One-dimensional Quasi-crystal Rod Vibration And Fluctuation Analysis

Since the experimental discovery of quasicrystals in 1984, this new kind of materials has attracted much attention and has been extensively studied. In this paper, the vibration problem and the wave problem in longitudinal direction of a bar, which is made of quasicrystals are mainly considered. It includes the bar's free vibration in longitudinal direction and wave problem, and the forced vibration problem when a force is applied to an end.Firstly, based on the constitutive equations of the one-dimensional hexagonal quasicrystals, the control equations are derived by energy variational principle. Then using the related boundary conditions and solving the control equations, we found the one-dimensional free vibration solution in longitudinal direction of phononfield and phasonfield. At the same time, we found all the frequencies of the free vibration.Secondly, considering the quasicrystal bar and applying two cycle forces to it's one end, which separately have an effect on the phononfield and phasonfield, we obtained the forced vibration equations by using the virtual work principle. Then, by using the Duhamel integral, we found the displacement responses of the phonon field and phason field. Simultaneity, we take a specific example, and give a chart to clearly show their infection relation.Finally, we make an analysis on elastic wave problem of the one-dimensional quasicrystals bar. We obtain the standard wave equations by transforming the control equations, and find four speeds of the wave. Based on the given initial conditions, we gained the general solutions of the one-dimensional wave equations. The displacement solutions are also figured.This dissertation is divided into six chapters. The research actuality of the related field of quasicrystals is briefly introduced in the first section. And the work to be done, the research methods and its significance are also explained in this section. The linear elastic theory and the physical methods used in this article are briefly introduced in chapter 2. In chapter 3 and chapter 4, we respectively analysed and studied the free vibration in longitudinal direction problem and the fored vibration problem of the one-dimensional quasicrystal bar. The wavecharacteristic of the one-dimensional quasicrystal bar is studied and analyzed as an emphasis in the fifth section. The last chapter includes the summary and prospect.