Formulation of Problem of Free Vibration Analysis of Isotropic Homogeneous Rectangular Mindlin Plates Having Variable Thickness With General Elastic Boundary Conditions

2017 ◽  
Author(s):  
Sangle Sourabh ◽  
Verma Shesha ◽  
Mali Kiran
Keyword(s):  
Shear Stress ◽  
Boundary Conditions ◽  
Variable Thickness ◽  
Ritz Method ◽  
Free Vibration Analysis ◽  
Free Vibrations ◽  
Element Analysis ◽  
Elastic Boundary ◽  
Constant Shear Stress ◽  
Transverse Deflection

This work presents a formulation for the free vibrations of isotropic homogeneous rectangular Mindlin plates with variable thickness. These plates are subjected to general boundary supports in present study. To obtain arbitrarily supported boundary conditions, new form of trigonometric series expansion functions is used as the admissible functions for transverse deflection and rotation due to bending. In order to account the constant shear stress assumption, a shear stress correction factor is taken into consideration. The Rayleigh-Ritz Method is employed in this formulation. The boundaries are assumed to have three set of springs to achieve required boundary condition. Thus the changes in boundary conditions can be easily obtained by varying the stiffness of these springs, without actually making any changes in the shape functions. In this study, FEA (Finite Element Analysis) has been carried out for the Mindlin plates, for simply supported and constrained on two opposite sides.

scholarly journals In-Plane free Vibration Analysis of an Annular Disk with Point Elastic Support

2011 ◽  
Vol 18 (4) ◽  
pp. 627-640 ◽  
Author(s):  
S. Bashmal ◽  
R. Bhat ◽  
S. Rakheja
Keyword(s):  
Boundary Conditions ◽  
Orthogonal Polynomials ◽  
Ritz Method ◽  
Free Vibration Analysis ◽  
Free Vibrations ◽  
Outer Edge ◽  
Mode Shapes ◽  
Annular Disk ◽  
Different Boundary Conditions ◽  
Boundary Characteristic

In-plane free vibrations of an elastic and isotropic annular disk with elastic constraints at the inner and outer boundaries, which are applied either along the entire periphery of the disk or at a point are investigated. The boundary characteristic orthogonal polynomials are employed in the Rayleigh-Ritz method to obtain the frequency parameters and the associated mode shapes. Boundary characteristic orthogonal polynomials are generated for the free boundary conditions of the disk while artificial springs are used to account for different boundary conditions. The frequency parameters for different boundary conditions of the outer edge are evaluated and compared with those available in the published studies and computed from a finite element model. The computed mode shapes are presented for a disk clamped at the inner edge and point supported at the outer edge to illustrate the free in-plane vibration behavior of the disk. Results show that addition of point clamped support causes some of the higher modes to split into two different frequencies with different mode shapes.

Free Vibration Analysis of a Simply Supported Rectangular Tank Partially Filled With a Liquid

2010 ◽  
Author(s):  
Kyeong-Hoon Jeong ◽  
Jin-Seok Park ◽  
Won-Jae Lee
Keyword(s):  
Boundary Conditions ◽  
Ritz Method ◽  
Free Vibration Analysis ◽  
Ideal Liquid ◽  
Rectangular Plates ◽  
Element Analysis ◽  
Displacement Potential ◽  
Simply Supported ◽  
Rectangular Tank ◽  
Liquid Displacement

This paper presents a theoretical analysis for the hydroelastic vibration of a rectangular tank partially filled with an ideal liquid. The wet dynamic displacement of the tank is approximated by combining the orthogonal polynomials satisfying the simply supported boundary conditions, since the rectangular tank is composed of four rectangular plates. As the facing rectangular plates are geometrically identical, the vibration modes of the facing plates can be divided into two categories: symmetric modes and asymmetric modes with respect to the vertical centerlines of the plates. The liquid displacement potential satisfying the boundary conditions is derived and the wet dynamic modal functions of the four plates are expanded by the finite Fourier transformation for a compatibility requirement along the contacting surface between the tank and the liquid. The natural frequencies of the rectangular tank in the wet condition are calculated by using the Rayleigh-Ritz method. The proposed analytical method is verified by observing an excellent agreement with three-dimensional finite element analysis results.

Vibration of Curvilinearly Stiffened Plates Using Ritz Method With Orthogonal Jacobi Polynomials

2018 ◽  
Author(s):  
Berkan Alanbay ◽  
Karanpreet Singh ◽  
Rakesh K. Kapania
Keyword(s):  
Boundary Conditions ◽  
Acoustic Analysis ◽  
Jacobi Polynomials ◽  
Ritz Method ◽  
Shear Deformation Theory ◽  
Free Vibration Analysis ◽  
Computational Time ◽  
Element Analysis ◽  
Stiffness Matrices ◽  
Geometric Boundary

This paper presents a general approach for the free vibration analysis of curvilinearly stiffened rectangular and quadrilateral plates using Ritz method employing classical orthogonal Jacobi polynomials. Both the plate and stiffeners are modeled using first-order shear deformation theory (FSDT). The displacement and rotations of the plate and a stiffener are approximated by separate sets of Jacobi polynomials. The ease of modification of the Jacobi polynomials enables the Jacobi weight function to satisfy geometric boundary conditions without loss of orthogonality. The distinctive advantage of Jacobi polynomials, over other polynomial-based trial functions, lies in that their use eliminates the well-known ill-conditioning issues when a high number of terms are used in the Ritz method; e.g., to obtain higher modes required for vibro-acoustic analysis. In this paper, numerous case studies are undertaken by considering various sets of boundary conditions. The results are verified both with the detailed Finite Element Analysis (FEA) using commercial software MSC.NASTRAN and for some cases, and with those available in the open literature for others. Convergence studies are presented for studying the effect of the number of terms used on the accuracy of the solution. The paper also discusses the effects of stiffener and plate geometric dimensions on the dynamic characteristics of the structure. The method also has an advantage of saving significant computational time during optimization of such structures as changing the placement and shape of stiffeners does not require repeated calculation of plate mass and stiffness matrices as the stiffener shapes are changed.

Free vibration analysis of rotating thin-walled cylindrical shells with hard coating based on Rayleigh-Ritz method

2020 ◽  
pp. 095441002096724
Author(s):  
Dongxu Du ◽  
Wei Sun ◽  
Xianfei Yan ◽  
Kunpeng Xu
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Vibration Suppression ◽  
Cylindrical Shells ◽  
Ritz Method ◽  
Free Vibration Analysis ◽  
Hard Coating ◽  
Element Analysis ◽  
Vector Method ◽  
Thin Walled

This paper focuses on the free vibration analyses of the rotating hard-coating cylindrical shells with various boundary conditions and the effects of hard coating on the vibration behaviors of the rotating shells. To accurately predict more natural characteristics, the characteristic orthogonal polynomials are taken as the admissible displacement functions. Considering the influences of Coriolis force, centrifugal force and initial hoop tension caused by rotation, the equations of motion of the shells are established by the use of the Rayleigh-Ritz method. Based on the state vector method, an efficient method is developed to solve the equations. By comparing with the results of both the finite element analysis and published literatures, the high accuracy and good convergence of the proposed model are verified. In addition, the effects of the boundary conditions, parameters of hard coating, rotating speed and number of circumferential waves on the vibration behaviors of the hard-coating shells are evaluated. This study may provide a reference for the application of hard-coating damping treatment to the vibration suppression of rotating thin-walled structures.

Free Vibration Analysis of Stiffened Conical Shell with Variable Thickness Distribution

2014 ◽  
Vol 614 ◽  
pp. 7-11 ◽  
Author(s):  
Wei Ning ◽  
Dong Sheng Zhang ◽  
Ji Ling Jia
Keyword(s):  
Variable Thickness ◽  
Conical Shell ◽  
Thickness Distribution ◽  
Ritz Method ◽  
Free Vibration Analysis ◽  
Free Vibrations ◽  
Conical Shells ◽  
Linear Eigenvalue Problem ◽  
Standard Linear ◽  
Energy Equations

The free vibrations of the stiffened hollow conical shells with different variable thickness distribution modes are investigated in detail in the context of Donnel-Mushtari conical shell theory. Two sets of boundary conditions have been considered. The algebraic energy equations of the conical shell and the stiffeners are established separately. The Rayleigh-Ritz method is used to equate maximum strain energy to maximum kinetic energy which leads to a standard linear eigenvalue problem. Numerical results are presented graphically for different geometric parameters. The parametric study reveals the characteristic behavior which is usefulis inuseful in selecting the shell thickness distribution modes and the stiffener type.

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.

Free vibration analysis of three-layer thin cylindrical shell with variable thickness two-dimensional FGM middle layer under arbitrary boundary conditions

2021 ◽  
pp. 109963622110204
Author(s):  
Xue-Yang Miao ◽  
Chao-Feng Li ◽  
Yu-Lin Jiang ◽  
Zi-Xuan Zhang
Keyword(s):  
Cylindrical Shell ◽  
Boundary Conditions ◽  
Volume Fraction ◽  
Ritz Method ◽  
Admissible Function ◽  
Free Vibration Analysis ◽  
Middle Layer ◽  
Two Dimensional ◽  
Arbitrary Boundary ◽  
Arbitrary Boundary Conditions

In this paper, a unified method is developed to analyze free vibrations of the three-layer functionally graded cylindrical shell with non-uniform thickness. The middle layer is composed of two-dimensional functionally gradient materials (2D-FGMs), whose thickness is set as a function of smooth continuity. Four sets of artificial springs are assigned at the ends of the shells to satisfy the arbitrary boundary conditions. The Sanders’ shell theory is used to obtain the strain and curvature-displacement relations. Furthermore, the Chebyshev polynomials are selected as the admissible function to improve computational efficiency, and the equation of motion is derived by the Rayleigh–Ritz method. The effects of spring stiffness, volume fraction indexes, configuration on of shell, and the change in thickness of the middle layer on the modal characteristics of the new structural shell are also analyzed.

Vibration Analysis of Laminated Composite Rectangular Plates With General Boundary Conditions

2018 ◽  
Author(s):  
Yu Fu ◽  
Jianjun Yao ◽  
Zhenshuai Wan ◽  
Gang Zhao
Keyword(s):  
Boundary Conditions ◽  
Vibration Analysis ◽  
Ritz Method ◽  
Laminated Composite ◽  
Free Vibration Analysis ◽  
Geometric Parameters ◽  
General Boundary ◽  
Rectangular Plates ◽  
General Boundary Conditions ◽  
Benchmark Solutions

In this investigation, the free vibration analysis of laminated composite rectangular plates with general boundary conditions is performed with a modified Fourier series method. Vibration characteristics of the plates have been obtained via an energy function represented in the general coordinates, in which the displacement and rotation in each direction is described as an improved form of double Fourier cosine series and several closed-form auxiliary functions to eliminate any possible jumps and boundary discontinuities. All the expansion coefficients are then treated as the generalized coordinates and determined by Rayleigh-Ritz method. The convergence and reliability of the current method are verified by comparing with the results in the literature and those of Finite Element Analysis. The effects of boundary conditions and geometric parameters on the frequencies are discussed as well. Finally, numerous new results for laminated composite rectangular plates with different geometric parameters are presented for various boundary conditions, which may serve as benchmark solutions for future research.

scholarly journals Vibrations of Composite Laminated Circular Panels and Shells of Revolution with General Elastic Boundary Conditions via Fourier-Ritz Method

2016 ◽  
Vol 3 (1) ◽  
Author(s):  
Qingshan Wang ◽  
Dongyan Shi ◽  
Fuzhen Pang ◽  
Qian Liang
Keyword(s):  
Boundary Conditions ◽  
Ritz Method ◽  
Shear Deformation Theory ◽  
Computational Effort ◽  
General Boundary Condition ◽  
Shells Of Revolution ◽  
Elastic Boundary ◽  
Boundary Force ◽  
Classical Boundary ◽  
The Common

AbstractA Fourier-Ritz method for predicting the free vibration of composite laminated circular panels and shells of revolution subjected to various combinations of classical and non-classical boundary conditions is presented in this paper. A modified Fourier series approach in conjunction with a Ritz technique is employed to derive the formulation based on the first-order shear deformation theory. The general boundary condition can be achieved by the boundary spring technique in which three types of liner and two types of rotation springs along the edges of the composite laminated circular panels and shells of revolution are set to imitate the boundary force. Besides, the complete shells of revolution can be achieved by using the coupling spring technique to imitate the kinematic compatibility and physical compatibility conditions of composite laminated circular panels at the common meridian with θ = 0 and 2π. The comparisons established in a sufficiently conclusive manner show that the present formulation is capable of yielding highly accurate solutions with little computational effort. The influence of boundary and coupling restraint parameters, circumference angles, stiffness ratios, numbers of layer and fiber orientations on the vibration behavior of the composite laminated circular panels and shells of revolution are also discussed.

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