Abstract: In this paper, we investigate certain spaces of
generalized functions for the Fourier and Fourier type integral
transforms. We discuss convolution theorems and establish certain
spaces of distributions for the considered integrals. The new Fourier
type integral is well-defined, linear, one-to-one and continuous with
respect to certain types of convergences. Many properties and an
inverse problem are also discussed in some details.
Abstract: To study the dynamic mechanics response of asphalt
pavement under the temperature load and vehicle loading, asphalt
pavement was regarded as multilayered elastic half-space system, and
theory analysis was conducted by regarding dynamic modulus of
asphalt mixture as the parameter. Firstly, based on the dynamic
modulus test of asphalt mixture, function relationship between the
dynamic modulus of representative asphalt mixture and temperature
was obtained. In addition, the analytical solution for thermal stress in
single layer was derived by using Laplace integral transformation and
Hankel integral transformation respectively by using thermal
equations of equilibrium. The analytical solution of calculation model
of thermal stress in asphalt pavement was derived by transfer matrix
of thermal stress in multilayer elastic system. Finally, the variation of
thermal stress in pavement structure was analyzed. The result shows
that there is obvious difference between the thermal stress based on
dynamic modulus and the solution based on static modulus. So the
dynamic change of parameter in asphalt mixture should be taken into
consideration when theoretical analysis is taken out.
Abstract: This article is concerned with the determination of the static interaction of a vertically loaded rigid circular disc embedded at the interface of a horizontal layer sandwiched in between two different transversely isotropic half-spaces called as tri-material full-space. The axes of symmetry of different regions are assumed to be normal to the horizontal interfaces and parallel to the movement direction. With the use of a potential function method, and by implementing Hankel integral transforms in the radial direction, the government partial differential equation for the solely scalar potential function is transformed to an ordinary 4th order differential equation, and the mixed boundary conditions are transformed into a pair of integral equations called dual integral equations, which can be reduced to a Fredholm integral equation of the second kind, which is solved analytically. Then, the displacements and stresses are given in the form of improper line integrals, which is due to inverse Hankel integral transforms. It is shown that the present solutions are in exact agreement with the existing solutions for a homogeneous full-space with transversely isotropic material. To confirm the accuracy of the numerical evaluation of the integrals involved, the numerical results are compared with the solutions exists for the homogeneous full-space. Then, some different cases with different degrees of material anisotropy are compared to portray the effect of degree of anisotropy.
Abstract: The general solution of the equations for a homogeneous isotropic microstretch thermo elastic medium with mass diffusion for two dimensional problems is obtained due to normal and tangential forces. The Integral transform technique is used to obtain the components of displacements, microrotation, stress and mass concentration, temperature change and mass concentration. A particular case of interest is deduced from the present investigation.
Abstract: Transient eddy current problem is solved in the
present paper by the method of the Laplace transform for the case of
a double conductor line located parallel to a conducting half-space.
The Fourier sine and cosine integral transforms are used in order to
find the Laplace transform of the solution. The inverse Laplace
transform of the solution is found in closed form. The integrated
electromotive force per unit length of the double conductor line is
calculated in the form of an improper integral.
Abstract: Oil debris signal generated from the inductive oil
debris monitor (ODM) is useful information for machine condition
monitoring but is often spoiled by background noise. To improve the
reliability in machine condition monitoring, the high-fidelity signal
has to be recovered from the noisy raw data. Considering that the noise
components with large amplitude often have higher frequency than
that of the oil debris signal, the integral transform is proposed to
enhance the detectability of the oil debris signal. To cancel out the
baseline wander resulting from the integral transform, the empirical
mode decomposition (EMD) method is employed to identify the trend
components. An optimal reconstruction strategy including both
de-trending and de-noising is presented to detect the oil debris signal
with less distortion. The proposed approach is applied to detect the oil
debris signal in the raw data collected from an experimental setup. The
result demonstrates that this approach is able to detect the weak oil
debris signal with acceptable distortion from noisy raw data.
Abstract: In this article an isotropic linear elastic half-space with
a cylindrical cavity of finite length is considered to be under the
effect of a ring shape time-harmonic torsion force applied at an
arbitrary depth on the surface of the cavity. The equation of
equilibrium has been written in a cylindrical coordinate system. By
means of Fourier cosine integral transform, the non-zero
displacement component is obtained in the transformed domain. With
the aid of the inversion theorem of the Fourier cosine integral
transform, the displacement is obtained in the real domain. With the
aid of boundary conditions, the involved boundary value problem for
the fundamental solution is reduced to a generalized Cauchy singular
integral equation. Integral representation of the stress and
displacement are obtained, and it is shown that their degenerated
form to the static problem coincides with existing solutions in the
literature.