Nonlinear and Dual-Parameter Chaotic Vibrations of Lumped Parameter Model in Blisk under Combined Aerodynamic Force and Varying Rotating Speed

In this paper, the nonlinear and dual-parameter chaotic vibrations are investigated for the blisk structure with the lumped parameter model under combined the aerodynamic force and varying rotating speed. The varying rotating speed and aerodynamic force are respectively simplified to the parametric and external excitations. The nonlinear governing equations of motion for the rotating blisk are established by using Hamilton’s principle. The free vibration and mode localization phenomena are studied for the tuning and mistuning blisks. Due to the mistuning, the periodic characteristics of the blisk structure are destroyed and uniform distribution of the energy is broken. It is found that there is a positive correlation between the mistuning variable and mode localization factor to exhibit the large vibration of the blisk in a certain region. The method of multiple scales is applied to derive four-dimensional averaged equations of the blisk under 1:1 internal and principal parametric resonances. The amplitude-frequency response curves of the blisk are studied, which illustrate the influence of different parameters on the bandwidth and vibration amplitudes of the blisk. Lyapunov exponent, bifurcation diagrams, phase portraits, waveforms and Poincare maps are depicted. The dual-parameter Lyapunov exponents and bifurcation diagrams of the blisk reveal the paths leading to the chaos. The influences of different parameters on the bifurcation and chaotic vibrations are analyzed. The numerical results demonstrate that the parametric and external excitations, rotating speed and damping determine the occurrence of the chaotic vibrations and paths leading to the chaotic vibrations in the blisk.

Bursting Oscillations as well as the Mechanism in a Filippov System with Parametric and External Excitations

The main purpose of this paper is to explore the bursting oscillations as well as the mechanism of a parametric and external excitation Filippov type system (PEEFS), in which different types of bursting oscillations such as fold/nonsmooth fold (NSF)/fold/NSF, fold/NSF/fold and fold/fold bursting oscillations can be observed. By employing the overlap of the transformed phase portrait and the equilibrium branches of the generalized autonomous system, the mechanisms of the bursting oscillations are investigated. Our results show that the fold bifurcation and the boundary equilibrium bifurcation (BEB) can cause the transitions between the quiescent states and repetitive spiking states. The oscillating frequencies of the spiking states can be approximated theoretically by their occurring mechanisms, which agree well with the numerical simulations. Furthermore, some nonsmooth evolutions are investigated by employing differential inclusions theory, which reveals that the positional relationship between the points of the trajectory interacting with the nonsmooth boundary and the related sliding boundary of the nonsmooth system may affect the nonsmooth evolutions.

A Re-Examination of Various Resonances in Parametrically Excited Systems1

The dynamics of parametrically excited systems are characterized by distinct types of resonances including parametric, combination, and internal. Existing resonance conditions for these instability phenomena involve natural frequencies and thus are valid when the amplitude of the parametric excitation term is zero or close to zero. In this paper, various types of resonances in parametrically excited systems are revisited and new resonance conditions are developed such that the new conditions are valid in the entire parametric space, unlike existing conditions. This is achieved by expressing resonance conditions in terms of “true characteristic exponents” which are defined using characteristic exponents and their non-uniqueness property. Since different types of resonances may arise depending upon the class of parametrically excited systems, the present study has categorized such systems into four classes: linear systems with parametric excitation, linear systems with combined parametric and external excitations, nonlinear systems with parametric excitation, and nonlinear systems with combined parametric and external excitations. Each class is investigated separately for different types of resonances, and examples are provided to establish the proof of concept. Resonances in linear systems with parametric excitation are examined using the Lyapunov–Poincaré theorem, whereas Lyapunov–Floquet transformation is utilized to generate a resonance condition for linear systems with combined excitations. In the case of nonlinear parametrically excited systems, nonlinear techniques such as “time-dependent normal forms” and “order reduction using invariant manifolds” are employed to express various resonance conditions. It is found that the forms of new resonance conditions obtained in terms of ‘true characteristic exponents’ are similar to the forms of existing resonance conditions that involve natural frequencies.