Vibration of Circular Rings With Local Deviation

1994 ◽  
Vol 61 (2) ◽  
pp. 317-322 ◽  
Author(s):  
Jin Sun Hong ◽  
Jang Moo Lee
Keyword(s):  
Sharp Decrease ◽  
Transformation Method ◽  
Step Function ◽  
Mode Shapes ◽  
Bending Vibration ◽  
Element Computation ◽  
Local Deviation ◽  
Circular Rings ◽  
Laplace Transformation Method ◽  
Heaviside Unit Step Function

An analytical method is presented for predicting the effects of local deviations such as a point mass, a sharp decrease in stiffness and variation of thickness on the free in-plane bending vibration characteristics of nearly axisymmetric rings. The approach is based on the Laplace transformation method and the expression of local deviation as the variation of heaviside unit step function. The effects of local deviation on the natural frequencies and mode shapes of the rings are predicted for the proposed nondimensional mass and stiffness parameters. The validity of the approach was examined through modal testings and/or finite element computation.

scholarly journals Exact Solutions of a Spatially Dependent Mass Dirac Equation for Coulomb Field plus Tensor Interaction via Laplace Transformation Method

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
M. Eshghi ◽  
M. Hamzavi ◽  
S. M. Ikhdair
Keyword(s):  
Dirac Equation ◽  
Laplace Transformation ◽  
Bound States ◽  
Energy Eigenvalue ◽  
Coulomb Field ◽  
Transformation Method ◽  
Arbitrary Spin ◽  
Tensor Interaction ◽  
Dependent Mass ◽  
Laplace Transformation Method

The spatially dependent mass Dirac equation is solved exactly for attractive scalar and repulsive vector Coulomb potentials including a tensor interaction potential under the spin and pseudospin (p-spin) symmetric limits by using the Laplace transformation method (LTM). Closed forms of the energy eigenvalue equation and wave functions are obtained for arbitrary spin-orbit quantum number κ. Some numerical results are given too. The effect of the tensor interaction on the bound states is presented. It is shown that the tensor interaction removes the degeneracy between two states in the spin doublets. We also investigate the effects of the spatially-dependent mass on the bound states under the conditions of the spin symmetric limit and in the absence of tensor interaction (T=0).

In-Plane Vibration Modes of Arbitrarily Thick Disks

1997 ◽  
Author(s):  
Kevin I. Tzou ◽  
Jonathan A. Wickert ◽  
Adnan Akay
Keyword(s):  
Natural Frequencies ◽  
Arbitrary Dimension ◽  
Three Dimensional ◽  
Mode Shapes ◽  
Vibration Modes ◽  
Three Dimensional Model ◽  
Bending Vibration ◽  
Annular Disk ◽  
Out Of Plane ◽  
Plane Bending

Abstract The three-dimensional vibration of an arbitrarily thick annular disk is investigated for two classes of boundary conditions: all surfaces traction-free, and all free except for the clamped inner radius. These two models represent limiting cases of such common engineering components as automotive and aircraft disk brakes, for which existing models focus on out-of-plane bending vibration. For a disk of significant thickness, vibration modes in which motion occurs within the disk’s equilibrium plane can play a substantial role in setting its dynamic response. Laboratory experiments demonstrate that in-plane modes exist at frequencies comparable to those of out-of-plane bending even for thickness-to-diameter ratios as small as 10−1. The equations for three-dimensional motion are discretized through the Ritz technique, yielding natural frequencies and mode shapes for coupled axial, radial, and circumferential deformations. This treatment is applicable to “disks” of arbitrary dimension, and encompasses classical models for plates, bars, cylinders, rings, and shells. The solutions so obtained converge in the limiting cases to the values expected from the classical theories, and to ones that account for shear deformation and rotary inertia. The three-dimensional model demonstrates that for geometries within the technologically-important range, the natural frequencies of certain in- and out-of-plane modes can be close to one another, or even identically repeated.

Two-Temperature Generalized Thermo-Elastic Medium Thermally Excited by Time Exponentially Decaying Laser Pulse

2016 ◽  
Vol 16 (03) ◽  
pp. 1450102 ◽  
Author(s):  
Hamdy M. Youssef ◽  
A. A. El-Bary
Keyword(s):  
Laser Pulse ◽  
Absorption Coefficient ◽  
Transformation Method ◽  
State Space Approach ◽  
Closed Form Solutions ◽  
Riemann Sum ◽  
Temperature Parameter ◽  
Laplace Transformation Method ◽  
Two Temperature ◽  
Space Approach

This paper deals with a two-temperature thermoelastic material subjected to a laser heating pulse as the heat source. Closed form solutions for the temperature and stress fields due to time exponentially decaying laser pulse are presented using the state-space approach. The Laplace transformation method is employed in deriving the governing equations. The inversion of Laplace transform is obtained numerically by using the Riemann-sum approximation method. The results have been presented in figures to show the effect of the time exponentially decaying laser pulse, the two-temperature parameter and the absorption coefficient on all the fields studied. The results show that the two-temperature parameter and the absorption coefficient parameter have significant effects on all the field parameters studied.

scholarly journals A new Laplace transformation method for dynamic testing of solar collectors

2015 ◽  
Vol 75 ◽  
pp. 448-458 ◽  
Author(s):  
Weiqiang Kong ◽  
Bengt Perers ◽  
Jianhua Fan ◽  
Simon Furbo ◽  
Federico Bava
Keyword(s):  
Laplace Transformation ◽  
Dynamic Testing ◽  
Transformation Method ◽  
Solar Collectors ◽  
Laplace Transformation Method

Solution of the Boundary Layer Problems for Calculating the Natural Modes of Riser-Type Slender Structures

2006 ◽  
Author(s):  
Ioannis K. Chatjigeorgiou
Keyword(s):  
Boundary Layer ◽  
Natural Frequencies ◽  
Mathematical Formulation ◽  
Mode Shapes ◽  
Perturbation Approach ◽  
Problem Solution ◽  
Bending Vibration ◽  
Boundary Layer Problem ◽  
Boundary Layer Problems ◽  
Slender Structures

The present work treats the problem of the calculation of the natural frequencies and the corresponding bending vibration modes of vertical slender structures. The originality of the study lies on fact that for the derivation of natural frequencies and the corresponding mode shapes, all physical properties that influence the bending vibration of the structure were considered including the aspect of the variation of tension. The resulting mathematical formulation incorporates all principal contributions such as the bending stiffness, the weight and the tension variation. The governing equation is treated using a perturbation approach. The application of this method results to the development of two boundary layer problems at the ends of the structure. These problems are treated properly using a boundary layer problem solution methodology in order to obtain asymptotic approximations to the shape of the vibrating riser-type structure. It should be noted that in this work the term ‘boundary layer’ is not connected with fluid flows but it is used to indicate the narrow region across which the dependent variable undergoes very rapid changes. Frequently these narrow regions adjoin the boundaries of the domain of intersect, especially when the small parameter multiplies the highest derivative.

Comparison of Adomian Decomposition Method and Differential Transformation Method for Vibration Problems of Euler-Bernoulli Beam

2012 ◽  
Vol 157-158 ◽  
pp. 476-483
Author(s):  
Zhi Feng Liu ◽  
Chun Hua Guo ◽  
Li Gang Cai ◽  
Wen Tong Yang ◽  
Zhi Min Zhang
Keyword(s):  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Transformation Method ◽  
Differential Transformation Method ◽  
Mode Shapes ◽  
Bernoulli Beam ◽  
Differential Transformation ◽  
Adomian Decomposition ◽  
Vibration Problems ◽  
Euler Bernoulli Beam

In this paper, we compare the Differential transformation method and Adomian decomposition method to solve Euler-Bernoulli Beam vibration problems. The natural frequencies and mode shapes of the clamped-free uniform Euler-Bernoulli equation are calculated using the two methods. The Adomian decomposition method avoids the difficulties and massive computational work inherent in Differential transformation method by determining the very rapidly convergent analytic solutions directly. We found the solution between the two methods to be quite close. According to calculation of eigenvalues, natural frequencies and mode shapes, we compare the convergence of Differential transformation method and Adomian decomposition method. The two methods can be alternative ways to solve linear and nonlinear higher-order initial value problems.

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