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CALCULATED AND EXPERIMENTAL RESULTS YIELDED BY

THE GENERATION OF TRAJECTORIES ALONG A MOBILE PLAN, AT A MECHANISM WITH GEARS

Professor Iulian Popescu

University of Craiova, member of the Academy of Technical Sciences in Romania

Associate Professor PhD Ludmila Sass

University of Craiova, Faculty of Mechanics Abstract. Experimental and calculated results yielded by the generation of trajectories along mobile plans (relative trajectories) corresponding to a mechanism consisting of a train of gears are presented. Two cases are analyzed: when the tracer point belongs to one gearwheel whilst the trajectory belongs to the other one and vice versa. Other cases are also approached. The calculated results are compared to those obtained from experiments. Keyword: relative trajectories, coupler curves along mobile plans 1. Introduction In many papers papers published along time, many researchers realized experimentaly plots generated by couples from various mechanisms in order to validate their theoretical researches. Many papers deal with trajectories described by various points belonging to different mechanisms. All of them refer to fixed plans. For example [7] is dedicated to the study of the influence of clearances from kinematic couples over the trajectories for a mechanism with stepping. Finally an optimization problem is approached, considering the minimization of errors. In [2] the movements from the solar system are considered as starting point. Afterward, mathematic procedures used for the generation of geometrical curves, based on the relative movement are considered.

In [1], using the multi-body dynamic, the optimum trajectory of the characteristic point from a R-R- robot is studied. Experimental tests were used to validate the algorithm. In [6] and [3] details were presented with respect to the results of plotting couple curves along mobile plans. Modalities to obtain cycloidal curves along mobile plans were reported by [4]. A paper from the same area of interest [6] is dedicated to a theoretical study on the generation of trajectories along a mobile plan. They correspond to a train of gears. This paper comes to continue the paper referred above. It describes experimental results and validations of theoretical nature for trajectories along mobile plans.

2. Studied mechanism Figs. 1 and 2 present the kinematic schematic and a picture of the mechanism

used for experiments.

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Fig. 1.

Fig. 2. 3. Results obtained when the trajectory is placed on the wheel z6 Fig. 3.a depicts the curve generated by computer to simulate the trajectory plotted by the wheel 6 from Fig. 1 if the point E belongs to the wheel z2,

(z1=64, z2=72, z3=40, z4=36, z5=96, z6=96, b=44). The experimental curve for the same data is represented by Fig. 4.

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Fig. 3.a is obtained for a full rotation of the crank. For the second rotation the

curve is not repeated. Instead it is continued as depicted by Fig. 3.b.

Fig. 3.a.

Fig. 3.b.

Fig. 4.

The theoretical curve has two arcs in extension. The rest of it is fitting almost perfect to the experimental curve (the errors are generated by the procedure of drawing with a pencil).

4. Other toothed wheels Other curves were obtained for different toothed wheels and experimental tests for their validation were also accomplished. Curves were drawn for two full rotations of the crank. The analyzed cases are given in: - fig. 5: (z1=64, z2=72, z3=40, z4=56, z5=96, z6=96, b=25;

- fig. 6: (z1=64, z2=72, z3=40, z4=28, z5=96, z6=96, b=25); - fig. 7: (z1=96, z2=96, z3=40, z4=28, z5=64, z6=72, b=50); - fig. 8: (z1=96, z2=96, z3=36, z4=40, z5=64, z6=36, b=25).

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Fig. 5.

Fig. 6.

Fig. 7.

Fig. 8.

The curves are not necessarily remarkable for their esthetic features, but

present interest because are drawn on mobile plans.

5. Results obtained when the trajectory appears on the wheel z2

When the tracer point is placed on the wheel z6, and the trajectory is obtained on the wheel z2 (as in Fig. 1), other curves will be obtained, because the speed of the wheel on which plots are made is different from that described above. The images corresponding to three full rotations of the crank are depicted by: - fig. 9: (z1=64, z2=72, z3=64, z4=24, z5=48, z6=96, b=40); - fig. 10: (z1=64, z2=72, z3=64, z4=24, z5=96, z6=96, b=50);

- fig. 11: (z1=96, z2=96, z3=36, z4=40, z5=64, z6=36, b=25); - fig. 12: (z1=64, z2=36, z3=40, z4=36, z5=96, z6=96, b=25); - fig. 13: (z1=96, z2=96, z3=40, z4 =28, z5=64, z6=72, b=50); - fig. 14: (z1=96, z2=96, z3=36, z4=40, z5=64, z6=36, b=25); - fig. 15: (z1=64, z2=72, z3=40, z4=56, z5=96, z6=96, b=25).

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Fig. 9.

Fig. 10.

Fig. 11.

Fig. 12.

Fig. 13.

Fig. 14.

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Some curves are closed, others remain open. The factor to influence this consists in the number of teeth. Differences of this parameter were noticed because different speeds were used.

Fig. 15.

6. Conclusions - A train of gears was designed and implemented in order to perform experiments related to plots of trajectories on mobile plans;

- The experimental results were in good agreement with the theoretical considerations; - The method was extended (in other papers) to the planar movement, for various types of mechanisms with bars.

References [1] Boscariol, P., Gasparetto, A. – Model-based trajectory planning for flexible-link mechanisms with bounded jerk. Journal Robotics and Computer-Integrated Manufacturing, Volume 29 Issue 4, august, 2013, pp. 90-99. [2] Guan-Ze Liao, Chun-Wang Sun – Geometric patterns design with recursive pursuit relative motions. www.mi. sanu.ac.rs/vismath/liao/index.html. [3] Popescu, Iulian, Sass, L. – Mecanisme generatoare de curbe. Editura „Scrisul Românesc”, Craiova, 2001 [4] Popescu, Iulian, Sass, L. - Recherches experimentales concernant l'engendrement des courbes epicicloydales, sur des plans mobiles. Univ. "Politehnica" Timişoara, Analele Fac. de Inginerie din Hunedoara, Anul 2, Fascicola 4, pg. 23- 26, 2000

[5] Popescu, Iulian, Sass, L. – Generarea traiectoriilor pe un plan mobil, la un mecanism cu angrenaje. Annals of Constantin Brancusi of Targu-Jiu, Engineering series, National Scientific Conference with international participation “CONFERENG 2014“, Tg-Jiu, November 14-15 2014 [6] Sass, L. – Contributii la analiza si sinteza mecanismelor generatoare de curbe. Teza de doctorat, Universitatea din Craiova, 2000 [7] Selçuk Erikaya – Trajectory optimization of a walking mechanism having revolute joints with clearance using ANFIS approache. Nonlinear Dinamics, january 2013, Volume 71, Issue 1-1, pp. 75-91.