Abstract
The tippedisk is a mathematical–mechanical archetype, showing a non-intuitive inversion behavior, if the disk is spun fast enough. By introducing a full 3D mechanical model and assuming set-valued force laws to account for normal and frictional contact forces, the dynamics of the tippedisk can be studied numerically. In addition, the model dimension can be reduced through model reduction techniques, yielding a lower dimensional model, being perfectly suited for the closed-form analysis of the qualitative dynamics. The previous work of the authors has shown that the bifurcation scenario contains a heteroclinic bifurcation, followed by a subcritical Hopf bifurcation. In this chapter, we derive a complete stability chart that characterizes various bifurcation scenarios in closed-form, which allows us to understand the qualitative dynamics of the tippedisk.
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Notes
- 1.
In fact, \(\omega \) is the ratio in which the square of the spinning speed \(\varOmega ^2\) appears in a reciprocal way. Still, we will refer to \(\omega \) as the normalized spinning speed as it gives a dimensionless measure for the magnitude of the spinning speed \(\varOmega \).
- 2.
To avoid long expressions we introduce the abbreviations \(\text{s}\) and \(\text{c}\) for the trigonometric functions \(\sin \) and \(\cos \). Likewise, \(\mathrm {s}^2\gamma \) is used as shorthand for \(\sin ^2\gamma \).
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Sailer, S., Leine, R.I. (2024). A Complete Stability Chart for the Tippedisk. In: Lacarbonara, W. (eds) Advances in Nonlinear Dynamics, Volume I. ICNDA 2023. NODYCON Conference Proceedings Series. Springer, Cham. https://doi.org/10.1007/978-3-031-50631-4_51
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DOI: https://doi.org/10.1007/978-3-031-50631-4_51
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