scholarly journals A Modified Lindstedt–Poincaré Method for a Strongly Nonlinear System with Quadratic and Cubic Nonlinearities

1996 ◽  
Vol 3 (4) ◽  
pp. 279-285 ◽  
Author(s):  
S.H. Chen ◽  
Y. K. Cheung
Keyword(s):  
Nonlinear System ◽  
Present Method ◽  
Nonlinear Oscillations ◽  
Perturbation Expansion ◽  
Nonlinear Characteristics ◽  
Strongly Nonlinear ◽  
Strongly Nonlinear System ◽  
Strongly Nonlinear Oscillations ◽  
Quadratic And Cubic Nonlinearities

A modified Lindstedt–Poincaré method is presented for extending the range of the validity of perturbation expansion to strongly nonlinear oscillations of a system with quadratic and cubic nonlinearities. Different parameter transformations are introduced to deal with equations with different nonlinear characteristics. All examples show that the efficiency and accuracy of the present method are very good.

Dynamics of Oscillator With Soft Impacts

2001 ◽  
Author(s):  
František Peterka
Keyword(s):  
Nonlinear System ◽  
Linear System ◽  
Mechanical System ◽  
Linear Model ◽  
Piecewise Linear ◽  
Impact Oscillator ◽  
Strongly Nonlinear ◽  
Strongly Nonlinear System ◽  
New Phenomena ◽  
The Impact

Abstract The impact oscillator is the simplest mechanical system with one degree of freedom, the periodically excited mass of which can impact on the stop. The aim of this paper is to explain the dynamics of the system, when the stiffness of the stop changes from zero to infinity. It corresponds to the transition from the linear system into strongly nonlinear system with rigid impacts. The Kelvin-Voigt and piecewise linear model of soft impact was chosen for the study. New phenomena in the dynamics of motion with soft impacts in comparison with known dynamics of motion with rigid impacts are introduced in this paper.

EFFECT OF ELEVATED TEMPERATURE ON CONCRETE STRUCTURES BY BOUNDARY ELEMENTS

2012 ◽  
Vol 09 (01) ◽  
pp. 1240014 ◽  
Author(s):  
PETR P. PROCHAZKA ◽  
TAT S. LOK
Keyword(s):  
Elevated Temperature ◽  
Boundary Element Method ◽  
Nonlinear System ◽  
Time Dependent ◽  
System Of Equations ◽  
Systems Of Equations ◽  
Strongly Nonlinear ◽  
Strongly Nonlinear System ◽  
Long Time ◽  
Large Systems

Extreme elevation of temperature principally threatens tunnel linings and may cause fatal disaster; the recovery of it may take a long time and significant traffic troubles. System of equations is to be described and solution in terms of boundary element method (BEM) is suggested. Moreover, a technique of time-dependent eigenparameters enables one to apply parallel computations and converts the strongly nonlinear system to pseudo-linear one using the influence and polarization tensors. Consequently, instead of repeated solution of large systems of equations, the multiplication of pre-calculated influence matrices has to be carried out instead. In order to properly create the above-outlined procedure, internal cells are selected in the regions primarily connected by the change of temperature. Some examples follow the theory.

scholarly journals Iterative Ensemble Smoother as an Approximate Solution to a Regularized Minimum-Average-Cost Problem: Theory and Applications

SPE Journal ◽  
2015 ◽  
Vol 20 (05) ◽  
pp. 962-982 ◽  
Author(s):  
Xiaodong Luo ◽  
Andreas S. Stordal ◽  
Rolf J. Lorentzen ◽  
Geir Nævdal
Keyword(s):  
Nonlinear System ◽  
Average Cost ◽  
History Matching ◽  
Estimation Problem ◽  
Matching Problem ◽  
Strongly Nonlinear ◽  
System A ◽  
Ensemble Smoother ◽  
Strongly Nonlinear System ◽  
Iterative Ensemble Smoother

Summary The focus of this work is on an alternative implementation of the iterative-ensemble smoother (iES). We show that iteration formulae similar to those used by Chen and Oliver (2013) and Emerick and Reynolds (2012) can be derived by adopting a regularized Levenberg-Marquardt (RLM) algorithm (Jin 2010) to approximately solve a minimum-average-cost (MAC) problem. This not only leads to an alternative theoretical tool in understanding and analyzing the behavior of the aforementioned iES, but also provides insights and guidelines for further developments of the smoothing algorithms. For illustration, we compare the performance of an implementation of the RLM-MAC algorithm with that of the approximate iES used by Chen and Oliver (2013) in three numerical examples: an initial condition estimation problem in a strongly nonlinear system, a facies estimation problem in a 2D reservoir, and the history-matching problem in the Brugge field case. In these three specific cases, the RLM-MAC algorithm exhibits comparable or better performance, especially in the strongly nonlinear system.

Study of Energy Methods to Strongly Nonlinear Vibrations in a Loudspeaker Cone-Shaped Shell

2005 ◽  
Author(s):  
F. D. Zong ◽  
Z. L. Zhang ◽  
J. W. Fang ◽  
Y. J. Yu ◽  
Q. Chen
Keyword(s):  
Magnetic Field ◽  
Limit Cycles ◽  
Nonlinear Oscillations ◽  
Low Frequency ◽  
Approximate Solutions ◽  
Energy Methods ◽  
Frequency Range ◽  
Strongly Nonlinear ◽  
Harmonic Vibrations ◽  
Strongly Nonlinear Oscillations

H. F. Olson points out that a loudspeaker cone-shaped shell, as a nonlinear oscillation system, can be described as the Classical Duffing Equation in low frequency range. Yoshinisa, a Japanese scholar, studied the nonlinear phenomena of the loudspeaker cone-shaped shell in low frequency range driven by a stable galvanic source, including the resonance frequency changing with amplitude and leap phenomena. But their research were not taken the influence of nonlinear magnetic field into account. Its work mostly related to getting solution of nonlinear differential equation by the Numerical Calculation, but it didn’t get approximate solutions. Through research and analysis of the experiment on the loudspeaker cone-shaped shell, we obtain the Generalized Duffing Equation that’s a strongly nonlinearity system which is used to describe the loudspeaker cone-shaped shell driven by a stable voltage source, it considers the nonlinearity of mechanical resilience and the magnetic field. This paper focuses on first finding the approximate solutions (limit cycles) of strongly nonlinear oscillations and nonlinear heteronomy of the loudspeaker cone-shaped shell in low frequency range by use of energy methods. They obtained the equation relating to the forced vibration amplitude with frequency and the corresponding relation about phase versus frequency, and analysed particularly complete stability of limit cycles belonged to the strongly nonlinear systems, and drew several important conclusions. (1) As to strongly nonlinear oscillations of the loudspeaker cone-shaped shell in low frequency range, it is only likely to appear main oscillation and odd-order sub-harmonic oscillations. But it cannot appear super-harmonic vibrations and even-order sub-harmonic vibrations. (2) As to strongly nonlinear oscillations of the loudspeaker cone-shaped shell in low frequency range, two cases about main oscillation and one third sub-harmonic oscillation whose approximate solutions accord with numerical solutions very well. (3) It is worthy to study strongly nonlinear oscillations of commonly thin shell structure such as a loudspeaker cone-shaped shell by use of energy methods, and we will continue to carry out this research.

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