98-133

249~257() Journal of the Chinese Society of Mechanical Engineers, Vol.30, No.3, pp.249~257 (2009)

-249-

5

1

P

1

2

3

4

Design of Variable Coupler Curve Four-bar Mechanisms

Ren-Chung Soong * and Sun-Li Wu **

Keywords: Variable coupler curve, Four-bar mechanism, Continuous path generation.

ABSTRACT

This paper presents a method for designing a

variable coupler curve four-bar mechanism with one link replaced by an adjustable screw-nut link and driven by a servomotor. Different desired coupler curves can be generated by controlling the angular displacement of the driving link and adjusting the length of the adjustable links for continuous path generation. This paper also presents a derivation of the adjustable link lengths and the specified angular displacement of the driving link corresponding to the desired coupler curves. The conditions for generable desired coupler curves are also described. The examples and experiments described in this paper confirm the feasibility and effectiveness of the proposed method.

INTRODUCTION

There are two types of path generation. One is point-to-point path generation, in which the coupler curves only specify discrete points on the desired path. The other is continuous path generation, in which the coupler curves specify the entire path, or at least many points on it. Because the coupler curves of linkage mechanisms are functions of their link lengths, the only way to generate different continuous coupler curves with a single linkage mechanism is to make the length of at least one of its links adjustable.

One way of doing this is to replace the normal links with screw-nut links driven by servomotors, as shown in Fig. 1. The different desired coupler curves can then be obtained by controlling the length of the adjustable links and the angular displacement of the driving link.

The investigation of new synthesis methods for path generation using linkage mechanisms has been the subject of some research attention in recent years. Tao and Krishnamoothy (1978) developed graphical synthesis procedures of adjustable mechanisms for generating variable coupler curves with cusps and

Paper Received February, 2009. Revised April, 2009. Accepted May, 2009. Author for Correspondence: Ren-Chung Soong * Associate Professor, Department of Mechanical and

Automation Engineering, Kao Yuan University, Kaohsiung 82141, TAIWAIN, R.O.C.

** Assistant Professor, Department of Electrical Engineering,

Kao Yuan University, Kaohsiung 82141, TAIWAIN, R.O.C.

1

2

3

4

5

P

1

(a)

(b)

Fig. 1. Adjustable mechanisms

J. CSME Vol.30, No.3 (2009)

-250-

symmetrical coupler curves with a double point. McGovern and Sandor (1973) used complex number methods to synthesize adjustable mechanisms for path generation. Kay and Haws (1975) developed a design procedure for a path generation mechanism with a cam link, which provided accuracy over a range of motion. Angeles et al. (1998) proposed an unconstrained nonlinear least-square optimal synthesis method for RRRR planar path generators. Hoeltzel and Chieng (1990) proposed a pattern matching synthesis method based on the classification of coupler curves according to moment variants. Watanabe (1992) presented a natural equation that expressed the curvature of the path as an equation of the arc length and was independent of the location and orientation of the path. Ullah and Kota (1994, 1997) presented an optimal synthesis method in which the objective function was expressed as Fourier descriptors. Shimojima et al. (1983) developed a synthesis method for straight-line and L-shaped path generation using fixed pivot positions as adjustable parameters. Unruh and Krishnaswami (1995) proposed a computer-aided design technique for infinite point coupler curve synthesis of four-bar linkages. Kim and Sodhi (1996) introduced a method of path generation that made the desired path pass exactly through five specified points and close to other points. Chuenchom and Kota (1997) presented a synthesis method for programmable mechanisms using adjustable dyads. Chang (2001) proposed a synthesis method for adjustable mechanisms to trace variable arcs with prescribed velocities. Zhou et al. (2002) proposed an optimal synthesis method with modified genetic optimization algorithms by adjusting the position of the driven side link for continuous path generation. Russell and Sodhi (2005) presented a design method for slider-crank mechanisms to achieve multiphase path and function generation.

In this paper, we propose a new design method for continuous path generation by four-bar mechanisms that incorporates a screw-nut link called an adjustable link. Different desired coupler curves can be generated by appropriately adjusting the length of the adjustable link and controlling the angular displacement of the driving link. Examples and experiments are provided to demonstrate this design method.

REQUIRED DRIVING LINK ANGULAR DISPLACEMENT

CORRESPONDING TO THE DESIRED COUPLER CURVE

The coordinate system of a four-bar linkage is

shown in Fig. 2.

The speed trajectory of the driving link, and the lengths of links 1 or 4 can be adjusted to generate new coupler curves. Figure 2 shows that the relationship between the angular displacement of the driving link 2 and the coordinate of the coupler point ( yx PP , ) can be written as

=+

x

y

PP1

2 tan)( (1)

and the vector loop equation can be written as R2 + R5 RP = 0. (2)

Separating Eq. (2) into two scalar component

equations in the x- and y-directions yields

0)cos(cos 3522 =++ xPrr and (3)

0)sin(sin 3522 =++ yPrr (4) where ir and i represent the length and angular displacement of the ith link, respectively. Adding Eqs. (3) and (4) after squaring both sides gives

22222

2225 )sincos(2 rPPrPPr yxyx +++= ,

(5) which, after rearrangement, gives

Fig. 2. The coordinate system of a four-bar linkage

2

3

4

P

X

Y

2r

1r

5rrp

3r

R2

R3

R4

R1

RP R5

R.C. Soong and S.L. Wu: Design of Variable Coupler Curve Four-bar Mechanisms.

-251-

02

)sincos(2

22

2225

22 =

++

rrPPr

PP yxyx .

(6)

To reduce Eq. (6) to a form that can be solved more

easily, we substitute the half angle identities to convert the 2cos and 2sin terms to

2tan terms:

+

=

)2

(tan1

)2

(tan1cos

22

22

2

;

+=

)2

(tan1

)2

tan(2sin

22

2

2

.

This results in the following simplified form, where the link lengths ( 2r and 5r ) and the known value ( yx PP , ) terms have been collected as constants A, B,

and C: 0)2

tan()2

(tan 222 =++ CBA

where

xyx P

rrPPr

A

=

2

22

2225

2, yPB 2= , and

xyx P

rrPPr

C +

=

2

22

2225

2. The angular

displacement of the driving link can then be calculated as

=

AACBB

24tan2

21

2 (7)

and the corresponding 3 can be obtained from Eq. (4):

= )sinPcosP(tan

22y

22x13 r

r. (8)

ADJUSTABLE LENGTH OF LINKS 1 AND 4

From Figure 2, the vector loop equation can be

written as R2 + R3 R1 R4 =0 . (9)

If we assume that the length of link 4 can be adjusted, then we separate Eq. (9) into two scalar component equations and rearrange as follows:

113322444 coscoscoscos)( rrrrr +=+ (10)

113322444 sinsinsinsin)( rrrrr +=+ (11) where 4r is the length of adjustable link 4.

By dividing Eq. (11) by Eq. (10) to eliminate )( 44 rr + , the angular displacement of link 4, 4 , can be expressed as

)coscoscossinsinsin

(tan113322

11332214

rrrrrr

++

= (12)

Then 4r can be calculated as

44

1133224 cos

coscoscosr

rrrr

+=

(13)

Assuming that the length of link 1 can be adjusted, we separate Eq. 9 into two scalar component equations and rearrange them as follows:

111332244 cos)(coscoscos rrrrr ++= (14) and

111332244 sin)(sinsinsin rrrrr ++= (15) We then square both equations and add them to eliminate one unknown, say 4. The adjustable length of link 1, denoted as 1r , can then be expressed as

1

2

1 24 rCBBr = (16)

where )sinsincos(cos2

)sinsincos(cos2

31313

21212

++=

rrB

and

)sinsincos(cos2 323232

23

22

24

++++=

rrrrrC

. The

corresponding 4 is

J. CSME Vol.30, No.3 (2009)

-252-

)cos)(coscossin)(sinsin

(tan1113322

111332214

rrrrrrrr

++++

=

(17)

CONDITIONS FOR GENERABLE COUPLER CURVES

The coupler curves that can be generated must

satisfy both the following conditions:

2525 rrrrr p + and (18) ( ) ( ) 25222222 sincos rrPrP yx =+ (19)

where 22 Yxp PPr += . In Fig. 2, we assume that

2r and 5r are not adjustable. Therefore, as long as the desired continuous coupler curves are in the area between the two concentric circles with radii

25 rr and 25 rr + , they can be generated by controlling the angular displacement of the driving link and adjusting the length of links l or 4.

EXAMPLES

Burrs have always been a problem for steel pipe manufacturers. Burrs frequently form on cross-sections when pipes, especially thick ones, are cut, as shown in Fig. 3. Eliminating burrs in pipes with circular cross-sections is relatively easy, but this is much more difficult for non-circular cross-sections. Since pipe manufacturers generally produce pipes with various different cross-sections, clearing burrs from pipes is very important.

(a)

(b)

(c)

(d) In following examples, we use the four-bar linkage

shown in Fig. 2 with the dimensions shown in Table 1 to generate the coupler curves shown in Fig. 4 by controlling the angular position of the input link and the length of links 1 or 4. The intended application is the removal of burrs from pipes.

Table 1 Four-bar linkage dimensions

1r 2r 3r 4r 5r

dimension 22.2 cm 10 cm 20.6 cm 23.3 cm 30.6 cm

(c)

Fig. 3. Burrs on the cross-section of steel pipes


-253-

0 5 10 15 20 25 30 35 40 45 500

5

10

15

20

25

30

35

40

45

50

The X coordinate of the desired coupler curves(cm)

The

Y c

oord

inat

e of

the

des

ired

coup

ler

curv

es(c

m)

The quarter circle with diameter(r5-r2)

The quarter circle with diameter(r5+r2)

Desired circle curveDesired ellipse coupler curve

Desired square coupler curve

Example 1.

Generation of a circular coupler curve with the center at (25, 18.5) and radius = 8.5 cm, as shown in Fig. 4.

Example 2.

Generation of an elliptical coupler curve with the center at (25, 18.5), long axis = 10 cm, and short axis = 6 cm, as shown in Fig. 4. Example 3.

Generation of a square coupler curve with four vertexes p1 (17.5, 22.5), p2 (17.5, 13), p3 (27.5, 13), and p4 (27.5, 23), as shown in Fig 4.

Figure 5 shows the desired coupler curves

generated for all examples and Fig. 6 shows the required angular displacements of the driving link corresponding to the desired coupler curves. Figures 7 and 8 show the required lengths of links 1 and 4, respectively, corresponding to the desired coupler curves for all examples.

10 15 20 25 30 35 405

10

15

20

25

30

X coordinate of coupler point (cm)

Y c

oord

inat

e of

cou

pler

poi

nt (

cm)

Example 1

Example 2

Example 3

0 10 20 30 40 50 60 70 80-120

-100

-80

-60

-40

-20

0

20

The number of points on the coupler curve

Ang

ular

dis

plac

emen

t of

the

driv

ing

link

(deg

ree)

Example 1

Example 2

Example 3

0 10 20 30 40 50 60 70 80-2

0

2

4

6

8

10

12


The

leng

th-a

djus

tabl

e m

agni

tude

of

the

link

1 (c

m)

Example 1

Example 2

Example 3

0 10 20 30 40 50 60 70 80-8

-7

-6

-5

-4

-3

-2

-1

0

1

2


The

leng

th-a

djus

tabl

e m

agni

tude

of

the

link

4 (c

m)

Example 1

Example 2

Example 3

Fig. 4. The desired coupler curves for examples

Fig. 5. The desired coupler curves in all examples

Fig. 6. The required angular displacement of the driving link for all examples

Fig. 7. The length-adjustable magnitude of the link 1 for all examples

Fig. 8. The length-adjustable magnitude of the link 4 for all examples

J. CSME Vol.30, No.3 (2009)

-254-

EXPERIMENTS

1 Experimental Setup

Figure 9 shows the schematic of a planar PC-based controlled variable coupler curve four-bar mechanism used in our experiments. The setup included the four-bar mechanism, two AC servomotors with encoders and drivers, and a belt. One AC servomotor was used to control the angular displacement of the driving link while the other drove the screw to adjust the length of link 1 simultaneously.

The hardware specifications of the control system were as follows:

(1) Intel Pentium IV 400-MHz microcomputer with 512 MB RAM;

(2) Motion control card (PCI-8164; Adlink Technology, Inc.);

(3) AC servomotors (400 W; Mitsubishi Co.) and drivers (MR-J2S-A; Mitsubishi Co.); and

(4) Incremental encoder (10,000 pulses per revolution).

2 Implementation

Three experiments corresponding to the examples listed in Section 6 were conducted, but only link 1 was adjusted. The desired (command) and actual coupler curves, the angular displacement of the driving link, and the corresponding length of link 1 are shown in Figs. 10, 11, and 12, respectively. The experimental results in this section agreed with the design results in Section EXAMPLES. These examples and experiments thus confirm the practical feasibility of the proposed design method.

Time (s)

0 5 10 15 20 25 30 35The

ang

ular

dis

plac

emen

t of t

he d

rivin

g lin

k (d

egre

e)

-100

-80

-60

-40

-20

0

20

Command Actual

(a) Angular displacement of the driving link

Time (s)

0 5 10 15 20 25 30 35

The

leng

th-a

djus

tabl

e m

agni

tude

of t

he li

nk 1

(cm

)

0

2

4

6

8

10

12

CommandActual

(b) The length-adjustable magnitude of the link 1


16 18 20 22 24 26 28 30 32 34

Y c

oord

inat

e of

cou

pler

poi

nt (c

m)

10

12

14

16

18

20

22

24

26

28

Command Actual

(c) The coupler curves

Fig. 9. The variable coupler curve mechanism

Fig. 10. The experimental results of the Example 1


-255-

Time (S)

0 5 10 15 20 25 30 35The

ang

ular

dis

plac

emen

t of t

he d

rivin

g lin

k (d

egre

e)

-120

-100

-80

-60

-40

-20

0

20

40

Command Actual


Time (s)

0 5 10 15 20 25 30 35

The

leng

th-a

djus

tabl

e m

agni

tude

of t

he li

nk 1

(cm

)

-4

-2

0

2

4

6

8

10

12

14

Command Actual



10 15 20 25 30 35 40

Y co

ordi

nate

of c

oupl

er p

oint

(cm

)

10

12

14

16

18

20

22

24

26

Command Actual


Time (s)

0 2 4 6 8 10 12 14The

ang

ular

dis

plac

emen

t of t

he d

rivin

g lin

k (d

egre

e)

-120

-100

-80

-60

-40

-20

0

Command Actual


Time (s)

0 2 4 6 8 10 12 14

The

leng

th-a

djus

tabl

e m

agni

tude

of t

he li

nk 1

(cm

)

-2

0

2

4

6

8

10

Command Actual



16 18 20 22 24 26 28 30

Y co

ordi

nate

of c

oupl

er p

oint

(cm

)

12

14

16

18

20

22

24

Command Actual


Fig. 11. The experimental results of the Example 2 Fig. 12. The experimental results of the Example 3

J. CSME Vol.30, No.3 (2009)

-256-

CONCLUSIONS

The proposed approach was based on a variable coupler curve four-bar mechanism in which one link was replaced by a screw-nut link driven by a servomotor. The different desired couple curves could be generated by controlling the angular displacement of the driving link and changing the length of the adjustable link. The derivations of the adjustable link length and the specified angular displacement of the driving link corresponding to the desired coupler curves were presented along with the conditions required to achieve the desired generable coupler curves. The examples and experiments confirmed the feasibility of this design method, which is suitable for cases that require several different coupler curves within a specific area for practical applications.

ACKNOWLEDGMENT

This research was supported by the National Science Council Taiwan, R.O.C, through the grant NSC 94-2212-E-244-003.

REFERENCES Angeles, J., Alivizators, A., and Akhras, A., An

Unconstrained Nonlinear Least-square Method of RRRR Planar Path Generators, Mechanism and Machine Theory, Vol. 23, No. 5, pp. 343353(1998).

Chuenchom, T., and Kota, S., Synthesis of Programmable Mechanisms Using Adjustable Dyads, ASME Journal of Mechanical Design, Vol. 29, No. 5, pp. 232237.

Chang, C.F., 2001, Synthesis of Adjustable Four-bar Mechanisms Generating Circular Arcs with Specified Tangential Velocities, Mechanism and Machine Theory, Vol. 36, No 3, pp. 387395(1997).

Hoeltzel, D.A., and Chieng, W.H., Pattern Matching Synthesis as An Automated Approach to Mechanism Design, ASME Journal of Mechanical Design, Vol. 112, No. 2, pp. 190199(1990).

Kay, F. J., and Haws, R. E., Adjustable Mechanisms for Exact Path Generation, ASME Journal of Engineering for Industry, Vol. 97, No. 2, pp. 702707(1975).

Kim, H.J,. and Sodhi, R.S., Synthesis of Planar Four-bar Mechanisms for Generating A Prescribed Coupler Curve with Five Precision Points, Journal of Applied Mechanisms and Robotics, Vol. 3, No 2, pp. 913(1996).

McGovern, J. F., and Sandor, G.N., Kinematic Synthesis of Adjustable Mechanism (Part 1:

Function Generation), ASME Journal of Engineering for Industry, Vol. 95, No. 2, pp. 417422(1973).

McGovern, J. F., and Sandor, G.N., Kinematic Synthesis of Adjustable Mechanism (Part 2: Path Generation), ASME Journal of Engineering for Industry, Vol. 95, No. 2, pp. 423429(1973).

Russell, K., and Sodhi, R. S., On The Design of Slider-crank Mechanisms. Part II: Multi-phase Path and Function Generation, Mechanism and Machine Theory, Vol. 40, No. 3, pp. 301317(2005).

Shimojima, H., Ogawa, K., and Sato, O., Kinematic synthesis of adjustable mechanisms, Bulletin of JSME, Vol. 26, No. 214, pp 627632(1983).

Tao, D. C., and Krishnamoothy, S., Linkage Mechanism Adjustable for Variable Coupler Curves with Cusps, Mechanism and Machine Theory, Vol. 13, No. 6, pp. 577583(1978).

Tao, D. C., and Krishnamoothy, S., Linkage Mechanism Adjustable for Variable Symmetrical Coupler Curves with a Double Point, Mechanism and Machine Theory, Vol. 13, No 6, pp. 585591(1978).

Ullah, I., and Kota, S., A more Effective Formulation of the Path Generation Mechanism Synthesis Problem, ASME Design Technical Conference, Minneapolis, USA, 11-14 September, DE-Vol. 70, pp. 239244(1994).

Ullah, I., and Kota, S., Optimal Synthesis of Mechanisms for Path Generation Using Fourier Descriptors and Global Search Methods, ASME Journal of Mechanical Design, Vol. 119, No. 4, pp. 504510(1997).

Unruh, V., and Krishnaswami, P., A computer-aided Design Technique for Semi-automated Infinite Point Coupler Curve Synthesis of Four-bar Linkages, ASME Journal of Mechanical Design, Vol. 117, No. 1, pp. 143149(1995).

Watanabe, K., Application of Nature Equations to The Synthesis of Curve Generating Mechanisms, Mechanism and Machine Theory, Vol. 27, No. 3, pp. 261273(1992).

Zhou, H., and Cheung, E.H.M., Analysis and Optimal Synthesis of Adjustable Linkages for Path Generation, Mechatronics, Vol. 12, No. 7, pp. 949961(2002).

Zhou, H., and Ting, K.L., Adjustable Slider-crank Linkages for Multiple Path Generation, Mechanism and Machine Theory, Vol. 37, No. 5, pp. 499509(2002).


-257-

Documents

98-133