Theoretical analysis of adaptive harmonic window and its application in frequency extraction of vibration signal

The goal of this paper is to find an excellent adaptive window function for extracting the weak vibration signal and high frequency vibration signal under strong noise.The relationship between windowing transform andfiltering is analyzed first in the paper.The advantage of adjustable time-frequency window of wavelet transform is introduced.Secondly the relationship between harmonic wavelet and multiple analytic band-pass filter is analyzed.The coherence of the multiple analytic band-pass filter and harmonic wavelet base function is discussed,and the characteristic that multiple analytic band-pass filter included in the harmonic wavelet transform is founded.Thirdly,by extending the harmonic wavelet transform,the concept of the adaptive harmonic window and its theoretical equation without decomposition are put forward in this paper.Then comparing with the Hanning window,the good performance of restraining side-lobe leakage possessed by adaptive harmonic window is shown,and the adaptive characteristics of window width changing and analytical center moving of the adaptive harmonic window are presented.Finally,the proposed adaptive harmonic window is applied to weak signal extraction and high frequency orbit extraction of high speed rotor under strong noise,and the satisfactory results are achieved.The application results show that the adaptive harmonic window function can be successfully applied to the actual engineering signal processing.

Nonlinear vibration analysis of a circular composite plate harvester via harmonic balance

Alumped parameter transversevibration model of a composite plate harvester is analyzed via harmonic balance approaches. The harvester is mainly composed of a piezoelectriccircular composite clamped by two steel rings and a proof mass on the plate.The lumped parameter model is a 1.5 degree-of-freedom strongly nonlinear system with a higher order polynomial stiffness. Aharmonic balance approach is developed to analyze the system, and the resulting algebraic equations are numerically solved by adopting an arc-length continuation technique. Anincremental harmonic balance approach is also developedfor the lumped parameter model. The two approaches yieldthe same results.The amplitude-frequency responses produced by the harmonic balance approach are validated by the numericalintegrations and the experimental data. The investigation reveals that there coexist hardening and softening characteristics in the amplitude-frequency response curves under sufficiently large excitations. The harvester with thecoexistenceof hardening and softening nonlinearitiescan outperform not only linear energy harvesters but also typical hardening nonlinear energy harvesters.

Nonlinear vibration analysis of a circular micro-plate in two-sided NEMS/MEMS capacitive system by using harmonic balance method

In this study, forced nonlinear vibration of a circular micro-plate under two-sided electrostatic, two-sided Casimir and external harmonic forces is investigated analytically. For this purpose, at first, von Karman plate theory including geometrical nonlinearity is used to obtain the deflection of the micro-plate. Galerkin decomposition method is then employed, and nonlinear ordinary differential equations (ODEs) of motion are determined. A harmonic balance method (HBM) is applied to equations and analytical relation for nonlineaT frequency response (F-R) curves are derived for two categories (including and neglecting Casimir force) separately. The analytical results for three cases:(1) semi-linear vibration;(2) weakly nonlinear vibration;(3) highly non linear vibration, are validated by comparing with the numerical solutio ns. After validation, the effects of the voltage and Casimir force on the natural frequency of two-sided capacitor system are investigated. It is shown that by assuming Casimir force in small gap distances, reduction of the natural frequency is considerable. The influences of the applied voltage, damping, micro-plate thickness and Casimir force on the frequency response curves have been presented too. The results of this study can be useful for modeling circular parallel-plates in nano /microelectromechanical transducers such as microphones and pressure sensors.

Nonlinear vibration of embedded single-walled carbon nanotube with geometrical imperfection under harmonic load based on nonlocal Timoshenko beam theory

Based on the nonlocal continuum theory, the nonlinear vibration of an embedded single-walled carbon nanotube （SWCNT） subjected to a harmonic load is in- vestigated. In the present study, the SWCNT is assumed to be a curved beam, which is unlike previous similar work. Firstly, the governing equations of motion are derived by the Hamilton principle, meanwhile, the Galerkin approach is carried out to convert the nonlinear integral-differential equation into a second-order nonlinear ordinary differ- ential equation. Then, the precise integration method based on the local linearzation is appropriately designed for solving the above dynamic equations. Besides, the numerical example is presented, the effects of the nonlocal parameters, the elastic medium constants, the waviness ratios, and the material lengths on the dynamic response are analyzed. The results show that the above mentioned effects have influences on the dynamic behavior of the SWCNT.

STEADY-STATE RESPONSES AND THEIR STABILITY OF NONLINEAR VIBRATION OF AN AXIALLY ACCELERATING STRING

The steady-state transverse vibration of an axially moving string with geometric nonlinearity was investigated. The transport speed was assumed to be a constant mean speed with small harmonic variations. The nonlinear partial-differential equation that governs the transverse vibration of the string was derived by use of the Hamilton principle. The method of multiple scales was applied directly to the equation. The solvability condition of eliminating the secular terms was established. Closed form solutions for the amplitude and the existence conditions of nontrivial steady-state response of the two-to-one parametric resonance were obtained. Some numerical examples showing effects of the mean transport speed, the amplitude and the frequency of speed variation were presented. The Liapunov linearized stability theory was employed to derive the instability conditions of the trivial solution and the nontrivial solutions for the two-to-one parametric resonance. Some numerical examples highlighting influences of the related parameters on the instability conditions were presented.

HELICOPTER VIBRATION REDUCTION TECHNIQUE WITH HARMONIC ACTIVE CONTROL

In forward flight, the helicopter performance is seriously affected by the airframe vibration caused by asymmetrical airflow through the rotor disk. so the higher harmonic rotor blade pitch control technique is becoming an active subject in the 1980's, which is one of the active control techniques of helicopter vibration reduction and whose effectivity has been demonstrated by flight testing. In the view of control theory, this paper interprets the basic principles of higher harmonic control(HHC) and proposes mathematical models of the rotor and of airframe vibration, as well as methods for identification of the parameters. A higher harmonic controller is designed and the results of numerical simulation show that the vibration level is reduced effectively.

Ab Initio Calculations on Nitramide and Its Methyl Derivatives( Ⅱ )——The Harmonic Force Fields and Vibrational Spectra of Nitramide and Its Isotopic Derivatives

The harmonic force field and the vibrational spectrum of nitramide were calculated by using the ab initio gradient program TEXAS at the Hartree-Fock level with a 4-21G basis set. The directly computed theoretical harmonic force field was scaled by using empirical scale factors which are transferred from other molecules and provided an a priori prediction of fundamental frequencies and intensities. The average deviations between predicted vibrational frequencies of nitramide and experimental IR spectrum in an argon matrix are 63 cm-1 for symmetric vibrations and 41 cm-1 for antisymmetric modes. A new set of scale factors was optimized in this paper. These scale factors reduced the average deviations to 2. 3 cm-1 for symmetric modes and 0. 8 cm-1 for antisymmetric ones. The vibrational spectra of three isotopic derivatives of nitramide were predicted by using the force field resulted from the optimized set of scale factors, which are in good agreement with their experimental data in an argon matrix.

Investigation of analytical harmonic frequency and potential energy function, vibrational levels for the X^2∑^＋ and A^2Л states of CN radical

This paper calculates the equilibrium structure and the potential energy functions of the ground state （X^2∑^＋） and the low lying excited electronic state （A^2Л） of CN radical are calculated by using CASSCF method. The potential energy curves are obtained by a least square fitting to the modified Murrell-Sorbie function. On the basis of physical theory of potential energy function, harmonic frequency （ωe） and other spectroscopic constants （ωeχe, βe and αe） are calculated by employing the Rydberg-Klei-Rees method. The theoretical calculation results are in excellent agreement with the experimental and other complicated theoretical calculation data. In addition, the eigenvalues of vibrational levels have been calculated by solving the radial one-dimensional SchrSdinger equation of nuclear motion using the algebraic method based on the analytical potential energy function.

Theoretical Analysis on the Vibrational Resonance in Two Coupled Overdamped Anharmonic Oscillators

Under both low- and high-frequency signals,the phenomenon of vibrational resonance in two coupled overdamped anharmonic oscillators with time delay feedback is investigated.By separating the slow and fast motions,the approximate solution of the response amplitude of the coupled oscillators is obtained,and the analytical results are in good agreement with the numerical ones.In the delay-free systems,the traditional vibrational resonance is presented,and in the delayed systems,the delay-induced periodic pattern of the vibrational resonance profile with respect to the delay parameter is revealed.

Vibrational Resonance in Fractional-Order Anharmonic Oscillators

The phenomenon of vibrational resonance in fractional-order anharmonic oscillators is investigated.Based on the method of separating slow and fast motions,the approximate solution of the response amplitude is obtained.Both analytical and numerical results show that not only the high-frequency signal but also the fractional-order damping can induce vibrational resonance.The present results provide a new way to control periodical signals in coupled systems.

High-order harmonic generation of CO_2 with different vibrational modes in an intense laser field

We apply the strong-field Lewenstein model to demonstrate the high-order harmonic generation of CO2 with three vibrational modes（balance vibration,bending vibration,and stretching vibration） driven by an intense laser field.The results show that the intensity of harmonic spectra is sensitive to molecular vibrational modes,and the high harmonic efficiency with stretching vibrational mode is the strongest.The underlying physical mechanism of the harmonic emission can be well explained by the corresponding ionization yield and the time-frequency analysis.Finally,we demonstrate the attosecond pulse generation with different vibrational modes and an isolated attosecond pulse with a duration of about 112 as is generated.

A novel way for vibration control of FGM fluid-conveying pipes via NiTiNOL-steel wire rope

In this study,a coupling model of fluid-conveying pipes made of functionally graded materials(FGMs)with NiTiNOL-steel(NiTi-ST)for vibration absorption is investigated.The vibration responses of the FGM fluid-conveying pipe with NiTi-ST are studied by the Galerkin truncation method(GTM)and harmonic balance method(HBM).The harmonic balance solutions and the numerical results are consistent.Also,the linearized stability of the structure is determined.The effects of the structure parameters on the absorption performance are also studied.The results show that the NiTi-ST is an effective means of vibration absorption.Furthermore,in studying the effect of the NiTi-ST,a closed detached response(CDR)is first observed.It is noteworthy that the CDR may dramatically change the vibration amplitude and that the parameters of the NiTi-ST may determine the emergence or disappearance of the CDR.This vibration absorption device can be extended to offer more general vibration control in engineering applications.

A novel adaptive harmonic balance method with an asymptotic harmonic selection

The harmonic balance method(HBM)is one of the most widely used methods in solving nonlinear vibration problems,and its accuracy and computational efficiency largely depend on the number of the harmonics selected.The adaptive harmonic balance(AHB)method is an improved HBM method.This paper presents a modified AHB method with the asymptotic harmonic selection(AHS)procedure.This new harmonic selection procedure selects harmonics from the frequency spectra of nonlinear terms instead of estimating the contribution of each harmonic to the whole nonlinear response,by which the additional calculation is avoided.A modified continuation method is proposed to deal with the variable size of nonlinear algebraic equations at different values of path parameters,and then all solution branches of the amplitude-frequency response are obtained.Numerical experiments are carried out to verify the performance of the AHB-AHS method.Five typical nonlinear dynamic equations with different types of nonlinearities and excitations are chosen as the illustrative examples.Compared with the classical HBM and Runge-Kutta methods,the proposed AHB-AHS method is of higher accuracy and better convergence.The AHB-AHS method proposed in this paper has the potential to investigate the nonlinear vibrations of complex high-dimensional nonlinear systems.

Highly adaptive and broadband triboelectric energy harvester with stretching silicone rubber strip for variable harmonic frequency vibration

An enormous number of wireless sensing nodes(WSNs)are of great significance for the Internet of Things(IoT).It is tremendously prospective to realize the in-situ power supply of WSNs by harvesting unutilized mechanical vibration energy.A harmonic silicone rubber triboelectric nanogenerator(HSR-TENG)is developed focusing on ubiquitous constant working frequency machinery.The unique design of the strip serving as a flexible resonator realizes both soft contact and high and broadband output.The significant factors influencing the 1^(st)-order vibration mode of the strip are developed for realizing the harmonic frequency adaptation to external vibration.The surface treatment of the strip improves the output performance of HSR-TENG by 49.1%as well as eliminates the adhesion effect.The HSR-TENG is able to achieve a voltage output bandwidth of 19 Hz under a vibration strength of 3.0,showing its broadband capability.The peak power density of 153.9 W/m^(3)is achieved and 12×0.5 W light-emitting diodes(LEDs)are successfully illuminated by the HSR-TENG.It can continuously power a temperature sensor by harvesting the actual compressor vibration energy.In brief,the HSR-TENG provides a promising way for constant frequency vibration energy harvesting,so as to achieve in-situ power supply for the WSNs in the vicinity.

Vibration harmonic suppression technology for electromagnetic vibrators based on an improved sensorless feedback control method

To realize low harmonic distortion of the vibration waveform output from electromagnetic vibrators,we propose a vibration harmonic suppression technology based on an improved sensorless feedback control method.Without changing the original driving circuit,the alternating current(AC)equivalent resistance of the driving coil is used to obtain high-precision vibration velocity information,and then a simple and reliable velocity feedback control system is established.Through the study of the effect of different values of key parameters on the system,we have achieved an effective expansion of the velocity characteristic frequency band of low-frequency vibration,resulting in an enhanced harmonic suppression capability of velocity feedback control.We present extensive experiments to prove the effectiveness of the proposed method and make comparisons with conventional control methods.In the frequency range of 0.01-1.00 Hz,without using any sensors,the method proposed in this study can reduce the harmonic distortion of the vibration waveform by about 40%compared to open-loop control and by about 20%compared to a conventional sensorless feedback control method.

Characteristic of Torsional Vibration of Mill Main Drive Excited by Electromechanical Coupling

In the study of electromechanical coupling vibration of mill main drive system, the influence of electrical system on the mechanical transmission is considered generally, however the research for the mechanism of electromechanical interaction is lacked. In order to research the electromechanical coupling resonance of main drive system on the F3 mill in a plant, the cycloconverter and synchronous motor are modeled and simulated by the MTLAB/SIMUL1NK firstly, simulation result show that the current harmonic of the cycloconverter can lead to the pulsating torque of motor output. Then the natural characteristics of the mechanical drive system are calculated by ANSYS, the result show that the modal frequency contains the component which is close to the coupling vibration frequency of 42Hz. According to the simulation result of the mechanical and electrical system, the closed loop feedback model including the two systems are built, and the mechanism analysis of electromechanical coupling presents that there is the interaction between the current harmonic of electrical system and the speed of the mechanical drive system. At last, by building and computing the equivalent nonlinear dynamics model of the mechanical drive system, the dynamic characteristics of system changing with the stiffness, damping coefficient and the electromagnetic torque are obtained. Such electromechanical interaction process is suggested to consider in research of mill vibration, which can induce strong coupling vibration behavior in the rolling mill drive system.

Steady-state responses of a belt-drive dynamical system under dual excitations

The stable steady-state periodic responses of a belt-drive system with a one-way clutch are studied. For the first time, the dynamical system is investigated under dual excitations. The system is simultaneously excited by the firing pulsations of the engine and the harmonic motion of the foundation. Nonlinear discrete-continuous equations are derived for coupling the transverse vibration of the belt spans and the rotations of the driving and driven pulleys and the accessory pulley. The nonlinear dynamics is studied under equal and multiple relations between the frequency of the fir- ing pulsations and the frequency of the foundation motion. Furthermore, translating belt spans are modeled as axially moving strings. A set of nonlinear piecewise ordinary differ- ential equations is achieved by using the Galerkin truncation. Under various relations between the excitation frequencies, the time histories of the dynamical system are numerically simulated based on the time discretization method. Further- more, the stable steady-state periodic response curves are calculated based on the frequency sweep. Moreover, the convergence of the Galerkin truncation is examined. Numer- ical results demonstrate that the one-way clutch reduces the resonance amplitude of the rotations of the driven pul- ley and the accessory pulley. On the other hand, numerical examples prove that the resonance areas of the belt spans are decreased by eliminating the torque-transmitting in the opposite direction. With the increasing amplitude of the foun- dation excitation, the damping effect of the one-way clutch will be reduced. Furthermore, as the amplitude of the firing pulsations of the engine increases, the jumping phenomena in steady-state response curves of the belt-drive system with or without a one-way clutch both occur.

STUDY ON SENSOR-BASED CONTROL METHOD FOR HARMONIC DRIVE SYSTEM

A new type of robust traje ctory tracking control for harmonic using joint torque sensor and joint acceleration sensor information is concerned with.Joint torque sensor information is used to compensate the uncertainty of link and load parameters. Joint acceleration feedback control will enhace the robustness of the driving system, resist the dynamic uncertainties and disturbing torque acted on the joint axis within definite bandwidth, improve the joint tracking performance, and resist the vibration of the load side of the harmonic drive system. Experimental studies are carried out and comparison of several controllers , such as PD and sensor- based control, the experimental results clearly illustrate the effectiveness of the proposed methods.

Nonlinear vibration of functionally graded graphene platelet-reinforced composite truncated conical shell using first-order shear deformation theory

In this study,the first-order shear deformation theory(FSDT)is used to establish a nonlinear dynamic model for a conical shell truncated by a functionally graded graphene platelet-reinforced composite(FG-GPLRC).The vibration analyses of the FG-GPLRC truncated conical shell are presented.Considering the graphene platelets(GPLs)of the FG-GPLRC truncated conical shell with three different distribution patterns,the modified Halpin-Tsai model is used to calculate the effective Young’s modulus.Hamilton’s principle,the FSDT,and the von-Karman type nonlinear geometric relationships are used to derive a system of partial differential governing equations of the FG-GPLRC truncated conical shell.The Galerkin method is used to obtain the ordinary differential equations of the truncated conical shell.Then,the analytical nonlinear frequencies of the FG-GPLRC truncated conical shell are solved by the harmonic balance method.The effects of the weight fraction and distribution pattern of the GPLs,the ratio of the length to the radius as well as the ratio of the radius to the thickness of the FG-GPLRC truncated conical shell on the nonlinear natural frequency characteristics are discussed.This study culminates in the discovery of the periodic motion and chaotic motion of the FG-GPLRC truncated conical shell.

Cascaded quasi-zero stiffness nonlinear low-frequency vibration isolator inspired by human spine

Human motion induced vibration has very low frequency,ranging from 2 Hz to 5 Hz.Traditional vibration isolators are not effective in low-frequency regions due to the trade-off between the low natural frequency and the high load capacity.In this paper,inspired by the human spine,we propose a novel bionic human spine inspired quasi-zero stiffness(QZS)vibration isolator which consists of a cascaded multi-stage negative stiffness structure.The force and stiffness characteristics are investigated first,the dynamic model is established by Newton’s second law,and the isolation performance is analyzed by the harmonic balance method(HBM).Numerical results show that the bionic isolator can obtain better low-frequency isolation performance by increasing the number of negative structure stages,and reducing the damping values and external force values can obtain better low-frequency isolation performance.In comparison with the linear structure and existing traditional QZS isolator,the bionic spine isolator has better vibration isolation performance in low-frequency regions.It paves the way for the design of bionic ultra-low-frequency isolators and shows potential in many engineering applications.