Stability of the wave equations on a tree with local Kelvin–Voigt damping (original) (raw)
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Stability of Nonlinear Oscillations of Stretched Strings
Journal of The Acoustical Society of America, 1969
The stability of damped forced vibrations and undamped free vibrations of stretched strings is investigated. The regions of stability for the forced vibrations are plotted in the amplitude-frequency plane. It is shown that undamped planar free vibrations are unstable. INTRODUCTION Nonlinear vibrations of a stretched string have been the subject of many investigations. 1-7 Analysis of the stability of periodic vibrations is one important aspect of the problem. Miles 4 analyzed the stability of undamped forced vibrations near resonance and charted the regions of stability in the frequency-amplitude plane. Narasimha 5 considered the effect of damping also, but he confined his stability analysis to the determination of the boundaries of onset of nonplanar motion. The regions of stability of forced oscillations in the presence of damping have not been determined so far. The stability of free vibrations of the string has also not been analyzed. The present paper deals with these unexplored aspects of the stability analysis. The problem is essentially one of determining the conditions under which all the solutions of two coupled Hill-type variational equations 8 are stable. For this purpose, a generalization of van der Pol's method ø can be employed. This technique was used by Miles, 4 and more recently by Williams, 1ø and Efstathiades and Williams. n Gilchrist 12 employed a different method to analyze the stability of free vibrations of a 2-degree-of-freedom system. In his method, periodic solutions appeared as singular points in a state plane. The problem was thus reduced to one of investigating the stability of singular points. We present here an analysis based on the theory of Hill's equation 8 and van der Pol's method. 9 The variational equations are obtained in Sec. I. Stability of forced vibrations is investigated in Sec. II, which is divided into three parts. Planar and nonplanar vibrations are investigated separately in Secs. II-A and II-B, while the results are interpreted in Sec. II-C. Stability analysis of free vibrations is carried out in Sec. III. Results of the present analysis confirm certain predictions made in Refs. 6 and 7, based on purely physical considerations. In particular, it is established that free planar vibrations of an undamped string are unstable. But they become stable in the presence of damping. I. FORMULATION OF THE PROBLEM The differential equations governing the vibrations of an elastic string were derived in an earlier paper. 7 The transverse and longitudinal vibrations of the string are coupled, and hence longitudinal vibrations exist even in the absence of an external longitudinal force. However, it was shown in Ref. 7 that, when there is no external longitudinal force, a pair of equations governing the transverse vibrations can be separated if the order of the transverse modes is small compared to the ratio of longitudinal-to transverse-wave speeds. When this condition is fulfilled, the equations of transverse motion of the string subject to a transverse,
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.
Uniform stability of damped nonlinear vibrations of an elastic string
Proceedings Mathematical Sciences, 2003
Here we are concerned about uniform stability of damped nonlinear transverse vibrations of an elastic string fixed at its two ends. The vibrations governed by nonlinear integro-differential equation of Kirchoff type, is shown to possess energy uniformly bounded by exponentially decaying function of time. The result is achieved by considering an energy-like Lyapunov functional for the system.
2020
In this paper, we study the stability problem of a star-shaped network of elastic strings with a local Kelvin-Voigt damping. Under the assumption that the damping coefficients have some singularities near the transmission point, we prove that the semigroup corresponding to the system is polynomially stable and the decay rates depends on the speed of the degeneracy. This result improves the decay rate of the semigroup associated to the system on an earlier result of Z. Liu and Q. Zhang in <cit.> involving the wave equation with local Kelvin-Voigt damping and non-smooth coefficient at interface.
Nonlinear stability criteria for tree-like structures composed of branched elastic rods
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2012
New necessary conditions for the nonlinear stability of branched tree-like structures composed of elastic rods are presented. The conditions, which are found by examining the second variation of an energy functional, are established by extending Legendre's classical work on this topic. For the branched tree-like structures of interest in this paper, one of the new necessary conditions is the existence of a bounded solution to a set of Riccati equations, which are coupled at branching points. In addition, a set of new necessary conditions are established in the event that the abscissa of the branching point is not fixed. The results are illustrated using a range of examples featuring planar configurations of branched elastic rods where each of the individual rods is modelled using Euler's theory of the elastica. Conditions of the form (2.13) 1 feature in adhesion problems where N = 1 (e.g. Majidi & Adams (2009) or Seifert (1991)). (c) Necessary conditions for an extremal Following O'Reilly & Tresierras (2011a), we can establish necessary conditions for an extremal by examining the first variation of I . We start by substituting Proc. R. Soc. A on August 20, 2018 http://rspa.royalsocietypublishing.org/ Downloaded from For those branches where y J (x J = L J ) is unspecified, the vanishing of the righthand side of equation (2.15) also yields the natural boundary conditions vf J vy J (L J , y * J (L J ), y * J (L J )) = 0. (2.17)
Stability of elastic transmission systems with a local Kelvin–Voigt damping
European Journal of Control, 2015
In this paper, we consider the longitudinal and transversal vibrations of the transmission Euler-Bernoulli beam with Kelvin-Voigt damping distributed locally on any subinterval of the region occupied by the beam and only in one side of the transmission point. We prove that the semigroup associated with the equation for the transversal motion of the beam is exponentially stable, although the semigroup associated with the equation for the longitudinal motion of the beam is polynomially stable. Due to the locally distributed and unbounded nature of the damping, we use a frequency domain method and combine a contradiction argument with the multiplier technique to carry out a special analysis for the resolvent.
Vibration Modes in a String of Three Branches
Journal of Sound and Vibration, 1999
The classic problem of the transverse vibrations of a string is of basic and technological interest, since it constitutes an acceptable model for the dynamic behavior of oceanographic cables and musical instruments among others . One of the important points in this area is the propagation of waves in systems with interferences or in mediums with different propagation speed. However, the literature shows, in general, very little information on the main aspects of stationary waves in these types of systems. In this work we studied the behavior of the transversal stationary waves in a mechanical system composed of three strings connected to a ramification point ( . First, the spectrum of natural frequencies of the system for a completely general case is presented. Then two particularly simple situations are studied: the case of a string with three identical branches and that of a string with two equal branches and the third of different length. Results obtained for a string with branches of different chain lengths are similar to those for a linear string of fixed ends in some particular situations.
On the boundedness of damped strings and beams with boundary and distributed inputs
2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601), 2004
This paper proves the bouudeduess of boundary and distributed damped strings and Euler-Bernoulli beams under combined distributed and boundary inputs. Distributed viscous or Kelvin-Voigt damping or a translational boundary damper stabilize strings and beams. Pointwise bounded re sponse is proven using the energy multiplier method. Without disturbances, the method proves strong exponential stability.