The Response of Infinite Strings and Beams to an Initially Applied Moving Force: Analytical Solution

2012 ◽  
Vol 134 (4) ◽  
Author(s):  
André Langlet ◽  
Ophélie Safont ◽  
Jérôme Renard
Keyword(s):  
Fourier Transform ◽  
Laplace Transform ◽  
Stationary Solution ◽  
Timoshenko Beam ◽  
Bernoulli Beam ◽  
Moving Force ◽  
The Laplace Transform ◽  
Bar Velocity ◽  
Infinite Strings ◽  
Euler Bernoulli Beam

This paper presents the analytical solutions for bilaterally infinite strings and infinite beams on which a point force is initially applied, which then moves on the structure at a constant velocity. The solutions are sought by first applying the Fourier transform to the spatial coordinate dependence, and then the Laplace transform to the time variable of dependence, of the governing equations of motion. For the strings, it is necessary to distinguish between the case of a sonic load (a force moving at the phase velocity of transverse waves) and the cases of subsonic and supersonic loads. This is achieved by a suitable expansion in polynomial ratios of the Laplace transform, before going back to the original Fourier transform, whose inverse is obtained by exact calculations of the integrals over the complex infinite domain. For the Euler-Bernoulli beam, the same process leads to the closed-form (exact) formula for the displacement, from which the stress can be deduced. The displacement consists of the sum of two integrals: one representing the transient part, and the other, the stationary part of the solution. The stationary part is observed in the vicinity of the force for a very long travel time. The transient part is observed at a finite position coordinate, in relative proximity to the starting point of the moving force. For the Timoshenko beam, the final step in the calculation of the displacement and rotation, which requires a numerical evaluation of the integrals, leads to Fourier cosine and sine transforms. The response of the beam depends on the load velocity, relative to the two characteristic velocities: those of shear waves and longitudinal waves. This demonstrates that the transient parts of the solutions, in the Euler-Bernoulli beam or in the Timoshenko beam, are quasi identical. However, classical theory fails to forecast high frequency responses, occurring with velocities of the load exceeding twenty per cent of the bar velocity. For a velocity greater than the velocity of the shear waves, classical theory wrongly forecasts the response. In addition, according to the Euler-Bernoulli beam theory, the flexural waves are able to exceed the bar velocity, which is not realistic. If the load moves for a long period, the solution in the vicinity of the load tends towards a stationary solution. It is important to note that the solution to the stationary problem must be completed by the solution to the associated homogeneous system to represent the physical stationary solution.

scholarly journals An Elastodiffusive Orthotropic Euler–Bernoulli Beam Considering Diffusion Flux Relaxation

2019 ◽  
Vol 24 (1) ◽  
pp. 23 ◽  
Author(s):  
Dmitry Tarlakovskii ◽  
Andrei Zemskov
Keyword(s):  
Fourier Series ◽  
Variational Principle ◽  
Laplace Transform ◽  
Diffusion Flux ◽  
Fourier Series Expansion ◽  
Bernoulli Beam ◽  
Elastic Diffusion ◽  
The Laplace Transform ◽  
Flux Relaxation ◽  
Euler Bernoulli Beam

This article considers an unsteady elastic diffusion model of Euler–Bernoulli beam oscillations in the presence of diffusion flux relaxation. We used the model of coupled elastic diffusion for a homogeneous orthotropic multicomponent continuum to formulate the problem. A model of unsteady bending for the elastic diffusive Euler–Bernoulli beam was obtained using Hamilton’s variational principle. The Laplace transform on time and the Fourier series expansion by the spatial coordinate were used to solve the obtained problem.

scholarly journals Application of ADM Using Laplace Transform to Approximate Solutions of Nonlinear Deformation for Cantilever Beam

2016 ◽  
Vol 2016 ◽  
pp. 1-5
Author(s):  
Ratchata Theinchai ◽  
Siriwan Chankan ◽  
Weera Yukunthorn
Keyword(s):  
Laplace Transform ◽  
Cantilever Beam ◽  
Adomian Decomposition Method ◽  
Small Deformation ◽  
Approximate Solutions ◽  
Beam Equation ◽  
Bernoulli Beam ◽  
Initial Value ◽  
Adomian Decomposition ◽  
Euler Bernoulli Beam

We investigate semianalytical solutions of Euler-Bernoulli beam equation by using Laplace transform and Adomian decomposition method (LADM). The deformation of a uniform flexible cantilever beam is formulated to initial value problems. We separate the problems into 2 cases: integer order for small deformation and fractional order for large deformation. The numerical results show the approximated solutions of deflection curve, moment diagram, and shear diagram of the presented method.

Medium Frequency Vibration Analysis of Beam Structures Modeled by the Timoshenko Beam Theory

2020 ◽  
Author(s):  
Yichi Zhang ◽  
Bingen Yang
Keyword(s):  
Shear Deformation ◽  
Timoshenko Beam ◽  
Vibration Analysis ◽  
Beam Theory ◽  
Timoshenko Beam Theory ◽  
Bernoulli Beam ◽  
Beam Structure ◽  
Medium Frequency ◽  
Beam Structures ◽  
Euler Bernoulli Beam

Abstract Vibration analysis of complex structures at medium frequencies plays an important role in automotive engineering. Flexible beam structures modeled by the classical Euler-Bernoulli beam theory have been widely used in many engineering problems. A kinematic hypothesis in the Euler-Bernoulli beam theory is that plane sections of a beam normal to its neutral axis remain normal when the beam experiences bending deformation, which neglects the shear deformation of the beam. However, as observed by researchers, the shear deformation of a beam component becomes noticeable in high-frequency vibrations. In this sense, the Timoshenko beam theory, which describes both bending deformation and shear deformation, may be more suitable for medium-frequency vibration analysis of beam structures. This paper presents an analytical method for medium-frequency vibration analysis of beam structures, with components modeled by the Timoshenko beam theory. The proposed method is developed based on the augmented Distributed Transfer Function Method (DTFM), which has been shown to be useful in various vibration problems. The proposed method models a Timoshenko beam structure by a spatial state-space formulation in the s-domain, without any discretization. With the state-space formulation, the frequency response of a beam structure, in any frequency region (from low to very high frequencies), can be obtained in an exact and analytical form. One advantage of the proposed method is that the local information of a beam structure, such as displacements, bending moment and shear force at any location, can be directly obtained from the space-state formulation, which otherwise would be very difficult with energy-based methods. The medium-frequency analysis by the augmented DTFM is validated with the FEA in numerical examples, where the efficiency and accuracy of the proposed method is present. Also, the effects of shear deformation on the dynamic behaviors of a beam structure at medium frequencies are illustrated through comparison of the Timoshenko beam theory and the Euler-Bernoulli beam theory.

New Laplace, z and Fourier-related transforms

2007 ◽  
Vol 463 (2081) ◽  
pp. 1179-1198 ◽  
Author(s):  
Michael J Corinthios
Keyword(s):  
Fourier Transform ◽  
Laplace Transform ◽  
Large Class ◽  
Large Family ◽  
Step Function ◽  
Dirac Delta ◽  
Mellin Transforms ◽  
Generalized Distributions ◽  
The Laplace Transform ◽  
The Fourier Transform

In this paper, the author uses his recently proposed complex variable generalized distribution theory to expand the domains of existence of bilateral Laplace and z transforms, as well as a whole new class of related transforms. A vast expansion of the domains of existence of bilateral Laplace and z transforms and continuous-time and discrete-time Hilbert, Hartley and Mellin transforms, as well as transforms of multidimensional functions and sequences are obtained. It is noted that the Fourier transform and its applications have advanced by leaps and bounds during the last century, thanks to the introduction of the theory of distributions and, in particular, the concept of the Dirac-delta impulse. Meanwhile, however, the truly two-sided ‘bilateral’ Laplace and z transforms, which are more general than Fourier, remained at a standstill incapable of transforming the most basic of functions. In fact, they were reduced by half to one-sided transforms and received no more than a passing reference in the literature. It is shown that the newly proposed generalized distributions expand the domains of existence and application of Laplace and z transforms similar to and even more extensively than the expansion of the domain of Fourier transform that resulted from the introduction, nearly a century ago, of the theory of distributions and the Dirac-delta impulse. It is also shown that the new generalized distributions put an end to an anomaly that still exists today, which meant that for a large class of basic functions, the Fourier transform exists while the more general Laplace and z transforms do not. The anomaly further manifests itself in the fact that even for the one-sided causal functions, such as the Heaviside unit step function u ( t ) and the sinusoid sin βtu ( t ), the Laplace transform does not exist on the j ω -axis, and the Fourier transform which does exist cannot be deduced thereof by the substitution s =j ω in the Laplace transform, which by definition it should. The extended generalized transforms are well defined for a large class of functions ranging from the most basic to highly complex fast-rising exponential ones that have so far had no transform. Among basic applications, the solution of partial differential equations using the extended generalized transforms is provided. This paper clearly presents and articulates the significant impact of extending the domains of Laplace and z transforms on a large family of related transforms, after nearly a century during which bilateral Laplace and z transforms of even the most basic of functions were undefined, and the domains of definition of related transforms such as Hilbert, Hartley and Mellin transforms were confined to a fraction of the space they can now occupy.

scholarly journals Macaulay's Method for a Timoshenko Beam

2007 ◽  
Vol 35 (4) ◽  
pp. 285-292 ◽  
Author(s):  
N. G. Stephen
Keyword(s):  
Timoshenko Beam ◽  
Bending Moment ◽  
Shearing Force ◽  
Bernoulli Beam ◽  
Deflection Curve ◽  
Axial Coordinate ◽  
Deflection Analysis ◽  
Euler Bernoulli Beam

The Macaulay bracket notation is familiar to many engineers for the deflection analysis of a Euler–Bernoulli beam subject to multiple or discontinuous loads. An expression for the internal bending moment, and hence curvature, is valid at all locations along the beam, and the deflection curve can be calculated by integrating twice with respect to the axial coordinate. The notation obviates the need for matching of multiple constants of integration for the various sections of the beam. Here, the method is extended to a Timoshenko beam, which includes the additional deflection due to shear. This requires an additional expression for the shearing force, also valid at all locations along the beam.

Mechanical Characters of Six Engineering Timoshenko Beams

2014 ◽  
Vol 668-669 ◽  
pp. 201-204
Author(s):  
Hong Liang Tian
Keyword(s):  
Timoshenko Beam ◽  
Analytic Solutions ◽  
Timoshenko Beams ◽  
Bernoulli Beam ◽  
Simply Supported ◽  
Normality Assumption ◽  
Length Direction ◽  
The Right ◽  
Euler Bernoulli Beam ◽  
Mechanical Characters

Timoshenko beam is an extension of Euler-Bernoulli beam to interpret the transverse shear impact. The more refined Timoshenko beam relaxes the normality assumption of plane section that remains plane and normal to the deformed centerline. The manuscript presents some exact concise analytic solutions on deflection and stress resultants of NET single-span Timoshenko beam with general distributed force and 6 kinds of standard boundary conditions, adopting its counterpart Euler-Bernoulli beam solutions. Engineering example shows that scale impact would not unveil itself for micro structure with micrometer μm-order length, yet will be prominent for nanostructure with nanometer nm-order length. When simply supported CNTs is undergone to a concentrative force at the median and complete bend moment, scale action is observed along the ensemble CNTs, while it unfurls itself the most at the position of the concentrated strength. When a clamped-free CNTs is exposed to a centralized force at the mesial and distributed force, there is no scale impact about the deflection at all positions on the left border of the concentrated strength position, while such operation inspires at once at all positions on the right margin of the concentrated strength position. When a clamped-clamped CNTs is lain under a concentrative strength at the middle, the deflection of NET Euler-Bernoulli CNTs reflects scale effect completely. Notable differences between the deflection of Euler-Bernoulli CNTs and that of Timoshenko CNTs are reflected at large ratio of diameter versus length. The deflection of NET clamped-free and simply supported Timoshenko beam doesn’t introduce surplus scale process in terms of its counterpart, NET Euler-Bernoulli beam. However, the deflection of NET clamped-clamped Timoshenko beam does involve additional scale impact solely including the method when the concentrated strength position is at the midway in the beam-length direction.

Effects of Different Models on Natural Frequencies of Short Span Bridges Used in High Speed Rail

2015 ◽  
Author(s):  
Said I. Nour ◽  
Mohsen A. Issa
Keyword(s):  
Timoshenko Beam ◽  
High Speed ◽  
Natural Frequencies ◽  
Bernoulli Beam ◽  
High Speed Rail ◽  
Vertical Stiffness ◽  
Short Span ◽  
Elastically Supported ◽  
Euler Bernoulli Beam ◽  
Track Modulus

The natural frequencies of vibration of short span bridges used in high-speed rail were investigated. Three different models of increasing complexity were evaluated and their effects on the vibration frequency were compared to the first basic model of simply supported Euler-Bernoulli beam. In the second and third cases, the bridge was modeled as an Euler-Bernoulli and Timoshenko beam supported at its two ends by identical spring elements with an equivalent vertical stiffness to simulate elastomeric bearings and soil foundation. The boundary value problem was solved numerically to extract the bridge eigenfrequencies. In the case of Euler-Bernoulli beam, curve fitting techniques were used to deduce accurate simple empirical formulae to calculate the first six natural frequencies of an elastically supported bridge. In the case of a Timoshenko beam, graphical solutions were proposed to compute the fundamental frequency. Results confirmed that the use of Timoshenko beam theory reduces the natural frequency and the consideration of flexible supports further decreases the natural frequency. In the fourth model, the interaction of the track and the bridge was included. The bridge was modeled as an elastically supported beam and the track was modeled as a spring-damper element with an equivalent vertical stiffness resulting from track components like rail pads, cross-ties and ballast. A parametric study was performed to analyze the effects of the track stiffness on the natural frequencies of the bridge. Graphical solutions were presented to quantify the change of the normalized natural frequencies of the system with the increase in the track modulus. Results indicated that the changes in the track modulus have no significant effects in models with rigid supports. A decrease in the fundamental frequency was noticeable with softer track modulus as the support flexibility increased.

scholarly journals On solutions of local fractional Schrodinger equation

2019 ◽  
Vol 23 (Suppl. 6) ◽  
pp. 1929-1934
Author(s):  
Resat Yilmazer ◽  
Neslihan Demirel
Keyword(s):  
Fourier Transform ◽  
Schrödinger Equation ◽  
Laplace Transform ◽  
Closed Form ◽  
Schrodinger Equation ◽  
Fractional Schrödinger Equation ◽  
The Laplace Transform ◽  
Fractional Schrodinger Equation

In this study, we obtain the solution of a local fractional Schrodinger equation. The solution is obtained by the implementation of the Laplace transform and Fourier transform in closed form in terms of the Mittag-Leffler function.

Fractional visco-elastic Timoshenko beam from elastic Euler–Bernoulli beam

2014 ◽  
Vol 226 (1) ◽  
pp. 179-189 ◽  
Author(s):  
Antonina Pirrotta ◽  
Stefano Cutrona ◽  
Salvatore Di Lorenzo
Keyword(s):  
Timoshenko Beam ◽  
Bernoulli Beam ◽  
Euler Bernoulli Beam

scholarly journals On the Equivalence between Static and Dynamic Railway Track Response and on the Euler-Bernoulli and Timoshenko Beams Analogy

2017 ◽  
Vol 2017 ◽  
pp. 1-13 ◽  
Author(s):  
Wlodzimierz Czyczula ◽  
Piotr Koziol ◽  
Dorota Blaszkiewicz
Keyword(s):  
Axial Force ◽  
Timoshenko Beam ◽  
Linear Models ◽  
Moving Load ◽  
Railway Track ◽  
Variable Load ◽  
Static Case ◽  
Timoshenko Beams ◽  
Bernoulli Beam ◽  
Euler Bernoulli Beam

The paper tries to clarify the problem of solution and interpretation of railway track dynamics equations for linear models. Set of theorems is introduced in the paper describing two types of equivalence: between static and dynamic track response under moving load and between the dynamic response of track described by both the Euler-Bernoulli and Timoshenko beams. The equivalence is clarified in terms of mathematical method of solution. It is shown that inertia element of rail equation for the Euler-Bernoulli beam and constant distributed load can be considered as a substitute axial force multiplied by second derivative of displacement. Damping properties can be treated as additional substitute load in the static case taking into account this substitute axial force. When one considers the Timoshenko beam, the substitute axial force depends additionally on shear properties of rail section, rail bending stiffness, and subgrade stiffness. It is also proved that Timoshenko beam, described by a single equation, from the point of view of solution, is an analogy of the Euler-Bernoulli beam for both constant and variable load. Certain numerical examples are presented and practical interpretation of proved theorems is shown.

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