Mode-locking in nonlinear rotordynamics
The paper presents a computer-assisted study of the dynamics of two nonlinearly coupled driven oscillators with rotational symmetry which arise in rotor dynamics (nonlinear Jeffcott rotor). In the variable \(x \in \mathbb{C}\) the system can be written as \[ \ddot x + \gamma \dot x + \alpha (x) \left( 1 - {\delta \over |x |} \right) x = \varepsilon \omega^2 e^{i \omega t}, \tag{1} \] where \(\alpha (x) : = 0\) for \(|x |\leq \delta\) and \(\alpha (x) : = 1\) otherwise. The unperturbed linear system \((\gamma = \delta = 0)\) has a twofold natural frequency 1. But the nonlinearity, although preserving the rotational symmetry, resolves this degeneracy, which leads to the interaction of two internal frequencies \(\omega_1\) and \(\omega_2\) and the forcing frequency \(\omega\). System (1) possesses two infinite series of attracting invariant 2-tori in the phase space \(\mathbb{R}^4 \times S^1\). The flow on the tori is quasiperiodic with frequencies \(\omega_1\) and \(\omega_2\) which are partially mode-locked \((n_1 \omega_1 + n_2 \omega_2 = \omega\) with \(n_1, n_2 \in \mathbb{N} \cup \{0\}\), \(|n_1 - n_2 |= 1)\). Under a perturbation that breaks the rotational symmetry a second rational relation between the frequencies can be established in certain regions of the parameter space (fully mode-locked periodic motion). In rotating coordinates system (1) becomes autonomous, and the 2-tori in \(\mathbb{R}^4 \times S^1\) correspond to limit cycles in \(\mathbb{R}^4\) with certain discrete symmetry properties.
scientific article; zbMATH DE number 798898
scientific article