Theory and engineering application of length fractal dimension on nonlinear vibration waveform

Using supercube as overlying cell, the fractal dimension computing method to waveform proposed by Katz is analyzed and its shortage is pointed out. Based on Hausdorff and divider dimensions and using vibration waveform's property of going forward and topology map,the definition and formula of length fractal dimension (D _L) are formed. Its strictness in math is proved and reliability in engineering application is assessed, so the conclusion is made that D _L is of characteristics of concept clear, computation simple, quick to the theory solution and suitable to the small data. It is proved by practical application that state space inspection of nonlinear dynamic system can be carried out through D _L and its distinctions of strong immunity to noise and high accuracy to fault detection are demonstrated. D _L approximates properly to theoretic value makes it an effective characteristic quantity of nonlinear dynamic system by which mode identification is done well.