Stability of an Inhomogeneous Damped Vibrating String

In this paper, we consider the vibrations of an inhomogeneous damped string under a distributed disturbing force which is clamped at both ends. The well-possedness of the system is studied. We prove that the amplitude of such vibrations is bounded under some restriction of the disturbing force. Finally, we establish the uniform exponential stabilization of the system when the disturbing force is insignificant. The results are established directly by means of an exponential energy decay estimate

NASA Workshop on Distributed Parameter Modeling and Control of Flexible Aerospace Systems

Although significant advances have been made in modeling and controlling flexible systems, there remains a need for improvements in model accuracy and in control performance. The finite element models of flexible systems are unduly complex and are almost intractable to optimum parameter estimation for refinement using experimental data. Distributed parameter or continuum modeling offers some advantages and some challenges in both modeling and control. Continuum models often result in a significantly reduced number of model parameters, thereby enabling optimum parameter estimation. The dynamic equations of motion of continuum models provide the advantage of allowing the embedding of the control system dynamics, thus forming a complete set of system dynamics. There is also increased insight provided by the continuum model approach

The string vibrator experiment-experimental and analytical determination of system fundamental natural frequencies

This thesis represents an experimental and intellectual journey of more than three years. The String Vibrator Experiment (SVE) is an experimental payload on the SumbandilaSat, satellite. The experiment was initially designed to compare low-earthorbitstretched string nonlinear dynamics with those found terrestrially, with particular emphasis on the effects of micro-gravity and near vacuum. Due to the damage of some internal components of the SVE, the terrestrial experiment had to be characterized by finding both transverse fundamental natural frequencies and the longitudinal system fundamental natural frequency. Forced and free vibration tests were performed on the SVE, with comparable results. The percentage frequency difference between measurement results obtained for the said natural frequencies was found to be acceptable. An analytical model for the string longitudinal system was developed using Energy Methods, assuming linear coupling of the stretched string and the flexible support. Bessel Functions were used to find the natural frequency of the flexible support, which is made of a thin-circular plate loaded centrally by the stretched string and fixed on its outer boundary. The natural frequency thus predicted, corresponds closely with that measured by Laser Doppler Vibrometry. It was found that the longitudinal system behaves in an uncoupled, manner. In this study use was made of piezo bi-morph sensors as vibration measuring devices, with promising results. A method called the peak amplitude discrimination method is introduced to find the respective resonant frequencies; this method is an adaptation of the sweptsine frequency method Nonlinear behavior of the SVE was found to exist exhibiting Sub-harmonics, Super-harmonics, Combination harmonics and Internal Resonances. It is hoped that the experimental methods used in this study would aid in the measurement and analysis of longitudinal vibrations found in flexible cables, rods and rigid bars.Thesis (PhD) -- Faculty of Science and Agriculture, 201

Magnetic suspension and vibration control of flexible structures for non-contact processing

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering, 2000.Includes bibliographical references (p. 365-372).This thesis presents the design, analysis, and experimental testing of systems for noncontact suspension and control of flexible structures. Our particular focus is on the use of such suspensions in manufacturing processes which can be facilitated by the ability to control workpiece motion without contact. This can be of significant utility in processes such as coating, painting, heat treating, and web handling. We develop a novel approach for the control of such non-contact suspensions through what we term sensor averaging and actuator averaging. The difficult stability and robustness problems imposed by the flexible dynamics of the workpiece can be overcome by taking a properly-weighted average of the outputs of a distributed array of N motion sensors (sensor averaging), and/or by applying a properly-weighted distributed array of M forces (actuator averaging) to the workpiece. The theory for these dual techniques is developed in detail in the thesis. These approaches are shown to be independent of the specific boundary conditions or the longitudinal dimensions of the workpiece. These approaches are thus generally applicable to a wide range of structural control problems. We present both analytical and numerical analyses of the structural dynamics for typical flexible workpieces such as strings, beams, membranes, and plates. The analyses include axial translation of the workpiece. We have experimentally demonstrated the utility of our theory by application in the successful magnetic suspension of a 3 m long, 6.35 mm diameter, 0.89 mm wall thickness steel tube with varying boundary conditions. This is a very challenging problem due to the extremely light damping of the modes (< 0.001 with free ends). The experiment uses a set of 8 sensors and 8 actuators to measure and control the motion of the tube in the two lateral degrees of freedom. We present the details of the developed electromagnetic actuators, position sensors, modeling of the structural dynamics, the relevant vibration control techniques, and develop the associated theory for choosing sensor and actuator locations. Our results experimentally confirm the value of our averaging techniques, and suggest the wide future application of these ideas in industrial processes which require non-contact handling of workpieces.by Ming-chih Weng.Ph.D

The dynamics of mine hoist catenaries.

A Thesis Submitted to the Faculty of Engineering, University of the Witwatersrand, Johannesburg, South Africa for the Degree of Doctor of Philosophy.The dynamic analysis of catenary vibration of mine hoist ropes on South African mines is examined. This research has been preceded by studies in the mining industry, which have laid the foundation fot the definition of design guidelines of hoist systems to avoid catenary vibrations or rope whip. These guidelines are based on a classical linear analysis of a taut string, and in essence rely on ensuring that the frequency of excitation at the winder drum due to the coilingmechanism, does not coincide with the linear transverse natural frequency of the taut catenary. Such an approach neglects the nonlinear coupling between the lateral catenary motion and the longitudinal systern response. Although previous research sug gested the possibility of autoparametric coupling between the catenary and vertical rope, this was not developed further on a theoretical level.. The possibility of such behaviour is defined by considering the equations of motion of the coupled system. A design methodology is developed for determining the parameters of a mine hoist systern so as to avoid rope whip. The methodology accounts for the nonlinear coupling between the catenary and longitudinal system. In order to implement the proposed methodology, two phases of the analysis are developed. In the first phase the stability of the linear steady state motion is examined in the context of the nonlinear equations of motion, by applying a harmonic balance method. The stability analysis defines regions of secondary resonance, where it is shown that such regions may arise at sum and difference combinations of the linear lateral and longitudinal natural frequencies due to autoparametric excitation. Prior to this research, this phenomenon had not been appreciated in the context of the mine hoist system. A laboratory experiment was conducted to confirm the existence of these regions experimentally. In reality, the system is non-stationary since the dynamic characteristics of the system change during the winding cycle, and hence the steady state stability analysis can only describe potential regions of nonlinear interaction on a qualitative basis. The second phase of the analysis deals with a non-linear numerical simulation of the hoist system, which accounts for the non-stationary nature of the systems dynamic characteristics, and includes transient excitations induced during the wind. The methodology developed is assessed by considering the Kloof mine rope system, where rope whip was observed. This study demonstrates that although an appreciation of the steady state system characteristics is useful, the stability analysis alone is not sufficient. It is necessary to account for the non-stationary aspects of the winding cycle if a realistic interpretation of the observed behavlcur is to be achieved. To compliment this study, a motion analysis system was developed to record catenary response on an existing mine hoist installation. Such data has not been recorded before. This data provides direct evidence of the autoparametric nature of the coupled catenary/vertical rope system.AC201

Nonlinear damping model for flexible structures

The study of nonlinear damping problem of flexible structures is addressed. Both passive and active damping, both finite dimensional and infinite dimensional models are studied. In the first part, the spectral density and the correlation function of a single DOF nonlinear damping model is investigated. A formula for the spectral density is established with O(Gamma(sub 2)) accuracy based upon Fokker-Planck technique and perturbation. The spectral density depends upon certain first order statistics which could be obtained if the stationary density is known. A method is proposed to find the approximate stationary density explicitly. In the second part, the spectral density of a multi-DOF nonlinear damping model is investigated. In the third part, energy type nonlinear damping model in an infinite dimensional setting is studied

Dynamics and Control of Very High Altitude Tethered Balloons

This research involves the development of a full 3-dimensional nonlinear model of a tethered aerostat, with the aim of conducting a preliminary feasibility assessment of Very High Altitude Tethered Balloons (VHATBs) under both steady-state and extreme operational conditions. A discretized lumped-parameter model is used to represent the tether. HATBs have a variety of potential applications ranging from telecommunications to solar power harvesting, but none have been practically implemented to date. The application considered in this research is Stratospheric Particle Injection for Climate Engineering for which the setup comprises of several unique features that are expected to have an impact on the system’s dynamics and therefore its technical feasibility. An operational altitude of 20km is recommended for this application, which makes it the largest scale tethered aerostat considered to date. A hollow pipe-like tether is required for the transportation of the high-pressure fluid to the stratosphere where it is expected to scatter radiation. In order to qualify for long-duration operation, HATBs must be able to withstand extreme weather conditions on top of the harsh steady-state winds the tether would be subjected to. In this research, a series of dynamic simulations are conducted to assess the system’s safety in the face of a range of potential 3-dimensional disturbances. The balloon’s altitude deviation and the tether’s maximum longitudinal stress relative to that of the system’s equilibrium are considered the critical parameters that are used to quantify the system’s response, as these are considered the primary risks of failure. The use of optimal control methods to minimise these is proposed and introduced into the dynamic simulations. Previous methods proposed for the altitude stabilization of tethered aerostats are not practical for the scales considered in this research, and so the horizontal motion of the ship the tether is mounted to is suggested as an alternative control input. A preliminary assessment of the potential use of this type of control to reduce failure risks in the face of extreme wind disturbances is provided. The closed-loop performance, practicality and robustness of the controller are considered in this analysis. The critical parameters are found to be reduced by ∼ 65% for in-plane disturbances and ∼ 90% for out of plane disturbances through the implementation of controllers that are deemed both practically feasible and sufficiently robust