PROBABILITY DENSITY EVOLUTION EQUATIONS — A HISTORICAL INVESTIGATION

2009 ◽  
Vol 03 (03) ◽  
pp. 209-226 ◽  
Author(s):  
LI JIE ◽  
CHEN JIANBING
Keyword(s):  
Evolution Equation ◽  
Probability Density ◽  
Evolution Equations ◽  
Planck Equation ◽  
Point Of View ◽  
Lagrangian Description ◽  
Density Evolution ◽  
Physical Point ◽  
Probability Density Evolution ◽  
Generalized Probability

The paper aims at clarifying the essential relationship between traditional probability density evolution equations and the generalized probability density evolution equation which is developed by the authors in recent years. Using the principle of preservation of probability as a uniform fundamental, the probability density evolution equations, including the Liouville equation, Fokker–Planck equation and the Dostupov–Pugachev equation, are derived from the physical point of view. It is pointed out that combining with Eulerian or Lagrangian description of the associated dynamical system will lead to different probability density evolution equations. Particularly, when both the principle and dynamical systems are viewed from Lagrangian description, we are led to the generalized probability density evolution equation.

Wheel–rail stochastic dynamics and rail wear analysis of small radius curved sections of a tram line based on generalized probability density evolution

2020 ◽  
pp. 095440972094753
Author(s):  
Zhi-ping Zeng ◽  
Xu-dong Huang ◽  
Jun-dong Wang ◽  
Fu-shan Liu ◽  
Wei-dong Wang ◽  
...  
Keyword(s):  
Dynamic System ◽  
Probability Density ◽  
Small Radius ◽  
Stochastic Dynamic ◽  
Density Evolution ◽  
Rail Wear ◽  
Sample Set ◽  
Probability Density Evolution ◽  
Generalized Probability ◽  
Track Irregularities

Compared with ordinary railways, the curve radius of tram lines tends to be smaller, with minimum values of only 30 m. Therefore, wheel–rail interaction is more intense and complicated in sections of small radius tram line curves. Using a stochastic variable sample set based on a generalized probability density evolution method, the stochastic variable–spectrum representation method was used to generate a time-domain sample set of stochastic track irregularities. By inputting the stochastic set of track irregularities into a tram-track coupled dynamic system model, the stochastic dynamic response of the coupled dynamic system can be obtained. Moreover, by substituting the stochastic dynamic system response into the generalized probability density evolution formula, the process of probability density evolution of each evaluation index can be obtained by the finite difference method. Finally, the dynamic response of the tram-track coupled dynamic system can be evaluated by the probability distribution of each index. By setting a series of specific groove rail wear values, a tram-track coupled dynamic analysis was carried out, and compared with the specification requirements, vehicle safety limits under different wear values were obtained. This research has great engineering value for guiding the routine maintenance of small radius curve sections of trams.

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