Non-Stationary Responses in Externally Excited Two-Degrees-of-Freedom Nonlinear Systems with 1: 2 Internal Resonance

2004 ◽  
Vol 10 (11) ◽  
pp. 1663-1697 ◽  
Author(s):  
Anil K. Bajaj ◽  
Patricia Davies ◽  
Bappaditya Banerjee
Keyword(s):  
Internal Resonance ◽  
Degrees Of Freedom ◽  
Numerical Study ◽  
Excitation Frequency ◽  
Primary Resonance ◽  
Single Mode ◽  
Two Degrees Of Freedom ◽  
Analytical Technique ◽  
Bifurcation Points ◽  
Coupled Mode

The dynamics of two-degrees-of-freedom dynamical systems with weak quadratic nonlinearities is analyzed in the neighborhood of bifurcation points when the excitation frequency varies slowly through the region of primary resonance. The two modes of vibration are in 1: 2 subharmonic internal resonance. The slowly evolving averaged equations are numerically studied for motions initiated in the vicinity of stationary responses, and observations are made about the nature of responses of the system near the transition from single-mode to coupled-mode solutions (pitchfork points), and near jump and Hopf bifurcations in the coupled-mode solutions. An analytical technique based on the dynamic bifurcation theory is developed to explain the numerical observations for passage through the bifurcations. A numerical study is carried out to determine the effects of system parameters on the dynamics near the pitchfork bifurcation points and results are compared with analytical and numerical descriptions of dynamics.

On Mode Coupling Analysis and Stability Regions to Energy Harvesting in a Two-Degrees-of-Freedom Portal Frame Platform

2017 ◽  
Author(s):  
Rodrigo T. Rocha ◽  
Jose M. Balthazar ◽  
Angelo M. Tusset ◽  
Vinicius Piccirillo ◽  
Frederic C. Janzen ◽  
...  
Keyword(s):  
Energy Harvesting ◽  
Piezoelectric Material ◽  
Internal Resonance ◽  
Degrees Of Freedom ◽  
Multiple Scales ◽  
Great Influence ◽  
Two Degrees Of Freedom ◽  
Stability Diagrams ◽  
Modal Coupling ◽  
Portal Frame

This work aims to study the modal coupling of a nonlinear two-degrees-of-freedom portal frame platform and a numerical analysis of the system with a nonlinear piezoelectric (PZT) material coupled to one of its columns, both externally base-excited. The nonlinear platform possesses two-to-one internal resonance between its two vibration modes and presenting the saturation phenomenon. The nonlinearities of the piezoelectric material are considered by a nonlinear mathematical relation. Here, it is considered an electro-dynamical shaker with harmonic output. The employed methodology to carry out the analysis of this work was: the application of the method of multiple scales to find the best configuration of the parameters, and to find some kind of phenomena due to the two-to-one internal resonance; several numerical simulations were carried out to optimize the energy harvesting through parametrical variations, bifurcation diagrams, stability diagrams. It will be analyzed: the influence of the nonlinearity of the piezoelectric material and of the electro-dynamical shaker on the energy harvesting. Results showed great influence of the nonlinearity of the material and using the electro-dynamical device. It was possible to gain considerably in energy harvesting and stability of the system.

Nonlinear Multi-Modal Panel Flutter Oscillations at Low Supersonic Speeds

2014 ◽  
Author(s):  
Vasily Vedeneev ◽  
Anastasia Shishaeva ◽  
Konstantin Kuznetsov ◽  
Andrey Aksenov
Keyword(s):  
Numerical Simulation ◽  
Limit Cycle ◽  
Time Domain ◽  
Internal Resonance ◽  
Gas Flow ◽  
Single Mode ◽  
Panel Flutter ◽  
Limit Cycle Oscillations ◽  
Plate Deformation ◽  
Coupled Mode

In this paper aeroelastic instability of a plate in a gas flow is investigated by direct time-domain numerical simulation. Plate deformation and gas flow are simulated in solid and fluid codes, respectively, with direct coupling between these codes. A series of simulations under different parameters has been conducted. Three types of the plate response have been observed: stability, static divergence and flutter. Depending on Mach number, two types of flutter were detected: single mode flutter and coupled mode flutter. At M = 1.8, a good correlation between the present study and the piston theory for coupled mode flutter has been obtained. At lower M, from 1 to 1.6, single mode flutter in 1st, 2nd and higher modes has been observed. Amplitudes and frequencies of flutter limit cycle oscillations have been studied. It is shown that limit cycle oscillations can occur in form of pure one-mode oscillations, or include 1:2 internal resonance, when fluttering mode excites another mode. In the region of Mach numbers from 1.3 to 1.5, where several plate modes are simultaneously unstable, transition from periodic to quasi-chaotic flutter oscillations occurs.

Transient Growth Before Coupled-Mode Flutter

2003 ◽  
Vol 70 (6) ◽  
pp. 894-901 ◽  
Author(s):  
P. J. Schmid ◽  
E. de Langre
Keyword(s):  
Degrees Of Freedom ◽  
Mechanical Energy ◽  
Linear Stability Theory ◽  
Transient Growth ◽  
Significant Discrepancy ◽  
Two Degrees Of Freedom ◽  
Coupled Mode ◽  
Flow Induced Vibrations ◽  
The One ◽  
Analytical Approaches

Transient growth of energy is known to occur even in stable dynamical systems due to the non-normality of the underlying linear operator. This has been the object of growing attention in the field of hydrodynamic stability, where linearly stable flows may be found to be strongly nonlinearly unstable as a consequence of transient growth. We apply these concepts to the generic case of coupled-mode flutter, which is a mechanism with important applications in the field of fluid-structure interactions. Using numerical and analytical approaches on a simple system with two degrees-of-freedom and antisymmetric coupling we show that the energy of such a system may grow by a factor of more than 10, before the threshold of coupled-mode flutter is crossed. This growth is a simple consequence of the nonorthogonality of modes arising from the nonconservative forces. These general results are then applied to three cases in the field of flow-induced vibrations: (a) panel flutter (two-degrees-of-freedom model, as used by Dowell) (b) follower force (two-degrees-of-freedom model, as used by Bamberger) and (c) fluid-conveying pipes (two-degree-of-freedom model, as used by Benjamin and Pai¨doussis) for different mass ratios. For these three cases we show that the magnitude of transient growth of mechanical energy before the onset of coupled-mode flutter is substantial enough to cause a significant discrepancy between the apparent threshold of instability and the one predicted by linear stability theory.

Performance Enhancement of an Autoparametric Vibration Absorber Using Multiple Pendulums

2000 ◽  
Author(s):  
Ashwin Vyas ◽  
Anil K. Bajaj ◽  
Arvind Raman
Keyword(s):  
Performance Enhancement ◽  
Excitation Frequency ◽  
Single Mode ◽  
Frequency Interval ◽  
Vibration Absorbers ◽  
Coupled Mode ◽  
Averaged Equations ◽  
Stability And Bifurcations ◽  
The Stability ◽  
Mode Response

Abstract This paper analyzes the dynamics of a resonantly excited single degree-of-freedom linear system coupled to an array of nonlinear autoparametric vibration absorbers (pendulums). The case of a 1:1:…:2 internal resonance between pendulums and the primary oscillator is studied. The method of averaging is used to obtain first-order approximations to the nonlinear response of the system. The stability and bifurcations of the equilibria of the averaged equations are computed. It is shown that the frequency interval of the unstable single-mode response, or the absorber bandwidth, can be enlarged substantially compared to that of a single pendulum absorber by adjusting individually the internal mistunings of the pendulums. Use of multiple pendulums is also shown to engender degenerate bifurcations as the coupled-mode response “switches” from one pendulum to the other with changing external excitation frequency. This results in a significant enhancement of the performance of autoparametric vibration absorbers.

Non-Ideal System With Quadratic Nonlinearities Containing a Two-to-One Internal Resonance

2016 ◽  
Author(s):  
Rodrigo T. Rocha ◽  
Jose M. Balthazar ◽  
D. Dane Quinn ◽  
Angelo M. Tusset ◽  
Jorge L. P. Felix
Keyword(s):  
Internal Resonance ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Excitation Frequency ◽  
Coupled System ◽  
Dynamical Behaviour ◽  
Main System ◽  
Weakly Coupled System ◽  
Ideal System ◽  
Quadratic Nonlinearities

The dynamical behaviour of a non-ideal three-degrees-of-freedom weakly coupled system associated with the quadratic nonlinearities in the equations of motion is investigated. The main system consists of two nonlinear mechanical oscillators coupling with quadratic nonlinearities and in which possess a 2:1 internal resonance between their translational movements. Under these conditions, we analyzed the response when a DC unbalanced motor with limited power supply (non-ideal system) excites the main system. When the excitation frequency is near to second natural frequency of the main system, saturation and jump phenomena are presented. Then, this work will analyze some torques of the motor, which causes the phenomena, and due to high amplitudes of motion will be possible to look for a way to harvest energy in a future work.

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