Kinematical and dynamical analysis of closed kinematic chains using graphs and contour equations

Abstract

The classical method of determination of velocities and accelerations of the closed kinematical chains consists in calculation of derivatives of the position vector in relation to time. In the paper, the methods of kinematical and dynamical analysis of the closed kinematical systems based upon the graph representation are presented. Planetary gears are the exemplary mechanical systems which have been considered. Two types of graphs are assigned to the gears. Both graphs allow for derivation of systems of equations which constitute the kinematical analysis. The results of classical and graph-based methods were compared. The graph based methods allow for further considerations like e.g. synthesis, enumeration of design solutions and even evolutionary design of mechanisms. Dynamical analysis is shown in graphical form, only.

Analysis of a planetary gear modelled with a contour graph considering the decision making complexity of game-tree structures

“…The mobility W (DOF, degree of freedom) for the considered mechanism, i.e. planetary gear, can be calculated from the following formula [5,6,7,19] (5):…”

“…Based upon the distinguished contours, the following system of equations may be developed [5,6,7,19] …”

“…In the second case, some summands are omitted when the arm length is equal to zero. The following relations are used for simplification of the system [5][6][7]19].…”

“…For other mechanisms enclosing cranks, pistons, cylinders, sliders, followers, etc, a detailed analysis of angles has to be carried out [5,6,7,19,20] to correctly simplify the system of equations. Assuming the cylindrical gear wheels, we have the relationships: 2 2…”

“…There have also been other attempts to model planetary gears via diagrams e.g. Wolf's pictograms [5], but those methods did not further evolve due to lack of generalization and lack of connections with other branches of mathematics [6,7]. Furthermore, a method based on the signal flow graph theory for modelling gears [8,9] has not been too frequently used.…”