Zones Of Parametric Instability Research Articles

Parametric Instability of an Asymmetric, Rotating Sandwich Beam

The parametric dynamic stability of an asymmetric, rotating sandwich beam and subjected to an axial pulsating load is investigated. A set of Hills equations are obtained from the non-dimensional equation of motion by the application of the general Galerkin method. The zones of parametric instability are obtained using Saito Otomi conditions. The influence of core-loss factor, geometric parameters and rotation parameters on the zones of instability are investigated.

Stability analysis of a two-layer elastic beam resting on a variable Pasternak foundation subjected to an axial pulsating load and thermal gradient

The stability analysis of a two-layer elastic beam resting on a variable Pasternak foundation subjected to an axial pulsating load and one dimensional thermal gradient is investigated for two different boundary condition. The equation of motion and boundary conditions are derived using extended Hamilton’s principle. A set of Hill’s equation is derived using generalized Galerkin’s method. The static buckling loads are determined from Hill’s equation. The zones of parametric instability are obtained using Saito-Otomi conditions. The effect of elastic foundation parameter, modulus ratio and thermal gradient parameter on the static buckling load and zones of parametric instability is analyzed. The analysis is done for two different boundary condition: Pinned-Pinned and Guided-pinned. The effects are presented as a series of graphs using required MATLAB program.

Parametric resonance in the dynamics of an elliptic vortex in a periodically strained environment

Abstract. The model of an elliptic vortex evolving in a periodically strained background flow is studied in order to establish the possible unbounded regimes. Depending on the parameters of the exterior flow, there are three classical regimes of the elliptic vortex motion under constant linear deformation: (i) rotation, (ii) nutation, and (iii) infinite elongation. The phase portrait for the vortex dynamics features critical points which correspond to the stationary vortex not changing its form and orientation. We demonstrate that, given superimposed periodic oscillations to the exterior deformation, the phase space region corresponding to the elliptic critical point experiences parametric instability leading to locally unbounded dynamics of the vortex. This dynamics manifests itself as the vortex nutates along the strain axis while continuously elongating. This motion continues until nonlinear effects intervene near the region associated with the steady-state separatrix. Next, we show that, for specific values of the perturbation parameters, the parametric instability is effectively suppressed by nonlinearity in the primal parametric instability zone. The secondary zone of the parametric instability, on the contrary, produces an effective growth of the vortex's aspect ratio.

Modal Interactions and Time Delay Control of a Parametrically Excited Nonlinear Composite Column

Modal interactions, stability analysis and time delay control of a composite column loaded by axial periodic force is presented in the paper. The simply supported column is composed of channel sections made of layers of a laminate with an arrangement which couples bending and twisting of the structure. The laminate coupling, together with a nonlinear geometrical displacement field result in a nonlinear model with parametric excitation. It is shown that for the selected configuration different mode couplings and parametric instability zones may occur. In order to stabilise the system a time delay displacement feedback control is proposed. An effect of the time delay control on the first principal parametric resonance zone is studied analytically and numerically.

Stability of an asymmetric tapered sandwich beam resting on a variable Pasternak foundation subjected to a pulsating axial load with thermal gradient

The static and parametric stability of an asymmetric tapered sandwich beam resting on a variable Pasternak foundation subjected to a pulsating axial load with thermal gradient under two different boundary conditions is investigated. The complete mathematical modeling of the system has been derived by the application of Hamilton’s principle which helps in getting the admissible path for the system. The equations of motion and boundary conditions obtained from the Hamilton’s equation are non-dimensionalized. A set of Hill’s equation are obtained from the non-dimensional equations of motion by the application of generalized Galerkin’s method. The zones of parametric instability are obtained using Saito–Otomi conditions. The effects of taper, elastic foundation, thermal gradient, core-loss factor, geometric parameter, modulus ratios and shear parameter on static buckling loads and parametric regions of instability are investigated.

Parametric Instability of an Asymmetric Sandwich Beam with Thermal Gradient under Various Boundary Conditions by Computational Method

Dynamic stability of a three-layered asymmetric sandwich beam with viscoelastic core subjected to a periodic axial load and a steady, one-dimensional thermal gradient is considered under five different boundary conditions. Equations of motion are derived by using Hamilton's principle. A generalized Galerkin's method is used to reduce the non-dimensional equations of motion to a set of coupled Hill's equations with complex coefficients. The zones of parametric instability are obtained using Saito-Otomi conditions. The effects of shear parameter, core loss factor, core thickness parameter, thermal gradient on the regions of parametric instability are studied.

Dynamic Stability Analysis of a Circularly Tapered Rotating Beam Subjected to Axial Pulsating Load and Thermal Gradient under Various Boundary Conditions

The dynamic stability of a circularly tapered rotating beam subjected to a pulsating axial external excitation with thermal gradient was studied for all possible combinations of clamped, guided, pinned, fixed, and free boundary conditions. The equations of motion and associated boundary conditions were obtained using the extended Hamilton’s principle. Then these equations of motion and the associated boundary conditions were non-dimensionalised. A set of Hill’s equations were obtained from the non-dimensional equations of motion by the application of the extended Galerkin method. The zones of parametric instability were obtained using Saito-Otomi conditions. The effects of various boundary conditions, thermal gradient, taper, and rotational speed on the regions of parametric instability were investigated and presented through a series of graphs. The results reveal that increasing rotational speed and taper have stabilizing effects, whereas increasing thermal gradient has a destabilizing effect for all boundary conditions of the beam.

Recurrence for motion of solitons of the Bose–Einstein condensate in a dynamic trap

We consider, analytically and numerically, the dynamics of an atomic Bose–Einstein condensate soliton bouncing in a magnetic or optical trap between two ideal barriers oscillating harmonically with coinciding frequency and opposite phases. There are zones of parametric instability of the soliton’s central position in the trap for certain ranges of the frequency of the barriers’ oscillations. We demonstrate the Fermi–Pasta–Ulam recurrence in the Fermi–Ulam scheme for solitons: the soliton’s deviations from the trap’s center grow and decay quasi-periodically in time. The results allow one to excite and investigate various structures of matter waves in traps and to manipulate their localization and dynamics.

Parametric instability analysis of deepwater top-tensioned risers considering variable tension along the length

Parametric instability of a riser is caused by fluctuation of its tension in time due to the heave motion of floating platform. Many studies have tackled the problem of parametric instability of a riser with constant tension. However, tension in the riser actually varies linearly from the top to the bottom due to the effect of gravity. This paper presents the parametric instability analysis of deepwater top-tensioned risers (TTR) considering the linearly varying tension along the length. Firstly, the governing equation of transverse motion of TTR under parametric excitation is established. This equation is reduced to a system of ordinary differential equations by using the Galerkin method. Then the parametric instability of TTR for three calculation models are investigated by applying the Floquet theory. The results show that the natural frequencies of TTR with variable tension are evidently reduced, the parametric instability zones are significantly increased and the maximum allowable amplitude of platform heave is much smaller under the same damping; The nodes and antinodes of mode shape are no longer uniformly distributed along the axial direction and the amplitude also changes with depth, which leads to coupling between the modes. The combination resonance phenomenon occurs as a result of mode coupling, which causes more serious damage.

Heating and thermal squeezing in parametrically driven oscillators with added noise

In this paper we report a theoretical model based on Green's functions, Floquet theory, and averaging techniques up to second order that describes the dynamics of parametrically driven oscillators with added thermal noise. Quantitative estimates for heating and quadrature thermal noise squeezing near and below the transition line of the first parametric instability zone of the oscillator are given. Furthermore, we give an intuitive explanation as to why heating and thermal squeezing occur. For small amplitudes of the parametric pump the Floquet multipliers are complex conjugate of each other with a constant magnitude. As the pump amplitude is increased past a threshold value in the stable zone near the first parametric instability, the two Floquet multipliers become real and have different magnitudes. This creates two different effective dissipation rates (one smaller and the other larger than the real dissipation rate) along the stable manifolds of the first-return Poincaré map. We also show that the statistical average of the input power due to thermal noise is constant and independent of the pump amplitude and frequency. The combination of these effects causes most of heating and thermal squeezing. Very good agreement between analytical and numerical estimates of the thermal fluctuations is achieved.

Signal-to-noise ratio in parametrically driven oscillators

We report a theoretical model based on Green's functions and averaging techniques that gives analytical estimates to the signal-to-noise ratio (SNR) near the first parametric instability zone in parametrically driven oscillators in the presence of added ac drive and added thermal noise. The signal term is given by the response of the parametrically driven oscillator to the added ac drive, while the noise term has two different measures: one is dc and the other is ac. The dc measure of noise is given by a time average of the statistically averaged fluctuations of the displacement from equilibrium in the parametric oscillator due to thermal noise. The ac measure of noise is given by the amplitude of the statistically averaged fluctuations at the frequency of the parametric pump. We observe a strong dependence of the SNR on the phase between the external drive and the parametric pump. For some range of the phase there is a high SNR, while for other values of phase the SNR remains flat or decreases with increasing pump amplitude. Very good agreement between analytical estimates and numerical results is achieved.

Squeezing of thermal noise in a parametrically-driven oscillator

In this article we report a theoretical model based on Green's functions and averaging techniques that describes the dynamics of parametrically-driven mechanical resonators under the action of thermal noise. Quantitative estimates for quadrature thermal noise squeezing near the first parametric instability zone of the oscillator are given. Furthermore, the parameter space where these phenomena occur is presented. Very good agreement between analytical estimates and numerical results is achieved.

Cooling, heating, and thermal noise squeezing in a parametrically driven resonator

In this paper we report a theoretical model based on Green’s functions and averagingtechniques that describes the dynamics of parametrically driven mechanical resonatorsunder the action of thermal noise. Quantitative estimates for cooling, heating, andquadrature thermal noise squeezing near the first parametric instability zone of theoscillator are given. Furthermore, the parameter space where these phenomena occur ispresented. Very good agreement between analytical estimates and numerical results isachieved.

Dynamic stability of a viscoelastically supported sandwich beam

The parametric dynamic stability of an asymmetric sandwich beam with viscoelastic core on viscoelastic supports at the ends and subjected to an axial pulsating load is investigated. A set of Hill`s equations are obtained from the non-dimensional equations of motion by the application of the general Galerkin method. The zones of parametric instability are obtained using Saito-Otomi conditions. The effects of shear parameter, support characteristics, various geometric parameters and excitation force on the zones of instability are investigated.

Parametric instability of a dual-cored sandwich beam

The present paper is concerned with the study of the parametric instability of a dual-cored symmetric sandwich beam subjected to an axial pulsating load. Numerical results are obtained for various combinations of the system parameters such as the thickness, core loss-factor, shear and elastic parameters and the effects of these on the zones of parametric instability are studied.

Parametric instability of a symmetric sandwich beam with higher order effects

The parametric instability of a pinned-pinned, three-layered symmetric sandwich beam with viscoelastic core, subjected to an axial pulsating load at one end, is considered. The analysis includes the higher order effects, viz. extensional and rotary inertias as well as bending and shear in all the layers. Numerical results are obtained with both the refined (including higher order effects) and simple (excluding these effects) analyses and the extent of applicability of the latter analysis is explored. The influence of the shear parameter upon the fundamental static buckling load as well as the effects of the relevant system parameters such as the shear, core thickness and static load parameters, the core loss factor etc., upon the regions of parametric instability are investigated. It is seen that to obtain the zones of parametric instability of short sandwich beams accurately, one must use the refined theory, since the simple theory predicts the number, as well as the locations, of the zones of instability along the frequency axis quite erroneously.

Forced Nonlinear Oscillations of an Autoparametric System—Part 1: Periodic Responses

Forced oscillations of a two degree-of-freedom autoparametric system are studied with moderately high excitations. The approximate results obtained by the method of harmonic balance are found to be satisfactory by comparing with those obtained by numerical integration. In the primary parametric instability zone, separate regions of stable and unstable harmonic solutions are obtained. In the regions of unstable harmonic solutions, depending on the forcing amplitude and frequency, the solutions may be amplitude modulated or completely nonperiodic. In the latter case the numerical integrations do not converge.

Parametric Resonance Oscillations of Flexible Slender Cylinders in Harmonically Perturbed Axial Flow—Part 1: Theory

This paper studies theoretically the dynamical behavior of a flexible slender cylinder in pulsating axial flow. The dynamics of the system in steady, unperturbed flow are examined first. For various sets of boundary conditions the eigenfrequencies of the system at any given flow velocity are determined, and the critical flow velocities are established, beyond which buckling (divergence) would occur. The behavior of the system in pulsating flow is examined next, establishing the existence of parametric resonances. The effects of the mean flow velocity, boundary conditions, dissipative forces, and virtual (hydrodynamic) mass on the extent of the parametric instability zones are then discussed.

Dynamic stability of pipes conveying fluid

This paper deals with the dynamics and stability of flexible pipes containing flowing fluid, where the flow velocity is either entirely constant, or with a small harmonic component superposed. An extensive historical review of the subject is given. In the case of constant flow velocity, the dynamics of the system is examined in a general way and it is shown that conservative systems are subject not only to buckling (divergence) at sufficiently high flow velocities, but also to oscillatory instabilities (flutter) at higher flow velocities. Also presented are some new results for cases of systems subjected to internal dissipative forces. In the case of harmonically varying flow velocity, the equation, of motion derived here exposes an error in a previous derivation. Stability, maps are presented for parametric instabilities, computed by Bolotin's method, for pipes with pined or clamped ends, as well as for cantilevered pipes. It is found that the extent of the instability regions increases with flow velocity for clamped-clamped and pinned-pinned pipes, while a more complex behaviour obtaines in the case of cantilevered pipes. In all cases, dissipation reduces the extent of, or entirely eliminates, parametric instability zones.