Parametric Resonance Condition Research Articles

Nonlinear dynamic analysis of a spring coupled double beam based piezoelectric energy harvester under parametric base excitation

This paper proposes a base excited, spring coupled double cantilever beam with attached tip mass based piezoelectric energy harvester. The dimensions of the beams are so chosen that the system experiences 1:1 internal resonance. The governing coupled electro-mechanical equations of motion of the system are obtained by using the Lagrange principle which is reduced to the temporal form by using generalized Galerkin’s method. The governing temporal equations of motion are in the form of that of a parametrically excited coupled system with geometric and inertial nonlinearities. These nonlinear equations are solved using the method of multiple scales and the results are compared with those solved by numerical method (4th order Runge Kutta method) for principal parametric resonance conditions. Parametric studies are conducted by varying the tip masses of the beams, coupled spring stiffness, load resistance, external excitation frequency and excitation amplitude. Results show that by introducing spring coupling, the system operational frequency range increases significantly and also energy transfer process takes place due to the internal resonance. Significant enhancement of voltage and power are observed in the present work in comparison to the earlier studied equivalent single beam-based energy harvester.

Parametric resonance of Alfvén waves driven by ionization-recombination waves in the weakly ionized solar atmosphere.

Parametric coupling of waves is one of the most efficient mechanisms of energy transfer that can lead to the growth or decay of waves. This transfer occurs at frequencies close to their natural frequencies. In partially ionized solar plasma, there are a multitude of waves that can undergo this process. Here, we study the parametric coupling of Alfvén waves propagating in a partially ionized solar plasma with ionization-recombination waves identified by our study to appear in a plasma in ionization non-equilibrium. Depending on the parameters that describe the plasma (density, temperature), coupling can lead to a parametric resonance. Our study determines the occurrence conditions of parametric resonance, by finding the boundaries between stable and unstable regions in the parameter space. Our results show that collisions and non-equilibrium recombination can both contribute to the onset of unstable behaviour of parametrically resonant Alfvén waves. This article is part of the theme issue 'Partially ionized plasma of the solar atmosphere: recent advances and future pathways'.

Resonant Scattering of Gravitational Waves with Electromagnetic Waves

A certain class of exact solutions of Einstein Maxwell spacetime in general relativity is discussed to demonstrate that at the level of theory, when certain parametric resonance condition is met, the interaction of electromagnetic field with a gravitational wave will display certain Lyapunov instability and lead to exponential amplification of a gravitational wave train described by certain Newman–Penrose component of the Weyl curvature. In some way akin to a free electron laser in electromagnetic theory, by the conversion of electromagnetic energy into gravitational energy in a coherent way, the feasibility of generating a pulsed-laser-like intense beam of gravitational wave is displayed.

Non-trivial solutions and their stability in a two-degree-of-freedom Mathieu–Duffing system

The Mathieu–Duffing equation represents a basic form for a parametrically excited system with cubic nonlinearities. In multi-degree-of-freedom systems, parametric resonances and the associated limit cycles take place at both principal and combination resonance frequencies. Furthermore, using asynchronous parametric excitation of coupling terms leads to a broadband destabilization of the trivial solution and the appearance of limit cycles at non-resonant frequencies. Regarding applications, the utilization of this excitation method has its significant importance in micro- and nanosystems. On the one hand, cubic nonlinearities are found to be abundant in these systems. On the other hand, parametric excitation is preferably utilized in these systems for better amplification leading to an enhanced sensitivity and for squeezing thermal noise, and thus, proved to be significantly useful in mechanical, optical and microwave systems. Therefore, this theoretical investigation should be of relevant importance to those small-scaled systems. Accordingly, a general two-degree-of-freedom Mathieu–Duffing system is studied. The non-trivial solutions are obtained at different parametric resonance conditions. A bifurcation analysis is carried out using the multiple scales method, followed by investigating the effect of the asynchronous parametric excitation on the existence of limit cycles at resonant and non-resonant frequencies.

Parametric resonance of fractional viscoelastic webs under time-dependent tension

The focus of this paper is on the parametric resonance of fractional viscoelastic webs under time-dependent tension. The novelty of this work lies in the introduction of a fractional order model for viscoelastic webs and the time-dependent tension in roll-to-roll manufacturing. The time-dependent tension is induced by the web treatment processes in roll-to-roll mass production. The rheological properties of viscoelastic webs are represented by the fractional Kelvin-Voigt relation. Considering the geometric nonlinearity, the governing equation is derived based on the Hamilton principle, and then the resulting equation is reduced to a fractional ordinary differential equation by using the Bubnov-Galerkin method. The multiple-scale method is applied to analyze the occurrence condition and dynamic responses of parametric resonance, and the solution stability is demarcated by the Routh-Hurwitz criterion. The effects of the tension variation coefficient, viscoelastic parameter and initial tension on the resonance responses are discussed. The results reveal that the increase in tension variation coefficient and initial tension not only widen the unstable zone of the response curve, but also enhance the resonance amplitude. Moreover, the resonance amplitude decreases with a larger fractional order. This investigation aims to explore the cause of the instability phenomena and thus avoid the unexpected failure of flexible fabrication.

On opto-thermally excited parametric oscillations of microbeam resonators. I

The present article is the first part of the work devoted to investigation of the nonlinear dynamics of parametrically excited flexural vibrations of a clamped-clamped microbeam - the basic sensitive element of a promising class of microsensors of various physical quantities - under laser thermooptical action in the form of periodically generated pulses acting on a certain part of the surface of the beam element. An analytical solution of the heat transfer problem is found for the steady harmonic distribution of temperature in the volume of the resonator. The static and dynamic components of temperature-induced axial force and bending moment are determined. Using the Galerkin method, the discretization of nonlinear coupled partial differential equations describing the longitudinal-flexural oscillations of the resonator is performed. Using the asymptotic method of multiple time-scales, an approximate analytical solution is obtained for the nonlinear dynamics problem under the conditions of primary parametric resonance.

Instability Regions of a laminated Sandwich Plate with MRE Core Under Parametric Excitation Using Finite Element Method

This paper is concerned with the instability regions of a laminated sandwich plate with magneto-rheological elastomer (MRE) core under periodic axial loads. Structures operating under this unstable region cause large response even if the excitation force is small in magnitude. The governing equations of motion of the system are obtained by using finite element method [FEM].The parametric stable and unstable regions of the MRE cored composite sandwich plate for different system parameters and end constraints are found using modified Hsu method for simple parametric resonance condition. This analysis will help to design passive attenuation by using sandwich plates of varied dimensions, material parameters of both skin and core. Further active vibration reduction is possible by applying suitable magnetic field and operating the system outside of the instability region.

Examination of the virtues of parametric energy coupling in wave energy conversion

Parametric resonance has been observed, both numerically and experimentally, in various studies of wave energy converters (WECs). Large heave motions induce a periodic variation in the metacentric height of a WEC body and, consequently, cause a harmonic variation in pitch/roll restoring coefficients, which can parametrically excite the pitch/roll modes. Current studies attempt to determine the onset conditions of parametric resonance, by detecting the boundaries between stable and unstable regions in the parameter space. In the literature, some studies aim to make use of parametric resonance for improving power capture. In contrast, some studies try to suppress the effect of parametric resonance, as it can reduce power capture efficiency in the primary degree of freedom. However, how energy transfers from one mode to another is not fully understood. This study aims to analyse energy transfer between heave and pitch/roll modes when parametric resonance occurs. A generic cylindrical point absorber is studied as a WEC floater to consider non-linear wave-structure interaction, including non-linear Froude-Krylov and viscous forces. A heave-pitch-roll three-degree-of-freedom model is derived for numerical study of the energy transfer between different operational modes.

Investigation of Nonlinear Thermo-Elastic Behavior of Fluid Conveying Piezoelectric Microtube Reinforced by Functionally Distributed Carbon Nanotubes on Viscoelastic-Hetenyi Foundation

In this paper, nonlinear and nonlocal thermo-elastic behavior of a microtube reinforced by Functionally Distributed Carbon Nanotubes, with internal and external piezoelectric layers, in the presence of nonlinear viscoelastic-Hetenyi foundation, and axial fluid flow inside the microtube is studied. Nonlinear partial differential equations governing the system are derived using Reddy’s third-order shear deformations theory along with the Von-Karman theory including the effect of fluid viscosity. Then, the equations are converted to time-dependent ordinary nonlinear equations using the Galerkin method. Afterward, the governing equations of the microtube’s lateral displacements are solved using the multiple scales method. The analysis of the piezoelectric’s parametric resonance is performed by obtaining trivial and nontrivial stationary solutions and plotting characteristic curves of the frequency response and voltage response. At the end, the effect of different parameters including the flow velocity, excitation voltage, parameters of the foundation, viscosity parameter, thermal loading and nanotubes’ volume fraction index on the nonlinear behavior of the system, under parametric resonance condition, is investigated.

Stochastic modulational instability in the nonlinear Schrödinger equation with colored random dispersion

We study modulational instability (MI) in optical fibers with random group-velocity dispersion (GVD). We consider Gaussian and dichotomous colored stochastic processes. We resort to different analytical methods (namely, the cumulant expansion and the functional approach) and assess their reliability in estimating the MI gain of stochastic origin. If the power spectral density (PSD) of the GVD fluctuations is centered at null wave number, we obtain low-frequency MI sidelobes which converge to those given by a white-noise perturbation when the correlation length tends to 0. If instead the stochastic processes are modulated in space, one or more MI sidelobe pairs corresponding to the well-known parametric resonance (PR) condition can be found. A transition from small and broad sidelobes to peaks nearly indistinguishable from PR-MI is predicted, in the limit of large perturbation amplitudes and correlation lengths of the random process. We find that the cumulant expansion provides good analytical estimates for small PSD values and small correlation lengths, when the MI gain is very small. The functional approach is rigorous only for the dichotomous processes, but allows us to model a wider range of parameters and to predict the existence of MI sidelobes comparable to those observed in homogeneous fibers of anomalous GVD.

Linear and nonlinear analysis of traditional and non-traditional piezoelectric vibration absorber with time delay feedback for simultaneous resonance conditions

In the present work, vibration suppression of a single degree of freedom nonlinear spring-mass primary (main) system subjected to simultaneous external harmonic and parametric excitation by using traditional and non-traditional active nonlinear vibration absorber (ANVA) is carried out. The traditional and non-traditional ANVA consists of a PZT (lead zirconate titanate) stack actuator in series with a nonlinear spring and a damper. In the traditional ANVA, the damper is connected to the main system whereas, in non-traditional ANVA, the damper is connected to the base of the main system. The governing nonlinear equations of motion for the system have been derived using weighted modal matrix approach considering novel actuation force by time delay in displacement, velocity and acceleration feedback of the main system by the PZT stack actuator and the nonlinear spring in series combination. The method of multiple scales (MMS) is used to obtain the reduced equations which are solved using Newton’s method. The responses of the system for the primary, principal parametric and simultaneous primary and principal parametric resonance conditions have been studied. The results are compared with those obtained by using the harmonic balance method (HBM), numerical method and with previous literature. From these results, two resonating peaks in the frequency response of the main system are shown by using the reduced equation of the MMS which was not reported in earlier literature. It is shown that for the active case the vibration of the undamped main system is completely reduced at the specified operating frequency for both the linear and nonlinear systems. Also, for a wider range of operating frequencies, the vibration of the main system is reduced by actively tuning the absorber for various mass ratios. Further, the efficacy of the active force is observed in reducing the instability regions and improving the critical value of excitations in the trivial state.

Out-of-plane dynamic instability of nonlocal shear deformable nanoplates made of polyvinylidene fluoride materials subjected to electromechanical forces

Nowadays, piezoelectric materials are used as smart nanostructures in many engineering applications. Polyvinylidene fluoride (PVDF) is one of the piezoelectric polymeric materials which has fascinated the attention of the scientific community due to its remarkable properties. In the present study, dynamic stability and the parametric resonance of a Mindlin PVDF nanoplate lying on an elastic medium under electromechanical loadings and a moving nanoparticle are investigated. First, Hamilton's principle is used to obtain the partial equations governing the transverse oscillations of the PVDF nanoplate. By utilizing the theory of nonlocal piezoelasticity, the small-scale effects are applied to the equations. Navier’s approach is employed to procure the solution for the simply supported nanoplate. Then, the boundaries of instability in the plane of parameters are obtained by applying the energy-rate method to the governing time-varying ODEs. The results demonstrate that the increase in the nonlocal parameter and reduction of the nanoparticle movement path decrease the dynamic stability of the PVDF nanoplate. Also, the nanoplate made of PVDF materials is dynamically more stable than the nanoplate made of PZT-4 piezoceramic materials. Moreover, the piezoelectric voltage can be used to control the parametric resonance conditions of nanostructures. The results of this study have the potential to be utilized in the accurate design of nanoscale piezoelectric mass sensors in nanoelectromechanical systems.

Analysis of Parametric and Subharmonic Excitation in Push-Pull Driven Disk Resonator Gyroscopes.

For micro-electromechanical system (MEMS) resonators, once the devices are fabricated and packaged, their intrinsic quality factors (Q) will be fixed and cannot be changed, which seriously limits the further improvement of the resonator’s performance. In this paper, parametric excitation is applied in a push-pull driven disk resonator gyroscope (DRG) to improve its sensitivity by an electrical pump, causing an arbitrary increase of the “effective Q”. However, due to the differential characteristics of the push-pull driving method, the traditional parametric excitation method is not applicable. As a result, two novel methods are proposed and experimentally carried out to achieve parametric excitation in the push-pull driven DRGs, resulting in a maximum “effective Q” of 2.24 × 106 in the experiment, about a 7.6 times improvement over the intrinsic Q. Besides, subharmonic excitation is also theoretically analyzed and experimentally characterized. The stability boundary of parametric excitation, defined by a threshold voltage, is theoretically predicted and verified by related experiments. It is demonstrated that, when keeping the gyroscope’s vibration at a constant amplitude, the fundamental frequency driving voltage will decrease with the increasing of the parametric voltage and will drop to zero at its threshold value. In this case, the gyroscope operates in a generalized parametric resonance condition, which is called subharmonic excitation. The novel parametric and subharmonic excitation theories displayed in this paper are proven to be efficient and tunable dynamical methods with great potential for adjusting the quality factor flexibly, which can be used to further enhance the resonator’s performance.

Generation of Internal Gravity Waves in the Thermosphere during Operation of the SURA Facility under Parametric Resonance Conditions

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.

Dynamic analysis of piezoelectric energy harvester under combination parametric and internal resonance: a theoretical and experimental study

In this work, theoretical and experimental analysis of a piezoelectric energy harvester with parametric base excitation is presented under combination parametric resonance condition. The harvester consists of a cantilever beam with a piezoelectric patch and an attached mass, which is positioned in such a way that the system exhibits 1:3 internal resonance. The generalized Galerkin’s method up to two modes is used to obtain the temporal form of the nonlinear electromechanical governing equation of motion. The method of multiple scales is used to reduce the equations of motion into a set of first-order differential equations. The fixed-point response and the stability of the system under combination parametric resonance are studied. The multi-branched non-trivial response exhibits bifurcations such as turning point and Hopf bifurcations. Experiments are performed under various resonance conditions. This study on the parametric excitation along with combination and internal resonances will help to harvest energy for a wider frequency range from ambient vibrations.

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.

ОСОБЕННОСТИ КОНТАКТНОГО РАЗРУШЕНИЯ ЛЬДА

The formation of an intermediate layer under hydrostatic compression at a shear appearing due to the action of converging and diverging fronts of stress momentums (pulses) is considered. Continuous monitoring of deformational changes in the structure of ice was carried out using acoustic methods. The features of contact ice breaking in the diverging fronts of stress pulses are considered by the example of the slow impact of a rigid spherical indenter on an ice plate simulating half-space. Using the piezoelectric accelerometer, an oscillogram of the impact was recorded and a generalized dependence of the reduced stress on the reduced instantaneous velocity of the impact (semi-cubic parabola) was obtained. It is established that under conditions of the experiment (smooth convex indenter surface and icy half-space) a thin intermediate layer is formed, the properties of which determine the physical similarity in the family of curves «instantaneous force-instantaneous velocity». A rheological model with due regard for the change in the microstructure of ice during the impact is proposed. Quantitative determinations of the deformation changes in structure of solid ice samples were performed under intensive plastic deformation in a matrix with a profile similar to the Laval nozzle. The deformations created by the piston caused forced vibrations in the ice. The working surface of the piston in the form of an ellipsoid together with the smooth walls of the matrix and the reverse cone created conditions for parametric resonance and the formation of fronts of highfrequency stress pulses. Under influence of these pulses, zones with a superplastic fine-crystalline structure of ice (cumulative effect) were formed in ice. In the outlet cylindrical channel, a flow around an obstacle of the ice with the structure of an intermediate layer (dynamic viscosity 20 MPa s) and the distribution of velocities of motion over the channel cross section were studied. The obtained results can be used to simulate the processes of contact destruction of deep rocks by a support or an ice-resistant platform loaded with an ice field.

Piezoelectric energy harvester under parametric excitation: A theoretical and experimental investigation

In the present work, both theoretical and experimental investigation of a vertical cantilever beam–based piezoelectric energy harvester are carried out under principal parametric resonance condition. A piezoelectric patch is attached near the fixed end of the cantilever beam along with an attached mass positioned at an arbitrary location. The extended Hamilton’s principle is used to derive the spatio-temporal equation of motion, and generalized Galerkin’s approximation is used to obtain the temporal nonlinear electromechanical governing equation of motion. The method of multiple scales is used to find the reduced modulation equations. Due to large transverse deflection and effect of rotary inertia of the attached mass, the system exhibits cubic and inertial nonlinearities. An experimental setup with slider crank mechanism–based shaker and a harvester consisting of a cantilever beam with piezoelectric patch and attached mass is designed and developed. The challenges posed by parametric resonance in crack development in the PZT and in the beam are reported. The theoretical and experimental output voltage and the power obtained are found to be in good agreement. Furthermore, a qualitative and quantitative comparative study of 17 energy harvesters has been carried out, and the normalized power densities have been compared.

Dynamic analysis of pneumatic artificial muscle (PAM) actuator for rehabilitation with principal parametric resonance condition

In this present work, the dynamic behavior of a pneumatic artificial muscle (PAM) has been studied by varying different muscle parameters and air pressure into it. The governing equation of motion has been derived using Newton’s law of motion to study the various responses in the system at principal parametric resonance condition. The temporal equation of motion contains various nonlinear parameters with forced and nonlinear parametric excitation. Then, the second-order method of multiple scales is used to find the approximate solutions and to study the dynamic stability and bifurcations of the system. The results are found to be in good agreement with the solutions obtained by solving the temporal equation of motion numerically. The instability regions by varying different system parameters have been plotted. The time responses and phase portraits have been plotted to study the system behavior with the nonlinearity. The influences of the different system parameters in the amplitude for the muscle have also been studied with the help frequency responses. In order to verify the solution, the basin of attraction has also been plotted. The obtained results will be very useful for designing the desired PAM use in different rehabilitation robotics and exoskeleton system.

Heating of Plasma Electrons by Laser Radiation under Parametric Resonance Conditions in Strong Magnetic Field

Heating of plasma electrons by an extraordinary laser wave at the double upper hybrid frequency in a strong magnetic field is studied numerically by the particle-in-cell method. It is shown that, in this case, significant heating takes place even at considerable detunings from the fundamental parametric resonance. This makes it possible to reduce the resonance magnitude of the magnetic field. The minimum value of the laser field amplitude at which parametric heating of this type can be implemented is found.