Dynamics of mechanical systems with nonlinear nonholonomic constraints – I The history of solving the problem of a material realization of a nonlinear nonholonomic constraint

Abstract

The paper brings forth a detailed analysis of the solution of the problem of the material realization of a nonlinear nonholonomic constraint (NNC). The existing models of the NNC are shown that can be classified into two groups: the first group comprises correctly realized physical models, while the second group contains the so‐called “quasinonlinear” nonholonomic constraints which in fact represent mathematical models. The correctness of the cited models is considered in detail, and the essential nature of such constraints, the basic of which is holonomic, is shown. The second group of models, i.e., the “quasinonlinear” NC (nonholonomic constraints) in fact represents the given program of motion, while the additional force, which carries out the program, has the analytical form of the reaction of the NNC. That is why are presented the models of the NNC which possess a clear physical sense, on the basis of which certain statements on the method of variation and the reaction of the NNC can be given. With regard to the clear physical sense and the nature of the models cited, the NNC that come out of them are used quite normally in the analysis of motion of such a system. The cited models, together with standard models oh nonholonomic Mechanics (sphere, disk, blade) make a group of basic nonholonomic constraints which can be classified, according to the three criteria, into certain types. Finally, it is shown that the cited model can be used for the construction of “nonholonomic chains”, both open and closed ones.

Analysis the brachistochronic motion of a mechanical system with nonlinear nonholonomic constraint

“…The axis of the non-stationary coordinate system A ξ is defined by the direction AB , that is, B Aξ  , whereas unit vectors of the non-stationary coordinate system axes are , λ μ   and ν  , respectively. The system is composed of two Chaplygin blades [3], of negligible masses and dimensions, which impose constraint to the motion of particles A and B, of equal masses m , in the form of perpendicularity of the velocities, as shown in Fig. 1.…”

“…q ,q ,q ,q , where 1 q x  and 2 q y  are Cartesian coordinates of the point A, 3 q φ  is the angle between the axis Ox and axis A ξ , while 4 q ξ  is the relative coordinate of point B relative to the nonstationary coordinate system. Further analysis relates to the case when the motion of point A is constrained in the Aξ axis direction, while the motion of point B is constrained in the Aη axis direction, that is, lateral slipping of the points A and B of the system is not permissible in the Aξ and Aη axis directions respectively.…”

“…respectively. In accordance with the constrained motion in the form of perpendicularity of the velocities of points A and B, the nonholonomic nonlinear homogeneous constraint has the form [3], [4]…”

“…Now, based on nonholonomic constraints (1) and (2), taking into account the relations (3) and 4, the angular velocity of the system is determined in the form…”

Analysis of the brachistochronic motion of a variable mass nonholonomic mechanical system

“…A nonholonomic mechanical system [3] is composed of two variable mass particles, A and B, whose motion is constrained by the imposition of perpendicularity of the velocities by means of the Chaplygin blades of negligible masses, as shown in Figure 1a. In order to develop the differential equations of motion of a variable mass nonholonomic mechanical system (henceforth referred to as 'the system'), as well as for the needs of further considerations, first, two Cartesian reference coordinate systems must be introduced: the stationary coordinate system Oxyz, whose coordinate plane Oxy coincides with the horizontal plane of motion, and the non-stationary coordinate system Aξηζ that is rigidly attached to point A of the system, so that the coordinate plane Aξη coincides with the plane Oxy (refer to Figure 1a).…”