The structures in engineering can be simplified into elastic beams with concentrated masses and elastic spring supports. Studying the law of vibration of the beams can be a help in guiding its design and avoiding resonance. Based on the Laplace transform method, the mode shape functions and the frequency equations of the beams in the typical boundary conditions are derived. A cantilever beam with a lumped mass and a spring is selected to obtain its natural frequencies and mode shape functions. An experiment was conducted in order to get the modal parameters of the beam based on the NExT-ERA method. By comparing the analytical and experimental results, the effects of the locations of the mass and spring on the modal parameter are discussed. The variation of the natural frequencies was obtained with the changing stiffness coefficient and mass coefficient, respectively. The findings provide a reference for the vibration analysis methods and the lumped parameters layout design of elastic beams used in engineering.
In recent years, as the composite laminated plates are widely used in engineering practice such as aerospace, marine and building engineering, the vibration problem of the composite laminated plates is becoming more and more important. Frequency, especially the fundamental frequency, has been considered as an important factor in vibration problem. In this paper, a calculation method of the fundamental frequency of arbitrary laminated plates under various boundary conditions is proposed. The vibration differential equation of the laminated plates is established at the beginning of this paper and the frequency formulae of specialty orthotropic laminated plates under various boundary conditions and antisymmetric angle-ply laminated plates with simply-supported edges are investigated. They are proved to be correct. Simple algorithm of the fundamental frequency for multilayer antisymmetric and arbitrary laminated plates under various boundary conditions is studied by a series of typical examples. From the perspective of coupling, when the number of laminated plates layers N〉8-10, some coupling influence on the fundamental frequency can be neglected. It is reasonable to use specialty orthotropic laminated plates with the same thickness but less layers to calculate the corresponding fundamental frequency of laminated plates. Several examples are conducted to prove correctness of this conclusion. At the end of this paper, the influence of the selected number of layers of specialty orthotropic laminates on the fundamental frequency is investigated. The accuracy and complexity are determined by the number of layers. It is necessary to use proper number of layers of special orthotropic laminates with the same thickness to simulate the fundamental frequency in different boundary conditions.
