Abstract
Based on the principle of energy variation, an improved Fourier series is introduced as an allowable displacement function. This paper constructs a calculation model that can study the in-plane and out-of-plane free and forced vibrations of curved beam structures under different boundary conditions. Firstly, based on the generalized shell theory, considering the shear and inertial effects of curved beam structures, as well as the coupling effects of displacement components, the kinetic energy and strain potential energy of the curved beam are obtained. Subsequently, an artificial spring system is introduced to satisfy the constraint condition of the displacement at the boundary of the curved beam, obtain its elastic potential energy, and add it to the system energy functional. Any concentrated mass point or concentrated external load can also be added to the energy function of the entire system with a corresponding energy term. In various situations including classical boundary conditions, the accuracy and efficiency of the method in this paper are proved by comparing with the calculation results of FEM. Besides, by accurately calculating the vibration characteristics of common engineering structures like slow curvature (whirl line), the wide application prospects of this method are shown.
Highlights
As a common structure in the engineering field, curved beam structure has attracted widespread attention for its dynamic characteristics
E order intercepted after the expansion of the improved Fourier series used in this paper has a direct effect on the accuracy of the calculation result. e more the expansion order, the closer to the accurate value, but it will affect the efficiency of the solution
The verification model adopts the free-free boundary condition, which corresponds to the value of 0 for the boundary spring groups in the model. e comparison content is the natural frequencies after ignoring rigid modes
Summary
As a common structure in the engineering field, curved beam structure has attracted widespread attention for its dynamic characteristics. Unlike the straight beam structure, due to the influence of deformation, there is considerable coupling between the vibration displacement components of the curved beam structure. For the problem of in-plane vibration of curved beams, if all influencing factors such as axial tension are taken into account, discrete Green function combined with numerical integration was used in early research by Kawakami et al [9]. As one of the most widely used numerical methods, FEM has been used for the analysis of in-plane vibration of curved beams almost since its birth [14]. Based on the extended Hamilton principle, Yang et al [2] developed a high-order Lagrangian-type element suitable for variable curvature curved beams. Cannarozzi and Molari [16]
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