Free in-plane vibration analysis of a curved beam (arch) with arbitrary various concentrated elements

In the existing literature, the information regarding the exact solutions for free in-plane vibrations of the curved beams (or arches) carrying various concentrated elements is rare, particularly for the case with multiple attachments including eccentricities and mass moments of inertias. For this reason, this paper aims at presenting an effective approach to tackle the title problem. First of all, the un-coupled equation of motion for the circumferential displacement of an arch segment is derived. Next, based on the value of the discriminate parameter for a cubic equation, the exact solutions for the three types of roots of the un-coupled equation are determined and, corresponding to each type of roots, all displacement functions for the arch segment in terms of the real numbers (instead of the complex ones) are obtained. Finally, use of the compatible equations for the displacements and slopes together with the equilibrium equations for the forces and moments at each intermediate node and two ends of the entire curved beam, a frequency equation of the form {divides}H(ω){divides}=0 is obtained. It is found that the conventional approach by using the condition "{divides}H(ω^t){divides}≤ε" to search for the approximate value of ω^t is difficult even if the convergence tolerance ε is greater than 10⁺³ (i.e., ε>10⁺³) instead of less than 10^-3 (i.e., ε<10^-3), however, the half-interval method is one of the effective tools for solving the problem if all coefficients of the determinant {divides}H(ω){divides} are the real numbers. In addition to comparing with the existing literature, most of the numerical results obtained from the presented method are compared with those obtained from the conventional finite element method (FEM) and good agreement is achieved.