An approach to account for interfering parametric resonances and anti-resonances applied to examples from rotor dynamics

  • Thomas BreunungEmail author
  • Fadi DohnalEmail author
  • Bastian Pfau
Original Paper


An initially unstable equilibrium position of a system can be stabilized by introducing a parametric excitation. This is especially of interest for suppressing self-excited vibrations, and the effect is known as parametric anti-resonance which can be observed close to the parametric combination frequencies. For the stability analysis, linearized mechanical systems with an arbitrary number of degrees of freedom and time-periodic damping and stiffness matrices are analyzed. To approximate stability maps analytically, the method of averaging is applied. A state-space representation is outlined which lifts the restriction of symmetric stiffness and damping matrices in the common approaches. The eigenvalues of the slow flow are used to determine stability. Close to a parametric combination frequency, these can change significantly. Restricting the analysis to a single parametric resonance frequency may lead to unsatisfactory and even contradictory stability maps due to the local approximation. Therefore, a novel approach, that is capable to account for the interference between the averaged eigenvalues, is outlined and motivated in an engineering manner. To verify the potential of this approach, two example systems from rotor dynamics are revisited.


Parametric excitation Averaging Stability Rotor dynamics 



The authors acknowledge Richard Markert for the fruitful discussions on this research topic.

Supplementary material

11071_2019_4761_MOESM1_ESM.mat (653 kb)
Supplementary material 1 (mat 653 KB)


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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Institute for Applied Dynamics, Otto-Berndt-Straße 2Technische Universität DarmstadtDarmstadtGermany
  2. 2.Division for Mechatronics LienzUMITLienzAustria
  3. 3.Institute for Mechanical SystemsETH ZürichZurichSwitzerland

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