scholarly journals A Unified Formulation for Free Vibration of Spherical Cap Based on the Ritz Method

2019 ◽  
Vol 2019 ◽  
pp. 1-18 ◽  
Author(s):  
Yuan Du ◽  
Liping Sun ◽  
Xuhong Miao ◽  
Fuzhen Pang ◽  
Haichao Li ◽  
...  
Keyword(s):  
Free Vibration ◽  
Jacobi Polynomials ◽  
Domain Decomposition Method ◽  
Ritz Method ◽  
Axial Direction ◽  
Displacement Function ◽  
Spherical Cap ◽  
Simple Boundary ◽  
Edge Conditions ◽  
Adjacent Segments

The free vibration characteristic of spherical cap with general edge constraints is studied by means of a unified method. The energy method and Kirchhoff hypothesis are adopted to derive the formulas. The displacement functions are improved based on the domain decomposition method, in which the unified Jacobi polynomials are introduced to represent the displacement function component along circumferential direction. The displacement function component along axial direction is still the Fourier series. In addition, the spring stiffness method forms a unified format to deal with various complex boundary conditions and the continuity conditions at two adjacent segments. Then, the final solutions can be obtained based on the Ritz method. To prove the validity of this method, the results of the same condition are compared with FEM, published literatures, and experiment. The results show that the present method has the advantages of fast convergence, high solution accuracy, simple boundary simulation, etc. In addition, some numerical results of uniform and stepped spherical caps with various geometric parameters and edge conditions are reported.

scholarly journals Free Vibration Analysis of Functionally Graded Spherical Torus Structure with Uniform Variable Thickness along Axial Direction

2019 ◽  
Vol 2019 ◽  
pp. 1-22
Author(s):  
Cong Gao ◽  
Xuhong Miao ◽  
Lin Lu ◽  
Ruidong Huo ◽  
Qiaolin Hu ◽  
...  
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Variable Thickness ◽  
Domain Decomposition Method ◽  
Ritz Method ◽  
Axial Direction ◽  
Functionally Graded ◽  
Spherical Torus ◽  
Different Boundary Conditions ◽  
Vibration Responses

Based on the Ritz method, this paper focused on the free vibration of functionally graded (FG) spherical torus with uniform variable thickness along axial direction under different boundary conditions. The first-order shear deformation theory (FSDT) is employed to formulate the analytical model. The method involves partitioning of the spherical torus structure into proper shell segments in order to satisfy the computing requirement of high-order vibration responses according to the domain decomposition method. The two adjacent segments are connected by using the penalty method, where penalty parameters are defined by the artificial springs; the continuity condition and different boundary conditions can be obtained by assigning the appropriate values of springs. The displacement functions’ components are double mixed series, in which Fourier series and unified Jacobi polynomials, respectively, represent displacement function along circumferential direction and axial direction. Then the Ritz method is used to obtain final solutions. The numerical results obtained by the proposed method show great agreement with previously published literatures and those from the finite element program ABAQUS. The effects of boundary conditions and geometric parameters on the vibration responses of the structure are also presented. The most novelty of this paper is to generalize the selection of admissible displacement functions by using Jacobi polynomial.

Combinations for the Free Vibration Behaviors of Anisotropic Rectangular Plates Under General Edge Conditions

1999 ◽  
Author(s):  
Yoshihiro Narita
Keyword(s):  
Free Vibration ◽  
Structural Design ◽  
Natural Frequencies ◽  
Ritz Method ◽  
Technical Information ◽  
Mode Shapes ◽  
Rectangular Plates ◽  
Plate Models ◽  
Edge Conditions ◽  
Anisotropic Rectangular Plates

Abstract The free vibration behavior of rectangular plates provides important technical information in structural design, and the natural frequencies are primarily affected by the boundary conditions as well as aspect and thickness ratios. One of the three classical edge conditions, i.e., free, simple supported and clamped edges, may be used to model the constraint along an edge of the rectangle. Along the entire boundary with four edges, there exist a wide variety of combinations in the edge conditions, each yielding different natural frequencies and mode shapes. For counting the total number of possible combinations, the present paper introduces the Polya counting theory in combinatorial mathematics, and formulas are derived for counting the exact numbers. A modified Ritz method is then developed to calculate natural frequencies of anisotropic rectangular plates under any combination of the three edge conditions and is used to numerically verify the numbers. In numerical experiments, the number of combinations in the free vibration behaviors is determined for some plate models by using the derived formulas, and are corroborated by counting the numbers of different sets of the natural frequencies that are obtained from the Ritz method.

scholarly journals Free vibration of spherical cap subjected to various boundary conditions

2019 ◽  
Vol 11 (9) ◽  
pp. 168781401987926
Author(s):  
Yuan Du ◽  
Ruidong Huo ◽  
Fuzhen Pang ◽  
Shuo Li ◽  
Yongming Huang ◽  
...  
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Jacobi Polynomials ◽  
Thickness Distribution ◽  
Current Method ◽  
Vibration Characteristics ◽  
Numerical Verification ◽  
Spherical Cap ◽  
Various Boundary Conditions ◽  
Spherical Caps

In this article, the free vibration characteristics of spherical caps with different thickness distribution subjected to general boundary conditions are investigated using a semi-analytical approach. Based on the theory of thin shell, the theoretical model of spherical cap is established. Spherical caps are partitioned into sections along the meridional orientation. The displacement components of spherical caps along the meridional direction are represented by Jacobi polynomials. Meanwhile, Fourier series are utilized to express displacement components in the circumferential direction. Various boundary conditions can be easily achieved by the penalty method of the spring stiffness technique. The vibration characteristics of spherical caps are derived by means of the Rayleigh–Ritz energy method. Reliability and validity of the current method are verified by convergence studies and numerical verification. The comparison of results between the current method, finite element method, and those published in the literature prove that the current method works well when handling free vibration of spherical caps. More results of spherical caps with different geometric specifications and edge conditions are displayed in the form of table and graphic, which may serve as a reference for future studies.

Combinations for the Free-Vibration Behaviors of Anisotropic Rectangular Plates Under General Edge Conditions

1999 ◽  
Vol 67 (3) ◽  
pp. 568-573 ◽  
Author(s):  
Y. Narita
Keyword(s):  
Free Vibration ◽  
Natural Frequencies ◽  
Numerical Study ◽  
Ritz Method ◽  
Mode Shapes ◽  
Rectangular Plates ◽  
Vibration Behavior ◽  
Edge Conditions ◽  
Important Field ◽  
Anisotropic Rectangular Plates

The free-vibration behavior of rectangular plates constitutes an important field in applied mechanics, and the natural frequencies are known to be primarily affected by the boundary conditions as well as aspect and thickness ratios. Any one of the three classical edge conditions, i.e., free, simply supported, and clamped edges, may be used to model the constraint along an edge of the rectangle. Along the entire boundary with four edges, there exist a wide variety of combinations in the edge conditions, each yielding different natural frequencies and mode shapes. For counting the total number of possible combinations the present paper introduces the Polya counting theory in combinatorial mathematics. Formulas are derived for counting the exact numbers. A modified Ritz method is then developed to calculate natural frequencies of anisotropic rectangular plates under any combination of the three classical edge conditions and is used to numerically verify the numbers. In this numerical study the number of combinations in the free-vibration behavior is determined for some plate models by using the derived formulas. Results are corroborated by counting the numbers of different sets of the natural frequencies that are obtained from the modified Ritz method. [S0021-8936(00)02203-0]

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.

NUMERICAL ANALYSIS ON NONLINEAR FREE VIBRATION OF CARBON NANOTUBE REINFORCED COMPOSITE BEAMS

2013 ◽  
Vol 14 (01) ◽  
pp. 1350056 ◽  
Author(s):  
F. LIN ◽  
Y. XIANG
Keyword(s):  
Carbon Nanotube ◽  
Free Vibration ◽  
Volume Fraction ◽  
Ritz Method ◽  
Axial Direction ◽  
Functionally Graded ◽  
Nonlinear Free Vibration ◽  
Third Order ◽  
First Order ◽  
Beam Theories

This paper investigates the free vibration of nanocomposite beams reinforced by single-walled carbon nanotubes (SWCNTs). The distribution of the SWCNTs may vary through the thickness of a beam and are aligned along the beam axial direction. The virtual strain and kinetic energies of the carbon nanotube (CNT) composite beam are obtained using the classical variational method of Hamilton's principle, and the geometric nonlinearity of von Kármán sense is also included. The eigenvalue equation for free vibration of the beam is derived by the p-Ritz method. Vibration frequency parameters for the uniformly distributed (UD) and functionally graded (FG) CNT beams based on the first-order and third-order beam theories are presented and the effects of CNT filler volume fraction, distribution, beam length to depth ratio and end support conditions on the nonlinear free vibration characteristics of the beams are discussed. Comparison studies for UD-CNT and FG-CNT beams based on the first-order and the third-order beam theories are also performed and the differences in vibration frequencies and the nonlinear to linear frequency ratios between these two theories are highlighted.

scholarly journals Free vibration analysis on axially graded beam resting on variable Pasternak foundation

2021 ◽  
Vol 1206 (1) ◽  
pp. 012016
Author(s):  
Saurabh Kumar
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Vibration Analysis ◽  
Ritz Method ◽  
Free Vibration Analysis ◽  
Axial Direction ◽  
Functionally Graded ◽  
Pasternak Foundation ◽  
Material Gradation ◽  
Euler Bernoulli Beam

Abstract Free vibration analysis is conducted on axially functionally graded Euler-Bernoulli beam resting on variable Pasternak foundation. The material properties of the beam and the stiffness of the foundation are considered to be varying linearly along the axial direction. Two types of boundary conditions namely; clamped and simply supported are used in the analysis. The problem is formulated using Rayleigh-Ritz method and governing equations are derived with the help of Hamilton’s principle. The numerical results are generated for different material gradation parameter, foundation parameter and boundary conditions and the effect of these parameters on the free vibration behaviour of the beam is discussed.

Free Vibration Analysis on Combined Cylindrical-Spherical Shell

2012 ◽  
Vol 226-228 ◽  
pp. 3-8 ◽  
Author(s):  
Shi Hao Wu ◽  
Ye Gao Qu ◽  
Xiu Chang Huang ◽  
Hong Xing Hua
Keyword(s):  
Free Vibration ◽  
Spherical Shell ◽  
Domain Decomposition Method ◽  
Equations Of Motion ◽  
Free Vibration Analysis ◽  
Axial Direction ◽  
Energy Functional ◽  
Displacement Boundary ◽  
Finite Element Software ◽  
Boundary Equations

Based upon the Reissner-Naghdi-Berry’s shell theory, a domain decomposition method (DDM) is utilized to investigate the vibration characteristics of the combined cylindrical-spherical shell with different boundary conditions. The combined shell was first apart from prescribed-displacement boundary and then divided into some cylindrical and spherical shell subdomains, respectively. The boundary equations were introduced into the energy functional of the combined shell as well as the constraint equations derived from interface continuity conditions between two adjacent shell subdomains. Fourier series and Chebyshev orthogonal polynomials were employed as the admissible displacement functions for each shell subdomain in the circumferential direction and axial direction in order to obtain the discretization equations of motion of the combined shell. Exact free vibration solutions of the combined shell has been performed via the DDM and were compared with those obtained by the finite element software ANSYS to confirm the reliability and accuracy.

scholarly journals Free and Forced Vibration Characteristics Analysis of a Multispan Timoshenko Beam Based on the Ritz Method

2021 ◽  
Vol 2021 ◽  
pp. 1-18
Author(s):  
Cong Gao ◽  
Fuzhen Pang ◽  
Haichao Li ◽  
Hongfu Wang ◽  
Jie Cui ◽  
...  
Keyword(s):  
Domain Decomposition ◽  
Forced Vibration ◽  
Decomposition Method ◽  
Jacobi Polynomials ◽  
Domain Decomposition Method ◽  
Ritz Method ◽  
Continuity Condition ◽  
Vibration Characteristics ◽  
Application Range ◽  
Various Boundary Conditions

The uniform formulation of dynamic vibration analysis of multispan beams is presented by using an efficient domain decomposition method in this paper. Firstly, the structure is divided into several equal sections based on domain decomposition method. Next, the artificial spring is used to simulate complex boundaries and continuity condition of multispan beam. Finally, the admissible displacement functions are expanded through Jacobi orthogonal polynomials, and the free and forced vibration characteristics of multispan beam structures can be obtained by using Rayleigh–Ritz method. Results for various boundary conditions, ratios of thickness to length (h/L), numbers, and stiffness of supporting springs are presented. It is clearly shown that accurate solutions can be obtained by using the proposed method, and this study extends the application range of the Jacobi polynomials-Ritz method. In addition, the research results of this paper can provide data support for engineers such as bridge designers to design multispan bridges.

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