Chaotic Dynamics Of Ship Propeller Singing Time Series

The phenomenon of the propeller singing has great influences on ship concealment, quietness and comfort. However, it has many problems which deserve further research. So far, the exact mechanism of propeller singing, which are so complex that it cannot be interpreted objectively whether by linear theory or by Kaman vortex street theory, has not been well described. Moreover, the effective control equation of propeller singing has not been established.In the past twenty years, the rapidly developing researches on chaotic dynamics theory have shown that many complicated phenomena which appear irregular, however have deterministic in lower dimensional space, such as for time series analysis, the irregular random signals having no definite physical significance might exist some kind of certainty or even be generated by simple nonlinear governing equations, that is so-called chaotic phenomena. Then, whether the propeller singing signal which appeared complicated is some kind of chaotic phenomena and can be studied by chaotic theory? Besides, the turbulence itself has been proved to be a kind of chaotic phenomena, and the ship-radiated noise and reverberant in underwater acoustic engineering have been certified to possess chaotic dynamic characteristics. Therefore, this article not only studied on the phase-space reconstruction of propeller singing system and its complexity but also evaluated its stability by recurrence plot technique. In this article, it is the first time to use surrogate data analysis to evaluate the nonlinear characteristics of this system and to further study its chaotic dynamics. We proved the existence of chaotic dynamics of this system based on phase-space reconstruction and the estimated result of chaotic invariants for the first time and provided theoretical basis for the signal processing, detection and recognition of the ship propeller singing. Then we also studied instantaneous energy spectrum and frequency spectrum of the time-series of propeller singing system using local wave analysis which proved technological support for the detection and recognition of the propeller singing.(1) Phase space reconstruction of time series in propeller singing system To study the property of phase-space reconstruction on the complex nonlinear system or chaotic dynamical system, the phase-space reconstruction topology of the system can be achieved by one certain dimension or several limited dimensions of the time series embedded in more dimensions of this system. In this article, we used mutural information and the modified auto-relativity method proposed by us to estimate the optimal embedded latency for propeller singing system and then we estimated the minimal dimensions embedded in propeller singing system by using the maximal eigenvalue invariant method and average false neighborhood and we used phase-space reconstruction dual-parameter joint estimation method for analyzing the reconstruction parameters of propeller singing system.(2) Symbolic dynamics study on time series of propeller singing systemThe symbolic sequence statistical analysis was applied to analyze the time series of propeller singing system. Based on two kinds of symbolic method (the partition method and the interpolation method), we found the propeller singing signal was almost not destroyed by the noise. Furthermore, our study showed that the propeller singing signal had the feature of intense nonlinearity and determination after calculating and analyzing the Shannon entropy and irreversible parameters.The complex sequences of propeller singing system were established by Lempel-Ziv complexity algorithm based on the Kolmogorov definition. The results showed that its normalization of complexity measures made narrow width fluctuations around 0.73. That meant, the complex sequences frequently appeared local maximum and minimum with the time series of propeller singing system jumping from left to right around zero point, however, it presented normalization of complexity measures around 0.73 (between 0 and 1) overall. All these indicated that the time series of propeller singing system had deterministic as well as complicated nonlinear characteristics.(3) Detecting Stationarity and Nonlinearity in Propeller Singing SignalThe stationarity of propeller singing signal is tested by recurrence plot technique for the first time which is based on graphic method to test stationarity of systemic time series.Moreover, we used surrogate data analysis to evaluate the nonlinear characteristics of this system and further studied features of three null hypotheses and four test statistics. According to surrogate data, the singing time series is proved to have nonlinearity character. (a) The time series of propeller singing system which appeared random and irregular were proven to have deterministic for the first time by null hypotheses 1 and test statistics T₁.(b) The time series of propeller singing system were proven the existence of nonlinear dynamical features for the first time by null hypotheses 2 and test statistics T₂ and T₃.(c) And the nonlinearity of time series is not caused by the static nonlinear measurement function but the intrinsic character itself based on further research by null hypotheses 3 and test statistics T_rev.It is the first time to prove the nonlinear dynamical characteristics of propeller singing system by surrogate data. That can be the necessary condition of chaotic dynamics and provide the theoretical foundation for further study. Moreover, the analysis results could provide the technical support for the target detection and identification of propeller singing system, and equations formation of systemic numerical simulation.(4) Study on chaotic dynamics of propeller singing systemApplying for phase-space reconstructing technique, the correlation dimension of propeller singing system was estimated as D₂ = 5.1579 by G-P algorithm. The non-integral results constituted the necessary condition of chaotic dynamics. In addition, the boundary of topological dimension was considered as 6, that is, the number of independent variable was not less than 6 that could generate this complicated system.We calculated the correlation function of propeller singing system time series by q-order Renyi entropy which had monotonic consistency. The estimation of Kolmogorov entropy was achieved as K₂ = 0.6478 by least square method, that is, Kolmogorov entropy of propeller singing system is about 0.6478. This positive finite value provided the sufficient condition to have chaotic dynamics.By the estimation of maximum Lyapunov exponent, we obtained a positive and finite maximum Lyapunov exponent withλ(l_D) = 0.0771. This result was considered as the evidence to judge the propeller singing system to have chaotic dynamics.According to the forementioned analysis, it is the first time to prove the chaotic dynamics of propeller singing system by estimating the chaotic invariant of its time series. It might lay a theoretical foundation for further study the propeller singing phenomenon.(5) Research on time-frequency characteristics of propeller singing system time series Propeller singing system time series have been analyzed by local wave decomposition. The fractional correlation dimension and maximum Lyapunov exponent of time series and its intrinsic mode functions (IMF) have been estimated. To solve the de-noising problem, the local wave decomposition method was adopted because of its adaptive filter characteristic for distinguishing the chaotic time series and noise one. The intrinsic energy spectrum and frequency spectrum of propeller singing time series has been got, which revealed the time series has chaotic dynamical character by triple section frequency phenomenon. So it is good for target detection and identification of propeller singing system.So the existence of chaos in propeller singing system is confirmed in experiments. The results in this dissertation are helpful to understand the nonlinear dynamics of propeller singing, and useful in the numerical simulation study. The method and algorithms about chaotic time series analysis and target detection may be referenced to dealing with other chaotic systems.