Parametric Resonance Region Research Articles

Parametric excitation of gyroscopic waves in Thomson-Delaney cells of the ocean of Jovian moon Europa

A possible mechanism (of parametric resonance type) of excitation of planetary gyroscopic waves in Thomson-Delaney cells of the ocean of Jovian moon Europa is considered. It is assumed that the basis of this mechanism is a variation of liquid depth in a cell caused by tidal oscillations under the action of gravitational perturbing influence of Galilean satellites of Jupiter. Such a model leads to a system of linear differential equations with periodic coefficients of the Hill’s type. Under some additional assumptions it changes over into a system of independent Mathieu equations. The regions of parametric resonance of this system are constructed.

Dynamics of an Autoparametric Pendulum‐Like System with a Nonlinear Semiactive Suspension

This paper presents vibration analysis of an autoparametric pendulum‐like mechanism subjected to harmonic excitation. To improve dynamics and control motions, a new suspension composed of a semiactive magnetorheological damper and a nonlinear spring is applied. The influence of essential parameters such as the nonlinear damping or stiffness on vibration, near the main parametric resonance region, are carried out numerically and next verified experimentally in a special experimental rig. Results show that the magnetorheological damper, together with the nonlinear spring can be efficiently used to change the dynamic behaviour of the system. Furthermore, the nonlinear elements applied in the suspension of the autoparametric system allow to reduce the unstable areas and chaotic or rotating motion of the pendulum.

Parametric Instability and Snap-Through of Partially Fluid-Filled Cylindrical Shells

The aim of the present paper is to study the parametric instability and snap-through buckling of an axially pre-loaded, partially fluid-filled cylindrical shell. The Donnell non-linear shallow shell theory is used to study the nonlinear vibrations of the shell. For this, the Galerkin method is used, together with a suitable expansion that takes into account the main nonlinear interactions, to discretize the shell. The resulting nonlinear equations of motion are solved by numerical integration. The fluid is assumed to be non-viscous and incompressible and its inertial effects on the shell surface are obtained by the potential flow theory. A detailed parametric analysis is carried out to demonstrate the influence of the fluid height within the shell on the parametric instability load and on the snap-through buckling load in the main parametric resonance region. Using bifurcations diagrams, the main bifurcation events associated with these stability boundaries are identified. The influence of the different types of bifurcation and fluid height on the safety is also discussed.

Extremum Seeking Speed and Heading Control Applied to Parametric Roll Resonance

Parametric roll is a dangerous resonance phenomenon for ships caused by time-varying restoring forces. The resulting large roll oscillations may endanger the ship's safety and possibly lead to capsizing. The susceptibility of container ships to parametric roll resonance has caused considerable research activities in the development of control and operational strategies to avoid large roll motions. In this paper, a novel approach to the active control of ships experiencing parametric roll resonance is presented. The methodology of extremum seeking control is used to iteratively determine the set-point for the wave encounter frequency in order to avoid the conditions for the onset and build-up of parametric roll. The desired trajectory for the wave encounter frequency is tracked by mapping it to the ship speed and heading angle by a control allocation, formulated as a constrained optimization problem, and consequently by implementing speed and heading controllers. The combined variation of the speed and heading angle is shown to efficiently drive the ship out of the parametric roll resonance region.

Stability regions for Mathieu equation with imperfect periodicity

We consider a mean square stability for Mathieu equation with a random phase modulation in parametric excitation. An efficient numerical scheme is proposed for obtaining the stability charts for this equation. The influence of the random phase modulation on the shape of parametric resonance regions is studied. It is found that this influence can lead to stabilization under some conditions. A comparison with a case of Gaussian parametric excitation is presented.

Conditions of Parametric Resonance in Periodically Time-Variant Systems With Distributed Parameters

Theoretical analysis of the stability problem for the control systems with distributed parameters shall be given. The approach to the analysis of such systems can be composed of two parts. First, the distributed parameter element is modeled by a frequency response function. Second, approximate conditions of parametric resonance are derived by a method of stationarization (describing functions of time-variant elements). The approach is illustrated by two examples. One is a robot-manipulator arm (distributed mechanical parameter system) controlled by a controller with a modulator/demodulator cascade (time-varying element). Another is an electromechanical transformer that consists of a constant current motor and a synchronous generator. Inductance between stator windings and the rotor of the synchronous generator serves as a periodical time-varying parameter, and a long electrical line plays the role of an element with distributed parameters. In both examples, dangerous (in terms of the first parametric resonance) regions for time-varying parameter are obtained theoretically and compared with simulation experiment.

Dynamical Casimir effect in oscillating media

We show that oscillations of a homogeneous medium with constant material coefficients produce pairs of photons. Classical analysis of an oscillating medium reveals regions of parametric resonance where the electromagnetic waves are exponentially amplified. The quantum counterpart of parametric resonance is an exponentially growing number of photons in the same parameter regions. This process may be viewed as another manifestation of the dynamical Casimir effect. However, in contrast to the standard dynamical Casimir effect, photon production here takes place in the entire volume and is not due to time dependence of the boundary conditions or material constants.

QUANTUM PARAMETRIC RESONANCE: ANALYTICAL STUDIES

The quantum version of parametric resonance of the harmonic oscillator is studied in terms of Feynman's propagator method and a discrete map. The eigenfunctions and eigenvalues of the Floquet operator are derived explicitly. The solutions follow the general laws: In the regions of parametric resonance the spectrum of the Floquet operator becomes continuous, while it is pointlike in the stable regimes. The Gaussian wave packet undergoes spreading exponentially and linearly in the two regimes, respectively.

The effect of vertical vibration on the onset of thermocapillary convection in a horizontal liquid layer

The effect of vertical vibration on the onset of Marangoni convection in a horizontal layer of a viscous incompressible uniform liquid with a free surface and a hard (solid) or soft (impermeable and stress-free) wall is investigated. In the case of harmonic vibration, a dispersion relation is constructed in explicit form using continued fractions. From this, equations are obtained for determining the critical values of the parameters for all three main types of loss of stability. Neutral curves of the monotonic and oscillatory instability are constructed, for fixed frequency and amplitude of the vibration, in the form of a graph of the Marangoni number against the wave number. The regions of parametric resonances, corresponding to synchronous and subharmonic modes are determined. The frequency values for which a high-frequency asymptotic form is reached are obtained. The long-wave Marangoni oscillatory instability is investigated, and it is shown that in this case the Marangoni numbers are negative and depend only on the Prandtl and Biot numbers.

Nonlinear Parametric Rolling in Regular Waves: An Approximate Analytical Solution for the Response Curve in the Region of First Parametric Resonance

An approximate analytical solution for the parametric roll response curve in the first parametric resonance zone is presented. Comparisons between analytical predictions and results obtained by means of numerical time domain simulations are performed in order to check the accuracy of the method. The use of a highly nonlinear 1.5-DOF model allows the disclosure of the presence of fold and pitchfork bifurcations. The method can be a simple and good aid both in the early design stage and for experimental planning. As an exercise showing the capabilities of the analytical methodology, the influence of damping parameters on the roll response is analyzed for a roll-on/roll-off (RO/RO) Pax.

Autoparametric vibrations of a nonlinear system with pendulum

Vibrations of a nonlinear oscillator with an attached pendulum, excited by movement of its point of suspension, have been analysed in the paper. The derived differential equations of motion show that the system is strongly nonlinear and the motions of both subsystems, the pendulum and the oscillator, are strongly coupled by inertial terms, leading to the so‐called autoparametric vibrations. It has been found that the motion of the oscillator, forced by an external harmonic force, has been dynamically eliminated by the pendulum oscillations. Influence of a nonlinear spring on the vibration absorption near the main parametric resonance region has been carried out analytically, whereas the transition from regular to chaotic vibrations has been presented by using numerical methods. A transmission force on the foundation for regular and chaotic vibrations is presented as well.

Characterization of electrostatically coupled microcantilevers

The use of tightly packed arrays of probes can achieve the much desirable goal of increasing the throughput of scanning probe devices. However the proximity of the probes induces coupling in their dynamics, which increases the complexity of the overall device. In this paper we analyze and model the behavior of a pair of electrostatically and mechanically coupled microcantilevers. For the common case of periodic driving voltage, we show that the underlying linearized dynamics are governed by a pair of coupled Mathieu equations. We provide experimental evidence that confirms the validity of the mathematical model proposed, which is verified by finite element simulations as well. The coefficients of electrostatic and mechanical coupling are estimated respectively by frequency identification methods and noise analysis. Finally, we discuss parametric resonance for coupled oscillators and include a mapping of the first order coupled parametric resonance region.

A Closure Procedure for Random Vibration Parametric Resonances

We investigate the mean-square stability for single-degree-of-freedom linear systems with random parametric excitation. The excitation is assumed to be of the form of a Gaussian stationary non-white process. We propose a new numerical approach to determine regions of parametric resonances based on a closure procedure for hierarchy of moment equations. Mean-square stability charts are obtained using the numerical analysis of eigenvalues for large-scale matrices. The results show three parametric resonances for narrow-band excitations.

Approximate analytical response curve for a parametrically excited highly nonlinear 1-DOF system with an application to ship roll motion prediction

The paper presents a model for a parametrically excited nonlinear 1-DOF damped mass-spring system. An approximate analytical technique is used to obtain the general polynomial whose roots give the response of the system in frequency domain when the parametric excitation acts in the first parametric resonance region. All the details needed for the implementation of the method are given. The model is suitable for the prediction of the parametrically excited roll motion of ships in longitudinal regular waves, thus some examples of application are shown to analyse the behaviour of two different hulls. The proposed approximate frequency domain analysis is shown to be able to disclose some nonlinear features of the ship response still not well known in the field of Naval Architecture.

Colored-noise-induced parametric resonance

The phenomenon of parametric resonance for harmonic oscillator is studied in the presence of a parametric colored Gaussian noise. Together with the main resonance, the higher-order ones are observed for the mean square of displacement. The influence of noise color on the shape of parametric resonance regions is investigated.

Parametric instability of a sphere immersed in a liquid and suspended from a string with varying tension

The regions of parametric resonance were determined for a sphere immersed in water and suspended from a string with varying tension.

Phase-induced stability in a parametric dimer.

We report results on a model of two coupled oscillators that undergo periodic parametric modulations with a phase difference straight theta. Being to a large extent analytically solvable, the model reveals a rich straight theta dependence of the regions of parametric resonance. In particular, the intuitive notion that antiphase modulations are less prone to parametric resonance is confirmed for sufficiently large coupling and damping. Some general results concerning synchronization properties in this system are presented. We also compare our results to a recently reported mean-field model of collective parametric instability, showing that the two-oscillator model captures much of the qualitative behavior of the infinite system.

Linear Dynamics of an Elastic Beam Under Moving Loads

The dynamic response of an Euler-Bernoulli beam under moving loads is studied by mode superposition. The inertial effects of the moving load are included in the analysis. The time-dependent equations of motion in modal space are solved by the method of multiple scales. Instability regions of parametric resonance are identified and the moving mass effect is shown to significantly affect the transient response of the beam. Importance of modal interaction arising out of the possible internal resonance is highlighted. While the external resonance is due to the gravity effects of the moving load, the parametric and internal resonance solely depends on the load mass parameter—ratio of the moving load mass to the beam mass. Numerical results show the influence of the load inertia terms on the beam response under either a single moving load or a series of moving loads. [S0739-3717(00)01703-7]

Resonances in linear one-degree-of-freedom systems with piecewise-constant parameters

A technique based on the composition of elementary phase fluxes is proposed for investigating parametric resonance in systems with “large” perturbations, described by second-order linear differential equations with periodic piecewise-constant coefficients. A monodromy matrix is given and a parametric resonance criterion is indicated, which takes into account the possibility of multiple multipliers and the action of dissipative forces. When there is a two-stage dependence of the coefficients on time during one period, regions of parametric resonance are obtained for different types of linear mechanical systems with one degree of freedom.

Width of parametric resonance regions for equations on a torus

We present new proofs of the theorem on the width of the forbidden regions for the Hill equation with a small potential and the theorem on the width of the parametric resonance regions for a first-order differential equation on a torus. These results are special cases of the theorem proved in this paper by the normal form method.