الفهرس 
Ch. 1.Only 14 pages are availabe for public view
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Abstract
Nonlinear dynamical systems play a crucial role in diverse fields like medicine and engineering. In medicine, they assist in diagnosis and prediction, and play a part in developing precise surgical robots. In engineering, these systems are indispensable for analyzing, designing, and enhancing a broad spectrum of engineering systems. They facilitate the exploration of system stability and control, allowing for the creation of stable and dependable systems in disciplines such as aerospace engineering, civil engineering, and mechanical engineering. Examples of mechanical systems include pendulum models, extensively utilized in scientific research and applications.
The thesis includes a list of the used symbols, a list of figures, and another of tables, along with an introduction and four chapters. In conclusion, the thesis contains a summary that explains and interprets the main research outcomes of the study, as well as a list of scientific references relevant to the topic of the research.
Introduction includes a comprehensive summary of the researchers’ main contributions in the field of thesis, closely related to the focal point of the research, along with a comprehensive presentation of the study’s topic.
Chapter 1 focuses on studying the dynamics for the motion of a system consisting of three interconnected rigid bodies, moving vertically under the influence of gravity. The study involved using Lagrange’s equations to derive the equations of motion and applying the multiple-scales method to obtain detailed analytical solutions. Resonance cases were accurately classified, and the effect of various paramters within the system on its dynamic behavior is examined. Types of stable points are carefully classified, and the time behavior of the system’s modified angles, capacities, and solutions is analyzed. Additionally, nonlinear stability assessment was conducted and discussed using the Routh-Hurwitz criteria, demonstrating the system’s stability under different operating conditions. The obtained results confirm the stability of the system under various operational conditions, enhancing our understanding of its dynamic behavior.
Chapter 2 presents a dynamic study for the motion of a parametrically driven dynamical system, notable for its ability to exhibit various resonance phenomena, where certain frequencies of external influences coincide with the natural frequencies of the system, leading to rapid responses. The study utilizes the method of multiple-scales to analyze this complex system and provide a precise understanding of its behavior. Additionally, it focuses on determining modulation equations for studied resonance scenarios, offering valuable insights into how the system responds to external forces at frequencies matching its natural frequencies. Moreover, the study meticulously examines
the system’s behavior and identifies regions of stability and instability using the Routh-Hurwitz criteria, contributing to our overall understanding of autonomous parametric dynamical systems.
Chapter 3 examines the complex motion of the double pendulum with two degrees of freedom, where the pendulum moves along a Lissajous curve. The system is analyzed using second-order Lagrangian equations, enabling a precise description of its behavior, employing the multiple-scales method to derive capacitance equations. The study also encompasses analyzing the solution and stability conditions for various resonance cases, demonstrating the impact of physical parameters on the system’s response through graphical representations such as bifurcation diagrams, Lyapunov exponent spectra, and Poincaré maps, aiding in a deeper understanding and classification of the system’s dynamics, thereby elucidating and enhancing the interpretability of the results.
Chapter 4 focuses on the dynamics of a system with four degrees of freedom. The nonlinear differential equations describing this system are derived using Lagrange’s equations. Numerical solutions are subsequently obtained using the fourth-order Runge-Kutta algrothim. The inquiry involves examining the relationships between angular solutions and their corresponding first-order derivatives, a technique commonly referred to as phase plane analysis. The aim is to explore bifurcation diagrams and Lyapunov exponent spectra to reveal the various modes of motion present within the system and generate Poincaré maps for visualization. Finally, the solvability conditions and characteristic exponents are determined based on resonance cases. The Routh-Hurwitz criteria are used to evaluate the stability of fixed points related to steady-state solutions and to illustrate frequency response curves. The nonlinear stability is analyzed across both stable and unstable ranges.
Conclusion: a comprehensive summary of all the research outcomes extracted from this study has been provided.
Keywords: Nonlinear dynamics; Vibrating motions; perturbation methods; Numerical solution; Bifurcation analyses; Poincare map; Stability.