The “paradoxical” mechanism of P. L. Chebyshev

The kinematics and dynamics of P. L. Chebyshev's “paradoxical” mechanism are considered. The point of interest in the dynamics of the “paradoxical” mechanism is connected with the fact that its configuration space contains six singular points. These points are successively passed through a full turn of the handle. Holonomic constraints which are imposed on the system become linearly dependent at singular points. Thus it is impossible to apply the standard methods of derivation of the motion equations at singular points. The properties of the “paradoxical” mechanism are based on the properties of the lambda mechanism. P. L. Chebyshev designed many mechanisms for particular types of motion by using the construction of the lambda mechanism. For example, it is possible to obtain an anti-rotation mechanism, a mechanism with two swings per revolution of the handle, or a mechanism with a stop of the driven link with certain parameters of the lambda mechanism. The trajectory of the vertex of the lambda mechanism in the “paradoxical” mechanism is located between two circles and touches each circle at three points. Therefore singular points arise in the configuration space. It is shown in the article that the configuration space consists of two “curves” that intersect at a nonzero angle in the neighbourhood of a singular point. In order to get a numerical and analytical model of the “paradoxical” mechanism, the main formulas from the works of P. L. Chebyshev are given. “Paradoxical” mechanism is represented as a combination of a lambda-mechanism and a singular pendulum, which motions are limited by two holonomic constraints. The equations of motion are written out and the reaction forces are found. It is shown that with a small increase in the length of the double pendulum rod the configuration space splits into two non-intersecting curves. The smaller the link perturbation becomes, the larger the Lagrange multipliers around singular configurations become.