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POSITION ANALYSIS OF A 5th CLASS MECHANISM WITH THREE

PRISMATIC JOINTS

IulianPopescu, Faculty of Mechanical Engineering, Craiova, ROMANIA Sevasti Mitsi , Mechanical Engineering Department, Aristotle University of

Thessaloniki, GREECE Mirela Cherciu , Faculty of Mechanical Engineering, Craiova, ROMANIA

ABSTRACT : It is studied the positional analysis of a mechanism with one degree of freedom, whose structure includes a pentad with five revolute and four prismatic joints. The solving was made by changing the driving element, leading to a mechanism whose structure includes three dyads. Diagrams and successive positions of the studied mechanism are given. It was found that the mechanism has breaks in operation due to the lengths and angles initially adopted. KEY WORDS : positional analysis, pentad, mechanism, the method of changing the driving element 1. INTRODUCTION Over time the kinematic analysis of Assur groups was studied in various works. There were some problems in solving the nonlinear algebraic systems, during the positional analysis of Assur groups with the number of elements greater than 2. We used different methods, developing algorithms that reached at useful solutions. Thus, for different versions of triads, Mitsi and others [1] were able to reach solutions by eliminating the unknowns sequentially, ultimately resulting in a high degree equation, solved by numerical methods. In other papers [2,3] Assur IVth class groups have been solved by the same method. Other kinematic groups were studied [4,5], with different methods. The kinematic analysis of kinematic groups with large number of elements has been studied in [6,7]. Kinematic analysis of the pentad with revolute joints was achieved in [8] through distances method. Matrix equations were established for velocities and accelerations without solving the positions.

Pentad is a Vth class Assur group that has six elements and nine revolute and / or prismatic joints. It has a closed contour formed by five sides, three ternary elements and three binary elements, existing three free joints that links into mechanism. Depending on the number of revolute and prismatic joints of it's structure, it can be obtained different aspects of the pentad. Thus, in fig. 1 it is shown a pentad only with revolute joints, where the free joints are the ones labeled B, E and H, indicating also the components of velocities for the input joints.

Figure 1. Pentad with revolute joints

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A pentad with five revolute and four prismatic joints is shown in fig. 2. The elements forming the closed contour are the elements 6, 5, 4, 3, 7 and the free joints are labeled K, F and B. Figure 2. Pentad with four prismatic joints

Below it is studied the positional analysis of a mechanism that has a pentad with six revolute and three prismatic joints in it's structure. 2. THE MECHANISM ANALYZED The kinematic scheme of the mechanism is shown in fig. 3. The driving element is the element 1 and the pentad binds through it's free joint B to the driving element and through the free joints K and F, at the base. In relation to the chosen cartesian coordinate system xy, there are given: the position of the driving element (angle φ), the coordinates of the revolute joints A, K and F. Also there are known the length and the geometry of each element. In relation to the chosen axis system (fig. 3) there can be written the following equations:

Figure 3. Mechanism with one degree of

freedom and a pentad with three prismatic joints

The system of equations (1) is a nonlinear algebraic system with six equations and six unknowns (angles α, β, ψ and displacements S 4, S 5, S 7 ). Determining all the solutions require special and complicated methods for solving. In order to avoid solving the system of equations, it is proposed the method of changing the driving element , presented in the next chapter. 3. DETERMINING THE POSITIONS USING THE METHOD OF CHANGING THE DRIVING ELEMENT If , for the mechanism in fig. 3, is changed the driving element from 1 to

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element 6, it is obtained a three dyads mechanism (fig. 4), with the following structure : R-RPR-PRP-RRR. By changing the driving element, the positional analysis of this mechanism is reduced to solving successively the three systems of two nonlinear equations with two unknowns, corresponding to RPR, PRP and RRR dyads.

Figure 4. The mechanism R-RPR-PRP-RRR

Considering the angle β known, one can write the equations for the kinematic chain KLHGF that includes the dyad RPR, as follows:

(2)

of which it determines the displacement S 5 and the angle . Let h 1 and h 2 sizes known, leading to a trigonometric equation that gives .

(3)

There are two solutions, selecting the nearest previous position, starting from the initial drawing of the mechanism. For KLDE chain that includes dyad PRP, there can be written the following equations:

(4)

By solving the system of equations (4), the displacements S 4 and S 7 are determined:

(5)

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0.0 100. 200 . 300 . 40 0.

B et a [gr d]

2 0.

4 0.

6 0.

8 0.

10 0.

12 0.

14 0.

S5 [m

m]

0.0 100. 200. 300. 400.

B et a [ grd]

0.0

20.

40.

60.

80.

Psi [g

rd]

0.0 100. 200. 300. 400.

B et a [grd]

70.

80.

90.

100.

110.

X EY E

In this case there are also two solutions, processed as above. For the chain CBA, common relations of the dyad RRR are to be used [8], resulting the angles and , existing two solutions here too. In total, for one position of the element 6 (angle ) eight solutions for the angles and are given. 4 RESULTS The method described above is applied to the mechanism (fig. 3) whose data are: XK = 24; YK = 41; XA = 119; YA = 10; XF = 70; YF = 122; KL = 38; LH = 34; FE = 33; DC = 35; BC = 42; AB = 30; C 1 = 108; C 2 = 346; C 3 = 23; C 5 = 75; C 6 = 284 (Ci - degrees). With an original program, based on the relationship of the three dyads, the results have been obtained for each dyad when the angle β ranges from 0 to 360 degrees. The positional analysis of the kinematic chain KLHGF gives the resulting variation diagrams for the displacement S 5 (fig. 5) and the angle (fig. 6) reported to the angle β .On this basis we determined the coordinates of point E (fig. 7), finding the movement in a full cycle (0 ... 360), being continuous.

Figure 5. Graph of variation of displacement S 5

Figure 6. Diagram of variation of the angle

Figure 7. Graph of variation of the coordinates of point E

The position of the kinematic

chain KLHGF for the angle 10 degrees, obtained by calculation

and checked by the graphical method is given in fig. 8. The successive positions of the chain for a complete rotation of the element 6 are given in fig. 9.

Figure 8. Position of the kinematic chain KLHGF for angle 10 degrees

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0.0 100. 200. 300. 400.

Bet [grd]

-1000.

-500.0

0.0

500.

1000.

S4S7

0 .0 100 . 200 . 300 . 400.

B et [ g rd ]

-500 .0

-250 .0

0 .0

250 .

500 .

750 .

X DY D

Figure 9. Successive positions of the kinematic chain KLHGF

From the analysis of the kinematic chain LDE positions, the displacements S 4, S 7 and the coordinates of the revolute joint D were determined. The diagrams of variation of these parameters reported to the angle β are given in figures 10 and 11, where there is discontinuity in two areas . These areas are areas of blockage, the mechanism does not work (straight lines in diagrams) because of the scale adopted initial (zero appears in the denominator relations in these positions). The values of Y D, were limited, otherwise the curves tended to infinity.

Figure 10. Diagram of variation of displacements S 4 and S 7

Figure 11. Diagram of variation of D point coordinates

The position of LDE kinematic chain for angle 10 degrees, obtained by calculation and checked by the graphical method is given in fig. 12 and the successive positions of the chain for a complete rotation of the element 6 are given in fig. 13.

Figure 12. The position of LDE kinematic chain for angle 10 degrees

Figure 13. Successive positions of LDE kinematic chain

From the positional analysis of the kinematic chain DCBA were determined the angles α and φ and the coordinates of the revolute joint B. The angle φ (fig. 14) and the coordinates of point B (fig. 15) variation diagrams show that the mechanism does not work for all cinematic cycle, existing subintervals in which movement of the driving element is blocked by the next dyads. On charts, the blocking areas are those in which appear straight lines on the curves (the program join the points where breaks occur in operation). In fig.14 the areas where element AB can rotate and is stationary

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0.0 1 00. 20 0. 30 0. 400 .

B et [grd]

- 100.0

- 50.00

0.0

50.

100.

Fi [gr

d]

0.0 100. 200. 300. 400.

B eta [ grd]

-50.00

0.0

50.

100.

150.

X BY B

(straight lines in the diagram) are observed.

Figure 14. Diagram of variation of the angle φ

Figure 15. Diagram of variation of coordinates of Β point

The position of DCBA kinematic

chain for angle 10 degrees, obtained by calculation and checked by the graphical method is given in fig. 16 and the successive positions of the chain for a complete rotation of the element 6 are given in fig. 17.

Figure 16. The position of DCBA

kinematic chain for angle 10 degrees

The trajectories of the points B and C are given in fig. 18 : arcs for point B and a crank open curve for point D. The arcs show the subintervals in witch the mechanism works.

Figure 18. Trajectories of points B and C Successive positions of the whole mechanism are given in fig. 19. It appears circular arcs described by points E, L, B, so emphasizing the subintervals that mechanism works.

Figure 19. Successive positions of the mechanism

5. CONCLUSIONS In this paper, using the contours method, there were developed the equations for analyzing the positions of a Vth class planar mechanism, with three prismatic joints. The system of equations obtained is a nonlinear algebraic system, with six equations and six unknowns , difficult to solve. To avoid these problems, the positions analysis is carried

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out by the method of changing the driving element, obtaining a three dyads mechanism. Thus, positional analysis of this mechanism is reduced to solving successively three systems, each of two nonlinear equations with two unknowns, corresponding to the dyads RPR, PRP and RRR, easier to solve. For the studied mechanism it was found that, as they move from one dyad to another, the movement is no longer continuous, but occur subintervals the mechanism jams. REFERENCES [1] Mitsi, S., Bouzakis, K. D., Mansour, G., Popescu, I., Position Analysis in Polynomial Form of Planar Mechanisms with Assur Groups of Class 3 Including Revolute and Prismatic Joints, Mechanism and Machine Theory, vol. 38, no. 12, pp. 1325 – 1344, 2003 [2] Mitsi, S., Position Analysis in Polynomial Form of Planar Mechanisms with a Closed Chain of the Assur Group of Class 4, Mechanism and Machine Theory, vol. 34, no. 8, pp. 1195-1209, 1999

[3] Mitsi, S., Bouzakis, K. D., Mansour, G., Position Analysis in Polynomial Form of Planar Mechanisms with a Assur Group of Class 4 including one prismatic joint, Mechanism and Machine Theory, vol. 39, pp. 237-245, 2004 [4] Kong, X., Gosselin, C.M., Forward displacement analysis of class-three analytic 3-RPR, planar parallel manipulators, Mechanism and Machine Theory, vol. 36, pp. 1009-1018, 2001 [5] Chung, W. Y., The position analysis of Assur kinematic chain with five links, Mechanism and Machine Theory, vol. 40, pp. 1015-1029, 2005 [6] Innocenti, C., Position Analysis in Analytical Form of the 7-Link Assur Kinematic Chain Featuring one Ternary Link Connected to Ternary Links only, Mechanism and Machine Theory, vol.32, no. 4, pp. 501-509, 1997 [7] Han, L., Liao, Q., Liang, Ch., Closed-form displacement analysis for a nine-link Barranov truss or a eight-link Assur group, Mechanism and Machine Theory, vol. 35, pp. 379-390, 1991 [8] Popescu, I., Mechanisms. Matrix Analysis. University of Craiova, 1977