Forced Mathieu Equation Research Articles

Responses of a Strongly Forced Mathieu Equation—Part 1: Cyclic Loading

AbstractThis work concerns the response of a damped Mathieu equation with hard cyclic excitation at the same frequency as the parametric excitation. A second-order perturbation analysis using the method of multiple scales unfolds resonances and stability. Superharmonic and subharmonic resonances are analyzed and the effect of different parameters on the responses are examined. While superharmonic resonances of order two have been captured by a first-order analysis, the second-order analysis improves the prediction of the peak frequency. Superharmonic resonances of order three are captured only by the second-order analysis. The order-two superharmonic resonance amplitude is of order ε0, and the order-three superharmonic amplitude is of order ε. As the parametric excitation level increases, the superharmonic resonance amplitudes increase. An nth-order multiple-scales analysis will indicate conditions of superharmonic resonances of order n + 1. At the subharmonic of order one-half, there is no steady-state resonance, but known subharmonic instability is unfolded consistently. Analytical expressions for resonant responses are presented and compared with numerical results for specific system parameters. The behavior of this system could be relevant to applications such as large wind-turbine blades and parametric resonators.

Responses of a Strongly Forced Mathieu Equation—Part 2: Constant Loading

AbstractThe present study deals with the response of a damped Mathieu equation with hard constant external loading. A second-order perturbation analysis using the method of multiple scales (MMS) unfolds resonances and stability. Non-resonant and low-frequency quasi-static responses are examined. Under constant loading, primary resonances are captured with a first-order analysis, but are accurately described with the second-order analysis. The response magnitude is of order ϵ0, where ϵ is the small bookkeeping parameter, but can become arbitrarily large due to a small denominator as the Mathieu system approaches the primary instability wedge. A superharmonic resonance of order two is unfolded with the second-order MMS. The magnitude of this response is of order ϵ and grows with the strength of parametric excitation squared. An nth-order multiple scales analysis under hard constant loading will indicate conditions of superharmonic resonances of order n. Subharmonic resonances do not produce a non-zero steady-state harmonic, but have the instability property known to the regular Mathieu equation. Analytical expressions for predicting the magnitude of responses are presented and compared with numerical results for a specific set of system parameters. In all cases, the second-order analysis accommodates slow time-scale effects, which enable responses of order ϵ or ϵ0. The behavior of this system could be relevant to applications such as large wind-turbine blades and parametric amplifiers.

Primary Parametric Amplification in a Weakly Forced Mathieu Equation

Abstract The present study deals with the response of a forced Mathieu equation with damping, with weak harmonic direct excitation at the same frequency as the parametric excitation. A second-order perturbation analysis using the method of multiple scales unfolds parametric amplification at primary resonance. The parametric effect on the primary resonance behavior occurs with a slow time scale of second-order, although the effect on the steady-state response is of order 1. As the parametric excitation level increases, the response at primary resonance stretches before becoming unbounded and unstable. Analytical expressions for predicting the response amplitudes are presented and compared with numerical results for a specific set of system parameters. Dependence of the amplification behavior, and indeed possible deamplification, on parameters is examined. The effect of parametric excitation on the response phase behavior is also presented.

Dynamic behavior of T-beam resonator with repulsive actuation

Electrostatic MEMS transducer driven by repulsive force is an attractive possibility and has advantages of avoiding the pull-in instability, tuning the natural frequency, and achieving high sensitivity by applying high enough voltages. In this work, a “T”-shaped beam, which is formed by attaching a secondary beam perpendicular to a primary cantilever at the tip, is introduced and its nonlinear dynamics is analyzed. A reduced-order model is derived from mode shapes formed from electromechanical coupling effects respectively. Generalized forms of forced Mathieu equation of motion are derived, and then, dynamic behaviors are investigated through the theory of multiple scales. The resonant responses, including both primary and principal parametric resonances, reveal softening behavior originating from quadratic and cubic nonlinearities in the governing equation. The behavior of the T-beam is compared with traditional cantilever structure. The resonance under repulsive force demonstrates that the T-beam has several advantages over a traditional cantilever: Lower natural frequency but higher resonant responses can improve the signal-to-noise ratio; with an attached micropaddle, the T-beam has a larger surface for absorption of targeted analytes for mass sensing. We conclude that an electrostatic MEMS resonator with a “T”-shaped beam is potentially appropriate for the new generations of sensors and actuators.

Subharmonics and superharmonics of the weak field in a driven two-level quantum system: Vibrational resonance enhancement.

We consider a quantum two-level system in bichromatic classical time-periodic fields, the frequency of one of which far exceeds that of the other. Based on systematic separation of timescales and averaging over the fast motion a reduced quantum dynamics in the form of a nonlinear forced Mathieu equation is derived to identify the stable oscillatory resonance zones intercepted by unstable zones in the frequency-amplitude plot. We show how this forcing of the dressed two-level system may generate the subharmonics and superharmonics of the weak field in the stable region, which can be amplified by optimization of the strength of the high frequency field. We have carried out detailed numerical simulations of the driven quantum dynamics to corroborate the theoretical analysis.

Response of linear parametric amplifiers with arbitrary direct and parametric excitations

The effect of parametric amplification, where parametric excitation is added to direct excitation, on the performance of a dynamical system is investigated. To model the system, the linear forced Mathieu equation is considered. In order to obtain the response of the system without restrictions such as requiring the system parameters to be small or the parametric excitation frequency to be exactly twice the direct excitation frequency, the method of varying amplitudes (MVA) is used. Employing the MVA, the response of the system and its maximum value for arbitrary values of the system parameters including the excitation frequencies are derived. It is shown that when the ratio between the excitation frequencies is rational, the maximum response of the system is highly dependent on the phase angle between the excitations. However, when the frequency ratio is irrational, the maximum response of the system is not dependent on the phase angle. Consequently, large values of response can be attained by using an irrational frequency ratio regardless of the phase angle. MVA results for peak responses show good agreement with numerical results obtained by direct integration of the equation of motion.

Bifurcation analysis of the levitation force MEMS actuators

This paper investigates the nonlinear dynamic behavior of a repulsive force actuator from the bifurcation perspective in the cases of primary and principal parametric resonances. The studied model consists of a specific configuration that produces out-of-plane repulsive (levitation) force. The governed equation of motion around static equilibrium is described in the form of a generalized forced Mathieu equation holding quadratic and cubic nonlinearities. While the particular system of interest is one with generated levitation force, the results can be applied to a wide class of MEMS in which the governing equation of motion is classified as the forced Mathieu equation. Coordinate transformation of the systems’ analytical solution from polar to Cartesian is done to preclude the singularity in origin. The frequency response is studied, and analytical expressions for equilibrium positions as fixed points are presented. Assessing the Jacobian matrix, the stability of each branch is examined, and the critical detuning values are defined in which the bifurcations occur. This criterion is then used to border the regions and illustrate the phase portraits of picked points within the regions. The results graphically show that inducing system by the initial conditions corresponding to the fixed points, triggers constant amplitude as well as angle, which ends up with periodic motion. Moreover, the outcome proves that perturbation analysis merely captures the periodic motions and fails to detect quasi-periodic motions. Our analysis provides a platform for further understanding of the repulsive electrostatic force, which is highly nonlinear. The results can be deployed in designing sensors with a wider detection range in which the amplitude jump is measured near the pitchfork bifurcation of principal parametric and cyclic fold bifurcation of primary resonances.

Theoretical and experimental investigations of the primary and parametric resonances in repulsive force based MEMS actuators

This work demonstrates an efficient approach to excite primary and parametric resonances of a Microelectromechanical system (MEMS) based repulsive force actuator. The electrostatic micro-actuator design consists of a particular arrangement of the actuating electrodes to create a repulsive like force, offering, therefore, a higher traveling range and eliminating pull-in instability. A generalized form of forced Mathieu equation of motion is first formulated and afterward a perturbation analysis, through the Method of Multiple Scales (MMS), is carried out to find an approximate analytical solution of the micro-actuator dynamic response. The steady-state response is subsequently acquired and then compared with experimental results in the vicinity of the microsystem's natural frequency (primary resonance) and then around twice the same frequency (principal parametric resonance). MMS based frequency, force, and transition curves are obtained for both cases with clearly shown distinct regions bordered by lines in which the bifurcation points are well traced. The results show that the parametric excitation actuation can be more efficient, requiring less power as compared to the primary resonance excitation. Moreover, unlike the classical method where the structure is vulnerable to the dynamic pull-in instability, this design provides a larger amplitude of motion while protecting the structure from any dynamic pull-in instability. The ability of this design in exciting both the parametric and primary resonances can be advantageous for several applications ranging from micro-resonator based logic system to memory-based micro-devices. Besides, this study provides an analytical tool for designing MEMS sensors with higher resolutions as the amplitude of the micro-system is no longer restricted to any geometrical limitation.

Estimation of Parameters in Forced Mathieu Equation with Damping Term

Estimation of Parameters in Forced Mathieu Equation with Damping Term

Stability of Ordinary Differential Equations with Colored Noise Forcing

We present a perturbation method for determining the moment stability of linear ordinary differential equations with parametric forcing by colored noise. In particular, the forcing arises from passing white noise through an $n$th order filter. We carry out a perturbation analysis based on a small parameter $\varepsilon$ that gives the amplitude of the forcing. Our perturbation analysis is based on a ladder operator approach to the vector Ornstein--Uhlenbeck process. We can carry out our perturbation expansion to any order in $\varepsilon$, for a large class linear filters, and for quite arbitrary linear systems. As an example we apply our results to the stochastically forced Mathieu equation.

Resonances of a Forced Mathieu Equation With Reference to Wind Turbine Blades

A horizontal axis wind turbine blade in steady rotation endures cyclic transverse loading due to wind shear, tower shadowing and gravity, and a cyclic gravitational axial loading at the same fundamental frequency. These direct and parametric excitations motivate the consideration of a forced Mathieu equation. This equation with cubic nonlinearity is analyzed for resonances by using the method of multiple scales. Superharmonic and subharmonic resonances occur. The effect of various parameters on the response of the system is demonstrated using the amplitude-frequency curve. The order-two superharmonic resonance persists for the linear system. While the order-two subharmonic response level is dependent on the ratio of parametric excitation and damping, nonlinearity is essential for the order-two subharmonic resonance. Order-three resonances are present in the system as well and, to first order, they are similar to those of the Duffing equation.

Energy-rate method and stability chart of parametric vibrating systems

The Energy-Rate method is an applied method to determine the transient curves and stability chart for the parametric equations. This method is based on the first integral of the energy of the systems. Energy-Rate method finds the values of parameters of the system equations in such a way that a periodic response can be produced. In this study, the Energy-Rate method is applied to the following forced Mathieu equation: + hy' + (1 - 2β + 2β cos (2rt)) y = 2β sin2 (rt) This equation governs the lateral vibration of a microcanilever resonator in linear domain. Its stability chart in the β-r plane shows a complicated map, which cannot be detected by perturbation methods.

Stability Region Control for a Parametrically Forced Mathieu Equation

We exhibit common features of how the size of parametric regions of stability for the Mathieu equation can be enlarged. The paper shows that the mechanisms for these changes via parametric forcing follow the pattern established earlier for the Arnold circle map which provides a discrete model for external forcing. The various types of behaviour of the standard Mathieu equation for a given set of parameters can be classified as having either (i) all solutions bounded, (ii) at least one unbounded solution, or (iii) periodic solutions of period -/-2π or -/-4π. The marginal case (iii) forms the boundary of the regions of stability and instability. We consider a parametric method for changing the shapes of the stability regions and show how maximally stable regions can be produced.

On acoustic cavitation of slightly subcritical bubbles

An analytic expression is obtained for the threshold acoustic pressure amplitude which causes gas bubbles that are slightly smaller than the critical size to undergo cavitation in a liquid. The derivation is based on the Rayleigh–Plesset equation which describes nonlinear bubble oscillations. In the limit where the steady part of the pressure field in the vicinity of the bubble is decreased to a value less than the vapor pressure inside the bubble, it is well known that there exists a critical bubble radius above which bubbles spontaneously cavitate. For bubbles which are below this critical size and would otherwise be stable, imposition of a time-harmonic acoustic pressure of the right frequency causes the bubbles to become unstable and undergo cavitation. The threshold value of this acoustic pressure is found by employing a nonlinear analysis which reduces the Rayleigh–Plesset equation to a damped and forced Mathieu equation for the case of such slightly subcritical bubbles.

On the kinematics of short waves in the presence of surface flows of larger scales

When the resonance condition is satisfied, i.e. that the local group velocity of short surface waves matches the local velocity associated with a larger-scale surface flow, it is known that the short waves are reflected or trapped by the flow. A typical example is the case of short surface waves propagating on long surface waves. By direct numerical resolution of the kinematic equations, some aspects of the reflection or trapping are first examined. Next, the effects of a second long wave on the trajectory of the short waves are considered. It is found that the trajectory is strongly distorted in general. Reflection still occurs, having a larger effect on the variation of the shortwave wavenumber than when only a single long wave is present. The entrapment becomes more sporadic. At short time intervals, a forced Mathieu equation is found to govern the short-wave development. This leads to a discussion on a more general physical context.