Internal Resonance Phenomena Research Articles

Measurement of High Frequency Impedance of ITER CS Model Coil and the Simulation Analysis of Internal Resonance Phenomenon

This paper describes the internal resonance phenomenon of a large scale superconducting coil. The frequency dependence of electric impedance defined at the coil terminal and the voltage distribution of the layer winding were measured at the ITER CS model coil under the superconducting state in advance of the pulse operation test. As a result, large change of the impedance and unbalance of the layer voltage were observed. The AC circuit analysis was carried out in order to evaluate the measurement results and to calculate the voltage of layer insulator. It was concluded that the higher voltage, which may appear on the layer insulation, is about a half of the coil terminal voltage in high frequency.

Three-dimensional non-linear coupling and dynamic tension in the large-amplitude free vibrations of arbitrarily sagged cables

This paper presents a model formulation capable of analyzing large-amplitude free vibrations of a suspended cable in three dimensions. The virtual work-energy functional is used to obtain the non-linear equations of three-dimensional motion. The formulation is not restricted to cables having small sag-to-span ratios, and is conveniently applied for the case of a specified end tension. The axial extensibility effect is also included in order to obtain accurate results. Based on a multi-degree-of-freedom model, numerical procedures are implemented to solve both spatial and temporal problems. Various numerical examples of arbitrarily sagged cables with large-amplitude initial conditions are carried out to highlight some outstanding features of cable non-linear dynamics by accounting also for internal resonance phenomena. Non-linear coupling between three- and two-dimensional motions, and non-linear cable tension responses are analyzed. For specific cables, modal transition phenomena taking place during in-plane vibrations and ensuing from occurrence of a dominant internal resonance are observed. When only a single mode is initiated, a higher or lower mode can be accommodated into the responses, making cable spatial shapes hybrid in some time intervals.

A GEOMETRICALLY NON-LINEAR MODEL OF ROTATING SHAFTS WITH INTERNAL RESONANCE AND SELF-EXCITED VIBRATION

A geometrically non-linear model of the rotating shaft is introduced, which includes Kárman non-linearity, non-linear curvature effects, large displacements and rotations as well as gyroscopic effects. Through applying Timoshenko-type assumptions, the shear effects are also included in the model. Convenient matrix descriptions are used in order to facilitate the analysis based on Galerkin and continuation methods. The model is used to analyze the phenomenon of internal resonance. The influence of some of the system's parameters on the amplitude and frequency of self-excited vibration is investigated.

Bifurcations and internal resonances in space-curved rods

The paper discusses a geometrically non-linear model of space-curved beams. The Timoshenko-type model is used, which includes the shear effects and rotatory inertia. The motion of the system is described by a non-linear matrix equation, which accounts for non-linearities up to the second order. To solve the non-linear problem, discretization methods based on the Galerkin method are used and the resulting non-linear eigenvalue problem is then solved by the continuation methods. The analysis of the discretized problem allows the study of the bifurcations of solutions. The post-bifurcation analysis is used to explain the phenomenon of internal resonance, which is of much importance in the prediction of the dynamic response of the analyzed systems. The approach is illustrated by numerical examples which consider the free vibration of beams and circular arches with different boundary conditions.

Chaotic Vibrations and Internal Resonance Phenomena in Rotor Systems. Case that the Critical Speeds of a Combination Resonance and Subharmonic Resonances of Order 1/2 are close.

In rotor systems, if the gyroscopic moment is small, a forward natural frequency and a backward natural frequency almost satisfy a condition of 1 : (-1) internal resonance. In such systems, the critical speeds of the forward and the backward subharmonic resonances of order 1/2 and the combination resonance are close in the vicinity of twice the major critical speed and internal resonance phenomena may appear. This study clarifies dynamical characteristics of nonlinear phenomena due to internal resonance, such as steady-state oscillations, almost periodic motions, bifurcation phenomena, and two kinds of chaotic vibrations, at the rotational speed of twice the major critical speed.

Effects of colored noise on internal stochastic resonance in a chemical model system

The phenomenon of internal stochastic resonance (SR) in a chemical model system was studied. The effects of colored noise with finite correlation time were compared with those of white noise without any time correlation. It was shown that the correlation time of noise could affect the peak value of internal SR and the location of the optimal noise intensity for resonance behavior. The application of correlation time in controlling internal SR was discussed.

Experimental Observation of Internal Signal Stochastic Resonance in the Belousov−Zhabotinsky Reaction

We show experimentally the phenomenon of internal signal stochastic resonance in the Belousov−Zhabotinsky reaction without external signal. The chemical reaction is placed in an excitable state near a Hopf bifurcation in a CSTR. When the flow rate is perturbed by stochastic noise, noise-induced oscillations are observed, and two closely spaced, large-amplitude spikes appear nearly periodically at the proper noise level. More importantly, analysis from the power spectra and interspike interval histogram of the time series output, the coherence of these noise-induced oscillations, is maximal at an optimal noise intensity, indicating the occurrence of internal signal stochastic resonance.

Harmonic Resonances of Continuous Rotor with Nonlinearity and Internal Resonances

Harmonic resonances in a continuous rotating shaft with distributed mass are discussed. The restoring force of the shaft has geometric stiffening nonlinearity due to the extension of the shaft centerline. The effect of a distributed lateral force, such as the gravity, is assumed. The possibility of the occurrences of harmonic resonances, the shapes of resonance curves, and internal resonance phenomena are investigated.

Chaotic Vibration and Internal Resonance Phenomena in Rotor Systems. (In the Vicinity of the Major Critical Speed).

Chaotic Vibration and Internal Resonance Phenomena in Rotor Systems. (In the Vicinity of the Major Critical Speed).

Internal Resonance Phenomena of an Asymmetrical Shaft. In the Vicinities of the Major Critical Speed and Critical Speed of Three Times the Major Critical Speed.

An asymmetrical shaft has been investigated for many years because unstable vibrations appear in this rotor system. However, when the natural frequencies hold simple integral relation exactly or approximately, an internal resonance may occur and the nonlinear phenomena change remarkably. In this study, the nonlinear oscillations and internal resonance phenomena of an asymmetrical shaft are investigated theoretically and experimentally in the vicinities of the major critical speed and critical speed of three times the major critical speed. Further, we clarified characteristics of nonlinear phenomena in the transition from the system with internal resonance to the system with no internal resonance.

Forced Oscillations of a Horizontal Continuous Rotor with Geometric Nonlinearity. Internal Resonance Phenomena at Harmonic Resonances and at Subharmonic Resonances of Order 1/3.

Harmonic resonances and subharmonic resonances of order 1/3 in a continuous rotating shaft with distributed mass are discussed. The restoring force of the shaft has geometric stiffening nonlinearity due to the extension of the shaft centerline. The effect of a distributed lateral force, such as the gravity, is assumed. The possibility of the occurrences of harmonic resonances and subharmonic resonances of order 1/3, the shapes of resonance curves, and internal resonance phenomena are investigated.

Internal Resonance of the Jeffcott Rotor. Critical Speed of Twice the Major Critical Speed.

The Jeffcott Rotor used widely in the analysis of resonancies is a two-degree-of-freedom model with a disk at the midspan of a massless-shaft. Natural frequency of a forward whirling mode pf and that of a backward whirling mode pb have the relation of ps:pb=1:-1. Consequently, subharmonic resonancies of a forward and a backward whirling modes and a combination resonance occur at the same rotational speed of twice the major critical speed and internal resonance phenomena appear. In this paper, nonlinear phenomena in the neighborhood of this rotational speed are investigated. The following are clarified theoretically and experimentally. (a) The shape of resonance curves becomes far more complex than that of a single resonance. (b) A quasi-periodic motion occurs. (c) These phenomena are influenced greatly by the misalignment in assembly and misattachment of rotor position.

Forced Oscillations of a Horizontal Continuous Rotor with Geometric Nonlinearity. Combination and Supercombination Resonances and Internal Resonance.

Combination and supercombination resonances in a continuous rotating shaft with distributed mass are discussed. The restoring force of the shaft has geometric stiffening nonlinearity due to the extension of the shaft center line. It is considered that a distributed lateral force works. The possibility of their occurrences, the shapes of resonance curves, and internal resonance phenomena are investigated.

Forced Oscillations of a Continuous Rotor with Geometric Nonlinearity. Subharmonic Resonances of Order 1/2 and Internal Resonance.

Subharmonic resonances of order 1/2 of a continuous rotating shaft with distributed mass are discussed. The restoring force of the shaft exhibits geometric stiffening nonlinearity due to the extension of the shaft center line. It is assumed that a distributed lateral force, such as the gravity, acts on the rotor. The possibility of the occurrence of subharmonic resonances, the shapes of resonance curves, and internal resonance phenomena are investigated.

Regulation of a Two-Degree-of-Freedom Structure Using Internal Resonance

In this paper, we present a first attempt at using an energy based control technique to regulate the oscillations of a flexible joint, flexible arm device, through computer simulation. This technique takes advantage of the Internal Resonance (IR) phenomenon. The plant is governed by two coupled linear differential equations. The control scheme is implemented by introducing two software based controllers which are coupled dynamically with the plant through a nonlinear feedback control law. At Internal Resonance, the nonlinear coupling generates an energy link between the plant and the controllers. Thus, energy is transferred from the plant to the controllers where two active damping mechanisms subsequently dissipate it. Here the response of the structure is regulated with a single input torque applied to one plant coordinate. The theoretical analysis is based on the two-variable expansion perturbation method. Thereafter, the analytical findings are verified numerically. Simulation results indicate that the IR control strategy is able to effectively quench the oscillations of the plant.

Internal resonance in a parametrically excited three-degree-of-freedom system with nonlinear spring characteristics. 1st Report. The case that the relation lpi+mpj=npk holds among the natural frequencies.

In a parametrically excited system whose natural frequencies are pi, pj, pk, and the parametric excitation frequency is Ω, an oscillation of a subharmonic type occurs when the relation pi∼- Ω/2 holds and that of a combination type occurs when the relation pi + pj ∼- Ω/2 holds. When the restoring forces are linear, these oscillations are unstable oscillations. But it is known that, in a nonlinear system, the nonlinearity modifies the growth of the unstable vibration and results in the steady oscillation with finite amplitude. In this paper, the internal resonance phenomena in such a system with nonlinear spring characteristics are investigated. Especially, we treat the case where the relation lpi + mpj = npk holds but each natural frequency is not an integer multiple of another natural frequency. By theoretical analysis, we clarified the followings : the kinds of parametric excitation terms and nonlinear terms which have an influence on oscillations ; induction of oscillation due to internal resonance ; occurrence of the almost periodic oscillations ; and so on.

Internal resonance in a parametrically excited two-degree-of-freedom system with nonlinear spring characteristics. (2nd report, On the combination type)

In a parametrically excited system with linear spring characteristics (pi, pj : natural frequency, Ω : frequency of parametric excitation), unstable vibration occur when the relations pi≒Ω/2 and pi+pj≒Ω hold. We call the former a subharmonic type and the latter a combination type. In the previous paper, we reported effects of internal resonance on parametrically excited oscillations of subharmonic type in a nonlinear two-degree-of-freedom system where pi : pj=1 : 2 and 1 : 3. In the present paper, we investigate the internal resonance phenomena in parametrically excited oscillations of combination type in the same system. It is clarified that the characteristics of this kind of oscillation change remarkably due to internal resonance. Especially, a different type of almost periodic oscillation which did not occur in the subharmonic type appears.

Nonlinear vibrations of a clamped rectangular plate with initial deflection and initial edge displacement— Part II: Experiment

To compare with the theoretical results of Part I, 1 detailed experimental results have been obtained on the nonlinear response of a clamped square plate under a uniformly distributed periodic load, which is subjected to various initial edge displacements. An aluminum test specimen with side length 200 mm and thickness 0·51 mm was used and tests were conducted by shaking the supporting frame with a constant peak acceleration and measuring the relative displacement of the plate to the frame. For the steady-state symmetric periodic responses, the theoretical results are found to be in good agreement with the experimental ones, with the exception of some quantitative differences in the large amplitude vibrations. Besides the responses theoretically predicted, a variety of non-symmetric and non-periodic responses were observed in connection with the internal resonance, combination resonance and dynamic snap-through phenomena, for which the present experimental results seem to provide effective data facilitating further theoretical analyses.

Non-linear vibrations of a clamped circular plate with initial deflection and initial edge displacement, part II: Experiment

To compare with the theoretical results of Part I, detailed experimental results have been obtained on the non-linear response of a clamped circular plate under a uniformly distributed periodic load, which is subjected to various initial edge displacements. An aluminum test specimen with radius 100 mm and thickness 0·51 mm was used and tests were conducted by shaking the supporting frame with a constant peak acceleration and measuring the relative displacement of the plate to the frame. For the steady state axisymmetric periodic responses, the theoretical results are found to be in good agreement with the experimental ones, with the exception of some quantitative differences in the large amplitude vibrations. Besides the responses theoretically predicted, a variety of non-axisymmetric and non-periodic responses were observed in connection with the internal resonance, combination resonance and dynamic snap-through phenomena, for which the present experimental results seem to provide effective data facilitating further theoretical analyses.

Non-linear vibrations of a clamped beam with initial deflection and initial axial displacement, Part II: Experiment

To compare with the theoretical results of Part I, detailed experimental results have been obtained on the non-linear response of a clamped beam under a uniformly distributed periodic load. A duralumin test specimen with breadth 26 mm, thickness 0·5 mm and length 150 mm was used and tests were conducted by pulsating the base frame with a constant peak accerelation and measuring the relative displacement of the beam to the frame. It is found that the theoretical results are generally is reasonable agreement with the experimental ones, the effects of initial axial displacements being included. Besides the results for comparison with theoretical predictions, various non-linear responses were observed in connection with the internal resonance, combination resonance and dynamic snap-through phenomena, for which the present experimental results seem to provide effective data facilitating further theoretical analyses.