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.

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 GENERALIZED SOLUTIONS FOR THE EULER–BERNOULLI MODEL WITH ZENER VISCOELASTIC FOUNDATIONS AND DISTRIBUTIONAL FORCES

2013 ◽  
Vol 11 (02) ◽  
pp. 1350017 ◽  
Author(s):  
GÜNTHER HÖRMANN ◽  
SANJA KONJIK ◽  
LJUBICA OPARNICA
Keyword(s):  
Differential Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Variational Problems ◽  
Generalized Solutions ◽  
Beam Model ◽  
Order Partial Differential Equation ◽  
Viscoelastic Foundation ◽  
Regularization Techniques ◽  
Initial Boundary Value

We study the initial-boundary value problem for an Euler–Bernoulli beam model with discontinuous bending stiffness laying on a viscoelastic foundation and subjected to an axial force and an external load both of Dirac-type. The corresponding model equation is a fourth-order partial differential equation and involves discontinuous and distributional coefficients as well as a distributional right-hand side. Moreover the viscoelastic foundation is of Zener-type and described by a fractional differential equation with respect to time. We show how functional analytic methods for abstract variational problems can be applied in combination with regularization techniques to prove existence and uniqueness of generalized solutions.

scholarly journals Existence and long-time behavior of solutions to the velocity-vorticity-Voigt model of the 3D Navier-Stokes equations with damping and memory

2021 ◽  
Author(s):  
Nguyen Toan
Keyword(s):  
Stokes Equations ◽  
Dynamical Behavior ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Navier Stokes ◽  
Time Behavior ◽  
Navier Stokes Equations ◽  
Long Time ◽  
Voigt Model ◽  
Initial Boundary Value

In this paper, we study the long-time dynamical behavior of the non-autonomous velocity-vorticity-Voigt model of the 3D Navier-Stokes equations with damping and memory. We first investigate the existence and uniqueness of weak solutions to the initial boundary value problem for above-mentioned model. Next, we prove the existence of uniform attractor of this problem, where the time-dependent forcing term $f \in L^2_b(\mathbb{R}; H^{-1}(\Omega))$ is only translation bounded instead of translation compact. The results in this paper will extend and improve some results in Yue, Wang (Comput. Math. Appl., 2020) in the case of non-autonomous and contain memory kernels which have not been studied before.

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.

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 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.

WELL-POSEDNESS OF AN INITIAL BOUNDARY VALUE PROBLEM FROM LASER DYNAMICS

2002 ◽  
Vol 12 (04) ◽  
pp. 593-606 ◽  
Author(s):  
F. JOCHMANN ◽  
L. RECKE
Keyword(s):  
Boundary Conditions ◽  
Semiconductor Lasers ◽  
Dynamical Behavior ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Weak Formulation ◽  
First Order ◽  
Laser Dynamics ◽  
Regularity Properties ◽  
Initial Boundary Value

In this paper a mathematical model, consisting of nonlinear first-order ordinary and partial differential equations with initial and boundary conditions, for the dynamical behavior of multisection DFB (distributed feedback) semiconductor lasers is investigated. We introduce a suitable weak formulation and prove existence, uniqueness and regularity properties of the solutions. The assumptions on the data are quite general, in particular, the physically relevant case of piecewise smooth, but discontinuous with respect to space and time coefficients in the equations and in the boundary conditions is included.

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