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Using the asymptotic method described in the monograph referred to in the title, expressions are obtained that determine the boundaries of three regions of parametric resonance of the damped homogeneous Mathieu equation. The formulas for the boundaries of the second and third regions, validated by solving the equation numerically, differ significantly from the known ones obtained in the monograph. It is shown that the very existence of resonance regions depends on the choice of orders of smallness of the three small parameters of the problem. [ABSTRACT FROM AUTHOR]

The differential quadrature method (DQM) is a numerical technique widely applied in structure mechanics problems. In this work, a top-tensioned riser conveying fluid is considered. The governing equation of this riser under parametric excitations is deduced. Through Galerkin's method, the partial differential governing equation with respect to time t and vertical coordinate z is reduced into a 1D differential equation with respect only to time. Moreover, the DQM is applied to discretize the governing equation to give solution schemes for the risers' parametric vibration problem. Furthermore, the instability region of Mathieu equation is studied by both the DQM and the Floquet theory to verify the effectiveness of the DQM, and the solutions of both methods show good consistency. After that, the influences of some factors such as damping coefficient, internal flow velocity, and wet-weight coefficient on the parametric instability of a top-tensioned riser are discussed through investigating the instability regions solved by the DQM solution scheme. Hence, conclusions are obtained that the increase of damping coefficient will save the riser from parametric resonance while increasing internal flow velocity, or the wet-weight coefficient will deteriorate the parametric instability of the riser. Finally, the time-domain responses of several specific cases in both stable region and unstable region are presented. [ABSTRACT FROM AUTHOR]

In this paper, we performed an asymptotic analysis for equations of the classical plate theory with the von K´arma´n strains under the assumption that the width of the plate is small compared with its length. A system of one-dimensional equations, which describes the nonlinear interaction of flexural and torsional vibrations of beams, was derived. This enables the possibility of exciting torsional vibrations by flexural vibrations. This possibility was analyzed for a model problem, when flexural vibrations occur in normal modes.

This paper proposes an analytic model of a resonant MEMS mirror with electrostatic actuation based on a Fourier series approximation for both the comb drive torque and the input waveform and verifies the model by measurements using rectangular input waveforms with various duty cycles. The analytic model is derived by the perturbation method, results in slow flow evolution in amplitude and phase with dynamic influence matrices and vectors and also provides the local dynamics for each equilibrium described by a Jacobian matrix. An analysis of the dynamic influence matrices and vectors provides understanding of the mirror dynamics by frequency components of the input waveform and the comb drive capacitance. The asymptotic behavior at zero amplitude provides the transition curve in an extended dynamic model, which corresponds to the well-known Mathieu’s equation solely with the constant and fundamental frequency components of the input waveform. The measurement results verify the proposed model, showing less than ±0.06 % frequency error for large amplitudes and ±0.47 % for small amplitudes, which corresponds to ±1.2 Hz and ±9.6 Hz for the case of a mirror with 2 kHz natural frequency, respectively. Measurements of local dynamics and transition curves also show a good agreement with the proposed model, which can be used for a fast and accurate analysis of resonant MEMS mirrors for high precision applications. [2020-0387] [ABSTRACT FROM AUTHOR]

The main load on a pile foundation in the ocean is the wave load. Presently, few reports exist on the dynamic stability of pile foundations in the ocean. This paper investigated the dynamic stability of pile foundations under wave loads. The foundation reaction force was calculated using a double-parameter model and considering pile side soil softening under a cyclic load. By establishing the energy equation of the entire pile, the Hamiltonian principle was used to obtain the dynamic differential equations of the pile in four different situations (vertical harmonic, transverse harmonic, longitudinal and transverse harmonic, and longitudinal and horizontal harmonic loads with different frequencies). Then, the nonhomogeneous Mathieu equation was obtained by arranging the dynamic differential equations, and the parametric resonance critical frequency and instability load were obtained by solving this equation. The analytical solution was verified by comparing the results obtained by the finite-element software simulation with the analytical solution and analyzing the influence of several different factors on the critical frequency and amplitude. The research indicated that the amplitude of the pile body increases linearly with an increase of wave height, the effect of wavelength on amplitude increases nonlinearly, and the amplitude increases much rapidly with an increase of wavelength. The critical frequency calculated by the double-parameter method was lower than that calculated by the Winkler model, and the result of double-parameter model was more practical and safe. [ABSTRACT FROM AUTHOR]

The differential quadrature method (DQM) is a numerical technique widely applied in structure mechanics problems. In this work, a top-tensioned riser conveying fluid is considered. The governing equation of this riser under parametric excitations is deduced. Through Galerkin’s method, the partial differential governing equation with respect to time t and vertical coordinate z is reduced into a 1D differential equation with respect only to time. Moreover, the DQM is applied to discretize the governing equation to give solution schemes for the risers’ parametric vibration problem. Furthermore, the instability region of Mathieu equation is studied by both the DQM and the Floquet theory to verify the effectiveness of the DQM, and the solutions of both methods show good consistency. After that, the influences of some factors such as damping coefficient, internal flow velocity, and wet-weight coefficient on the parametric instability of a top-tensioned riser are discussed through investigating the instability regions solved by the DQM solution scheme. Hence, conclusions are obtained that the increase of damping coefficient will save the riser from parametric resonance while increasing internal flow velocity, or the wet-weight coefficient will deteriorate the parametric instability of the riser. Finally, the time-domain responses of several specific cases in both stable region and unstable region are presented.

Exploiting parametric resonance may enable increased performance for wave energy converters (WECs). By designing the geometry of a heaving WEC, it is possible to introduce a heave-to-heave Mathieu instability that can trigger parametric resonance. To evaluate the potential of such a WEC, a mathematical model is introduced in this paper for a heaving buoy with a non-constant waterplane area in monochromatic waves. The efficacy of the model in capturing parametric resonance is verified by a comparison against the results from a nonlinear Froude–Krylov force model, which numerically calculates the forces on the buoy based on the evolving wetted surface area. The introduced model is more than 1000 times faster than the nonlinear Froude–Krylov force model and also provides the significant benefit of enabling analytical investigation techniques to be utilised.

The performance of bimorph cantilever energy harvester subjected to horizontal and vertical excitations is investigated. The energy harvester is simulated as an inextensible piezoelectric beam with the Euler–Bernoulli assumptions. A horizontal base excitation along the axis of the beam is converted into the parametric excitation. The governing equations include geometric, inertia and electromechanical coupling nonlinearities. Using the Galerkin method, the electromechanical coupling Mathieu–Duffing equation is developed. Analytical solutions of the frequency response curves are presented by using the method of multiple scales. Some analytical results are obtained, which reveal the influence of different parameters such as the damping, load resistance and excitation amplitude on the output power of the energy harvester. In the case of parametric excitation, the effect of mechanical damping and load resistance on the initiation excitation threshold is studied. In the case of combination of parametric and direct excitations, the dynamic characteristics and performance of the nonlinear piezoelectric energy harvesters are studied. Our studies revealed that the bending deformation generated by direct excitation pushes the system out of axial deformation and overcomes the limitation of initial threshold of parametric excitation system. The combination of parametric and direct excitations, which compensates and complements each other, can be served as a better solution which enhances performance of energy harvesters. [ABSTRACT FROM AUTHOR]

Abstract In the present work, the version of homotopy perturbation included time-scales is applied to the governing equation of time-periodic delay Mathieu equation. Periodical structure for the amplitude of the zero-order perturbation is constructed. The stability analysis is accompanied by considering three-time-scales. Approximate periodic solutions are derived to the second accuracy of perturbations at the harmonic resonance case as well as at the non-harmonic resonance case. Stability conditions are derived in both cases. Numerical calculations have been done to illustrate the stability behavior at both resonance and non-resonance case. It is shown that the time-delay has a destabilizing influence. We note that the delayed of the parametric excitation has a great interested and application to the design of nuclear accelerators. [ABSTRACT FROM AUTHOR]

The equations of the oscillator motion are considered. The exact solutions are given in the form of exponents with an additional parameter that characterizes the asymmetry of the oscillations. It is shown that these equations are the special case of the Hill’s equation. The equations for the three types of exponents, including having the property of unitarity are obtained. Lagrangians and Hamiltonians are found for these equations. It is proved that all the equations are associated by canonical transformations and essentially are the same single equation, expressed in different generalized coordinates and momenta. Moreover, the solutions of linear homogeneous equations of the same type are both solutions of inhomogeneous linear equations of another one. A quantization possibility of such systems is discussed.

The problem of excitation of internal gravity waves (IGWs) in the upper atmosphere by an external source of a limited duration of operation is investigated. An isothermal atmosphere was chosen as the propagation environment of IGWs in the presence of a uniform wind that changes over time according to the harmonic law. For the vertical component of the displacement of an environment, the Mathieu equation with zero initial conditions was solved with the right part simulating the effect of a powerful heating facility on the ionosphere. In the case of a small amplitude of the variable component of the wind, the time dependence of the vertical displacement under parametric resonance conditions using the perturbation method is obtained. The obtained dependence of the solution of the differential equation on the parameters allows us to perform a numerical analysis of the problem in the case of variable wind of arbitrary amplitude. For practical estimations of the obtained values, data on the operating modes of the SURA heating facility (56.15° N, 46.11° E) with periodic (15–30 min) switching on during of 2–3 h for ionosphere impact were used.

Model allowing analytical and numerical studies of a dusty plasma system is used to describe the dynamics of a monolayer of dusty particles. The mechanism of energy transfer between horizontal and vertical particle motion based on parametric resonance is described by the extended Mathieu equation. The resonance regions and growth rates of dust particles energy are obtained. The conditions for the occurrence of resonance and the initial stage of energy transfer are described more precisely based on the analysis of the obtained data. It is shown that a wide spectrum of dust particle oscillations participates in the system heating. The harmonics contributing the most to this process are determined. [ABSTRACT FROM AUTHOR]

The main torsional mode of electrostatically actuated micromirrors is known to be dominated by parametric resonance when the actuation is performed via in-plane comb fingers. Here, we show that, for specific geometrical features of the mirror, parametric resonance simultaneously activates a spurious yaw mode. Due to the large torsional rotations, the two modes are nonlinearly coupled, inducing mutual stiffness variations and an unexpected temperature dependence of the main mode. After presenting an experimental evidence of the coupling, we develop and discuss a numerical model capable of capturing the key phenomena and of providing guidelines for a robust design. [ABSTRACT FROM PUBLISHER]

In this paper, parametric excitation of a repulsive force electrostatic resonator is studied. A theoretical model is developed and validated by experimental data. A correspondence of the model to Mathieu's Equation is made to prove the existence and location of parametric resonance. The repulsive force creates a combined response that shows parametric and subharmonic resonance when driven at twice its natural frequency. The resonator can achieve large amplitudes of almost 24 μm and can remain dynamically stable while tapping on the electrode. Because the pull-in instability is eliminated, the beam bounces off after impact instead of sticking to the electrode. This creates larger, stable trajectories that would not be possible with traditional electrostatic actuation. A large dynamic range is attractive for MEMS resonators that require a large signal-to-noise ratio. [ABSTRACT FROM AUTHOR]

The parametric instability of a spar platform in irregular waves is analyzed. Parametric resonance is a phenomenon that may occur when a mechanical system parameter varies over time. When it occurs, a spar platform will have excessive pitch motion and may capsize. Therefore, avoiding parametric resonance is an important design requirement. The traditional methodology includes only a prediction of the Mathieu stability with harmonic excitation in regular waves. However, real sea conditions are irregular, and it has been observed that parametric resonance also occurs in non-harmonic excitations. Thus, it is imperative to predict the parametric resonance of a spar platform in irregular waves. A Hill equation is derived in this work, which can be used to analyze the parametric resonance under multi-frequency excitations. The derived Hill equation for predicting the instability of a spar can include non-harmonic excitation and random phases. The stability charts for multi-frequency excitation in irregular waves are given and compared with that for single frequency excitation in regular waves. Simulations of the pitch dynamic responses are carried out to check the stability. Three-dimensional stability charts with various damping coefficients for irregular waves are also investigated. The results show that the stability property in irregular waves has notable differences compared with that in case of regular waves. In addition, using the Hill equation to obtain the stability chart is an effective method to predict the parametric instability of spar platforms. Moreover, some suggestions for designing spar platforms to avoid parametric resonance are presented, such as increasing the damping coefficient, using an appropriate RAO and increasing the metacentric height. [ABSTRACT FROM AUTHOR]

Two dimensions Mathieu equation containing periodic terms as well as the delayed parameters has been investigated in the present work. The present system represents to a generalized form of the one-dimension delay Mathieu equation. The mathematical difficulty for delay the coupled Mathieu equation has been overcome by using the matrices method. Properties of inverse complex matrices enable us to transform the vector form of the solvability conditions to the scalar form. Small oscillation about a marginal state is introduced by using the method of multiple scales. Stability criteria for the complex matrices have been established and lead to obtain resonance curves. The analysis has been extended so that the delay 2-dimensions Mathieu equation containing weak complex damping part. Stability conditions and the transition curves that included the in uence of both the delayed as well the complex damping terms has been obtained. The transition curves are analyzed using the method of harmonic balance. We note that the delayed higher dimension of the parametric excitation has a great interest and application to the design of nuclear accelerators. [ABSTRACT FROM AUTHOR]

We consider the numerical integration of the matrix Hill equation. Parametric resonances can appear and this property is of great interest in many different physical applications. Usually, Hill’s equations originate from a Hamiltonian function and the fundamental matrix solution is a symplectic matrix. This is a very important property to be preserved by the numerical integrators. In this work we present new sixth-and eighth-order symplectic exponential integrators that are tailored to Hill’s equation. The methods are based on an efficient symplectic approximation to the exponential of high dimensional coupled autonomous harmonic oscillators and yield accurate results for oscillatory problems at a low computational cost. The proposed methods can also be used for solving general second order linear differential equations where their performance will depend on how the methods are finally adapted to each particular problem or the qualitative properties one is interested to preserve. Several numerical examples illustrate the performance of the new methods. [ABSTRACT FROM AUTHOR]

Electrostatically actuated torsional micromirrors are key elements in Micro-Opto-Electro-Mechanical-Systems. When forced by means of in-plane comb-fingers, the dynamics of the main torsional response is known to be strongly non-linear and governed by parametric resonance. Here, in order to also trace unstable branches of the mirror response, we implement a simplified continuation method with arc-length control and propose an innovative technique based on Finite Elements and the concepts of material derivative in order to compute the electrostatic stiffness; i.e., the derivative of the torque with respect to the torsional angle, as required by the continuation approach. [ABSTRACT FROM AUTHOR]

We consider an electrostatically actuated torsional micromirror, a key element of recent optical microdevices. The mechanical response is analyzed with specific emphasis on its nonlinear features. We show that the mirror motion is an example of parametric resonance, activated when the drive frequency is twice the natural frequency of the system. The numerical model, solved with a continuation approach, is validated with very good accuracy through an extensive experimental campaign. [ABSTRACT FROM AUTHOR]

Parametric resonance is a phenomenon caused by time-varying changes in the parameters of a system which may result in undesirable motion responses and instability. Floating bodies like ships and spar-buoys are prone to Mathieu instability mainly due to the instantaneous change of the metacentric height. With the fast-growing developments in Ocean Renewable Energy systems, spar-buoys are commonly used for wave energy convertors and floating wind turbines. Undesirable, unstable motions as a result of the parametric resonance can be problematic as it may cause inefficiency in operations and structural risk integrity. In this research, a new approach has been developed to investigate these nonlinear oscillations and analyze the conditions when parametric resonance occurs. The hydrodynamic loads are calculated using the linear approach, and the motion responses of the floating body coupled in heave, pitch and surge are determined. It is shown that the eigen values obtained from Floquet Theory can be used as indicators of stability under different wave conditions. This procedure can be practically used with little computational cost to determine factors affecting the equilibrium status of a system in regular waves. • The response amplitude obtained by dynamic equation are in good agreement with experiments. • The eigen values obtained from Floquet theory are correlated with bounded or unbounded response. • Transition from stable to unstable are accurately predicted by the non-dimensional parameters. • Extra damping shifts the stability parameters far from the transition curves of instability. [ABSTRACT FROM AUTHOR]

In this study, the parametric resonances of pinned intrinsic localized modes (ILMs) were investigated by computing the unstable regions in parameter space consisting of parametric excitation amplitude and frequency. In the unstable regions, the pinned ILMs were observed to lose stability and begin to fluctuate. A nonlinear Klein–Gordon, Fermi–Pasta–Ulam-like, and mixed lattices were investigated. The pinned ILMs, particularly in the mixed lattice, were destabilized by parametric resonances, which were determined by comparing the shapes of the unstable regions with those in the Mathieu differential equation. In addition, traveling ILMs could be generated by parametric excitation. [ABSTRACT FROM AUTHOR]

We consider the dynamic response of systems subjected to principal parametric resonant excitation in which the full potential, including nonlinearities, is modulated in a time-periodic manner. This work was motivated by the model of Rhoads et al. [J. Sound Vibration, 296 (2006), pp. 797-829], which was used to describe an interesting bifurcation structure that was experimentally observed in a microelectro- mechanical system. The goal of the present investigation is to more fully explore this class of systems, described by a generalized nonlinear Mathieu equation, and ascertain general features of their response. The method of averaging is used to derive equations governing the slowly varying amplitude and phase for parametrically excited systems with weak nonlinearity, small damping, and near resonant excitation, allowing for a detailed analysis of the system steady-state response. Results for a general class of models are presented first, followed by details for two examples: (i) generalized parametric resonance with cubic nonlinearities, for which the model of Rhoads et al. [J. Sound Vibration, 296 (2006), pp. 797-829] is extended by including nonlinear damping and the system response is more fully described, and (ii) a vertically excited inverted pendulum with a stiff linear torsional spring, for which the time-periodic modulation acts on the full gravitational potential. The analysis, which is supported by numerical simulations, shows that nonlinear damping and higher-order nonlinearities lead to a bounded primary response, in contrast to the unbounded primary response found in some parameter regions in the Rhoads model. More interesting is the fact that the steady-state frequency response can exhibit a sequence of isolas along the nonlinear response branches associated with the usual parametric resonance, where the number of isolas depends on the level of damping, the form of the potential, and the level of excitation. These results indicate that a relatively simple generalization of the Mathieu equation is able to capture a variety of responses that are important in applications. [ABSTRACT FROM AUTHOR]

Consider a second order differential linear periodic equation. The friction coefficient is real positive constant. Some transformation of the solution and its first derivative allow writing two-order differential equations with void friction coefficients. The solutions of these equations are periodic functions or sum of periodic function and an oscillating function with monotone linear increasing amplitude. The second order equation with linear friction is recast as a first order system. The coefficients of the principal fundamental matrix solution of the system are explicit analytical functions. [ABSTRACT FROM AUTHOR]

Consider a second order differential linear periodic equation. The periodic coefficient is an approximation of the Mathieu's coefficient. This equation is recast as a first-order homogeneous system. For this system we obtain analytical solutions in an explicit form. The first solution is a periodic function. The second solution is a sum of two functions, the first is a continuous periodic function, but the second is an oscillating function with monotone linear increasing amplitude. We give a formula to directly compute the slope of this increase, without knowing the second numeric solution. The periodic term of the second solution may be computed directly. The coefficients of fundamental matrix of the system are analytical functions. [ABSTRACT FROM AUTHOR]

Electrostatically actuated torsional micromirrors are key elements in Micro-Opto-Electro- Mechanical-Systems. When forced by means of in-plane comb-fingers, the dynamics of the main torsional response is known to be strongly non-linear and governed by parametric resonance. Here, in order to also trace unstable branches of the mirror response, we implement a simplified continuation method with arc-length control and propose an innovative technique based on Finite Elements and the concepts of material derivative in order to compute the electrostatic stiffness; i.e., the derivative of the torque with respect to the torsional angle, as required by the continuation approach.

Various phenomena in science, physics, and engineering result in the Mathieu equation with cubic nonlinear term, known as the Mathieu–Duffing equation. In previous works, different perturbation methods have been used to investigate the stability and bifurcation of this equation in the vicinity of the first unstable tongue and for relatively small values of natural frequency. The primary goal of this paper is to adapt the Strained Parameters Method to investigate the stability and bifurcation associated with stability change around the second unstable tongue. In addition, this work shows that the Strained Parameters Method is able to obtain the same results previously obtained by other perturbation techniques with minimum computational effort. An inductive approach is used to express the multipliers of the transition curves and the location of the newborn equilibria as a function of the parametric frequency. Lastly, the Floquet theory and Poincaré map are used to validate the analytical results. [ABSTRACT FROM AUTHOR]

This paper investigates the dynamical behaviour of the fractional delayed damped Mathieu equation. This system includes three different phenomena (fractional order, time delay, parametric resonance). The method of harmonic balance is employed to achieve approximate expressions for the transition curves in the parameter plane. Then= 0 andn= 1 transition curves (both lower and higher order approximations) are obtained. The dependencies of these curves on the system parameters and fractional orders are determined. Previous results for the transition curves reported for the damped Mathieu equation, delayed second-order oscillator, and fractional Mathieu equation are confirmed as special cases of the results for the current system. [ABSTRACT FROM PUBLISHER]

In this brief, a vibrational control law is designed and applied to an unstable and parametrically excited mechanical system, namely, a gantry crane. The control law cancels the parametric resonance by reshaping the stable regions. The parameters of the control law are tuned by using a graphic scheme based on the symbolic computation of the monodromy matrix. The control law is implemented experimentally in the laboratory model of a gantry crane, confirming the theoretical results. [ABSTRACT FROM AUTHOR]

We investigate the classical problem of motion of a mathematical pendulum with an oscillating pivot. This simple mechanical setting is frequently used as the prime example of a system exhibiting the parametric resonance phenomenon, which manifests itself by surprising stabilisation/destabilisation effects. In the classical case the pivot oscillations are described by a cosine wave, and the corresponding stability analysis requires one to investigate the behaviour of solutions to the Mathieu equation. This is not a straightforward procedure, and it does not lead to exact and simple analytical results expressed in terms of elementary functions. Consequently, the explanation of the parametric resonance phenomenon can be in this case obscured by the relatively involved technical calculations. We show that the stability analysis is much easier if one considers the pivot motion described by a non-smooth function—a triangular or a nearly rectangular wave. The non-smooth pivot motion leads to the presence of singularities (Dirac distributions) in the corresponding Mathieu type equation, which seemingly further complicates the analysis. Fortunately, this is only a minor technical difficulty. Once the mathematical setting for the non-smooth forcing is settled down, the corresponding stability diagram is indeed straightforward to obtain, and the stability boundaries are, unlike in the classical case, given in terms of simple analytical formulae involving only elementary functions. • Parametric resonance induced by a singular parameter modulation is investigated. • The modulation under consideration is given as a time-periodic sequence of impulses. • Results are compared to the classical modulation by a sine wave. • Unlike in the classical case the results are given in terms of elementary functions. • Results provide simple and instructive analysis of the inverted pendulum system. [ABSTRACT FROM AUTHOR]

Aimed at the axial disturbed load to the head of supercavitating projectile at high speed motion underwater, its indefiniteness was considered, and the structure reliability of supercavitating projectile was analyzed on the basis of dynamic stability. The partial differential dynamic equation of the projectile's cut-off-cone structure was established, then it was transformed to a second-order ordinary differential Mathieu equation, and the numerical calculation for the dynamic stability of supercavitating structure was performed with Bolotin method, finally, the dynamic stability safety margin equations of supercavitating structure was given. The influences of length-diameter ratio, velocity and length of supercavitating projectile on the dynamic stability were analyzed. Based on those, the structural non-probabilistic reliability was analyzed and obtained, the effectiveness of the method was demonstrated by using some examples. [ABSTRACT FROM AUTHOR]

We show that finite-amplitude shearing motions superimposed on an ‘unsteady’ simple extension are admissible for incompressible isotropic viscoelastic materials of differential kind. The amplitude of these motions is determined by solving a non-linear non-autonomous differential equation for which we show that limit cycles are possible. [ABSTRACT FROM PUBLISHER]

Abstract: We use a simple model of Bullard-type disc dynamo, in which the disc rotation rate is subject to harmonic oscillations, to analyze the generation of magnetic field by the parametric resonance mechanism. The problem is governed by a damped Mathieu equation. The Floquet exponents, which define the magnetic field growth rates, are calculated depending on the amplitude and frequency of the oscillations. Firstly, we show that the dynamo can be excited at significantly subcritical disc rotation rate when the latter is subject to harmonic oscillations with a certain frequency. Secondly, at supercritical mean rotation rates, the dynamo can also be suppressed but only in narrow frequency bands and at sufficiently large oscillation amplitudes. [Copyright &y& Elsevier]

Abstract: 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. [Copyright &y& Elsevier]

Abstract: We show that finite amplitude shearing motions superimposed on an unsteady simple extension are admissible in any incompressible isotropic elastic material. We show that the determining equations for these shearing motions admit a general reduction to a system of ordinary differential equations (ODEs) in the remarkable case of generalized circularly polarized transverse waves. When these waves are standing and the underlying unsteady simple extension is composed of a harmonic perturbation of a static stretch it is possible to reduce the determining ODEs to linear or non-linear Mathieu equations. We use this property for a detailed study of the phenomenon of parametric resonance in non-linear elastodynamics. [Copyright &y& Elsevier]

Abstract: The present paper describes a non-linear third order coupled mathematical model of parametric resonance of ships in head seas. Coupling is contemplated by considering the restoring modes of heave, roll and pitch motions. Numerical simulations employing this new model are compared to experimental results corresponding to excessive motions of a transom stern fishing vessel in head seas. It is shown that this enhanced model matches its results with the experiments more closely than a second order model. It is shown that the new model, due to the introduction of the third order terms, entails qualitative differences when compared to the more commonly used second order model. The variational equation of the roll motion will not be in the form of a Mathieu equation. In fact, it is shown in the paper that the associated time-dependent equation falls into the category of a Hill equation. Additionally, a hardening effect is analytically derived, related to the third order coupling of modes and wave passage effects. Limits of stability corresponding to the linear variational equation of the coupled roll motion are analytically derived. Numerical limits of stability corresponding to the non-linear equations are computed and compared to the analytically derived limits. [Copyright &y& Elsevier]

This work is devoted to the investigation of the possibility of controlling of ion motion inside the Paul trap. It has been shown that by proper selection of the parameters controlling the electric fields, stable localization of ions inside the Paul trap is possible. Quantum consideration of this problem is reduced to the investigation of the Mathieu–Schrodinger equation. It has been shown that quantum consideration is appreciably different from the classical one that leads to stronger limitations of the values of the parameters of stable motion. Connection between the problem under study and the possibility of experimental observation of quantum chaos has been shown. [ABSTRACT FROM AUTHOR]

Parametric resonance has been well established in many areas of science, including the stability of ships, the forced motion of a swing and Faraday surface wave patterns on water. We have previously investigated a linear parametrically driven torsional oscillator and along with other groups have mentioned applications including mass sensing, parametric amplification, and others. Here, we thoroughly investigate the design of a highly sensitive mass sensor. The device we use to carry out this study is an in-plane parametrically resonant oscillator. We show that in this configuration, the nonlinearities (electrostatic and mechanical) have a large impact on the dynamic response of the structure. This result is not unique to this oscillator—many MEMS oscillators display nonlinearities of equal importance (including the very common parallel plate actuator). We report the effects of nonlinearity on the behavior of parametric resonance of a micro-machined oscillator. A nonlinear Mathieu equation is used to model this problem. Analytical results show that nonlinearity significantly changes the stability characteristics of parametric resonance. Experimental frequency response around the first parametric resonance is well validated by theoretical analysis. Unlike parametric resonance in the linear case, the jumps (very critical for mass sensor application) from large response to zero happen at additional frequencies other than at the boundary of instability area. The instability area of the first parametric resonance is experimentally mapped. Some important parameters, such as damping co-efficient, cubic stiffness and linear electrostatic stiffness are extracted from the nonlinear response of parametric resonance and agree very well with normal methods. [Copyright &y& Elsevier]

Alternating or rotating magnetic fields are often used to stir, shape and support masses of liquid metal. The periodic electromagnetic force may parametrically excite growing disturbances which can be beneficial or detrimental to the process, and the aim of this paper is to develop the analysis of such effects. In many cases parametric excitation is described by a system of coupled simple harmonic oscillators with small periodic forcing and damping terms, and we use Floquet theory to derive a recursion formula for a matrix whose eigenvalues determine the growth rates. We consider two applications in detail – low-frequency magnetic stirring in a circular tank and the instability of a free-surface in the presence of an alternating field. [Copyright &y& Elsevier]

Exploiting parametric resonance may enable increased performance for wave energy converters (WECs). By designing the geometry of a heaving WEC, it is possible to introduce a heave-to-heave Mathieu instability that can trigger parametric resonance. To evaluate the potential of such a WEC, a mathematical model is introduced in this paper for a heaving buoy with a non-constant waterplane area in monochromatic waves. The efficacy of the model in capturing parametric resonance is verified by a comparison against the results from a nonlinear Froude–Krylov force model, which numerically calculates the forces on the buoy based on the evolving wetted surface area. The introduced model is more than 1000 times faster than the nonlinear Froude–Krylov force model and also provides the significant benefit of enabling analytical investigation techniques to be utilised. [ABSTRACT FROM AUTHOR]

Among the phenomena attributable to the Moon's actions on living organisms, one of them seems to be related to analytical fluid mechanics: along the route of the International Space Station around the Earth, experiments on plants have revealed leaf oscillations. A parametric resonance due to a short period of microgravitational forces could explain these oscillations. Indeed, Rayleigh-Taylor's instabilities occurring at the interfaces between liquid-water and its vapor verify a second-order Mathieu differential equation. This is the case of interfaces existing in the xylem channels of plant stems filled with sap and air-vapor. The magnitude of the instabilities depends on the distances between the Moon, the Sun, and the Earth. They are analogous, but less spectacular, to those that occur during ocean tides. [ABSTRACT FROM AUTHOR]

The problem of excitation of internal gravity waves (IGWs) in the upper atmosphere by an external source of a limited duration of operation is investigated. An isothermal atmosphere was chosen as the propagation environment of IGWs in the presence of a uniform wind that changes over time according to the harmonic law. For the vertical component of the displacement of an environment, the Mathieu equation with zero initial conditions was solved with the right part simulating the effect of a powerful heating facility on the ionosphere. In the case of a small amplitude of the variable component of the wind, the time dependence of the vertical displacement under parametric resonance conditions using the perturbation method is obtained. The obtained dependence of the solution of the differential equation on the parameters allows us to perform a numerical analysis of the problem in the case of variable wind of arbitrary amplitude. For practical estimations of the obtained values, data on the operating modes of the SURA heating facility (56.15° N, 46.11° E) with periodic (15–30 min) switching on during of 2–3 h for ionosphere impact were used. [ABSTRACT FROM AUTHOR]