International Refereed Journal of Engineering and Science (IRJES) ISSN (Online) 2319-183X, (Print) 2319-1821 Volume 6, Issue 7 (July 2017), PP.08-20 www.irjes.com 8 | Page Active Control and Dynamical Analysis of two Coupled Parametrically Excited Van Der Pol Oscillators * Y.A.Amer 1 ,N. M. Ali 2 ,Manar M. Dahsha 3 ,S. M. Ahmed 3 1 Mathematics Department, Faculty of Science, Zagazig University, Egypt. 2 Mathematics Department, Faculty of Science, Suez canal University, Egypt 3 Mathematics Department, Faculty of Science, El-Arish University, Egypt. . Abstract: The dynamical behavior of two coupled parametrically excited van der pol oscillator is investigated by using perturbation method. Resonance cases were obtained, the worst one has been chosen to be discussed. The stability of the obtained numerical solution is investigated using both phase plane methods and frequency response equations. Effect of the different parameters on the system behavior is studied numerically. Comparison between the approximate solution and numerical solution is obtained. Keywords: Vibration control, nonlinear oscillation, perturbation technique, Resonance cases, Frequency response curves. I. INTRODUCTION Vibrations at most time are non-desirable, humans suffered from these bad vibrations, so it must be eliminating or at least controlled. The dynamic absorber is the most common methods for reduced the vibrations. Its importance tends to as it is need low coast, and it is a simple operation at one modal frequency. El-Badawy and Nayfeh [1] adopted linear velocity feedback and cubic velocity feedback control laws.Yang,Cao and Morris[2] use Mat lab for applying numerical methods. Amer [3] investigation the coupling of two nonlinear oscillators of the main system and absorber representing ultrasonic cutting process subjected to parametric excitation force. Non-linearities necessary introduce a whole range of phenomena that are not found in linear system [4], including jump phenomena, occurrence of multiple solutions, modulation, shift in natural frequencies, the generation of combination resonances, evidence of period multiplying bifurcations and chaotic motion [5-8]. In these systems the vibrations are needed to be controlled to minimize or eliminating the hazard of damage or destruction. There are two types for vibration control, active and passive control. Pinto and Goncalves [9] investigated the active control of the nonlinear vibration of a simply-supported buckled beam under lateral loading. One of most effective tools of passive control is dynamic absorber or the neutralizer [10]. Nabergoj et al [11] studied the stability of auto- parametric resonance in an external excited system. Abdel Hafz and Eissa [12] study the effect of nonlinear elastomeric torsion absorber to control the vibrations of the crank shaft in internal combustion engines, when subject to external excitation torque. Fuller, Elliot and Nelson [13] investigate the active control of vibrations which give many ideas and approaches for controlling chaos. Abe et al. [14] investigate the nonlinear responses of clamped laminated shallow shells with 1:1 internal resonance. Eissa et al. [15-16] investigated saturation phenomena in non-linear oscillating systems subjected to multi-parametric and external excitation. Gerald [17] apply the numerical analysis to find out the solutions for the vibrations problems. Sayed and Kamel [18-19] investigated the effect of different controllers on the vibrating system and saturation control of a linear absorber to reduce vibrations due to rotor blade flapping motion. Kamel et al [20] studied the vibration suppression in ultrasonic machining described by non-linear differential equations via passive controller. Elena et al. [21] studied the formal analysis and description of the steady-state behavior of an electrostatic vibration energy harvester operating in constant-charge mode and using different types of electromechanical transducers .Orhan and Peter [22] investigate the effect of excitation and damping parameters on the super harmonic and primary resonance responses of a slender cantilever beam undergoing flapping motion. In this paper we studied vibration control of a nonlinear system under tuned excitation force. The method of multiple scale method is applied to obtain the approximate solution of the system. Vibration method is used to reduce the amplitude of vibration at the worst resonance case. The effect of different parameters are investigated, the comparison between the numerical solution and approximation solution obtained. II. MATHEMATICAL MODELING The considered system is described by the equations: 2 2 2 2 2 3 1 1 1 1 1 ( 2 cos( )) ( ) ( ) X f t X X Y X X aY X GX               (1)