On Energy Estimates for Damped String-Like Equation Considering Dirichlet, Neumann and Robin Boundary Conditions

This article provides detailed construction of energy estimates of the viscous damping aspects for axially moving string, which is modeled by a linear homogeneous sting-like equation, will be studied. The nine different boundary conditions are considered for the axially moving continua. The problem at hand describes the damped vertical vibrations of string-like equations, for example, a conveyor belt system and a band-saw blade. In this work, the velocity and coefficient of damping are kept positive and fixed. The stability of the system substantially depends upon change in boundary and subsequently boundary conditions. Also a decay in oscillatory energy is observed in all the considered cases of boundary conditions due to viscous damping. In some cases, the belt energy may increase or may decrease due to variations in different parameters . This exposes the uncertainty in these cases. Keywords: Belt Conveyor, String, Axially translating, Viscous dampin

On applicability of truncation method for damped axially moving string

In this paper, the detailed study of the transversal vibrations of a damped axially moving string is considered. Two end pulleys of the string are taken to be fixed and the initial conditions are assumed to be of general displacement field and the general velocity field. The axial speed of the string is considered to be sinusoidal, time-dependent and small compared to wave-velocity. A two timescales perturbation method with a combination of Fourier-sine series which fits the boundary conditions is employed in order to formulate the valid and uniform asymptotic approximations of the exact solutions for the equation. It is found that there are infinitely many values of frequency parameter Ω which cause the internal resonances in system. The fundamental resonant frequency, the non-resonant frequency and the detuning cases have been discussed and analyzed in detail. It has been found explicitly that the total mechanical energy of the infinite dimensional system decreases for two cases of the damping parameter, that is, for δ=2 and for δ>2. By truncation method it has been shown that the mode-amplitude response for first few modes is stable. So, Galerkin’s truncation method may be possible for these two cases of the parameter δ. But for case δ<2 the total mechanical energy of belt system is increasing exponentially. Therefore, it is evident that the Galerkin’s truncation method cannot be applied in order to obtain valid approximations on long timescales, that is, on timescales of O1/ε

Modelling and simulation of a stationary high-rise elevator system to predict the dynamic interactions between its components

In a high-rise elevator system lateral vibrations of the suspension and compensating ropes, coupled with vertical motions of the car and counterweight are induced by the building structure motions. When the frequency of the building coincides with the fundamental natural frequency of the ropes, large resonance whirling motions of the ropes result. This phenomenon leads to impacts of the ropes on the elevator walls, large displacements of the car and counterweight making the building and elevator system unsafe. This paper presents a comprehensive mathematical model of a high-rise elevator system taking into account the combined lateral stiffness of the roller guides and guide rails. The results and analysis presented in the paper demonstrate frequency curve veering phenomena and a wide range of resonances that occur in the system. A case study is presented when the car is parked at a landing level where the fundamental natural frequencies of the car, suspension and compensating rope system coincide with one of the natural frequencies of the high-rise building. The results show a range of nonlinear dynamic interactions between the components of the elevator system that play a significant role in the operation of the entire installation

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

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 approximateanalytic 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 -

On Aspects of Asymptotics for Axially Moving Continua

In this thesis two models for axially moving continua have been studied: a string-like model and a beam-like model. Mathematically, a string-like model is described by a wave equation and a beam-like model is usually described by the Euler-Bernoulli beam equation. A string-like model has been used to describe the transversal vibrations of a vertically moving elevator cable system with time-varying length and a beam-like model has been used to describe the transversal vibrations of a constant length conveyor belt system between two supports. For the string model, a rigid body is attached to the lower end of the string, and the suspension of this body against the guide rails is assumed to be rigid. For the string model, it is assumed that the length changes linearly in time or that the length changes harmonically about a constant mean length. For the linear length-variations it is assumed that the axial velocity of the string is small compared to nominal wave velocity and the string mass is small compared to the mass of the rigid body and, for the harmonically length variations small oscillation amplitudes are assumed. The case with boundary excitations has also been investigated in detail, and interesting resonance conditions have been found. For the beam model, the axial velocity is assumed to be constant and relatively small compared to the wave speed. The case with boundary damping has been investigated in detail for the beam equation and interesting damping properties have been obtained. The corresponding initial boundary value problems have been formulated, and in all cases formal asymptotic approximations of the analytic solutions have been constructed by using the multiple timescales perturbation methods. For the string-like problem, it has been shown that Galerkin's truncation method can not be applied in order to obtain asymptotic approximations valid on long timescales. For boundary excitations the interesting phenomenon of autoresonance occurs when there is passage through (dynamic) resonance. The maximal amplitude of the autoresonant solution and the time of autoresonant growth of the amplitude of the modes of fast oscillations have been determined. Interior layer analysis has been provided systematically and it has been shown that there exists an unexpected timescale of order (1/?(?). For this reason three timescales have been introduced when constructing asymptotic results. For the beam-like problem, by using the energy integral, it has been shown that the solutions are bounded for times t of order 1/?. It has also been analytically and numerically shown that all solutions (up to order ?) are uniformly damped.Applied mathematicsElectrical Engineering, Mathematics and Computer Scienc

On applicability of truncation method for damped axially moving string

In this paper, the detailed study of the transversal vibrations of a damped axially moving string is considered. Two end pulleys of the string are taken to be fixed and the initial conditions are assumed to be of general displacement field and the general velocity field. The axial speed of the string is considered to be sinusoidal, time-dependent and small compared to wave-velocity. A two timescales perturbation method with a combination of Fourier-sine series which fits the boundary conditions is employed in order to formulate the valid and uniform asymptotic approximations of the exact solutions for the equation. It is found that there are infinitely many values of frequency parameter Ω which cause the internal resonances in system. The fundamental resonant frequency, the non-resonant frequency and the detuning cases have been discussed and analyzed in detail. It has been found explicitly that the total mechanical energy of the infinite dimensional system decreases for two cases of the damping parameter, that is, for δ=2 and for δ&gt;2. By truncation method it has been shown that the mode-amplitude response for first few modes is stable. So, Galerkin’s truncation method may be possible for these two cases of the parameter δ. But for case δ&lt;2 the total mechanical energy of belt system is increasing exponentially. Therefore, it is evident that the Galerkin’s truncation method cannot be applied in order to obtain valid approximations on long timescales, that is, on timescales of O1/ε. </jats:p

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

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</jats:p