Differential transformation method for vibration of membranes (original) (raw)
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A New Analytical Approach For Free Vibration Of Membrane From Wave Standpoint
2008
In this paper, an analytical approach for free vibration analysis of rectangular and circular membranes is presented. The method is based on wave approach. From wave standpoint vibration propagate, reflect and transmit in a structure. Firstly, the propagation and reflection matrices for rectangular and circular membranes are derived. Then, these matrices are combined to provide a concise and systematic approach to free vibration analysis of membranes. Subsequently, the eigenvalue problem for free vibration of membrane is formulated and the equation of membrane natural frequencies is constructed. Finally, the effectiveness of the approach is shown by comparison of the results with existing classical solution.
International Journal for Numerical Methods in Engineering, 1988
In this paper a boundary element approach is presented to establish the eigenfrequencies and the mode shapes of the free vibrations of flexible membranes with arbitrary shape. The membranes may be of homogeneous, non-homogeneous or composite material, while their boundaries may be fixed or elastically supported. The implementation of the method is mainly based on the development of both a boundary element technique to compute numerically the Green function for the Laplace equation and a numerical procedure to solve domain integral equations using two-dimensional Gauss integration on regions of arbitrary shape. The efficiency of the method presented is demonstrated by applying it to study the free vibrations of membranes consisting of homogeneous, non-homogeneous and composite material, fixed or elastically supported on the boundary, and having various shapes, such as circular, rectangular, triangular, or elliptical, or being L-shaped or of arbitrary shape.
JOURNAL OF ADVANCES IN MATHEMATICS, 2015
The aim of this paper is to present a reliable and efficient algorithm Elzaki projected differential transform method (EPDTM) to obtain the explicit solution of vibration equation for a very large membrane with given initial conditions. By using initial conditions, explicit series solutions for six different cases have been derived for the fast convergence of the solution. Numerical results show the reliability, efficiency and accuracy of Elzaki projected differential transform method (EPDTM). Numerical results for the six different cases are presented graphically.
2015
One of the powerful analytical methods to solve partial differential equation is the Adomian Decomposition Method (ADM). In t his paper, a general approach based on the generalized Fourier series expansion is applied to obtain an analytical solution. The solution is simplified in terms of a given orthogonal basis functions that these functions satisfy the boundary conditions. Based on the success of Fourier analysis and Hilbert space theory, orthogonal expansions undoubtedly count as fundamental concept of mathematical analysis. For the first time, We solved this equation using ADM and the results are compared with classical methods to demonstrate the accuracy of the scheme.
Vibration analysis of continuous systems by differential transformation
Applied Mathematics and Computation, 1998
This paper demonstrates the application of the technique of differential transformation to free vibration of continuous systems. The specific problem chosen for this purpose is that of thin beams. The transformation technique is employed for deriving frequency equations and mode functions. The frequencies and mode shapes obtained from the differential-transformation solutions are compared with the exact analytical solutions. It is shown that the solutions obtained from the technique have a very high degree of accuracy.
This paper provides a new technique for solving free vibration problems of composite arbitrarily shaped membranes by using Generalized Differential Quadrature Finite Element Method (GDQFEM). The proposed technique, also known as Multi-Domain Differential Quadrature (MDQ), is an extension of the classic Generalized Differential Quadrature (GDQ) procedure. The multi-domain method can be directly applied to regular sub-domains of rectangular shape, as well as to elements of general shape when a coordinate transformation is considered. The mapping technique is used to transform both the governing differential equations and the compatibility conditions between two adjacent sub-domains into the regular master element in the parent space, called computational space. The numerical implementation of the global algebraic system obtained by the technique at issue is simple and straightforward. Computer investigations concerning a large number of membrane geometries have been carried out. GDQFEM results are compared with those presented in the literature and a perfect agreement is observed. Membranes of complex geometry with a material inhomogeneity are also carefully examined. Numerical results referring to some new unpublished geometric shapes are reported to let comparisons with further research on this subject.
Analytical and experimental investigation on vibrating membranes with a central point support
Journal of Sound …, 1999
This investigation arose from the practical necessity of placing a centrifugal pump rigidly attached to a thin, circular cover plate of a water tank in a medium size ocean vessel. Due to lack of space, it was necessary to locate the system off-center of the circular configuration. It was considered necessary to calculate the fundamental frequency of the coupled system.
A FINITE DIFFERENCES SOLUTION TO THE VIBRATING MEMBRANE PROBLEM
A realistic approach to the solution of mechanical systems containing multiple parameters must take into account the fact that dependent variables depend not only on , but on more space variables. The modelling of such problems leads to partial differential equations (P.D.Es), rather than Ordinary Differential Equations (O.D.Es). Here, the wave equation, a P.D E that governs the vibrating membrane problem is considered. A finite difference method (F.D.M) is provided as an alternative to the analytic methods. F.D.Ms basically involve three steps; dividing the solution into grids of nodes, approximating the given differential equation by finite difference equivalences that relate the solutions to grid points and solving the difference equations subject to the prescribed boundary and/or initial conditions. It is shown here that the error in the result is relatively negligible, and the conclusion made that the method developed can further be used to solve certain non-linear P.D.Es.