Resonance velocities for systems with half-closed chain

I.A. Bolgrabskaya

e-mail: ignat@iamm.ac.donetsk.ua

The motion of the system of N symmetric rigid bodies is studied. Rigid bodies are connected by elastic hinges with two degrees of freedom (universal joints). It is assumed, that the body S₁ has a fixed point and one of points of the symmetry axis of the body S_N moves along a fixed line. These systems are called as the systems with half-closed chain by analogy with J. Wittenburg’s classification. Such systems can be used for studying of small oscillations of elastic rotating rods with two supports at the ends. The resonance velocities for which the characteristic equation for the system of first approximation has a multiple roots are determined. Two groups of these velocities are found. The determination of resonance velocities is necessary because the motion of the system with small asymmetry is unstable in neighborhood of such velocities.

Before the stability of stationary regimes for the system of rigid bodies connected by elastic joints was studied. It was established that the presence of small asymmetry in the systems lead to appearance of instability area in the neighborhood of some frequencies. These frequencies and velocities of rotations at which they are appearanced was called resonance frequencies and resonance velocities respectively. It was established that the characteristic equation of a first approximation for symmetrical system of interconnected rigid bodies (SIRB) motion equations which was written in rotation axes has multiple roots at the resonance velocities of rotations. As a result of investigations it was shown that in absence of external forces and moments the reason of resonance velocities appearance is elastic vibrations in SIRB. The interval of instability depends on parameters of system asymmetry. Received results have not only theoretical but practical interest because SIRB can be used as a discrete model of real technical construction. The knowledge of resonance velocities is necessary for choice of constructions working regimes with presence of elasticity and small asymmetry at any real object.

Thus, it is necessary to determine the resonance velocities in the systems with different boundary conditions. In the present paper the resonance velocities for the systems with half-closed chain are founded. Peculiarity of this system is the possibility of analytical determination of resonance velocity spectrum.