Longitudinal vibration analysis of nonlocal nanorods with elastic end restraints by an improved Fourier series method

In this paper, one newly developed method named the Improved Fourier Series method is applied to the vibration analysis of a beam elastically supported at the both end. The flexural displacement of the beam is supposed to be one set of Fourier Series coupled with four appended terms. Based on the Rayleigh-Ritz procedure and and the vibration characteristics of the beam are also acquired by solving these two matrices. In the end, the frequencies calculated are also compared with those from references and Results ar the Hamilton’s equation, the mass matrices and the stiffness matrices of the beam are obtained e all proved excellent.

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Vibration Characteristics Analysis of Cylindrical Shell-Plate Coupled Structure Using an Improved Fourier Series Method

Shock and Vibration ◽

10.1155/2018/9214189 ◽

2018 ◽

Vol 2018 ◽

pp. 1-19 ◽

Cited By ~ 2

Author(s):

Yipeng Cao ◽

Runze Zhang ◽

Wenping Zhang ◽

Jinzhao Wang

Keyword(s):

Cylindrical Shell ◽

Fourier Series ◽

Boundary Conditions ◽

Accurate Solution ◽

Vibration Characteristics ◽

Series Method ◽

Mechanical Coupling ◽

Fourier Series Method ◽

Characteristics Analysis ◽

Improved Fourier Series Method

A simple yet accurate solution procedure based on the improved Fourier series method (IFSM) is applied to the vibration characteristics analysis of a cylindrical shell-circular plate (S-P) coupled structure subjected to various boundary conditions. By applying four types of coupling springs with arbitrary stiffness at the junction of the coupled structure, the mechanical coupling effects are completely considered. Each of the plate and shell displacement functions is expressed as the superposition of a two-dimensional Fourier series and several supplementary functions. The unknown series-expansion coefficients are treated as the generalized coordinates and determined using the familiar Rayleigh-Ritz procedure. Using the IFSM, a unified solution for the S-P coupled structure with symmetrical and asymmetrical boundary conditions can be derived directly without the need to change either the equations of motion or the expressions of the displacements. This solution can be verified by comparing the current results with those calculated by the finite-element method (FEM). The effects of several significant factors, including the restraint stiffness, the coupling stiffness, and the situation of coupling, are presented. The forced vibration behaviors of the S-P coupled structure are also illustrated.

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Nonlinear forced vibration analysis of the rectangular plates by the Fourier series method

Computational Mechanics ◽

10.1007/s004660050380 ◽

1999 ◽

Vol 23 (1) ◽

pp. 1-7 ◽

Cited By ~ 6

Author(s):

T.-L. Teng ◽

C.-C. Liang ◽

C.-C. Liao

Keyword(s):

Fourier Series ◽

Forced Vibration ◽

Vibration Analysis ◽

Rectangular Plates ◽

Series Method ◽

Fourier Series Method ◽

Nonlinear Forced Vibration ◽

Forced Vibration Analysis

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Free Vibration Analysis of Laminated Composite Double-Plate Structure System with Elastic Constraints Based on Improved Fourier Series Method

Shock and Vibration ◽

10.1155/2021/8811747 ◽

2021 ◽

Vol 2021 ◽

pp. 1-25

Author(s):

Ying Zhang ◽

Dongyan Shi ◽

Dongze He ◽

Dong Shao

Keyword(s):

Fourier Series ◽

Free Vibration ◽

Boundary Conditions ◽

Laminated Composite ◽

Mode Shapes ◽

Series Method ◽

Plate Structure ◽

Fourier Series Method ◽

Linear Algebraic Equations ◽

Improved Fourier Series Method

An analytical model of laminated composite double-plate system (LCDPS) is established, which efficiently analyzes the common 3D plate structure in engineering applications. The proposed model combines the first-order shear deformation theory (FSDT) and the classical delamination theory, and then the LCDPS’s vibration characteristics are investigated. In the process of analysis, the improved Fourier series method (IFSM) is used to describe the displacement admissible function of the LCDPS, which can remove the potential discontinuities at the boundaries. Five sets of artificial springs are introduced to simulate the elastic boundary constraints, and the restraints of the Winkler elastic layer can be adjustable. The improved Fourier series is substituted into the governing equations and boundary conditions; then, applying the Rayleigh–Ritz method, we take all the series expansion coefficients as the generalized coordinates. After that, a set of standard linear algebraic equations was obtained. On this basis, the natural frequency and mode shapes of the LCDPS can be obtained by solving the standard eigenvalue problem. By the discussion of numerical examples and the comparison with those of the reports in the literature, the convergence and the reliability of the present approach are validated. Finally, the parametric investigations of the free vibration with complex boundary conditions are carried out, including the influence of boundary conditions, lamination scheme, plate geometric parameters, and elastic coefficient between two plates.

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