A Lyapunov Controller for a Varying Length Flexible Cable System to Supress Transverse Vibration

2004 ◽  
Author(s):  
Yuhong Zhang ◽  
Sunil Agrawal
Keyword(s):  
Differential Equations ◽  
Transverse Vibration ◽  
Axial Velocity ◽  
Controller Design ◽  
Lyapunov Stability Theory ◽  
Lyapunov Theory ◽  
Governing Equations ◽  
Flexible Cable ◽  
Linear Partial Differential Equations ◽  
Lyapunov Controller

Modeling and controller design for a flexible cable transporter system, with arbitrarily varying cable length is presented using Hamilton’s principle and Lyapunov theory. The axial velocity of the system is assumed to be arbitrary in the model. This is different from existing literature where the axial velocity is assumed either constant or is prescribed. The governing equations are coupled non-linear partial differential equations (PDEs) and ordinary differential equations (ODEs), and boundary conditions. The interactions between the cables and the slider, pulleys, and motors are included in the model. Numerical solution of the governing equations is obtained using Galerkin’s method. Based on the Lyapunov stability theory, we propose boundary controllers and the control law for an actuator within the domain to suppress the transverse vibration of the cables, while achieving the slider goal. The proposed controllers dissipate the vibratory energy and guarantee asymptotic stability of the closed-loop system. Simulation results demonstrate the effectiveness of the proposed controllers.

Modeling and Control of Flexible Transporter System With Arbitrarily Time-Varying Cable Lengths

2003 ◽  
Author(s):  
Yuhong Zhang ◽  
Sunil K. Agrawal ◽  
Peter Hagedorn
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Ritz Method ◽  
Nonlinear Partial Differential Equations ◽  
Time Varying ◽  
Governing Equations ◽  
Partial Differential ◽  
Modeling And Control ◽  
Lyapunov Controller ◽  
And Control

A systematic procedure for deriving the system model of a cable transporter system with arbitrarily time-varying lengths is presented. Two different approaches are used to develop the model, namely, Newton’s Law and Hamilton’s Principle. The derived governing equations are nonlinear partial differential equations. The same results are obtained using the two methods. The Rayleigh-Ritz method is used to obtain an approximate numerical solution of the governing equations by transforming the infinite order partial differential equations into a finite order discretized system. A Lyapunov controller which can both dissipate the vibratory energy and assure the attainment of the desired goal is derived. The validity of the proposed controller is verified by numerical simulation.

Mathematical Model of a Railway Pneumatic Brake System With Varying Cylinder Capacity Effects

1990 ◽  
Vol 112 (3) ◽  
pp. 456-462 ◽  
Author(s):  
S. Bharath ◽  
B. C. Nakra ◽  
K. N. Gupta
Keyword(s):  
Differential Equations ◽  
Volume Effect ◽  
Brake System ◽  
Governing Equations ◽  
Linear Ordinary Differential Equations ◽  
Linear Partial Differential Equations ◽  
Reservoir Volume ◽  
State Approximation ◽  
Piston Displacement ◽  
Non Linear

Governing equations for the analysis of pressure transient are derived from the principle of conservation of mass and momentum for a pneumatic brake system, which consists of a train pipe connected to a number of linear actuators (brake cylinders with piston displacement). The governing one-dimensional non-linear partial differential equations for the train pipe, non-linear ordinary differential equations for the brake cylinders, and second-order differential equation of motion for piston displacement are solved to determine the pressure transients in the brake system for a step change in pressure at the inlet. The governing equations are nondimensionalized and reduced to a set of ordinary nonlinear differential difference equations and integrated by standard numerical methods. The flow is considered isothermal, and the friction effects for turbulent and laminar flow are evaluated by quasi-steady state approximation. The auxiliary reservoir volume effect is also included. The results are compared with the experimental data obtained on a brake test rig.

Mathematical Solution on MHD Stagnation Point Flow of Ferrofluid

2020 ◽  
Vol 399 ◽  
pp. 38-54
Author(s):  
Siti Hanani Mat Yasin ◽  
Muhammad Khairul Anuar Mohamed ◽  
Zulkhibri Ismail ◽  
Mohd Zuki Salleh
Keyword(s):  
Heat Transfer ◽  
Differential Equations ◽  
Stagnation Point ◽  
Volume Fraction ◽  
Stagnation Point Flow ◽  
Governing Equations ◽  
Linear Ordinary Differential Equations ◽  
Linear Partial Differential Equations ◽  
Keller Box Method ◽  
Non Linear

This study presents a numerical investigation on the magnetohydrodynamic (MHD) stagnation point flow of a ferrofluid with Newtonian heating. The black oxide of iron, magnetite (Fe3O4) which acts as magnetic materials and water as a base fluid are considered. The two dimensional stagnation point flow of cold ferrofluid against a hot wall under the influence of the uniform magnetic field of strength is located some distance behind the stagnation point. The effect of magnetic and volume fraction on the velocity and temperature boundary layer profiles are obtained through the formulated governing equations. The governing equations which are in the form of dimensional non-linear partial differential equations are reduced to dimensionless non-linear ordinary differential equations by using appropriate similarity transformation. Then, they are solved numerically by using the Keller-box method which is programmed in the Matlab software. It is found that the cold fluid moves towards the magnetic source that is close to the hot wall. Hence, leads to the better cooling rate and enhances the heat transfer rate. Meanwhile, an increase of the magnetite nanoparticles volume fraction, increases the ferrofluid capabilities in thermal conductivity and consequently enhances the heat transfer.

Multi-parameter Singular Integrals. (AM-189)

2017 ◽  
Author(s):  
Brian Street
Keyword(s):  
Partial Differential Equations ◽  
Graduate Students ◽  
General Theory ◽  
Differential Equations ◽  
Singular Integrals ◽  
Main Tool ◽  
Classical Theorem ◽  
Quantitative Version ◽  
Linear Partial Differential Equations

This book develops a new theory of multi-parameter singular integrals associated with Carnot–Carathéodory balls. The book first details the classical theory of Calderón–Zygmund singular integrals and applications to linear partial differential equations. It then outlines the theory of multi-parameter Carnot–Carathéodory geometry, where the main tool is a quantitative version of the classical theorem of Frobenius. The book then gives several examples of multi-parameter singular integrals arising naturally in various problems. The final chapter of the book develops a general theory of singular integrals that generalizes and unifies these examples. This is one of the first general theories of multi-parameter singular integrals that goes beyond the product theory of singular integrals and their analogs. This book will interest graduate students and researchers working in singular integrals and related fields.

Lyapunov-type inequalities for partial differential equations with 𝑝-Laplacian

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Robert Stegliński
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Nonlinear Partial Differential Equations ◽  
Dirichlet Boundary Conditions ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
Lyapunov Inequalities ◽  
Funct Anal

Abstract The aim of this paper is to extend results from [A. Cañada, J. A. Montero and S. Villegas, Lyapunov inequalities for partial differential equations, J. Funct. Anal. 237 (2006), 1, 176–193] about Lyapunov-type inequalities for linear partial differential equations to nonlinear partial differential equations with 𝑝-Laplacian with zero Neumann or Dirichlet boundary conditions.

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