Subharmonic Parametric Resonance Research Articles

Nonlinear Vibration Analysis for Stiffened Cylindrical Shells Subjected to Electromagnetic Environment

The nonlinear vibration behaviors of stiffened cylindrical shells under electromagnetic excitations, transverse excitations, and in‐plane excitations are studied for the first time in this paper. Given the first‐order shear deformation theory and Hamilton principle, the nonlinear partial differential governing equations of motion are derived with considering the von Karman geometric nonlinearity. By employing the Galerkin discretization procedure, the partial differential equations are diverted to a set of coupled nonlinear ordinary differential equations of motion. Based on the case of 1 : 2 internal resonance and principal resonance‐1/2 subharmonic parametric resonance, the multiscale method of perturbation analysis is employed to precisely acquire the four‐dimensional nonlinear averaged equations. From the resonant response analysis and nonlinear dynamic simulation, we discovered that the unstable regions of stiffened cylindrical shells can be narrowed by decreasing the external excitation or increasing the magnetic intensity, and their working frequency range can be expanded by reducing the in‐plane excitation. Moreover, the different nonlinear dynamic responses of the stiffened cylindrical shell are acquired by controlling stiffener number, stiffener size, and aspect ratio. The presented approach in this paper can provide an efficient analytical framework for nonlinear dynamics theories of stiffened cylindrical shells and will shed light on complex structure design in vibration test engineering.

Nonlinear Parametric Vibration Analysis of Radial Gates

In this work, based on dynamic characteristics of radial gates, a nonlinear differential equation of motion for the radial gate arm is established. Excitation conditions of principal parametric resonance and subharmonic parametric resonance are obtained by using multi-scale method and numerical method. The parameter analysis shows that: comparing with traditional calculation method of dynamic instability region division, the presented method is more suitable to analysis of parametric vibration for radial gates as considering the end moment, vibration duration and amplitude. The vibration amplitudes of arm increase with the increase of its length and excitation amplitude, as well as the decrease of arm inclination angle. Moreover, the parametric resonance is easier to be excited and its resonance region become wider with the initial end moment increasing. Since the vibration response of the arm is influenced by the nonlinear term in the equation, the damping effect is limited. Thus, energy transfer method (e.g. tuned mass damper) can be adopted to achieve vibration control.

Nonlinear vibrations of axially moving simply supported viscoelastic nanobeams based on nonlocal strain gradient theory

The lateral nonlinear vibration of an axially moving simply supported viscoelastic nanobeam is analysed based on nonlocal strain gradient theory. The proposed model includes the nonlocal parameters and material characteristic length parameters, investigating the two kinds of size effects of micro-nano beam structures. Firstly, the steady-state amplitude-frequency response of the subharmonic parametric resonance is analysed by a direct multiscale method, and the stability of the (non-) zero equilibrium solution determined by the Routh–Hurwitz criterion. Subsequently, the nonlinear frequencies of the nanobeams are calculated. Finally, several numerical examples are used to illustrate the influence of the scale parameters on the nonlinear vibration characteristics of nanobeams. The results show that when subharmonic parametric resonance occurs in the system, the (non-) zero equilibrium solution and the boundary of the instability region are markedly affected by the scale parameters. In addition, the nonlocal parameters soften the system, the material characteristic length parameters harden the system, and these softening and hardening effects are strengthened (or weakened) to varying degrees in the presence of nonlinearity.

Subharmonic parametric resonance of simply supported buckled beams

This study presents the nonlinear vibrations of simply supported beams that are subjected to a subharmonic resonance excitation of order two. The applied axial force has a static component to buckle the beam and a dynamic component for parametric excitation. The equation governing the nonlinear transverse deformations is an integral partial differential equation. The static counter part of the governing equation represents the nonlinear buckling problem that is solved for the critical buckling loads and exact buckled configurations. A reduced-order model based on the Galerkin’s discretization is obtained where a multi-mode discretization is assumed. The significance of the number of modes retained in the discretization is examined. It is found out that for simply supported beams, the second and higher modes have no contribution on either the static or the dynamic response of the beam. To study the nonlinear dynamics of the problem, a shooting method is used to locate periodic orbits and long-time numerical integration using fifth-order Runge–Kutta method is used to find the fast Fourier transforms of the response. A forward sweep, where the amplitude of the force is increased, while the frequency is kept fixed, is used to drive the beam. We find local attractors coexisting with snapthrough motion. Nonlinear instabilities such as period-doubling leading to chaos, symmetry-breaking, cyclic-fold, period-demultiplying, and jumping are reported. A bifurcation diagram showing all possible attractors near and far from the equilibrium positions is presented. This diagram is a helpful tool to design beams for a desired response especially for energy harvesting applications where high-amplitude vibration is needed.

High-order subharmonic parametric resonance of multiple nonlinearly coupled micromechanical nonlinear oscillators

High-order subharmonic parametric resonance of coupled microbeams is studied based on a model of multiple nonlinearly coupled micromechanical nonlinear oscillators. Of particular interest are the effects of nonlinear coupling between adjacent oscillators and elastic nonlinearity of individual oscillators. First, we analyze parametric resonance of a nonlinearly coupled nonlinear oscillator, and conduct instability analysis of steady-state solutions. Then we study parametric resonance of three nonlinearly coupled nonlinear oscillators. It is found that the nonlinear coupling between adjacent oscillators leads to the appearance of high-order subharmonic parametric resonance which, to the best of our knowledge, has received little attention in previous related works. In the present analysis, the effects of elastic nonlinearity of individual oscillators, damping and some loading parameters (such as dc and ac voltages) on high-order subharmonics are also examined in detail. Our results suggest that the range of excitation frequency for parametric resonance of coupled oscillators can be much larger than what the previous studies predicted, due to the appearance of high-order subharmonic parametric resonance. These results are believed to offer new and interesting insights into the ongoing research on nonlinear dynamics of coupled microbeams or nanobeams in MEMS or NEMS.

Parametric Resonances of Clamped-clamped Pipes Conveying Fluid by Incremental Harmonic Balance Method

Parametric resonances of clamped-clamped pipes conveying pulsating fluid are investigated by the incremental harmonic balance (IHB) method. The discretized differential equation of motion is solved and the amplitude-frequency response curves of the subharmonic parametric resonance are obtained by using the IHB method. The effect of the physical parameters of the system on the critical frequency of instability is discussed. The amplitude of the pipe obtained by IHB method is compared with the result of numerical simulation and it is shown that they are in good agreement. The typical example demonstrates that IHB is a precise and effective semi-analytical method for the fluid-structure interaction systems with strong nonlinearity.

High-order subharmonic parametric resonance of nonlinearly coupled micromechanical oscillators

This paper studies parametric resonance of coupled micromechanical oscillators under periodically varying nonlinear coupling forces. Different from most of previous related works in which the periodically varying coupling forces between adjacent oscillators are linearized, our work focuses on new physical phenomena caused by the periodically varying nonlinear coupling. Harmonic balance method (HBM) combined with Newton iteration method is employed to find steady-state periodic solutions. Similar to linearly coupled oscillators studied previously, the present model predicts superharmonic parametric resonance and the lower-order subharmonic parametric resonance. On the other hand, the present analysis shows that periodically varying nonlinear coupling considered in the present model does lead to the appearance of high-order subharmonic parametric resonance when the external excitation frequency is a multiple or nearly a multiple (≥3) of one of the natural frequencies of the oscillator system. This remarkable new phenomenon does not appear in the linearly coupled micromechanical oscillators studied previously, and makes the range of exciting resonance frequencies expanded to infinity. In addition, the effect of a linear damping on parametric resonance is studied in detail, and the conditions for the occurrence of the high-order subharmonics with a linear damping are discussed.

Equatorial inertial-parametric instability of zonally symmetric oscillating shear flows

This study revisits the problem of the zonally symmetric instability on the equatorial β-plane. Rather than treating the classical problem of a steady basic flow, it treats a sequence of problems of increasing complexity in which the basic flow is oscillatory in time with a frequency ω 0 . First, for the case of a homogeneous fluid, a time-oscillating barotropic shear forcing may excite a subharmonic parametric resonance of inertial oscillations. Because of the continuous distribution of inertial oscillation frequencies, this resonance occurs at critical inertial latitudes y c such that βy c = ±ω 0 /2. Next the effects of stratification, characterized by Brunt-Vaisala frequency N, are taken into account. It is shown analytically (in the asymptotic limit of a weak shear) that the forced temporal oscillation leads to an inertial-parametric instability, when a resonance condition between the basic flow frequency and the sum of two inertio-gravity free-mode frequencies is met. This inertial-parametric instability has a well-defined inviscid vertical scale selection favouring the high-vertical mode m c ∼ 7.45m 0 , where m 0 =βN/ω 2 0 is the equatorial vertical mode characteristic of frequency ω 0 . The viscous critical shear of inertial-parametric instability is lower than the steady inertial instability one. Finally, this type of setting naturally arises when the basic flow is considered to be an equatorial wave, so the problem is recast with the nonlinear adjustment of the vertically sinusoidal basic state of a zonally symmetric mixed Rossby-gravity (MRG) wave. Initial-value numerical simulations show that the same inertial-parametric instability exists leading to a resonant subharmonic excitation of free modes with vertical scales 7 and 8 times smaller than the basic-state wave. A simplified dynamical model of the instability is introduced, demonstrating that the oscillatory nature of the shear with height for the MRG wave necessarily implies a resonance between distinct vertical modes, the most unstable ones being modes 7 and 8 for a large enough Froude number of the MRG wave. The nonlinear action of the instability is described in terms of angular momentum and potential vorticity changes: a significant mixing due to the breaking of the excited high vertical modes creates a vertically averaged westward flow at the equator and extra-equatorial eastward flows. The ideas exposed may play a part in explaining layering phenomena and the latitudinal structure of the zonal flow in the equatorial oceans below the thermocline.

The Subharmonic Bifurcation of a Viscoelastic Circular Cylindrical Shell

In this paper the nonlinear dynamic behavior of a viscoelastic circular cylindrical shell under a harmonic excitation applied at both ends is studied. The modified Flugge partial differential equations of motion are reduced to a system of finite degrees of freedom using the Galerkin method. The equations are solved by the Liapunov–Schmidt reduction procedure. In order to study 1/2 and 1/4 subharmonic parametric resonance of the shell, the transition sets in parameter plane and bifurcation diagrams are plotted for a number of situations. Results indicate that, for certain static loads, the shell may display jumps due to the presence of dynamic periodic load with small amplitude. Additionally, different physical situations are identified in which periodic oscillating phenomena can be observed, and where 1/4 subharmonic parametric resonance is simpler than the 1/2-one.

Study of the parametric oscillator driven by narrow-band noise to model the response of a fluid surface to time-dependent accelerations

A stochastic formulation is introduced to study the large amplitude and high-frequency components of residual accelerations found in a typical microgravity environment (or g-jitter). The linear response of a fluid surface to such residual accelerations is discussed in detail. The analysis of the stability of a free fluid surface can be reduced in the underdamped limit to studying the equation of the parametric harmonic oscillator for each of the Fourier components of the surface displacement. A narrow-band noise is introduced to describe a realistic spectrum of accelerations, that interpolates between white noise and monochromatic noise. Analytic results for the stability of the second moments of the stochastic parametric oscillator are presented in the limits of low-frequency oscillations, and near the region of subharmonic parametric resonance. Based upon simple physical considerations, an explicit form of the stability boundary valid for arbitrary frequencies is proposed, which interpolates smoothly between the low frequency and the near resonance limits with no adjustable parameter, and extrapolates to higher frequencies. A second-order numerical algorithm has also been implemented to simulate the parametric stochastic oscillator driven with narrow-band noise. The simulations are in excellent agreement with our theoretical predictions for a very wide range of noise parameters. The validity of previous approximate theories for the particular case of Ornstein–Uhlenbeck noise is also checked numerically. Finally, the results obtained are applied to typical microgravity conditions to determine the characteristic wavelength for instability of a fluid surface as a function of the intensity of residual acceleration and its spectral width.