scholarly journals Forced Vibrations of Circular Plates: From Periodic to Chaotic Motions

2010 ◽  
Author(s):  
Cyril Touze´ ◽  
Olivier Thomas ◽  
Marco Amabili
Keyword(s):  
Internal Resonance ◽  
Numerical Study ◽  
Excitation Frequency ◽  
Free Edge ◽  
Forced Vibrations ◽  
Chaotic Regime ◽  
Circular Plates ◽  
Chaotic Motions ◽  
Resonant Modes ◽  
Transition Scenario

A numerical study of the transition from periodic to chaotic motions in forced vibrations of circular plates, is proposed. A pointwise harmonic forcing of constant excitation frequency Ω and increasing values of the amplitude is considered. Perfect and imperfect circular plates with a free edge are studied within the von Ka´rma´n assumptions for large displacements (geometric non-linearity). The transition scenario is observed for different excitation frequencies in the range of the first eigenfrequencies of the plate. For perfect plate with no specific internal resonance relationships, a direct transition to chaos is at hand. For imperfect plate tuned so as to fulfill specific internal resonance relations, a coupling between internally resonant modes is first observed. The chaotic regime shows an attractor of large dimension, and thus is studied within the framework of wave turbulence.

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.

scholarly journals ASYMMETRIC NON-LINEAR FORCED VIBRATIONS OF FREE-EDGE CIRCULAR PLATES. PART 1: THEORY

2002 ◽  
Vol 258 (4) ◽  
pp. 649-676 ◽  
Author(s):  
C. TOUZÉ ◽  
O. THOMAS ◽  
A. CHAIGNE
Keyword(s):  
Free Edge ◽  
Forced Vibrations ◽  
Circular Plates ◽  
Non Linear

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.

scholarly journals Measuring the modal parameters of a cultural heritage tower by using strong-motion signals

ACTA IMEKO ◽  
2018 ◽  
Vol 7 (3) ◽  
pp. 86 ◽  
Author(s):  
Mariella Diaferio ◽  
Dora Foti ◽  
Nicola Ivan Giannoccaro ◽  
Salvador Ivorra Ivorra
Keyword(s):  
Phase Shift ◽  
Environmental Conditions ◽  
Excitation Frequency ◽  
Strong Motion ◽  
Forced Vibrations ◽  
Mode Shapes ◽  
Modal Parameters ◽  
Vibration Modes ◽  
Experimental Campaign ◽  
Different Levels

This paper presents the dynamic experimental campaign carried out on a stocky masonry clock tower situated in the Swabian Castle of Trani (Italy). The main objective of this paper is, after estimating the main frequencies and vibration modes of the considered structure, defining the transmission of vibrations along the height of the tower by varying the forced frequency at the base. At this aim, short acceleration records have been acquired simultaneously in 20 points of the tower at different levels, due to a series of sinusoidal forced vibrations applied at the base by using a pneumatic shaker device specify designed for the tests. The proposed procedure permit to extract for each monitored point the amplitude of the sinusoidal component related to the excitation frequency and the phase shift due to the structure damping. The results of the proposed procedure are compared with the results of a classical operational modal analysis in environmental conditions in order to demonstrate that the short forced tests permit to classify the typology of the structure mode shapes.

Dynamics of high-Deborah-number entry flows: a numerical study

2011 ◽  
Vol 677 ◽  
pp. 272-304 ◽  
Author(s):  
A. M. AFONSO ◽  
P. J. OLIVEIRA ◽  
F. T. PINHO ◽  
M. A. ALVES
Keyword(s):  
Finite Volume Method ◽  
Finite Volume ◽  
Newtonian Fluid ◽  
Numerical Study ◽  
Chaotic Regime ◽  
Deborah Number ◽  
Weissenberg Number ◽  
Local Flow ◽  
Abrupt Contraction ◽  
Volume Method

High-elasticity simulations of flows through a two-dimensional (2D) 4 : 1 abrupt contraction and a 4 : 1 three-dimensional square–square abrupt contraction were performed with a finite-volume method implementing the log-conformation formulation, proposed by Fattal & Kupferman (J. Non-Newtonian Fluid Mech., vol. 123, 2004, p. 281) to alleviate the high-Weissenberg-number problem. For the 2D simulations of Boger fluids, modelled by the Oldroyd-B constitutive equation, local flow unsteadiness appears at a relatively low Deborah number (De) of 2.5. Predictions at higher De were possible only with the log-conformation technique and showed that the periodic unsteadiness grows with De leading to an asymmetric flow with alternate back-shedding of vorticity from pulsating upstream recirculating eddies. This is accompanied by a frequency doubling mechanism deteriorating to a chaotic regime at high De. The log-conformation technique provides solutions of accuracy similar to the thoroughly tested standard finite-volume method under steady flow conditions and the onset of a time-dependent solution occurred approximately at the same Deborah number for both formulations. Nevertheless, for Deborah numbers higher than the critical Deborah number, and for which the standard iterative technique diverges, the log-conformation technique continues to provide stable solutions up to quite (impressively) high Deborah numbers, demonstrating its advantages relative to the standard methodology. For the 3D contraction, calculations were restricted to steady flows of Oldroyd-B and Phan-Thien–Tanner (PTT) fluids and very high De were attained (De ≈ 20 for PTT with ϵ = 0.02 and De ≈ 10000 for PTT with ϵ = 0.25), with prediction of strong vortex enhancement. For the Boger fluid calculations, there was inversion of the secondary flow at high De, as observed experimentally by Sousa et al. (J. Non-Newtonian Fluid Mech., vol. 160, 2009, p. 122).

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.

Nonlinear Breathing Vibrations and Chaos of a Circular Truss Antenna with 1:2 Internal Resonance

2016 ◽  
Vol 26 (05) ◽  
pp. 1650077 ◽  
Author(s):  
W. Zhang ◽  
J. Chen ◽  
Y. Sun
Keyword(s):  
Differential Equation ◽  
Cylindrical Shell ◽  
Internal Resonance ◽  
Multiple Scales ◽  
Thermal Environment ◽  
Circular Cylindrical Shell ◽  
Numerical Results ◽  
Thermal Excitation ◽  
Response Curves ◽  
Chaotic Motions

This paper investigates the nonlinear breathing vibrations and chaos of a circular truss antenna under changing thermal environment with 1:2 internal resonance for the first time. A continuum circular cylindrical shell clamped by one beam along its axial direction on one side is proposed to replace the circular truss antenna composed of the repetitive beam-like lattice by the principle of equivalent effect. The effective stiffness coefficients of the equivalent circular cylindrical shell are obtained. Based on the first-order shear deformation shell theory and the Hamilton’s principle, the nonlinear governing equations of motion are derived for the equivalent circular cylindrical shell. The Galerkin approach is utilized to discretize the nonlinear partial governing differential equation of motion to the ordinary differential equation for the equivalent circular cylindrical shell. The case of the 1:2 internal resonance, primary parametric resonance and 1/2 subharmonic resonance is taken into account. The method of multiple scales is used to obtain the four-dimensional averaged equation. The frequency-response curves and force-response curves are obtained when considering the strongly coupled of two modes. The numerical results indicate that there are the hardening type and softening type nonlinearities for the circular truss antenna. Numerical simulation is used to investigate the influences of the thermal excitation on the nonlinear breathing vibrations of the circular truss antenna. It is demonstrated from the numerical results that there exist the bifurcation and chaotic motions of the circular truss antenna.

Sign in / Sign up

Export Citation Format

Share Document