS software for kinematic simulation, and the output displacement is shown in Fig. 7.

6. EXPERIMENT

To verify the feasibility of the proposed design, a prototype shown in Fig. 8 is designed andfabricated. In order to have higher precision, an AMT linear guideway with a carriage (MSB 15TS-BH FC H) and a rail (MSB 15 R 460 - 20 /20 H) is used to replace the slider and theguideway, respectively, in the prototype. The slots and ribs are made of S45C Steel, and theirprofiles are machined and then grinded with a CNC machine. Two 6903ZZ ball bearings are

Fig. 6. Kinematic sketch.


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 andtesting. The system comprises an AC servo motor (Mitsubishi, HC-KFS73, 750W, 3000 rpm)with a servo driver (MR-J2S-70A), a flexible coupling (TSD, MHC-1218L48), an linear encoder(Easson, GS100200, ¡ 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 thepower source. In practical application, a motor runs at constant speed can be used instead. Byconnecting the motor shaft and the input shaft with the flexible coupling, then the servo motordrives the input shaft of the prototype at constant speed with open loop control. Thedisplacement of the slider (carriage) is measured by the linear encoder. The pulses generated bythe linear encoder are counted by the servo amplifier, and then transmitted to the Elmocomposer 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 inputspeed of 60 rpm are shown in Fig. 11. It can be observed from the figure that both the outputsagree fairly with each other, the experimental output has only a slight lag occurred. Also, thetime ratio of the simulation and the experiment can be found by calculating the time for eachstroke. For example, the maximum displacement occurred in the simulation is at the time of0.624 sec in Fig. 11(a), then the time for the working stroke is tw 5 0.624 sec, and the time forthe return stroke is tr 5 0.376 sec. Therefore, by definition, its time ratio is TR 5 1.659.

Fig. 7. Kinematic simulation.


(a) Assembly drawing (b) Photo

Fig. 8. Prototype.

Fig. 9. Schematic diagram of the experimental system.

Fig. 10. Experimental set up.


Similarly, in the experiment, tw 5 0.621 sec, and tr 5 0.379 sec, therefore TR 5 1.638. It can beeasily found that the error is 0.021 which is 1.27 % to the simulation. Therefore, the proposeddesign 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 ageneralized Oldham coupling, has been proposed. The design equations for time ratio have beenderived from its geometry. Kinematic analysis has been performed. Then, a design example hasbeen given for illustrating the design process. Moreover, a computer simulation using ADAMSsoftware has been carried out. Finally, to verify the feasibility and accuracy, a prototype hasbeen made, and then an experiment has been conducted. The experimental result shows that theproposed mechanism can produce the designed time ratio with reasonable accuracy. Inaddition, the new design will consume less space and can be balanced more easily, compared toa conventional quick return linkage. Hence it provides a good alternative for the quick returnmechanism that requires more compact volume and better dynamical balance.

ACKNOWLEDGEMENT

The support of the National Science Council, Republic of China (Taiwan), under GrantsNSC 96-2221-E-150-049 and NSC 97-2221-E-150-019, is gratefully acknowledged.

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A STUDY ON A NOVEL QUICK RETURN MECHANISM