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This article is devoted to the design and engineering calculation of the dynamic stability of the saw blade of a new type of sawing equipment with a saw module performing a circular flat rotational–translational motion. The presented equipment relates to the field of sawing hard materials, mainly wood, but also metal, plastics, stone, bone, glass, and others. The main assembly unit of the equipment is the saw unit, consisting of six saw modules that ensure its dynamic balance without additional balancing (correction) elements. [ABSTRACT FROM AUTHOR]

The stability of the parametric resonance of a controlled stay cable with time delay is investigated. The in-plane nonlinear equations of motion are initially determined via the Hamilton principle. Then, utilizing the method of multiple scales, the modulation equations that govern the nonlinear dynamics are obtained. These equations are then utilized to investigate the effect of time delays on the amplitude and frequency-response behavior and, subsequently, on the stability of the parametric resonance of the controlled cable, that it is shown to depend on the excitation amplitude and the commensurability of the delayed-response frequency to the excitation frequency. The stability region of the parametric resonance is shifted, and the effects of control on the cable become worse by increasing time delay. The work plays a guiding role in the parametric design of the control system for stay cables. [ABSTRACT FROM AUTHOR]

This contribution highlights that a linear periodic Hamiltonian system preserves a symplectic structure if a particular dissipation is present. This specific structure is defined by the algebraic properties of μ -symplectic matrices and symmetry of its eigenvalues. A method is established for the stability analysis of this class of systems consisting of damped and coupled Mathieu equations. It enables an efficient computation of the corresponding stability chart. One main strength of the method is the calculation of the stability chart even for large parameter values, especially for the amplitude of the parametric excitation and the system response itself. The proposed stability analysis is applied in detail on two examples consisting of two coupled equations. [ABSTRACT FROM AUTHOR]

Due to their simplicity, relatively high efficiency, and scalability, point wave energy absorbers (PWAs) have emerged as one of the most popular and promising solutions for wave energy harvesting. Their fundamental operation principle is based on activating linear resonance to pump energy from the waves to the PWA. Thus, their effective operation bandwidth is restricted to the range of incident wave frequencies that are close to the natural frequency of the PWA. This key constraint is very challenging to meet given the high stiffness of the hydrostatic buoyancy force and the variability of the wave conditions. A direct result of not meeting this constraint is a substantial reduction in the efficacy and power transduction capabilities of the PWA. To overcome this problem, deliberate introduction of nonlinearities into the PWA design has been recently proposed and exploited in two forms: (i) introduction of a nonlinear multi-stable restoring force, and (ii) introducing and/or exploiting parametric instabilities. The premise is that such approaches may be able to (i) shift the response frequency of the PWA towards the energetic low-frequency waves, and (ii) reduce the sensitivity of the PWA to the uncontrollable spatio-temporal variations in the incident waves. This review critically assesses the feasibility of leveraging nonlinear phenomena to improve the performance of PWAs. Our findings strongly point towards the conclusion that a nonlinear restoring element does not improve the capture width ratio or effective bandwidth of the PWA when compared to an optimal linear design. Furthermore, since the nonlinearity often results in aperiodic and coexisting competing responses, it adds additional layers of complexity during performance optimization and full-scale implementation. Nonetheless, the nonlinearity can be utilized as an effective means to passively shift the effective bandwidth of the PWA towards the energetic low-frequency wave content, and to decrease the sensitivity of the PWA to the wave parameters under irregular waves whose parameters drift with time. [ABSTRACT FROM AUTHOR]

We apply perturbative techniques to a driven undamped sinusoidal oscillator at resonance. The angular displacement, θ, obeys the dynamics θ ¨ + ω 2 sin ⁡ θ = H cos ⁡ ω t . The linearized approximation gives a divergent response (at long times) but the nonlinear terms make the response finite. We address the nonlinearity-induced finiteness in two ways by separately treating the short and long time scales. At long times, we use the traditional perturbative techniques to extract two drive dependent behaviours—one, the amplitude of oscillation scales as (H / ω 2 ) 1 / 3 and, two, the time period of the slow mode varies as (H / ω 2 ) − 2 / 3 . For the early time behaviour, on the other hand, we devise an alternate perturbative expansion where the successive terms get larger with the order of evaluation but have alternating signs. The alternating signs (phase differences) between these terms leads to adestructive interference like effect. A careful consideration of this destructive interference like effect between successive terms leads to a finite response which describes the initial behaviour of the amplitude of the response reasonably correctly. We further note that for larger drive values, the system seems to undergo a first order transitional behaviour with a sudden jump in the largest Lyapunov exponent [ABSTRACT FROM AUTHOR]

Parametric resonance is a complicated phenomenon that manifests itself in many areas. When subjected to parametric resonance, the amplitude of the flexible manipulator will increase abruptly, resulting in the rapid deterioration of working performance. Most conventional control methods are ineffective when approaching resonance. Because of this, a new method for suppressing the parametric resonance of the flexible manipulator is proposed. A novel parametric resonance absorber, characterized by controllable vibration parameters and adjustable nonlinear coupling parameters, is designed to strengthen quadratic modal coupling with the flexible manipulator and to construct a transfer tunnel for exchanging and dissipating parametric resonance energy. Dynamics equations of the controlled flexible arm mode and the parametric resonance absorber mode are derived, and the corresponding steady-state solutions of parametric resonance are solved. The saturation principle is revealed and implemented based on the stability analysis of the steady-state response of parametric resonance. With the help of the saturation principle, the parametric resonance response of the flexible manipulator can be effectively suppressed to a small amplitude by the proposed parametric resonance absorber. A series of numerical simulations and experiments have verified the proposed method’s effectiveness in suppressing the flexible manipulator’s parametric resonance. [ABSTRACT FROM AUTHOR]

Nanomechanical resonators provide a versatile platform for nanoscale mass sensing and force microscopy, as well as for enhancing light‐matter interaction offering unique functionality for optomechanical applications. In this way, discovering new approaches for coupling light with the mechanical degrees of freedom opens the strong desire paths for further developing of nanomechanical technology. Here, the parametric optothermal modulation of hybrid nanomechanical systems consisting of carbon nanowire with a silicon nanoparticle on its top, is reported. The mechanism of the modulation is based on the periodic optical heating of the nanowire and further modulation of the elasticity parameters. Utilizing the silicon nanoparticle provides additional functionality owing to optical absorption enhanced with Mie resonance and the unique feature of optical Raman thermometry enabling optical monitoring of local temperature. It is shown that the parametric mechanism of modulation allows for a significant increase of the optomechanical coupling strength. [ABSTRACT FROM AUTHOR]

The stability diagram for the parametric resonance in the mechanical tank–free surface fluid system in the classical Faraday problem has been derived analytically. Unlike the previous studies, the stability diagram is based on the nonlinear equation of the fluid free surface oscillations, which is transformed into the Ince–Strutt form. Perturbation theory, specifically the method of small parameters, has been employed to derive the stability boundary equations. The findings have shown that the cubic nonlinearities in the equation of the fluid parametric oscillations significantly influence only the location of the first instability domain (parametric resonance). To accurately evaluate the effect of third-order nonlinearities on the boundaries of the second and third resonance domains, higher-order nonlinear terms need to be considered. [ABSTRACT FROM AUTHOR]

Introducing a time-periodicity into a system parameter leads to parametric excitation, which in general, may cause a parametric resonance with exponentially increased vibration. Applying a parametric excitation but carefully tuning its frequencies to multiple parametric anti-resonance frequencies is investigated here. The parametric excitation here is realized by an open-loop control at the system boundary that allows for an energy flow into or from the system. A parametric anti-resonance successfully triggers an energy transfer between specific vibration modes of the system and occurs in systems with at least two degrees of freedom. Such an energy transfer increases the overall dissipation of kinetic energy of a lightly damped system. This contribution presents an approach to accelerate the mitigation of transient vibrations by applying a multi-frequency parametric excitation with two or more parametric anti-resonance frequencies. The potential application in a MEMS sensor arrangement consisting of two and more coupled flexible beams exemplifies the method. Starting from the minimum system with two degrees of freedom, the averaging method is applied to analyze the transient slow flow, leading to an analytical approximation of the transition time response of a pulsed multi-frequency parametric excitation system. For a specific example, a reduction of 96.7% of the transient vibrations is achievable. [ABSTRACT FROM AUTHOR]

Parametric excitation is a unique type of excitation that arises from the time-varying parameters of a dynamical system, typically induced by the periodic movements of the system components. However, the structure-moving subsystems problem creates a new type of parametric excitation where the subsystems do not have exact periodic movements. Under the circumstances, dimension of the mathematical model of such a system exhibits time dependency, which prohibits the employment of conventional analytical methods to predict its stability. In this paper, we propose a novel Generalized Sequential State Equation Method that provides a distinct solution framework for analysis of the subsystem-induced parametric resonance. With the time-domain system partition and state mapping, the proposed method reconfigures the coupled system with a sequence of state space equations and solves them chronologically. The proposed method relaxes the single-subsystem-assumption in its original form and applies to a general scenario where the subsystems travel with arbitrary characteristic period. With the proposed method, occurrences of the subsystem-induced parametric resonance are precisely predicted via a set of analytical stability criteria and the steady state response is determined in an exact analytical form. • Novel generalized sequential state equation method for parametric resonance with arbitrary characteristic period. • The proposed method advances the existing method by lifting the constraints on the inter-distance between subsystems. • Determining the stability of a moving subsystem induced parametric vibration via analytical stability criteria. • Delivering the exact waveform of the steady state of a parametrically excited structure in a general manner. • Investigating the characteristics of the subsystem induced parametric resonance in a multidimensional parameter space. [ABSTRACT FROM AUTHOR]

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The dynamics of a vibrating feeder at Raman parametric resonance are studied. Mathematical and dynamic models of a vibrating feeder with a parametric vibration exciter are given. Stationary solutions are obtained by the averaging method, taking dissipative forces, nonlinear components of restoring forces, and resistance forces as small parameters. The amplitude–frequency and phase–frequency characteristics of the feeder and the dependences of the generation frequencies are plotted. The trajectory of the center of mass of spherical rolling elements relative to the axis of rotation of the vibration exciter is obtained. [ABSTRACT FROM AUTHOR]

The instability and breakup of liquid jets under static or alternating electric fields are involved in numerous industrial applications. Unlike under electrostatic fields, far fewer investigations have been conducted to analyze the instability of liquid jets in alternating electric fields. Thus, the electric and viscous correction of viscous potential flow (EVCVPF) is applied here to describe the linear instability of leaky-dielectric liquid jets subjected to alternating electric fields. The effects of alternating electric fields, fluid electric properties, and other parameters are investigated. The capillary instability response is like that of the jets under electrostatic fields. Under a sufficiently strong alternating electric field, the resonance instability dominates surface disturbances, leading to the resonant atomization. Viscous damping makes the resonance weaker–even vanishing with the increasing frequency. Furthermore, the conductive charge–largely dependent on fluid conductivities–has the opposite effect of the surface charge. Thus, when the charge relaxation time approaches the imposed period, the parametric resonance is strongly inhibited. In addition, when aerodynamic effects are sufficiently strong, the resonance is covered. [ABSTRACT FROM AUTHOR]

Based on the generalized Hamilton's principle, the nonlinear governing equation of an axially functionally graded (AFG) pipe is established. The non-trivial equilibrium configuration is superposed by the modal functions of a simply supported beam. Via the direct multi-scale method, the response and stability boundary to the pulsating fluid velocity are solved analytically and verified by the differential quadrature element method (DQEM). The influence of Young's modulus gradient on the parametric resonance is investigated in the subcritical and supercritical regions. In general, the pipe in the supercritical region is more sensitive to the pulsating excitation. The nonlinearity changes from hard to soft, and the non-trivial equilibrium configuration introduces more frequency components to the vibration. Besides, the increasing Young's modulus gradient improves the critical pulsating flow velocity of the parametric resonance, and further enhances the stability of the system. In addition, when the temperature increases along the axial direction, reducing the gradient parameter can enhance the response asymmetry. This work further complements the theoretical analysis of pipes conveying pulsating fluid. [ABSTRACT FROM AUTHOR]

In the article the parametric oscillations of an unreformed disk located in a contactless electromagnetic suspension with an electrostatic loop of the effective control stiffness were investigated analytically. The analytical expressions for the transition curves of the stationary position of the levitated object were obtained based on the asymptotical method of nonlinear dynamics for the areas of the main and secondary parametric resonances. The system parameters were estimated for which the contactless suspension with quasi-zero electromagnetic stiffness is asymptotically stable. [ABSTRACT FROM AUTHOR]

The current study aims to analyze the dynamic characteristics of the nonlinear system excited parametrically in the presence of internal resonance. The method of multiple time scales (MMS) is directly adopted to simplify the higher-order integro-partial differential equation of motion to get an approximate solution that leads to a set of first-order partial differential equations. To develop a suitable model for a moving beam, the parameters incorporated are viscoelasticity and viscous damping, geometric nonlinearity, Coriolis acceleration, harmonically varying velocity, and axially varying tension. The stability and bifurcations of the steady-state solution are examined under a subcritical speed regime. The investigation focuses on the changes in the stability and bifurcation features of steady-state solutions accounting for the effects of variations in the system parameters like internal and parametric frequency detuning parameters, the amplitude of fluctuating speed, and axial stiffness. The outcomes of this investigation are unique, interesting, and not available in the existing literature, which may provide theoretical insight in designing a traveling system. [ABSTRACT FROM AUTHOR]

Axions can be stimulated to decay into photons by ambient photons of the right frequency or by photons from the decay of neighboring axions. If the axion density is high enough, the photon intensity can be amplified, which is a type of lasing or an axion maser. Here, we review the astrophysical situations where axion lasing can appear and possibly be detected. [ABSTRACT FROM AUTHOR]

We analyse the spectral content and parametric resonant dynamics of an array of elastically and electrostatically coupled interdigitated micro cantilevers assembled into two identical half-arrays. In this uncommon arrangement, within each of the half-arrays, the beams are coupled only elastically. The half-arrays are intercoupled only electrostatically, through fringing fields. First, by using the reduced order (RO) model, we analyse the voltage-dependent evolution of the eigenvalues and the eigenvectors of the equivalent mass-spring system, starting from the small two, three and four beams arrays and up to large beams assemblies. We show that at the coupling voltages below a certain critical value, the shape of the eigenvectors, the frequencies of the veering and of the crossing are influenced by the electrostatic coupling and can be tuned by the voltage. Next, by implementing the assumed modes techniques we explore the parametric resonant behavior of the array. We show that in the case of the sub critical electrostatic coupling the actuating voltages required to excite parametric resonance in the damped system can be lower than in a strongly coupled array. The results of the work may inspire new designs of more efficient resonant sensors. [ABSTRACT FROM AUTHOR]

The paper contains a parametric analysis of tensegrity structures subjected to periodic loads. The analysis focuses on determining the main region of dynamic instability. When load parameters fall within this region, the resulting vibration amplitudes increase, posing a risk to the durability of structures. The study considers structures built using commonly used modules. The influence of the initial prestress on the distribution of the instability regions is examined. Additional prestress can significantly reduce the extent of instability regions, potentially narrowing them by up to 99%. A nondimensional parameter is introduced to accurately assess changes in the extent of the instability region. A geometrically non-linear model is employed to evaluate the behavior of the analyzed structures. [ABSTRACT FROM AUTHOR]

A general self-consistent theory of adaptive properties of a ferromagnetic single-domain grain in alternating magnetic fields is constructed. Thresholds of new family of parametric instabilities of all orders are calculated. It is shown that the level of excitation of the magnetic moment and the phase of forced oscillations with respect to the pump field serve as a convenient tool for describing emerging nonequilibrium states. An analysis was made of the overthreshold excited state determined by third- and fourth-order nonlinear interactions in terms of deviations of the ferromagnetic moment. Adaptive nonequilibrium states describe energy flows from the pump field to the thermal bath and are characterized by non-linear damped oscillations. Near the frequency of these oscillations, the effect of mixing of weak RF signals with the microwave pump field arises, as well as the effect amplification of the RF modulation of the pump field. Using mathematical analogies, the developed theory can be transferred to other physical systems.

In this paper, parametric forced tuned solid ball dampers (TSBD) are considered for vibration control of engineering structures in an untypical way. The special feature of the presented investigation is to evaluate the potential application of parametric forcing of the rolling cylindrical or spherical body in the runway for reducing the vertical vibrations of a vibration-prone main system. Typically, tuned solid ball dampers are applied to structures that are prone to horizontal vibrations only. The coupled nonlinear differential equations of motion are derived and the phenomenon of parametric resonance of the rolling body in the runway is analyzed. A criterion for avoiding parametric resonance is given to achieve the optimal damping effect of the TSBD. In the second part of the article, a method for the targeted use of parametric resonance to reduce the vertical vibrations of engineering structures is presented and verified, considering a biaxially harmonic excited pedestrian bridge. It is shown that, with a suitable choice of damper parameters, a stable vibration of the rolling body in the runway is formed over the course of the vibration despite the occurrence of parametric resonance and that the maximum vertical vibration amplitudes of the main system can be reduced up to 93%. Hence, the here presented untypical application of parametric forced TSBD for reducing the vertical forced vibrations of vibration-prone main systems could be successfully demonstrated. [ABSTRACT FROM AUTHOR]

In this study, we analyze a planar mathematical pendulum with a suspension point that oscillates harmonically in the vertical direction. The bob of the pendulum is electrically charged and is located between two wires with a uniform distribution of electric charges, both equidistant from the suspension point. The dynamics of this phenomenon is investigated. The system has three parameters, and we analyze the parametric stability of the equilibrium points, determining surfaces that separate the regions of stability and instability in the parameter space. In the case where the parameter associated with the charges is equal to zero, we obtain boundary curves that separate the regions of stability and instability for the Mathieu equation. [ABSTRACT FROM AUTHOR]

The review aims to present the current state of research on the resonance plasma heating by high-power laser radiation in the presence of a strong magnetic field. It is demonstrated that modulation instability with a modulation period equal to the wavelength of the driven plasma wave occurs upon propagation of a low-amplitude pulse parallel to the magnetic field in a subcritical-density plasma in the ECR region. The magnitude of the driven longitudinal electric field grows substantially with increase in the pulse amplitude. In the process, the energy imparted to plasma electrons by the laser radiation increases severalfold relative to the case of an isotropic plasma. The physical mechanism of strong heating of electrons consists in transformation of the modulation instability into stochastic regime at high amplitudes. Propagation of an extraordinary laser wave in plasma in the region of parametric resonance at twice the upper hybrid frequency is analyzed in the case of wave propagation perpendicular to the magnetic field. Considerable auxiliary heating of electrons due to laser-wave decay into upper hybrid plasmons and excitation of the Bernstein waves takes place under such interaction as well. It follows from the electric-field distribution at the time of laser pulse arriving to the right boundary of the plasma layer that strong absorption of the transverse electric field of the laser pulse and enhancement of the longitudinal field occur in the region of parametric upper-hybrid resonance, which is accompanied by the appearance of the reflected electromagnetic wave at the upper hybrid frequency. [ABSTRACT FROM AUTHOR]

The parametric resonance, found in a single floating body, discloses that the kinetic energy could be transferred from heave mode to roll mode and causes motion instability if there is an integer multiple relationship between the two mode natural frequencies. For multi-module floating structures, the event of parametric resonance has not been investigated, but important for the stability and safety design of the floating platforms. In this paper, an experimental test is carried out using five box-type floating modules in a wave flume and observes the existence of the parametric resonance between the heave mode and roll mode. A mathematical model, validated by the experiment data, is built up for the theoretical analysis of the influential factors of the parametric resonance. The effects on the motion instability of wave condition, connector stiffness and number of modules are analyzed. It reveals that an appropriate stiffness setting of the connectors could eliminate the parametric resonance of multi-module floating structures. This theoretical finding is confirmed in a further experiment test on a five-module floating structure in the wave flume. [ABSTRACT FROM AUTHOR]

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.

Axions can be stimulated to decay into photons by ambient photons of the right frequency or by photons from the decay of neighboring axions. If the axion density is high enough, the photon intensity can be amplified, which is a type of lasing or an axion maser. Here, we review the astrophysical situations where axion lasing can appear and possibly be detected.

Railway vehicle wheelsets experience hunting motion when the running speed exceeds a critical value. While it is well known that hunting motion is a self-excited oscillation due to a nonconservative contact force acting between the rail and wheel, this study investigates another type of parametric resonance also caused by wheelset mass imbalance in cases of large tread angle. Because the critical speed of the self-excited oscillation decreases or large tread angles, it can be induced simultaneously with parametric resonance. Focusing on such situations, we theoretically and experimentally clarify the following wheelset nonlinear dynamics, which depend on the running speed. The wheelset is first destabilized through a pitchfork bifurcation and undergoes parametric resonance with a stable nontrivial steady-state amplitude. As the running speed increases, the stable steady-state amplitude is destabilized through a subcritical Hopf bifurcation by a nonconservative contact force effect related to the self-excitation. In addition to a constant amplitude oscillation, it is theoretically indicated that an oscillation with a beating amplitude can be produced in the neighborhood of the Hopf bifurcation point, depending on the disturbance magnitude. A simple apparatus with a roller rig has experimentally confirmed these nonlinear phenomena that depend on the running speed and the disturbance. [ABSTRACT FROM AUTHOR]

Spillway radial gate, a spatial frame structure, plays an important role in flood prevention and sustainable energy system. Dynamic instability mechanism of the spillway radial gate, excited by quasi-periodic hydrodynamic loads, remains unclear. In this work, an efficient numerical method, for analyzing dynamic stability of discretized model, is developed. Furthermore, a comprehensive investigation on the dynamic instability regions of struts and frame structure in the spillway radial gate, performed in discrete model, is provided. Specifically, the presented discrete finite element model and developed numerical method are validated by using numerical simulation in ANSYS software and definitive works from Bolotin. The out-of-plane vibration of struts and frame structure render dynamic instability of the spillway radial gate. The mode-coupling dynamic instability regions of frame structure are revealed, which is ignored in the dynamic instability analysis of struts and the analytical solution of radial gate structure as well. Long-term quasi-stable state exists in parametric resonance response, threatening the reliability of structural health diagnosis based on monitoring data for radial gates structure. From the comparison of strut and frame structure, it is shown that the dynamic instability regions of frame structure are more dense and complicated, close to the dominant frequency of external hydrodynamic loads during water discharging. The proposed insights highlight the importance of analyzing frame structure's dynamic stability in evaluating the safety of radial gate structure, as the corresponding dynamic stability characteristics are generally ignored in current works. [ABSTRACT FROM AUTHOR]

• The subcritical and supercritical resonances of a 3-D cantilevered pipe conveying pulsating fluid are investigated. • The instability regions of the corresponding to the linear system are predicted by parametric analyses. • The bifurcation routes in two oscillating directions are constructed by varying the pulsating frequencies. • The planar and nonplanar vibrations are detected within several ranges of pulsating frequencies. This paper investigates the stability and three-dimensional (3-D) nonlinear dynamics of a cantilevered pipe with an internal fluid having a harmonic component of flow velocity superposed on a constant mean value. The nonlinear equations of motion for an inextensible cantilevered pipe with the consideration of internal pulsating flow are presented. The nonlinear inertial terms in the governing equations are replaced by equivalent displacement and velocity terms by using a perturbation method. The partial differential equations are then transformed into a set of ordinary differential equations (ODEs) by using the Galerkin method. The instability regions of the subcritical and supercritical resonances of a linear system are determined via the Floquet theory. The effects of mean flow velocity and mass ratio are investigated. The resulting coupled nonlinear differential equations are numerically solved using a fourth-order Runge-Kutta integration scheme for the subcritical and supercritical flow velocities. The nonlinear dynamical responses are presented in the form of bifurcation diagrams, time histories, phase portraits, power spectral densities (PSDs) and Poincaré maps. Some interesting and sometimes unexpected results have been observed with different flow velocities. The analytical model is found to exhibit rich and variegated dynamical behaviors which include 2-D or 3-D periodic, quasiperiodic and chaotic motions. The convergence analysis of the number of truncating modes in the Galerkin approach is also conducted. [ABSTRACT FROM AUTHOR]

• Highly accurate and efficient analytical instability boundaries are derived and validated. • The parametric resonances of difference type do not exist. • The effects of anisotropy in bearing supports on rotor instability are determined. • Anisotropy leads to interactions within a single whirl mode or between two whirl modes. Ensuring rotor stability is a major concern in engineering, as instabilities can lead to catastrophic failures. Existing literature shows that anisotropic boundary conditions significantly affect the parametric instability characteristics of rotors under periodical axial loads. However, there is little literature systematically analyzing the formation mechanism of parametric resonance under these boundary conditions or providing a detailed classification of the parametric instability regions. Therefore, this paper presents a comprehensive parametric instability analysis of a rotor subjected to periodic axial loads under anisotropic boundary conditions. A novel approach based on the multiple scales method is proposed to address anisotropy in the boundary conditions. Using this approach, the analytical boundaries of the parametric instability regions are derived, and a proof regarding the absence of certain parametric resonances is presented. These analytical solutions are validated by numerical results obtained from the discrete transition matrix method, which form the basis for systematically investigating the effects of anisotropy in direct or cross-coupling stiffness/damping coefficients on the rotor instability. The key scientific contributions of this work include: Deriving analytical instability boundaries, providing a more efficient alternative to purely numerical methods while maintaining high accuracy; Demonstrating the absence of parametric resonance of difference type under both isotropic or anisotropic boundary conditions; Discovering that anisotropy in stiffness coefficients can induce self-interaction within a given forward or backward whirl mode, as well as interaction between two forward or two backward whirl modes, leading to additional instability regions; Reducing anisotropy in direct damping coefficients may increase critical dynamic load coefficients, potentially enhancing rotor safety; If the cross-coupling stiffness coefficients exceed the threshold for triggering intrinsic instability, the rotor may become unstable in all operating conditions. All these findings offer insights into the stability management of rotors under various operating conditions and provide valuable guidance for designing and operating safer, more efficient rotor systems. [Display omitted] [ABSTRACT FROM AUTHOR]

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. [Display omitted] • Pendulum pitching wave energy converter exploits hydrodynamic parametric resonance. • Fast nonlinear Froude–Krylov force integrals using Bezier curves for prismatic hulls. • Instantaneous frequency variability proxy of the onset and severity of instability. • Second region of power extraction, also with PTO damping and panchromatic waves. • Relative importance to power extraction decreases by 75% with panchromatic waves. [ABSTRACT FROM AUTHOR]

• Design a micromechanical resonator with inherent high-order nonlinearities. • Unveil the pseudo-linear behavior in resonators with high-order nonlinearities. • Frequency stabilization is achieved in the pseudo-linear region. • The tailored inherent high-order nonlinearities make the frequency stabilization scheme more robust. • Reveal the effect of parametric excitation on the suppression of phase noise. Micromechanical oscillator sustains oscillation through a velocity-proportional feedback force. Stabilization of the oscillation frequency is typically achieved by increasing the amplitude of the resonator while remaining in its linear regime. However, nonlinear effects and parasitic cross-talk unavoidably emerge with the reduction of oscillator size, severely limiting its performance. To overcome these limitations, in this paper, a novel and robust frequency stabilization scheme that employs the interaction of intrinsic high-order nonlinearities in a parametrically driven arch beam resonator is proposed and verified. Firstly, a micromechanical resonator with intrinsic high-order nonlinearities is specifically designed. Upon parametrically driving the resonator into resonance at large vibrational amplitudes, the bending of the resonance curve changes direction. This mixed behavior forms a pseudo-linear region without the need for auxiliary electrostatic tuning, simplifying the circuits and enhancing the robustness of the system. Secondly, significant frequency stability enhancement is observed when driving the resonator into self-sustained parametric oscillation in the pseudo-linear regime where output frequency decouples from the amplitude noise. Besides, the phase noise suppression effect of the parametrically driven oscillator is experimentally demonstrated when compared with the conventional directly driven oscillator. Finally, a theoretical phase noise model is developed, which well reproduces the observed frequency stabilization in the pseudo-linear regime and phase noise suppression effect of parametric oscillations. [Display omitted] [ABSTRACT FROM AUTHOR]

We analyse the spectral content and parametric resonant dynamics of an array of elastically and electrostatically coupled interdigitated micro cantilevers assembled into two identical half-arrays. In this uncommon arrangement, within each of the half-arrays, the beams are coupled only elastically. The half-arrays are intercoupled only electrostatically, through fringing fields. First, by using the reduced order (RO) model, we analyse the voltage-dependent evolution of the eigenvalues and the eigenvectors of the equivalent mass-spring system, starting from the small two, three and four beams arrays and up to large beams assemblies. We show that at the coupling voltages below a certain critical value, the shape of the eigenvectors, the frequencies of the veering and of the crossing are influenced by the electrostatic coupling and can be tuned by the voltage. Next, by implementing the assumed modes techniques we explore the parametric resonant behavior of the array. We show that in the case of the sub critical electrostatic coupling the actuating voltages required to excite parametric resonance in the damped system can be lower than in a strongly coupled array. The results of the work may inspire new designs of more efficient resonant sensors.

To suppress the nonlinearity of an excited Van der Pol–Duffing oscillator (VdPD), time-delayed position and velocity are used throughout this study. The time delay is supplemental to prevent the nonlinear vibration of the considered system. The topic of this work is extremely current because technologies with a time delay have been the subject of several studies in the latest days. The classical homotopy perturbation method (HPM) is utilized to extract an approximate systematic explanation for the system at hand. Furthermore, a modification of the HPM reveals a more accurate approximate solution. This accuracy is tested through a comparison with the numerical solution. The practical approximate analytical methodology makes the work possible to qualitatively evaluate the results. The time histories of the obtained solutions are drawn for various values of the natural frequency and the time delay parameters. Discussion of the results is presented in light of the plotted curves. On the other hand, the multiple scale procedure examines the organized nonlinear prototypical approach. The influence of the diverse regulatory restrictions on the organization's vibration performances is explored. Two important cases of resonance, the sub-harmonic and super-harmonic, are examined according to the cubic nonlinearity. The modulation equations achieved for these cases are examined graphically according to the impact of the used parameters. [ABSTRACT FROM AUTHOR]

• A new approach is proposed for the parametrically excited linear gyroscopic system. • The analytical instability boundaries match well with numerical ones. • Only the sum type instability regions can be observed. • The primary difference type instability regions do not exist. The present work investigates the existence/nonexistence of instability regions for a parametrically excited linear gyroscopic system. To achieve this goal, a new approach is proposed to determine the boundaries of parametric instability regions. As long as the gyroscopic system's undamped equations of motion are derived, one can easily use this approach to determine its instability regions. For convenience, a rotor-bearing system with periodic axial loaded is used as a parametrically excited representative gyroscopic system. The approach rewrites the second order differential equations in the state space form. Then the generalized eigenvalue problem is solved to derive the left and right eigenvectors. They are used to decouple the governing equations and reduce the order. In the subsequent theoretical derivation, the multiple scale method is applied to obtain the analytical solutions of the boundaries of instability regions. The numerical simulation is also carried out to validate the analytical boundaries. Wherein the numerical instability regions are obtained by applying the discrete state transition matrix method. From the theoretical and numerical analysis, we find out: (1) the analytical boundaries match well with the numerical results; (2) only the sum type instability regions can be observed; (3) the primary difference type instability regions do not exist; (4) the secondary or higher order difference type instability regions also may not exist. [ABSTRACT FROM AUTHOR]

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]

A model of the oscillations of a sensitive element of a hemispherical rate-integrating gyroscope is considered taking into account the geometric and electric nonlinearities of the system. The equations of motion of the sensitive element under parametric excitation of oscillations are examined. The transients are plotted taking into account the viscous friction. Analytical expressions for the steady-state amplitude and phase in the region of parametric resonance at the eigenfrequency of the resonator are obtained. The effect of the factor of negative electrostatic stiffness of the electrode excitation system on the process of generation of parametric oscillations is investigated. [ABSTRACT FROM AUTHOR]