Out-of-plane vibration analysis of circular curved beam with attachments

Generalized functions are widely used in structural mechanics to address discontinuities in beam-like structures. However, for circular curved beams, bending torsion coupling and intrinsic curvature often prevent the modal functions from being expressed as simple linear combinations of standard trigonometric and hyperbolic functions. This limitation restricts the applicability of the generalized function method to vibration analysis of circular curved beams. This paper presents an analytical solution for the out-of-plane free and forced vibrations of circular curved beams with various attachments under arbitrary boundary conditions. These discrete attachments include translational dampers, torsional-rotational dampers, bending-rotational dampers and attached masses. By successfully utilizing generalized function theory to address discontinuities in response variables of circular curved beam, the proposed method overcomes the limitations of traditional methods, such as beam segmentation, Green’s functions, and Lagrange multipliers. This study provides exact closed-form expressions for natural frequencies, mode shapes, and frequency response functions (FRFs) through a simple process. Furthermore, regardless of the number of attachments, the characteristic matrices remain 6 × 6 in size, offering significant computational advantages and eliminating the repetitive and complex matrix inversion required by some traditional methods. The accuracy and versatility of the proposed approach are validated by comparison with experimental and theoretical results from the literature and with finite element method (FEM) results developed in this study. Additionally, parametric studies on selected examples reveal the influence of various parameters on the system's dynamic behavior.