# Dimensional synthesis of a planar five-bar mechanism for motion between two extreme positions.

1. Introduction

The reviews of papers reveal that the syntheses of variable topology mechanisms are available for various multi-link mechanisms. Many researchers focused on the synthesis of mechanisms that has variable configuration but their synthesis work is confined for those mechanisms whose couplers have a single tracing point. The dimensional synthesis of those mechanisms that has more than one tracing point is not dealt. The five-bar mechanism with variable topology may be used to provide double tracing point motion transmission. This fulfils the motion requirements in the industry and overcomes the limitations of existing syntheses work. Also, sometimes the mechanism is to be designed to fit a particular work envelop for any given application. In such applications, consideration of extreme positions of various links is important.

The novelty of the present work lies in the fact that it suggests synthesis of a five-bar mechanism for simultaneous tracing of two points that transmits motion between two extreme positions. Moreover, the present work provides solution to design problem of mechanism synthesis for a particular work envelop by considering extreme positions of various links. The mechanism operates in two stages. In each stage, a link adjacent to fixed link of the mechanism is temporarily fixed thereby converting it into a four-bar mechanism. The original five-bar mechanism has two degrees of freedom but after conversion, it has a single degree of freedom in each stage. The dimensional synthesis of the mechanism has been carried out for a standard kinematic task. The analytical equations have been written using dyadic and triadic approaches of mechanism synthesis. Finally, a numerical problem has been solved and verified to illustrate the double stage synthesis of five-bar mechanism for a standard kinematic task.

2. Mathematical modelling for dimensional synthesis

The mathematical model for conducting the dimensional synthesis of a 2 DOF 5-bar planar kinematic linkage is explained with the help of flow chart as shown in Figure 1. It involves modelling of a planar kinematic linkage in the form of complex number followed by the generation of loop closure and standard dyad equations. The dimensional synthesis of 2 DOF 5-bar planar kinematic a linkage is considered in two stages. In each stage, standard dyad equations are generated by alternately temporarily fixing a link adjacent to permanent fixed link. The final solution is determined by solving the standard dyad equations for each stage.

The various members or links of any planar kinematic linkage can be modelled in the form of complex numbers as discussed below:

Consider a planar kinematic linkage comprising 'n' number of links each forming a revolute pair as shown in Figure 2. When the linkage transmits motion, the various links are rotated at different orientation angles.

Suppose in the kinematic linkage, the starting position of the link [X.sub.m] is [[alpha].sub.1] measured from the real axis of a fixedly oriented rectangular coordinate system. As the linkage transmits motion, the link [X.sub.m] is rotated from its home position to primed position by an orientation angle of [[beta].sub.j] as shown in Figure 3. The primed position of the link [X.sub.m] measured with respect to the fixedly oriented rectangular coordinate system is [[alpha].sub.j].

The home or first position of the mth bar in planar kinematic linkage (as shown in Figure 3) can be expressed as

[mathematical expression not reproducible] (1)

where i = [square root of -1

m = mth bar of the kinematic linkage

[X.sub.m] = |[X.sub.m]| = length between the joints of the link. [[alpha].sub.1] = arg [Z.sub.k] = angle of vector [X.sub.m] measured with respect to the real axis of a fixedly oriented rectangular coordinate system (counter-clockwise rotations are positive).

The length of the mth bar in the kinematic linkage (as shown in Figure 3) at the primed position (jth) is expressed as

[mathematical expression not reproducible] (2)

where [[beta].sub.j] = [[alpha].sub.j]-[[alpha].sub.1].

2.2. Generation of loop closure equation

The mathematical technique used for synthesising function generators is called as the loop closure equation technique. To explain how the loop closure equations are formed, consider a four-bar mechanism as shown in Figure 4 with link vectors [X.sub.m], representing the mechanism loop. The loop closure equation is generated as follows:

Now, applying the principle 'Sum of all the link vectors in a clockwise closed loop is zero'

i-e. [4.summation over (m=1)] [Z.sub.m] = 0 (3)

The equation of closure for the four-bar linkage in its first position will be

[X.sub.2] + [X.sub.3] + [X.sub.4] - [X.sub.1] = 0 (4)

Similarly, the equation of closure for the four-bar linkage in its th position will be

[mathematical expression not reproducible] (5)

Using [X.sub.m]' = [X.sub.m][e.sup.i[[alpha].sub.j]], Equation (5) can be rewritten as

[mathematical expression not reproducible] (6)

The above equation is known as a linear non-homogeneous displacement loop closure equation in complex unknowns [X.sub.1], [X.sub.2], [X.sub.3], [X.sub.4] and has complex coefficients [e.sup.i[[theta].sub.j]], [e.sup.i[[omega].sub.j]] and [e.sup.i[[lambda].sub.j]].

2.3. Generation of standard dyad loop closure equations

The standard dyad loop closure equations are formed to determine the unknown link lengths of a given linkage.

To explain how the dyad equation is generated, consider the right side of a four-bar linkage as shown in Figure 5. The home position of the links [O.sub.2]B and B[C.sub.1] is represented by X and Y, respectively. Now, as the link X is rotated counterclockwise by an angle of [[beta].sub.j], the vector Y is rotated by an angle of [[alpha].sub.j] with respect to home position. During the rotation of link X, coupler tracing point [C.sub.1] is also moved to Primed position [C.sub.j]. Now, the primed position of vector X and Y is represented by X[e.sup.i[[beta].sub.j]] and Y[e.sup.i[[alpha].sub.j]] respectively (Refer Figure 5).

Now, writing the loop closure equation for the closed loop [O.sub.2][B.sub.j][C.sub.j][C.sub.1]B[O.sub.2], we get

X[e.sup.i[[beta].sub.j]] + Y[e.sup.i[[alpha].sub.j]] - [P.sub.j] + [P.sub.1] - Y - X = 0

X([e.sup.i[[beta].sub.j]] - 1) + Y([e.sup.i[[alpha].sub.j]] - 1) = [P.sub.j] - [P.sub.1] (7)

The displacement vector along the prescribed path from [P.sub.1] to [P.sub.j] is given by

[P.sub.j] - [P.sub.1] = [[delta].sub.j] (8)

Substituting the value of [P.sub.j] - [P.sub.1] from Equation (8) to Equation (7), we get

X([e.sup.i[[beta].sub.j]] - 1) + Y([e.sup.i[[alpha].sub.j]] - 1) = [[delta].sub.j] (9)

Equation (9) is the standard dyad equation form and is used to determine the link dimensions of a given linkage depending upon the kinematic tasks to be performed, i.e. path generation, motion generation or function generation.

3. Configuration of the proposed five-bar mechanism

The mechanism consists of five binary links with a permanently fixed link [O.sub.1][O.sub.2]. The vector representation of the links of the mechanism is shown in Figure 6. Two binary links, i.e. ABC and CDE, are offset at points B and D, respectively. The points B and D are called tracing points that trace the trajectory for desired precision points.

This proposed mechanism can be operated in two different stages as shown in Figures 7 and 8. These two stages are discussed in the following paragraphs.

3.1. Stage-I

In this stage, the binary link [O.sub.2][E.sub.1] is temporarily fixed which converts the original five-bar mechanism into four-bar mechanism. This reduces the degree of freedom from two to one. The input link [O.sub.1][A.sub.1] rotates by an angle [[alpha].sub.1] while B and D are the tracing points that trace from home position [B.sub.1] and [D.sub.1] to an extreme position [B.sub.2] and [D.sub.2], respectively. The letter suffixes 1 and 2 represent the finitely separated positions of the reduced four-bar mechanism as shown in Figure 7.

3.2. Stage-II

When the reduced four-bar mechanism of stage-I reaches the position 2, the link [O.sub.2][E.sub.1] is released to move and the binary link [O.sub.1][A.sub.2] is fixed temporarily. At this instant, the mechanism switches from stage-I to stage-II. In this stage, binary link [O.sub.2][E.sub.2] becomes the input link and rotates by an angle [[phi].sub.2]; tracing points B and D trace from one extreme position [B.sub.2] and [D.sub.2] to other extreme position [B.sub.3] and [D.sub.3] respectively. The letter suffixes 2 and 3 represent the finitely separated positions of the reduced four-bar mechanism as shown in Figure 8.

4. Methodology

The problem to be solved consists of the following steps:

(i) Identify the input link, output link and the link which is to be fixed temporarily in each stage.

(ii) Recognise the mechanism type in each stage.

(iv) Identify and specify the unknown parameters, prescribed parameters and freely chosen parameters.

(v) The parameters which can be arbitrarily chosen without affecting the functionality of the mechanism are referred as freely chosen parameters. E.g. the link lengths adjacent to permanent or temporarily fixed links can be considered as freely chosen parameters. In stage-I, the parameters [Z.sub.2], [Z.sub.8] are the examples of freely chosen parameters. The free choices are also chosen so as to form the design equations system solvable and consistent. The parameters [[beta].sub.1], [[gamma].sub.1] chosen in stage-I are such examples.

(vi) Determine some of the link lengths by solving the equations written in step (iii) for stage-I.

(vii) While retaining the link lengths of stage-I, solve the equations written in stage-II.

(viii) Determine the total number of solutions.

5. Function generation

In function generation problem, the relative motion of those links is considered which are hinged to ground. For this purpose, the relation is coordinated between input and output links for two prescribed positions.

5.1. Stage-I synthesis

In function generation stage-I synthesis, the input and output crank motions ([[alpha].sub.1], [[gamma].sub.1]) and [Z.sub.7] are prescribed. [[beta].sub.1], [[gamma].sub.1], [Z.sub.2], [Z.sub.8] are free choices. Therefore, there will be [[infinity].sup.4] number of solutions. Then, the unknowns [Z.sub.4], [Z.sub.5], [Z.sub.6] are determined by generating triad equations for stage-I, Sandoor and Erdman (1984a, 1984b).

[mathematical expression not reproducible] (10)

[mathematical expression not reproducible] (11)

[mathematical expression not reproducible] (12)

[mathematical expression not reproducible] (13)

where [[delta].sub.[B.sub.1] [B.sub.2]]' and [delta].sub.[D.sub.1] [D.sub.2]]' are the displacement vectors [B.sub.1][B.sub.2] and [D.sub.1][D.sub.2], respectively. Therefore, Equations (10) and (11) are reduced to the forms of standard dyad equations as follows:

[mathematical expression not reproducible] (14)

[mathematical expression not reproducible] (15)

[mathematical expression not reproducible] (16)

[mathematical expression not reproducible] (17)

[Z.sub.6] could be determined using the following loop closure equation

[Z.sub.6] = [Z.sub.7] + [Z.sub.8] (18)

Table 1 illustrates the parameters for stage-I synthesis.

5.2. Stage-II synthesis

In function generation stage-II synthesis, the input and output crank motions ([[phi].sub.2] and [[beta].sub.2]) and [Z.sub.3] are prescribed. [[gamma].sub.2] is a free choice. Therefore, there will be [infinity] number of solutions. Then, the unknown [Z.sub.9] is determined by generating triad equation for stage-II, Sandoor and Erdman (1984a, 1984b).

[mathematical expression not reproducible] (19)

[mathematical expression not reproducible] (20)

where [[delta]'.sub.[B.sub.2][B.sub.3]] is the displacement vector [B.sub.2][B.sub.3].

Therefore, Equation (19) is reduced to the form of standard dyad equation as follows:

[mathematical expression not reproducible] (21)

[mathematical expression not reproducible] (22)

[Z.sub.1] ([O.sub.1][O.sub.2]) could be determined using the following loop closure equation

[Z.sub.1] = [O.sub.1][O.sub.2] = [Z.sub.2] + [Z.sub.5] + [Z.sub.8] - [Z.sub.9] (23)

Table 2 illustrates the parameters for stage-II synthesis.

6. Solution algorithm

The following steps are followed to determine the solutions of various equations discussed in article 5:

Step 1. Determine the displacements for stage-I and stage-II for the two given tracing points represented by [[delta].sub.[B.sub.1][B.sub.2]], [[delta].sub.[D.sub.1][D.sub.2]] and [[delta].sub.[B.sub.2][B.sub.3]], [[delta].sub.[D.sub.2][D.sub.3]], respectively.

Step 2. Consider the values for prescribed and free choice Parameters.

Step 3. Compute the value of [[delta]'.sub.[D.sub.1][D.sub.2]] using Equation (13).

Step 4. Determine the link length [Z.sub.6] using Equation (18).

Step 5. Determine the link length [Z.sub.5] using Equation (17).

Step 6. Compute the value of [[delta]'.sub.[B.sub.1][B.sub.2]] using Equation (12).

Step 7. Determine the link length [Z.sub.4] using Equation (16).

Step 8. Compute the value of [[delta]'.sub.[B.sub.2][B.sub.3]] using Equation (20).

Step 9. Determine the link length [Z.sub.9] using Equation (22).

Step 10. Determine the link length [Z.sub.1] using Equation (23).

7. Numerical example

Synthesise a planar five-bar mechanism that transmits motion between two extreme positions. The mechanism has two point displacements specified by the following two stages:

Stage-I: From point (143.0, 121.5) to the point (90.0, 220.5) and from point (172.5, 149.0) to the point (180.5, 218.0).

Stage-II: From point (90.0, 220.5) to the point (73.5, 52.5) and from point (180.5, 218.0) to the point (101.5, 64.5).

The synthesised dimensions for standard kinematic task of function generation are determined under the following subheading.

7.1. Solution

The displacements in stage-I are calculated as

[[delta].sub.[B.sub.1][B.sub.2]] = (90.0 + 220.50i) - (143.0 + 121.501) = -53.0 + 99.0i

[[delta].sub.[D.sub.1][D.sub.2]] = (180.5 + 218.0i) - (172.5 + 149.0i) = 8.0 + 69.0i

The displacements in stage-II are calculated as

[[delta].sub.[B.sub.2][B.sub.3]] = (73.5 + 52.51) - (90.0 + 220.51) = -16.5 - 168.0i

[[delta].sub.[D.sub.2][D.sub.3]] = (101.5 + 64.5i) - (180.5 + 218.0i) = -79.0 - 153.5i

The prescribed and free choice parameters in stage-I and stage-II for function generation are illustrated in Tables 3 and 4, respectively.

Now, considering CCW motion as positive and CW motion negative,

|[Z.sub.7]| = 23.7013

|[Z.sub.2]| = 71.4493

|[Z.sub.8]| = 109.6358

|[Z.sub.3]| = 90.1130

From Equation (13), [[delta]'.sub.[D.sub.1][D.sub.2]] = 5.8637 + 7.3164 i

From Equation (18), [Z.sub.6] = 83.1973 + 67.8123 i i.e. |[Z.sub.6]| = 107.3327

From Equation (17), [Z.sub.5] = -6.1880 - 16.0892 i i.e. |[Z.sub.5]| = 17.2382

From Equation (12), [[delta]'.sub.[B.sub.1][B.sub.2]] = -148.9400 + 1.1487 i

From Equation (16), [Z.sub.4] = 95.1462 + 1.0789 i i.e. |[Z.sub.4]| = 95.1616

From Equation (20), [[delta]'.sub.[B.sub.3][B.sub.2]] = 86.8674 + 53.5129 i

From Equation (22), [Z.sub.9] = 122.8900 - 77.6880 i i.e. |[Z.sub.9]| = 145.3870

From Equation (23), [Z.sub.1] = -12.0800 + 199.6000 i i.e. |[Z.sub.1]| = 199.9640

8. Result and discussion

In this paper, we summarise the state-of-the-art synthesis of mechanism by performing the dimensional synthesis of a five-bar one-degree of freedom planar kinematic linkage. The linkage transmits motion between two extreme positions in two different stages by alternately fixing the link adjacent to the permanently fixed link in each stage. The final dimensions determined for various links of the kinematic linkage based on function generation are given in Table 5. Also, the graphical representation of these dimensions is shown in Figure 9. Many researchers focused on the synthesis of mechanisms that has variable configuration but their synthesis work is confined to a single tracing point. The dimensional synthesis of a five-bar mechanism that has more than one tracing point was not dealt with. The given kinematic linkage with variable topology is capable of providing a double tracing point motion transmission. This fulfils the motion requirements in the industry. Also, the linkage can be designed to fit a particular work envelop for any given application. In such applications, consideration of extreme positions of various links is important.

9. Conclusion

The present work suggests dimensional synthesis of a five-bar mechanism that transmits motion between two extreme positions in two different stages by alternately fixing the link adjacent to the permanently fixed link in each stage. The mechanism consists of two binary links which are offset at two points called as tracing points. These tracing points trace the trajectory for desired precision points. The synthesis task employs writing and subsequently solving the dyad and triad equations formed using complex numbers. It is synthesised for multiple standard kinematic tasks of function generation. The results of a synthesis example verify the effectiveness of the proposed method. This technique is simple, non-iterative, offers reduced solution space with increased accuracy and overcomes the drawback of graphical techniques of limited accuracy. The solution obtained is consistent with multiple standard kinematic tasks of function generation and can be further extended for a combination of function generation and path generation in each stage.
```Nomenclature

DOF                     Degree of Freedom
[Z.sub.i]               Dimensional parameters of various links (i = 1,
2,... 9)
[[alpha].sub.1]         Rotation of link [O.sub.1]A from [O.sub.1]
[A.sub.1] to [O.sub.1][A.sub.2]
[[beta].sub.1]          Rotation of link ABC from [A.sub.1][B.sub.1]
[C.sub.1] to [A.sub.2][B.sub.2][C.sub.2]
[[gamma].sub.1]         Rotation of link CDE from [C.sub.2][D.sub.2]
[E.sub.2] to [C.sub.3][D.sub.3][E.sub.3]
[[beta].sub.2]          Rotation of link ABC from [A.sub.2][B.sub.2]
[C.sub.2] to [A.sub.3][B.sub.3][C.sub.3]
[[gamma].sub.2]         Rotation of link CDE from [C.sub.2][D.sub.2]
[E.sub.2] to [C.sub.3][D.sub.3][E.sub.3]
[[phi].sub.2]           Rotation of link [O.sub.2]E from [O.sub.2]
[E.sub.2] to [O.sub.2][E.sub.3]
[[delta].sub.[B.sub.1]
[B.sub.2]               Displacement of tracer point B from position
[B.sub.1] to [B.sub.2].
[[delta].sub.[D.sub.1]
[D.sub.2]               Displacement of tracer point D from position
[D.sub.1] to [D.sub.2]
[[delta].sub.[B.sub.2]
[B.sub.3]               Displacement of tracer point B from position
[B.sub.2] to [B.sub.3]
[[delta].sub.[D.sub.2]
[D.sub.3]               Displacement of tracer point D from position
[D.sub.2] to [D.sub.3]
```

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes on contributors

Khalid Nafees received his BE and MTech with Gold Medal from the Department of Mechanical Engineering, Jamia Millia Islamia, New Delhi, India. Currently, he is a research scholar at the Department of Mechanical Engineering, Jamia Millia Islamia, New Delhi. His research interests include kinematic synthesis of mechanisms, design of machine elements and machine drawing.

Aas Mohammad, PhD, is a professor at Department of Mechanical Engineering, Jamia Millia Islamia, New Delhi, India. He has 21 years of teaching experience and guided many MTech and PhD students in their research work. He has extensive publications in various reputed National and International journals and conferences. His research interests include kinematic analysis and kinematic synthesis of mechanisms, FEA (Finite Element Analysis) of mechanical components and robotic manipulators.

References

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Faculty of Engineering & Technology, Department of Mechanical Engineering, Jamia Millia Islamia, New Delhi, India

CONTACT Khalid Nafees khalidnafees@gmail.com

ARTICLE HISTORY

Accepted 19 May 2017

https://doi.org/10.1080/14484846.2017.1335456
```Table 1. Stage-I parameters for function generation.

S. No.       Description                Parameters

1       input motion              [[alpha].sub.1]
2       output motion             [[gamma].sub.1]
4       Prescribed parameters     [[alpha].sub.1], [[gamma].sub.1],
[Z.sub.7]
5       Parameters chosen freely  [[beta].sub.1], [[gamma].sub.1],
[Z.sub.2], [Z.sub.8]
6       unknown parameters        [Z.sub.4], [Z.sub.5], [Z.sub.6]

Table 2. Stage-II parameters for function generation.

S. No.       Description               Parameters

1       input motion              [[phi].sub.2]
2       output motion             [[beta].sub.2]
4       Prescribed parameters     [[phi].sub.2], [[beta].sub.2],
[Z.sub.3]
5       Parameters chosen freely  [[gamma].sub.2]
6       unknown parameters        [Z.sub.1], [Z.sub.9]

Table 3. Stage-I parameter values for function generation.

Prescribed parameters                        Free choice parameters

[[alpha].sub.1] = 5.5[degrees] (CCW)  [[beta].sub.1] = 72[degrees] (CCW)
[[gamma].sub.1] = 75[degrees] (CW)    [Z.sub.2] = 49.0000 + 52.0000 i
[Z.sub.7] = 15.1973 - 18.1877 i       [Z.sub.8] = 68.0000 + 86.0000 i

Table 4. Stage-II parameter values for function generation.

Prescribed parameters                   Free choice parameters

[[phi].sub.2] = 87[degrees] (CCW)  [[gamma].sub.2] = 39[degrees] (CCW)
[[beta].sub.2] =152[degrees] (CW)
[Z.sub.3] = 88.9582 - 14.3803 i

Table 5. Final dimensions of each link of a given five-bar mechanism.

|[Z.sub.1]| = 199.9640        [angle][Z.sub.1] = 93.46[degrees]
|[Z.sub.2]| = 71.4493         [angle][Z.sub.2] = 46.70[degrees]
|[Z.sub.3]| = 90.1130         [angle][Z.sub.3] = -9.18[degrees]
|[Z.sub.4]| = 95.1616         [angle][Z.sub.4] = 0.65[degrees]
|[Z.sub.5]| = 17.2382         [angle][Z.sub.5] = -111.04[degrees]
|[Z.sub.6]| = 107.3327        [angle][Z.sub.6] = 39.18[degrees]
|[Z.sub.7]| = 23.7013         [angle][Z.sub.7] = -50.12[degrees]
|[Z.sub.8]| = 109.6358        [angle][Z.sub.8] = 51.67[degrees]
|[Z.sub.9]| = 145.3870        [angle][Z.sub.9] = -32.30[degrees]
```
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