Narrow Results

This paper presents a unified method for analyzing the dynamic behavior of spinning beams under elastic constraints. Based on the Timoshenko beam theory, a dynamic model of a spinning beam with elastic constraints is established. The displacement and bending angle are represented by a modified Fourier series. Compared with the traditional Fourier series, the improved Fourier series eliminates the discontinuity of the derivative at the boundary by introducing auxiliary polynomials, making it more suitable for elastic constraints. The governing equations and boundary conditions are coupled together using the energy method to form a set of standard linear equations. By solving this equation, the modes of the spinning beam structure under elastic constraints can be concisely and quickly obtained. Finally, by comparing with other methods, it is proved that the method has good convergence and practicability. Then, the effects of spinning speed, boundary stiffness and slenderness ratio on the whirling characteristics are analyzed. The results show that the linear spring has a more pronounced effect on the whirl frequency than the torsion spring. Different boundary constraints produce different elastic intervals. Mode exchange was found with increasing spinning speed. In the case of elastic constraints, the mode exchange occurs at lower spinning speed. This method has a certain universal applicability to the dynamic analysis of spinning beams under elastic constraints, and the research results can provide theoretical reference for subsequent research.

Free vibration analysis of moderately thick laminated functionally graded rectangular plates with elastic restraints is presented using the modified Fourier–Ritz method in conjunction with the first-order shear deformation plate theory. The material properties are assumed to change continuously through the lamina thickness according to a power-law distribution of volume fractions of the constituents. Each of the displacements and rotations of the laminated functionally graded plates, regardless of boundary conditions, is represented by a modified Fourier series which is constructed as the linear superposition of a standard Fourier cosine series and several closed-form auxiliary functions. The accuracy, convergence and reliability of the current solutions are demonstrated by numerical examples and comparison of the present results with those available in the literature. New results for free vibration of laminated functionally graded plates are presented, which may serve as benchmark solutions. The effects of the boundary conditions, volume fractions and lamina thickness ratios on the frequencies of the plates are investigated.

This paper presents the dynamic response of a rotating beam with elastic restraints in forward flight. The motion equations of the system are established by Hamilton’s principle. The structure model is established by Euler-Bernoulli beam theory. The influence of elastic restraints and centrifugal force is considered in the form of potential energy. The aerodynamic model is established by Greenberg theory and considered in the form of an external force. A modified Fourier series method is used to expand the displacement field. The stiffness intervals of boundary springs corresponding to the elastic restraints are determined by natural frequencies. The effects of advance ratios on the dynamic response of the system are studied. Then, the effects of spring stiffness in different directions are compared. The results show that the displacements and velocities of the rotating beam increase with the advance ratio. The displacement amplitudes of the rotating beam increase as the stiffness of boundary springs decrease.