الفهرس 
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Abstract
The main object of this work is the mathematical study of the vibration behavior in non-linear systems with time delay. These systems are described by non-linear differential equation of the second order including quadratic and cubic non-linearities. The multiple scale perturbation technique (MSPT) to the first order approximations and the averaging method are applied to study the analytical solution and analyze the response of these systems near the deduced worst resonance cases to obtain the frequency response equations for these systems. These worst resonance cases are investigated numerically, using the method of Runge-Kutta. The stability of the steady state solution near these resonance cases for the above different types of nonlinear systems are investigated and studied applying frequency response equations (Using MAPLE 13 software). The numerical solutions are focused on both the effects of the time delay and some system parameters on the vibrating systems behavior. The simulation results are achieved using MATLAB 7.0 programs. Comparison between analytical and numerical results is illustrated and there are good agreement between the analytical and numerical results.
The conclusions of the system with time delay are reported in the following sections:
1) Study the nonlinear vibrations of a parametric excited Duffing oscillator with time delay feedback. At some values of the time delay can be used to suppress the vibration of the nonlinear system, which are studied in chapter two.
2) Study parametric excited of a bilinear system with nonlinear velocity time-delayed feedback. With the time delay varying for a fixed gain, it is seen that the vibration can be suppressed. Both external (forcing) and parametric excitations have been included, which are studied in chapter three
3) Investigated the position and velocity time delay control of external and parametric resonance in modified Rayleigh-Duffing oscillator. The effect of time delay studied and finds the vibration suppression region, which are studied in chapter four.