scholarly journals Derivation and Application of the Fokker‐Planck Equation to Discrete Nonlinear Dynamic Systems Subjected to White Random Excitation

1963 ◽  
Vol 35 (11) ◽  
pp. 1683-1692 ◽  
Author(s):  
Thomas K. Caughey
Keyword(s):  
Dynamic Systems ◽  
Nonlinear Dynamic ◽  
Planck Equation ◽  
Random Excitation ◽  
Nonlinear Dynamic Systems ◽  
Fokker Planck Equation ◽  
Fokker Planck

Approximate Solution of the Fokker–Planck Equation for a Multidegree of Freedom Frictionally Damped Bladed Disk Under Random Excitation

2018 ◽  
Vol 141 (1) ◽  
Author(s):  
Alwin Förster ◽  
Lars Panning-von Scheidt ◽  
Jörg Wallaschek
Keyword(s):  
Approximate Solution ◽  
Weighting Function ◽  
Gaussian White Noise ◽  
Planck Equation ◽  
Random Excitation ◽  
Stationary Response ◽  
Bladed Disk ◽  
Fokker Planck Equation ◽  
Finite Differences Method ◽  
Fokker Planck

Bladed disks are subjected to different types of excitations, which cannot, in any, case be described in a deterministic manner. Fuzzy factors, such as slightly varying airflow or density fluctuation, can lead to an uncertain excitation in terms of amplitude and frequency, which has to be described by random variables. The computation of frictionally damped blades under random excitation becomes highly complex due to the presence of nonlinearities. Only a few publications are dedicated to this particular problem. Most of these deal with systems of only one or two degrees-of-freedom (DOFs) and use computational expensive methods, like finite element method or finite differences method (FDM), to solve the determining differential equation. The stochastic stationary response of a mechanical system is characterized by the joint probability density function (JPDF), which is driven by the Fokker–Planck equation (FPE). Exact stationary solutions of the FPE only exist for a few classes of mechanical systems. This paper presents the application of a semi-analytical Galerkin-type method to a frictionally damped bladed disk under influence of Gaussian white noise (GWN) excitation in order to calculate its stationary response. One of the main difficulties is the selection of a proper initial approximate solution, which is applicable as a weighting function. Comparing the presented results with those from the FDM, Monte–Carlo simulation (MCS) as well as analytical solutions proves the applicability of the methodology.

Approximate Solution of the Fokker-Planck Equation for a Multi-Degree of Freedom Frictionally Damped Bladed Disk Under Random Excitation

2018 ◽  
Author(s):  
Alwin Förster ◽  
Lars Panning-von Scheidt ◽  
Jörg Wallaschek
Keyword(s):  
Approximate Solution ◽  
Weighting Function ◽  
Gaussian White Noise ◽  
Planck Equation ◽  
Random Excitation ◽  
Stationary Response ◽  
Bladed Disk ◽  
Fokker Planck Equation ◽  
Finite Differences Method ◽  
Fokker Planck

Bladed Disks are subjected to different types of excitations, which cannot in any case be described in a deterministic manner. Fuzzy factors, such as slightly varying airflow or density fluctuation, can lead to an uncertain excitation in terms of amplitude and frequency, which has to be described by random variables. The computation of frictionally damped blades under random excitation becomes highly complex due to the presence of nonlinearities. Only a few publications are dedicated to this particular problem. Most of these deal with systems of only one or two degrees of freedom and use computational expensive methods, like finite element method (FEM) or finite differences method (FDM), to solve the determining differential equation. The stochastic stationary response of a mechanical system is characterized by the joint probability density function (JPDF), which is driven by the Fokker-Planck equation (FPE). Exact stationary solutions of the FPE only exist for a few classes of mechanical systems. This paper presents the application of a semi-analytical Galerkin-type method to a frictionally damped bladed disk under influence of Gaussian white noise (GWN) excitation in order to calculate its stationary response. One of the main difficulties is the selection of a proper initial approximate solution, which is applicable as a weighting function. Comparing the presented results with those from the FDM, Monte-Carlo Simulation (MCS) as well as analytical solutions proves the applicability of the methodology.

Response of a Loaded Nonlinear String to Random Excitation

1962 ◽  
Vol 29 (3) ◽  
pp. 483-485 ◽  
Author(s):  
S. T. Ariaratnam
Keyword(s):  
Planck Equation ◽  
Nonlinear Effects ◽  
String Tension ◽  
Random Excitation ◽  
Markov Random ◽  
Fokker Planck Equation ◽  
Markov Random Process ◽  
The Mean ◽  
Nonlinear String ◽  
Fokker Planck

The response to white noise excitation of a light elastic string loaded at equal intervals by a number of equal masses is examined using the theory of the Markov random process and the associated Fokker-Planck equation. Taking nonlinear effects due to variation in string tension into account, an exact solution of the Fokker-Planck equation is obtained and used in an approximate analytical evaluation of the mean squared response of the system.

AN APPLICATION OF FOKKER–PLANCK EQUATION TO JOSEPHSON JUNCTION

1989 ◽  
Vol 9 (1) ◽  
pp. 109-120
Author(s):  
G. Liao ◽  
A.F. Lawrence ◽  
A.T. Abawi
Keyword(s):  
Josephson Junction ◽  
Planck Equation ◽  
Fokker Planck Equation ◽  
Fokker Planck
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