scholarly journals On the Internal Resonance of a Spinning Disk Under Space-Fixed Pulsating Edge Loads

2001 ◽  
Vol 68 (6) ◽  
pp. 854-859 ◽  
Author(s):  
Jen-San Chen
Keyword(s):  
Natural Frequency ◽  
Trivial Solution ◽  
Internal Resonance ◽  
Excitation Frequency ◽  
Single Mode ◽  
Multiple Scale ◽  
Hopf Bifurcations ◽  
Solution Path ◽  
Nontrivial Solutions ◽  
Spinning Disk

Internal resonance between a pair of forward and backward modes of a spinning disk under space-fixed pulsating edge loads is investigated by means of multiple scale method. It is found that internal resonance can occur only at certain rotation speeds at which the natural frequency of the forward mode is close to three times the natural frequency of the backward mode and the excitation frequency is close to twice the frequency of the backward mode. For a light damping case the trivial solution can lose stability via both pitchfork as well as Hopf bifurcations when frequency detuning of the edge load is varied. On the other hand, nontrivial solutions experience both saddle-node and Hopf bifurcations. When the damping is increased, the Hopf bifurcations along the trivial solution path disappear. Furthermore, there exists a certain value of damping beyond which no nontrivial solution is possible. Single-mode resonance is also briefly discussed for comparison.

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.

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.

Indeterminate trans-critical bifurcations in parametrically excited systems

1992 ◽  
Vol 439 (1907) ◽  
pp. 601-610 ◽  
Keyword(s):  
Natural Frequency ◽  
Trivial Solution ◽  
Recent Article ◽  
Excitation Frequency ◽  
Companion Paper ◽  
Subharmonic Resonance ◽  
Fractal Boundary ◽  
Parametrically Excited ◽  
Harmonic Resonance

This paper may be regarded as a companion paper to a recent article (M. S. Soliman & J. M. T. Thompson Proc . R . Soc . Lond A438, 511–518 (1992)) that addressed the phenomena of indeterminate sub-critical bifurcations in typical parametrically excited systems. This earlier work focused on the principal (subharmonic) resonance which occurs when the excitation frequency is almost twice the linear natural frequency. In contrast we here focus on the fundamental harmonic resonance that arises when the two frequencies are almost the same. We show that under this condition indeterminate trans-critical bifurcations with unpredictable outcome can arise when the trivial solution is located on a fractal boundary as it loses its stability at the bifurcation. The outcome is then unpredictable even under infinitesimal perturbations from the trivial solution.

scholarly journals On the Secondary Resonance of a Spinning Disk Under Space-Fixed Excitations

2004 ◽  
Vol 126 (3) ◽  
pp. 422-429 ◽  
Author(s):  
Jen-San Chen ◽  
Chin-Yi Hua ◽  
Chia-Min Sun
Keyword(s):  
Free Oscillation ◽  
Equations Of Motion ◽  
Excitation Frequency ◽  
Multiple Scale ◽  
Combination Resonance ◽  
Secondary Resonance ◽  
Spinning Disk ◽  
Final Response ◽  
Steady State Solutions ◽  
Special Case

We investigate the possibility of secondary resonance of a spinning disk under space-fixed excitations. Von Karman’s plate model is employed in formulating the equations of motion of the spinning disk. Galerkin’s procedure is used to discretize the equations of motion, and the multiple scale method is used to predict the steady state solutions. Attention is focused on the nonlinear coupling between a pair of forward (with frequency ωmn¯) and backward (with frequency ωmn) traveling waves. It is found that combination resonance may occur when the excitation frequency is close to 2ωmn+ωmn¯,ωmn+2ωmn¯, or 1/2ωmn¯+ωmn. When the combination resonance does occur, the frequencies of the free oscillation components are shifted slightly from the respective natural frequencies ωmn¯ and ωmn. The final response is therefore quasiperiodic. However, in the case when the excitation frequency is close to 1/2ωmn¯−ωmn, no combination resonance is possible. In the case when the excitation frequency is close to 1/3ωmn and 1/2ωmn¯−ωmn simultaneously, internal resonance between the forward and backward modes can occur. The frequencies of the free oscillation components are exactly three times and five times that of the excitation frequency. In this special case both saddle-node and Hopf bifurcations are observed.

scholarly journals Parametric Resonance of a Spinning Disk Under Space-Fixed Pulsating Edge Loads

1997 ◽  
Vol 64 (1) ◽  
pp. 139-143 ◽  
Author(s):  
Jen-San Chen
Keyword(s):  
Parametric Resonance ◽  
Single Mode ◽  
Multiple Scale ◽  
The Other ◽  
Gyroscopic System ◽  
First Order ◽  
Reflected Wave ◽  
Spinning Disk ◽  
Difference Type ◽  
Hill’S Equations

The parametric resonance of a spinning disk under a space-fixed pulsating edge load is investigated analytically. We assume that the radial edge load can be expanded in a Fourier series. With use of the orthogonality properties among the eigenfunctions of a gyroscopic system, the partial differential equation of motion is discretized into a system of generalized Hill’s equations in the first-order form. The method of multiple scale is employed to determine the conditions for single mode as well as combination resonances to occur. For any two modes, with s and υ nodal diameters, respectively, combination resonance occurs only when there exists a specific Fourier component cos kθ in the edge load, where s + υ = ±k. Sum type resonance occurs when both modes are reflected or both modes are nonreflected. On the other hand, difference type resonance occurs when one mode is reflected and the other is nonreflected. In applying this rule, the number of nodal diameters of a forward and a reflected wave is considered as negative. Several typical loadings are discussed, including uniform and concentrated edge loads.

A fifth-order multiple-scale solution for Hopf bifurcations

2004 ◽  
Vol 82 (31-32) ◽  
pp. 2723-2731 ◽  
Author(s):  
D. Dessi ◽  
F. Mastroddi ◽  
L. Morino
Keyword(s):  
Multiple Scale ◽  
Hopf Bifurcations ◽  
Fifth Order

Slow–Fast Behaviors and Their Mechanism in a Periodically Excited Dynamical System with Double Hopf Bifurcations

2021 ◽  
Vol 31 (08) ◽  
pp. 2130022
Author(s):  
Miaorong Zhang ◽  
Xiaofang Zhang ◽  
Qinsheng Bi
Keyword(s):  
Dynamical System ◽  
Hopf Bifurcation ◽  
Frequency Domain ◽  
Autonomous System ◽  
Single Mode ◽  
Phase Portraits ◽  
Hopf Bifurcations ◽  
Double Hopf Bifurcation ◽  
Exciting Frequency ◽  
Slow Passage

This paper focuses on the influence of two scales in the frequency domain on the behaviors of a typical dynamical system with a double Hopf bifurcation. By introducing an external periodic excitation to the normal form of the vector field with double Hopf bifurcation at the origin and taking the exciting frequency far less than the natural frequency, a theoretical model with two scales in the frequency domain is established. Regarding the whole exciting term as a slow-varying parameter leads to a generalized autonomous system, in which the equilibrium branches and their bifurcations with the variation of the slow-varying parameter can be derived. With the increase of the exciting amplitude, different types of bifurcations may be involved in the generalized autonomous system, resulting in several qualitatively different forms of bursting attractors, the mechanism of which is presented by overlapping the transformed phase portraits and the bifurcations of the equilibrium branches. It is found that the single mode 2D torus may evolve to the bursting attractors with mixed modes, in which the trajectory alternates between the single mode oscillations and the mixed mode oscillations. Furthermore, the transitions between the quiescent states and the spiking states may not occur exactly at the bifurcation points because of the slow passage effect, while Hopf bifurcations may cause different forms of repetitive spiking oscillations.

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.

Nonlinear Vibrations in an Elastic Structure Subjected to Vertical Excitation and Coupled With Liquid Sloshing in a Rectangular Tank

1997 ◽  
Author(s):  
Takashi Ikeda
Keyword(s):  
Theoretical Analysis ◽  
Natural Frequency ◽  
Liquid Surface ◽  
Excitation Frequency ◽  
Harmonic Balance Method ◽  
Liquid Sloshing ◽  
Elastic Structure ◽  
Vertical Excitation ◽  
Rectangular Tank ◽  
Resonance Curves

Abstract The nonlinear coupled vibrations of an elastic structure and liquid sloshing in a rectangular tank, partially filled with liquid, are investigated. The structure containing the tank is vertically subjected to a sinusoidal excitation. In the theoretical analysis, the resonance curves for the responses of the structure and liquid surface are presented by the harmonic balance method, when the natural frequency of the structure is equal to twice the natural frequency of one of the sloshing modes. From the theoretical analysis, the following predictions have been obtained: (a) Due to the nonlinearity of the fluid force, harmonic oscillations appear in the structure, while subharmonic oscillations occur on the liquid surface, (b) the shapes of the resonance curves markedly change depending on the liquid depth, and (c) when the detuning condition is slightly deviated, almost periodic oscillations and chaotic oscillations appear at certain intervals of the excitation frequency. These were qualitatively in good agreement with the experimental results.

Natural frequency modulation of a kinematically redundant planar parallel robot

2021 ◽  
pp. 095440622110524
Author(s):  
Jiawei Gu ◽  
Zhijiang Xie ◽  
Jian Zhang ◽  
Yangjun Pi
Keyword(s):  
Natural Frequency ◽  
Frequency Modulation ◽  
Natural Frequencies ◽  
Vibration Suppression ◽  
Excitation Frequency ◽  
Parallel Robot ◽  
Modulation Method ◽  
Kinematic Redundancy ◽  
Resonance Amplitude ◽  
Searching Method

When a parallel robot is equipped with kinematic redundancy, it has sufficient capabilities of natural frequency modulation through adjusting geometric configuration. To reduce resonance of a mechanism, this paper investigates the natural frequency modulation of a kinematically redundant planar parallel robot. A double-threshold searching method is proposed for controlling the inverse kinematics solution and keeping the natural frequencies away from the excitation frequency. The effectiveness of modulating the natural frequencies is demonstrated by comparing it with a non-modulation method. The simulation results indicate that, in all directions, the responses are coupled, and every order should be taken into consideration during natural frequency modulation. Compared to the non-modulation method, the proposed method can reduce the resonance amplitude to a certain extent, and the effect of vibration suppression is remarkable.

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