scholarly journals An infinite plate loaded with a normal force moving along a complex open trajectory

2021 ◽  
Vol 21 (3) ◽  
pp. 239-246
Author(s):  
A. V. Galaburdin
Keyword(s):  
Fourier Transform ◽  
Fundamental Solution ◽  
Elastic Waves ◽  
Normal Force ◽  
Moving Load ◽  
Infinite Plate ◽  
Elastic Base ◽  
Concentrated Force ◽  
The Fourier Transform ◽  
Propagation Of Elastic Waves

Introduction. A method for solving the problem on the action of a normal force moving on an infinite plate according to an arbitrary law is considered. This method and the results obtained can be used to study the effect of a moving load on various structures.Materials and Methods. An original method for solving problems of the action of a normal force moving arbitrarily along a freeform open curve on an infinite plate resting on an elastic base, is developed. For this purpose, a fundamental solution to the differential equation of the dynamics of a plate resting on an elastic base is used. It is assumed that the movement of force begins at a sufficiently distant moment in time. Therefore, there are no initial conditions in this formulation of the problem. When determining the fundamental solution, the Fourier transform is performed in time. When the Fourier transform is inverted, the image is expanded in terms of the transformation parameter into a series in Hermite polynomials.Results. The solution to the problem on an infinite plate resting on an elastic base, along which a concentrated force moves at a variable speed, is presented. A smooth open curve, consisting of straight lines and arcs of circles, was considered as a trajectory. The behavior of the components of the displacement vector and the stress tensor at the location of the moving force is studied, as well as the process of wave energy propagation, for which the change in the Umov-Poynting energy flux density vector is considered. The effect of the speed and acceleration of the force movement on the displacements, stresses and propagation of elastic waves is investigated. The influence of the force trajectory shape on the stress-strain state of the plate and on the nature of the propagation of elastic waves is studied. The results indicate that the method is quite stable within a wide range of changes in the speed of force movement.Discussion and Conclusions. The calculations have shown that the most significant factor affecting the stress-strain states of the plate and the propagation of elastic wave energy near the concentrated force is the speed of its movement. These results will be useful under studying dynamic processes generated by a moving load.

Fundamental Solutions of a Normal Force in Semi-Infinite Coating Materials Part II: Solution of Force Acting in the Interior of Substrate and at Interface

2011 ◽  
Vol 243-249 ◽  
pp. 5935-5940
Author(s):  
Zhen Yang ◽  
Shi Qun Guo
Keyword(s):  
Numerical Analysis ◽  
Fundamental Solution ◽  
Explicit Solution ◽  
Normal Force ◽  
Fundamental Solutions ◽  
Theoretical Solution ◽  
Image Point ◽  
Coating Materials ◽  
Concentrated Force ◽  
Normal Forces

In this work, the spatial theoretical solutions of concentrated normal forces acting in the substrate and at the interface of semi-infinite coating materials have been deduced based on the image point method. The explicit solution is given in the series of the displacement functions corresponding to each image point. It is found that all displacement functions can be deduced by the order of the image point. Numerical analysis has been carried out to verify the theoretical deductions. It is found that the accurate enough theoretical solution can be obtained by only taking the displacement functions corresponding to the first several image points into account. The fundamental solutions for a concentrated force acting at interface of two bonded semi-infinite bodies is also deduced, which is necessary in the analysis of coating materials by using these theoretical solution as the fundamental solution.

scholarly journals Dynamic Analysis of an Infinitely Long Beam Resting on a Kelvin Foundation under Moving Random Loads

2017 ◽  
Vol 2017 ◽  
pp. 1-13
Author(s):  
Y. Zhao ◽  
L. T. Si ◽  
H. Ouyang
Keyword(s):  
Fourier Transform ◽  
Random Vibration ◽  
Moving Load ◽  
Closed Form Solution ◽  
Form Solution ◽  
Integration Scheme ◽  
Random Loads ◽  
Numerical Integration Scheme ◽  
Vibration Responses ◽  
The Fourier Transform

Nonstationary random vibration analysis of an infinitely long beam resting on a Kelvin foundation subjected to moving random loads is studied in this paper. Based on the pseudo excitation method (PEM) combined with the Fourier transform (FT), a closed-form solution of the power spectral responses of the nonstationary random vibration of the system is derived in the frequency-wavenumber domain. On the numerical integration scheme a fast Fourier transform is developed for moving load problems through a parameter substitution, which is found to be superior to Simpson’s rule. The results obtained by using the PEM-FT method are verified using Monte Carlo method and good agreement between these two sets of results is achieved. Special attention is paid to investigation of the effects of the moving load velocity, a few key system parameters, and coherence of loads on the random vibration responses. The relationship between the critical speed and resonance is also explored.

scholarly journals A Note on the Generalized Relativistic Diffusion Equation

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1009
Author(s):  
Luisa Beghin ◽  
Roberto Garra
Keyword(s):  
Fourier Transform ◽  
Fundamental Solution ◽  
Diffusion Equation ◽  
Fractional Derivatives ◽  
Stable Process ◽  
Caputo Fractional Derivatives ◽  
The Fourier Transform ◽  
Derivatives Of

We study here a generalization of the time-fractional relativistic diffusion equation based on the application of Caputo fractional derivatives of a function with respect to another function. We find the Fourier transform of the fundamental solution and discuss the probabilistic meaning of the results obtained in relation to the time-scaled fractional relativistic stable process. We briefly consider also the application of fractional derivatives of a function with respect to another function in order to generalize fractional Riesz-Bessel equations, suggesting their stochastic meaning.

Propagations of Elastic Waves Generated by Dynamical Loads on a Circular Cavity

1961 ◽  
Vol 28 (2) ◽  
pp. 218-222 ◽  
Author(s):  
A. C. Eringen
Keyword(s):  
Fourier Transform ◽  
Elastic Body ◽  
Elastic Waves ◽  
Dynamic Problem ◽  
Polar Angle ◽  
Circular Cavity ◽  
Moving Loads ◽  
The Fourier Transform

The Fourier-transform technique has been employed to solve the exterior elasto-dynamic problem concerning the region outside a circular cavity in a plane elastic body. The normal and tangential tractions acting at the surface of the circular cavity are prescribed as arbitrary functions of the polar angle θ, and the time t. The case of impact, blast, and moving loads are studied in detail.

BENDING OF A SUBTLE-FINISHED PLATE ON ELASTIC BASE WITH SPECIAL CONDITIONS OF ITS WORK

2019 ◽  
pp. 424-430
Author(s):  
A. T. Marufiy ◽  
A. S. Kalykov
Keyword(s):  
Analytical Solution ◽  
Working Conditions ◽  
Fourier Transforms ◽  
Infinite Plate ◽  
Elastic Base ◽  
Generalized Solutions ◽  
Middle Plane ◽  
Actual Working

In this article, an analytical solution is obtained for the problem of bending a semi-infinite plate on an elastic Winkler base, taking into account incomplete contact with the base and the influence of longitudinal forces applied in the middle plane of the plate. The analytical solution is obtained by the method of generalized solutions using integral Fourier transforms. Any analytical solution is the result, approaching the actual working conditions of the designed structures.

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