Stochastic dynamic analysis of a Kaplan turbine system considering synergistic regulation

A stochastic dynamic model of a Kaplan turbine is established in this paper during the transient process. When the Kaplan turbine operates with fluctuating load, the synergistic relationship between the guide vanes and blades experiences random fluctuation resulting from the mechanical, hydraulic and signal factors. To study the effect of stochastic fluctuations of the synergistic relationship, Chebyshev polynomial approximation method is adopted to analyze the stochastic dynamic characteristics of the Kaplan turbine during the transient process. Using Chebyshev polynomial approximation, the stochastic model of the Kaplan turbine is simplified to its equivalent deterministic model, and the stochastic dynamic characteristics of the model are investigated in the transient process. The effects of stochastic intensity on the dynamic behaviors of the Kaplan turbine are analyzed by means of numerical simulation. Moreover, the influences of PID parameters on the stochastic dynamic characteristics of the Kaplan turbine are studied through bifurcation diagrams. Analysis of stochastic characteristics and dynamic behaviors suggests that transient performance improvement can be obtained by controlling the synergistic stochastic intensity and PID parameters.

Response analysis of the nonlinear vibration energy harvester with an uncertain parameter

In this paper, the Chebyshev polynomial approximation is firstly utilized to analyze the dynamical characteristics of the nonlinear vibration energy harvester with an uncertain parameter. First, the stochastic energy harvester is transformed into a high-dimensional equivalent deterministic system by the Chebyshev polynomial approximation. And the ensemble mean response of the stochastic energy harvester is introduced to discuss the stochastic response. Then, the effectiveness of the approximation method is verified by numerical results. Furthermore, the bifurcation property of the displacement and voltage is analyzed, which is also consistent with the results derived by the top Lyapunov exponent. It is found that random factor can induce the appearance of multi-periodic phenomena and lead to appear the behavior of the periodic bifurcation. The strong random factor induces the fluctuation of the output voltage. In addition, the existence of the random factor greatly influences the property of the sub-harmonics and super-harmonics of the spectrum. Overall, the response mechanism of the nonlinear vibration energy harvester with an uncertain parameter is revealed.

Dynamic Modelling and Control of a Compressor Using Chebyshev Polynomial Approximation

The aim of this study is to apply a Chebyshev polynomial approximation of the compressor map for dynamic modelling and control of centrifugal compressors. The results are compared to those from an approximation based on the third order polynomials and a compressor map derived from first principles. In the analysis of centrifugal compressors, a combination of dynamic conservation laws and static compressor map provides an insight into the surge phenomenon, whose avoidance remains one of the objectives of compressor control. The compressor maps based on the physical laws provide accurate results, but require a detailed knowledge about the properties of the system, such as the geometry of the compressor and gas quality. Third order polynomials are usually used as an approximation for the compressor map, providing simplified models at the expense of accuracy. Chebyshev polynomial approximation provides a trade-off between the accuracy of physical modelling with the ease of use provided by third order polynomial approximation.

Period-Doubling Bifurcation of Stochastic Fractional-Order Duffing System via Chebyshev Polynomial Approximation

Fractional-order calculus is more competent than integer-order one when modeling systems with properties of nonlocality and memory effect. And many real world problems related to uncertainties can be modeled with stochastic fractional-order systems with random parameters. Therefore, it is necessary to analyze the dynamical behaviors in those systems concerning both memory and uncertainties. The period-doubling bifurcation of stochastic fractional-order Duffing (SFOD for short) system with a bounded random parameter subject to harmonic excitation is studied in this paper. Firstly, Chebyshev polynomial approximation in conjunction with the predictor-corrector approach is used to numerically solve the SFOD system that can be reduced to the equivalent deterministic system. Then, the global and local analysis of period-doubling bifurcation are presented, respectively. It is shown that both the fractional-order and the intensity of the random parameter can be taken as bifurcation parameters, which are peculiar to the stochastic fractional-order system, comparing with the stochastic integer-order system or the deterministic fractional-order system. Moreover, the Chebyshev polynomial approximation is proved to be an effective approach for studying the period-doubling bifurcation of the SFOD system.