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Bearing Monitoring and Diagnostics
Published in Maurice L. Adams, Bearings, 2018
Figure 5.8 presents simulation results of the rotor unbalance excited rub-impact model illustrated in Figure 5.7a, showing a confluence of rotor orbit trajectories and their signal chaos-content mappings using some typical chaos signal processing tools. The central portion of Figure 5.8 is a bifurcation diagram, plotting the orbit's x-position coordinate (with a dot) once for each shaft revolution as a reference mark (keyphaser) on the rotor passes the same fixed position point on the stator. If the rotor orbit is strictly rotational-speed synchronous, then the same dot appears once per rev, repeatedly. If a half-synchronous (N/2) subharmonic component is superimposed, then only the same two dots repeatedly appear. Similarly, the Poincaré maps in Figure 5.8 contain a dot deposited for the rotor orbit's (x, y) position at each shaft revolution as the reference keyphaser mark fixed on the rotor passes the same fixed stator point. The term quasiperiodic is used by chaos specialists and others to label non-periodic signals that are comprised of incommensurate (non-integer-related) periodic signals.
On the Transition to Phase Synchronized Chaos
Published in Christos H. Skiadas, Charilaos Skiadas, Handbook of Applications of Chaos Theory, 2017
Erik Mosekilde, Jakob Lund Laugesen, Zhanybai T. Zhusubaliyev
This function has been taken over by a cascade of torus bifurcation curves that stretch up along the saddle-node bifurcation curves, but inside the resonance zone. In this way, a region is formed where stable quasiperiodic oscillations coexist with doubly and triply unstable resonance modes. The torus bifurcation curves emerge from the points of intersection between the corresponding period-doubling curve and the former torus bifurcation curve. They are supported, close to the transition to chaos, by points on the corresponding saddle-node bifurcation curves.
Bifurcation and Chaos
Published in Wai-Kai Chen, Feedback, Nonlinear, and Distributed Circuits, 2018
Michael Peter Kennedy, Vandenberghe Lieven
A quasiperiodic function is one that may be expressed as a countable sum of periodic functions with incommensurate frequencies, i.e., frequencies that are not rationally related. For example, X(t) = sin(t) + sin(2πt) is a quasiperiodic signal. In the time domain, a quasiperiodic signal may look like an amplitude-or phase-modulated waveform.
Dynamic Characterization of a Ducted Inverse Diffusion Flame Using Recurrence Analysis
Published in Combustion Science and Technology, 2018
Uddalok Sen, Tryambak Gangopadhyay, Chandrachur Bhattacharya, Achintya Mukhopadhyay, Swarnendu Sen
On increasing x beyond 46 cm, the flame-acoustic coupling is again lost. Such a behavior was also reported by Kabiraj et al. (2015), where the variation of the control parameter led to a region of strong flame-acoustic coupling sandwiched between two regions of weak coupling. A representative case of this region (x = 50 cm, = 20 lpm) is shown in Figure 3. The waveform of the signal (Figure 3c1) suggests intermittent behavior, but having much less time span of intermittency as compared to Figure 3a1. The PSD (power spectral density) plot (Figure 3c2) shows both the cold natural frequency (100 Hz) and the instability-generated frequency (237 Hz) having comparable strengths, along with another significant peak at 155 Hz. Such incommensurate frequencies often characterize quasiperiodic systems. Such an emergence of quasiperiodicity from limit cycles is often a result of a secondary Hopf bifurcation, referred to as a Neimark–Sacker bifurcation in literature (Nayfeh and Balachandran, 2004). The phase plot (Figure 3c3) in this case does resemble an experimental quasiperiodic system, and the recurrence plot (Figure 3c4) also lacks any discernible dense black patch that would have otherwise suggested intermittency. However, the frequency distribution of the vertical lines in the recurrence plot (Figure 3c5) showed the familiar structure for type-II and type-III intermittencies—a skewed distribution with an exponential tail. However, since the dense black patches in the recurrence plot are not very discernible, the precise nature of the intermittency (type-II or type-III) cannot be inferred from the available data.
Dynamic characteristics analysis of a vertical Jeffcott rotor-brush seal system
Published in Journal of the Chinese Institute of Engineers, 2022
Yuan Wei, Xuhe Ran, Tao Sun, Shulin Liu, Hongli Zhang, Dongfang Zhao
The amount of system damping affects the dynamic characteristics and amplitude of the vertical system. Figure 10 shows the bifurcation diagram and LLE of the vertical system with respect to the level of rotor damping. The system alternately experiences chaotic, quasi-periodic, and periodic motions. The bifurcation diagrams in the x and y directions are basically the same, and the system amplitude decreases significantly as the damping level increases. Obviously, increasing the system damping effectively enhances the stability of the rotor-seal system.