Out-of-Plane Free Vibration Analysis of a Horizontally Circular Curved Beam Carrying Arbitrary Sets of Concentrated Elements

For convenience, a horizontally circular curved beam without any attachments is called a “bare” curved beam and the one carrying any attachments is called a “loaded” curved beam, in this paper. For the out-of-plane free vibrations of bare curved beams, one can find some exact solutions from the existing literature, but this is not true for those of the loaded curved beams. One of the main reasons for the last situation is due to the difficulty of solving a complex-variable eigenvalue equation. It is well known that the half-interval method is one of the simplest techniques for searching the roots of an eigenvalue equation. However, it suffers difficulty if the eigenvalue equation is a determinant form $\left(\left|H\left(\omega \right)\right|=0\right)$ with some (or all) of its coefficients $\left[H_{i,j}\left(\omega \right)\right]$ being the complex numbers, because it is difficult to find a trial root $\left({\omega }^t\right)$ so that both the real part $H_R$ and imaginary part $H_I$ of the associated determinant value $\left|H\left({\omega }^t\right)\right|$ are equal to zero simultaneously (i.e., $H_R=H_I=0$ ). Furthermore, the magnitude of the determinant value is greater than or equal to zero (i.e., $\overline{H}=\sqrt{H_R^2+H_R^2}\ge 0$ ). To overcome the last difficulty, this paper presents a technique to replace all complex coefficients of the eigenvalue equation by the real ones, so that the conventional half-interval method may be easily applied to determining the “exact” solution for the natural frequencies and mode shapes of out-of-plane free vibrations of a uniform curved Euler-Bernoulli beam carrying arbitrary sets of concentrated elements in various boundary conditions, where each set of concentrated elements includes a lumped mass, a linear spring, a bending spring and a twisting (torsional) spring. To confirm the reliability of the presented theory and the developed computer program, most of the exact solutions for natural frequencies and mode shapes obtained from the presented approach are compared with the “approximate” ones obtained from the conventional finite-element method and good agreements are achieved.