In-plane vibrations of curved beams having variable curvature and cross-section

Abstract

In this study, vibration characteristics of curved beams having variable curvatures and cross-sections are investigated. For convenience and progressive requirements, vibration characteristics of curved beams having; constant curvature and cross-section, variable curvature and constant cross-section, constant curvature and variable cross-section are also examined. The governing differential equations have derivatives with variable coefficients except for constant curvature and cross-sectioned case. Due to the fact that the solutions of differential equations with variable coefficients are analytically impossible except for special combinations of coefficients, in the investigation of eigenvalues of differential equations with variable coefficients usage of a numerical solution technique becomes necessary. At this point, the Finite Difference Method (FDM) is used to have the eigenvalues by converting continuous eigenvalue problem into discrete eigenvalue problem. Numerical solutions of the equations of motion with variable coefficients based on Finite Difference Method are carried out by using a symbolic program developed in Mathematica. The accuracy and numerical precisions of the developed program are evaluated by comparing the results with the analytical results given in literature. Good agreement is obtained in the comparisons of the present results with analytical results given in tabular form. Then, the effects of selected taper and curvature functions of beams on natural frequencies are found. The results are presented in tabular and graphical forms.