We consider the Hill equation with damping describing the parametric oscillations of a torsional pendulum excited by varying the moment of inertia of the rotating body. Using the method of a small parameter, we analytically calculate a fundamental system of solutions of this equation in the form of power series in the excitation amplitude \epsilon with accuracy O(\epsilon^2) and verify conditions for its stability. In the first order approximation in \epsilon, we prove that the resonance domain exists only if the excitation frequency \Omega is sufficiently close to the double natural frequency of the pendulum; the corresponding equation of the stability boundary is obtained.

### On the stability of the Hill's equation with damping

#### Abstract

We consider the Hill equation with damping describing the parametric oscillations of a torsional pendulum excited by varying the moment of inertia of the rotating body. Using the method of a small parameter, we analytically calculate a fundamental system of solutions of this equation in the form of power series in the excitation amplitude \epsilon with accuracy O(\epsilon^2) and verify conditions for its stability. In the first order approximation in \epsilon, we prove that the resonance domain exists only if the excitation frequency \Omega is sufficiently close to the double natural frequency of the pendulum; the corresponding equation of the stability boundary is obtained.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11386/1067211
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