scholarly journals The dynamical problem on acting concentrated load on the elastic quarter space

2021 ◽  
Vol 25 (2(36)) ◽  
pp. 7-25
Author(s):  
A. A. Fesenko ◽  
K. S. Bondarenko
Keyword(s):  
Fourier Transforms ◽  
Equations Of Motion ◽  
Compressive Load ◽  
Integral Transforms ◽  
Dynamical Problem ◽  
Asymptotic Formulas ◽  
Initial Moment ◽  
Concentrated Load ◽  
Quarter Space ◽  
Matrix Differential

The wave field of an elastic quarter space is constructed when one face is rigidly fixed and a dynamic normal compressive load acts on the other along a rectangular section at the initial moment of time. Integral Laplace and Fourier transforms are applied sequentially to the equations of motion and boundary conditions in contrast to traditional approaches when integral transforms are applied to solutions' representations through harmonic functions. This leads to a one-dimensional vector homogeneous boundary value problem with respect to unknown displacement's transformants. The problem was solved using matrix differential calculus. The original displacement field was found after applying inverse integral transforms. For the case of stationary vibrations a method of calculating integrals in the solution in the near loading zone was indicated. For the analysis of oscillations in a remote zone the asymptotic formulas were constructed. The amplitude of vertical vibrations was investigated depending on the shape of the load section, natural frequencies of vibrations and the material of the medium.

Extending the unified transform: curvilinear polygons and variable coefficient PDEs

2018 ◽  
Vol 40 (2) ◽  
pp. 976-1004 ◽  
Author(s):  
Matthew J Colbrook
Keyword(s):  
Fourier Transforms ◽  
Elliptic Boundary ◽  
Neumann Boundary ◽  
Integral Transforms ◽  
Variable Coefficients ◽  
Central Component ◽  
Unknown Boundary ◽  
Boundary Values ◽  
Global Relation ◽  
Chebyshev Interpolation

Abstract We provide the first significant extension of the unified transform (also known as the Fokas method) applied to elliptic boundary value problems, namely, we extend the method to curvilinear polygons and partial differential equations (PDEs) with variable coefficients. This is used to solve the generalized Dirichlet-to-Neumann map. The central component of the unified transform is the coupling of certain integral transforms of the given boundary data and of the unknown boundary values. This has become known as the global relation and, in the case of constant coefficient PDEs, simply links the Fourier transforms of the Dirichlet and Neumann boundary values. We extend the global relation to PDEs with variable coefficients and to domains with curved boundaries. Furthermore, we provide a natural choice of global relations for separable PDEs. These generalizations are numerically implemented using a method based on Chebyshev interpolation for efficient and accurate computation of the integral transforms that appear in the global relation. Extensive numerical examples are provided, demonstrating that the method presented in this paper is both accurate and fast, yielding exponential convergence for sufficiently smooth solutions. Furthermore, the method is straightforward to use, involving just the construction of a simple linear system from the integral transforms, and does not require knowledge of Green’s functions of the PDE. Details on the implementation are discussed at length.

Wavelet packets associated with linear canonical transform on spectrum

2021 ◽  
pp. 2150030
Author(s):  
M. Younus Bhat ◽  
Aamir H. Dar
Keyword(s):  
Fourier Transform ◽  
Multiresolution Analysis ◽  
Fourier Transforms ◽  
Fractional Fourier Transform ◽  
Integral Transforms ◽  
Wavelet Packets ◽  
Linear Canonical Transform ◽  
Fresnel Transform ◽  
The Fourier Transform ◽  
Vector Valued

The linear canonical transform (LCT) provides a unified treatment of the generalized Fourier transforms in the sense that it is an embodiment of several well-known integral transforms including the Fourier transform, fractional Fourier transform, Fresnel transform. Using this fascinating property of LCT, we, in this paper, constructed associated wavelet packets. First, we construct wavelet packets corresponding to nonuniform Multiresolution analysis (MRA) associated with LCT and then those corresponding to vector-valued nonuniform MRA associated with LCT. We investigate their various properties by means of LCT.

Derivation by a Transform Method of Integral Equations of Unsteady Lifting Surface Theory in Subsonic and Supersonic Flow

1979 ◽  
Vol 30 (4) ◽  
pp. 529-543
Author(s):  
Shigenori Ando ◽  
Akio Ichikawa
Keyword(s):  
Fourier Transforms ◽  
Integral Transforms ◽  
Physical Models ◽  
Streamwise Direction ◽  
Fourier Inversion ◽  
Transform Method ◽  
Inviscid Flows ◽  
Surface Theory ◽  
Method Of Integral Equations ◽  
Lifting Surface

SummaryApplications of “integral transforms of in-plane coordinate variables” in order to formulate unsteady planar lifting surface theories are demonstrated for both sub- and supersonic inviscid flows. It is concise and pithy. Fourier transforms are exclusively used, except for only Laplace transform in the supersonic streamwise direction. It is found that the streamwise Fourier inversion in the subsonic case requires some caution. Concepts based on the theory of distributions seem to be essential, in order to solve the convergence difficulties of integrals. Apart from this caution, the method of integral transforms of in-plane coordinate variables makes it be pure-mathematical to formulate the lifting surface problems, and makes aerodynamicist’s experiences and physical models such as vortices or doublets be useless.

scholarly journals Response Due To Impulsive Force In Generalized Thermomicrostretch Elastic Solid

2015 ◽  
Vol 20 (3) ◽  
pp. 487-502
Author(s):  
V. Kumar ◽  
R. Singh
Keyword(s):  
Fourier Transforms ◽  
Normal Force ◽  
Elastic Solid ◽  
Integral Transforms ◽  
Normal Displacement ◽  
Impulsive Force ◽  
Field Equations ◽  
Laplace And Fourier Transforms ◽  
Eigen Value ◽  
Plain Strain

Abstract A two dimensional Cartesian model of a generalized thermo-microstretch elastic solid subjected to impulsive force has been studied. The eigen value approach is employed after applying the Laplace and Fourier transforms on the field equations for L-S and G-L model of the plain strain problem. The integral transforms have been inverted into physical domain numerically and components of normal displacement, normal force stress, couple stress and microstress have been illustrated graphically.

scholarly journals Thermal Disturbances in Twinned Orthotropic Thermoelastic Material

2018 ◽  
Vol 23 (4) ◽  
pp. 897-910 ◽  
Author(s):  
L. Rani ◽  
V. Singh
Keyword(s):  
Fourier Transforms ◽  
Crystallographic Axis ◽  
Physical Domain ◽  
Matrix Differential Equation ◽  
Special Cases ◽  
Eigen Value ◽  
Basic Equations ◽  
Matrix Differential ◽  
Thermally Conducting ◽  
Vector Matrix

Abstract This paper deals with deformation in homogeneous, thermally conducting, single-crystal orthotropic twins, bounded symmetrically along a plane containing only one common crystallographic axis. The Fourier transforms technique is applied to basic equations to form a vector matrix differential equation, which is then solved by the eigen value approach. The solution obtained is applied to specific problems of an orthotropic twin crystal subjected to triangular loading. The components of displacement, stresses and temperature distribution so obtained in the physical domain are computed numerically. A numerical inversion technique has been used to obtain the components in the physical domain. Particular cases as quasi-static thermo-elastic and static thermoelastic as well as special cases are also discussed in the context of the problem.

scholarly journals Effect of Deformation on Semi–infinite Viscothermoelastic Cylinder Based on Five Theories of Generalized Thermoelasticity

2017 ◽  
Vol 6 (1) ◽  
pp. 17-35 ◽  
Author(s):  
D. K. Sharma ◽  
Himani Mittal ◽  
Sita Ram Sharma ◽  
Inder Parkash
Keyword(s):  
Heat Conduction ◽  
Laplace Transformation ◽  
Numerical Technique ◽  
Equations Of Motion ◽  
Generalized Thermoelasticity ◽  
Dynamical Problem ◽  
Infinite Cylinder ◽  
Matlab Software ◽  
Voigt Model ◽  
Laplace Transformations

We considera dynamical problem for semi-infinite viscothermoelastic semi infinite cylinder loaded mechanically and thermally and investigated the behaviour of variations of displacements, temperatures and stresses. The problem has been investigated with the help of five theories of the generalized viscothermoelasticity by using the Kelvin – Voigt model. Laplace transformations and Hankel transformations are applied to equations of constituent relations, equations of motion and heat conduction to obtain exact equations in transformed domain. Hankel transformed equations are inverted analytically and for the inversion of Laplace transformation we apply numerical technique to obtain field functions. In order to obtain field functions i.e. displacements, temperature and stresses numerically we apply MATLAB software tools. Numerically analyzed results for the temperature, displacements and stresses are shown graphically.

Scattering of a Rayleigh Wave by an Elastic Quarter Space

1985 ◽  
Vol 52 (3) ◽  
pp. 664-668 ◽  
Author(s):  
A. K. Gautesen
Keyword(s):  
Steady State ◽  
Surface Wave ◽  
Fourier Transforms ◽  
Two Dimensional ◽  
Rayleigh Surface Wave ◽  
State Problem ◽  
Free Surfaces ◽  
Linear Elastic ◽  
Quarter Space ◽  
Steady State Problem

We study the two-dimensional, steady-state problem of the scattering of waves in a homogeneous, isotropic, linear-elastic quarter space. We derive decoupled equations for the Fourier transforms of the normal and tangential displacements on the free surfaces. For incidence of a Rayleigh surface wave, we plot the amplitudes and phases of the surface waves reflected and transmitted by the corner. These curves were obtained numerically.

Nonlinear Transverse Vibration of a Hyperelastic Beam Under Harmonically Varying Axial Loading

2021 ◽  
Vol 16 (3) ◽  
Author(s):  
Yuanbin Wang ◽  
Weidong Zhu
Keyword(s):  
Governing Equation ◽  
Transverse Vibration ◽  
Harmonic Balance ◽  
Fourier Transforms ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Multiple Scales Method ◽  
Frequency Responses ◽  
Runge Kutta ◽  
Nonlinear Transverse Vibration

Abstract Nonlinear transverse vibration of a hyperelastic beam under a harmonically varying axial load is analyzed in this work. Equations of motion of the beam are derived via the extended Hamilton's principle, where transverse vibration is coupled with longitudinal vibration. The governing equation of nonlinear transverse vibration of the beam is obtained by decoupling the equations of motion. By applying the Galerkin method, the governing equation transforms to a series of nonlinear ordinary differential equations (ODEs). Response of the beam is obtained via three different methods: the Runge–Kutta method, multiple scales method, and harmonic balance method. Time histories, phase-plane portraits, fast Fourier transforms (FFTs), and amplitude–frequency responses of nonlinear transverse vibration of the beam are obtained. Comparison of results from the three methods is made. Results from the multiple scales method are in good agreement with those from the harmonic balance and Runge–Kutta methods when the amplitude of vibration is small. Effects of the material parameter and geometrical parameter of the beam on its amplitude–frequency responses are analyzed.

The Role of Modal Coupling on the Non-Linear Response of Cylindrical Shells Subjected to Dynamic Axial Loads

2000 ◽  
Author(s):  
Paulo B. Gonçalves ◽  
Zenón J. G. N. Del Prado
Keyword(s):  
Dynamic Instability ◽  
Equations Of Motion ◽  
Cylindrical Shells ◽  
Parametric Instability ◽  
Compressive Load ◽  
Basins Of Attraction ◽  
Dynamic Component ◽  
Linear Ordinary Differential Equations ◽  
Non Linear ◽  
Low Dimensional

Abstract This paper discusses the dynamic instability of circular cylindrical shells subjected to time-dependent axial edge loads of the form P(t) = P0+P1(t), where the dynamic component p1(t) is periodic in time and P0 is a uniform compressive load. In the present paper a low dimensional model, which retains the essential non-linear terms, is used to study the non-linear oscillations and instabilities of the shell. For this, Donnell’s shallow shell equations are used together with the Galerkin method to derive a set of coupled non-linear ordinary differential equations of motion which are, in turn, solved by the Runge-Kutta method. To study the non-linear behavior of the shell, several numerical strategies were used to obtain Poincaré maps, stable and unstable fixed points, bifurcation diagrams and basins of attraction. Particular attention is paid to two dynamic instability phenomena that may arise under these loading conditions: parametric instability and escape from the pre-buckling potential well. The numerical results obtained from this investigation clarify the conditions, which control whether or not instability may occur. This may help in establishing proper design criteria for these shells under dynamic loads, a topic practically unexplored in literature.

Integral Limit Laws for Additive Functions

1973 ◽  
Vol 25 (1) ◽  
pp. 194-203
Author(s):  
J. Galambos
Keyword(s):  
Characteristic Function ◽  
Fourier Transforms ◽  
Mean Value ◽  
Characteristic Functions ◽  
Asymptotic Formulas ◽  
Mean Value Theorem ◽  
Multiplicative Functions ◽  
Additive Functions ◽  
Limit Laws ◽  
Integral Limit

In the present paper a general form of integral limit laws for additive functions is obtained. Our limit law contains Kubilius’ results [5] on his class H. In the proof we make use of characteristic functions (Fourier transforms), which reduces our problem to finding asymptotic formulas for sums of multiplicative functions. This requires an extension of previous results in order to enable us to take into consideration the parameter of the characteristic function in question. We call this extension a parametric mean value theorem for multiplicative functions and its proof is analytic on the line of [4].

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