Nonlinear Vibration And Stability Analysis Of Axially Moving Strings/Cables

Axially moving systems are extensively applicable to manufacturing, transportation, aerospace and national defense industries for mass transfer and power transmission. The dy-namical theory and application on traveling systems are challenging and of great scientific merit, and have already been international frontiers for dynamical systems. The fundamental issue about axially moving systems is vibration analysis and dynamic stability which plays an important role in designing practical mechanisms of transportation. To avoid accidents and to provide guidance for engineers, it is important to accurately predict parametrically stable re-gions of dynamical responses, especially the critical transport speed. So far rapid progress has been made in the studies of axially moving systems and fruitful achievement has been re-ported in literature regarding linear and nonlinear oscillations of axially moving materials. Transverse vibration problems of strings developed due to various excitations such as support motions, moving forces, constraints from discrete or distributed elastic foundations and dry friction of eyelets have been extensively studied. New analysis approaches and phenomenon are emerging one after another. The understanding of what causes oscillation to the system is increasingly deep and accurate, and provides a strong boost to the development of nonlinear dynamics. However, few investigations have been found regarding the dynamics and stability of axially moving materials with complicated environmental excitations, constraints and coupling conditions. Generally, axially moving systems are still designed empirically based on experience from non-traveling systems to this date. It remains difficult to analyze the in-fluence of moving speed on the amplitude-frequency characteristics and capture significant dynamics of the traveling strings, such as principal resonances and bifurcations with varying design parameters. It is indeed an urgent need to initiate the related investigations.Axially moving strings and cables belong to gyroscopic continuous systems with infinite degrees of freedoms. They are loaded by Colioris force induced by the flow of relative refer-ence. Mathematically, the motion of strings and cables is governed by a second-order partial differential equation. Notice that the partial derivatives with respect to both temporal and spa-tial variables, known as the term of convective acceleration, render complex, speed-dependent modes, which brings great challenge in finding closed-form solutions for nonlinear systems. In this dissertation, the Galerkin's procedure is adopted to discretize the partial differential equation. Combining the analytical and numerical techniques, the nonlinear transverse vibra- tions of axially moving strings and cables with complicated loadings and constraints are in-vestigated in time-domain, such as periodic motion and its stability, resonance response and bifurcation. The main contents are as follows:(1) Based on the principles of quasi-steady aerodynamics, the dynamical model of trans-verse vibration of axially moving strings with aerodynamic forces is established. The wind loadings are introduced into the partial differential equation by nonlinear curve-fitting. The stability of static configuration of the string is considered. Explicit conditions are derived for stability region of equilibrium of the string based on the Routh-Hurwitz criterion and for gen-eration of stable limit cycles via the Hopf bifurcation in multi-parameter spaces. The periodic motions for self-excited and forced self-excited vibration are determined using the Incremen-tal Harmonic Balance method, with stability analysis carried out by eigenvalue computation of Floquet multipliers. It is demonstrated that for forced systems the stability of the periodic motion may be lost due to more successive bifurcations. The periodic motion becomes qua-si-periodic after secondary Hopf bifurcations. Further, continuation software MatCont is adopted to analyze the bifurcation scenario of the limit cycle, including fold bifurcation, Neimark-Sacker (NS) bifurcation as well as codim-2 bifurcations of NS-NS and the 1:1,1:3 and 1:4 strong resonance. The effects of excitation amplitude and frequency on the quench and synchronized quasi-periodic motions are illustrated. In addition, it is found that low-dimensional chaotic motions exist for the string when the moving speed is supercritical. It is demonstrated that the explicit criterion based on the Melnikov's method can be mislead-ing when the criteria is used to predict the occurrence of chaotic motions. In fact, the chaotic motions are restricted in a chaotic belt region. Based on the perturbation theory of Hamilto-nian systems, the analytical conditions for global transversality and tangency of the periodic motions to the homoclinic orbits are presented and verified by numerical simulation. To un-derstand the geometric structure of steady-state responses for a periodically forced axially moving string, the Poincare map, the largest Lyapunov exponent and the Z(t0,kT)-function (i.e. increment of the first integral) are used. The periodic, quasi-periodic and chaotic motions are illustrated in phase space. The grazing periodic solutions are analytically predicted. It is shown that an increasing mean velocity of the wind decreases the possibility of chaotic res-ponses. Further, The string exhibits quasi-periodic solutions for large wind speed. As the moving speed or excitation amplitude is increased, axially moving string in the supercritical regime has both period-doubling and intermittent routes to chaos. To reveal the effect of time-varying of wind velocity on the dynamical behavior, the slow transition across the point of Hopf bifurcation between trivial solution and steady-state solution is analyzed by means of the matched asymptotic expansions method by taking the wind speed as the slowly varying parameter. The slowly varying equilibrium solution and the size of boundary layer are deter- mined. It is indicated that for self-excited system, there are no periodic motions but slowly varying quasi-periodic motions for almost all initial conditions.(2) The coupled nonlinear response of the in-plane and out-of-plane modes is investi-gated for a small sagged axially moving cable by using the modified Lindstedt-Poincare me-thod. The frequency analysis shows that internal resonance occurs for a range of small sag-to-span ratio and cable tension. Under an in-plane harmonic excitation, the three-to-one internal resonance between the in-plane and out-of-plane primary modes is analyzed using the modified Lindstedt-Poincare method. Several branches of large frequency-amplitude response curves of the out-of-plane motion are found in the region of internal resonance. The influence of excitation amplitude and transport speed on the energy exchange between the in-plane and the out-of-plane modes is discussed.(3) The eigenvalue problem of an axially moving string attached by multiple mass-spring oscillators is solved through the Green's function method. The explicit Green's function is obtained by the Construction Theorem of Green's Function and the closed-form transcenden-tal equations of the eigenvalues are presented. The maximum variance rate is adopted to ex-press the dynamical interaction between subsystems of the string and the oscillators. It is found out that the dynamical interaction between the subsystems mainly happens to the first two modes of the string when the eigen-frequency of the oscillator is close to the first fre-quency of the string. The Galerkin's discretization method is analyzed so as to determine the approximate eigenvalues for large numbers of oscillators. It is found that the solution with Galerkin's method tends to be accurate with increasing expansion number. In this way, an ex-planation in terms of eigenvalues is given to show the validity of large series of Flourier ex-pansion in solving transient dynamical responses of axially moving string with attached mass-spring oscillators. Numerical results also illustrate how the eigenvalues are affected by spring stiffness and transport speed.(4) The local dynamics of an axially moving string under aerodynamic forces is investi-gated with a collocated time-delayed velocity feedback controller. The Belair Theorem is ad-vanced to a more generalized theorem for any polynomial-exponential equations with con-stant time delay. It is proved that as the time delay varies, the number of solutions of the cha-racteristic equation can only be changed when the eigenvalue passes through the imaginary axis. Thus, it is inferred that the static configuration of the string losses its stability only through Hopf bifurcation for sub-critically moving strings. The Hopf bifurcation curves are presented in the space of controlling parameter. With the aid of the center manifold reduction, a functional analysis is carried out to reduce the modal equation to a single ordinary differen-tial equation in one complex variable on the center manifold. The approximate analytical so-lutions in the vicinity of Hopf bifurcations are derived in the case of primary resonance. The curves of excitation-responses and frequency-response are shown with the effect of time delay. The stability analysis for steady-state periodic solutions of the reduced system indicates the onset of local control parameter for vibration control and response suppression. Moreover, the Poincare-Bendixon theorem and energy-like function are used to investigate the existence and characteristics of quasi-periodic solutions when the periodic solution becomes unstable. The validity of the analytical prediction is demonstrated through numerical results. Two different kinds of quasi-periodic solutions are reported.