Stability and free vibration analysis of compound column with piezoelectric rod

Abstract

This paper deals with the divergence and flutter instability of a slender geometrically nonlinear two-member column under eccentric partially follower load. The external force direction is governed by follower parameter η, which changes over the interval 0 ≤ η ≤ 1, hence both the conservative (η = 0) and nonconservative load may be taken into account. To eliminate the static column deflection, the host aluminium member is associated with an auxiliary piezoceramic rod subjected to an electric field. The governing equations of motion and boundary conditions are derived using extended variational Hamilton’s principle. Constitutive equations expressing the piezoelectric coupling between the electric and elastic fields, the von Kármán theory of moderately large rotations but small strains and the Euler–Bernoulli beam theory are used in the formulation. Due to the nonlinearity of the governing equations, their solutions are attempted by a regular perturbation technique leading to asymptotic expansions of displacements, internal forces and the natural vibration frequency. The stability analysis of the column is investigated by using the kinetic criterion, i.e. by studying the evolution of the eigenfrequencies for an increasing external load. Both the divergence and flutter instability phenomena are detected and the latter exists only for the pure tangential load.It has been found that the courses of frequency curves, which are strictly dependent on the load eccentricity and follower parameter, may be modified by the piezoelectric actuation. As a result, both the divergence and flutter forces of the column are affected by the electric field. When η ≤ 0.5, the tensile piezoelectric force increases the vibration frequency, whereas the compressive force acts in the opposite way. For such η, the limit points of column instability are manifested, which are reflected by the additional branches of the eigenfrequency curves existing over a narrow range of the external load.For the pure follower load, the piezoelectric force may increase or decrease the frequency, which depends on both the external force level and its eccentricity. On the other hand, the piezoelectric actuation may be efficiently used to increase the flutter force. Specific mode shapes of the column are observed, which are characterized by an uneven number of nodes for particular members. Additionally, it is observed that the sequence of vibration modes with the growing values of the external load is reorganized. Finally, the present analysis reveals that the piezoelectric actuation not only counteracts the column bending, but also modifies the transversal vibration frequency of the system and enhances its critical forces for the conservative or nonconservative external load.