Design of Nonlinear Sectors with Comparison functions

2021 ◽  
pp. 1-1
Author(s):  
Ankit Sachan ◽  
Kranthi Kumar Deveerasetty ◽  
Sandeep Kumar Soni
Keyword(s):  
Comparison Functions

An Investigation Into the Dynamics of a Pipe Aspirating Fluid

2018 ◽  
Author(s):  
Muhammad Faisal Javed Butt ◽  
Michael P. Paidoussis ◽  
Meyer Nahon
Keyword(s):  
Numerical Study ◽  
Fluid Velocity ◽  
Dominant Frequency ◽  
Dynamical Behaviour ◽  
Working Fluid ◽  
Flexible Pipe ◽  
Recovery Processes ◽  
Underground Caverns ◽  
Comparison Functions ◽  
Cantilevered Beam

Pipes aspirating fluid have applications in the filling and recovery processes for underground caverns — large subterranean cavities used to store hydrocarbons, such as natural gas and oil. This paper deals with the dynamics of a vertical cantilevered flexible pipe, immersed in fluid. Fluid is aspirated from its bottom free end up to the fixed upper end. In this study, the working fluid is assumed to be water. An existing analytical model is used to predict the dynamical behaviour of the aspirating pipe. This model is then discretized with Galerkin’s method, using Euler-Bernoulli eigen-functions for cantilevered beam as comparison functions. Once solved, the model results show a unique kind of flutter comprising three regions, denoted regions 01–03. These regions are delineated by two critical flow velocities, Ucf1 and Ucf2. In addition, two frequencies of oscillation, f1 and f2, are found to characterize the aforementioned flutter. The dominant frequency of oscillation changes from f1 to f2 as the flow velocity is increased from approximately 3 to 6 m/s — a frequency exchange phenomenon observed and reported here for the first time for this system. The analytical/numerical study was followed by a corresponding experimental study. Experiments were performed on a flexible (Silastic) pipe that was completely submerged in water. The behaviour observed experimentally was similar to the numerical study, as the aspirating fluid velocity was increased from zero to 7 m/s.

A Multi-Mode Approach for the Nonlinear Vibrations of Circular Cylindrical Shells

2001 ◽  
Author(s):  
Francesco Pellicano ◽  
Marco Amabili ◽  
Michael P. Païdoussis
Keyword(s):  
Degrees Of Freedom ◽  
Nonlinear Vibrations ◽  
Shell Theory ◽  
Cylindrical Shells ◽  
Nonlinear Behavior ◽  
Symbolic Manipulation ◽  
Comparison Functions ◽  
Circular Cylindrical Shells ◽  
Multi Mode

Abstract The nonlinear vibrations of simply supported, circular cylindrical shells, having geometric nonlinearities is analyzed. Donnell’s nonlinear shallow-shell theory is used, and the partial differential equations are spatially discretized by means of the Galerkin procedure, using a large number of degrees of freedom. A symbolic manipulation code is developed for the discretization, allowing an unlimited number of modes. In the displacement expansion particular care is given to the comparison functions in order to reduce as much as possible the dimension of the dynamical system, without losing accuracy. Both driven and companion modes are included, allowing for traveling-wave response of the shell. The fundamental role of the axisymmetric modes, which are included in the expansion, is confirmed and a convergence analysis is performed. The effect of the geometric shell characteristics, radius, length and thickness, on the nonlinear behavior is analyzed.

Bounds on the critical times for the Fisher-KPP equation

2021 ◽  
Vol 63 ◽  
pp. 448-468
Author(s):  
Marianito Rodrigo
Keyword(s):  
Upper And Lower Solutions ◽  
Critical Time ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Kpp Equation ◽  
Diffusive Process ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Comparison Functions

The Fisher–Kolmogorov–Petrovsky–Piskunov (Fisher–KPP) equation is one of the prototypical reaction–diffusion equations and is encountered in many areas, primarily in population dynamics. An important consideration for the phenomena modelled by diffusion equations is the length of the diffusive process. In this paper, three definitions of the critical time are given, and bounds are obtained by a careful construction of the upper and lower solutions. The comparison functions satisfy the nonlinear, but linearizable, partial differential equations of Fisher–KPP type. Results of the numerical simulations are displayed. Extensions to some classes of reaction–diffusion systems and an application to a spatially heterogeneous harvesting model are also presented. doi:10.1017/S1446181121000365

Phragmèn-Lindelöf and Comparison Theorems for Elliptic-Parabolic Differential Equations

1967 ◽  
Vol 19 ◽  
pp. 864-871
Author(s):  
J. K. Oddson
Keyword(s):  
Maximum Principle ◽  
Differential Equations ◽  
Parabolic Equations ◽  
Comparison Theorems ◽  
Parabolic Differential Equations ◽  
The Maximum Principle ◽  
Comparison Functions

Theorems of Phragmèn-Lindelöf type and other related results for solutions of elliptic-parabolic equations have been given by numerous authors in recent years. Many of these results are based upon the maximum principle and the use of auxiliary comparison functions which are constructed as supersolutions of the equations under various conditions on the coefficients. In this paper we present an axiomatized treatment of these topics, replacing specific hypotheses on the nature of the coefficients of the equations by a single assumption concerning the maximum principle and another concerning the existence of positive supersolutions, in terms of which the theorems are stated.

scholarly journals Contraction conditions using comparison functions on b-metric spaces

2014 ◽  
Vol 2014 (1) ◽  
pp. 135 ◽  
Author(s):  
Wasfi Shatanawi ◽  
Ariana Pitea ◽  
Rade Lazović
Keyword(s):  
Metric Spaces ◽  
Comparison Functions

scholarly journals Validation of a quantitative method for denning CAD/CAM socket modifications

1999 ◽  
Vol 23 (1) ◽  
pp. 30-44 ◽  
Author(s):  
E. D. Lemaire ◽  
P. Bexiga ◽  
F. Johnson ◽  
S. E. Solomonidis ◽  
J. P. Paul
Keyword(s):  
Clinical Application ◽  
Ground Reaction Force ◽  
Laboratory Testing ◽  
Quantitative Method ◽  
Reaction Force ◽  
General Representation ◽  
Comparison Functions ◽  
Cad Cam ◽  
Apex Point ◽  
Cam Systems

A quantitative method was developed for defining manual socket modifications, averaging these modifications over a series of amputees, and using the average modifications as a template in commercial CAD/CAM systems. The CADVIEW programme (i.e. software for viewing and analysing CAD sockets) was rewritten to provide comparison functions for aligning sockets to a common axis, visualising the differences between sockets, generating modification outlines, assigning apex point values, and averaging the modification outlines. A CAD template generated in this manner should be the best general representation of a prosthetist's modification style. To test this hypothesis, 13 people with trans-tibial amputations were fitted with both a manual and a CAD/CAM socket. Questionnaires were completed by the subjects and by the prosthetist to obtain information on prosthetic comfort, function, and overall satisfaction. Ground reaction force and stride parameter data were also collected for each prosthesis during gait laboratory testing. No significant differences were found between the manually designed socket and the CAD/CAM designed socket for all data except the vertical peak forces on the amputated side. These results support the clinical application of this quantitative technique for making the transition from manual to CAD/CAM prosthetic modification procedures.

Modeling Controlled Articulated Flexible Systems Using Assumed Modes: Part I — Theory

2001 ◽  
Author(s):  
Theodore G. Mordfin ◽  
Sivakumar S. K. Tadikonda
Keyword(s):  
Closed Form ◽  
Structural Model ◽  
Numerical Solutions ◽  
Closed Form Solution ◽  
Flexible Multibody Dynamics ◽  
Companion Paper ◽  
Form Solution ◽  
Control Laws ◽  
Comparison Functions ◽  
Flexible Behavior

Abstract Guidelines are sought for generating component body models for use in controlled, articulated, flexible multibody dynamics system simulations. In support of this effort, exact closed-form and numerical solutions are developed for the small elastic motions of a planar, flexible, single link system, in which the link is represented as an Euler-Bernoulli bar in transverse vibration. The link is connected to ground by a pin joint, and the articulation is controlled by proportional and proprotional/derivative (PD) feedback control laws. The characteristics of the closed-form solution are shown to consist of combinations of the characteristic expressions associated with classical end conditions. A large-articulation flexible body model of a controlled-articulation flexible link is then developed and linearized about an arbitrary reference angle. This model uses the method of assumed modes to represent the flexible behavior of the link. It is shown the model is analytically equivalent to a purely structural model which uses a hybrid set of assumed modes, and that numerical convergence can be investigated in terms of admissible functions and quasi-comparison functions. Numerical evaluation of the use of various types of assumed modes is presented in a companion paper.

Asymptotic Behavior of The Unique Solution to a Singular Elliptic Problem With Nonlinear Convection Term And Singular Weight

2008 ◽  
Vol 8 (2) ◽  
Author(s):  
Zhijun Zhang
Keyword(s):  
Asymptotic Behavior ◽  
Local Solution ◽  
Variation Theory ◽  
Convection Term ◽  
Unique Classical Solution ◽  
Nonlinear Convection ◽  
Comparison Functions ◽  
Exact Asymptotic Behavior ◽  
Nonlinear Convection Term ◽  
Singular Dirichlet Problem

AbstractBy Karamata regular variation theory, we first derived the exact asymptotic behavior of the local solution to the problem -φʹʹ(s) = g(φ(s)), φ(s) > 0, s ∈ (0, a) and φ(0) = 0. Then, by a perturbation method and constructing comparison functions, we derived the exact asymptotic behavior of the unique classical solution near the boundary to a singular Dirichlet problem -Δu = b(x)g(u) + λ|▽u|

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