Keywords

8.1 Introduction

The purpose of synthesis of mechanisms is to achieve motion transformation from input to output of such mechanisms, which can generate specifically desired motions at certain positions with specified postures [1]. The main outputs of the mechanisms include function, path, and motion and so on.

The approximate mechanism synthesis refers to that if the mechanism can produce accurate positions and postures at some positions, but which can produce approximate positions and postures at other positions, and the differences between actual and provided positions and postures must be within an allowable limit. The geometric method and analytic method are two basic methods commonly used [2]. The geometric method is simple and visualised, but inaccurate, less repeatable, and provides only a few numbers of synthesis positions. The analytic method is accurate and repeatable, but complex and not visualised [3,4,5]. Thus the CAD geometric approach was developed for mechanism synthesis. This method is very useful and powerful to research all kinds of mechanisms for researchers.

In this study, the CAD geometric approach is developed and utilised for conducting the approximate position and motion syntheses for Stephenson six-bar mechanism via using the CAD geometric approach as a showcase. Firstly using the geometric constraint and dimension driving techniques, a primary simulated mechanism is constructed. Then based on different tasks of path and motion generations for the dimensional synthesis, the simulated mechanisms of the Stephenson six-bar mechanism are developed from the primary simulated model. The computer model and simulations are further used for the path synthesis and motion synthesis to extract approximated solutions, which are further validated by comparing with those accurate solutions availiable.

8.2 Simulation of Basic Stephenson Six-Bar Mechanism

The Stephenson six-bar mechanism is shown in Fig. 8.1a, which is used to construct a mechanism synthesis at the k prescribed position via using the CAD geometric approach. When the mechanism is at the ith position, two triangles Δ 1 and Δ2, and four lines O1O2, Lbi, Ldi and Lei constitute the mechanism. Line O1O2 is fixed, and line Lei is a floating bar. In order to produce an approximate orientation of the bar Lei, the two lines Mi and Ni are constructed using the suitable geometric constraint and dimension driving. In order to produce an approximate trajectory positions of the bar Lei, the line Si is constructed using the suitable geometric constraint and dimension driving. The multiple approximate positions and postures of bar Lei are then produced.

Fig. 8.1
figure 1

Stephenson six-bar mechanism and its basic simulation

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First, the basic mechanism at the kth position is constructed using the geometric constraint and dimension driving, before constructing the Stephenson six-bar simulation mechanism, as shown in Fig. 8.1b. The construction process is described as follows:

  1. 1.

    First, construct one line L a1 and a dimension D a1, then the same k lines are moved and copied using ‘Move’ and ‘Copy’ commands, where k is the prescribed number of positions. The dimensions Dai are used as driving dimensions, setting Dai = Da1 (i = 2, 3…k), and the dimensions Dai of the line Lai (i = 2, 3…k) are changed as driven dimensions using the constraint function of the dimension equation. In order to simplify the view, all the driven dimensions Dai (i = 2, 3…k) are hided using the ‘Hide’ command. All the driven dimensions Dai (i = 2, 3…k) change accordingly when modifying the driving dimension Da1.

  2. 2.

    Second, construct two circles C a and C b with the radii R a and R b, respectively. The two centres of circles are connected to the O 1 using a geometric constraint command. The two ends of the set of lines are respectively connected to the point a i on the Ca and the point bi on Cb (i = 2, 3…k) using coincide command. Thus three sets of lines Ra, R b and Lai can made up k identical equivalent triangle Δ1i (i = 1, 2, … k), and the triangle Δ1i can rotate with O1.

  3. 3.

    Repeat Steps 1 and 2, construct a line Lci, two circles Cc and Cd, and respectively dimensioned Dci, Rc and Rd, O2 as the center of the circle. Thus three sets of lines Rc, Rd and Lci can made up k identical equivalent triangle Δ2i (i = 1, 2, … k), and the triangle Δ2i can rotate with respect to O2.

  4. 4.

    Repeat Step 1, construct three sets of lines Lbi, Ldi and Lei (i = 1, 2, 3 … k).

  5. 5.

    Lastly, the two lines Lei and Ldi are connected to the point ei at the ith position using the ‘Coincide’ command. The free ends of Lei and Ldi are connected to ai on the Ca and the point di on Cd using the ‘Coincide’ command, respectively. The two ends of the Lbi are connected to bi on the Cb and the point ci on Cc.

8.3 Path Synthesis of the Simulated Stephenson Six-Bar Mechanism

The one point at the floating bar should move along the prescribed path relative to the base following the rule of the path synthesis. According to the requirements of the path synthesis, the six-bar mechanism of the approximate path synthesis is constructed on the basis of the basic simulation mechanism, as shown in Fig. 8.2. The input function is determined by the accurate angle Φi of the crank (the triangle Δ1). The position of the output path is determined by the approximate position of the point ei in the floating bar Lei. At the ith position in Fig. 8.1, the end point pi of the line Si is as the accurate position point of the output path, the other point ei of the line Si is as the approximate position point of the output path. Obviously, the point ei is restricted the arc Cs of the radius Si, so any point on the Cs may be the approximate position of the bar Lei. Thus not only the number of the mechanism constraint is reduced, but also the number of the prescribed synthesis position is added. The path synthesis of the Stephenson six-bar mechanism can be achieved at more positions. When the length of the radius Si is gradually reduced to the allowable limit, the ideal simulation mechanism of approximate path synthesis can be obtained, as shown in Fig. 8.2b.

Fig. 8.2
figure 2

Path synthesis of the Stephenson six-bar simulated mechanism

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The construction process is described as follows:

  1. 1.

    Construct one datum line B, the two ends B1 and B2 of the line B are fixed, as shown in Fig. 8.1a.

  2. 2.

    Construct k lines Pi (i = 1, 2, 3 … k), the end of the line Pi is connected to point B2, as shown in Fig. 8.1b.

  3. 3.

    Construct k lines Rai (i = 1, 2, 3 … k), the two ends of line Rai are connected to the points O1 and ai. The angle of the lines Ra1 and Rai+1 (i = 1, 2, 3 … k) is as input function, and label prescribed value.

  4. 4.

    Construct k lines Si and dimension Dsi, the two ends of the line Si are connected to the other end of the line Pi and point ei of the bar Lei.

  5. 5.

    Gradually reduce the dimension Ds1 until Ds1 = 1 mm, the point pi and the point ei are almost coincident.

  6. 6.

    Label the prescribed value for angle θi and length of the line Pi. The maximum difference between the accurate k position and the actual position of the floating bar is defined by the dimension Ds1.

The results can be obtained from the approximate path synthesis of the six-bar simulation mechanism as follows:

  • When the dimension of line S1 reduced to zero (Ds1 = 0), the approximate path synthesis and the accurate path synthesis of the simulated mechanism are the same, as shown in Fig. 8.2a. When the angle Φi is given with a prescribed value of the input function, and the angle θi and the length of the line Pi are given with the prescribed value for the output path, the maximum number of positions is found as 6 in the path synthesis of the simulated mechanism.

  • When the length of the line S1 is reduced to 1 mm (Ds1 = 1 mm), the actual path position of the floating bar is close to accurate prescribed position, their difference is within the limit. Thus the simulated mechanism can achieve the approximate synthesis of the path position and orientation. Now the number k of the floating bar can be increased to 11, all the angle Φi, the angle θi and the length of the lines Pi can be given with initial values of driving dimensions. This shows that the maximum number of prescribed positions can be increased to 11, which are significantly more than 6 of the prescribed positions of the accurate path synthesis of the simulated mechanism. These results show that the six-bar mechanism can complete a path synthesis with more prescribed positions.

8.4 Motion Synthesis of the Simulated Stephenson Six-Bar Mechanism

According to the requirements of the motion synthesis, two kinds of motion syntheses of the six-bar mechanism are constructed for the basic simulated mechanism, as shown in Figs. 8.3 and 8.4, respectively.

Fig. 8.3
figure 3

Motion synthesis of the first Stephenson six-bar simulated mechanism

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Fig. 8.4
figure 4

Motion synthesis of the second Stephenson six-bar simulated mechanism

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In the first motion synthesis of the simulated mechanism, the azimuth angle of the floating bar Lei is given with the prescribed accurate value at the k prescribed position, and the path position of the floating bar is given with an approximate value of the allowable limit. The constructed method of the first motion synthesis of the simulated mechanism is similar to the approximate path synthesis for the simulated mechanism. The prescribed angle αi (i = 1, 2, 3 … k−1) between the line B and the bar Lei will be replaced by using accurate input angles Φi, as shown in Fig. 8.3b.

The result can be obtained from the approximate motion synthesis of the simulated six-bar mechanism as follows:

  1. 1.

    When the dimension of the line S1 is reduced to zero (Ds1 = 0), the actual path position of the floating bar Lei is the same as the accurate path position at the k prescribed positions, as shown in Fig. 8.3a. The maximum number of positions is 4 in the motion synthesis of the simulated mechanism.

  2. 2.

    When the dimension of the line S1 is reduced to 1 mm (Ds1 = 1 mm), the actual path position of the floating bar Lei is very close to the accurate path position at k prescribed position, as shown in Fig. 8.3b. Using the first six-bar simulation mechanism of approximate motion synthesis, which shows that the maximum number of prescribed positions can be increased to 11, which is significantly more than 4 of the prescribed positions of the accurate motion synthesis of the simulated mechanism.

In the second motion synthesis of the simulated mechanism, the azimuth angle of the floating bar Lei is given prescribed approximate value at k prescribed positions, and the path position of the floating bar is given with an accurate value of the allowable limit. When the dimension of line S1 reduced to zero (Ds1 = 0), the floating bar Lei can obtain accurate position. In order to obtain approximate azimuth angle, the two sets of lines Mi and Ni (i = 1, 2, 3 … k−1) are increased at the basic simulation mechanism in Fig. 8.1, as shown in Fig. 8.4.

The construction process is described as follows:

The first end of the line Mi is connected to the point ei of the floating bar Lei, and the second end of the line Mi is connected to the first end of line Ni. The second end of the line Ni is connected to the point qi of the floating bar Lei (i = 1, 2, 3 … k−1), which is equivalent to a sliding pair of planar mechanism. Because the geometric constraints can make point qi sliding in line Lei, the line Mi can rotate with point ei. When the azimuth angle ψi between line B and line Mi is given prescribed accurate value, the actual approximate azimuth angle Φi (i = 1, 2, 3 … k−1) of the bar Lei can be obtained between line B and line Lei, as shown in Fig. 8.4. The results show that it not only reduces the constraint number of the second motion synthesis simulation mechanism, but also increases prescribed position number.

When the dimension of line Ni (i = 1, 2, 3 … k−1) are gradually reduced to small enough, which can construct the motion synthesis of the second Stephenson simulated mechanism as shown in Fig. 8.4. The maximum difference Δψi (i  = 1, 2, 3 … k) between the angle ψi and angle φi can be obtained using the equation Δψ = |φi−ψi| = Ni/Mi at k prescribed positions. The motion synthesis of the second Stephenson simulated mechanism is under ideal conditions, so long as the Δψ is less than the allowable value.

The result can be obtained from the approximate motion synthesis of the second six-bar simulated mechanism as follows:

When the dimension of line S1 is reduced to zero (Ds1 = 0), and the dimension of line N1 is reduced to small enough, and the dimension of the line M1 is given with an equal value (700 mm) with the floating bar, the actual path position of the floating bar is the same as its exact position, as shown in Fig. 8.4. Based on the approximate motion synthesis of the second six-bar simulated mechanism, the maximum number of the prescribed positions can be increased to 11, which is significantly more than 4 of the prescribed positions of the accurate motion synthesis of the simulated mechanism, as shown in Fig. 8.4.

8.5 Conclusion

The path and motion simulation mechanism have been constructed by using the CAD geometric approach. The six-bar approximate simulation mechanism obtained by using this approach can be used for path and motion syntheses at 11 prescribed positions, which are more than the number of the accurate simulation mechanisms. The results show that the CAD geometric approach is equivalent to the geometric method and the analytic method but the CAD geometric approach is more cost-effective, visualised, accurate and repeatable. Therefore this method can be not only used in simple mechanism, but also can be used to analyse complex mechanisms.