Design of Variable Coupler Curve Four-bar MECHANISM

8/18/2019 Design of Variable Coupler Curve Four-bar MECHANISM

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中國機械工程學刊第三十卷第三期第249~257頁(民國九十八年)Journal of the Chinese Society of Mechanical Engineers, Vol.30, No.3, pp.249~257 (2009)

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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 avariable coupler curve four-bar mechanism with onelink replaced by an adjustable screw-nut link anddriven by a servomotor. Different desired couplercurves can be generated by controlling the angulardisplacement of the driving link and adjusting thelength of the adjustable links for continuous pathgeneration. This paper also presents a derivation ofthe adjustable link lengths and the specified angular

displacement of the driving link corresponding to thedesired coupler curves. The conditions for generabledesired coupler curves are also described. Theexamples 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 couplercurves only specify discrete points on the desired

path. The other is continuous path generation, inwhich the coupler curves specify the entire path, or at

least many points on it. Because the coupler curves oflinkage mechanisms are functions of their linklengths, the only way to generate different continuouscoupler curves with a single linkage mechanism is tomake the length of at least one of its links adjustable.

One way of doing this is to replace the normal linkswith screw-nut links driven by servomotors, as shownin Fig. 1. The different desired coupler curves can

then be obtained by controlling the length of theadjustable links and the angular displacement of thedriving link.

The investigation of new synthesis methods for path generation using linkage mechanisms has beenthe subject of some research attention in recent years.

Tao and Krishnamoothy (1978) developed graphicalsynthesis procedures of adjustable mechanisms forgenerating variable coupler curves with cusps and

Paper Received February, 2009. Revised April, 2009.

Accepted May, 2009. Author for Correspondence: Ren-ChungSoong

* 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


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

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symmetrical coupler curves with a double point.McGovern and Sandor (1973) used complex numbermethods to synthesize adjustable mechanisms for path generation. Kay and Haws (1975) developed adesign procedure for a path generation mechanism

with a cam link, which provided accuracy over arange of motion. Angeles et al. (1998) proposed anunconstrained nonlinear least-square optimalsynthesis method for RRRR planar path generators.Hoeltzel and Chieng (1990) proposed a patternmatching synthesis method based on theclassification of coupler curves according to momentvariants. Watanabe (1992) presented a natural

equation that expressed the curvature of the path asan equation of the arc length and was independent of

the location and orientation of the path. Ullah andKota (1994, 1997) presented an optimal synthesismethod in which the objective function was

expressed as Fourier descriptors. Shimojima et al.(1983) developed a synthesis method for straight-lineand L-shaped path generation using fixed pivot positions as adjustable parameters. Unruh andKrishnaswami (1995) proposed a computer-aideddesign technique for infinite point coupler curvesynthesis of four-bar linkages. Kim and Sodhi (1996)introduced a method of path generation that made thedesired path pass exactly through five specified points and close to other points. Chuenchom andKota (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 geneticoptimization algorithms by adjusting the position ofthe driven side link for continuous path generation.Russell and Sodhi (2005) presented a design methodfor slider-crank mechanisms to achieve multiphase path and function generation.

In this paper, we propose a new design method forcontinuous path generation by four-bar mechanismsthat incorporates a screw-nut link called an adjustablelink. Different desired coupler curves can begenerated by appropriately adjusting the length of the

adjustable link and controlling the angulardisplacement of the driving link. Examples andexperiments 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 thelengths of links 1 or 4 can be adjusted to generatenew coupler curves. Figure 2 shows that therelationship between the angular displacement of the

driving link 2θ and the coordinate of the coupler

point ( y x PP , ) can be written as

⎟⎟

⎠

⎞⎜⎜

⎝

⎛ =+ −

x

y

P

P1

2 tan)( μ θ (1)

and the vector loop equation can be written as

R2 + R5 – RP = 0. (2)

Separating Eq. (2) into two scalar componentequations in the x- and y-directions yields

0)cos(cos3522

=−++ xPr r β θ θ and (3)

0)sin(sin 3522 =−++ yPr r β θ θ (4)

whereir and iθ represent the length and angular

displacement of the ith link, respectively. Adding Eqs.(3) and (4) after squaring both sides gives

2

2222

222

5 )sincos(2 r PPr PPr y x y x ++−+= θ θ ,

(5)

which, after rearrangement, gives

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

θ2

θ3

θ4

β

μ

P

X

Y

2r

1r

5r

r p3r

R2

R3

R4

R1

RP

R5


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

-251-

02

)sincos(2

2

2

222

5

22 =⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ −−−++

r

r PPr PP

y x

y x θ θ .

(6)

To reduce Eq. (6) to a form that can be solved moreeasily, we substitute the half angle identities to

convert the 2cosθ and 2sinθ terms to

2tanθ terms:

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

+

−=

)2

(tan1

)2

(tan1

cos22

22

2 θ

θ

θ ;

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

+=

)2

(tan1

)2

tan(2

sin22

2

2 θ

θ

θ .

This results in the following simplified form, where

the link lengths ( 2r and 5r ) and the known value

( y x PP , ) terms have been collected as constants A, B,

and C : 0)2

tan()2

(tan 222 =++ C B A θ θ

where

x

y xP

r

r PPr A −

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ −−−=

2

2

2

222

5

2,

yP B 2= , and

x

y xP

r

r PPr C +

⎟⎟

⎠

⎞⎜⎜

⎝

⎛ −−−=2

2

2

222

5

2. The angular

displacement of the driving link can then becalculated as

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ −±−= −

A

AC B B

2

4tan2

21

2θ (7)

and the corresponding 3θ can be obtained from Eq.

(4):

β θ

θ θ −

−

−= − )

sinP

cosP(tan

22y

22x1

3r

r . (8)

ADJUSTABLE LENGTH OF LINKS 1

AND 4

From Figure 2, the vector loop equation can bewritten as

R2 + R3 – R1 – R4 =0 . (9)

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

113322444 coscoscoscos)( θ θ θ θ r r r r r −+=Δ+ (10)

113322444 sinsinsinsin)( θ θ θ θ r r r r r −+=Δ+ (11)

where 4r Δ is the length of adjustable link 4. By dividing Eq. (11) by Eq. (10) to

eliminate )( 44 r r Δ+ , the angular displacement of link

4,4θ , can be expressed as

)coscoscossinsinsin(tan

113322

11332214

θ θ θ θ θ θ θ

r r r r r r

−+ −+= − (12)

Then 4r Δ can be calculated as

4

4

113322

4cos

coscoscosr

r r r r −⎟⎟

⎠

⎞⎜⎜⎝

⎛ −+=Δ

θ

θ θ θ (13)

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

111332244 cos)(coscoscos θ θ θ θ r r r r r Δ+−+= (14)

and

111332244 sin)(sinsinsin θ θ θ θ r r r r r Δ+−+= (15)

We then square both equations and add them to

eliminate one unknown, say θ4. The adjustable lengthof link 1, denoted as 1r Δ , can then be expressed as

1

2

12

4r

C B Br −

−±−=Δ (16)

where)sinsincos(cos2

)sinsincos(cos2

31313

21212

θ θ θ θ

θ θ θ θ

+−

+−=

r

r B and

)sinsincos(cos2 323232

2

3

2

2

2

4

θ θ θ θ ++

++−=

r r

r r r C . The

corresponding 4θ is


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

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)cos)(coscos

sin)(sinsin(tan

1113322

11133221

4θ θ θ

θ θ θ θ

r r r r

r r r r

Δ+−+

Δ+−+= −

(17)

CONDITIONS FOR GENERABLE

COUPLER CURVES

The coupler curves that can be generated mustsatisfy both the following conditions:

2525 r r r r r p +≤≤− and (18)

( ) ( ) 252

22

2

22 sincos r r Pr P y x =−+− θ θ (19)

where22

Y x p 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 r r − and 25 r r + , they can be generated by

controlling the angular displacement of the drivinglink and adjusting the length of links l or 4.

EXAMPLES

Burrs have always been a problem for steel pipe

manufacturers. Burrs frequently form oncross-sections when pipes, especially thick ones, arecut, as shown in Fig. 3. Eliminating burrs in pipeswith circular cross-sections is relatively easy, but thisis much more difficult for non-circular cross-sections.Since pipe manufacturers generally produce pipeswith various different cross-sections, clearing burrsfrom pipes is very important.

(a)

(b)

(c)

(d)

In following examples, we use the four-bar linkageshown in Fig. 2 with the dimensions shown in Table 1to generate the coupler curves shown in Fig. 4 bycontrolling the angular position of the input link andthe length of links 1 or 4. The intended application isthe 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


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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)

T h e Y

c o o r d i n a t e o f t h e d e s i r e d c o u p

l e r c u r v e s ( c m )

The quarter circle with diameter(r5-r

2)

The quarter circle with diameter(r5+r

2)

Desired circle curve

Desired 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 inFig. 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 curvesgenerated for all examples and Fig. 6 shows therequired angular displacements of the driving linkcorresponding 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 o o r d i n a t e o f c o u p l e r p o i n t ( c m )

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

A n g u l a r d i s p l a c e m e n t o f t h e d r i v

i n g l i n k ( d e g r e e )

Example 1

Example 2

Example 3

0 10 20 30 40 50 60 70 80-2

0

2

4

6

8

10

12


T h e l e n g t h - a d j u s t a b l e m a g n i t u d e o f t h e l i n k 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


T h e l e n g t h - a d j u s t a b l e m a g n i t u d e o f t h e l i n

k 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


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

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EXPERIMENTS

1 Experimental Setup

Figure 9 shows the schematic of a planarPC-based controlled variable coupler curve four-barmechanism used in our experiments. The setupincluded 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 drovethe screw to adjust the length of link 1simultaneously.

The hardware specifications of the control systemwere as follows:

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

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

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

(4) Incremental encoder (10,000 pulses perrevolution).

2 Implementation

Three experiments corresponding to the exampleslisted in Section 6 were conducted, but only link 1was adjusted. The desired (command) and actualcoupler curves, the angular displacement of thedriving link, and the corresponding length of link 1are shown in Figs. 10, 11, and 12, respectively. Theexperimental results in this section agreed with thedesign results in Section EXAMPLES. Theseexamples and experiments thus confirm the practical

feasibility of the proposed design method.

Time (s)

0 5 10 15 20 25 30 35 T h e a n g u l a r d i s p l a c e m e n t o f t h e d

r i v i n g l i n k ( d e g r e 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 T

h e l e n g t h - a d j u s t a b l e m a g n i t u d e o f t h e l i n k 1 ( c m )

0

2

4

6

8

10

12

Command

Actual

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


16 18 20 22 24 26 28 30 32 34

Y c o o r d i n a t e o f c o u

p l e r p o i n t ( 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


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Time (S)

0 5 10 15 20 25 30 35 T h e a n g u l a r d i s p l a c e m e n t o f t h e

d r i v i n g l i n k ( d e g r e e )

-120

-100

-80

-60

-40

-20

0

20

40

Command

Actual


Time (s)

0 5 10 15 20 25 30 35 T h e l e n g t h - a d j u s t a b l e m a g n i t u d e o f t h e l i n k 1 ( c m )

-4

-2

0

2

4

6

8

10

12

14

Command

Actual



10 15 20 25 30 35 40

Y c o o r d i n a t e o f c o u p l e r p o i n t ( c m )

10

12

14

16

18

20

22

24

26

Command

Actual


Time (s)

0 2 4 6 8 10 12 14 T h e a n g u l a r d i s p l a c e m e n t o f t h e

d r i v i n g l i n k ( d e g r e e )

-120

-100

-80

-60

-40

-20

0

Command

Actual


Time (s)

0 2 4 6 8 10 12 14 T h e l e n g t h - a d j u s t a b l e m a g n i t u d e o f t h e l i n k 1 ( c m )

-2

0

2

4

6

8

10

Command

Actual



16 18 20 22 24 26 28 30

Y c o o r d i n a t e o f c o u p l e r

p o i n t ( c m )

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


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

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CONCLUSIONS

The proposed approach was based on a variablecoupler 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 displacementof the driving link and changing the length of the

adjustable link. The derivations of the adjustable linklength and the specified angular displacement of thedriving link corresponding to the desired couplercurves were presented along with the conditionsrequired to achieve the desired generable couplercurves. The examples and experiments confirmed thefeasibility of this design method, which is suitable forcases that require several different coupler curveswithin a specific area for practical applications.

ACKNOWLEDGMENT

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

REFERENCES

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Chuenchom, T., and Kota, S., “Synthesis of

Programmable Mechanisms Using AdjustableDyads, ” ASME Journal of Mechanical Design,

Vol. 29, No. 5, pp. 232–237.Chang, C.F., 2001, “Synthesis of Adjustable Four-bar

Mechanisms Generating Circular Arcs withSpecified Tangential Velocities,” Mechanismand Machine Theory, Vol. 36, No 3, pp.387–395(1997).

Hoeltzel, D.A., and Chieng, W.H., “Pattern Matching

Synthesis as An Automated Approach toMechanism Design,” ASME Journal of Mechanical Design, Vol. 112, No. 2, pp.190–199(1990).

Kay, F. J., and Haws, R. E., “Adjustable Mechanismsfor Exact Path Generation,” ASME Journal of Engineering for Industry, Vol. 97, No. 2, pp.

702–707(1975).Kim, H.J,. and Sodhi, R.S., “Synthesis of Planar

Four-bar Mechanisms for Generating APrescribed Coupler Curve with Five PrecisionPoints,” Journal of Applied Mechanisms and Robotics, Vol. 3, No 2, pp. 9–13(1996).

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

Function Generation),” ASME Journal of Engineering for Industry, Vol. 95, No. 2, pp.417–422(1973).

McGovern, J. F., and Sandor, G.N., “KinematicSynthesis of Adjustable Mechanism (Part 2:

Path Generation),” ASME Journal of Engineering for Industry, Vol. 95, No. 2, pp.423–429(1973).

Russell, K., and Sodhi, R. S., “On The Design ofSlider-crank Mechanisms. Part II: Multi-phasePath and Function Generation,” Mechanismand Machine Theory, Vol. 40, No. 3, pp.301–317(2005).

Shimojima, H., Ogawa, K., and Sato, O., “Kinematicsynthesis of adjustable mechanisms,” Bulletin

of JSME , Vol. 26, No. 214, pp 627–632(1983).Tao, D. C., and Krishnamoothy, S., “Linkage

Mechanism Adjustable for Variable Coupler

Curves with Cusps,“ Mechanism and MachineTheory, Vol. 13, No. 6, pp. 577–583(1978).

Tao, D. C., and Krishnamoothy, S., “LinkageMechanism Adjustable for VariableSymmetrical Coupler Curves with a DoublePoint,” Mechanism and Machine Theory, Vol.13, No 6, pp. 585–591(1978).

Ullah, I., and Kota, S., “A more EffectiveFormulation of the Path GenerationMechanism Synthesis Problem,” ASME Design Technical Conference, Minneapolis,

USA, 11-14 September, DE-Vol. 70, pp.239–244(1994).

Ullah, I., and Kota, S., “Optimal Synthesis ofMechanisms for Path Generation Using

Fourier Descriptors and Global SearchMethods,” ASME Journal of Mechanical

Design, Vol. 119, No. 4, pp. 504–510(1997).Unruh, V., and Krishnaswami, P., “A computer-aided

Design Technique for Semi-automated InfinitePoint Coupler Curve Synthesis of Four-barLinkages,” ASME Journal of Mechanical Design, Vol. 117, No. 1, pp. 143–149(1995).

Watanabe, K., “Application of Nature Equations toThe Synthesis of Curve GeneratingMechanisms,” Mechanism and Machine

Theory, Vol. 27, No. 3, pp. 261–273(1992).Zhou, H., and Cheung, E.H.M., “Analysis and

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變耦桿點曲線四連桿

機構設計

宋仁羣高苑科技大學機械與自動化工程系

吳上立高苑科技大學電機工程系

摘要

本文提出一種可變耦桿點曲線之四連桿機構

設計方法，將四連桿機構中之一根連桿由一根可調

整桿長以伺服馬驅動之螺桿取代。藉由控制輸入桿之角位置及調整可調整桿之桿長，產生所欲之不同

型態的連續耦桿點曲線。本文並提出對應不同之期

望耦桿點曲線，輸入桿之角位置及可調整桿之桿長

調整量的數學推導結果。本文亦提出可被產生之耦

桿點曲線之條件。最後以設計實例及實驗驗證本設

計方法之可行性及有效性。

Design of Variable Coupler Curve Four-bar MECHANISM

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