scholarly journals FORCED RESPONSE VIBRATION OF SIMPLY SUPPORTED BEAMS WITH AN ELASTIC PASTERNAK FOUNDATION UNDER A DISTRIBUTED MOVING LOAD

2020 ◽  
Vol 4 (2) ◽  
pp. 1-7
Author(s):  
Fatai Hammed ◽  
M. A. Usman ◽  
S. A. Onitilo ◽  
F. A. Alade ◽  
K. A. Omoteso
Keyword(s):  
Shear Modulus ◽  
Moving Load ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Forced Response ◽  
Dynamic Responses ◽  
Moving Force ◽  
Pasternak Elastic Foundation

In this study, the response of two homogeneous parallel beams with two-parameter Pasternak elastic foundation subjected to a constant uniform partially distributed moving force is considered. On the basis of Euler-Bernoulli beam theory, the fourth order partial differential equations of motion describing the behavior of the beams when subjected to a moving force were formulated. In order to solve the resulting initial-boundary value problem, finite Fourier sine integral technique and differential transform scheme were employed to obtain the analytical solution. The dynamic responses of the two beams obtained was investigated under moving force conditions using MATLAB. The effects of speed of the moving force, layer parameters such as stiffness (K_0) and shear modulus (G_0 ) have been conducted for the moving force. Various values of speed of the moving load, stiffness parameters and shear modulus were considered. The results obtained indicates that response amplitudes of both the upper and lower beams increases with increase in the speed of the moving load. Increasing the stiffness parameter is observed to cause a decrease in the response amplitudes of the beams. The response amplitudes decreases with increase in the shear modulus of the linear elastic layer.

A NEW NONLOCAL BEAM THEORY WITH THICKNESS STRETCHING EFFECT FOR NANOBEAMS

2013 ◽  
Vol 12 (04) ◽  
pp. 1350025 ◽  
Author(s):  
ABDELOUAHED TOUNSI ◽  
SOUMIA BENGUEDIAB ◽  
MOHAMMED SID AHMED HOUARI ◽  
ABDELWAHED SEMMAH
Keyword(s):  
Shear Deformation ◽  
Constitutive Relations ◽  
Equations Of Motion ◽  
Nonlocal Parameter ◽  
Beam Theory ◽  
Dynamic Responses ◽  
Theoretical Development ◽  
Small Scale ◽  
Small Scale Effect ◽  
Nonlocal Beam

This paper presents a new nonlocal thickness-stretching sinusoidal shear deformation beam theory for the static and vibration of nanobeams. The present model incorporates the length scale parameter (nonlocal parameter) which can capture the small scale effect, and it accounts for both shear deformation and thickness stretching effects by a sinusoidal variation of all displacements through the thickness without using shear correction factor. Based on the nonlocal differential constitutive relations of Eringen, the equations of motion of the nanobeam are derived using Hamilton's principle. The effects of nonlocal parameter, aspect ratio and the thickness stretching on the static and dynamic responses of the nanobeam are discussed. The theoretical development presented herein may serve as a reference for nonlocal theories as applied to the bending and dynamic behaviors of complex-nanobeam-system such as complex carbon nanotube system.

Nonlinear Dynamics of Simple-Supported Beam under Concentrated Moving Load

2012 ◽  
Vol 226-228 ◽  
pp. 541-545 ◽  
Author(s):  
Dong Xing Cao ◽  
Bao Chen ◽  
Wei Zhang
Keyword(s):  
Nonlinear Dynamics ◽  
Double Layer ◽  
Vibration Suppression ◽  
Structural Parameters ◽  
Moving Load ◽  
Equations Of Motion ◽  
Single Layer ◽  
Dynamic Responses ◽  
Beam Model ◽  
Layer Model

The dynamic responses of two kinds of simple-supported beams with single layer and double-layer under a moving load were analyzed based on the theory of nonlinear dynamics. The equations of motion are derived by using Hamilton’s principle and von Karman type equations for the two models. Galerkin’s method was employed to obtain the ordinary differential equations of motion. First we obtain the periodic motion waveforms in the mid-point of the beams at the same initial velocity, and the result show that the amplitude of the double-layer model is much smaller then that of the single-layer model. Then for the two models, the vibration response and critical velocity were studied considering the effect of the structural parameters, the magnitude and velocity of moving load. The results of numerical simulation show that double-layer beam model has better vibration suppression performance than single-layer beam model.

On the Vibrations of an Axially Moving String With a Time-Dependent Velocity

2015 ◽  
Author(s):  
Rajab A. Malookani ◽  
Wim T. van Horssen
Keyword(s):  
Perturbation Method ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Approximate Solutions ◽  
Axially Moving String ◽  
Fourier Sine Series ◽  
Axially Moving ◽  
Initial Boundary Value

The transverse vibrations of an axially moving string with a time-varying speed is studied in this paper. The governing equations of motion describing an axially moving string is analyzed using two different techniques. At first, the initial-boundary value problem is discretized using the Fourier sine series, and then the two timescales perturbation method is employed in search of infinite mode approximate solutions. Secondly, a new approach based on the two timescales perturbation method and the method of characteristics is used. It is found that there are infinitely many values of the velocity fluctuation frequency yielding infinitely many resonance conditions in the system. The response of the system with harmonically varying velocity function is computed for particular harmonic initial conditions.

Vibration analysis of stiffened multi-plate structure based on a modified variational principle

2015 ◽  
Vol 23 (17) ◽  
pp. 2767-2781 ◽  
Author(s):  
Guo-qing Yuan ◽  
Wei-Kang Jiang
Keyword(s):  
Variational Principle ◽  
Plate Theory ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Energy Functional ◽  
Dynamic Responses ◽  
Published Data ◽  
Discrete Elements ◽  
Displacement Fields ◽  
Analysis And Design

In order to study the vibration characteristics of flow-induced open cavity structures, the dynamic model of stiffened multi-plate is established. The first-order shear deformable plate theory and the Timoshenko beam theory are used to model the displacement fields of isotropic plates and stiffeners, respectively. A modified variational principle combined with a multi-segment partitioning procedure is employed to formulate the discretized equations of motion. The stiffeners are considered as discrete elements, and the energy contributions are included into the system energy functional by using the displacement compatibility conditions. The displacement and rotation components of each plate segment are expanded by a duplicate series of Chebyshev orthogonal polynomials of first kind. The convergence and accuracy of the present results for isotropic stiffened plates with different boundary conditions have been validated using comparisons with the published data and those obtained from the finite element analyses. Free vibration and dynamic responses of stiffened multi-plates with either longitudinal or orthogonally oriented stiffeners are discussed. The mathematical model and methodology presented in this paper may be used as an appropriate numerical tool in the analysis and design of stiffened multi-plate structures.

scholarly journals INVESTIGATION OF HYPERELASTIC SOFT SHELLS NONSTATIONARY DYNAMICS PROBLEMS SOLUTION FEATURES

2021 ◽  
Vol 83 (2) ◽  
pp. 151-159
Author(s):  
E.A. Korovaytseva
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Ordinary Differential Equations ◽  
Boundary Value ◽  
Equations Of Motion ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Finite Difference Approximation ◽  
Analytical Studies ◽  
Initial Boundary Value

Results of hyperelastic soft shells nonlinear axisymmetric dynamic deforming problems solution algorithm testing are represented in the work. Equations of motion are given in vector-matrix form. For the nonlinear initial-boundary value problem solution an algorithm which lies in reduction of the system of partial differential equations of motion to the system of ordinary differential equations with the help of lines method is developed. At this finite-difference approximation of partial time derivatives is used. The system of ordinary differential equations obtained as a result of this approximation is solved using parameter differentiation method at each time step. The algorithm testing results are represented for the case of pressure uniformly distributed along the meridian of the shell and linearly increasing in time. Three types of elastic potential characterizing shell material are considered: Neo-hookean, Mooney – Rivlin and Yeoh. Features of numerical realization of the algorithm used are pointed out. These features are connected both with the properties of soft shells deforming equations system and with the features of the algorithm itself. The results are compared with analytical solution of the problem considered. Solution behavior at critical pressure value is investigated. Formulations and conclusions given in analytical studies of the problem are clarified. Applicability of the used algorithm to solution of the problems of soft shells dynamic deforming in the range of displacements several times greater than initial dimensions of the shell and deformations much greater than unity is shown. The numerical solution of the initial boundary value problem of nonstationary dynamic deformation of the soft shell is obtained without assumptions about the limitation of displacements and deformations. The results of the calculations are in good agreement with the results of analytical studies of the test problem.

Dynamic Responses of Functionally Graded Sandwich Beams Resting on Elastic Foundation Under Harmonic Moving Loads

2018 ◽  
Vol 18 (09) ◽  
pp. 1850112 ◽  
Author(s):  
Wachirawit Songsuwan ◽  
Monsak Pimsarn ◽  
Nuttawit Wattanasakulpong
Keyword(s):  
Boundary Conditions ◽  
Elastic Foundation ◽  
Natural Frequencies ◽  
Moving Load ◽  
Beam Theory ◽  
Equation Of Motion ◽  
Functionally Graded ◽  
Dynamic Responses ◽  
Harmonic Load ◽  
Sandwich Beams

This paper investigates the free vibration and dynamic response of functionally graded sandwich beams resting on an elastic foundation under the action of a moving harmonic load. The governing equation of motion of the beam, which includes the effects of shear deformation and rotary inertia based on the Timoshenko beam theory, is derived from Lagrange’s equations. The Ritz and Newmark methods are employed to solve the equation of motion for the free and forced vibration responses of the beam with different boundary conditions. The results are presented in both tabular and graphical forms to show the effects of layer thickness ratios, boundary conditions, length to height ratios, spring constants, etc. on natural frequencies and dynamic deflections of the beam. It was found that increasing the spring constant of the elastic foundation leads to considerable increase in natural frequencies of the beam; while the same is not true for the dynamic deflection. Additionally, very large dynamic deflection occurs for the beam in resonance under the harmonic moving load.

scholarly journals Vibrations of a Timoshenko beam on an inertial foundation under a moving load

2018 ◽  
Vol 196 ◽  
pp. 01056
Author(s):  
Magdalena Ataman
Keyword(s):  
Timoshenko Beam ◽  
Moving Load ◽  
Equations Of Motion ◽  
Free Vibrations ◽  
Forced Vibrations ◽  
Governing Equations ◽  
Concentrated Load ◽  
Numerical Example ◽  
Moving Force ◽  
Speed Response

In the paper vibrations of the Timoshenko beam on an inertial foundation subjected to a moving force are discussed. Considered model of the inertial foundation is described by three parameters. They take into account elasticity, shear and inertia of the subgrade. In the literature such model of the subgrade is called Vlasov or Vlasov-Leontiev model. The Timoshenko beam is traversed by a concentrated load, moving with uniform speed. Response of the beam is found from the governing equations of motion of the problem. Problem of forced vibrations and problem of free vibrations of the beam are solved. Damping of the system is taken into consideration. Solution of the problem is illustrated by numerical example.

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