Vibration analysis of rotating toroidal shell by the Rayleigh-Ritz method and Fourier series

2018 ◽  
Vol 173 ◽  
pp. 870-891 ◽  
Author(s):  
Ivo Senjanović ◽  
Neven Alujević ◽  
Ivan Ćatipović ◽  
Damjan Čakmak ◽  
Nikola Vladimir
Keyword(s):  
Fourier Series ◽  
Vibration Analysis ◽  
Ritz Method ◽  
Toroidal Shell

A Finite Strip for the Vibration Analysis of Rotating Toroidal Shell Under Internal Pressure

2018 ◽  
Vol 141 (2) ◽  
Author(s):  
Ivo Senjanović ◽  
Ivan Áatipović ◽  
Neven Alujević ◽  
Damjan Čakmak ◽  
Nikola Vladimir
Keyword(s):  
Internal Pressure ◽  
Cross Sections ◽  
Stiffness Matrix ◽  
Vibration Analysis ◽  
Ritz Method ◽  
Toroidal Shell ◽  
Beam Shape ◽  
Finite Strip ◽  
Finite Strip Method ◽  
Toroidal Shells

In this paper, a finite strip for vibration analysis of rotating toroidal shells subjected to internal pressure is developed. The expressions for strain and kinetic energies are formulated in a previous paper in which vibrations of a toroidal shell with a closed cross section are analyzed using the Rayleigh–Ritz method (RRM) and Fourier series. In this paper, however, the variation of displacements u, v, and w with the meridional coordinate is modeled through a discretization with a number of finite strips. The variation of the displacements with the circumferential coordinate is taken into account exactly by using simple sine and cosine functions of the circumferential coordinate. A unique argument nφ+ω t is used in order to be able to capture traveling modes due to the shell rotation. The finite strip properties, i.e., the stiffness matrix, the geometric stiffness matrix, and the mass matrices, are defined by employing bar and beam shape functions, and by minimizing the strain and kinetic energies. In order to improve the convergence of the results, also a strip of a higher-order is developed. The application of the finite strip method is illustrated in cases of toroidal shells with closed and open cross sections. The obtained results are compared with those determined by the RRM and the finite element method (FEM).

scholarly journals Influence of boundary conditions on three-dimensional vibration characteristics of thick rectangular plates

2020 ◽  
Vol 103 (4) ◽  
pp. 003685042096954
Author(s):  
Yufei Zhang ◽  
Jingtao Du
Keyword(s):  
Fourier Series ◽  
Boundary Conditions ◽  
Vibration Analysis ◽  
Ritz Method ◽  
Three Dimensional ◽  
Design Procedure ◽  
Vibration Characteristics ◽  
Rectangular Plates ◽  
Element Analysis ◽  
Elastically Restrained

Vibration analysis of the classical elastic structures is not only essential for the study of vibration reduction by predicting the dynamic behavior, but also important to ensure a reliable, safe, and lasting structural performance through the proper design procedure. In this paper, the influence of boundary conditions on the free and forced three-dimensional vibration analysis of thick rectangular plates has been performed using the improved Fourier series method. For the elastically restrained thick rectangular plate, the three-dimensional improved Fourier series displacement forms are used to model the vibration field. The energy formula is employed to describe the three-dimensional dynamics of the plate. All the unknown Fourier series coefficients are then solved by the Rayleigh-Ritz method. In order to validate the proposed model, several numerical examples are provided and compared against the results from the literature and Finite Element Analysis (FEA). In addition, the effects of the boundary restraining spring stiffness and the thickness ratios of thick rectangular plates are analyzed under elastically restrained boundary conditions to develop an in-depth understanding of the three-dimensional vibration characteristics of thick rectangular plates.

scholarly journals A Unified Spectro-Geometric-Ritz Method for Vibration Analysis of Open and Closed Shells with Arbitrary Boundary Conditions

2016 ◽  
Vol 2016 ◽  
pp. 1-30 ◽  
Author(s):  
Dongyan Shi ◽  
Yunke Zhao ◽  
Qingshan Wang ◽  
Xiaoyan Teng ◽  
Fuzhen Pang
Keyword(s):  
Boundary Conditions ◽  
Vibration Analysis ◽  
Ritz Method ◽  
Free Vibration Analysis ◽  
Analysis Model ◽  
Arbitrary Boundary ◽  
Fourier Cosine ◽  
Auxiliary Functions ◽  
Arbitrary Boundary Conditions ◽  
Closed Shells

This paper presents free vibration analysis of open and closed shells with arbitrary boundary conditions using a spectro-geometric-Ritz method. In this method, regardless of the boundary conditions, each of the displacement components of open and closed shells is represented simultaneously as a standard Fourier cosine series and several auxiliary functions. The auxiliary functions are introduced to accelerate the convergence of the series expansion and eliminate all the relevant discontinuities with the displacement and its derivatives at the boundaries. The boundary conditions are modeled using the spring stiffness technique. All the expansion coefficients are treated equally and independently as the generalized coordinates and determined using Rayleigh-Ritz method. By using this method, a unified vibration analysis model for the open and closed shells with arbitrary boundary conditions can be established without the need of changing either the equations of motion or the expression of the displacement components. The reliability and accuracy of the proposed method are validated with the FEM results and those from the literature.

The Surface Variational Principle: A Different Perspective for Structural Acoustic Modeling

1997 ◽  
Author(s):  
J. H. Ginsberg
Keyword(s):  
Fourier Series ◽  
Variational Principle ◽  
Vibration Analysis ◽  
Basis Functions ◽  
Acoustic Modeling ◽  
Normal Velocity ◽  
Convergence Properties ◽  
Variational Functional ◽  
Acoustic Interaction ◽  
System Equations

Abstract This paper surveys the development and application of the surface variational principle (SVP) governing the acoustic interaction between surface pressure and normal velocity. SVP is analogous to the method of assumed modes for vibration analysis, in that it represents the response in terms of a sequence of basis functions that are globally defined. The system equations governing the series coefficients are obtained by requiring that the value of the variational functional be stationary. In the wavenumber-based version of SVP, the pressure and velocity are represented by dual range Fourier series. A brief description of the steps entailed in formulating the SVP equations and coupling them to the equations for an elastic structure is provided. Then the computational requirements of an SVP analysis relative to conventional boundary element and finite element techniques are discussed. This is followed by an example illustrating the convergence properties of SVP. Another example is used to highlight the physical interpretation of the SVP representation of surface response. The evolution of the present version of SVP is surveyed, along with some of its applications. The paper closes with a brief discussion of possible future applications of the method.

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