scholarly journals Flexural Vibration Analysis of Nonuniform Double-Beam System with General Boundary and Coupling Conditions

2018 ◽  
Vol 2018 ◽  
pp. 1-8 ◽  
Author(s):  
Lujun Chen ◽  
Deshui Xu ◽  
Jingtao Du ◽  
Chengwen Zhong
Keyword(s):  
Fourier Series ◽  
Boundary Conditions ◽  
Vibration Analysis ◽  
Admissible Function ◽  
Flexural Vibration ◽  
Vibration Characteristics ◽  
Modeling Framework ◽  
Varying Thickness ◽  
Beam System ◽  
Double Beam

In this paper, an analytical modeling approach for the flexural vibration analysis of the nonuniform double-beam system is proposed via an improved Fourier series method, in which both types of translational and rotational springs are introduced to account for the mechanical coupling on the interface as well as boundary restraints. Energy formulation is employed for the dynamic description of the coupling system. With the aim to treat the varying thickness across the beam in a unified pattern, the relevant variables are all expanded into Fourier series. Supplementary terms with the smoothed characteristics are introduced to the standard Fourier series for the construction of displacement admissible function for each beam. In conjunction with the Rayleigh–Ritz procedure, the transverse modal characteristics of nonuniform double-beam system can be obtained by solving a standard eigenvalue problem. Instead of solving the certain value of nonideal boundary conditions, the continuous spring stiffnesses of the boundary conditions are considered, and the rotational restrains are introduced in the coupling beam interface. Numerical results are then presented to demonstrate the reliability of the current model and study the influence of various parameters, such as taper ratio, boundary, and coupling strength on the free vibration characteristics, with the emphasis put on the rotational restraining coefficients on the beam interface. This work can provide an efficient modeling framework for the vibration characteristics study of the complex double-beam system, especially with arbitrary varying thickness and coupling stiffness.

scholarly journals A Modified Fourier–Ritz Formulation for Vibration Analysis of Arbitrarily Restrained Rectangular Plate with Cutouts

2018 ◽  
Vol 2018 ◽  
pp. 1-22
Author(s):  
Yiming Liu ◽  
Zhuang Lin ◽  
Hu Ding ◽  
Guoyong Jin ◽  
Sensen Yan
Keyword(s):  
Boundary Conditions ◽  
Vibration Analysis ◽  
Ritz Method ◽  
Flexural Vibration ◽  
Vibration Characteristics ◽  
Geometric Parameters ◽  
Arbitrary Boundary ◽  
Arbitrary Boundary Conditions ◽  
Stiffness Constant ◽  
Proper Values

A modified Fourier–Ritz method is developed for the flexural and in-plane vibration analysis of plates with two rectangular cutouts with arbitrary boundary conditions, aiming to provide a unified solving process for cases that the plate has various locations or sizes of cutout, and different kinds of boundary conditions. Under the current framework, modifying the position of the cutout or the boundary conditions of the plate is just as changing the geometric parameters of the plate, and there is no need to change the solution procedures. The arbitrary boundary conditions can be obtained by setting the stiffness constant of the boundary springs which are fixed uniformly along the edges of the plate at proper values. The strain and kinetic energy functions of a plate with rectangular cutout are derived in detail. The convergence and accuracy of the present method are demonstrated by comparing the present results with those obtained from the FEM software. In this paper, free in-plane and flexural vibration characteristics of the plate with rectangular cutout under general boundary conditions are studied. From the results, it can be found that the geometric parameters and positions of the cutout and the boundary conditions of the plate will obviously influence the natural vibration characteristics of the structures.

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 In-Plane Vibration Analysis of Annular Plates with Arbitrary Boundary Conditions

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Xianjie Shi ◽  
Dongyan Shi ◽  
Zhengrong Qin ◽  
Qingshan Wang
Keyword(s):  
Fourier Series ◽  
Boundary Conditions ◽  
Vibration Analysis ◽  
Wide Spectrum ◽  
Arbitrary Boundary ◽  
Different Boundary Conditions ◽  
Plane Vibration ◽  
Arbitrary Boundary Conditions ◽  
Annular Plates ◽  
Generalized Fourier Series

In comparison with the out-of-plane vibrations of annular plates, far less attention has been paid to the in-plane vibrations which may also play a vital important role in affecting the sound radiation from and power flows in a built-up structure. In this investigation, a generalized Fourier series method is proposed for the in-plane vibration analysis of annular plates with arbitrary boundary conditions along each of its edges. Regardless of the boundary conditions, the in-plane displacement fields are invariantly expressed as a new form of trigonometric series expansions with a drastically improved convergence as compared with the conventional Fourier series. All the unknown expansion coefficients are treated as the generalized coordinates and determined using the Rayleigh-Ritz technique. Unlike most of the existing studies, the presented method can be readily and universally applied to a wide spectrum of in-plane vibration problems involving different boundary conditions, varying material, and geometric properties with no need of modifying the basic functions or adapting solution procedures. Several numerical examples are presented to demonstrate the effectiveness and reliability of the current solution for predicting the in-plane vibration characteristics of annular plates subjected to different boundary conditions.

Tables of Characteristic Functions for Uniform Beams with Sliding Ends

1957 ◽  
Vol 61 (559) ◽  
pp. 494-499 ◽  
Author(s):  
R. E. D. Bishop
Keyword(s):  
Fourier Series ◽  
Vibration Analysis ◽  
Flexural Vibration ◽  
Characteristic Functions ◽  
Elastic Instability ◽  
Series Representations ◽  
In Series ◽  
Plates And Shells ◽  
Circular Functions

Tables of the characteristic functions of certain types of uniform beams were published by Young and Felgar in 1949. The functions concerned represent (to suitable scales) the principal modes of free flexural vibration, although their uses are far more extensive than is implied by this. Some of the uses to which they can be put are mentioned in Ref. 1 and others are discussed by Felgar and Bishop; they cover both statical and dynamical problems and relate to beams, plates and shells and include problems of elastic instability. In fact these functions can be used in series representations in the same way as circular functions are used in Fourier series.One of the main uses to which these characteristic functions may be put does not appear to have been described in print, although the relevant formulae are quoted in Ref. 2. The functions can be used in vibration analysis to shorten calculations, sometimes leading to very considerable savings of labour.

Sign in / Sign up

Export Citation Format

Share Document