An Analysis of the Effect of a Change in the Support Point Location on the Vibration of Thin-Walled Beams

Abstract

This paper deals with an analysis of the free vibration of nonprismatic thin-walled beams, with a special focus on the effect of a change in the support point location on the eigenfrequencies of the systems. A change in the support point location is understood here as occurring within the same fixed cross-section of the beam where the latter is supported. The original elements of this study are a thin-walled beam model and a method of solving differential equations, not previously used by other authors. The equations describing the model used in this paper were derived using the momentless theory of shells and the Vlasov theory assumptions. The displacement equations were derived relative to an arbitrary rectilinear reference axis. In most works known to the authors the equations describing the vibration of the nonprismatic thin-walled beam do not take into account the effects due to the curvature of the axis formed by the shear centers. The recursive algorithm was used to solve the obtained differential equations with variable coefficients. The algorithm enables one to solve the analyzed displacement equations in the form of series relative to the orthogonal Gegenbauer polynomials. For special values of the parameter defining the order of the polynomials the Gegenbauer polynomials became Chebyshev or Lagrange polynomials. In the provided numerical example, the effect of a change in the support point location within the fixed cross-section is examined. It is also analyzed which of the approximation polynomials (Chebyshev or Lagrange polynomials) yield more precise results for a small approximation base. In order to verify the model and the effectiveness of the adopted solution method, the results obtained using this method are compared with the results yielded by FEM. The results obtained showed a significant effect of the position of the support point, within a fixed section, on the eigenfrequencies values of the thin-walled systems.