The Study of the Effect of Static Axial Loads on Vertically-Mounted Tapered Cantilever Transverse Vibrations Using the Cauchy Function

Abstract

The Cauchy function and characteristic series were applied to solve the boundary value problem of free transverse vibrations of vertically mounted, elastically supported tapered cantilever beams. A concentrated mass was attached at the same distance from the base. The beams were subjected to universal axial loads - conservative and follow wing tangential forces - and distributed loads along the cantilever length. The general form of characteristic equation was obtained taking into account the shape of the tapered cantilever, elastic foundation and nonhomogeneous material. Bernstein-Kieropian double estimators of natural frequency were calculated based on free coefficients of the characteristic series. Good agreement was obtained between the calculated natural frequency results and the exact values available in the literature.

Analysis of transverse vibration and stability issues of discrete-continuous elastic systems with nonlinearly variable parameters

“…Based on the above equation (28), it is possible to determine the characteristic indices, i.e., the customary frequencies of a plurality of suitable three, two, one degrees of freedom, critical shaft speed with a disk, and also with respect to (24) and exemplary (25) discrete Number of degrees of freedom / up to six [11]/. Let, for example, α 1 = α 2 = , instead of (28) we obtain the following equation:…”

“…Among the numerical methods, finite element method and finite difference method are widely used [11,12]. Examples of analytical methods are variation methods that, although they do not guarantee high accuracy; because of the specific form of vibrations, they give a very simple form of solution [10].…”

“…The proposed method was then applied to the typical boundary problems described by the second and fourth order ordinary differential equations. The authors of this work [10,11] have used a series of characteristic methods to solve the boundary free-floating edge problem of a cantilever beam with a freely varying cross section. General formulas have been obtained which characterize the characteristic coefficients used in [5] for vibration analysis of cantilever beam of different shapes.…”