Objectives  The localization effect produced by a disordered structure can be quantitatively described by localizing factors. In order to study the mechanism of the effect and the expression of localizing factors, the rules of the localizing factors must be ascertained.

  Methods  From the perspective of waves, this paper analyzes the generation principle of the pass band and stop band of periodic structures, and explains the reason for the localization of disordered periodic structures. It compares the transfer characteristics of periodic and disordered periodic structures, and gives the calculation method for localizing factors based on Furstenberg's Theorem. Taking the mass-spring chain model as an example, this paper derives the pass-band and stop band of periodic systems, as well as the mathematical expressions of localizing factors for disordered periodic systems, studies the frequency variation of localizing factors and verifies them via numerical simulation based on the Monte Carlo method.

  Results  The results show that the degree of localization is related to the perturbation variance. The localizing factors have random distribution characteristics, but they approach the limit with the increase in the number of units, which can be predicted.

  Conclusions  This paper can provide references for understanding the vibration localization of generally disordered periodic structures, and guiding vibration control in engineering.