Internal Detuning Parameter Research Articles

Time-delayed control of a continuous flexible rotor via the saturation phenomenon

This paper aims to analyze and control nonlinear vibrations of a continuous spinning shaft, including a nonlinear saturation-based controller with time delays near the major critical speed and the presence of 2:1 internal resonance. The nonlinear modal interactions between the rotor and controller modes, as well as the primary and internal resonances, cause an energy bridge between the rotor and controller, so energy is transferred from the rotor to the controller, and a saturation phenomenon occurs. The nonlinear delay equations of motion are touched on analytically using the perturbation technique. Using the NSC gain coefficients, the saturation region and the frequency resonance bandwidth of the rotor are controlled. It is shown that, for negative values of the detuning parameter, as the internal detuning parameter increases, lower energies from the rotor are transferred to the controller, and the controller will be activated sooner. However, for positive values of the detuning parameter, by increasing the internal detuning parameter, the controller amplitude is increased, and the NSC will be later activated. Moreover, the stable zone of the rotor and NSC is studied.

Internal Resonances for the Heave Roll and Pitch Modes of a Spar Platform Considering Wave and Vortex Exciting Loads in Heave Main Resonance

The objective of this paper is to study the nonlinear coupling internal resonance of the heave, roll, and pitch response performance of a spar platform when their frequencies are in the ratio of 2:1:1 under wave and vortex exciting loads. The three degree-of-freedom (DOF) nonlinear coupled equations are established by considering a time-varying wet surface with a first-order wave force in heave and pitch and a vortex-induced force in roll. The first-order steady-state response is solved using the multi-scale method in heave main resonance. The multiple solutions of the motion equations are discussed using the analytic method and a numerical simulation. A sensitivity analysis is conducted to test the influence of the damping and internal detuning parameter. The regions of multiple solutions are found, and the jump phenomenon exists with the changes of the wave excitation. The regions of multiple solutions depend on the values of damping and detuning parameter.

Internal resonances for heave, roll and pitch modes of a spar platform considering wave and vortex-induced loads in the main roll resonance

We present a study of the nonlinear coupling internal resonance for the heave roll and pitch performance of a spar platform under the wave and vortex-induced loads when the ratio of the frequencies of heave, roll and pitch are approximately 2:1:1. In consideration of varying wet surface, the three DOFs nonlinear coupled equations are established for the spar platform under the effect of the first-order wave loads in the heave and pitch, and vortexinduced loads in the roll. By utilizing the method of multi-scales when the vortex-induced frequency is close to the natural roll frequency, the first-order perturbation solution is obtained analytically and further validated by the numerical integration. Sensitivity analysis is performed to understand the influence of the damping and the internal detuning parameter. Two cases with internal resonance are shown. The first case is that no saturation phenomenon exists under small vortex-induced loads. The first order perturbation solution illustrates that only the vortex-induced frequency motion in roll and the super-harmonic frequency motion in heave are excited. The second case is that the vortex-induced loads are large enough to excite the pitch and a saturation phenomenon in the heave mode follows. The results show that there is no steady response occurrence for some cases. For these cases chaos occurs and large amplitudes response can be induced by the vortex-induced excitation.

Triad mode resonant interactions in suspended cables

A triad mode resonance, or three-wave resonance, is typical of dynamical systems with quadratic nonlinearities. Suspended cables are found to be rich in triad mode resonant dynamics. In this paper, modulation equations for cable’s triad resonance are formulated by the multiple scale method. Dynamic conservative quantities, i.e., mode energy and Manley-Rowe relations, are then constructed. Equilibrium/dynamic solutions of the modulation equations are obtained, and full investigations into their stability and bifurcation characteristics are presented. Various bifurcation behaviors are detected in cable’s triad resonant responses, such as saddle-node, Hopf, pitchfork and period-doubling bifurcations. Nonlinear behaviors, like jump and saturation phenomena, are also found in cable’s responses. Based upon the bifurcation analysis, two interesting properties associated with activation of cable’s triad resonance are also proposed, i.e., energy barrier and directional dependence. The first gives the critical amplitude of high-frequency mode to activate cable’s triad resonance, and the second characterizes the degree of difficulty for activating cable’s triad resonance in two opposite directions, i.e., with positive or negative internal detuning parameter.

Primary resonance of a beam-rigid body microgyroscope

We present the non-linear dynamical features of a beam-rigid-body microgyroscope operating under the electrostatic force. A comprehensive system of discretized governing equations is solved using the method of multiple scales and the shooting method. For an amplitude-modulation gyro, the effects of the quality factor, AC voltage, and the rotation rate on the response are investigated and the fold-bifurcation points are identified. A frequency-modulation gyro is examined and in addition to the bifurcation points, the entrainment region is identified. The effects of a positive and a negative internal detuning parameter on the frequency-response of a frequency-modulation gyro are investigated.

Nonlinear dynamics of rotating box FGM beams using nonlinear normal modes

In this work an analysis is performed on the nonlinear planar vibrations of a functionally graded beam subjected to a combined thermal and harmonic transverse load in the presence of internal resonance. Adopting the direct perturbation MMS technique, the partial differential equations of motion of the beam are reduced to sets of first-order nonlinear modulation equations in terms of the complex modes of the beam. The assumption of steady-state values of centrifugal loads is evaluated. It has to be said that there is a lack of information about modeling of rotating box beams made of functionally graded materials (FGMs) under thermo-mechanical loads. The influence of the transverse load amplitude and the internal detuning parameter on the strength of nonlinear modal interaction is illustrated. It is also shown that the system exhibits periodic and quasiperiodic responses for a typical range of parameter values.

Autoparametric interaction of a liquid surface in a rectangular tank with an elastic support structure under 1:1 internal resonance

Autoparametric interaction of a liquid free surface in a rectangular tank with an elastic support structure, which is subjected to vertical excitation, is investigated. When the natural frequency of the structure is equal to the lowest natural frequency of liquid sloshing, this system is categorized as an autoparametric system with an internal resonance ratio 1:1. The structure is elastically supported so there is a higher possibility that the 1:1 internal resonance can be observed. The nonlinear theoretical analysis is conducted for a fluid assumed to be perfect in a tank with a finite liquid depth. The equations of motion for the first three sloshing modes are derived employing Galerkin’s technique and considering both the nonlinearity of the fluid motion, and the viscous damping effect. Then the theoretical frequency response curves for the harmonic oscillations of the structure and sloshing are determined using van der Pol’s method. The frequency response curves show that high amplitudes of the structure’s vibrations facilitate the liquid sloshing. Furthermore, the influence of the internal detuning parameter is investigated by showing the frequency response curves and bifurcation sets. Hopf bifurcations may occur followed by amplitude-modulated motions. The theoretical results are in quantitative agreement with the experimental data.

ONE-TO-ONE AUTO-PARAMETRIC RESONANCE IN SERPENTINE BELT DRIVE SYSTEMS

The coupled non-linear equations of motion of serpentine belt drive systems are solved using the direct multiple scales method. The entire hybrid system, which includes continuous belt spans, discrete pulleys, and tensioner arm, is divided into two subsystems. A one-to-one internal resonance combined with a primary external resonance is investigated. Solutions for the amplitude of non-trivial limit cycles are obtained. A stability analysis is performed on the steady state solutions. The quadratic non-linearity terms in the equations of motion are found to affect the belt drive systems significantly. It is shown that the direct multiple scales method yields better results for the system than the ordinary discretization multiple scales method. The effects of excitation frequencies, excitation amplitudes and the internal detuning parameter on dynamic responses are investigated.

SIMULTANEOUS PARAMETRIC AND INTERNAL RESONANCES IN SYSTEMS INVOLVING STRONG NON-LINEARITIES

The non-linear dynamic interaction between the impact of the first asymmetric liquid sloshing mode, represented by an equivalent pendulum, and the elastic structural dynamics is examined in the neighborhood of simultaneous occurrence of parametric and internal resonance conditions. The analytical modelling of the impact force is described by a power function. The present work considers both weak and strong non-linear forces on interaction. The method of multiple scales is used to determine the system response in the neighborhood of three sets of different resonance conditions. Under first- or mixed-mode parametric excitation, the normal modes interact through internal resonance condition. The system response is found to be strongly dependent on initial conditions. Depending on the initial conditions and internal detuning parameter, the response can be quasi-periodic or chaotic with irregular jumps between two unstable equilibria. In the presence of impact forces, the system preserves fixed response amplitude response within a small range of internal detuning parameter. Beyond that range, the response exhibits quasi-periodic motion mainly governed by the initial conditions, internal detuning parameter, damping ratios and excitation level. Under second- mode parametric excitation, the second mode reaches fixed response, depending on initial conditions, with no energy sharing with the first mode. However, the phase angles are found to vary with time. Under combination parametric resonance, and in the absence of impact forces, the response is found to be sensitive to initial conditions.

Nonlinear response of an initially buckled beam with 1:1 internal resonance to sinusoidal excitation

The nonlinear response of an initially buckled beam in the neighborhood of 1:1 internal resonance is investigated analytically, numerically, and experimentally. The method of multiple time scales is applied to derive the equations in amplitudes and phase angles. Within a small range of the internal detuning parameter, the first mode; which is externally excited, is found to transfer energy to the second mode. Outside this region, the response is governed by a unimodal response of the first mode. Stability boundaries of the unimodal response are determined in terms of the excitation level, and internal and external detuning parameters. Boundaries separating unimodal from mixed mode responses are obtained in terms of the excitation and internal detuning parameters. Stationary and non-stationary solutions are found to coexist in the case of mixed mode response. For the case of non-stationary response, the modulation of the amplitude depends on the integration increment such that the motion can be periodically or chaotically modulated for a choice of different integration increments. The results obtained by multiple time scales are qualitatively compared with those obtained by numerical simulation of the original equations of motion and by experimental measurements. Both numerical integration and experimental results reveal the occurrence of multifurcation, escaping from one well to the other in an irregular manner. and chaotic motion.

Structural Modal Multifurcation With International Resonance: Part 1—Deterministic Approach

The bifurcation and multifurcation in multimode interaction of nonlinear continuous structural systems is investigated. Under harmonic excitation the nonstationary response of multimode interaction is considered in the neighborhood of fourth-order internal resonance condition. The response dynamic characteristics are examined via three different approaches. These are the multiple scales method, numerical simulation, and experimental testing. The model considered is a clamped-clamped beam with initial static axial load. Under certain values of the static load the first three normal modes are nonlinearly coupled and this coupling results in a fourth-order internal resonance. The method of multiple time scales yields nonstationary response in the neighborhood of internal resonance. Within a small range of internal detuning parameter the third mode, which is externally excited, is found to transfer energy to the first two modes. Outside this region, the response is governed by a unimodal response of the third mode which follows the Duffing oscillator characteristics. The bifurcation diagram which represents the boundaries that separate unimodal and mixed mode responses is obtained in terms of the excitation level, damping ratios, and internal resonance detuning parameter. The domains of attraction of the two response regimes are also obtained. The numerical simulation of the original equations of motion suggested the occurrence of complex response characteristics for certain values of damping ratios and excitation amplitude. Both numerical integration and experimental results reveal the occurrence of multifurcation as reflected by multi-maxima of the response probability density curves.

Stochastic modal interaction in linear and nonlinear aeroelastic structures

The linear and autoparametric modal interactions in a three defree-of-freedom structure under wide band random excitation are examined. For a structure with constant parameters the linear response is obtained in a closed form. When the structure stiffness matrix involves random fluctuations, the governing equations of motion, in terms of the normal coordinates, are found to be coupled through parametric terms. The structural response is mainly governed by the condition of mean square stability. The boundary of stable-unstable responses is obtained as a function of the internal detuning parameter. The results of the linear system with constant parameters are used as a reference to measure the deviation of the system response when the nonlinear inertia coupling is included. In the neighbourhood of combination internal resonance the system random response is determined by using the Fokker Planck equation approach together with the Gaussian closure scheme. This approach results in 27 coupled first order differential equations in the first and second response moments. These equations are solved numerically. The response is found to deviate significantly from the linear solution when the system internal detuning is close to the exact internal resonance. The autoparametric interaction is found to depend significantly on the system damping ratios and a nonlinear coupling parameter. In the vicinity of combination internal resonance, the second normal mode mean square exhibits an increase associated with a corresponding decrease in the first and third normal modes.

Broad band random excitation of a two-degree-of-freedom system with autoparametric coupling

The response of a two-degree-of-freedom system with autoparametric coupling under the action of broad band random excitation is investigated. The system corresponds to the autoparametric vibration absorber and is also typical of many common structural configurations. A method based upon the Markov vector approach, together with an approximate treatment of third and higher statistical moments, is used to derive a set of fourteen coupled non-linear equations for the first and second moments of the system responses. A numerical integration procedure is used to obtain quantitative results for the system mean and mean square responses over a range of system parameters. The results show that large random motions of the coupled system may occur when the internal detuning parameter is close to the principal internal resonance, and that these motions may give rise to a suppression effect on the random motions of the main system. A feature of the results is that under conditions of internal resonance the random motions are found to be quasi-stationary, with steady oscillatory terms in the response moments. This suggests the possibility of entrainment of regular harmonic responses by the system random motions.