Nonlinear vibrations of nano-beams accounting for nonlocal effect using a multiple scale method

2009 ◽  
Vol 52 (3) ◽  
pp. 617-621 ◽  
Author(s):  
XiaoDong Yang ◽  
C. W. Lim
Keyword(s):  
Nonlinear Vibrations ◽  
Multiple Scale ◽  
Multiple Scale Method ◽  
Nonlocal Effect ◽  
Scale Method

scholarly journals Analisa Kestabilan dan Solusi Pendekatan Pada Persamaan Van der Pol

2019 ◽  
Vol 3 (2) ◽  
pp. 156
Author(s):  
Yuni Yulida ◽  
Muhammad Ahsar Karim
Keyword(s):  
Periodic Solution ◽  
Limit Cycle ◽  
Multiple Scale ◽  
Multiple Scale Method ◽  
Van Der Pol Equation ◽  
Damping Force ◽  
Van Der Pol ◽  
Scale Method ◽  
Existence And Stability ◽  
The Stability

Abstrak: Di dalam tulisan ini disajikan analisa kestabilan, diselidiki eksistensi dan kestabilan limit cycle, dan ditentukan solusi pendekatan dengan menggunakan metode multiple scale dari persamaan Van der Pol. Penelitian ini dilakukan dalam tiga tahapan metode. Pertama, menganalisa perilaku dinamik persamaan Van der Pol di sekitar ekuilibrium, meliputi transformasi persamaan ke sistem persamaan, analisa kestabilan persamaan melalui linearisasi, dan analisa kemungkinan terjadinya bifukasi pada persamaan. Kedua, membuktikan eksistensi dan kestabilan limit cycle dari persamaan Van der Pol dengan menggunakan teorema Lienard. Ketiga, menentukan solusi pendekatan dari persamaan Van der Pol dengan menggunakan metode multiple scale. Hasil penelitian adalah, berdasarkan variasi nilai parameter kekuatan redaman, daerah kestabilan dari persamaan Van der Pol terbagi menjadi tiga. Untuk parameter kekuatan redaman bernilai positif mengakibatkan ekuilibrium tidak stabil, dan sebaliknya, untuk parameter kekuatan redaman bernilai negatif mengakibatkan ekuilibrium stabil asimtotik, serta tanpa kekuatan redaman mengakibatkan ekuilibrium stabil. Pada kondisi tanpa kekuatan redaman, persamaan Van der Pol memiliki solusi periodik dan mengalami bifurkasi hopf. Selain itu, dengan menggunakan teorema Lienard dapat dibuktikan bahwa solusi periodik dari persamaan Van der Pol berupa limit cycle yang stabil. Pada akhirnya, dengan menggunakan metode multiple scale dan memberikan variasi nilai amplitudo awal dapat ditunjukkan bahwa solusi persamaan Van der Pol konvergen ke solusi periodik dengan periode dua. Abstract: In this paper, the stability analysis is given, the existence and stability of the limit cycle are investigated, and the approach solution is determined using the multiple scale method of the Van der Pol equation. This research was conducted in three stages of method. First, analyzing the dynamic behavior of the equation around the equilibrium, including the transformation of equations into a system of equations, analysis of the stability of equations through linearization, and analysis of the possibility of bifurcation of the equations. Second, the existence and stability of the limit cycle of the equation are proved using the Lienard theorem. Third, the approach solution of the Van der Pol equation is determined using the multiple scale method. Our results, based on variations in the values of the damping strength parameters, the stability region of the Van der Pol equation is divided into three types. For the positive value, it is resulting in unstable equilibrium, and contrary, for the negative value, it is resulting in asymptotic stable equilibrium, and without the damping force, it is resulting in stable equilibrium. In conditions without damping force, the Van der Pol equation has a periodic solution and has hopf bifurcation. In addition, by using the Lienard theorem, it is proven that the periodic solution is a stable limit cycle. Finally, by using the multiple scale method with varying the initial amplitude values, it is shown that the solution of the Van der Pol equation is converge to a periodic solution with a period of two.

scholarly journals PEMBENTUKAN PERSAMAAN VAN DER POL DAN SOLUSI MENGGUNAKAN METODE MULTIPLE SCALE

2021 ◽  
Vol 14 (2) ◽  
pp. 104
Author(s):  
Farohatin Na'imah ◽  
Yuni Yulida ◽  
Muhammad Ahsar Karim
Keyword(s):  
Applied Mathematics ◽  
Multiple Scale ◽  
Equation System ◽  
Multiple Scale Method ◽  
Nonlinear Ordinary Differential Equations ◽  
Van Der Pol Equation ◽  
Van Der Pol ◽  
Scale Method ◽  
Rlc Circuit ◽  
Mathematical Equations

Mathematical modeling is one of applied mathematics that explains everyday life in mathematical equations, one example is Van der Pol equation. The Van der Pol equation is an ordinary differential equation derived from the Resistor, Inductor, and Capacitor (RLC) circuit problem. The Van der Pol equation is a nonlinear ordinary differential equations that has a perturbation term. Perturbation is a problem in the system, denoted by ε which has a small value 0<E<1. The presence of perturbation tribe result in difficulty in solving the equation using anlytical methode. One method that can solve the Van der Pol equation is a multiple  scale method. The purpose of this study is to explain the constructions process of  Van der Pol equation, analyze dynamic equations around equilibrium, and determine the solution of Van der Pol equation uses a multiple scale method. From this study it was found that the Van der Pol equation system has one equilibrium. Through stability analysis, the Van der Pol equation system will be stable if E= 0 and  -~<E<=-2. The solution of the Van der Pol equation with the multiple scale method is Keywords: Van der Pol equation, equilibrium, stability, multiple scale. 

scholarly journals On the use of the multiple scale method in solving ‘difficult’ bifurcation problems

2015 ◽  
Vol 22 (5) ◽  
pp. 988-1004 ◽  
Author(s):  
Angelo Luongo
Keyword(s):  
Singular Systems ◽  
Autonomous Systems ◽  
Fractional Power ◽  
Harmonic Balance Method ◽  
Multiple Scale ◽  
Multiple Scale Method ◽  
Continuous Systems ◽  
Scale Method ◽  
Suitable Combination ◽  
Bifurcation Problems

Several algorithms consisting in ‘non-standard’ versions of the Multiple Scale Method are illustrated for ‘difficult’ bifurcation problems. Preliminary, the ‘easy’ case of bifurcation from a cluster of distinct eigenvalues is addressed, which requires using integer power expansions, and it leads to bifurcation equations all of the same order. Then, more complex problems are studied. The first class concerns bifurcation from a defective eigenvalue, which calls for using fractional power expansions and fractional time-scales, as well as Jordan or Keldysh chains. The second class regards the interaction between defective and non-defective eigenvalues. This problem also requires fractional powers, but it leads to differential equations which are of a different order for the involved amplitudes. Both autonomous and parametrically excited non-autonomous systems are studied. Moreover, the transition from a codimension-3 to a codimension-2 bifurcation is explained. As a third class of problems, singular systems possessing an evanescent mass, as Nonlinear Energy Sinks, are considered, and both autonomous systems undergoing Hopf bifurcation and non-autonomous systems under external resonant excitation, are studied. The algorithm calls for a suitable combination of the Multiple Scale Method and the Harmonic Balance Method, the latter is applied exclusively to the singular equations. Several applications are shown, to test the effectiveness of the proposed methods. They include discrete and continuous systems, autonomous, parametrically and externally excited systems.

Turing Patterns of a Lotka–Volterra Competitive System with Nonlocal Delay

2018 ◽  
Vol 28 (07) ◽  
pp. 1830021 ◽  
Author(s):  
Bang-Sheng Han ◽  
Zhi-Cheng Wang
Keyword(s):  
Theoretical Analysis ◽  
Numerical Simulations ◽  
Dynamical Behavior ◽  
Multiple Scale ◽  
Multiple Scale Method ◽  
Turing Patterns ◽  
Amplitude Equations ◽  
Competitive System ◽  
Scale Method ◽  
Nonlocal Delay

This paper focuses on the dynamical behavior of a Lotka–Volterra competitive system with nonlocal delay. We first establish the conditions of Turing bifurcation occurring in the system. According to it and by using multiple scale method, the amplitude equations of the different Turing patterns are obtained. Then, we observe when these patterns (spots pattern and stripes pattern) arise in the Lotka–Volterra competitive system. Finally, some numerical simulations are given to verify our theoretical analysis.

scholarly journals DYNAMICAL CHAOS IN 6  -RAYLEIGH OSCILLATOR WITH THREE WELLS DRIVEN AN AMPLITUDE MODULATED FORCE.

2016 ◽  
Vol 12 (8) ◽  
pp. 6545-6552
Author(s):  
Y A Amer ◽  
S M Ahmed ◽  
ManarM Dahshan ◽  
N M Al
Keyword(s):  
Chaotic Behavior ◽  
Order Approximation ◽  
Multiple Scale ◽  
Multiple Scale Method ◽  
Dynamical Chaos ◽  
Worst Case ◽  
Scale Method

Chaotic behavior of6  -Rayleigh oscillator with three wells is investigated. The method of multiple scale method is usedto solve the system up to 3rd order approximation. Effect of parameters is studied numerically; all resonance cases arestudied numerically to obtain the worst case. Stability of the system is investigated using both phase

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