Energy decay estimates for an axially travelling string damped at one end

We study the small vibrations of an axially travelling string with a dashpoint damping at one end. The string is modelled by a wave equation in a time-dependent interval with two endpoints moving at a constant speed $v$. For the undamped case, we obtain a conserved functional equivalent to the energy of the solution. We derive precise upper and lower estimates for the exponential decay of the energy with explicit constants. These estimates do not seem to be reported in the literature even for the non-travelling case $v=0$.Comment: 14 Pages, 2 Figure

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 -

Supercritical stability of an axially moving beam part II: Vibration and stability analyses

This paper focuses on the stability of axially moving beam-like materials (e.g., belts, bands, paper and webs) which translate at speeds near to and above the so-called "critical speed stability limit." In the companion paper, a theoretical model for an axially moving beam was presented which accounted for geometrically non-linear beam deflections and the initial beam curvature generated by supporting wheels and pulleys. In that paper, analysis of steady response revealed that the beam possesses multiple, non-trivial equilibrium states when translating at supercritical speeds. The equations of motion are presently linearized about these equilibria and their stability is predicted from the eigenvalue problem for free response. Asymptotic and numerical solutions to the eigenvalue problem are presented for the respective cases of small and arbitrary equilibrium curvature. The solutions illustrate that the translating beam has multiple stable equilibrium states in the supercritical speed regime. The solutions confirm that the critical speed behavior for axially moving materials is extremely sensitive to system imperfections, such as initial curvature.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/30051/1/0000419.pd

Nonlinear vibration analysis of axially moving strings in thermal environment

In this study, nonlinear vibration of axially moving strings in thermal environment is investigated. The vibration haracteristics of the system such as natural frequencies, time domain response and stability states are studied at different temperatures. The velocity of the axial movement is assumed to be constant with minor harmonic variations. It is presumed that the system and the environment are in thermal equilibrium. Using Hamilton’s principle, the system equation of motion, and t[1]he boundary conditions are derived and then solved by applying Multiple Time Scales (MTS) method. The effect of temperature on the vibration characteristics of the system such as linear and nonlinear natural frequencies, stability, and critical speeds is investigated. Considering ideal and non-ideal boundary conditions for the supports, nonlinear vibration of the system is discussed for three different excitation frequencies. The bifurcation diagrams for ideal and non-ideal boundary conditions are presented under the influence of temperature at various speeds

Supercritical stability of an axially moving beam part I: Model and equilibrium analysis

Axially moving material problems consider the dynamic response, vibration and stability of long, slender members which are in a state of translation. This study focuses on the response of axially moving beam-like elements at translation speeds that exceed the classical "critical speed stability limit". A non-linear model for an axially moving beam is derived that accounts for the initial beam curvature induced by supporting pulleys or wheels. Presently, the model is used to determine steady responses at critical and supercritical translation speeds. The properties of the equilibrium problem are examined using an approximate linear solution and an exact, non-linear solution. The deficiency of the linear solution is illustrated by its inability to capture essential features of the equilibrium problem particularly near and above the critical speed. In this high-speed region, the translating beam undergoes large overall buckling deformations leading to multiple and bifurcated equilibrium states. The stability of the equilibria is assessed in Part II.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/30050/1/0000418.pd

Experimental and numerical investigation of effect of stick-slip friction transition on lateral vibration of magnetic tapes

Magnetic tape storage which is primarily used for long-term archival and backup of digital data, has historically been the most efficient, high-capacity and least expensive storage technology for huge quantities of data. Tape storage applications are in diverse fields such as corporate and government financial records, satellite imagery, credit card databases and patient medical records. Recently, the Linear Tape-Open (LTO) Ultrium format has emerged as the most dominant tape technology option in the mid-range tape drive market, with the LTO generation 5 (LTO-5) being capable of holding as much as 1.5 TB of uncompressed data on a single cartridge. Tape storage however has been traditionally challenged by competing technologies like hard disk drives (HDD), consumer optical storage devices which include CDs, DVDs and Blu-ray disk technologies, optical library systems and holographic storage systems. Thus, one of main goals of the tape industry is to design and manufacture advanced tape storage technologies that aim at reducing the price per unit data storage ($/GB). In commercial tape drives, a flexible magnetic tape is transported between the supply and take-up packs at a fixed axial tension and transport speed and over edge and surface guides and read/write heads. The tape decks must assure accurate guiding and transport of the tape while it accelerates and decelerates by holding the axial tension constant. During transport, lateral in-plane vibration of tape\u27s narrow edge causes misalignment between data tracks on the tape and position of read/write head and leads to reduced storage capacity. Lateral vibration (low and high frequency) is caused by excitation sources viz. pack run-out, flange impacts, pack tilts and tape edge weave. High frequency lateral vibration is more detrimental as it is difficult to move the read/write head to follow the tape\u27s high frequency motion. To attenuate this vibration, surface guides (rollers or stationary guides) which control the lateral displacement of tape by applying friction on its wider surface, are used. Choice of an appropriate surface guide (or roller) is possible with an understanding of the physics involved in the surface friction between magnetic tape\u27s wide surface and the roller. This thesis is motivated by the need to conduct a detailed investigation into the frictional interaction between roller surface and magnetic tape and contribute towards the advancement of tape technology to meet the growing market needs. A parametric study is carried out with respect to the tape\u27s axial tension and axial velocity in the following two aspects: * An experimental setup is used to control these tape parameters and obtain lateral vibration measurements at two points equidistant from the tape-roller interface to understand the effect of stick-slip friction at the interface on tape\u27s lateral vibration * A numerical model is developed to study stick-slip friction between the roller surface and the tape that travels over it. The tape is modeled as an axially moving, tensioned, viscoelastic Euler-Bernoulli beam subjected to boundary disturbances arising from supply and take-up pack run-out and stick-slip friction between tape and roller surface. These analyses are used to predict the possibility of sticking or slipping between the surfaces in contact, as a function of parameters viz. axial tension, axial velocity, surface roughness of roller and span length. A `dynamic phase diagram\u27 is constructed to determine the regions in the stiffness-velocity phase-space where steady stick-slip motion occurs and its effects on lateral vibration of the magnetic tape

Propagation of longitudinal tension in a slender moving web

To date, most of the theoretical work on longitudinal web behavior has been directed at the problem of controlling average tension. Very little attention has been given to the subject of this paper - propagation of tension within a span.The model presented here is based on the one-dimensional wave equation, modified for a moving medium. Boundary conditions are developed that, for the first time, incorporate tension and mass transfer on rolling supports. The P.D.E. is solved analytically using Laplace transforms.A number of phenomena are described that will be of interest to process designers and troubleshooters. These can be used to explain existing tension problems, whose causes may have been unrecognized in the past, and to anticipate problems that will appear as line speeds are increased. Among these are:1. Propagation of strain discontinuities when draw is increased suddenly.2. Amplification of repetitive strain disturbances due to strain reflection and reinforcement.3. Damping of solitary strain disturbances.4. Alteration of longitudinal resonant frequencies by transport motion.Another important use of the model is to serve as a necessary step toward more advanced models that include out-of-plane motion, viscoelasticity and aerodynamics.The model is tested by comparing it to the currently accepted O.D.E. model. At large time scales, where propagation phenomena are imperceptible, the two models are in good agreement

Viscous fluid buckling: a theoretical and experimental analysis with extensions to general fluid stability

An experimental and theoretical study of the spontaneous oscillations observed when a very viscous fluid flows from an orifice vertically against a flat surface was carried out. Silicone oils of very high viscosities were the main working fluids;Both plane and axisymmetric jets were studied experimentally. The main theoretical considerations, such as the basic stable jet shape and the equations governing the response of the jet to external disturbances were, however, carried out only for the one-dimensional, axisymmetric jet;The moving threadline equation is shown to also govern the oscillations of the fluid column and it is suggested that certain conclusions reached in the study of the moving threadline have direct application to the problem of fluid buckling;Results are presented in non-dimensional form for the buckling height , the plate-orifice distance at which the spontaneous oscillations first occur and the behavior of these oscillations as functions of geometrical, fluid and flow properties for both axisymmetric and plane jets;Analytical solutions for the equations governing the jet radius along the stable jet for the viscous-gravity jet are provided. Based on the one-dimensional assumptions and continuity considerations, the velocity along the jet can easily be determined from these solutions;The analytic solutions for the viscous-gravity jet profile are compared with the experimentally determined jet profile and with the numerically obtained solutions of the complete one-dimensional equations (including inertia and surface tension);The existence of a jump or discontinuity in the oscillations of the jet is postulated and a one-dimensional model of the unstable jet, based on the concept of a discontinuity, is used to show that there may be a net energy loss across the discontinuity;Finally, it is suggested that other fluid stability problems might be governed by the same mechanisms responsible for fluid buckling, and a hypothesis for the determination of the regions of instability in fluids based on a one-dimensional linearized approach is offered