TY - JOUR

T1 - The Stability of a Flexible Vertical Rod on a Vibrating Support

AU - Belyaev, A. K.

AU - Morozov, N. F.

AU - Tovstik, P. E.

AU - Tovstik, T. P.

PY - 2018/7/1

Y1 - 2018/7/1

N2 - The classical Kapitsa problem of the inverted flexible pendulum is generalized. We consider a thin homogeneous vertical rod with a free top end and pivoted or rigid attached lower end under the weight of the pendulum’s action and vertical harmonic vibrations of the support. In both cases of attachment, we have stability conditions for the vertical rod position. We take the influence of axial and bending rod vibrations and describe the bending vibrations using the Bernoulli–Euler beam model. The solution is built as a Fourier expansion by eigenfunctions of auxiliary boundary-value problems. As a result, the problem is reduced to the set of ordinary differential equations with periodic coefficients and a small parameter. The asymptotic method of two-scale expansions is used for its solution and to determine the critical level of vibration. The influence of longitudinal waves in the rod essentially decreases the critical load. The single-mode approximation has an acceptable accuracy. With pivoting support at the lower end of the rod, we find the explicit approximate solution. For the rigid attachment, we conduct numerical analysis of the critical level of vibrations depending on the problem parameters.

AB - The classical Kapitsa problem of the inverted flexible pendulum is generalized. We consider a thin homogeneous vertical rod with a free top end and pivoted or rigid attached lower end under the weight of the pendulum’s action and vertical harmonic vibrations of the support. In both cases of attachment, we have stability conditions for the vertical rod position. We take the influence of axial and bending rod vibrations and describe the bending vibrations using the Bernoulli–Euler beam model. The solution is built as a Fourier expansion by eigenfunctions of auxiliary boundary-value problems. As a result, the problem is reduced to the set of ordinary differential equations with periodic coefficients and a small parameter. The asymptotic method of two-scale expansions is used for its solution and to determine the critical level of vibration. The influence of longitudinal waves in the rod essentially decreases the critical load. The single-mode approximation has an acceptable accuracy. With pivoting support at the lower end of the rod, we find the explicit approximate solution. For the rigid attachment, we conduct numerical analysis of the critical level of vibrations depending on the problem parameters.

KW - bending vibrations

KW - compressed rod

KW - flexible Kapitsa pendulum

KW - longitudinal vibrations

KW - two-scale asymptotic expansions

UR - http://www.scopus.com/inward/record.url?scp=85052687599&partnerID=8YFLogxK

U2 - 10.3103/S1063454118030020

DO - 10.3103/S1063454118030020

M3 - Article

AN - SCOPUS:85052687599

VL - 51

SP - 296

EP - 304

JO - Vestnik St. Petersburg University: Mathematics

JF - Vestnik St. Petersburg University: Mathematics

SN - 1063-4541

IS - 3

ER -