On applicability of truncation method for damped axially moving string

Abstract

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 2 and for 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. But for case 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 1⁄ .

Dynamic behavior of axially moving web in multi-span roll-to-roll microcontact printing system

“…23–27 Nonlinear dynamics of strings having varying axial tension and constant axial speed were analyzed in some studies. 28–31…”

“…[23][24][25][26][27] Nonlinear dynamics of strings having varying axial tension and constant axial speed were analyzed in some studies. [28][29][30][31] The study of the vibration of a moving membrane is vital when the web width is large enough to ensure that the full dynamics of axially moving materials is taken into account. A numerical study on the dynamics of an axially moving membrane was performed by Koivurova and Pramila, 32 who accounted for membrane viscosity, finite strain values and the geometric non-linearities.…”

An analytical wave solution for the vibrational response and energy of an axially translating string in any propagation cycle

“…As a classical solution, the Fourier series method has been used for stationary continuous systems such as beams [10] and shells [11], as well as for axially traveling string with fixed boundaries [12][13][14], yet it has not been applied to nonclassical boundaries with mass, damping or spring, due to the difficulty of satisfying the complex boundary conditions. In addition, Fourier series method is approximate since the limited number of terms to be expanded.…”