Vectorial method of the kinetic parameters analysis of the rotor with many axes and nonlinear dynamics

Abstract

By using examples of the rotor system which rotates about two axes with section or without section, we build the vectorial method of the kinetic parameters analysis of the rotors with many axes. The vectors connected for the pole and the axis are used for the analysis of kinetic parameters, by the use of which the rotation properties of the mass configurations are interpreted introducing the mass moment vectors for the pole and the axis, as well as the kinematic vectors rotators. Expressions for the corresponding linear momentum and angular momentum, as well as their derivatives in time are derived. By these expressions vectorial equations of the rotor system dynamics are derived, as well as the expression for the kinetic pressures on the rotor system bearings. By using vectorial equations, we composed two scalar differential equations of the heavy rotor system nonlinear dynamic. For the case when one rotation about axis is controlled by constant angular velocity the nonlinear dynamics of the rotation about other axis is studied. Nonlinear rotor system dynamics are presented by phase portrait in the phase plane, with trigger of the singularities, as well as with homoclinic orbits and homoclinic points of the nonstable type saddle. For the case rotor system dynamics under the action of the perturbed couple the sensitive dependence in the vicinity of the equilibrium nonstable position which corresponds to homoclinic point of the type nonstable saddle, the possibility of the chaotic character behavior is pointed out.