Abstract
This paper investigates the nonlinear dynamics of a flexible tyre coupling via computer modelling and simulation. The research mainly focused on identifying basins of attraction of coexisting solutions of the formulated phenomenological coupling model. On the basis of the derived mathematical model, and by assuming ranges of variability of the control parameters, the areas in which chaotic clutch movement takes place are determined. To identify multiple solutions, a new diagram of solutions (DS) was used, illustrating the number of coexisting solutions and their periodicity. The DS diagram was drawn based on the fixed points of the Poincaré section. To verify the proposed method of identifying periodic solutions, the graphic image of the DS was compared to the three-dimensional distribution of the largest Lyapunov exponent and the bifurcation diagram. For selected values of the control parameter ω, coexisting periodic solutions were identified, and basins of attraction were plotted. Basins of attraction were determined in relation to examples of coexistence of periodic solutions and transient chaos. Areas of initial conditions that correspond to the phenomenon of unstable chaos are mixed with the conditions of a stable periodic solution, to which the transient chaos is attracted. In the graphic images of the basins of attraction, the areas corresponding to the transient and periodic chaos are blurred.
Highlights
In flexible couplings, the nonlinearity of the system is mainly due to the mechanical characteristics of the flexible connector
Assuming that the clutch is affected by an external drive torque with a frequency of ω = 1.9013, which is located at the end of the tested variation range of the control parameter, three identical 2 T-periodic solutions coexist in the case study ( u = 0.3978, u = 0.1911 ), 4 T-periodic ( u = 0.1757, u = −0.0992 ) and 4 T-periodic solutions attracted to the attractor, corresponding to transient chaos (u = 0.1523, u = 0.1186 )
Based on diagram of solutions (DS) diagrams, it is possible to identify solutions reflecting the phenomenon of transient chaos
Summary
The nonlinearity of the system is mainly due to the mechanical characteristics of the flexible connector. An important aspect of nonlinear dynamics research is the identification of coexisting multiple (subharmonic) solutions for different initial conditions, which in consequence boils down to seeking attractors In general terms, this issue studies time series defining trajectories aimed at specific attractors. In contrast to the procedure for estimating the largest Lyapunov exponent, the entire set of initial conditions located at different points of the phase plane are considered The results of these studies are most often presented in the form of multicoloured maps showing the structure of the basins of attraction, which were plotted with reference to the Duffing equation [24], magnetic pendulum [6] and other dynamic systems [25]. Regardless of the applied method of transforming the signal from the time domain to the frequency domain, it is possible to determine at what moment the chaotic solution turns into a periodic solution
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