Coupled Mathieu Equations: <em>γ</em>-Hamiltonian and <em>μ</em>-Symplectic

Several theoretical studies deal with the stability transition curves of coupled and damped Mathieu equations utilizing numerical and asymptotic methods. In this contribution, we exploit the fact that symplectic maps describe the dynamics of Hamiltonian systems. Starting with a Hamiltonian system, a particular dissipation is introduced, which allows the extension of Hamiltonian and symplectic matrices to more general γ -Hamiltonian and μ -symplectic matrices. A proof is given that the state transition matrix of any γ -Hamiltonian system is μ -symplectic. Combined with Floquet theory, the symmetry of the Floquet multipliers with respect to a μ -circle, which is different from the unit circle, is highlighted. An attempt is made for generalizing the particular dissipation to a more general form. The methodology is applied for calculation of the stability transition curves of an example system of two coupled and damped Mathieu equations

Damping by Parametric Stiffness Excitation: Resonance and Anti-Resonance

The goal of this work is to develop deeper insight into the method of damping vibrations by means of parametric excitation in mechanical systems - employing the concept of the so-called parametric anti-resonance. Mechanical systems with simultaneously varying time-periodic stiffness, damping and inertia coefficients are examined. At least two vibration modes are necessary to achieve damping. For these minimum systems a thorough stability analysis is carried out using a perturbation technique. Parametric excitation may lead to a coupling of just two modes of a vibrating system, while the remaining modes are not affected. This coupling enables to transfer energy between modes and its subsequent mitigation. The results demonstrate that parametric excitation can be employed to extend significantly the area of stability in the parameter space of the system parameters. The proposed method shows potential in practical applications when a destabilization due to self-excitation occurs or when the damping of weakly damped systems shall be enhanced. The present book can be used as a comprehensive guide for designing a device for vibration suppression by parametric excitation

Stability improvement of a flexible rotor in active magnetic bearings by time-periodic stiffness variation

Previous theoretical studies have shown analytically and numerically that a vibrating system can be stabilised and its vibrations can be suppressed by an open-loop control of a stiffness parameter, a stabilisation by parametric stiffness excitation. In this paper this approach is investigated further numerically, analytically and experimentally for a flexible rotor with multiple disks supported by active bearings. A periodic, open-loop control of the stiffness coefficients of a bearing is realised by periodically changing the control parameters of an active magnetic bearing. This periodic variation of control parameters is regulated at fixed frequency and amplitude in such a way that it acts like a parametric excitation on the rotor system. As it was shown for simple vibrating structures (chain mass system, cantilever, Jeffcot rotor), the periodic variation may lead to an effective vibration reduction in a complex rotor system. Since this control is open-loop, it can operate in parallel to existing and well-established controllers already used in active magnetic bearings. In this paper, the method of damping by parametric excitation is realised for the first time experimentally in a rotor system. Both direct numerical simulations and analytical predictions are performed to calculate ranges for control and system parameters where damping by parametric excitation is effective. Results frombothmethods are compared with first experimental results. Finally, analytical conditions are given under which the damping by parametric stiffness is effective and the damping effect is guaranteed