الفهرس 
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Abstract
The main objective of this thesis is a mathematical study for mechanical vibrations of some dynamical systems in the presence of time-delay in some dynamical systems described by nonlinear ordinary differential equations. We studied the mechanical vibrations of the cantilever beam with time-delay. Different control algorithms were proposed to suppress these nonlinear vibrations. All different mathematical models were analyzed analytically applying homotopy multiple scales perturbation method. The acquired analytical results were validated numerically utilizing the appropriate standard Matlab solvers. Finally, a list of references regarding this discipline was cited, where the thesis is outlined as follows:
Chapter 1: This chapter is concerned with introducing the background necessary to understand the mechanical vibration problem, identifying time delay in the control system, some of important researches that deal with the nonlinear vibrations of the cantilever beam, time-delay, and the study of different methods of perturbation.
Chapter 2: The nonlinear transversal oscillations of a cantilever beam system at primary, superharmonic, and subharmonic resonance cases are investigated within this chapter. Time-delayed position-velocity controller is proposed to suppress the considered system nonlinear vibrations. The multiple scales homotopy approach is employed to analyze the controlled nonlinear model. The amplitude-phase modulating equations that govern the system dynamics at the different resonance cases are extracted. The stability charts of the loop-delay are obtained. The influence of the different controller parameters on the system vibration behaviors is explored. The acquired analytical results revealed that the loop-delay has a great influence on the controller efficiency. Accordingly, the optimal values of the loop-delay are reported and utilized to enhance the applied controller performance. Finally, numerical validations of the accomplished analytical results are performed, which illustrated an excellent agreement with the obtained analytical ones.
Chapter 3: In this chapter, Six different time-delayed controllers are introduced to explore their efficiencies in suppressing the nonlinear oscillations of a parametrically excited system. The
applied control techniques are the linear and nonlinear versions of the position, velocity, and acceleration of the considered system. The time-delay of the closed-loop control system is included in the proposed model. As the model under consideration is a nonlinear time-delayed dynamical system, the multiple scales homotopy method is utilized to derive two nonlinear algebraic equations that govern the vibration amplitude and the corresponding phase angle of the controlled system. Based on the obtained algebraic equations, the stability charts of the loop-delays are plotted. The influence of both the control gains and loop-delays on the steady-state vibration amplitude is examined. The obtained results illustrated that the loop-delays can play a dominant role in either improving the control efficiency or destabilizing the controlled system. Accordingly, two simple objective functions are introduced in order to design the optimum values of the control gains and loop-delays in such a way that improves the controllers’ efficiency and increases the system robustness against instability. The efficiency of the proposed six controllers in mitigating the system vibrations is compared. It is found that the cubic-acceleration feedback
controller is the most efficient in suppressing the system vibrations, while the cubic-velocity feedback controller is the best in bifurcation control when the loop-delay is neglected. However, the analytical and numerical investigations confirmed that the cubic-acceleration controller is the best either in vibration suppression or bifurcation control when the optimal time-delay is considered. It is worth mentioning that this may be the first article that has been dedicated to introducing an objective function to optimize the control gains and loop-delays of nonlinear time-delayed feedback controllers.
Chapter 4: In this chapter, a nonlinear integral resonant controller is utilized for the first time to suppress the principal parametric excitation of a nonlinear dynamical system. The whole system is modeled as a second-order nonlinear differential equation (i.e., main system) coupled to a nonlinear first-order differential equation (i.e., controller). The control loop time-delays are included in the studied model. The multiple scales homotopy approach is employed to obtain an approximate solution for the proposed time-delayed dynamical system. The nonlinear algebraic equation that governs the steady-state oscillatxion amplitude has been extracted. The effects of the time-delays, control gain, and feedback gains on the performance of the suggested controller
have been investigated. The obtained results indicated that the controller performance depends on the product of the control and feedback signal gains as well as the sum of the time-delays in the control loop. Accordingly, two simple objective functions have been derived to design the optimum values of the loop-delays, control gain, and feedback gains in such a way that enhances the efficiency of the proposed controller. The analytical and numerical simulations illustrated that the proposed controller could eliminate the system vibrations effectively at specific values of the control and feedback signal gains. In addition, the selection method of the loop-delays that either enhances the control performance or destabilizes the system motion has been explained in detail