Nonlinear Parametric Resonance Research Articles

The onset of instability in a parametric resonance energy harvester under panchromatic excitations

This paper further explores the potential of parametric resonance to enhance wave energy conversion in a pendulum-based pitching floating energy harvester, considering power extraction and panchromatic excitation. Unlike traditional parametric systems that focus on inertial mechanical instability induced by floater motion, this concept, firstly proposed in a previous recent publication, leverages the parametric resonance induced by nonlinear hydrodynamic coupling between heave and pitch. By deliberately integrating parametric instability into the floater design with a 2:1 ratio of natural periods in two hydrodynamic degrees of freedom, the frequency bandwidth of the mechanical system is expanded, leading to two distinct regions of significant power production. The research advances current understanding in three key areas: (i) the use of Bezier curves to advance a computationally efficient nonlinear hydrodynamic model based on nonlinear Froude–Krylov force calculations tailored for prismatic floaters, (ii) investigation of the impact of power extraction on the severity of nonlinear parametric resonance, and (iii) consideration of more realistic panchromatic waves to quantify the onset of instability and the decrease in the severity of parametric resonance response, also addressing practical implications for numerical simulations and signal processing. Findings indicate that beneficial parametric resonance persists across various damping and wave conditions, although the relative importance of the contribution of the 2:1 parametric resonance region to power extraction decreases by 75%, on average.

Vector form intrinsic finite element analysis for nonlinear parametric resonances of planar beam structures

The parametric resonance of a beam structure may lead to a disastrous consequence. To assess the safety of a structure, structural engineers need a straightforward and practical method to calculate the large-amplitude nonlinear parametric resonance responses of the beam structure. This paper proposes a newly developed vector form intrinsic finite element (VFIFE) calculation scheme to analyse the nonlinear parametric resonance of planar beam structures. By taking into account the geometric stiffness effect of the internal axial force for the frame element, a new relationship between internal nodal forces and deformation components is first established based on the VFIFE principle and the parametric resonance mechanism of frame structures. A numerical example of a portal frame is presented to demonstrate the efficacy of this new calculation scheme. A parametric resonance test of the cantilever beam is conducted to verify the applicability of the VFIFE method. The numerical results show that the proposed scheme can well simulate the nonlinear phenomena of parametric resonances. The numerical predictions are consistent with the experimental observations. A filtering-based finite-time Lyapunov exponent approach is presented to determine the stability boundary of the structure. The numerical stability boundary of parametric resonance agrees well with the experimental boundary. Possible causes of deviations between the numerical and experimental results are discussed. The proposed VFIFE scheme is a simple and effective means of analysing the stability and nonlinear responses of the parametric resonance of planar beam structures.

Nonlinear parametric resonances of a rotating composite thin-walled beam under aerodynamic force and hygrothermal environment

Considering combined effects of aerodynamic force and hygrothermal environment, the nonlinear parametric resonance of a composite thin-walled beam subjected to a time-dependent rotating speed is investigated in this work. The rotating speed is presumed to comprise a small disturbance harmonically changing one with a steady one. The dynamical model of the system is obtained with the help of established dynamic model in existing research. Then the nonlinear partial differential equation of the system is discretized by the Galerkin method. Afterwards, the low-frequency and high-frequency resonances of the system are analyzed via the multiple scales method to discuss the combined effect of rotating speed, pitch angle, setting angle, fiber orientation angle, temperature-humidity and inflow ratio on the amplitude-frequency responses and the unstable region. In addition, the Routh-Hurwitz criterion is employed to determine the stability conditions of the trivial and non-trivial solutions. The results show that both sub- and super-critical pitchfork bifurcations will occur to the system in parametric resonances regime, accompanied with stable and unstable non-trivial solutions simultaneously. Particularly, the mechanism of the influence of hygrothermal environment is deeply studied, which reveals a competitive relation between the change of mechanical properties and additional stress brought by the change of temperature and humidity.

Nonlinear Parametric Resonance in the Simplest Model of a Solar Dynamo

The properties of nonlinear parametric resonance are investigated using the example of the low-mode Parker dynamo model. This model is a system of four ordinary differential equations and in the simplest approximation describes the processes of generation and oscillation of large-scale magnetic fields in stellar systems. In the absence of nonlinear effects, the problem under consideration, by analogy with a system of harmonic oscillations, admits an asymptotic division of multiple resonant frequencies. However, despite the fact that at first glance at these frequencies it is reasonable to expect an amplification of the amplitude in the nonlinear case, it is demonstrated that in the presence of nonlinear terms, the behavior of the system is significantly more complex. In particular, generation suppression can be observed at resonant or low frequencies, while amplification occurs in the immediate vicinity of the resonance or at sufficiently high frequencies. The reasons are discussed for this behavior, as well as the possibility of the influence of parametric resonance on the establishment of planetary dynamo cycles.

Parametric resonance of shear deformable nanotubes: A novel higher-order model incorporating nonlinearity from both curvature and inertia

The research on nonlinear parametric resonance is of significance in the application of nanotubes in nanoelectromechanical systems. To capture the nonlinear behaviors of the nanotube in an accurate way, a size-dependent nonlinear model fully considering the transverse shear deformation, the nonlinearity from curvature and inertia is proposed for the first time. Specifically, the size-dependency is modeled by the nonlocal strain gradient theory coupled with surface effect. By introducing the accurate curvature, Zhang Fu's higher-order beam theory is developed to establish the refined displacement field. Besides, the axial nonlinear inertial force caused by transverse vibration is derived by introducing the hypothesis of “non-extensible beam”. The two-step perturbation-incremental harmonic balance method (TSP-IHBM) is developed to obtain both the stability boundary and bifurcation diagram. The accuracy is confirmed through convergence analysis and the validation by Runge Kutta method. Results provide four bifurcation topologies with different stiffness parameters and reveal the relationships between the stability boundary and the bifurcation diagram. It is found that the inertia nonlinearity may change the bifurcation diagram from stiffness hardening characteristics to softening characteristics. Also, parameter analysis provides the influence law of different size-dependent effects on the stability boundary and bifurcation diagram.

Instabilities of a vortex-ring-bright soliton in trapped binary three-dimensional Bose-Einstein condensates

Instabilities of vortex-ring-bright coherent structures in harmonically trapped two-component three-dimensional Bose-Einstein condensates are studied numerically within the coupled Gross-Pitaevskii equations and interpreted analytically. Interestingly, the filled vortex core with a sufficiently large amount of the bright component is observed to reduce the parametric interval of stability of the vortex ring. We have identified the mechanisms of several linear instabilities and one nonlinear parametric instability in this connection. Two of the linear instabilities are qualitatively different from ones reported earlier, to our knowledge, and are associated with azimuthal modes of $m=0$ and $m=1$, i.e., deviations of the vortex from the stationary ring shape. Our nonlinear parametric resonance instability occurs between the $m=0$ and $m=2$ modes and signals the exchange of energy between them.

Nonlinear Parametric Resonance Behavior of Cables in a Long Cantilever Bridge for Sightseeing Platform

In order to study the parametric vibration of stayed cables in a long cantilever bridge for a sightseeing platform, nonlinear parametric vibration equations of the stayed cables excited by the vibration of bridge deck and tower are derived. Then, a second‐order differential equation is transformed into a first‐order ordinary differential equation, which is solved by using the Runge–Kutta method. A finite element model of cables was also built to verify the solution of the Runge–Kutta method. Then, the inherent dynamic characteristics of the full structure and all the cables with different lengths were analyzed to discuss the potential risk of parametric vibration. The longest and shortest cables were taken as examples to explore their nonlinear vibration performance. The effects of damping ratio, excitation position, and amplitude on cables’ nonlinear vibration performance were investigated. The results show that it will be more efficient and convenient to use the Runge–Kutta method to calculate cables’ nonlinear vibration amplitude without loss of accuracy. In addition, short cables have more resonance zones compared to long cables. Especially, with the cable length shortening, the dominant frequencies of the dynamic response and its amplitude increase significantly, and the number of resonance zones also increases. However, excessive excitation amplitude will also cause multiple resonance zones in the cable. The parametric analysis results show that it is effective and efficient to mitigate the nonlinear vibration by adjusting the frequency relationship between the bridge and the cables, rather than by increasing the damping ratio.

Three-dimensional parametric resonance of fluid-conveying pipes in the pre-buckling and post-buckling states

The paper presents a model of describing three-dimension parametric vibrations of a simply supported pipe conveying pulsating fluid. The three-dimensional motion equation of the system is a set of two nonlinear partial differential equations developed on the basis of Euler-Bernoulli beam theory, geometric nonlinearity and Kelvin-Voigt damping model. The three-dimensional motion equation is discretized by the Galerkin method and the nonlinear responses are solved by a fourth order Runge-Kutta integration algorithm. The three-dimensional model is validated for the natural frequencies from the pre-buckling state to the post-buckling state. Results of the nonlinear parametric resonance responses are shown in the form of time histories, spectrograms, phase-plane portraits, Poincaré map sections and motion trajectories. The vibration mode of sub-harmonic resonance is similar to a certain order free vibration mode, and the vibration mode of combination resonance is a superposition of two adjacent modes. The motion trajectories indicate the vibration of sub-harmonic resonance is planar but the vibration of combination resonance is non-planar. The lock-in phenomenon occurs in the pre-buckling and post-buckling states when the natural frequencies are locked to the pulsation frequency. The amplitudes of parametric resonance in the post-buckling state is much larger in comparison to parametric resonance in the pre-buckling state. Jump phenomenon is highlighted for the parametric resonance in the post-buckling state in comparison to the pre-buckling state.

Parametric resonances of rotating composite laminated nonlinear cylindrical shells under periodic axial loads and hygrothermal environment

This paper focuses on nonlinear parametric resonance behaviors of rotating composite laminated cylindrical shells subjected to periodic axial loads and hygrothermal environment. With effects of the time-varying axial loads, hygrothermal expansion deformation, Coriolis and centrifugal forces as well as the rotation-induced initial hoop tension taken into account, the nonlinear dynamic equations of the shell are obtained on the base of Love’s nonlinear shell theory and Hamilton’s principle. Then, an analytical formulation on the steady state response of the shell is derived by the method of multiple scales, and the stability conditions of trivial and nontrivial solutions are determined by the Routh–Hurwitz criterion. Some numerical results are utilized to conduct detailed parametric studies on vibration characteristics, amplitude–frequency response curves and instability regions of forward and backward travelling waves of the shell. Of particular interest in the process is the combined effect of dynamic axial loads and hygrothermal effects on the resonance behaviors of the rotating nonlinear cylindrical shell.

Extremely large dynamics of axially excited cantilevers

Abstract The nonlinear parametric resonance of a cantilever under axial base excitation is examined while capturing extremely large oscillation amplitudes for the first time. A geometrically exact model is developed for the cantilever based on the Euler-Bernoulli beam theory and inextensibility condition. In order to be able to capture extremely large oscillation amplitudes accurately, the equation of motion is derived for centreline rotation while keeping trigonometric terms intact. The developed model is verified for the static case through comparison to a three-dimensional nonlinear finite element model. The internal energy dissipation model of Kelvin-Voigt is used to model the system damping in large amplitudes more accurately. The Galerkin modal decomposition scheme is utilised for discretisation procedure while keeping the trigonometric terms intact. It is shown that in parametric resonance region, the oscillation amplitudes grow extremely large even for smallest possible amplitudes of the base excitation, which highlights the significant importance of employing a geometrically exact model to examine the parametric resonance response of a cantilever.

Parametric interaction of gravity-capillary wave triads under radiation pressure of ultrasound

Nonlinear parametric coupling of gravity-capillary waves (GCW) by ultrasound wave impinging on a liquid surface is studied experimentally. Modulation of the plane wave radiation pressure by dynamically curved surface of the liquid is considered as a mechanism of the coupling. The standing GCW modes are excited near antinodes by ultrasound beam of carrier frequency 1.030 MHz and coupled parametrically by a plane ultrasound wave with a frequency 1.035 MHz. The beam was modulated at resonance frequencies of GCW overtones f2=5.265 Hz and f4=7.873 Hz while intensity of the plane wave was modulated at frequency fp=2.657 Hz that corresponds to the nonlinear parametric resonance fp=2f2-f4. The amplitudes of the parametric interaction of GCW triads are measured and compared with critical condition for GCWs explosive instability.

A Compact 6-DoF Nonlinear Wave Energy Device Model for Power Assessment and Control Investigations

High accuracy at a low computational time is likely to be a fundamental trait for mathematical models for wave energy converters, in order to be effective tools for reliable motion prediction and power production assessment, device and controller design, and loads estimation. Wave energy converters are particularly prone to exhibit complex and nonlinear behaviors, which are difficult to be modeled efficiently. Highly nonlinear effects, related to nonlinear Froude–Krylov forces, are nonlinear coupling, instability, and parametric resonance, which may damage or improve the power production. It is therefore fundamental to be able to describe such nonlinearities, in order to assess their repercussion on the performance of the device, and eventually design the system in order to exploit them. This paper provides a computationally efficient, compact, and flexible modeling approach for describing nonlinear Froude–Krylov forces for axisymmetric wave energy devices, in 6-DoFs. Unlike other similar models, based on a mesh discretization of the geometry, the analytical formulation of the wetted surface allows the dynamical model to run almost in real time.

Cubic–quintic nonlinear parametric resonance of a simply supported beam

We investigated the cubic–quintic nonlinear response in a parametrically excited simply supported beam subjected to a spring force in the axial direction. Taking into account the cubic and quintic geometric nonlinearities of curvature of the beam, the governing equation of the parametrically excited beam was derived based on Hamilton’s principle. The fifth-order approximate solution was analytically obtained using the method of multiple scales; with this calculation, the third-order nonlinear normal mode was also obtained. Its associated amplitude revealed saddle-node bifurcation and a hysteresis in the frequency response curve, which could not be predicted using the third-order approximate solution for the governing equation that included only cubic nonlinearity. Experimental results taken using a simple apparatus qualitatively verify the theoretically predicted nonlinear features in the parametric resonance caused by the cubic–quintic geometric nonlinearity of the beam.

Nonlinear parametric resonance of relativistic electrons with a linearly polarized laser pulse in a plasma channel

The direct laser-acceleration mechanism, nonlinear parametric resonance, of relativistic electrons in a linearly polarized laser-produced plasma channel is examined by a self-consistent model including the relativistic laser dispersion in plasmas. Nonlinear parametric resonance can be excited, and the oscillation amplitude of electrons grows exponentially when the betatron frequency of electron motion varies roughly twice the natural frequency of the oscillator. It is shown analytically that the region of parametric resonance is defined by the self-similar parameter nenca0. The width of this region decreases with nenca0, but the energy gain and oscillation amplitude increases. In this regime, the electron transverse momentum grows faster than that in the linear classical resonance regime.

Influence of the longitudinal displacement on the nonlinear principal parametric resonance of the woodworking bandsaw

The transverse vibrations of axially moving Timoshenko beam, as suitable mathematical models for woodworking bandsaws, are investigated. Special attention is paid to the influence of longitudinal displacement effect, as opposed to most models which can be usually encountered in the literature. This influence is introduced through the integro-partial differential equations. The expressions for the mode shapes in the case of hybrid supports with different torsion spring stiffness on the support points are also derived. The influence of mean beam velocity and axial tension on its natural frequencies and mode shapes is also investigated. Based on the nonlinear model, the amplitudes of the steady-state response are calculated for the case of principal parametric resonance. Developed program solution was tested on a number of earlier known examples. Present theoretical considerations, with the help of the program solution, is also used to analyse an example from industrial practice.

Considerations on frequency characteristics of an electromechanical vibration energy harvesting converter with nonlinear parametric resonance

The paper aims to determine and analyse the frequency characteristics of an electromechanical vibration energy harvesting converter that comprises of a metal cantilever-beam spring driving a linear-motion permanent-magnet generator with a single-phase cored armature winding. The nonlinear parametri c resonance phenomenon is analysed that the system exhibits due to action of the electromagnetic forces developed by the generator. In order to determine the steady-state frequency characteristics of the system a 1-D complex quasi-dynamic structural finite element model is coupled with 2-D magnetostatic finite element and circuit models. The nonlinearity and electric loading are taken into account. The region of feasible operation is determined involving physical limitations of the system that result from the assumed system topology and physical properties of materials used. The observations on bistability of the frequency characteristics are carried out. The results show that the converter should operate loaded due to risk of spring devastation at the bistable operation point. The quasi-optimal value of loading resistance providing maximum power at average value of the external excitation force is determined. The considerations are supported by laboratory experiments.

Magnetoelastic Principal Parametric Resonance of a Rotating Electroconductive Circular Plate

Nonlinear principal parametric resonance and stability are investigated for rotating circular plate subjected to parametric excitation resulting from the time-varying speed in the magnetic field. According to the conductive rotating thin circular plate in magnetic field, the magnetoelastic parametric vibration equations of a conductive rotating thin circular plate are deduced by the use of Hamilton principle with the expressions of kinetic energy and strain energy. The axisymmetric parameter vibration differential equation of the variable-velocity rotating circular plate is obtained through the application of Galerkin integral method. Then, the method of multiple scales is applied to derive the nonlinear principal parametric resonance amplitude-frequency equation. The stability and the critical condition of stability of the plate are discussed. The influences of detuning parameter, rotation rate, and magnetic induction intensity are investigated on the principal parametric resonance behavior. The result shows that stable and unstable solutions exist when detuning parameter is negative, and the resonance amplitude can be weakened by changing the magnetic induction intensity.

Theoretical and Numerical Analysis of 1 : 1 Main Parametric Resonance of Stayed Cable Considering Cable-Beam Coupling

For the 1 : 1 main parametric resonances problems of cable-bridge coupling vibration, a main parametric resonances model considering cable-beam coupling is developed and dimensionless parametric resonances differential equations are derived. The main parametric resonances characteristics are discussed by means of multiscale approximation solution methods. Using an actual cable of cable-stayed bridge project for research object, numerical simulation analysis under a variety of conditions is illustrated. The results show that when the coupling system causes 1 : 1 parametric resonance, nonlinear main parametric resonances in response are unrelated to initial displacement of the cable, but with the increase of deck beam end vertical initial displacement increases, accompanied with a considerable “beat” vibration. When the vertical initial displacement of deck beam end is 10−6 m order of magnitude or even smaller, “beat” vibration phenomenon of cable and beam appears. Displacement amplitude of the cable is small and considerable amplitude vibration may not occur at this time, only making a slight stable “beat” vibration in the vicinity of the equilibrium position, which is different from 2 : 1 parametric resonance condition of cable-bridge coupling system. Therefore, it is necessary to limit the initial displacement excitation amplitude of beam end and prevent the occurrence of amplitude main parametric excitation resonances.

Nonlinear analysis of dynamic stability for the thin cylindrical shells of supercavitating vehicles

The dynamic stability of supercavitating vehicles under periodic axial loading is investigated in this article. The supercavitating vehicle is simulated as a long and thin cylindrical shell subjected to periodic axial loading and simply supported boundary conditions. The nonlinear transverse vibration differential equation is obtained in terms of nonlinear geometric equations, physical equations, and balance equations of cylindrical shells. Mathieu equation with periodic coefficients and nonlinear term is derived by employing Galerkin variational method and Bolotin method. The analytical expressions of the steady-state amplitudes of vibrations in the first- and second-order instable regions are obtained by solving nonlinear Mathieu equation derived in this article. Numerical results are presented to analyze the influence of the sailing speed, ratio of loads, the frequency of axial loads, and the mode of vibration on parametric resonance curves and to show the nonlinear parametric resonance curves incline toward the side where it is greater than the excitation frequency, which significantly extends the range of the exciting region. The presented results indicate the enlargement of the exciting region will cause shrinkage of the safe frequency range of external loads and decrease in dynamic stability of supercavitating vehicle.

Effects of Supporting Member on the Nonlinear Parametric Resonance of a Cable

The cables in cable-supported structures are commonly subjected to potentially damaging large amplitude motions mainly arising from parametric vibrations of the cables. This paper presents an analysis of the effects of supporting member on the nonlinear parametric vibration of a cable using a coupled cable-beam model. The proposed model considers the parameters of the beam and geometric nonlinearities of the cable. First, the multiple scales method is applied directly to the model, and by using the first-order equation, the frequency response and stability conditions are obtained. The effects of the mass ratio, stiffness ratio, and inclined angle of the coupled model are then evaluated. Then, the effects of these parameters on the parametric vibration characteristics of cable are investigated in terms of the maximum mid-span displacement and dynamic amplification factor. The results obtained represent an extension of the previous studies, which provide some useful insights into the design of cable-supported structures.