scholarly journals An Approximate Analytical Solution of Sloshing Frequencies for a Liquid in Various Shape Aqueducts

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Yuchun Li ◽  
Zhuang Wang
Keyword(s):  
Analytical Solution ◽  
Approximate Method ◽  
Approximate Analytical Solution ◽  
Ritz Method ◽  
Approximate Solutions ◽  
The Other ◽  
Engineering Applications ◽  
Tuned Liquid Dampers ◽  
Liquid Dampers ◽  
Experimental Values

An approximate analytical solution of sloshing frequencies for a liquid in the various shape aqueducts is formulated by using the Ritz method. The present approximate method is, respectively, applied to find the sloshing frequencies of the liquid in rectangular, trapezoid, oval, circular, U-shaped tanks (aqueducts), and various shape tuned liquid dampers (TLD). The first three antisymmetric and symmetric frequencies by the present approach are within 5% accuracy compared to the other analytical, numerical, and experimental values. The approximate solutions of this paper for the various shape aqueducts are acceptable to the engineering applications.

Vibration of Rectangular Plates by the Ritz Method

1950 ◽  
Vol 17 (4) ◽  
pp. 448-453 ◽  
Author(s):  
Dana Young
Keyword(s):  
Ritz Method ◽  
Normal Modes ◽  
Approximate Solutions ◽  
The Other ◽  
Rectangular Plates ◽  
Elastic Plates ◽  
Modes Of Vibration ◽  
Ritz’S Method ◽  
Square Plate ◽  
Set Up

Abstract Ritz’s method is one of several possible procedures for obtaining approximate solutions for the frequencies and modes of vibration of thin elastic plates. The accuracy of the results and the practicability of the computations depend to a great extent upon the set of functions that is chosen to represent the plate deflection. In this investigation, use is made of the functions which define the normal modes of vibration of a uniform beam. Tables of values of these functions have been computed as well as values of different integrals of the functions and their derivatives. With the aid of these data, the necessary equations can be set up and solved with reasonable effort. Solutions are obtained for three specific plate problems, namely, (a) square plate clamped at all four edges, (b) square plate clamped along two adjacent edges and free along the other two edges, and (c) square plate clamped along one edge and free along the other three edges.

scholarly journals Wrinkling of Tympanic Membrane Under Unbalanced Pressure

2017 ◽  
Vol 84 (4) ◽  
Author(s):  
Bo Wang ◽  
Pravarsha Ghanta ◽  
Sandra Vinnikova ◽  
Siyuan Bao ◽  
Junfeng Liang ◽  
...  
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
Analytical Solution ◽  
Tympanic Membrane ◽  
Middle Ear ◽  
Approximate Analytical Solution ◽  
Ritz Method ◽  
Transmission Function ◽  
Acoustic Transmission ◽  
Element Analysis

Mechanics of tympanic membrane (TM) is crucial for investigating the acoustic transmission through the ear. In this study, we studied the wrinkling behavior of tympanic membrane when it is exposed to mismatched air pressure between the ambient and the middle ear. The Rayleigh–Ritz method is adopted to analyze the critical wrinkling pressure and the fundamental eigenmode. An approximate analytical solution is obtained and validated by finite element analysis (FEA). The model will be useful in future investigations on how the wrinkling deformation of the TM alters the acoustic transmission function of the ear.

Dynamics of an isolated barotropic eddy on a beta-plane

1994 ◽  
Vol 264 ◽  
pp. 277-301 ◽  
Author(s):  
Gennady K. Korotaev ◽  
Alexander B. Fedotov
Keyword(s):  
Kinetic Energy ◽  
Analytical Solution ◽  
Numerical Experiment ◽  
Point Vortex ◽  
Vortex Sheet ◽  
The Other ◽  
Beta Plane ◽  
The Difference ◽  
Experimental Values ◽  
Good Agreement

The dynamics of a Gaussian isolated barotropic eddy on a β-plane is considered. The analytical solution of the evolution of an isolated vortex is constructed by analogy to the theory of a point vortex. The results of a numerical experiment are compared with the conclusions of the theory for the case of the Gaussian vortex. Characteristics of the vortex such as its radius, trajectory of movement, kinetic energy, residual vorticity, and the structure of the vortex are discussed. The analysis of the numerical results shows that the experimentally determined radius of the vortex, its energy, and residual vorticity are in good agreement with the theory. On the other hand there is a difference between analytical and experimental values of velocity components, and hence in the trajectory of the centre of the vortex. The location of the separatrix of the streak function and its saddle point are considered as important characteristics of the structure of the vortex. We consider the phenomenon of the generation of the vortex sheet connected with the separatrix location as a cause of the difference between the experimental and analytical estimates of the velocity of the vortex.

Vibration of Rectangular and Skew Cantilever Plates

1951 ◽  
Vol 18 (2) ◽  
pp. 129-134 ◽  
Author(s):  
M. V. Barton
Keyword(s):  
Ritz Method ◽  
Normal Modes ◽  
Approximate Solutions ◽  
Equal Length ◽  
Uniform Beam ◽  
Analytical Results ◽  
Modes Of Vibration ◽  
Experimental Values ◽  
Skew Angles ◽  
Plate Deflection

Abstract The Ritz method is used to determine approximate solutions for the frequencies and modes of vibration of uniform rectangular and skew cantilever plates. The functions used to represent the plate deflection correspond to those which define the normal modes of vibration of a uniform beam. Solutions are obtained for (a) uniform rectangular cantilever plates with cantilever span-to-breadth ratios of 1/2, 1, 2, and 5; (b) uniform skew cantilever plates with sides of equal length and skew angles of 15 deg, 30 deg, and 45 deg. Experimental values are presented and the correlation with analytical results discussed.

Flexural Vibration of Rectangular Plates with Stiffeners Parallel to the Edges

1963 ◽  
Vol 67 (634) ◽  
pp. 664-668 ◽  
Author(s):  
S. Mahalingam
Keyword(s):  
Natural Frequencies ◽  
Ritz Method ◽  
Flexural Vibration ◽  
Approximate Solutions ◽  
The Other ◽  
Rectangular Plates ◽  
Equivalent System ◽  
Simply Supported ◽  
Finite Difference Equations ◽  
Four Corners

SummaryThe basis of the procedure described in the paper is the replacement of the stiffeners by an approximately equivalent system of line springs. One of two methods may then be used to determine the natural frequencies. A rectangular plate with edge stiffeners, point-supported at the four corners, is used to demonstrate the application of the Rayleigh-Ritz method. Numerical results obtained are compared with known approximate solutions based on finite difference equations. A Holzer-type iteration is employed in the case of a plate with parallel stiffeners, where the two edges perpendicular to the stiffeners are simply supported, the other two edges having any combination of conditions.

Approximate Analytical Solutions of the Pull-In Behavior of an Electrostatically Actuated Clamped-Clamped Micro-Beam

2013 ◽  
Vol 300-301 ◽  
pp. 419-422
Author(s):  
Sheng Li Kong
Keyword(s):  
Analytical Solution ◽  
Analytical Solutions ◽  
Approximate Analytical Solution ◽  
Ritz Method ◽  
High Accuracy ◽  
Electrostatically Actuated ◽  
Deformation Problem ◽  
Approximate Analytical Solutions ◽  
Analytical Expressions

For the deformation problem of electrostatically actuated clamped-clamped micro-beams, pull-in behaviors of the micro-beams have been analyzed by using Rayleigh-Ritz method. Approximate analytical expressions for pull-in voltage and normalized pull-in displacement of the micro-beam have been obtained. When the pull-in occurs, the pull-in voltage and normalized pull-in displacement of the micro-beam at the mid-span position are 38.6V and 0.398, respectively. The results show that the approximate analytical solution possesses high accuracy.

Experimental and Numerical Studies of Axisymmetric Tuned Liquid Dampers

2015 ◽  
Author(s):  
Bernard Molin ◽  
Fabien Remy ◽  
Julien Bonnici
Keyword(s):  
Test Bench ◽  
Added Mass ◽  
Analytical Models ◽  
Hydrodynamic Coefficients ◽  
Earthquake Excitation ◽  
Tuned Liquid Dampers ◽  
Numerical Studies ◽  
Liquid Dampers ◽  
Experimental Values ◽  
Good Agreement

Tuned Liquid Dampers (TLDs) have proved their efficiency to mitigate the vibratory response of slender buildings under wind or earthquake excitation. Simple semi-analytical models are proposed here to derive the hydrodynamic coefficients (added mass and damping) of axisymmetric TLDs fitted with circular or radial perforated screens. Comparisons are made with experimental values obtained with an Hexapode test bench. Good agreement is observed.

A new algorithm for finding CRM-model coefficients

2019 ◽  
Vol 5 (3) ◽  
pp. 164-185 ◽  
Author(s):  
Alexander D. Bekman ◽  
Sergey V. Stepanov ◽  
Alexander A. Ruchkin ◽  
Dmitry V. Zelenin
Keyword(s):  
Approximate Solution ◽  
Optimization Problem ◽  
Optimal Solution ◽  
Complex Form ◽  
Approximate Solutions ◽  
The Other ◽  
Programming Algorithm ◽  
Stochastic Algorithms ◽  
Large Set ◽  
Flow Rates

The quantitative evaluation of producer and injector well interference based on well operation data (profiles of flow rates/injectivities and bottomhole/reservoir pressures) with the help of CRM (Capacitance-Resistive Models) is an optimization problem with large set of variables and constraints. The analytical solution cannot be found because of the complex form of the objective function for this problem. Attempts to find the solution with stochastic algorithms take unacceptable time and the result may be far from the optimal solution. Besides, the use of universal (commercial) optimizers hides the details of step by step solution from the user, for example&nbsp;— the ambiguity of the solution as the result of data inaccuracy.<br> The present article concerns two variants of CRM problem. The authors present a new algorithm of solving the problems with the help of “General Quadratic Programming Algorithm”. The main advantage of the new algorithm is the greater performance in comparison with the other known algorithms. Its other advantage is the possibility of an ambiguity analysis. This article studies the conditions which guarantee that the first variant of problem has a unique solution, which can be found with the presented algorithm. Another algorithm for finding the approximate solution for the second variant of the problem is also considered. The method of visualization of approximate solutions set is presented. The results of experiments comparing the new algorithm with some previously known are given.

Numerical Approximations to Solution of Biochemical Reaction Model

2011 ◽  
Vol 9 (1) ◽  
Author(s):  
Ahmet Yildirim ◽  
Ahmet Gökdogan ◽  
Mehmet Merdan
Keyword(s):  
Analytical Solution ◽  
Approximate Analytical Solution ◽  
Transformation Method ◽  
Reaction Model ◽  
Biochemical Reaction ◽  
Graphical Form ◽  
Kutta Method ◽  
Transform Method ◽  
Runge Kutta Method ◽  
Differential Transform

In this paper, approximate analytical solution of biochemical reaction model is used by the multi-step differential transform method (MsDTM) based on classical differential transformation method (DTM). Numerical results are compared to those obtained by the fourth-order Runge-Kutta method to illustrate the preciseness and effectiveness of the proposed method. Results are given explicit and graphical form.

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