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

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

Oscillating Flows of Fractionalized Second Grade Fluid with Slip Effects

This paper examines the fractionalized second grade fluid dueto oscillating plate under slip condition. The discrete Laplace transformtechnique is employed to compute the analytical solutions for the equa-tions of motion. The velocity field and shear stress are computed. In orderto write them in compact form, the Wright generalized hyper geometricfunction is used and written as addition of slip and no slip contributions.The closed-form solutions for ordinary second grade and Newtonian flu-ids carrying out the similar motion are achieved. The computations forfractional and ordinary second grade fluids without slip effect are alsoachieved as a special case. Furthermore, the impact of various parameterssuch as the slip, fractional and material parameters on the motion of frac-tionalized second grade fluid will be explained through graphs. Finally,a comparison among the Newtonian fluids, ordinary second grade fluidsand fractionalized second grade fluids is also carried out