94 Analysis of Vibrational Dampers

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ANALYSIS OF VIBRATIONAL DAMPERS FOR TRANSMISSION LINES VIA SCATTERING THEORY Technical Note Submitted to IEEE Transactions on Power Systems August 19, 1994 Dr. Jacob S. Glower, Member ASME, IEEE Department of Electrical Engineering North Dakota State University, Fargo, North Dakota 58105 Abstract In this paper, the problem of adding damping to a vibrating string is considered. Instead of using feedback control or frequency domain techniques, a wave-analysis approach proposed by von-Flotow for be
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  ANALYSIS OF VIBRATIONAL DAMPERS FOR TRANSMISSIONLINES VIA SCATTERING THEORY Technical Note Submitted to IEEE Transactions on Power Systems August 19, 1994 Dr. Jacob S. Glower, Member ASME, IEEEDepartment of Electrical EngineeringNorth Dakota State University, Fargo, North Dakota 58105 Abstract  In this paper, the problem of adding damping to a vibrating string is considered. Instead of using feedback control or frequency domain techniques, a wave-analysis approach proposed by von-Flotow for beams and trusses is used. This technique is based upon the premise that resonances are caused by waves reflecting off of discontinuities. By eliminating thereflections, the resonances will dissappear. The purpose of the controller is therefore tominimize reflections.This paradigm is used to analyze two different controllers commonly used to dampenvicrations in high-voltage transmission lines: a mass damper tied to an absolute referencesimilar to winglets and a mass damper similar to a Stockbridge damper. Using scatteringtheory, it is shown that both designs can be effective at minimizing reflections of travelingwaves ad thereby minimize oscillations. Further, this analyzsis approach gives good insight asto why each controller works and what the effect of varying different parameters will be - features which are often lacking in other design and analysis approaches. Hence, the waveanalysis approach of von Flotow appears to be highly appropriate to the analysis of suchcontrollers. 1  Introduction A large number of systems have lightly damped modes that are undesirable. Such systems rangefrom long bridges during an earthquake to tall buildings swaying in the breeze to lightweight roboticarms that flex when rapidly slewed to high-voltage transmission lines which vibrate in the wind. In allthese cases, a better system could be obtained if the resonances could be reduced. The objective of a controller for each of these systems is likewise the same: reduce the resonance.While the objective for the controller in each case may be similar, the approach used to design thecontroller often varies considerably. Often times, the approach used is dictated by the point of viewtaken to model the system.One modeling approach uses modern control techniques along with a state-space formulation (Balas1979, Joshi 1989). With this approach, the differential equation describing the system is expressed interms of an infinite-series whose elements consist of eigenvectors and eigenvalues - often referred toas mode shapes and modal frequencies. The resonances results in the poles of the system beinglightly damped. The objective of the controller is then to increase the damping of these poles.An implicit assumption behind using a state-space formulation is that some form of state-feedback will be used to control the system. Those who take this design approach likewise use similarcontroller structures with the gains computed using different algorithms ranging from LQR techniques(Lim, 1992), root locus techniques (Maghami 1992, Yang 1992), inverse plant and sliding-modes(Watkins 1992, Yurkovich 1990), and others. This approach is extremely popular and has been2  used on trusses (Watkins, 1992), beams (Ferri, 1992), simulations of space structures, (Balas 1978,Lim 1992), one-link (Kotnik, 1986) and multilink robotic arms (Yurkovich 1992, Xu 1992).While these approaches all work well in simulations, they often have problems when implemented(Lim, 1992). One explanation for this may be the modelling errors introduced by using a finite-orderstate-space model of the system rather than the infinite-order model. Gibson (1980) demonstratedthat this alone will destabilize any feedback control law that does not use collocated feedback unlessthe system has natural damping. Another possible explanation could be the sensitivity of the modeland likewise the controllers based upon them. Since lightly damped structures often are time-varying(as the load changes) or are uncertain, this sensitivity may be a problem (Desanghre 1986, Joshi1989). The design of a controller than is robust to model uncertainty has likewise come intoprominence recently.A second approach for modeling an oscillatory system is to look at the transfer function from theapplied force to the resulting displacement - termed the input impedance (Snowdon, 1968). Usingthis point of view, resonances result in the impedance becoming large. The design of a controller,therefore, should be to reduce the amplitude of the input impedance.Ormondroyd (1928) first used this notion to design a mass damper for a flexible beam. By tuning themass and spring to the first resonance of the beam, the resonance could be shifted to a higherfrequency. The natural damping of the beam at higher frequencies would then reduce the impedanceat this resonance and likewise reduce the resonance. Brock (1946) extended this work by adding a3  damping to the mass damper and showing how the damper could be tuned to minimize theimpedance of the shifted resonance. Snowdon (1968) followed this by showing how several massdampers could be used to dampen several resonances. These techniques have been used to designdampers for flag poles, power lines, and tall buildings with some degree of success (Hunt 1972).Unfortunately, these controllers are often difficult to tune and often are not as effective as predicted(Iwuchukwu 1985). Some adaptive techniques to automatically tune these controllers has likewisebeen proposed (Desahghre, 1986).A third approach as proposed by von Flotow takes an entirely different point of view towards thecause of (and hence control of) resonances. To understand this point of view, consider the followingthought study:First, consider an system which extends to infinity such as a large pond or an infinite string. Such asystem would have no resonances since any initial conditions would disappear as the energy radiatesout to infinity similar to ripples in a pond.Next, suppose that you wanted to create a resonance. To do this, you must prevent the energy fromradiating out to infinity - i.e. the energy must be confined somehow. To confine the energy, somemethod of reflecting it back into the region when it gets too far away is required. Since reflections arecaused by discontinuities in the medium, this means that you must surround a region with adiscontinuity. This is analogous to replacing the pond with a glass of water - the waves now reflectoff the edges back into the glass, containing the energy and creating resonances.4
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