VIBRATION AND WAVE LOCALIZATION IN A NEARLY PERIODIC BEADED STRING

The dynamics of an exact, linear model of a string with attached beads are examined. A study of the perfectly periodic system is performed u sing a transfer matrix formulation. A closed-form solution for the nat ural frequencies of the finite system is obtained. The relationship be tween natural frequencies and the passbands and stop bands of propagat ing waves is studied. The effect of random disorder of (a) bead spacin g and (b) bead mass is examined and interesting, fundamental differenc es are observed, In the case of bead-spacing disorder both weak and st rong localization occur, whereas bead-mass disorder only causes weak l ocalization. Localization, the spatial confinement of vibration energy due to periodicity breaking disorder, is quantified by the localizati on factor. Analytical approximations of the localization factor are de rived, in the limits of large and small coupling to disorder ratios. T he results are verified through Monte Carlo simulations. The natural f requencies and modes of the disordered system are examined. A dispersi on of the natural frequencies is evidenced, presented as a curve veeri ng phenomenon. The existence of localized mode shapes is demonstrated. (C) 1997 Acoustical Society of America.