A Study on the Characteristics of a Nonlinear Oscillatory System With Dry Friction

2003 ◽  
Author(s):  
Liming Dai ◽  
Liang Xu
Keyword(s):  
Nonlinear System ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Oscillatory System ◽  
Approximate Solutions ◽  
Weakly Nonlinear ◽  
System Parameters ◽  
Weakly Nonlinear System ◽  
Highly Nonlinear ◽  
Nonlinear Oscillatory System

Nonlinear oscillatory system involved with friction is very common in nonlinear dynamics of engineering fields. This paper is to investigate the motions a nonlinear oscillatory system with involvement of dry friction. The cases of weakly and highly nonlinearity of the system are considered. Approximate and numerical solutions for the system are developed via the author’s newly developed P-T method. As demonstrated in the present work, the properties of the weakly and highly nonlinear systems exhibit great differences, though the governing equations of the two systems employ identical system parameters. The approximate solutions developed for the system are continuous everywhere on the time range considered. Under the conditions of weakly nonlinearity, the approximate solutions developed can therefore be conveniently implemented for the purpose of an analytical studying the properties of the system with numerous system parameters and various initial conditions. Taking this advantage, the behavior of motion of the weakly nonlinear system is analyzed and compared with the corresponding solutions developed with Van der Pol’s method. It is found in the present work, the system may undergo a self-excited oscillation under certain conditions. The highly nonlinear system is a physically much involved one. Its behavior is thus much complex in comparing with that of the weakly nonlinear system. Based on the approximate solutions developed for the highly nonlinear system, recurrence relations are generated for numerical calculations. For the sake of comparison with the oscillation of the weakly nonlinear system, numerical simulations for the highly nonlinear system are performed under the same initial conditions and identical system parameters. The conditions of convergence and divergence of the weakly nonlinear system are also established for application. Behavior of the oscillatory motion of the highly nonlinear system is investigated on the basis of the corresponding numerical solutions developed.

Investigating Dynamics of One Weakly Nonlinear System with Delay Argument

2018 ◽  
Vol 50 (1) ◽  
pp. 20-38 ◽  
Author(s):  
Denis Ya. Khusainov ◽  
Jozef Diblik ◽  
Jaromir Bashtinec ◽  
Andrey V. Shatyrko
Keyword(s):  
Nonlinear System ◽  
Weakly Nonlinear ◽  
Delay Argument ◽  
System With Delay ◽  
Weakly Nonlinear System

Modeling and Simulation of Stochastic Lorenz Systems by Polynomial Chaos Approach

2007 ◽  
Author(s):  
Lin Li ◽  
Corina Sandu
Keyword(s):  
Polynomial Chaos ◽  
Initial Conditions ◽  
Nonlinear Problems ◽  
System Response ◽  
Structural Vibrations ◽  
System Parameters ◽  
Highly Nonlinear ◽  
Body Dynamics ◽  
The Lorenz System ◽  
Multi Body

The Lorenz problem is one of the paradigms of the chaotic systems, which are sensitive to initial conditions and for which the performance is hard to predict. However, in many cases and dynamic systems, the initial conditions of a dynamic system and the system parameters can’t be measured accurately, and the response of the system must indeed be explored in advance. In this study, the polynomial chaos approach is used to handle uncertain initial conditions and system parameters of the Lorenz system. The method has been successfully applied by the authors and co-workers in multi-body dynamics and terrain profile and soil modeling. Other published studies illustrate the benefits of using the polynomial chaos, especially for problems involving large uncertainties and highly nonlinear problems in fluid mechanics, structural vibrations, and air quality studies. This study is an attempt to use the polynomial chaos approach to treat the Lorenz problem, and the results are compared with a classical Monte Carlo approach. Error bars are used to illustrate the standard deviation of the system response. Different meshing schemes are simulated, and the convergence of the method is analyzed.

scholarly journals Soft Computing Paradigms to Find the Numerical Solutions of a Nonlinear Influenza Disease Model

2021 ◽  
Vol 11 (18) ◽  
pp. 8549
Author(s):  
Zulqurnain Sabir ◽  
Ag Asri Ag Ibrahim ◽  
Muhammad Asif Zahoor Raja ◽  
Kashif Nisar ◽  
Muhammad Umar ◽  
...  
Keyword(s):  
Nonlinear System ◽  
Local Search ◽  
Objective Function ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Disease Model ◽  
Numerical Results ◽  
Active Set ◽  
System Equations ◽  
Global And Local

The aim of this work is to present the numerical results of the influenza disease nonlinear system using the feed forward artificial neural networks (ANNs) along with the optimization of the combination of global and local search schemes. The genetic algorithm (GA) and active-set method (ASM), i.e., GA-ASM, are implemented as global and local search schemes. The mathematical nonlinear influenza disease system is dependent of four classes, susceptible S(u), infected I(u), recovered R(u) and cross-immune individuals C(u). For the solutions of these classes based on influenza disease system, the design of an objective function is presented using these differential system equations and its corresponding initial conditions. The optimization of this objective function is using the hybrid computing combination of GA-ASM for solving all classes of the influenza disease nonlinear system. The obtained numerical results will be compared by the Adams numerical results to check the authenticity of the designed ANN-GA-ASM. In addition, the designed approach through statistical based operators shows the consistency and stability for solving the influenza disease nonlinear system.

scholarly journals PIECEWISE OPTIMAL FRACTIONAL REPRODUCING KERNEL SOLUTION AND CONVERGENCE ANALYSIS FOR THE ATANGANA–BALEANU–CAPUTO MODEL OF THE LIENARD’S EQUATION

Fractals ◽  
2020 ◽  
Vol 28 (08) ◽  
pp. 2040007
Author(s):  
SHAHER MOMANI ◽  
OMAR ABU ARQUB ◽  
BANAN MAAYAH
Keyword(s):  
Reproducing Kernel ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Reproducing Kernel Hilbert Space ◽  
Weakly Nonlinear ◽  
Analytical Technique ◽  
Wide Range ◽  
Fractional Models ◽  
Weakly Nonlinear Waves ◽  
Fractional Solution

In this paper, an attractive reliable analytical technique is implemented for constructing numerical solutions for the fractional Lienard’s model enclosed with suitable nonhomogeneous initial conditions, which are often designed to demonstrate the behavior of weakly nonlinear waves arising in the oscillating circuits. The fractional derivative is considered in the Atangana–Baleanu–Caputo sense. The proposed technique, namely, reproducing kernel Hilbert space method, optimizes numerical solutions bending on the Fourier approximation theorem to generate a required fractional solution with a rapidly convergent form. The influence, capacity, and feasibility of the presented approach are verified by testing some applications. The acquired results are numerically compared with the exact solutions in the case of nonfractional derivative, which show the superiority, compatibility, and applicability of the presented method to solve a wide range of nonlinear fractional models.

Capillary Forces Modeling in Micro/Nano Interactions

2007 ◽  
Author(s):  
Bo Chang ◽  
Jingrong Wang ◽  
Quan Zhou ◽  
Heikki Koivo
Keyword(s):  
Numerical Methods ◽  
Numerical Solutions ◽  
Capillary Condensation ◽  
Initial Conditions ◽  
Capillary Forces ◽  
Incline Angle ◽  
Second Derivative ◽  
System Parameters ◽  
Existence And Stability ◽  
Numerical Approaches

This paper introduces two numerical approaches to model the capillary forces under two different initial conditions: given volume of the liquid and under the capillary condensation. The paper thoroughly analyzes the solutions of both numerical methods. Due to multiple numerical solutions may exist for a given set of parameters, criteria based on the derivative and the second derivative of the solution are proposed to determine the existence and stability of those numerical solutions. The features of those numerical solutions are also carefully discussed. Moreover, the results of two numerical methods are compared in different system parameters for several configurations, including two plates with different volume of liquid between them, a plate and a cone of different incline angle, and a plate and spheres of different radius. Suggestions of the applicability of both methods are given based on the results. To allow calculation of capillary forces between arbitrary shaped objects, the paper proposes an early approach to calculate the capillary forces for discretized surfaces.

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