Spectral Analysis Of A String System With Viscoelasticity

The purpose of this report is to study the string system with local and global viscoelasticity which was done by author during postgraduate student studying. The properties of spectrum of this system such as spectral distribution are studied. By the technique of asymptotic analysis, the spectrum of this kind of system becomes clear.Smart materials, such as piezoelectric, electrostrictive, magnetostrictive, are applied widely on vibrating control. The advent of these materials as sensors and actuators provides the opportunity for designing systems with improved performance while reducing weight, hardware and power requirements. They also don't affect the elastomeric structure distinctly. So the sta-bilizability of distributed-parameter system is absorbed attention of scholars. More and more scholars started to study them. However, the spectrum of system has not been clear. For discussing the change of the system's behavior, it is necessary to study the structure of system from the point of mathematics. So the properties of spectrum of operator in system ,such as the change of spectral distribution should be analyzed. But there are few results about them. In this report, spectral analysis of a string system with local and global viscoelasticity is gaven . Then by the spectral analysis of this system, some properties of stability can be obtained.Although the purpose of this paper is to study two special systems, this method can be generalized in the study of other models, such as Timoshenko beam system with local or global viscoelasticity and Euler-Bernoulli beam system with local or global viscoelasticity.