On Boundary Damping for an Axially Moving Tensioned Beam

2011 ◽  
Vol 134 (1) ◽  
Author(s):  
Sajad H. Sandilo ◽  
Wim T. van Horssen
Keyword(s):  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Conveyor Belt ◽  
Beam Equation ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Boundary Damping ◽  
Axially Moving ◽  
Vertical Vibrations ◽  
Initial Boundary Value

In this paper, an initial-boundary value problem for a linear-homogeneous axially moving tensioned beam equation is considered. One end of the beam is assumed to be simply-supported and to the other end of the beam a spring and a dashpot are attached, where the damping generated by the dashpot is assumed to be small. In this paper only boundary damping is considered. The problem can be used as a simple model to describe the vertical vibrations of a conveyor belt, for which the velocity is assumed to be constant and relatively small compared to the wave speed. A multiple time-scales perturbation method is used to construct formal asymptotic approximations of the solutions, and it is shown how different oscillation modes are damped.

On the Transversal Vibrations of a Conveyor Belt Using a String-Like Equation

2001 ◽  
Author(s):  
Gede Suweken ◽  
W. T. van Horssen
Keyword(s):  
Wave Speed ◽  
Dynamical Behavior ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Conveyor Belt ◽  
Linear Wave ◽  
Long Time ◽  
Vertical Vibrations ◽  
Transversal Vibrations ◽  
Initial Boundary Value

Abstract In this paper an initial-boundary value problem for a linear wave (string) equation is considered. This problem can be used as a simple model to describe the vertical vibrations of a conveyor belt, for which the velocity is small with respect to the wave speed. In this paper the belt is assumed to move with varying speed. Formal asymptotic approximations of the solutions are constructed to show the complicated dynamical behavior of the conveyor belt. It also will be shown that for this problem, the truncation method is not valid on long time scales.

scholarly journals ESTIMATES OF THE EIGENVALUES AND EIGENFUNCTIONS OF OPERATOR ARISING IN SWELLING PRESSURE MODEL

2020 ◽  
Vol 71 (3) ◽  
pp. 59-66
Author(s):  
Zh. Kaiyrbek ◽  
◽  
G. Auzerkhan ◽  
L.K. Zhapsarbaeva ◽  
◽  
...  
Keyword(s):  
Boundary Value Problem ◽  
Nonlinear Problem ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Beam Equation ◽  
Force Model ◽  
Initial Attempt ◽  
Highly Nonlinear ◽  
Initial Boundary Value

Swelling forces from materials confined by structures can cause structural deformations and instability. Due to the complexity of interactions between expansive solid and solid-liquid equilibrium, the forces exerting on retaining structures from swelling are highly nonlinear. This work is our initial attempt to study a simplistic initial/boundary value problem based on the Euler-elastic beam theory and some swelling force model. In this paper, we study a nonlinear problem for the equation of a beam. The self-adjointness of the operator corresponding to the nonlinear problem for the Euler equations is proved. Two-sided estimates of the eigenvalues of the operator in question are established. Two-sided estimates of the eigenfunctions of the operator of the initial-boundary value problem for the beam equation are also obtained.

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.

scholarly journals Existence and Uniqueness of Weak Solutions for the Model Representing Motions of Curves Made of Elastic Materials

2021 ◽  
Vol 36 ◽  
pp. 44-56
Author(s):  
T. Aiki ◽  
◽  
C. Kosugi ◽  
Keyword(s):  
Mathematical Model ◽  
Weak Solutions ◽  
Existence And Uniqueness ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Beam Equation ◽  
Elastic Materials ◽  
Existence Of Weak Solutions ◽  
Uniqueness And Existence ◽  
Initial Boundary Value

We consider the initial boundary value problem for the beam equation with the nonlinear strain. In our previous work this problem was proposed as a mathematical model for stretching and shrinking motions of the curve made of the elastic material on the plane. The aim of this paper is to establish uniqueness and existence of weak solutions. In particular, the uniqueness is proved by applying the approximate dual equation method.

On the Transverse, Low Frequency Vibrations of a Traveling String With Boundary Damping

2015 ◽  
Vol 137 (4) ◽  
Author(s):  
Nick V. Gaiko ◽  
Wim T. van Horssen
Keyword(s):  
Low Frequency ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Oscillatory Behavior ◽  
Transform Method ◽  
Accurate Picture ◽  
The Laplace Transform ◽  
Conveyor Belts ◽  
Axially Moving ◽  
Initial Boundary Value

In this paper, we study the free transverse vibrations of an axially moving (gyroscopic) material represented by a perfectly flexible string. The problem can be used as a simple model to describe the low frequency oscillations of elastic structures such as conveyor belts. In order to suppress these oscillations, a spring–mass–dashpot system is attached at the nonfixed end of the string. In this paper, it is assumed that the damping in the dashpot is small and that the axial velocity of the string is small compared to the wave speed of the string. This paper has two main objectives. The first aim is to give explicit approximations of the solution on long timescales by using a multiple-timescales perturbation method. The other goal is to construct accurate approximations of the lower eigenvalues of the problem, which describe the oscillation and the damping properties of the problem. The eigenvalues follow from a so-called characteristic equation obtained by the direct application of the Laplace transform method to the initial-boundary value problem. Both approaches give a complete and accurate picture of the damping and the low frequency oscillatory behavior of the traveling string.

On the Transversal Vibrations of an Axially Moving Continuum With a Time-Varying Velocity: Transient From String to Beam Behavior

2007 ◽  
Author(s):  
S. V. Ponomareva ◽  
W. T. van Horssen
Keyword(s):  
Natural Frequencies ◽  
Dynamical Behavior ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Beam Model ◽  
Time Varying ◽  
Order Of Magnitude ◽  
Axially Moving ◽  
Transversal Vibrations ◽  
Initial Boundary Value

In this paper an initial-boundary value problem for a linear equation describing an axially moving stretched beam will be considered. The velocity of the beam is assumed to be time-varying. since the order of magnitude of the bending stiffness terms depends on the vibrations modes and the frequencies involved a that combination of two simplified models (a string equation and a beam with string effect equation) will be used to describe the transversal vibrations of the system accurately. Based on the calculations of the natural frequencies the regions of applicability of these models will be determined. A two time-scales perturbation method will be used to construct formal asymptotic approximations of the solutions. It will be shown that the linear axially moving “string to beam” model already has complicated dynamical behavior.

Existence, Uniqueness and Asymptotic Behaviour for a Non-Linear Beam Equation

1995 ◽  
Author(s):  
G. Judith Boertjens ◽  
Wim T. van Horssen
Keyword(s):  
Perturbation Methods ◽  
Flexible Structures ◽  
Initial Boundary ◽  
Beam Equation ◽  
Multiple Time ◽  
Initial Boundary Value Problems ◽  
Initial Boundary Value ◽  
The Right ◽  
Multiple Time Scale Method ◽  
Asymptotic Validity

Abstract The use of perturbation methods for fourth order PDE’s has not yet been examined extensively. Usually approximating power series are applied, which are truncated to one or two modes. Very little — or nothing — is said about the relation between this approximation and the exact solution. In this paper initial boundary value problems for the following equation will be discussed: w t t + w x x x x + ϵ ( u ( π , t ) − u ( 0 , t ) + ∫ 0 π w x 2 d x ) w x x = ϵ g ( x , t , w , w t ) . This equation can be regarded as a model describing wind-induced oscillations of flexible structures like elastic beams, where the small term on the right hand side of the equation represents the windforce acting on the structure. Existence and uniqueness for solutions of these problems will be discussed, as well as finding approximations using a multiple time-scale method. Finally the asymptotic validity of these approximations will be considered.

scholarly journals On (Non) Applicability of a Mode-Truncation of a Damped Traveling String

2019 ◽  
Vol 38 (2) ◽  
pp. 471-478 ◽  
Author(s):  
Rajab Ali Malookani ◽  
Sajid Hussain Sandilo ◽  
Abdul Hanan
Keyword(s):  
Initial Conditions ◽  
Numerical Technique ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Analytic Solutions ◽  
Amplitude Response ◽  
Lateral Vibrations ◽  
Axially Moving ◽  
Initial Boundary Value ◽  
Mode Truncation

This study investigates a linear homogeneous initial-boundary value problem for a traveling string under linear viscous damping. The string is assumed to be traveling with constant speed, while it is fixed at both ends. Physically, this problem represents the vertical (lateral) vibrations of damped axially moving materials. The axial belt speed is taken to be positive, constant and small in comparison with a wave speed, and the damping is also considered relatively small. A two timescale perturbation method together with the characteristic coordinate’s method will be employed to establish the approximate analytic solutions. The damped amplitude-response of the system will be computed under specific initial conditions. The obtained results are compared with the finite difference numerical technique for justification. It turned out that the introduced damping has a significant effect on the amplitude-response. Additionally, it is proven that the mode-truncation is applicable for the damped axially traveling string system on a timescale of order ε -1

Convergence of Finite Fifference Method for Parabolic Problem

2003 ◽  
Vol 3 (1) ◽  
pp. 45-58 ◽  
Author(s):  
Dejan Bojović
Keyword(s):  
Convergence Rate ◽  
Boundary Value Problem ◽  
Heat Equation ◽  
Parabolic Problem ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Variable Coefficients ◽  
Rate Estimate ◽  
Convergence Rate Estimate ◽  
Initial Boundary Value

Abstract In this paper we consider the first initial boundary-value problem for the heat equation with variable coefficients in a domain (0; 1)x(0; 1)x(0; T]. We assume that the solution of the problem and the coefficients of the equation belong to the corresponding anisotropic Sobolev spaces. Convergence rate estimate which is consistent with the smoothness of the data is obtained.

ON THE SOLVABILITY OF A MIXED PROBLEM WITH DEGREE LAPLACE OPERATORS WITH NONLOCAL BOUNDARY CONDITIONS

2020 ◽  
pp. 85-104
Author(s):  
Shakirbai G. Kasimov ◽  
◽  
Mahkambek M. Babaev ◽  
◽  
Keyword(s):  
Boundary Conditions ◽  
Spectral Problem ◽  
Nonlocal Boundary ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Nonlocal Boundary Conditions ◽  
Laplace Operators ◽  
Initial Boundary Problem ◽  
Initial Boundary Value ◽  
Delayed Time

The paper studies a problem with initial functions and boundary conditions for partial differential partial equations of fractional order in partial derivatives with a delayed time argument, with degree Laplace operators with spatial variables and nonlocal boundary conditions in Sobolev classes. The solution of the initial boundary-value problem is constructed as the series’ sum in the eigenfunction system of the multidimensional spectral problem. The eigenvalues are found for the spectral problem and the corresponding system of eigenfunctions is constructed. It is shown that the system of eigenfunctions is complete and forms a Riesz basis in the Sobolev subspace. Based on the completeness of the eigenfunctions system the uniqueness theorem for solving the problem is proved. In the Sobolev subspaces the existence of a regular solution to the stated initial-boundary problem is proved.

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