Stability and vibration of an elastic helical rod with circular cross section

Abstract

The dynamical behavior of an elastic helical rod with circular cross section are discussed on the basis of Kirchhoff's theory. The dynamical equations of the rod described by the Euler's angles are established in the Frenet coordinates of the centerline. The helical state without twisting of the rod under the action of axial force and torque is discussed. The stability of the helical equilibrium is analyzed in the fields of statics and dynamics respectively. The difference and relationship between Lyapunov's and Euler's stability concepts of the rod equilibrium are discussed. We proved in the sense of first approximation that the Euler's stability conditions of the helical rod in the space domain are the necessary conditions of Lyapunov's stability in the time domain. The free frequency of three-dimensional flexural vibration of the helical rod is derived in analytical form as a function of the pitch angle of the helix and the wave number of the perturbed elastica.