Influence Of Internal Resonance Research Articles

Nonlinear dynamics of a viscoelastic sandwich beam with parametric excitations and internal resonance

Nonlinear dynamical behaviors of an axially accelerating viscoelastic sandwich beam subjected to three-to-one internal resonance and parametric excitations resulting from simultaneous velocity and tension fluctuations are investigated. The direct method of multiple scales is adopted to obtain a set of first-order ordinary differential equations and associated boundary conditions. The frequency and amplitude response curves along with their stability and bifurcation are numerically studied. A great number of dynamic behaviors are presented in the form of phase portraits, time traces, Poincare sections, and FFT power spectra. Due to modal interaction, various periodic, quasiperiodic, and chaotic behaviors are displayed, depending on the initial conditions. The largest Lyapunov exponent is carried out to determine the midly chaotic response by the convergent form of exponents. Numerical results show various oscillatory behaviors indicating the influence of internal resonance and coupled effects of fluctuating axial velocity and tension.

Flapwise non-linear dynamics of wind turbine blades with both external and internal resonances

An investigation on the non-linear aeroelastic behavior of a wind turbine blade with both external and internal resonances is presented. The external resonance is a primary resonance that appears at the first flapwise mode; it can cause severe damage to blade. The internal resonance happens at the first two flapwise modes; it can enhance the energy transfer between two modes, and change blade dynamics in primary resonance. Three aspects including blade behavior in pure primary resonance (abbr. PPR; only considering external resonance), blade behavior in combination resonance (abbr. CR; including both external and internal resonances), and the influence of internal resonance (i.e. modal interaction) on external resonance are examined. A simple Bernoulli–Euler beam model, in which geometric non-linearity and unsteady aerodynamic force are considered, is used to describe the flapwise motion of blade. The perturbation method is applied to the infinite-degree-of-freedom discrete system, which is obtained from the original continuous system via Galerkin׳s method, to get dynamic responses. Amplitude–frequency curves of resonance modes in CR and PPR are derived, and the stability of the steady state motion of blade is judged. The strongest modal interaction between two resonance modes is taken into account, and then effects of modal interaction, excitation amplitude, damping and non-linearity on non-linear vibration properties of blade are analyzed. Also, influences of three designing parameters (inflow ratio, setting angle and coning angle) and two detuning on the non-linear behavior of blade are discussed for a concrete downwind turbine.

Internal Resonance Phenomena of the Jeffcott Rotor With Nonlinear Spring Characteristics

The Jeffcott rotor is a two-degree-of-freedom linear model with a disk at the midspan of a massless elastic shaft. This model, executing lateral whirling motions, has been widely used in the linear analyses of rotor vibrations. In the Jeffcott rotor, the natural frequency of a forward-whirling mode pf>0 and that of a backward-whirling mode pb<0 have the relation of internal resonance pf:pb=1:−1. Recently, many researchers analyzed nonlinear phenomena by using the Jeffcott rotor with nonlinear elements. However, they did not take this internal resonance relationship into account. Furthermore in many practical rotating machines, the effect of gyroscopic moments are relatively small. Therefore, the one-to-one internal resonance relationship holds approximately between forward and backward natural frequencies in such machinery. In this paper, nonlinear phenomena in the vicinity of the major critical speed and the rotational speeds of twice and three times the major critical speed are investigated in the Jeffcott rotor and rotor systems with a small gyroscopic moment. The influences of internal resonance on the nonlinear resonances are studied in detail. The following were clarified theoretically and experimentally: (a) the shape of resonance curves becomes far more complex than that of a single resonance; (b) almost periodic motions occur; (c) these phenomena are influenced remarkably by the asymmetrical nonlinearity and gyroscopic moment; and (d) the internal resonance phenomena are strongly influenced by the degree of the discrepancies among critical speeds. The results teach us that the usage of the Jeffcott rotor in nonlinear analyses of rotor systems may induce incorrect results.

Nonstationary Oscillations of a Nonlinear Rotor during Acceleration through the Major Critical Speed. (Influence of Internal Resonance).

The Jeffcott rotor is the most widely used theoretical model in the analysis of rotor systems.This model satisfies the condition of 1:(-1)internal resonance, and we clarified the effect of this internal resonance on nonlinear steady-state oscillations at the major, twice the major and three times the major critical speeds in previous papers.In this study, we investigated nonstationary oscillations during acceleration through the major critical speed in the Jeffcott rotor and other rotor systems which almost but not exactly satisfy the 1:(-1)internal resonance relation, and investigated the influence of internal resonance.The following are clarified theoretically and experimentally.(a)Nonstationary oscillations of the Jeffcott rotor and other rotor systems whose natural frequencies almost satisfy the 1:(-1)relation are influenced by the internal resonance.(b)It is easier to pass the major critical speed of a rotor system with internal resonance than that of a rotor system with no internal resonance.(c)When the system has slight discrepancies among critical speeds, that is, almost satisfies the relation of internal resonance, the rotor can pass the major critical speed very easily with small amplitude even if the acceleration is small.

Internal Resonance Phenomena of the Jeffcott Rotor. Major Critical Speed and Three Times the Major Critical Speed.

The Jeffcott rotor is the most widely used model in the analysis of nonlinear resonances of rotors. This model satisfies a condition of internal resonance. since a forward natural frequency and a backward natural frequency always satisfy the ratio of 1: (-1). Due to this internal resonance relation, some unique phenomena may appear in the resonances. However, in most of the previous studies on nonlinear resonances in the Jeffcott rotor, the authors paid no attention to this internal resonance relation. In the previous report, we clarified the effect of internal resonance at twice the maj or critical speed in the Jeffcott rotor. In this study, we deal those phenomena at the major critical speed and at three times the major critical speed theoretically and experimentally. Especially, the influences of internal resonance on the nonlinear oscillations of the Jeffcott rotor are investigated. The result shows that internal resonance relation must be considered in dealing with nonlinear vibrations of rotating machines by the Jeffcott rotor model.

Detection of the Point Where Rubbing Occurs in a Multisupported Rotor System

This paper describes a new method that detects directly the point where rubbing occurs during operation using data of vibrations. This method is based on the analysis of the nonlinear vibration caused by rubbing. A particular feature of the nonlinear vibration is that the amplitude of higher harmonics in the response becomes large in the vicinity of the point where rubbing occurs. The point where rubbing occurs can, thus, be detected by investigating the distribution of the amplitude of higher harmonics included in the response along the rotor. In the case of a rotor system having an overhanging disk or excited by internal resonance, the distribution of the amplitude occasionally shows another large peak in addition to the peak at the point where rubbing occurs. Even in this case, the presented method can be used to detect the point where rubbing occurs by investigating the eigenvalues and the mode of the rotor system and taking the influence of the overhanging disk into consideration, because the influence of internal resonance on the amplitude of higher harmonics is not great and, furthermore, the influence of the overhanging disk appears only in the vicinity of the overhanging disk.

Influence of internal resonance on the non-linear oscillations of a circular ring under primary resonance conditions

An accurate set of equations of motion, developed previously by the authors, is used to analyze the large amplitude forced oscillations of a thin circular ring. Qualitative and quantitative information about the primary response of a ring subjected to either a distributed harmonic in-plane or a distributed harmonic out-of-plane load is obtained by the method of multiple scales. The response due to the in-plane excitation of a ring in the presence of an internal resonance indicates that purely planar oscillations are unstable over a major portion of the primary resonant peak, and that non-planar oscillations exist over certain ranges of excitation frequencies. For certain values of excitation frequencies, the presence of unsteady motions with no exchange of energy between the in-plane and the out-of-plane modes is also indicated. On the other hand, internal resonance is shown to have no effect on the primary resonant peak due to the out-of-plane excitation. However, over certain other ranges of excitation frequencies, the in-plane bending mode is also excited (at the same order) and the resulting response is non-planar.