Harmonic Parametric Excitation Research Articles

Controlling the Chaotic Motions of an Airfoil with a Nonlinear Stiffness Using Closed-Loop Harmonic Parametric Excitation

The purpose of this research is to conduct a preliminary investigation into the possibility of suppressing the flutter and post-flutter (chaotic) responses of a two-dimensional self-excited airfoil with a cubic nonlinear stiffness (in torsion) and linear viscous damping via closed-loop harmonic parametric excitation. It was found that the initial configuration of the proposed control scheme caused the torsional/pitch dynamics to act as a nonlinear energy sink; as a result, it was identified that the mechanisms of vibration suppression are the resonance capture cascade and the short duration or isolated resonance capture. It is the isolated resonance capture that is responsible for the second-order-like damping and full vibration suppression of the aeroelastic system. The unforced and closed-loop system was subjected to random excitation to simulate aerodynamic turbulence. It was found that the random excitation suppresses the phase-coherent chaotic response, and the closed-loop system is susceptible to random excitation.

Stochastic stability of Duffing oscillator with fractional derivative damping under combined harmonic and Poisson white noise parametric excitations

The stochastic stability of a Duffing oscillator with fractional derivative damping under combined harmonic and Poisson white noise parametric excitations is investigated. A stochastic averaging method and the Khasminskii’s procedure are applied to evaluate the largest Lyapunov exponent. Then the asymptotic Lyapunov stability with probability one of the original system is determined approximately by using the largest Lyapunov exponent. Finally, the analytical results are verified by those from the Monte Carlo simulation of the original system.

Parametric excitation of an axisymmetric flow of a thin liquid film down a vertical fiber

We investigate axisymmetric flow of thin liquid films falling down the outer side of a vertical cylindrical surface oscillating harmonically in the axial direction. A set of two coupled nonlinear partial differential evolution equations in terms of the local instantaneous film thickness and volumetric flow rate is derived using the Galerkin method. A linear stability analysis of the time-periodic thickness-uniform base state is performed using the Floquet theory. We carry out numerical investigation of the nonlinear dynamics of the films in the framework of the set of evolution equations derived here. We find that the film evolution exhibits strange attractor dynamics ranging from coherent to fully irregular regimes. We show that harmonic parametric excitation affects the spatial topological structure of the interfacial wave and may modify its type from depression wave to elevation wave. The harmonic parametric excitation may also result in a significant decrease in the interfacial wave amplitude.

Nonlinear properties of the lattice network-based nonlinear CRLH transmission lines**Project supported by the National Natural Science Foundation of China (Grant Nos. 61201030, 60990322, and 60990320), the China Postdoctoral Science Foundation (Grant No. 2012M521048), and the Collegiate Natural Science Fund of Jiangsu Province, China (Grant No. 12KJB140010).

The nonlinear properties of lattice network-based (LNB) composite right-/left-handed transmission lines (CRLH TLs) with nonlinear capacitors are experimentally investigated. Harmonic generation, subharmonic generation, and parametric excitation are clearly observed in an unbalanced LNB CRLH TL separately. While the balanced design of the novel nonlinear TL shows that the subharmonic generation and parametric processes can be suppressed, and almost the same power level of the higher harmonics can be achieved over a wide bandwidth range, which are difficult to find in conventional CRLH TLs.

Stabilization via parametric excitation of multi-dof statically unstable systems

The problem of re-stabilization via parametric excitation of statically unstable linear Hamiltonian systems is addressed. An n-degree-of-freedom dynamical system is considered, at rest in a critical equilibrium position, possessing a pair of zero-eigenvalues and n−1 pairs of distinct purely imaginary conjugate eigenvalues. The response of the system to a small static load, making the zero eigenvalues real and opposite, simultaneous to a harmonic parametric excitation of small amplitude, is studied by the Multiple Scale perturbation method, and the stability of the equilibrium position is investigated. Several cases of resonance between the excitation frequency and the natural non-zero frequencies are studied, calling for standard and non-standard applications of the method. It is found that the parametric excitation is able to re-stabilize the equilibrium for any value of the excitation frequencies, except for frequencies close to resonant values, provided a sufficiently large excitation amplitude is enforced. Results are compared with those provided by a purely numerical approach grounded on the Floquet theory.

Stochastic response of a [formula omitted] oscillator subjected to combined harmonic and Poisson white noise excitations

The transient and stationary probability density functions (PDFs) of stochastic response of the ϕ6 Duffing oscillator under combined harmonic and external and parametric Poisson white noises excitations are investigated by the generalized cell mapping method in this paper. Based on the digraph analysis method, the global qualitative properties are obtained such as attractors, basins of attraction, basin boundaries, saddles and invariant manifolds. The evolutionary process of transient and stationary PDFs are shown based on the matrix analysis method. It is observed that there is a close relationship between evolutionary direction of PDF and the unstable manifold. Monte Carlo (MC) simulation is used to verify the accuracy of the matrix analysis method.

Synchronizations of rotating pendulums via self-learning terminal sliding-mode control subject to input nonlinearity

In this study, under the presentations of system uncertainties, external disturbances, and input nonlinearity in control, a self-learning terminal sliding-mode control scheme is adopted to control the synchronization and anti-synchronization between two chaotic rotating pendulums with different periods of harmonic parametric excitation. Without known bounds of the unmatched nonlinear dynamics, system uncertainties, and external disturbances, the proposed controller, which is associated with time-varying feedback gains, can achieve the control goals. Meanwhile, the feedback gains are not determined beforehand but are self-learning according to the learning rules. Some sufficient conditions for stable synchronizations are performed in the sense of the Lyapunov stability theorem. Besides, numerical simulations are provided to demonstrate the effectiveness of the present scheme.

DISORDER INDUCED ORDER IN AN ARRAY OF CHAOTIC DUFFING OSCILLATORS

This paper addresses the issue of disorder induced order in an array of coupled chaotic Duffing oscillators which are excited by harmonic parametric excitations. In order to investigate the effect of phase disorder on dynamics of the array, we take into account that individual uncoupled Duffing oscillator with a parametric excitation is chaotic no matter what the initial phase of the excitation is. It is shown that phase disorder by randomly choosing the initial phases of excitations can suppress spatio-temporal chaos in the system coupled by chaotic Duffing oscillators. When all the phases are the same and deterministic, the oscillators remain chaotic and asynchronous no matter what the common phase is. When driven asynchronously by introducing phase disorder, the oscillators coupled in the array appear more regular with increase of the amplitude of random phase, and the highest level of synchrony between them is induced by intermediate phase disorder, displaying a resonance like phenomenon caused from the transition of the coupled oscillators from chaos to periodic motion. Since varying the initial phases of excitations is more feasible than altering parameters intrinsic to the oscillators coupled in an array, this study provides a practical method for control and synchronization of chaotic dynamics in high-dimensional, spatially extended systems, which might have potential applications in engineering, neuroscience and biology.

A parametrically amplified MEMS rate gyroscope

This paper reports a parametrically amplified MEMS rate gyroscope. A practical parametric excitation scheme implemented using Digital Signal Processing (DSP) has been developed to enable either amplification of the primary mode of the gyroscope or amplification of the response to the applied angular velocity. The primary objective of this work is to improve the scale-factor and the signal to noise performance of the gyroscope. Parametric amplification of the primary mode of the gyroscope is achieved by frequency tracking and regulation of the amplitudes of the harmonic forcing and parametric excitation to maintain a desired parametric gain by closed loop PID control. Stable parametric amplification of the primary mode by a factor of 20 is demonstrated experimentally. This has important benefits regarding the minimisation of electrical feedthrough of the drive signal to the sense electrodes of the secondary mode. By taking advantage of the phase dependence of parametric amplification and the orthogonality of the Coriolis force and quadrature forcing, the response to the applied angular velocity may be parametrically amplified by applying excitation of a particular phase directly to the sensing mode. The major advantage of parametric amplification applied to MEMS gyroscopes is that it can mechanically amplify the Coriolis response before being picked off electrically. This is particularly advantageous for sensors where electronic noise is the major noise contributor. In this case parametric amplification can significantly improve the signal to noise ratio of the secondary mode by an amount approximately equal to the parametric amplification. Preliminary rate table tests performed in open loop demonstrate a magnification of the signal to noise ratio of the secondary mode by a factor of 9.5 and the scale-factor by 11. The excitation and control scheme have been implemented digitally using a DSP development board enabling very high sampling precision and execution speed necessary for gyroscopic applications.

TAMING CHAOTIC ARRAYS BY PARAMETRIC EXCITATION WITH PROPER RANDOM PHASE

A weak harmonic parametric excitation with random phase has been introduced to tame chaotic arrays. It has been shown that when the amplitude of random phase properly increases, two different kinds of chaotic arrays, unsynchronized and synchronized, can be controlled by the criterion of top Lyapunov exponent. The Lyapunov exponent was computed based on Khasminskii's formulation and the extension of Wedig's algorithm for linear stochastic systems. In particular, it was found that with stronger coupling the synchronized chaotic arrays are more controllable than the unsynchronized ones. The bifurcation analysis, the spatiotemporal evolution, and the Poincaré map were carried out to confirm the results of the top Lyapunov exponent on the dynamical behavior of control stability. Excellent agreement was found between these results.

Stability and bifurcations of hysteretic systems subjected to principal parametric excitation

Dynamic behavior of smooth hysteretic systems subjected to harmonic parametric excitation is investigated. Wen's differential equation model for hysteresis, which can describe a large class of hysteretic systems, is used. Employing the piecewise power series expression for hysteresis proposed in the previous paper, the method of multiple scales is applied for the case of principal parametric resonance to obtain the second-order approximate solutions and their stability. It is shown that the hysteretic systems are unstable inside the principal resonance region but the responses are bounded having symmetric periodic solutions. It is also shown that the systems are stable outside the region even if viscous damping does not exist because of Hopf bifurcation. Numerical integrations are also performed to illustrate the nonlinear resonant characteristics of the systems and confirmed that trivial stationary solutions and stable symmetric periodic solutions in principal resonance region bifurcate to produce a pair of non-symmetric periodic solutions. Stability regions with regard to excitation parameters are also illustrated. The theoretical and numerical results are compared to examine the validity of the present analysis.

NONLINEAR RESPONSES OF A STRING-BEAM COUPLED SYSTEM WITH FOUR-DEGREES-OF-FREEDOM AND MULTIPLE PARAMETRIC EXCITATIONS

The nonlinear dynamic responses of a string-beam coupled system subjected to harmonic external and parametric excitations are studied in this work in the case of 1:2 internal resonance between the modes of the beam and string. First, the nonlinear governing equations of motion for the string-beam coupled system are established. Then, the Galerkin's method is used to simplify the nonlinear governing equations to a set of ordinary differential equations with four-degrees-of-freedom. Utilizing the method of multiple scales, the eight-dimensional averaged equation is obtained. The case of 1:2 internal resonance between the modes of the beam and string — principal parametric resonance-1/2 subharmonic resonance for the beam and primary resonance for the string — is considered. Finally, nonlinear dynamic characteristics of the string-beam coupled system are studied through a numerical method based on the averaged equation. The phase portrait, Poincare map and power spectrum are plotted to demonstrate that the periodic and chaotic motions exist in the string-beam coupled system under certain conditions.

Experimental investigation of parametric and externally forced motion in resonant MEMS sensors

In this paper an excitation method employing both harmonic forcing and parametric excitation is applied to a resonant MEMS sensor in order to investigate and characterize the phenomena of parametric resonance and parametric amplification. The motivation for this research is that parametric excitation may be used to significantly reduce the total damping in MEMS sensors in a controllable manner. This is extremely pertinent to devices where the Q-factor is a principal factor in determining sensor performance. In this paper it is shown that, by adjusting the parametric excitation parameters (frequency, amplitude and phase) of an electrostatically actuated and sensed device, the gain of the frequency response function of a mode of vibration may be amplified. The amplification is quantified by the gain factor which is characterized experimentally. The instability regions defining the regions for parametric resonance are also characterized experimentally and compared to theoretical predictions. The boundaries of these instability regions define the thresholds for parametric resonance and play a crucial role in the design of the parametric amplifier.

Stochastic stability of Duffing oscillator with fractional derivative damping under combined harmonic and white noise parametric excitations

The stochastic stability of a Duffing oscillator with fractional derivative damping of order α (0 < α < 1) under parametric excitation of both harmonic and white noise is studied. First, the averaged Ito equations are derived by using the stochastic averaging method for an SDOF strongly nonlinear stochastic system with fractional derivative damping under combined harmonic and white noise excitations. Then, the expression for the largest Lyapunov exponent of the linearized averaged Ito equations is obtained and the asymptotic Lyapunov stability with probability one of the original system is determined approximately by using the largest Lyapunov exponent. Finally, the analytical results are confirmed by using those from a Monte Carlo simulation of the original system.

Stochastic averaging of strongly nonlinear oscillators with small fractional derivative damping under combined harmonic and white noise excitations

A stochastic averaging method for strongly nonlinear oscillators with lightly fractional derivative damping of order α (0<α<1) under combined harmonic and white noise external and (or) parametric excitations is proposed and then applied to study the first passage failure of Duffing oscillator with lightly fractional derivative damping of order 1/2 under combined harmonic and white noise excitations in the case of primary parametric resonance. Numerical results show that the proposed method works very well.

Simple parametric resonance in an electrostatically actuated microelectromechanical gyroscope: Theory and experiment

One of the major issues facing electrostatically actuated and sensed microelectromechanical systems (MEMS) sensors is electrical feed-through between the drive and the sense electrodes due to parasitic capacitances. This feed-through, in the case of a ‘tuned’ MEMS gyroscope, limits the sensor sensitivity. In the current paper, the first practical step towards demonstrating reduced feed-through using a combined harmonic forcing and parametric excitation scheme is demonstrated. The equation of motion for the primary mode of vibration of the electrostatically actuated MEMS ring gyroscope is shown to contain a stiffness modulating term which, when modulated at a frequency near twice the natural frequency of the mode, results in parametric resonance. A solution for the equation of motion is assumed, based on Floquet theory, and the method of harmonic balance is employed for analysis. Regions of stability and instability and the stability boundary demarcating the stable and unstable regions are determined. Frequency sweeps, centred on twice the measured resonant frequency of the primary mode, were performed at various values of voltage amplitudes of the parametric excitation and the parametric resonance was observed electrically at half the excitation frequency. This data were used to map the stability boundary of the parametric resonance. The theoretical and experimental stability boundaries are shown to demonstrate significant similarity.

Horizontal fast excitation in delayed van der Pol oscillator

This paper investigates the interaction effect of horizontal fast harmonic parametric excitation and time delay on self-excited vibration in van der Pol oscillator. We apply the method of direct partition of motion to derive the main autonomous equation governing the slow dynamic of the oscillator. The method of averaging is then performed on the slow dynamic to obtain a slow flow which is analyzed for equilibria and periodic motion. This analysis provides analytical approximations of regions in parameter space where periodic self-excited vibrations can be eliminated. A numerical study is performed on the original oscillator and compared to analytical approximations. It was shown that in the delayed case, horizontal fast harmonic excitation can eliminate undesirable self-excited vibrations for moderate values of the excitation frequency. In contrast, the case without delay requires large excitation frequency to eliminate such motions. This work has application to regenerative behavior in high-speed milling.

Effect of fast harmonic excitation on a self-excited motion in Van der Pol oscillator

Abstract This paper studies the effect of horizontal and vertical fast harmonic parametric excitation on periodic self-excited motion in Van der Pol pendulum. The method of direct partition of motion is first applied to derive the equation governing the slow motion of the oscillator which is the main equation of vibrational mechanics. Then, a perturbation method is used on the slow motion equation to obtain analytical approximation for the periodic solution and its frequency in the slow dynamic. A relationship between the frequency of the limit cycle and the frequency of the fast harmonic excitation is derived. It is shown that in the case where the point of suspension of the pendulum is subjected to horizontal fast harmonic excitation, the periodic self-excitation can disappear. In contrast, this periodic solution persists in the case of the vertical fast harmonic excitation. Numerical simulations on the slow flow and on the original equation confirms the analytical prediction.

Global synchronization of two parametrically excited systems using active control

Abstract In this paper, we apply an active control technique to synchronize a kind of two parametrically excited chaotic systems. Based on Lyapunov stability theory and Routh–Hurwitz criteria, some generic sufficient conditions for global asymptotic synchronization are obtained. Illustrative examples on synchronization of two Duffing systems subject to a harmonic parametric excitation and that of two parametrically excited chaotic pendulums are considered here. Numerical simulations show the validity and feasibility of the proposed method.

Triplon Modes of Puddles

Free fluctuations of the contact line of large drops ("puddles") of wavelength lambda > kappa(-1), the capillary length, cannot be seen on a solid substrate because even a small but finite hysteresis is enough to block these slow modes. We show here that vertical vibrations of the substrate (at frequency omegaE, acceleration Lambda) above a threshold amplitude Lambda(c) release the line and excite contour oscillations (triplons). We observe harmonic modes and parametric excitations at omegaE/2. We construct the phase diagram (Lambda, omegaE) of these subharmonic modes and we study their growth dynamics: they slow down near the threshold of the contour instability.