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Table of contents
1 Introduction
1.1 Research Field
1.2 Research goals
1.3 Reading guide
2 State of the art
2.1 Time-periodic systems
2.1.1 Relation between nonlinear and time periodic systems
2.1.2 Time-periodicity in Ordinary Differential Equations
2.1.3 Time-periodic vibrations in engineering
2.2 Examples: vibrations of beams in periodic elastic states
2.2.1 Ziegler column
2.2.2 Finite element discretization of the cantilever beam with periodic prestress
2.3 Stability of time-periodic systems
2.3.1 Floquet theory
2.3.2 Time domain
2.3.3 Frequency domain
2.3.4 Stability types
2.3.5 Stability analysis of the 2-DoF Ziegler column
2.4 Classic Modal Analysis
2.4.1 Modal projection of the cantilever beam with periodic prestress
2.5 Conclusions
3 Frequency domain analysis of Floquet forms
3.1 Introduction
3.2 Time-domain method
3.2.1 STM eigenvectors
3.3 Hill Matrix
3.3.1 Complex Hill Matrix Derivation
3.3.2 Real Hill Matrix Derivation
3.4 Periodically conservative case (η = 0)
3.4.1 Constant elastic state (β = 0)
3.4.2 Periodic elastic state (β 6= 0)
3.4.3 Asymptotic cases (β → +∞) and (β → 0)
3.5 Non-Conservative case (η = 1)
3.5.1 Constant elastic state (β = 0)
3.5.2 Periodic elastic state (β 6= 0)
3.6 Spectral convergence of the stability analysis
3.7 Discrete dynamical stabilization above buckling load
3.7.1 Conservative case
3.7.2 Finding stability zones
3.7.3 Stability Zone Characteristics
3.7.4 Nonconservative case
3.8 Conclusions
4 Time-periodic modal analysis
4.1 Modal-Floquet Transformation
4.2 Free vibration
4.2.1 Floquet form computation and visualisation
4.2.2 Projection on Floquet Forms
4.2.3 Numerical application
4.3 Forced vibrations of time-periodic systems
4.3.1 Projecting the force vector
4.3.2 Example of a harmonic external force
4.3.3 Frequency Response Spectrum
4.4 Conclusions
5 Conclusions
Appendices
A Equation of motion of the Ziegler column
B Hill matrix implementation
B.1 Floquet transform
B.2 Fourier transform
B.3 Harmonic Balance
B.4 Hill matrices
B.5 Numerical implementation
C High Frequency Averaging of the statically diverging Ziegler Column in periodic elastic state
C.1 Averaged equations of motion
