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.

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

2019 ◽  
Vol 142 (1) ◽  
Author(s):  
Berkan Alanbay ◽  
Karanpreet Singh ◽  
Rakesh K. Kapania
Keyword(s):  
Boundary Conditions ◽  
Weight Function ◽  
Jacobi Polynomials ◽  
Ritz Method ◽  
Shear Deformation Theory ◽  
Free Vibration Analysis ◽  
Computational Techniques ◽  
Higher Modes ◽  
Jacobi Weight ◽  
Benchmark Solutions

Abstract This paper presents a general approach for the free vibration analysis of curvilinearly stiffened rectangular and quadrilateral plates using the Ritz method by 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 stiffeners 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 with those available in the open literature. New formulation and results include: (i) exact boundary condition enforcement through Jacobi weight function for FSDT, (ii) formulation of quadrilateral plates with curvilinear stiffeners, and (iii) use of higher order Gauss quadrature scheme for required integral evaluations to obtain higher modes. It is demonstrated that the presented method provides good numerical stability and highly accurate results. The given new numerical results and convergence studies may serve as benchmark solutions for validating the new computational techniques.

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.

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.

Free vibration analysis of rectangular sandwich plates with compressible core and various boundary conditions

2020 ◽  
pp. 109963622097927
Author(s):  
Sajjad Riahi Farsani ◽  
Arash Ramian ◽  
Ramazan-Ali Jafari-Talookolaei ◽  
Paolo S Valvo ◽  
Maryam Abedi
Keyword(s):  
Boundary Conditions ◽  
Plate Theory ◽  
Ritz Method ◽  
Shear Deformation Theory ◽  
Free Vibration Analysis ◽  
Thickness Ratio ◽  
Quadratic Functions ◽  
Sandwich Plates ◽  
Various Boundary Conditions ◽  
Rectangular Sandwich Plates

Extended higher-order sandwich plate theory is used to analyze the free vibrations of rectangular sandwich plates with compressible core. Accordingly, first-order shear deformation theory is used to model the laminated face sheets. Besides, the in-plane and transverse displacements of the core are assumed to be cubic and quadratic functions of the thickness coordinate, respectively. To deduce the governing equations, Hamilton’s principle is used. Then, based on the Rayleigh-Ritz method, single series expansions with two-variable orthogonal polynomials – namely, the orthogonal plate functions – are considered to approximate the displacement components. Lastly, a generalized eigenvalue problem is solved to obtain the free vibrational characteristics of sandwich plates with both symmetric and anti-symmetric lay-ups subjected to various boundary conditions. The method is validated against the results obtained by different methods in the literature. Finally, the effects of the plate side-to-thickness ratio, in-plane aspect ratio, and core-to-face sheets thickness ratio on the natural frequencies are discussed.

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.

scholarly journals Free Vibration Characteristics of Moderately Thick Spherical Shell with General Boundary Conditions Based on Ritz Method

2020 ◽  
Vol 2020 ◽  
pp. 1-20
Author(s):  
Bing Hu ◽  
Cong Gao ◽  
Hang Zhang ◽  
Haichao Li ◽  
Fuzhen Pang ◽  
...  
Keyword(s):  
Boundary Conditions ◽  
Spherical Shell ◽  
Jacobi Polynomials ◽  
Domain Decomposition Method ◽  
Ritz Method ◽  
Admissible Function ◽  
Shear Deformation Theory ◽  
Vibration Characteristics ◽  
General Boundary ◽  
General Boundary Conditions

In this paper, the Ritz method is adopted to investigate the vibration characteristics of isotropic moderately thick annular spherical shell with general boundary conditions. The energy expressions of the annular spherical shell were established based on the first-order shear deformation theory (FSDT). The spring stiffness method is introduced to guarantee continuity and simulate various boundary conditions on the basis of the domain decomposition method. Under the current framework, the displacement admissible function along axial direction and circumferential direction of the shell structure are, respectively, expanded as the unified Jacobi polynomials and Fourier series. The final solutions can be obtained according to the Ritz method. The validity of the proposed method is proved by comparing the results of the same condition with those obtained by the finite element method (FEM) and published literatures. The results show that the current method has fast convergence and delightful accuracy through the comparative study. On this basis, the vibration characteristics of isotropic moderately thick annular spherical shell are further studied by a series of numerical examples.

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 Free Vibration Analysis of Curved Laminated Composite Beams with Different Shapes, Lamination Schemes, and Boundary Conditions

Materials ◽  
2020 ◽  
Vol 13 (4) ◽  
pp. 1010 ◽  
Author(s):  
Bin Qin ◽  
Xing Zhao ◽  
Huifang Liu ◽  
Yongge Yu ◽  
Qingshan Wang
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Variational Approach ◽  
Shear Deformation Theory ◽  
Laminated Composite ◽  
Free Vibration Analysis ◽  
Composite Beams ◽  
Arbitrary Boundary ◽  
Correlated Parameters ◽  
Laminated Composite Beams

A general formulation is considered for the free vibration of curved laminated composite beams (CLCBs) with alterable curvatures and diverse boundary restraints. In accordance with higher-order shear deformation theory (HSDT), an improved variational approach is introduced for the numerical modeling. Besides, the multi-segment partitioning strategy is exploited for the derivation of motion equations, where the CLCBs are separated into several segments. Penalty parameters are considered to handle the arbitrary boundary conditions. The admissible functions of each separated beam segment are expanded in terms of Jacobi polynomials. The solutions are achieved through the variational approach. The proposed methodology can deal with arbitrary boundary restraints in a unified way by conveniently changing correlated parameters without interfering with the solution procedure.

Sign in / Sign up

Export Citation Format

Share Document