Free Vibration Analysis of a Circular Plate With Multiple Circular Holes by Using Indirect BIEM and Addition Theorem

2010 ◽  
Vol 78 (1) ◽  
Author(s):  
W. M. Lee ◽  
J. T. Chen
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Circular Plate ◽  
Free Vibration Analysis ◽  
Addition Theorem ◽  
Boundary Integral ◽  
Kernel Functions ◽  
Mode Shapes ◽  
Linear Algebraic Equations ◽  
Circular Holes

In this paper, natural frequencies and natural modes of a circular plate with multiple circular holes are theoretically derived and numerically determined by using the indirect boundary integral formulation, the addition theorem, and the complex Fourier series. Owing to the addition theorem, all kernel functions are expanded into degenerate forms and further expressed in the same polar coordinates centered at one circle where the boundary conditions are specified. Not only the computation of the principal value is avoided but also the calculation of higher-order derivatives can be easily determined. By matching boundary conditions, a coupled infinite system of linear algebraic equations is derived as an analytical model for the free vibration of a circular plate with multiple circular holes. The direct-searching approach is utilized in the truncated finite system to determine the natural frequency through singular value decomposition. After determining the unknown Fourier coefficients, the corresponding mode shapes are obtained by using the indirect boundary integral formulations. Some numerical eigensolutions are presented and then utilized to explain some physical phenomenon such as the beating and the dynamic stress concentration. Good accuracy and fast rate of convergence are the main features of the present method, thanks to the analytical approach.

scholarly journals Wave Based Method for Free Vibration Analysis of Ring Stiffened Cylindrical Shell with Intermediate Large Frame Ribs

2013 ◽  
Vol 20 (3) ◽  
pp. 459-479 ◽  
Author(s):  
Meixia Chen ◽  
Jianhui Wei ◽  
Kun Xie ◽  
Naiqi Deng ◽  
Guoxiang Hou
Keyword(s):  
Cylindrical Shell ◽  
Free Vibration ◽  
Boundary Conditions ◽  
Natural Frequencies ◽  
Free Vibration Analysis ◽  
Structure Type ◽  
Vibration Characteristics ◽  
Mode Shapes ◽  
Arbitrary Boundary ◽  
Stiffened Cylindrical Shell

Wave based method which can be recognized as a semi-analytical and semi-numerical method is presented to analyze the free vibration characteristics of ring stiffened cylindrical shell with intermediate large frame ribs for arbitrary boundary conditions. According to the structure type and the positions of discontinuities, the model is divided into different substructures whose vibration field is expanded by wave functions which are exactly analytical solutions to the governing equations of the motions of corresponding structure type. Boundary conditions and continuity equations between different substructures are used to form the final matrix to be solved. Natural frequencies and vibration mode shapes are calculated by wave based method and the results show good agreement with finite element method for clamped-clamped, shear diaphragm – shear diaphragm and free-free boundary conditions. Free vibration characteristics of ring stiffened cylindrical shells with intermediate large frame ribs are compared with those with bulkheads and those with all ordinary ribs. Effects of the size, the number and the distribution of intermediate large frame rib are investigated. The frame rib which is large enough is playing a role as bulkhead, which can be considered imposing simply supported and clamped constraints at one end of the cabin and dividing the cylindrical shell into several cabins vibrating separately at their own natural frequencies.

Free vibration analysis of a rectangular plate carrying multiple various concentrated elements

2003 ◽  
Vol 217 (2) ◽  
pp. 171-183 ◽  
Author(s):  
J-S Wu ◽  
H-M Chou ◽  
D-W Chen
Keyword(s):  
Finite Element ◽  
Free Vibration ◽  
Boundary Conditions ◽  
Rectangular Plate ◽  
Normal Mode ◽  
Vibration Analysis ◽  
Natural Frequencies ◽  
Free Vibration Analysis ◽  
Equation Of Motion ◽  
Mode Shapes

The dynamic characteristic of a uniform rectangular plate with four boundary conditions and carrying three kinds of multiple concentrated element (rigidly attached point masses, linear springs and elastically mounted point masses) was investigated. Firstly, the closed-form solutions for the natural frequencies and the corresponding normal mode shapes of a rectangular ‘bare’ (or ‘unconstrained’) plate (without any attachments) with the specified boundary conditions were determined analytically. Next, by using these natural frequencies and normal mode shapes incorporated with the expansion theory, the equation of motion of the ‘constrained’ plate (carrying the three kinds of multiple concentrated element) were derived. Finally, numerical methods were used to solve this equation of motion to give the natural frequencies and mode shapes of the ‘constrained’ plate. To confirm the reliability of previous free vibration analysis results, a finite element analysis was also conducted. It was found that the results obtained from the above-mentioned two approaches were in good agreement. Compared with the conventional finite element method (FEM), the approach employed in this paper has the advantages of saving computing time and achieving better accuracy, as can be seen from the existing literature.

Asymmetric free vibration analysis of first-order shear deformable functionally graded magneto-electro-thermo-elastic circular plates

2019 ◽  
Vol 233 (16) ◽  
pp. 5659-5675 ◽  
Author(s):  
Ehsan Arshid ◽  
Ali Kiani ◽  
Saeed Amir ◽  
Mustafa Zarghami Dehaghani
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Differential Quadrature Method ◽  
Shear Deformation Theory ◽  
Free Vibration Analysis ◽  
Functionally Graded ◽  
Mode Shapes ◽  
Circular Plates ◽  
First Order ◽  
Shear Deformable

The current study aims to analyze the asymmetric free vibration behavior of shear deformable functionally graded magneto-electro-thermo-elastic circular plates. The plate’s displacements are described by employing the first-order shear deformation theory and based on the von Karman assumptions, the strains and displacements are related together. Using Hamilton’s principle and variational formulation, the governing motion equations and also the associated boundary conditions have been derived. The generalized differential quadrature method is applied to discretize and solve them. The effects of the most important parameters such as material gradient index, electromagnetic loads, boundary conditions, and also aspect ratio of the plate on the natural frequencies and mode shapes of the plate are considered and discussed in details. The results show that the effect of electric potential on the natural frequency is the opposite of the magnetic one. In other words as the magnetic potential increases, the rigidity of the plate increases too and the frequency enhances. The results are compared and verified with the simpler states in literature. The findings of this study are useful for designing more efficient sensors and actuators used in smart or intelligent structures.

scholarly journals Free Vibration Analysis of L-Shaped Folded Thin Plates

2019 ◽  
Vol 2 (1) ◽  
pp. 67-73
Author(s):  
Koji Sekine
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Vibration Analysis ◽  
Ritz Method ◽  
Free Vibration Analysis ◽  
Energy Functional ◽  
Thin Plates ◽  
Mode Shapes ◽  
Width Ratio ◽  
Various Boundary Conditions

Free vibration analysis of L-shaped folded thin plates having various boundary conditions is presented. Vibration properties of the folded plates are analyzed by means of the Ritz method. Displacement functions satisfying the geometric boundary conditions are assumed in the form of double power series. The interconnection of plate elements of the folded plates is defined by translational and rotational coupling springs. The generalized eigenvalue problem, which is derived by means of minimizing the energy functional, is solved to determine the natural frequencies and mode shapes. The accuracy and validity of the present solutions are demonstrated through convergence studies and comparisons with the results from the literature and FEM (finite element method) analysis solutions. Numerical results are presented for different conditions, such as width ratio, length ratio and the four types of boundary condition.

scholarly journals Free Vibration Analysis of Laminated Composite Double-Plate Structure System with Elastic Constraints Based on Improved Fourier Series Method

2021 ◽  
Vol 2021 ◽  
pp. 1-25
Author(s):  
Ying Zhang ◽  
Dongyan Shi ◽  
Dongze He ◽  
Dong Shao
Keyword(s):  
Fourier Series ◽  
Free Vibration ◽  
Boundary Conditions ◽  
Laminated Composite ◽  
Mode Shapes ◽  
Series Method ◽  
Plate Structure ◽  
Fourier Series Method ◽  
Linear Algebraic Equations ◽  
Improved Fourier Series Method

An analytical model of laminated composite double-plate system (LCDPS) is established, which efficiently analyzes the common 3D plate structure in engineering applications. The proposed model combines the first-order shear deformation theory (FSDT) and the classical delamination theory, and then the LCDPS’s vibration characteristics are investigated. In the process of analysis, the improved Fourier series method (IFSM) is used to describe the displacement admissible function of the LCDPS, which can remove the potential discontinuities at the boundaries. Five sets of artificial springs are introduced to simulate the elastic boundary constraints, and the restraints of the Winkler elastic layer can be adjustable. The improved Fourier series is substituted into the governing equations and boundary conditions; then, applying the Rayleigh–Ritz method, we take all the series expansion coefficients as the generalized coordinates. After that, a set of standard linear algebraic equations was obtained. On this basis, the natural frequency and mode shapes of the LCDPS can be obtained by solving the standard eigenvalue problem. By the discussion of numerical examples and the comparison with those of the reports in the literature, the convergence and the reliability of the present approach are validated. Finally, the parametric investigations of the free vibration with complex boundary conditions are carried out, including the influence of boundary conditions, lamination scheme, plate geometric parameters, and elastic coefficient between two plates.

Free Vibration Analysis of Uniform and Asymmetric Composite Pretwisted Rotating Thin Walled Beam

2017 ◽  
Author(s):  
Touraj Farsadi ◽  
Özgün Şener ◽  
Altan Kayran
Keyword(s):  
Finite Element ◽  
Free Vibration ◽  
Boundary Conditions ◽  
Wind Turbine ◽  
Vibration Analysis ◽  
Structural Model ◽  
Free Vibration Analysis ◽  
Mode Shapes ◽  
Analysis Tool ◽  
Thin Walled

Composite pretwisted rotating thin walled beams (TWB) can be used as the structural model for composite helicopter and wind turbine blades for the study of aeroelastic response of the blades. In the present study, semi-analytical solution is performed for the free vibration analysis of uniform and asymmetric composite pretwisted rotating TWB. The approximation of the Green-Lagrange strain tensor is adopted to derive the strain field of the system. The Euler–Lagrange governing equations of the dynamic system and the related boundary conditions are derived via Hamilton’s principle. In order to solve the governing set of equations, the Extended Galerkin’s Method (EGM) is employed. For this purpose, the structural variables are separated in space and time and the assumed mode shapes are defined to satisfy the essential boundary conditions. For the purpose of validating the TWB model developed, the commercial finite element analysis tool, MSC Nastran is used to compare the results of modal analysis obtained by the present structural model with the finite element solution. With the results obtained in this paper, it is aimed to ascertain the effect of various coupling in circumferentially asymmetric stiffness (CAS) and circumferentially uniform stiffness CUS configurations, pretwist, angular velocity and fibre orientation, on the natural frequencies and the mode shapes of the rotating thin-walled composite beams. The results are expected to propose better predictions of the vibrational behavior of thin walled structures in general, and in the design of rotor blades of turbomachinery, rotorcraft and wind turbine systems, in particular.

scholarly journals Application of Adomian Modified Decomposition Method to Free Vibration Analysis of Rotating Beams

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Qibo Mao
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Vibration Analysis ◽  
Decomposition Method ◽  
Free Vibration Analysis ◽  
Mode Shapes ◽  
Rotating Beams ◽  
Arbitrary Boundary ◽  
Various Boundary Conditions ◽  
Governing Differential Equation

The Adomian modified decomposition method (AMDM) is employed in this paper for dynamic analysis of a rotating Euler-Bernoulli beam under various boundary conditions. Based on AMDM, the governing differential equation for the rotating beam becomes a recursive algebraic equation. By using the boundary condition equations, the dimensionless natural frequencies and corresponding mode shapes can be easily obtained simultaneously. The computed results for different boundary conditions as well as different offset length and rotational speeds are presented. The accuracy is assured from the convergence and comparison published results. It is shown that the AMDM offers an accurate and effective method of free vibration analysis of rotating beams with arbitrary boundary conditions.

Dynamic characteristics of horizontally curved bridges

2017 ◽  
Vol 24 (19) ◽  
pp. 4465-4483 ◽  
Author(s):  
Mohsen Amjadian ◽  
Anil K Agrawal
Keyword(s):  
Free Vibration ◽  
Dynamic Characteristics ◽  
Natural Frequencies ◽  
Response Spectrum ◽  
Translational Motion ◽  
Free Vibration Analysis ◽  
Mode Shapes ◽  
Curved Bridges ◽  
Horizontally Curved ◽  
Horizontally Curved Bridges

Horizontally curved bridges have complicated dynamic characteristics because of their irregular geometry and nonuniform mass and stiffness distributions. This paper aims to develop a simplified and practical method for the calculation of the natural frequencies and mode shapes of horizontally curved bridges that would be of interest to bridge engineers for the estimation of the seismic response of these types of bridges. For this purpose, a simple three-degree-of-freedom (3DOF) dynamic model for free vibration equation of this type of bridge has been developed. It is shown that the translational motion of the deck of horizontally curved bridges in the direction that is perpendicular to their axis of symmetry is always coupled with the rotational motion of the deck, regardless of the location of the stiffness center. The model is further exploited to develop closed-form formulas for the estimation of the maximum displacements of the corners of the deck of one-way asymmetric horizontally curved bridges. The accuracy of the model is verified by finite-element model of a horizontally curved bridge prototype in OpenSEES. Finally, the model is utilized to study the influence of the location of the stiffness center with respect to the deck curvature center on the natural frequency and the maximum displacements of the corners of the deck for different curvatures of the deck. The results of free vibration analysis show that the natural frequencies of one-way asymmetric horizontally curved bridges, in general, increase with the increase of the subtended angle of the deck. The results of earthquake response spectrum analysis show that the increase in the subtended angle of one-way asymmetric horizontally curved bridges decreases the radial displacements of the corners of the deck but increases the azimuthal displacement. These two responses both increase with the increase in the distance between the stiffness center and the curvature center.

Sign in / Sign up

Export Citation Format

Share Document