Subharmonic Behaviors Research Articles

Parametric resonance and suppression for L-shaped pipe conveying pulsating fluid

This work establishes a new dynamic model for solving parametric resonance and suppression of a L-shaped pipe conveying pulsating fluid, which can be extended to arbitrary pipe configurations. Firstly, the linear dynamic analysis for the L-shaped pipe conveying steady-state fluid is performed. The new results show that there are more wave numbers of the mode shape than those for the straight pipe. With increasing the flow velocity, the natural frequency is decreased and the mode shape is changed from symmetry to asymmetric characteristics. Subsequently, the nonlinear dynamic analysis is conducted for the L-shaped pipe conveying pulsating fluid. It is found that primary and subharmonic resonance behaviors are newly produced due to parametric excitations. At the same time, the subharmonic resonance occurs for the pipe to have much higher vibration amplitudes than the primary resonance. Finally, the strategy of adding an intermediate support is proposed to suppress vibrations of the L-shaped pipe conveying pulsating fluid. Results indicate that there is an optimal position for the L-shaped pipe to add the intermediate support, which can significantly reduce the vibration amplitude. This optimal position is exactly where the fundamental frequency of the pipe system is maximized.

1/2 and 1/3 Subharmonic Resonance Behaviors of a Rotating FG-CNTRC Beam with Geometric Imperfections

This paper aims at investigating 1/2 and 1/3 subharmonic resonances of a rotating functionally graded carbon nanotube-reinforced composite (FG-CNTRC) beam in the presence of geometric imperfections. A general function is introduced to denote three types of imperfections containing sine, global, and local modes. At first, based on the von-Kármán geometric nonlinearity assumption, the forced vibration equation is set up. Then, the Galerkin method and multiple scale method are utilized to study subharmonic resonance behaviors. Finally, coupled effects of FG-CNTRCs, rotating motion, and geometric imperfections on subharmonic resonance responses are investigated. It is found that a specific range of the excitation amplitude and frequency motivates the subharmonic resonance behavior. The coupling of geometric imperfections and geometric nonlinearities generates the 1/2 resonance phenomena, and geometric nonlinearities result in the 1/3 resonance behavior. Both the imperfection mode and amplitude can affect the resonance response and resonance regions. Moreover, in contrast with the 1/3 subharmonic resonance, geometric imperfections produce greater influence on the 1/2 subharmonic resonance.

Nonlinear phenomena in vibrations of embedded carbon nanotubes conveying viscous fluid

Various nonlinear phenomena such as bifurcations and chaos in the responses of carbon nanotubes (CNTs) are recognized as being major contributors to the inaccuracy and instability of nanoscale mechanical systems. Therefore, the main purpose of this paper is to predict the nonlinear dynamic behavior of a CNT conveying viscous fluid and supported on a nonlinear elastic foundation. The proposed model is based on nonlocal Euler–Bernoulli beam theory. The Galerkin method and perturbation analysis are used to discretize the partial differential equation of motion and obtain the frequency-response equation, respectively. A detailed parametric study is reported into how the nonlocal parameter, foundation coefficients, fluid viscosity, and amplitude and frequency of the external force influence the nonlinear dynamics of the system. Subharmonic, quasi-periodic, and chaotic behaviors and hardening nonlinearity are revealed by means of the vibration time histories, frequency-response curves, bifurcation diagrams, phase portraits, power spectra, and Poincaré maps. Also, the results show that it is possible to eliminate irregular motion in the whole range of external force amplitude by selecting appropriate parameters.

An improved time-varying mesh stiffness calculation method and dynamic characteristic analysis for helical gears under variable torque conditions

Based on the slice theory and numerical calculation methods, the time varying meshing stiffness (TVMS) calculation method of the gear pair was constructed for the helical gear pair of a electric continuously variable transmission (ECVT), and the TVMS of the helical gear were calculated under different constant and time-varying torque in this study. The obtained stiffness values were introduced into the established dynamics model of helical gear system, and the influence of changed TVMS, resulting from the variable speed and torque, on the nonlinear dynamic characteristics of gear pair was analyzed using the Runge-Kutta method. The results show that the proposed TVMS calculation method is very effective in computing the TVMS of helical gear pair under time-varying condition. The TVMS and dynamic transmission error (DTE) increase as the torque increase, and vice versa. Meanwhile, the system exhibits a diverse range of periodic, sub-harmonic, and chaotic behaviors at high speed and at low speed when torque is low, whereas not appear at steady high torque under low speed range. This study offers a method for gaining the TVMS of helical gear pair to analyze the dynamic characteristics of gear pair under actual working condition, and the influence of torque that change with the speed ratio on the dynamic performance of a gear system should be considered in the industrial applications.

Load Resistance Optimization of a Broadband Bistable Piezoelectric Energy Harvester for Primary Harmonic and Subharmonic Behaviors

In this study, optimization of the external load resistance of a piezoelectric bistable energy harvester was performed for primary harmonic (period-1T) and subharmonic (period-3T) interwell motions. The analytical expression of the optimal load resistance was derived, based on the spectral analyses of the interwell motions, and evaluated. The analytical results are in excellent agreement with the numerical ones. A parametric study shows that the optimal load resistance depended on the forcing frequency, but not the intensity of the ambient vibration. Additionally, it was found that the optimal resistance for the period-3T interwell motion tended to be approximately three times larger than that for the period-1T interwell motion, which means that the optimal resistance was directly affected by the oscillation frequency (or oscillation period) of the motion rather than the forcing frequency. For broadband energy harvesting applications, the subharmonic interwell motion is also useful, in addition to the primary harmonic interwell motion. In designing such piezoelectric bistable energy harvesters, the frequency dependency of the optimal load resistance should be considered properly depending on ambient vibrations.

Numerical and experimental investigation of a spur gear pair with unloaded static transmission error

PurposeThe purpose of this paper is to investigate the dynamic responses of a spur gear pair with unloaded static transmission error (STE) excitation numerically and experimentally and the influences of the system factors including mesh stiffness, error excitation and torque on the dynamic transmission error (DTE).Design/methodology/approachA simple lumped parameters dynamic model of a gear pair considering time-varying mesh stiffness, backlash and unloaded STE excitation is developed. The STE is calculated from the measured tooth profile deviation under the unloaded condition. A four-square gear test rig is designed to measure and analyze the DTE and vibration responses of the gear pair. The dynamic responses of the gear transmission are studied numerically and experimentally.FindingsThe predicted numerical DTE matches well with the experimental results. When the real unloaded STE excitation without any approximation is used, the dynamic response is dominated by the mesh frequency and its high order harmonic components, which may not be result caused by the assembling error. The sub-harmonic and super-harmonic resonant behaviors are excited because of the high order harmonic components of STE. It will not certainly prevent the separations of mesh teeth when the gear pair is under the condition of high speed and heavy load.Originality/valueThis study helps to improve the modeling method of the dynamic analysis of spur gear transmission and provide some reference for the understanding of the influence of mesh stiffness, STE excitation and system torque on the vibration behaviors.

Nonlinear superharmonic and subharmonic resonances of piezoelectric elliptic thin plates under thermoelastic coupling effect

This paper aims at investigating the nonlinear vibration characteristics of the piezoelectric elliptic thin plate under the combined action of external harmonic excitation and temperature field. Based on the Von-Karman plate theory of large deflection, the nonlinear governing equation is derived by using the Bubnov-Galerkin method. Then, the amplitude-frequency response equations are further obtained using the multi-scale method, and the stability conditions of the steady solution are determined according to the Routh-Hurwitz criterion. The effects of the temperature difference at the plate center, the effective damping coefficient, the external excitation amplitude, and the ratio of the semi-major axis to the semi-minor axis on superharmonic and subharmonic resonances behaviors of the piezoelectric elliptic thin plate are analyzed. It is found that the coexistence region of multiple solutions and the resonant amplitude decrease for superharmonic resonance, as the temperature difference at the plate center increases. For subharmonic resonance, with the increase in the temperature difference at the plate center, the response amplitude increases, while the distance between two branches of the amplitude-frequency response curve has no obvious change. The softened/hardened nonlinear characteristics of the system are determined by the ratio of the semi-major axis to the semi-minor axis of the piezoelectric elliptic thin plate.

Harmonic resonances of graphene-reinforced nonlinear cylindrical shells: effects of spinning motion and thermal environment

This work investigates nonlinear harmonic resonance behaviors of graded graphene-reinforced composite spinning thin cylindrical shells subjected to a thermal load and an external excitation. The volume fraction of graphene platelets varies continuously in the shell’s thickness direction, which generates position-dependent useful material properties. Natural frequencies of shell traveling waves are derived by considering influences of the initial hoop tension, centrifugal and Coriolis forces, thermal expansion deformation, and thermal conductivity. A new Airy stress function is introduced. Harmonic resonance behaviors and their stable solutions for the spinning cylindrical shell are analyzed based on an equation of motion which is established by adopting Donnell’s nonlinear shell theory. The necessary and sufficient conditions for the existence of the subharmonic resonance of the spinning composite cylindrical shell are given. Besides the shell’s intrinsic structural damping, the Coriolis effect due to the spinning motion has a contribution to the damping terms of the system as well. Comparisons between the present analytical results and those in other papers are made to validate the existing solutions. Influences of main factors on vibration characteristics, primary resonance, and subharmonic resonance behaviors of the novel composite cylindrical shell are discussed. Furthermore, the mechanism of how the spinning motion affects the amplitude–frequency curves of harmonic resonances of the cylindrical shell is analyzed.

Parametric analysis for optimized piezoelectric bistable vibration energy harvesters

This study focuses on vibration energy harvesting with piezoelectric bistable inertial generators in order to provide an alternative or a support to chemical batteries in the power supply of isolated wireless sensors (leading to a better autonomy of these devices). The objective of the present work is to provide guidelines to optimize this kind of generator through the investigation of the influence of different parameters (load resistance, mass, stiffness, buckling level) on its frequency response. These guidelines will be obtained exploiting a new analytical model for piezoelectric bistable harvesters which will be constructed based on a recent model introduced for electromagnetic bistable harvesters. This model includes the study of subharmonic behaviors which can be used to enhance the global bandwidth of the harvester as well as a stability robustness criterion which allows a better prediction of experimental observations. The new model will be validated by experimental data and finally exploited to provide guidelines for piezoelectric bistable harvester optimization.

Bistable vibration energy harvester and SECE circuit: exploring their mutual influence

The present work focuses on vibration energy harvesting to replace chemical batteries as power supplies in wireless autonomous sensors. More particularly, this article deals with nonlinear inertial oscillators and more precisely bistable harvesters, which offer a wide bandwidth compared to linear structures. These harvesters require AC–DC electronic interface to be able to supply wireless sensors or small intermediate storage elements (used to smooth the power demand). However, few studies have investigated the influence of AC–DC circuits on bistable harvesters. This work therefore focuses on developing the first analytical model able to predict the frequency response of a bistable harvester coupled to one of these interfaces: the SECE (synchronized electric charge extraction) circuit. This mathematical model includes subharmonic behaviors, stability analyses to small disturbances and the analyses of the stability robustness of high orbits characterizing their capability to handle external disturbances in real conditions without falling on a low orbit. The model is then validated experimentally with a buckled beam bistable harvester with piezoelectric conversion. The latter shows that the use of SECE circuit leads to multiply the mean extracted power from 1.3 to 2.2 compared to a direct connection to the load. The reachable frequency range is nevertheless divided by 1.55.

Subharmonic resonance and critical eccentricity for the classical hydrogen atomic system

Subharmonic resonance behaviors are investigated for the classical hydrogen atom, with classical radiation damping and circularly polarized light acting on the classical electron. This study is intended for both potential experimental applications as well as for deeper theoretical purposes. Long resonant states are predicted for realistic Rydberg atoms and highly excited hydrogen states. Several previously undiscovered physical effects are predicted. First, the semimajor axis remains relatively constant when in subharmonic resonance; second, the eccentricity steadily increases until a maximum, critical value is reached, at which point orbital decay sets in. If the initial orbit is circular, this critical eccentricity value is shown to always be the same for each subharmonic condition, regardless of the initial orbital radius. An analytic derivation for this result is presented. The illustrated dynamics are of interest for the classical theory of stochastic electrodynamics (SED) regarding whether SED can fundamentally describe more of quantum phenomena, particularly atomic excited state behavior and related emission and absorption spectra. Also of interest are how classical resonances can be imposed on a near continuum of quantum states. Finally, there may be future technological applications, such as “reading” and “writing” information into Rydberg atoms in the form of subharmonic resonances.

Drastic bandwidth enhancement of bistable energy harvesters: Study of subharmonic behaviors and their stability robustness

In order to provide a serious alternative to chemical batteries for the energy supply of isolated sensors, bistable generators have been enthusiastically highlighted in recent years for their ability to harvest vibration energy over a wider frequency range compared to linear generators. Nevertheless, these bistable harvesters are generally characterized through a frequency sweep which does not reveal all the steady-state behaviors they can reach and therefore their full energy harvesting potential. Among such behaviors, subharmonic motions are hidden by this classical characterization and therefore had not received a lot of attention. This study proposes an original complete analytical analysis of subharmonic orbits for energy harvesting to predict their contribution to the global bandwidth of bistable generators. In addition, a new criterion, referred as stability robustness, is introduced to estimate the sensitivity of those behaviors to disturbances of different levels, allowing to finely and accurately estimate suitable behaviors for energy harvesting purposes in realistic conditions (behaviors easy to reach and maintain in time). Experimental results conducted with a buckled beam based electromagnetic generator confirm the relevance of this criterion showing good agreement with the analytical predictions. Subharmonic behaviors finally appear, both theoretically and experimentally, to be of significant interest, as exploiting them leads to a 180% increase of the global operating frequency range of the considered bistable energy harvester, for which more than 100 µW are generated on a 70 Hz bandwidth at 0.5 g.

Bifurcation and Chaos Analysis of Gear Pair System Based on Crack Rotor-Bearing System with Rub-Impact Effect

This study performs a systematic analysis of the dynamic behavior of a crack rotor-bearing system with rub-impact effect. The dynamic orbits of the system are observed using bifurcation diagrams plotted using the dimensionless damping coefficient, the dimensionless unbalance parameter and the dimensionless rotational speed ratio as control parameters. The analysis methods employed in this study include the dynamic trajectories of the crack rotor-bearing system, power spectra, Poincaré maps and bifurcation diagrams. Lyapunov exponent and fractal dimensional analysis are also used to identify the onset of chaotic motion. The numerical results reveal that the system exhibits a diverse range of periodic, sub-harmonic, quasi-periodic and chaotic behaviors. The results presented in this study provide an understanding of the operating conditions under which undesirable dynamic motion takes place in a crack rotor-bearing system and therefore serve as a useful source of reference for engineers in designing and controlling such systems.

Bifurcation and chaos analysis for multi-freedom gear-bearing system with time-varying stiffness

This study performs a systematic dynamic analysis of a gear-bearing system with time varying stiffness, gear backlash and surface friction. At first the period expansion method is proposed to build a six-degree-of freedom nonlinear dynamic model of a spur gear pair. Then the dynamic orbits of the system are observed using the bifurcation diagrams with the mesh stiffness and the rotational speed ratio as control parameters. And the onset of chaotic motion is identified from the phase diagrams, FFT spectra, Poincaré maps and the largest Lyapunov exponents of the gear-bearing system. The numerical results reveal that the system enters into chaotic motion under several frequency jumps with the increase of excitation frequency. When the support stiffness is raised, the number of frequency jumps increases, and the system exhibits a diverse range of periodic, sub-harmonic, and chaotic behaviors. In this study the results provide a useful source of reference for engineers and technicians in designing and controlling such systems.

OS0412-318 疲労き裂の性状とサブハーモニック波の発生挙動

Subharmonic ultrasound technique has been expected to improve the conventional ultrasonic inspection especially for SN ratio in crack tip echo measurement. We have developed a SPACE (Subharmonic Phased Array for Crack Evaluation) system and have confirmed the advantage for some industrial cracks inspection. In this study, we planned to observe subharmonic wave generation behavior with varied combinations of the crack openings and the amplitudes of incident ultrasound to investigate the mechanism of subharmonic wave at crack. For this purpose, we improved the angle transducer and ultrasonic pulser to increase the incident ultrasonic amplitude. The closed fatigue crack specimens were prepared and basic subharmonic behaviors were observed using the improved SPACE system by changing the applied slight load to control the crack opening and the incident ultrasonic amplitude.

Nonlinear behavior analysis of geared rotor bearing system featuring confluence transmission

In this paper, the nonlinear vibration characteristics of geared rotor bearing system and the interactions among gears, shafts, and plain journal bearings were studied. First, with the consideration of backlash, transmission error, time-varying mesh stiffness, and layout parameters, the dynamic model of geared rotor bearing system featuring confluence transmission was proposed. The nonlinear oil-film forces were computed with the Reynolds equation for finite-length journal bearings. Second, the responses of meshing vibration and bearing vibration were discussed. The numerical results revealed that the system exhibited a diverse range of periodic, sub-harmonic, and chaotic behaviors. Under different ranges of rolling frequency, the system got into chaos state through different roads. Moreover, in lower frequency, meshing vibration showed coexist of different periodic motions. Lastly, couplings of nonlinear oil-film force and nonlinear gear mesh force were discussed through a range of rolling frequencies. Gear-bearing dynamic interactions were demonstrated through the analysis of dynamic gear loads and dynamic bearing loads, and the coupling effect behaved different when rolling frequency changed.

Peak current control strategy with extended-state tracking compensator for DC-DC in hybrid energy storage system

Because variations of ultra-capacitor voltage and battery voltage generate subharmonic and chaotic behaviors in hybrid energy storage system (HESS) application when a DC-DC converter is under the peak current control, a novel digital control strategy, i.e., peak current control with extended-state tracking compensator, is introduced to deal with the stability. The gains of the control algorithm are selected based on pole locations formulated from the Bessel filter. The simulation results validate that under the peak current control strategy with compensator, the DC-DC converter does not have the subharmonic and chaotic behaviors. The response time under the peak current control with compensator is the same as that under the peak current control. The ripple voltage and ripple current of battery are less. The tracking error of inductor current tends to zero.

Bifurcation and Chaos of Gear Pair System Supported by Long Journal Bearings Based on Turbulent Flow Effect and Nonlinear Suspension Effect

A systematic analysis of the dynamic behavior of a gear-bearing system with nonlinear suspension, turbulent flow effect, long journal bearing approximation, nonlinear oil-film force and nonlinear gear mesh force is performed in the present study. The dynamic orbits of the system are observed by bifurcation diagrams plotted using the dimensionless unbalance coefficient and the dimensionless rotational speed ratio as control parameters. The onset of chaotic motion is identified from the phase diagrams, power spectra, Poincaré maps, Lyapunov exponents and fractal dimension of the gearbearing system. The numerical results reveal that the system exhibits a diverse range of periodic, sub-harmonic, quasiperiodic and chaotic behaviors. The results presented in this study provide some useful insights into the design and development of a gear-bearing system for rotating machinery that operates in highly rotational speed and highly nonlinear regimes.

Chaos suppression of uncertain gyros in a given finite time

The gyro is one of the most interesting and everlasting nonlinear dynamical systems, which displays very rich and complex dynamics, such as sub-harmonic and chaotic behaviors. We study the chaos suppression of the chaotic gyros in a given finite time. Considering the effects of model uncertainties, external disturbances, and fully unknown parameters, we design a robust adaptive finite-time controller to suppress the chaotic vibration of the uncertain gyro as quickly as possible. Using the finite-time control technique, we give the exact value of the chaos suppression time. A mathematical theorem is presented to prove the finite-time stability of the proposed scheme. The numerical simulation shows the efficiency and usefulness of the proposed finite-time chaos suppression strategy.

Couple‐Stress Fluid Improves Dynamic Response of Gear‐Pair System Supported by Journal Bearings

A systematic analysis of the dynamic behavior of a gear‐bearing system with nonlinear suspension, couple‐stress fluid flow effect, nonlinear oil‐film force, and nonlinear gear mesh force is performed in the present study. The dynamic orbits of the system are observed using bifurcation diagrams plotted using the dimensionless rotational speed ratio as a control parameter. The onset of chaotic motion is identified from the phase diagrams, power spectra, Poincaré maps, Lyapunov exponents and fractal dimension of the gear‐bearing system. The numerical results reveal that the system exhibits a diverse range of periodic, subharmonic, quasiperiodic, and chaotic behaviors. The couple‐stress fluid would be a useful lubricating fluid to suppress nonlinear dynamic responses and improve the steady of the systems. The results presented in this study provide some useful insights into the design and development of a gear‐bearing system for rotating machinery that operates in highly rotational speed and highly nonlinear regimes.