Andronov-Hopf bifurcations, Pomeau-Manneville intermittent chaos and nonlinear vibrations of large deployable space antenna subjected to thermal load and radial pre-stretched membranes with 1:3 internal resonance

2021 ◽  
Vol 144 ◽  
pp. 110719
Author(s):  
T. Liu ◽  
W. Zhang ◽  
Y. Zheng ◽  
Y.F. Zhang
Keyword(s):  
Internal Resonance ◽  
Thermal Load ◽  
Nonlinear Vibrations ◽  
Hopf Bifurcations ◽  
Space Antenna ◽  
Intermittent Chaos

Nonlinear Vibrations in a 2-Dimensional Protein Cluster Model with Linear Bonds

2000 ◽  
Vol 214 (1) ◽  
Author(s):  
E.G. Shidlovskaya ◽  
L. Schimansky-Geier ◽  
Yu.M. Romanovsky
Keyword(s):  
Active Site ◽  
Internal Resonance ◽  
Cluster Model ◽  
Active Sites ◽  
Nonlinear Vibrations ◽  
Nonlinear Phenomena ◽  
Two Dimensional ◽  
Dimensional Model ◽  
Two Dimensional Model

A two dimensional model for the substrate inside a pocket of an active site of an enzyme is presented and investigated as a vibrational system. The parameters of the system are evaluated for α-chymotrypsin. In the case of internal resonance it is analytically and numerically shown that the energy concentrated on a certain degree of freedom might be several times larger than in the non-resonant case. Additionally, the system is driven by harmonic excitations and again energy due to nonlinear phenomena is redistributed inhomogeneously. These results may be of importance for the determination of the rates of catalytic events of substrates bound in pockets of active sites.

scholarly journals Theoretical and Experimental Investigation of Two-to-One Internal Resonance in MEMS Arch Resonators

2018 ◽  
Vol 14 (1) ◽  
Author(s):  
Feras K. Alfosail ◽  
Amal Z. Hajjaj ◽  
Mohammad I. Younis
Keyword(s):  
Internal Resonance ◽  
Nonlinear Response ◽  
Multiple Scales ◽  
Hopf Bifurcations ◽  
Chaotic Motions ◽  
Different Types ◽  
Quadratic Nonlinearities ◽  
Multiple Scales Perturbation Method ◽  
Good Agreement ◽  
Perturbation Solutions

We investigate theoretically and experimentally the two-to-one internal resonance in micromachined arch beams, which are electrothermally tuned and electrostatically driven. By applying an electrothermal voltage across the arch, the ratio between its first two symmetric modes is tuned to two. We model the nonlinear response of the arch beam during the two-to-one internal resonance using the multiple scales perturbation method. The perturbation solution is expanded up to three orders considering the influence of the quadratic nonlinearities, cubic nonlinearities, and the two simultaneous excitations at higher AC voltages. The perturbation solutions are compared to those obtained from a multimode Galerkin procedure and to experimental data based on deliberately fabricated Silicon arch beam. Good agreement is found among the results. Results indicate that the system exhibits different types of bifurcations, such as saddle node and Hopf bifurcations, which can lead to quasi-periodic and potentially chaotic motions.

Bifurcations in Dynamical Systems With Internal Resonance

1978 ◽  
Vol 45 (4) ◽  
pp. 895-902 ◽  
Author(s):  
P. R. Sethna ◽  
A. K. Bajaj
Keyword(s):  
Dynamical Systems ◽  
Internal Resonance ◽  
Excitation Frequency ◽  
The Other ◽  
Hopf Bifurcations ◽  
System Of Equations ◽  
Jump Phenomena ◽  
Quadratic Nonlinearities

Dynamical systems with quadratic nonlinearities and exhibiting internal resonance under periodic excitations are studied. Two types of transition from stable to unstable motions are shown to occur. One kind are shown to be associated with jump phenomena while the other kind are shown to be associated with Hopf bifurcations of the averaged system of equations. In the case of the latter, the motions are shown to be amplitude modulated motions at the excitation frequency with the amplitude of modulation determined by the motion of a point on a torus.

Internal Resonance and Nonlinear Vibrations of Eccentric Rotating Composite Laminated Circular Cylindrical Shell

2019 ◽  
Author(s):  
Tao Liu ◽  
Wei Zhang ◽  
Yan Zheng ◽  
Yufei Zhang
Keyword(s):  
Cylindrical Shell ◽  
Differential Equations ◽  
Internal Resonance ◽  
Nonlinear Vibrations ◽  
Multiple Scales ◽  
Circular Cylindrical Shell ◽  
Equations Of Motion ◽  
Nonlinear Partial Differential Equations ◽  
Shear Deformation Theory ◽  
Internal Resonances

Abstract This paper is focused on the internal resonances and nonlinear vibrations of an eccentric rotating composite laminated circular cylindrical shell subjected to the lateral excitation and the parametric excitation. Based on Love thin shear deformation theory, the nonlinear partial differential equations of motion for the eccentric rotating composite laminated circular cylindrical shell are established by Hamilton’s principle, which are derived into a set of coupled nonlinear ordinary differential equations by the Galerkin discretization. The excitation conditions of the internal resonance is found through the Campbell diagram, and the effects of eccentricity ratio and geometric papameters on the internal resonance of the eccentric rotating system are studied. Then, the method of multiple scales is employed to obtain the four-dimensional nonlinear averaged equations in the case of 1:2 internal resonance and principal parametric resonance-1/2 subharmonic resonance. Finally, we study the nonlinear vibrations of the eccentric rotating composite laminated circular cylindrical shell systems.

scholarly journals Experimental Study on Nonlinear Vibrations of Fixed-Fixed Curved Beams

2016 ◽  
Vol 3 (1) ◽  
Author(s):  
Ajay Kumar ◽  
B. P. Patel
Keyword(s):  
Experimental Study ◽  
Internal Resonance ◽  
Nonlinear Vibrations ◽  
Excitation Amplitude ◽  
Curved Beam ◽  
Symmetric Mode ◽  
Curved Beams ◽  
Direct Excitation ◽  
First Time ◽  
Good Agreement

AbstractNonlinear dynamic behavior of fixed-fixed shallow and deep curved beams is studied experimentally using non-contact type of electromagnetic shaker and acceleration measurements. The frequency response obtained from acceleration measurements is found to be in fairly good agreement with the computational response. The travellingwave phenomenon along with participation of higher harmonics and softening nonlinearity are observed. The experimental results on the internal resonance of curved beams due to direct excitation of anti-symmetric mode are reported for the first time. The deep curved beam depicts chaotic response at higher excitation amplitude.

Phase-Locking and Chaotic Phenomena of Circular Mesh Antenna With 1:3 Internal Resonance

2019 ◽  
Author(s):  
Tao Liu ◽  
Wei Zhang ◽  
Yan Zheng ◽  
Xiangying Guo
Keyword(s):  
Fourth Order ◽  
Internal Resonance ◽  
Chaotic Dynamics ◽  
Circular Cylindrical Shell ◽  
Phase Locking ◽  
State Parameter ◽  
The Finite Element Method ◽  
First Order ◽  
Intermittent Chaos ◽  
Locking Phenomena

Abstract We study chaotic dynamics and the phase-locking phenomenon of the circular mesh antenna with 1:3 internal resonance subjected to the temperature excitation in this paper. Firstly, the frequencies and modes of the circular mesh antenna are analyzed by the finite element method, it is found that there is an approximate threefold relationship between the first-order and the fourth-order vibrations of the circular mesh antenna. Considering a composite laminated circular cylindrical shell clamped along a generatrix and with the radial pre-stretched membranes at both ends subjected to the temperature excitation, we study the nonlinear dynamic behaviors of the equivalent circular mesh antenna model based on the fourth-order Runge-Kutta algorithm, which are described by the bifurcation diagrams, waveforms, phase plots and Poincaré maps in the state-parameter space. It is found that there appear the Pomeau-Manneville type intermittent chaos. According to the topology evolution of phase trajectories, the phase-locking phenomena are found.

Experiments and Simulations in Travelling Wave and Non-Stationary Nonlinear Vibrations of Circular Cylindrical Shells

2016 ◽  
Author(s):  
Marco Amabili ◽  
Prabakaran Balasubramanian ◽  
Giovanni Ferrari
Keyword(s):  
Cylindrical Shell ◽  
Traveling Wave ◽  
Internal Resonance ◽  
Nonlinear Vibrations ◽  
Circular Cylindrical Shell ◽  
Cylindrical Shells ◽  
Excitation Amplitude ◽  
Added Mass ◽  
Travelling Wave ◽  
Wave Response

The nonlinear vibrations of a water-filled circular cylindrical shell subjected to radial harmonic excitation in the spectral neighborhood of the lowest resonances are investigated numerically and experimentally by using a seamless aluminum sample. The experimental boundary conditions are close to a simply supported circular cylindrical shell. Modal analysis reveals the presence of predominantly radial driven and companion modes in the low frequency range, implying the existence of a traveling wave phenomenon in the nonlinear field. Experimental studies previously carried out on cylindrical shells did not permit the complete identification of the characteristic traveling wave response and of its non-stationary nature. The added mass of the internal quiescent, incompressible and inviscid fluid results in an increase of the weakly softening behavior of the shell, as expected. The minimization of the added mass due to the excitation system and the negligible entity of the geometric imperfections of the shell allow the appearance of an exact one-to-one internal resonance between driven and companion modes. This internal resonance gives rise to a travelling wave response around the shell circumference and non-stationary, quasi-periodic vibrations, which are experimentally verified by means of stepped-sine testing with feedback control of the excitation amplitude. The same phenomenon is observed in the nonlinear response obtained numerically. The traveling wave is measured by means of state-of-the-art laser Doppler vibrometry applied to multiple points on the structure simultaneously. Previous studies present in literature did not show if this vibration can be chaotic for relatively small vibration amplitudes. Chaos is here observed in the frequency region where the travelling wave response is present for vibrations amplitudes smaller than the thickness of the shell. The relevant nonlinear reduced order model of the shell is based on the Novozhilov nonlinear shell theory retaining in-plane inertia and on an expansion of the displacements in terms of a properly chosen base of linear modes. An energy approach is used to obtain the nonlinear equations of motion, which are numerically studied (i) by using a code based on arc-length continuation and collocation method that allows bifurcation analysis in case of stationary vibrations, (ii) by a continuation code based on direct integration and Poincaré maps, which also evaluates the maximum Lyapunov exponent in case of non-stationary vibrations. The comparison of experimental and numerical results is particularly satisfactory throughout the various excitation amplitude levels considered. The two methods concur in describing the progressive development of the companion mode into a fully developed traveling wave and the subsequent appearance of quasi-periodic and eventually chaotic vibrations.

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