Analytical Solutions for In-Plane Dominated Vibrations of Transversely Isotropic Circular Plates Based on High-Order Theories

2021 ◽  
pp. 116110
Author(s):  
Ming Ji ◽  
Yi-Chuang Wu ◽  
Chien-Ching Ma
Keyword(s):  
Analytical Solutions ◽  
Transversely Isotropic ◽  
High Order ◽  
Circular Plates

Pseudoacoustic tilted transversely isotropic modeling with optimal k-space operator-based implicit finite-difference schemes

Geophysics ◽  
2018 ◽  
Vol 83 (3) ◽  
pp. T139-T157 ◽  
Author(s):  
Shigang Xu ◽  
Yang Liu
Keyword(s):  
Finite Difference ◽  
Spatial Modeling ◽  
Wave Equations ◽  
Transversely Isotropic ◽  
Dispersion Relations ◽  
High Order ◽  
Space Operator ◽  
Modeling Accuracy ◽  
Tti Media ◽  
Temporal And Spatial

Current temporal high-order finite-difference (FD) stencils are mainly designed for isotropic wave equations, which cannot be directly extended to pseudoacoustic wave equations (PWEs) in tilted transversely isotropic (TTI) media. Moreover, it is difficult to obtain the time-space domain FD coefficients for anisotropic PWEs based on nonlinear dispersion relations in which anisotropy parameters are coupled with FD coefficients. Therefore, a second-order FD for temporal derivatives and a high-order FD for spatial derivatives are commonly used to discretize PWEs in TTI media. To improve the temporal and spatial modeling accuracy further, we have developed several effective FD schemes for modeling PWEs in TTI media. Through combining the [Formula: see text] (wavenumber)-space operators with the conventional implicit FD stencils (i.e., regular-grid [RG], staggered-grid [SG], and rotated SG [RSG]), we establish novel dispersion relations and determine FD coefficients using least-squares (LS). Based on [Formula: see text]-space operator compensation, we adopt the modified LS-based implicit RG-FD, implicit SG-FD, and implicit RSG-FD methods to respectively solve the second- and first-order PWEs in TTI media. Dispersion analyses indicate that the modified implicit FD schemes based on [Formula: see text]-space operator compensation can greatly increase the numerical accuracy at large wavenumbers. Modeling examples in TTI media demonstrate that the proposed FD schemes can adopt a short FD operator to simultaneously achieve high temporal and spatial modeling accuracy, thus significantly improve the computational efficiency compared with the conventional methods.

scholarly journals Piecewise Approximate Analytical Solutions of High-Order Singular Perturbation Problems with a Discontinuous Source Term

2016 ◽  
Vol 2016 ◽  
pp. 1-12 ◽  
Author(s):  
Essam R. El-Zahar
Keyword(s):  
Asymptotic Expansion ◽  
Singular Perturbation ◽  
Analytical Solutions ◽  
Source Term ◽  
High Order ◽  
Singular Perturbation Problems ◽  
Weakly Coupled System ◽  
Approximate Analytical Solutions ◽  
Discontinuous Source Term ◽  
Perturbation Problems

A reliable algorithm is presented to develop piecewise approximate analytical solutions of third- and fourth-order convection diffusion singular perturbation problems with a discontinuous source term. The algorithm is based on an asymptotic expansion approximation and Differential Transform Method (DTM). First, the original problem is transformed into a weakly coupled system of ODEs and a zero-order asymptotic expansion of the solution is constructed. Then a piecewise smooth solution of the terminal value reduced system is obtained by using DTM and imposing the continuity and smoothness conditions. The error estimate of the method is presented. The results show that the method is a reliable and convenient asymptotic semianalytical numerical method for treating high-order singular perturbation problems with a discontinuous source term.

scholarly journals INVESTIGATION OF THE DYNAMIC BEHAVIOUR OF NON-UNIFORM THICKNESS CIRCULAR PLATES RESTING ON WINKLER AND PASTERNAK FOUNDATIONS

2020 ◽  
Vol 60 (2) ◽  
pp. 127-144
Author(s):  
Saheed Salawu ◽  
Gbeminiyi Sobamowo ◽  
Obanishola Sadiq
Keyword(s):  
Natural Frequency ◽  
Circular Plate ◽  
Analytical Solutions ◽  
Dynamic Behaviour ◽  
Circumferential Stress ◽  
Thickness Variation ◽  
Circular Plates ◽  
Uniform Thickness ◽  
Weighted Residuals ◽  
Elastic Foundations

The study of the dynamic behaviour of non-uniform thickness circular plate resting on elastic foundations is very imperative in designing structural systems. This present research investigates the free vibration analysis of varying density and non-uniform thickness isotropic circular plates resting on Winkler and Pasternak foundations. The governing differential equation is analysed using the Galerkin method of weighted residuals. Linear and nonlinear case is considered, the surface radial and circumferential stresses are also determined. Thereafter, the accuracy and consistency of the analytical solutions obtained are ascertained by comparing the existing results available in pieces of literature and confirmed to be in a good harmony. Also, it is observed that very accurate results can be obtained with few computations. Issues relating to the singularity of circular plate governing equations are handled with ease. The analytical solutions obtained are used to determine the influence of elastic foundations, homogeneity and thickness variation on the dynamic behaviour of the circular plate, the effect of vibration on a free surface of the foundation as well as the influence of radial and circumferential stress on mode shapes of the circular plate considered. From the results, it is observed that a maximum of 8.1% percentage difference is obtained with the solutions obtained from other analytical methods. Furthermore, increasing the elastic foundation parameter increases the natural frequency. Extrema modal displacement occurs due to radial and circumferential stress. Natural frequency increases as the thickness of the circular plate increases, Conversely, a decrease in natural frequency is observed as the density varies. It is envisioned that; the present study will contribute to the existing knowledge of the classical theory of vibration.

Exact solutions for free vibration of transversely isotropic piezoelectric circular plates

2000 ◽  
Vol 16 (2) ◽  
pp. 141-147 ◽  
Author(s):  
Ding Haojiang ◽  
Xu Rongqiao ◽  
Chen Weiqui
Keyword(s):  
Free Vibration ◽  
Exact Solutions ◽  
Transversely Isotropic ◽  
Circular Plates

Exact solutions of axisymmetric free vibration of transversely isotropic magnetoelectroelastic laminated circular plates

2006 ◽  
Vol 23 (2) ◽  
pp. 115-127 ◽  
Author(s):  
Jiangying Chen ◽  
Rongqiao Xu ◽  
Xusheng Huang ◽  
Haojiang Ding
Keyword(s):  
Free Vibration ◽  
Exact Solutions ◽  
Transversely Isotropic ◽  
Circular Plates
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