BREAKUP OF RESONANCE DOMAINS FOR MEISSNER'S EQUATION WITH SMALL DAMPING

摘要

A linear oscillatory system with one degree of freedom and a piecewise constant periodic excitation function is considered. Instability of the trivial solution (the parametric resonance) is studied on the parameter plane "frequency-amplitude of excitation". The so-called "overtwisting" of instability domains is the characteristic feature of the system under consideration. The phenomenon of breakup and formation of finite instability domains under small damping is discussed. The coordinates of overtwisting points are found. Some expressions for three-dimensional resonance domains (the halves of double-napped cones) near the above points are obtained. These expressions explain the phenomenon of breakup of the resonance domains. For the case of small excitation amplitudes and damping coefficients, asymptotic formulas describing the formation of instability domains for an arbitrary resonance number are derived. As an example, the domains of parametric resonance for a pendulum with a vertically oscillating point of suspension are examined.