A new Fourier series method and displacement function for in-plane vibration and power flow characteristics analysis of isotropic rectangular plates

2019 ◽  
Vol 50 (6) ◽  
pp. 176-194
Author(s):  
Kavikant Mahapatra ◽  
SK Panigrahi
Keyword(s):  
Fourier Series ◽  
Boundary Conditions ◽  
Power Flow ◽  
Flow Characteristics ◽  
Displacement Function ◽  
Rectangular Plates ◽  
Series Method ◽  
Plane Vibration ◽  
Fourier Series Method ◽  
Beam Displacement

The generation of in-plane vibration in plates is an important issue and frequently occurs due to the presence of excitations in the ship’s hull due to turbulent fluid flows, turbulent airflow excitation on aerospace structures, gear system subjected to axial excitation, assemblies housing piezoelectric crystals and sandwiched plates, and so on. The present analysis aims to establish a universal and numerically efficient method for determination of in-plane vibration characteristics of isotropic rectangular plates both for conventional and general boundary conditions. The new in-plane Fourier series and displacement function of the plate have been developed using beam displacement functions in x and y directions, respectively, under in-plane condition. A modified Fourier series assumption for the in-plane beam displacement has been utilised and further developed as plate displacement function. The computational efficiency of the present method is compared in terms of convergence of natural frequency parameter, speed of execution and manual convenience to reduce human errors with the frequently used Fourier series method by various researchers. Rayleigh–Ritz procedure has been applied to determine the in-plane natural frequencies. The mode shapes for few conventional and generally varying boundary conditions have been presented and analysed. The dynamic response has been obtained and analysed in terms of the in-plane mobility and power flow characteristics of the plate under varying boundary conditions. The validity of results obtained by the current method has shown excellent accuracy and faster convergence with the existing results. The present results can provide a benchmark to analyse the dynamic in-plane response of plate systems being used for built-up structures in real engineering applications.

An Exact Fourier Series Method for the Vibration Analysis of Multispan Beam Systems

2009 ◽  
Vol 4 (2) ◽  
Author(s):  
Wen L. Li ◽  
Hongan Xu
Keyword(s):  
Fourier Series ◽  
Boundary Conditions ◽  
Series Expansion ◽  
Vibration Analysis ◽  
Series Solution ◽  
Series Representation ◽  
Displacement Function ◽  
Fourier Series Expansion ◽  
Series Method ◽  
Fourier Series Method

An exact Fourier series method is developed for the vibration analysis of multispan beam systems. In this method, the displacement on each beam is expressed as a Fourier series expansion plus an auxiliary closed-form function such as polynomials. The auxiliary function is used to deal with all the possible discontinuities, at the end points, with the original displacement function and its derivatives when they are periodically extended over the entire x-axis as implied by a Fourier series representation. As a result, not only is it always possible to expand the beam displacements into Fourier series under any boundary conditions, but also the series solution will be substantially improved in terms of its accuracy and convergence. Mathematically, the current Fourier series expansion represents an exact solution to a class of beam problems in the sense that both the governing equations and the boundary/coupling conditions are simultaneously satisfied to any specified degree of accuracy. In the multispan beam system model, any two adjacent beams are generally connected together via a pair of linear and rotational springs, allowing a better modeling of many real-world joints. Each beam in the system can also be independently and elastically restrained at its ends so that all boundary conditions including the classical homogeneous boundary conditions at the end and intermediate supports can be universally dealt with by simply varying the stiffnesses of the restraining springs accordingly, which does not involve any modification of basis functions, formulations, or solution procedures. The excellent accuracy and convergence of this series solution is demonstrated through numerical examples.

scholarly journals The interactive bending wrinkling behaviour of inflated beams

2016 ◽  
Vol 472 (2193) ◽  
pp. 20160504 ◽  
Author(s):  
Y. P. Liu ◽  
C. G. Wang ◽  
H. F. Tan ◽  
M. K. Wadee
Keyword(s):  
Fourier Series ◽  
Boundary Conditions ◽  
Experimental Tests ◽  
Equilibrium Path ◽  
Series Method ◽  
Simply Supported ◽  
Fourier Series Method ◽  
Three Stages ◽  
Interactive Buckling ◽  
Significant Support

A model is proposed based on a Fourier series method to analyse the interactive bending wrinkling behaviour of inflated beams. The whole wrinkling evolution is tracked and divided into three stages by identifying the bifurcations of the equilibrium path. The critical wrinkling and failure moments of the inflated beam can then be predicted. The global–local interactive buckling pattern is elucidated by the proposed theoretical model and also verified by non-contact experimental tests. The effects of geometric parameters, internal pressure and boundary conditions on the buckling of inflated beams are investigated finally. The results reveal that the interactive buckling characteristics of an inflated beam under bending are more sensitive to the dimensions of the structure and boundary conditions. We find that for beams which are simply supported at both ends or clamped and simply supported, boundary conditions may prevent the wrinkling formation. The results provide significant support for our understanding of the bending wrinkling behaviour of inflated beams.

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

2018 ◽  
Vol 2018 ◽  
pp. 1-19 ◽  
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.

scholarly journals Free vibration of functionally graded parabolic and circular panels with general boundary conditions

2017 ◽  
Vol 4 (1) ◽  
pp. 52-84 ◽  
Author(s):  
Hong Zhang ◽  
Dongyan Shi ◽  
Qingshan Wang ◽  
Bin Qin
Keyword(s):  
Fourier Series ◽  
Free Vibration ◽  
Boundary Conditions ◽  
Power Law ◽  
Admissible Function ◽  
Functionally Graded ◽  
General Boundary ◽  
General Boundary Conditions ◽  
Series Method ◽  
Fourier Series Method

AbstractThe purpose of this content is to investigate the free vibration of functionally graded parabolic and circular panels with general boundary conditions by using the Fourier-Ritz method. The first-order shear deformation theory is adopted to consider the effects of the transverse shear and rotary inertia of the panel structures. The functionally graded panel structures consist of ceramic and metal which are assumed to vary continuously through the thickness according to the power-law distribution, and two types of power-law distributions are considered for the ceramic volume fraction. The improved Fourier series method is applied to construct the new admissible function of the panels to surmount the weakness of the relevant discontinuities with the original displacement and its derivatives at the boundaries while using the traditional Fourier series method. The boundary spring technique is adopted to simulate the general boundary condition. The unknown coefficients appearing in the admissible function are determined by using the Ritz procedure based on the energy functional of the panels. The numerical results show the present method has good convergence, reliability and accuracy. Some new results for functionally graded parabolic and circular panels with different material distributions and boundary conditions are provided, which may serve as benchmark solutions.

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

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.

Free Vibration Analysis of Moderately Thick Coupled Plates with Elastic Boundary Conditions and Point Supports

2019 ◽  
Vol 19 (12) ◽  
pp. 1950150
Author(s):  
Qiuhong Li ◽  
Joey Sanchez ◽  
Haym Benaroya ◽  
Jiufa Wang ◽  
Kai Xue
Keyword(s):  
Finite Element ◽  
Fourier Series ◽  
Boundary Conditions ◽  
Series Method ◽  
Plate Model ◽  
Coupling Conditions ◽  
Elastic Boundary ◽  
Fourier Series Method ◽  
Coupled Plates ◽  
Elastic Boundary Conditions

Plates are applied to a wide array of structural applications of varying complexity. Each application requires rigorous analysis to determine the viability of the proposed model. One such application involves modeling a larger structure as a collection of smaller flat plates connected at the plate boundaries. Previous research into these types of structures has led to varying levels of accuracy. It has been dependent on the applications and assumptions involved. To improve the accuracy of these types of structures in a more general context, we propose expanding on current models of coupled plates by modeling the plates using Mindlin plate theory. We analyze the vibration of the improved model with general elastic boundary conditions, point supports and coupling conditions using the Fourier series method and finite element software. When the Fourier series method is applied directly, continuity issues arise at the plate coupling boundaries. To resolve these issues, the Fourier series solution of the vibration displacements is amended to include auxiliary functions. This improved coupled plate model is analyzed and numerically simulated for a variety of elastic boundary conditions and coupling conditions. The numerical results are produced using the Fourier series method and a finite element solution to demonstrate the validity of the improved coupled plate model.

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