Buckling and Vibration of Elastically Restrained Standing Vertical Plates

2011 ◽  
Vol 134 (1) ◽  
Author(s):  
S. K. Lai ◽  
Y. Xiang
Keyword(s):  
Numerical Technique ◽  
Eigenvalue Equation ◽  
Mode Shapes ◽  
Body Forces ◽  
Discrete Singular Convolution ◽  
Dsc Method ◽  
Elastically Restrained ◽  
Contour Mode ◽  
Elastic Edge

This paper investigates the buckling and vibration of heavy standing plates with rotational elastic edge constraints. The discrete singular convolution (DSC) method as a powerful numerical technique is applied to derive the governing eigenvalue equation. Convergence and comparison studies are conducted to authenticate the correctness and accuracy of the DSC approach. Accurate first-known vibration solutions for elastically restrained vertical plates subjecting to body forces/self-weight are presented. Some contour mode shapes for the vibration of elastically restrained vertical plates are also depicted for illustration.

DSC ANALYSIS FOR BUCKLING AND VIBRATION OF RECTANGULAR PLATES WITH ELASTICALLY RESTRAINED EDGES AND LINEARLY VARYING IN-PLANE LOADING

2009 ◽  
Vol 09 (03) ◽  
pp. 511-531 ◽  
Author(s):  
S. K. LAI ◽  
Y. XIANG
Keyword(s):  
Numerical Stability ◽  
The Other ◽  
Rectangular Plates ◽  
Dsc Analysis ◽  
Vibration Problems ◽  
Discrete Singular Convolution ◽  
Dsc Method ◽  
Elastically Restrained Edges ◽  
Elastically Restrained ◽  
Two Sides

This paper presents the discrete singular convolution (DSC) method for solving buckling and vibration problems of rectangular plates with all edges transversely supported and restrained by uniform elastic rotational springs. The opposite plate edges are subjected to a linearly varying uni-axial in-plane loading. The rationale for using DSC method stems from its numerical stability and flexible implementation for structural analysis. To verify the present approach, convergence and comparison studies for rectangular plates with different combinations of elastically restrained and classical edges are carried out. Accurate buckling and vibration solutions of plates having two opposite edges elastically restrained and the other two sides clamped, or all edges elastically restrained are presented.

Vibration of Beam with Elastically Restrained Ends and Rotational Spring-Lumped Rotary Inertia System at Mid-Span

2015 ◽  
Vol 15 (02) ◽  
pp. 1450040 ◽  
Author(s):  
Seyed Mojtaba Hozhabrossadati ◽  
Ahmad Aftabi Sani ◽  
Masood Mofid
Keyword(s):  
Technical Note ◽  
Rotary Inertia ◽  
Mode Shapes ◽  
Bernoulli Beam ◽  
Mass System ◽  
Rotational Spring ◽  
Beam System ◽  
Sample Frequency ◽  
Elastically Restrained ◽  
Euler Bernoulli Beam

This technical note addresses the free vibration problem of an elastically restrained Euler–Bernoulli beam with rotational spring-lumped rotary inertia system at its mid-span hinge. The governing differential equations and the boundary conditions of the beam are presented. Special attention is directed toward the conditions of the intermediate spring-mass system which plays a key role in the solution. Sample frequency parameters of the beam system are solved and tabulated. Mode shapes of the beam are also plotted for some spring stiffnesses.

Free Vibration of a Circular Cylindrical Shell Elastically Restrained by Axially Spaced Springs

1983 ◽  
Vol 50 (3) ◽  
pp. 544-548 ◽  
Author(s):  
T. Irie ◽  
G. Yamada ◽  
Y. Muramoto
Keyword(s):  
Cylindrical Shell ◽  
Free Vibration ◽  
Circular Cylindrical Shell ◽  
Frequency Equation ◽  
Mode Shapes ◽  
Governing Equations ◽  
Structure Matrix ◽  
The Matrix ◽  
Circular Cylindrical Shells ◽  
Elastically Restrained

An analysis is presented for the free vibration of a circular cylindrical shell restrained by axially spaced elastic springs. The governing equations of vibration of a circular cylindrical shell are written as a coupled set of first-order differential equations by using the transfer matrix of the shell. Once the matrix has been determined, the entire structure matrix is obtained by the product of the transfer matrices and the point matrices at the springs, and the frequency equation is derived with terms of the elements of the structure matrix under the boundary conditions. The method is applied to circular cylindrical shells supported by axially equispaced springs of the same stiffness, and the natural frequencies and the mode shapes of vibration are calculated numerically.

Free vibration analysis of partially connected parallel beams with elastically restrained ends

2016 ◽  
Vol 230 (16) ◽  
pp. 2851-2864 ◽  
Author(s):  
Alborz Mirzabeigy ◽  
Reza Madoliat
Keyword(s):  
Free Vibration ◽  
Natural Frequencies ◽  
Flexural Rigidity ◽  
Free Vibration Analysis ◽  
Mode Shapes ◽  
Computational Stability ◽  
Transform Method ◽  
Practical Engineering ◽  
Elastically Restrained ◽  
Connection Length

In the present paper, the problem of transverse free vibration of two parallel beams partially connected to each other by a Winkler-type elastic layer is investigated. Euler–Bernoulli beam hypothesis has been applied, and translational and rotational elastic springs in each end considered as support. The motion of the system is described by coupled, piece-wise differential equations. The differential transform method (DTM) is employed to derive natural frequencies and mode shapes. DTM is a semi-analytical approach based on Taylor expansion series which does not require any admissible functions and yields rapid convergence and computational stability. After validation of the DTM results with results reported by well-known references and finite elements solution, the influences of the inner layer connection length, boundary conditions, the coefficient of elastic inner layer and ratio of beam’s flexural rigidity on natural frequencies as well as influences of the inner layer connection length on mode shapes are discussed. This problem is treated for the first time, and results are completely new which candidate them to being considered for practical engineering applications.

An efficient approach for dynamic analysis of a rotating beam using the discrete singular convolution

2016 ◽  
Vol 230 (20) ◽  
pp. 3642-3654 ◽  
Author(s):  
JunWei Chen ◽  
Ye Ding ◽  
Han Ding
Keyword(s):  
Dynamic Analysis ◽  
Differential Quadrature Method ◽  
Equations Of Motion ◽  
Time Integration ◽  
Computational Effort ◽  
Algebraic Equations ◽  
Rotating Beam ◽  
Efficient Approach ◽  
Discrete Singular Convolution ◽  
Dsc Method

This paper proposes an efficient approach for dynamic analysis of a rotating beam using the discrete singular convolution (DSC). By spatially discretizing the nonlinear equations of motion of the rotating beam using the DSC method, natural frequencies of the rotating beam are obtained. Numerical results show that the DSC method accurately captures not only the low-order but also the high-order frequencies of the beam rotating at a high angular velocity in very short time, compared with the classical finite element method. Moreover, by combining the DSC method and the differential quadrature method, the dynamic equations are reduced to a set of algebraic equations. Thus the dynamic response of the rotating beam is resolved accurately and efficiently with much less computational effort, and is able to be numerically stable for long-time integration.

scholarly journals Free Vibrations of a Trapezoidal Plate with an Internal Line Hinge

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
María Virginia Quintana ◽  
Ricardo Oscar Grossi
Keyword(s):  
Plate Theory ◽  
Ritz Method ◽  
Free Vibrations ◽  
Rectangular Domain ◽  
Internal Line ◽  
Thin Plates ◽  
Mode Shapes ◽  
Approximate Analytical Solutions ◽  
Elastically Restrained Edges ◽  
Elastically Restrained

This paper deals with a general variational formulation for the determination of natural frequencies and mode shapes of free vibrations of laminated thin plates of trapezoidal shape with an internal line hinge restrained against rotation. The analysis was carried out by using the kinematics corresponding to the classical laminated plate theory (CLPT). The eigenvalue problem is obtained by employing a combination of the Ritz method and the Lagrange multipliers method. The domain of the plate is transformed into a rectangular domain in the computational space by using nonorthogonal triangular coordinates and the transverse displacements are approximated with a set of simple polynomials automatically generated and expressed in the triangular coordinates. The developed algorithm allows obtaining approximate analytical solutions for mentioned plate with different geometries, aspect ratio, position of the line hinge, and boundary conditions including translational and rotational elastically restrained edges. It allows studying the influence of the mentioned line on the vibration frequencies and respective mode shapes. The algorithm can easily be programmed and it is numerically stable. Additionally, as a particular case, the results of triangular plates can be easily generated.

scholarly journals Vibration Analysis of Piezoelectric Composite Plate Resting on Nonlinear Elastic Foundations Using Sinc and Discrete Singular Convolution Differential Quadrature Techniques

2020 ◽  
Vol 2020 ◽  
pp. 1-22
Author(s):  
Ola Ragb ◽  
Mohamed Salah ◽  
M. S. Matbuly ◽  
R. B. M. Amer
Keyword(s):  
Composite Plate ◽  
Nonlinear Eigenvalue Problem ◽  
Differential Quadrature ◽  
Mode Shapes ◽  
Piezoelectric Composite ◽  
Sensors And Actuators ◽  
Elastic Foundations ◽  
Discrete Singular Convolution ◽  
Nonlinear Elastic ◽  
Foundation Parameters

In this work, free vibration of the piezoelectric composite plate resting on nonlinear elastic foundations is examined. The three-dimensionality of elasticity theory and piezoelectricity is used to derive the governing equation of motion. By implementing two differential quadrature schemes and applying different boundary conditions, the problem is converted to a nonlinear eigenvalue problem. The perturbation method and iterative quadrature formula are used to solve the obtained equation. Numerical analysis of the proposed schemes is introduced to demonstrate the accuracy and efficiency of the obtained results. The obtained results are compared with available results in the literature, showing excellent agreement. Additionally, the proposed schemes have higher efficiency than previous schemes. Furthermore, a parametric study is introduced to investigate the effect of elastic foundation parameters, different materials of sensors and actuators, and elastic and geometric characteristics of the composite plate on the natural frequencies and mode shapes.

scholarly journals A hybrid procedure to identify the optimal stiffness coefficients of elastically restrained beams

2015 ◽  
Vol 25 (2) ◽  
pp. 245-257 ◽  
Author(s):  
Tiago Silva ◽  
Maria Loja ◽  
Nuno Maia ◽  
Joaquim Barbosa
Keyword(s):  
Optimization Technique ◽  
Gaussian Process Regression ◽  
Torsional Stiffness ◽  
Mode Shapes ◽  
Euler Beam ◽  
Bending Vibration ◽  
Hybrid Strategy ◽  
Stiffness Properties ◽  
Elastically Restrained ◽  
Optimal Stiffness

Abstract The formulation of a bending vibration problem of an elastically restrained Bernoulli-Euler beam carrying a finite number of concentrated elements along its length is presented. In this study, the authors exploit the application of the differential evolution optimization technique to identify the torsional stiffness properties of the elastic supports of a Bernoulli-Euler beam. This hybrid strategy allows the determination of the natural frequencies and mode shapes of continuous beams, taking into account the effect of attached concentrated masses and rotational inertias, followed by a reconciliation step between the theoretical model results and the experimental ones. The proposed optimal identification of the elastic support parameters is computationally demanding if the exact eigenproblem solving is considered. Hence, the use of a Gaussian process regression as a meta-model is addressed. An experimental application is used in order to assess the accuracy of the estimated parameters throughout the comparison of the experimentally obtained natural frequency, from impact tests, and the correspondent computed eigenfrequency.

scholarly journals Free Vibrations of an Elastically Restrained Euler Beam Resting on a Movable Winkler Foundation

2019 ◽  
Vol 2019 ◽  
pp. 1-15
Author(s):  
Qiang Zhou ◽  
Tong Wang
Keyword(s):  
Elastic Foundation ◽  
Free Vibrations ◽  
Wave Number ◽  
Mode Shapes ◽  
Traditional Theory ◽  
Winkler Foundation ◽  
Euler Beam ◽  
Elastically Restrained ◽  
Elastic Restraints ◽  
Natural Characteristics

The traditional theory of beam on elastic foundation implies a hypothesis that the elastic foundation is static with respect to the inertia reference frame, so it may not be applicable when the foundation is movable. A general model is presented for the free vibration of a Euler beam supported on a movable Winkler foundation and with ends elastically restrained by two vertical and two rotational springs. Frequency equations and corresponding mode shapes are analytically derived and numerically solved to study the effects of the movable Winkler foundation as well as elastic restraints on beam’s natural characteristics. Results indicate that if one of the beam ends is not vertically fixed, the effect of the foundation’s movability cannot be neglected and is mainly on the first two modes. As the foundation stiffness increases, the first wave number, sometimes together with the second one, firstly decreases to zero at the critical foundation stiffness and then increases after this point.

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