Self-excited vibrations in time-variant systems

• Applicants: Professor Dr. Peter Hagedorn; Professor Dr. Michael Schäfer
• Subject Area: Mechanics ; Mathematics
• Term: from 2015 to 2019
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 267028366

• Final Report Year: 2019

Final Report Abstract

The project was overall successful following the general outline of the project proposal and each of the work packages. First, fundamental stability phenomena were investigated semianalytically on low-dimensional models. In particular the influence of circulatory terms in context of parametric excitation was studied. By the formulation of a general two-degrees-of-freedom system with asynchronous parametric excitation it was possible to fill some of the gaps in the studies on time-periodic systems. Novel global stabilizing and destabilizing effects were discovered showing that the previously known total instability is only a special case. Further, it was shown that in general there are simultaneously resonance and anti-resonance at each combination resonance frequency, while the effects were believed to occur separately. These findings considerably expand the theoretical knowledge on time-periodic systems but also help to understand some complex dynamical phenomena like the brake squeal. Further, Floquet theory was successfully applied in conjunction with the finite element method to two paradigmatic mechanical examples featuring time-periodic coefficients. In order to achieve this, an in house FEM software suite was developed, capable of running in parallel to utilize modern CPU architectures. Several algorithms for the numerical determination of the maximal Floquet multipliers were compared and expanded upon. The software was first used to examine the buckling problem of a harmonically forced 2D and 3D column, and determined that stability could be improved by use of composite materials. The problem of a rotating disk in frictional contact with pads yielded a complex mechanical model of a disk brake with a complex stability behavior. Even though there are still some open issues concerning e.g. the order reduction techniques, both examples proved the general applicability of the chosen methods to problems with a large number of degrees of freedom.

Publications

• “Atypical parametric instability in linear and nonlinear systems.” In: Procedia Engineering, 199, p. 657-662, 2017
  Peter Hagedorn, Artem Karev, and Daniel Hochlenert
  
• "The FEM for a loaded column with harmonic axial forcing using a large number of solid elements" IUTAM Symposium on Recent Advances in Moving Boundary Problems in Mechanics, Christchurch, New Zealand (2018-02-15)
  Markus Rieken and Eoin Clerkin
  
• “Asynchronous parametric excitation, total instability and its occurrence in engineering structures.” In: Journal of Sound and Vibration, 428, p. 1-12, 2018
  Artem Karev, Peter Hagedorn, and Daniel Hochlenert
  
• “Some remarks on parametric excitation in circulatory systems.” In: PAMM, 18(1), 2018
  Artem Karev, Lara De Broeck, and Peter Hagedorn
  
• (2019) "FEM with Floquet Theory for Non-slender Elastic Columns Subject to Harmonic Applied Axial Force Using 2D and 3D Solid Elements." In: Gutschmidt S., Hewett J., Sellier M. (eds) IUTAM Symposium on Recent Advances in Moving Boundary Problems in Mechanics. IUTAM Bookseries, vol 34. Springer, Cham
  Eoin Clerkin and Markus Rieken
  
• “Global stability effects of parametric excitation.” In: Journal of Sound and Vibration, 448, p. 34-52, 2019
  Artem Karev and Peter Hagedorn