Effect Of Parametric Excitation Research Articles

Nonlinear vibration and amplitude ratio output performance of a resonant micro-gyroscope based on bifurcation detection

In this paper, we design a 3-degree-of-freedom (3-DOF) nonlinear resonant micro-gyroscope, which innovatively utilizes the bifurcation phenomenon of the nonlinear resonant beam as a detection method and uses the amplitude ratio before and after bifurcation as the sensitivity output of the system. The steady-state response of the driving equation is first solved by the complex exponential method. Coriolis force is amplified by the lever mechanism and transmitted to the axial direction of the resonant beam. The dimensions of the resonant beam are designed so that the frequency of Coriolis force is in a 2:1 relationship with the natural frequency of the resonant beam to enhance the parametric excitation effect. Subsequently, Hamilton principle and Galerkin method are used to derive and discretize the dynamical equations of the resonant beam containing axial force, respectively. The multi-scale method is used to perturbation analysis of discrete equations. Finally, the bifurcation characteristics and the amplitude-frequency response with different input angular velocities are studied. The results show that the comprehensive performance of the micro-gyroscope system using backward frequency sweep (BFS) is better than forward frequency sweep (FFS). Furthermore, by using the BFS, the relative sensitivity of the nonlinear resonant micro-gyroscope based on the amplitude ratio variation rises by about 168 times compared with that based on the frequency variation in the linear case. In addition, when considering the input angular velocity with the same magnitude but different directions, the bifurcation frequency of resonant beam is closely related to the direction of the input angular velocity, and the direction of the input angular velocity can be further identified by utilizing this phenomenon.

Coupling effect of time-varying axial tension and combined unsteady flow on vortex-induced vibration of flexible risers

The surge and heave motions of the floating platform have a non-negligible effect on vortex-induced vibration (VIV) of marine risers, which generates time-varying axial tension of risers, i.e., parametric excitation, and unsteady flow and complicates the prediction of VIV. A semi-empirical model is improved to simulate the VIV of flexible risers with parametric excitation under combined unsteady flow. Both the flow velocity and the stiffness matrix are updated with time. Fifty-five cases are simulated to investigate the separate and coupling effects of parametric excitation and combined unsteady flow on VIV. The phenomena such as sum and difference frequencies, Mathieu-type resonance, intermittent vibration and frequency fluctuation are observed. The effects of parameters such as excitation amplitude dT, excitation frequency fp, combined ratio r and oscillation period P on VIV are discussed in detail. Some interesting conclusions are drawn: VIV may be significantly suppressed when transition-low-frequency parametric excitation and short-period combined unsteady flow act together. Larger r makes the effect of combined unsteady flow more significant. VIV is mainly dominated by the parametric excitation when fp is twice the natural frequency of the riser corresponding to the dominant mode.

Parametric resonance and bifurcation analysis of thin-walled asymmetric gyroscopic composite shafts: An asymptotic study

Nonlinear geometries and cross-sectional asymmetry can be among the most critical contributing factors affecting the performance and stability of composite shafts. The influence of these factors, which should be devoted to particular attention in the design of these systems and have not been investigated yet, are evaluated analytically in this study. The shaft is simply supported, made of orthotropic multi-layers, and spinning at a constant speed. To express the nonlinear system’s behavior, which is due to the large amplitude of the vibrations, it is assumed that the shaft is under the stretching assumption. Moreover, a rectangular cross-section is used to model the asymmetry that results in parametric excitation in the system. To accurately investigate the behavior of composite material, an optimal lay-up is employed. The gyroscopic coupling is included because of Rayleigh beam theory, and Euler’s angles are employed to achieve the angular velocities. The analytical study of the parametrically excited system, obtained by the method of multiple scales, is performed in two categories of resonant and nonresonant cases. In the nonresonant case, the analytical investigation suggests that the asymmetric shaft behaves like a symmetric one therefore, the parametric excitations do not have a significant impact. This claim is confirmed by numerical results. Also, the presence of the gyroscopic coupling and hollowness of the shaft causes the beating phenomenon in the system. However, in the resonant case, the presence of parametric excitation plays a pivotal role. The results also show that under certain conditions and despite the presence of damping, asymmetric balanced rotors can have a nontrivial stable amplitude. This response exists as long as the parametric excitation effects dominate the damping effects. Although damping reduces the vibrations’ amplitude, it can improve the stability of the system and eliminates unstable responses. Furthermore, the time response and frequency response curve of the system is carefully evaluated for various geometric design parameters and operation speed. Depending on the operating speed, the system can experience supercritical or subcritical pitchfork bifurcation. In addition, a detailed description of the system’s Campbell’s diagram, damping effect, and bifurcation is provided. Furthermore, it is proved that internal resonance cannot occur in the system. The accuracy of the analytical responses of the resonant case is compared with numerical ones. Stability of trivial and nontrivial responses is discussed in the time response and phase portrait of the system simultaneously. Finally, in the undamped system, multi-frequency responses are appeared as homoclinic and heteroclinic closed orbits.

Quasi-Periodic Solutions of a Damped Nonlinear Quasi-Periodic Mathieu Equation by the Incremental Harmonic Balance Method With Two Time Scales

Abstract Quasi-periodic (QP) solutions of a damped nonlinear QP Mathieu equation with cubic nonlinearity are investigated by using the incremental harmonic balance (IHB) method with two time scales. The damped nonlinear QP Mathieu equation contains two incommensurate harmonic excitation frequencies, one is a small frequency while the other nearly equals twice the linear natural frequency. It is found that Fourier spectra of QP solutions of the equation consist of uniformly spaced sidebands due to cubic nonlinearity. The IHB method with two time scales, which relates to the two excitation frequencies, is adopted to trace solution curves of the equation in an automatical way and find all frequencies of solutions and their corresponding amplitudes. Effects of parametric excitation are studied in detail. Based on approximation of QP solutions by periodic solutions with a large period, Floquet theory is used to study the stability of QP solutions. Three types of QP solutions can be obtained from the IHB method, which agree very well with results from numerical integration. However, the perturbation method using the double-step method of multiple scales (MMS) obtains only one type of QP solutions since the ratio of the small frequency to the linear natural frequency of the first reduced-modulation equation is nearly 1 in the second perturbation procedure, while the other two types of QP solutions from the IHB method with two time scales do not need the ratio. Furthermore, the results from the double-step MMS are different from those numerical integration and the IHB method with two time scales.

Primary Parametric Amplification in a Weakly Forced Mathieu Equation

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

The Vibration of a Transversely Cracked Rotor Supported by Anisotropic Journal Bearings with Speed-Dependent Characteristic

This paper presents the vibration of a transversely cracked rotor supported by anisotropic journal bearings, where the speed-dependent characteristic of bearing is considered. A 3D finite element model and the contact-based approach are employed for the shaft and crack. The governing differential equations of the whole cracked rotor-bearing system were obtained by synthesizing the equations of the cracked shaft, the breathing crack and the journal bearings. In order to solve the computational difficulties caused by the high dimensions of model, the free-interface complex component mode synthesis method (CMS) is employed to reduce the order of the model. On this basis, the eigenvalue and the steady-state forced response of the cracked rotor-bearing system are obtained by the Hill’s method. Finally, the effects of the anisotropic and speed-dependent characteristics of bearings on the vibration of the system are studied. Numerical results show that both the two characteristics can significantly affect the response of the system. The anisotropy in the bearing leads to the split of resonant peaks and influence the amplitudes of the peaks. The speed-dependent characteristic mainly affects the responses at the speeds close to the resonant regions, because the parametric excitation effect of the resonance region is greater than other speeds.

Effect of parametric excitation on a bifractional-order damped system with a fractional-power nonlinearity.

Investigation on linear/nonlinear resonance phenomena and supercritical/subcritical pitchfork bifurcation mechanism is reported in a complex bifractional-order damped system which endures a high-frequency parametric excitation and contains fractional-power nonlinearity. The approximate theoretical expression of the linear response amplitude at the primary frequency and the superharmonic response amplitude at the second and third harmonic frequencies are obtained by utilizing an analytical method and an iterative formula. A numerical approximation scheme based on the Caputo derivative for the simulation of the system is introduced, showing sufficient precision. Due to the parametric excitation, analytical approximation expressions of the stable equilibrium points are given explicitly when the exponent is not an integer so that the pitchfork bifurcation, nonlinear resonance can be studied in an analytical way, exhibiting much more operability than the external excitation case. It is found that the fractional-order derivative may bring new multibifurcation and new multiresonance phenomena, which have not yet been reported before. With the variation of different control parameters of the system, the equivalent slow-varying system can be converted from bistability to monostability and finally to bistability. Unlike the cases of the system excited by bifrequency external excitation, the optimum response amplitude of the parametric excited system is not monotonous with respect to the values of the exponent. For a range of parameters of the system, it is also found that the superharmonic resonance at the second and third harmonic frequencies is affected deeply by the parametric excitation.

Parametric Excitation of Pre-Stressed Graphene Sheets under Magnetic Field: Nonlinear Vibration and Dynamic Instability

Motivated by the lack of sufficient accuracy in investigation of nonlinear dynamics of graphene sheets (GS), nonlinear dynamic instability and frequency response of the pre-stressed single layered GS (SLGS) are investigated in the present paper. To achieve this aim, in the first step, SLGS embedded on a visco-Pasternak foundation is modeled while it is under an initial stress and subjected to a parametric axial force and magnetic field. Then, based on Eringen’s theory, nonlinear von Karman relations and Kelvin–Voigt model, the nonlinear governing equation of motion is derived. In the next step, Galerkin technique and multiple time scales method are employed to analyze and solve the equation of motion. Emphasizing the effect of parametric excitation, for considering the instability regions, bifurcation points are discussed. As a result, a parametric study is conducted to show the importance of damping coefficient and parametric excitation in dynamic instability of the system. Numerical examples are also treated which show various discontinuous bifurcations. Also, infinitely stable and unstable solutions are addressed.

On pre-stressed functionally graded anisotropic nanoshell in magnetic field

Elastic bulk wave characteristics of doubly curved nanoshell made of functionally graded (FG) anisotropic material are studied. The effective properties of FG anisotropic material vary along the thickness direction. A higher-order shear deformation shell theory and the Bi-Helmholtz nonlocal strain gradient theory are, respectively, utilized to model the nanoshell as a continuum model and predicting the size-dependent behavior. The Hamiltonian principle is adopted to obtain the governing equations of wave motion. These equations are solved analytically to evaluate the wave characteristics of the nanoshell. Emphasizing the effect of parametric excitation, the influences of small-scale parameters, exponential factor, magnetic field intensity, initial stress, elastic foundation parameters, and wave number are assessed on the wave dispersion response of FG anisotropic nanoshell.

Nonlinear vibration and dynamic instability analysis nanobeams under thermo-magneto-mechanical loads: a parametric excitation study

The nonlinear vibration behavior and dynamic instability of Euler–Bernoulli nanobeams under thermo-magneto-mechanical loads is the main objective of the present paper. Firstly, a short Euler–Bernoulli nanobeam is modeled and exposed to an external parametric excitation. Based on the nonlocal continuum theory and nonlinear von Karman beam theory, the nonlinear governing differential equation of motion is derived. Secondly, to transport the partial differential equation to the ordinary differential equation, Galerkin method is applied. Then, multiple scales method, as an analytical approach, is used to solve the equation. At the end, modulation equation of Euler–Bernoulli nanobeams is obtained. Then, to evaluate the dynamic instability of the system, trivial and nontrivial steady-state solutions are discussed. Emphasizing the effect of parametric excitation, for considering the instability regions, bifurcation points are studied and investigated. As a result, it can be observed that the damping coefficient plays an effective role as well as parametric excitation in stability and frequency response of the system.

Parametric excitation of Euler–Bernoulli nanobeams under thermo-magneto-mechanical loads: Nonlinear vibration and dynamic instability

The nonlinear vibration behavior and dynamic instability of Euler–Bernoulli nanobeams under thermo-magneto-mechanical loads is the main objective of present paper. Firstly, a short Euler–Bernoulli nanobeam is modeled and exposed to an external parametric excitation. Based on the nonlocal continuum theory and nonlinear von Karman beam theory, the nonlinear governing differential equation of motion is derived. Secondly, to transport the partial differential equation to the ordinary differential equation, Galerkin method is applied. Then, multiple scales method, as an analytical approach, is used to solve the equation. At the end, modulation equation of Euler–Bernoulli nanobeams is obtained. Then, to evaluate the dynamic instability of the system, trivial and nontrivial steady-state solutions are discussed. Emphasizing the effect of parametric excitation, for considering the instability regions, bifurcation points are studied and investigated. As a results, it can be observed that the damping coefficient plays an effective role as well as parametric excitation in stability and frequency response of the system.

Global stability effects of parametric excitation

Stability investigations of general non-conservative parametrically excited systems with asynchronous excitation are presented. Focusing on the global stability effects outside of the traditional resonance areas, systems with two degrees of freedom are considered featuring displacement- and/or velocity-proportional parametric excitation with variable phase relations. In particular, facing the lack of studies on this subject, special attention is paid to time-periodic systems containing gyroscopic and circulatory terms. Through the application of the semi-analytical method of normal forms, general conditions for the appearance of possible global effects are derived. Apart from the “total instability” – presently the only known global effect – new stabilizing and destabilizing effects affecting the stability over the whole range of excitation frequencies are discovered. The derived conditions show, that such global effects are expected to be rather common in complex mechanical system, especially those, featuring circulatory terms. The qualitative analytical results are also confirmed by numerical stability analysis based on Floquet theory. As a mechanical example a minimal model of a squealing disk brake is examined. It is shown that this complex model is indeed subject to parametric excitation leading to global stability effects. These findings may contribute to a better understanding of the squealing phenomenon. Further, the newly obtained knowledge may as well be utilized for extended vibration suppression in mechanical systems in the context of parametric anti-resonance.

The Nonlinear Properties of a Piezoelectric Cantilevered Plate under Parametric Excitation

In this paper, we investigate the nonlinear dynamics characteristics of the composite piezoelectric cantilevered plate forced under parametric excitation. Based on the Reddy’s classic deformation plate theory and the von Karman nonlinear strain–displacement equations, we deduce the nonlinear partial differential equations by applying the Hamilton’s principle. Then the dimensionless ordinary differential equations of the system with three-degree-of-freedom are derived. We choose a set of parameters of governing equation and employ numerical simulations to investigate the effects of parametric excitation on the nonlinear responses of the system. The results indicate that there exist the periodic and chaotic motions.

Nonlinear Parametric Excitation Effect Induces Stability Transitions in Swimming Direction of Flexible Superparamagnetic Microswimmers.

Microscopic artificial swimmers have recently become highly attractive due to their promising potential for biomedical microrobotic applications. Previous pioneering work has demonstrated the motion of a robotic microswimmer with a flexible chain of superparamagnetic beads, which is actuated by applying an oscillating external magnetic field. Interestingly, they have shown that the microswimmer's orientation undergoes a 90°-transition when the magnetic field's oscillation amplitude is increased above a critical value. This unexpected transition can cause severe problems in steering and manipulation of flexible magnetic microrobotic swimmers. Thus, theoretical understanding and analysis of the physical origins of this effect are of crucial importance. In this work, we investigate this transition both theoretically and experimentally by using numerical simulations and presenting a novel flexible microswimmer with an anisotropic superparamagnetic head. We prove that this effect depends on both frequency and amplitude of the oscillating magnetic field, and demonstrate existence of an optimal amplitude achieving maximal swimming speed. Asymptotic analysis of a minimal two-link model reveals that the changes in the swimmer's direction represent stability transitions, which are induced by a nonlinear parametric excitation.

The effect of parametric excitation on a dynamic system of cutting process

The problem of ensuring stability of movements of the tool about workpiece taking into account parametric self-excitation is considered. The factors causing periodic changes of parameters in dynamic system when cutting are analyzed. They are connected with spatial anisotropy of properties of elasticity in a subsystem of the processed workpiece with allowance variations on workpiece rotation perimeter, with kinematic perturbation from mechanical part of drives of executive elements of the machine, with periodic processes in a cutting zone. The generalized dynamic model of system with periodically changing parameters is given. Influence of periodically changing parameters on stability of trajectories is studied. The features of loss of stability specific to dynamic systems of cutting are disclosed. In particular, it is shown that due to parametrical effects with increasing cutting speed at increase of rotational frequency of spindle there is always a critical frequency at which the system loses stability.

Controllable parametric excitation effect on linear and nonlinear vibrational resonances in the dynamics of a buckled beam

In this study, the effect of a controllable parametric excitation on both linear and nonlinear vibrational resonances on the dynamic of a buckled beam excited by a combination of uncontrollable low- and high-frequency periodic forces are investigated. First of all, the beam dynamic is assumed to be constrained by two periodic and independent ambient solicitations, such as wind and earthquake. An axial load of the beam represented by a periodic and parametric excitation is used to control the vibrational resonance phenomenon, induced by the presence of the two external excitations. Approximate analytical expressions for the linear response and the high-frequency force amplitude at which linear vibrational resonance occurs are obtained. An analytical expression of the amplitude of the nonlinear response at the superharmonic equal to the double of the low-frequency, is obtained. For all these expressions, we show the effect of the parametric excitation. We compare all the obtained results with the ones of the case where, the parametric force is absent. It is shown that, the presence of the parametric excitation permit the suppression of both linear and nonlinear vibrational resonances. Moreover, the vibration amplitudes of the buckled beam are significantly reduced, around certain threshold values for the amplitude and the frequency of the parametric excitation.

Raman parametric excitation effect upon the third harmonic generation by a metallic nanoparticle lattice

This work is a theoretical study on third harmonic generation in the nonlinear propagation of an intense laser pulse through a periodic three-dimensional lattice of nanoparticles. Using a perturbative method, the nonlinear equations that describe the laser–nanoparticle interaction in the weakly relativistic regime are derived. Additionally, the nonlinear dispersion relation and the amplitude of the third harmonic are obtained. Finally, the effects of the nanoparticle radius and separation length, the distribution of the nanoparticle electron density, and the laser frequency upon the third harmonic efficiency are investigated. In addition to the expected resonance that occurs when the third harmonic resonates with the plasmon wave, another resonance appears when the nonlinear interaction of the fundamental mode with the third harmonic excites a longitudinal collective plasmon wave via the parametric Raman mechanism.

Suppressing boring bar vibrations by parametric excitation

Machine tool chatter causes an instability that can occur in tool-workpiece-machine system. Hence surface roughness and tool wear in metal cutting processes. This instability caused by interaction between current tool positions and surface left by previous tool passes. Many methods have been developed to minimize the effects of regenerative chatter in machine tools and to enhance the damping capability minimizing the loss in static stiffness through implementation of device uses PZT actuators or MR fluid. The possibility of suppressing selfexcited vibrations of boring process using parametric excitation is discussed. We consider a two-mass system of which the main mass is excited by a self excited force (cutting force). A single mass which acts as a dynamic absorber is attached to the main mass and, by varying the stiffness between the main mass and the absorber mass, represents a parametric excitation. Cutting experiments with an excitation current of different waveforms and diverse frequencies show that chatter can be significantly suppressed by the effect of parametric excitation.

Parametrically excited nonlinear piezoelectric compact wind turbine

Abstract A nonlinear piezoelectric rotary transducer is developed that makes compact low speed wind generators realizable. Compact wind generators provide power to sensor nodes in remote or hard to reach locations. Since the locations of the sensor nodes are not optimized in terms of wind speed, the compact wind generators should be able to produce power from low speed wind. At smaller scales piezoelectric transduction becomes more effective than electromagnetic transduction. Therefore one way of realizing the compact wind turbines is by replacing the electromagnetic generator with a piezoelectric transducer. This work presents a novel piezoelectric transducer where the rotation of the blades results in large oscillations of piezoelectric beams. The piezoelectric bimorphs are made bi-stable by incorporation of repelling magnetic force. The Magnetic force is due to interaction of permanent magnets at the tip of the beams with permanent magnets rotating with the blades. Since the magnetic force changes with blade rotation, the dynamics of the beams changes in time and the system is thus parametrically excited. Two configurations are presented one called tangential configuration and the other is radial configuration. An 80 mm × 80 mm × 175 mm nonlinear piezoelectric wind generator can generate milliwatts of power from wind as slow as 2 ms −1 . The proposed compact wind generators are experimentally investigated in two steps. First the piezoelectric transducer is examined through constant rotational speed tests. Second wind tunnel experiments are performed to characterize the entire wind generator. An analytical model is developed for the piezoelectric rotational transducer. The model is verified with the experimental results. The nonlinear phenomena captured by the experimental investigations are explained using the analytical model. The model is also used for more case studies identifying specifically the effect of parametric excitations.

System-level circuit simulation of nonlinearity in micromechanical resonators

We here present an equivalent circuit model capable of capturing the effects of nonlinearity in micromechanical resonators, particularly those driven and sensed by capacitive transducers. A charge controlled capacitor and a variable capacitor are developed to capture the nonlinear nature of the mechanical restoring force and the capacitive transducer. Distinctive features of the frequency response (like spring softening, hysteresis) induced by mechanical and capacitive nonlinearity are highlighted by illustrating the simulation results of a well-studied longitudinal beam resonator, which are verified against analytical solution of the Duffing equation. The circuit simulation approach is capable of capturing prominent parametric excitation effects in narrow-gap capacitive transducers which are commonly excluded in the approximations of the analytical model for simplicity. Finally, we constructed the full equivalent circuit for a fully differential Lamé bulk mode microresonator. The comparison between experimental and simulation results shows the proposed model can well capture the nonlinear behavior of this resonator which illustrates an approach to realize full system-level circuit simulation of microresonator based devices, such as precision resonant sensors, mechanical RF filters and MEMS oscillators.