Stability of а flexible vertical rod on the vibrating support

Abstract

The classic Kapitsa problem is generalized to the case of inverted flexible pendulum. A thin homogeneous vertical elastic rod with a free upper end is considered for (i) simply supported and (ii) clamped support. In the both support cases a rod motion under action of its weight and vertical harmonic vibrations of the support is studied. The stability conditions of the vertical rod position are obtained. The influence of both axial and bending vibrations is taken into account, with the Bernoulli—Euler beam model being applied for modeling the bending effects. The solution is presented as a Fourier expansion using the eigen-functions of auxiliary boundary value problems. As a result the problem is reduced to a system of ordinary differential equations with periodic coefficients containing a small parameter. The asymptotic method of two-scale expansions is utilized for its solution and for determining the critical level of the support vibration. It is established that the longitudinal waves influence essentially decrease the critical level of vibrations. Calculations show that the one-mode approximation has an acceptable exactness. The explicit approximate solution is obtained in the case of simply support of a lower rod end. For the clamper end a numerical analysis of the critical level of vibrations on the problem parameters is performed.