Analysis, Synthesis, and Design of Nonlinear Vibrations of Inverted Flexible Pendulum

Chaos encountered in many nonlinear physical and engineering problems often leads to an undesirable effect due to the nature of the irregular and unpredictable behavior. Hence, the challenge here is to control the chaotic behaviors of these systems. Problems and challenges arise from this are due to the complexity of nonlinear dynamic behaviors such as structural nonlinearities and couplings, the unpredictable motion, and in some cases may lead the chaotic behavior.</br> Many practical applications of deterministic chaos have been developed in various fields of engineering and technology. Recent research has shown that chaos can be useful under certain circumstances, such as enhancing the mixing of chemical reactants and in domestic appliances. On the other hand, chaos should be weakened or completely suppressed when it is undesirable and may become harmful. Therefore, it is helpful and important to encourage further investigations on chaos and chaos control in nonlinear dynamical systems.</br> Control of nonlinear and highly flexible systems is often effected by design requirements and also manufacturing aspects. The dynamics and control of such systems are challenging, especially when a nonlinear mechanical system is considered. The experimental study of the dynamical behavior of an inverted flexible pendulum system showing jumping phenomenon between three equilibria is not considered in detail in literature so far. In this thesis, the dynamical behavior of a nonlinear elastic mechanical system is considered, namely an inverted flexible pendulum excited at the base by a motor-driven cart.</br> Therefore, this thesis focuses on a combination of two aspects: an analysis on the dynamics of a nonlinear elastic system, and how to design the nonlinear vibration in this system in terms of controlling towards chaotic states between different equilibria. In the next step of the research, a transiently induced vibration is discussed. As in the initial experimental procedure, the chaotic motion of the flexible pendulum tip was identified, in combination with a specific range of parameters.</br> Verified sufficient conditions and parameters are tested experimentally for such chaotic vibration. The first part focuses on studying the dynamics of the flexible pendulum. By varying the excitation parameters, control parameters, as well as other distinguished mechanical parameters, different phenomena are observed in experiments are discussed. For this observation, a custom-built inverted flexible pendulum on cart system under PID-controlled harmonic excitation is considered. Data are collected from both cart excitation signal and displacement of the flexible pendulum, also to observe their correlation towards jumping behavior. Effects of the variation of the parameters leading to changes in chaotic jumping patterns. Multiple equilibria are observed and analyzed. It can be concluded that depending on the excitation amplitudes, frequencies, and controller parameters, the minimum of two equilibria with an unstable third equilibrium can be detected while jumping phenomena between the equilibria are observed. Questions about the stimulation of the jumping by impulses resulting from imperfect sinusoidal excitation due to control limitations later are discussed.</br> Additionally, this thesis further provides insight on the application of time-frequency energy (TFE) analysis for experimental modeling of the state transition between the equilibria of the chaotic systems during nonlinear vibration. In this part, the effect of impulses realized by non-perfect feedback to an inverted flexible pendulum system is studied. Using experimental data from chaotic jumping in an inverted flexible pendulum, several techniques of signal processing and time-frequency representation are carried out. These methods are used to observe the changes in the nonlinear dynamic properties and time behavior of the system before and during the jumping between equilibria.</br> The evaluation of the experimentally realized inverted flexible pendulum system for specific control parameters shows that 'chaotic' jumping behavior between the three equilibria (with different attraction regions) are depending on the existence of impulses. The results from a detailed analysis show that this is caused by feedback imperfectness. Considering the time-frequency energy (TFE) analysis conducted, it can be stated that the (uncontrolled) impulses causing a temporarily disturbance in the energy distribution between the 'modes'/frequency bands the system vibrates with. This will finally leads to a new and more stable distribution in energy, followed by related jumping between the equilibria. Here the energy is concentrated in different varying frequency ranges.</br> Using the data, the jumping phenomenon during the chaotic vibration are collected and analyzed. Here, time-frequency energy (TFE) analysis method can effectively show the characteristics of energy in the time domain and perform the component analysis in the specific frequency range. Applying a comparative study of jumping phenomenon discussing different equilibria, frequency range recognition, and energy characterization, the jumping phenomenon of the flexible pendulum induced by chaotic vibration is characterized. A state transition model is then established. Further, an additive impulsive control on the elastic system is considered to validate the model.</br> Further, an additive impulsive control on the sinusoidal moving inverted flexible pendulum is considered. An impulsive control method to effect the chaos and equilibrium position of this pendulum system is designed and developed. Some sufficient conditions for driving the chaotic states between different equilibria are presented. This part provides an experimental analysis, design, and validation to support the established model for state transition of chaos in the nonlinear flexible system. Controlling the chaotic behavior of the system is realized using impulsive control method, where additive impulses designed with specific impulses energy content at specific frequency band are injected into the system, under the specified threshold of the vibrating system.</br> The experimental results concentrate on the effect of the designed impulses which are injected into the system, in terms of the transition between states of equilibria. Results are presented and discussed in detail, concentrating on how the designed impulses injected affecting the system, specifically the transition between states of equilibria. The results from the experimental validation show that both additive impulse design and frequency filtering of the injected additive impulses are able to stimulate the equilibrium shift and therefore to control the chaotic behavior of the system.



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