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.

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.

Brownian Motion on Cantor Sets

2020 ◽  
Vol 21 (3-4) ◽  
pp. 275-281
Author(s):  
Ali Khalili Golmankhaneh ◽  
Saleh Ashrafi ◽  
Dumitru Baleanu ◽  
Arran Fernandez
Keyword(s):  
Brownian Motion ◽  
Classical Method ◽  
Mean Square Displacement ◽  
Planck Equation ◽  
Cantor Sets ◽  
Mean Square ◽  
Fokker Planck Equation ◽  
Fractal Time ◽  
The Mean ◽  
Fokker Planck

AbstractIn this paper, we have investigated the Langevin and Brownian equations on fractal time sets using Fα-calculus and shown that the mean square displacement is not varied linearly with time. We have also generalized the classical method of deriving the Fokker–Planck equation in order to obtain the Fokker–Planck equation on fractal time sets.

scholarly journals Model of metameric locomotion in smooth active directional filaments with curvature fluctuations

2021 ◽  
Author(s):  
Guangle Du ◽  
Sunita Kumari ◽  
Fangfu Ye ◽  
Rudolf Podgornik
Keyword(s):  
Theoretical Model ◽  
Planck Equation ◽  
The Body ◽  
Whole Body ◽  
Langevin Equations ◽  
Mean Square ◽  
Fokker Planck Equation ◽  
Directional Motion ◽  
The Mean ◽  
Fokker Planck

Abstract Locomotion in segmented animals, such as annelids and myriapods (centipedes and millipedes), is generated by a coordinated movement known as metameric locomotion, which can be also implemented in robots designed to perform specific tasks. We introduce a theoretical model, based on an active directional motion of the head segment and a passive trailing of the rest of the body segments, in order to formalize and study the metameric locomotion. The model is specifically formulated as a steered Ornstein-Uhlenbeck curvature process, preserving the continuity of the curvature along the whole body filament, and thus supersedes the simple active Brownian model, which would be inapplicable in this case. We obtain the probability density by analytically solving the Fokker-Planck equation pertinent to the model. We also calculate explicitly the correlators, such as the mean-square orientational fluctuations, the orientational correlation function and the mean-square separation between the head and tail segments, both analytically either via the Fokker-Planck equation or directly by either solving analytically or implementing it numerically from the Langevin equations. The analytical and numerical results coincide. Our theoretical model can help understand the locomotion of metameric animals and instruct the design of metameric robots.

Cosmic Ray Particle Diffusion and the Fokker-Planck Equation

1974 ◽  
Vol 2 (5) ◽  
pp. 306-307
Author(s):  
I. H. Urch
Keyword(s):  
Magnetic Field ◽  
Diffusion Coefficients ◽  
Interplanetary Medium ◽  
Planck Equation ◽  
Cosmic Ray ◽  
Particle Diffusion ◽  
Fokker Planck Equation ◽  
The Mean ◽  
Number Of Authors ◽  
Fokker Planck

The Fokker-Planck equation has been used by a number of authors (Jokipii 1966, 1971; Hall and Sturrock 1967; Hasselmann and Wibberentz 1968; Roelof 1968) to deduce the diffusion coefficients of cosmic-ray particles in the interplanetary magnetic field. However, these calculations suggest that the diffusion of particles perpendicular to the mean magnetic field is implausibly large; so large that the validity of a Fokker-Planck approach as applied to the interplanetary medium must be doubted.

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.

ON A DISPERSIVE MODEL FOR THE UNZIPPING OF DOUBLE-STRANDED DNA MOLECULES

2013 ◽  
Vol 24 (03) ◽  
pp. 495-511 ◽  
Author(s):  
J. CALVO ◽  
J. NIETO ◽  
J. SOLER ◽  
M. O. VÁSQUEZ
Keyword(s):  
Kinetic Equations ◽  
Planck Equation ◽  
Nonlinear Effects ◽  
Transcendental Equation ◽  
Applied Force ◽  
Equation Modeling ◽  
Order Kinetic ◽  
Double Stranded Dna ◽  
Fokker Planck Equation ◽  
Fokker Planck

The paper deals with the analysis of a nonlinear Fokker–Planck equation modeling the mechanical unzipping of double-stranded DNA under the influence of an applied force. The dependent variable is the probability density of unzipping m base pairs. The nonlinear Fokker–Planck equation we propose here is obtained when we couple the model proposed in [D. K. Lubensky and D. R. Nelson, Pulling pinned polymers and unzipping DNA, Phys. Rev. Lett.85 (2000) 1572–1575] with a transcendental equation for the applied force. The resulting model incorporates nonlinear effects in a different way than the usual models in kinetic theory. We show the well-posedness of this model. For that we require a combination of techniques coming from second-order kinetic equations and compensated compactness arguments in conservation laws.

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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