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.

Harmonic balance simulation of the influence of component uniformity and reliability on the performance of a Josephson traveling wave parametric amplifier

A Josephson traveling wave parametric amplifier(JTWPA),which is a quantum-limited amplifier with high gain and large bandwidth,is the core device of large-scale measurement and control systems for quantum computing.A typical JTWPA consists of thousands of Josephson junctions connected in series to form a transmission line and hundreds of shunt LC resonators periodically loaded along the line for phase matching.Because the variation of these capacitors and inductors can be detrimental to their high-frequency characteristics,the fabrication of a JTWPA typically necessitates precise processing equipment.To guide the fabrication process and further improve the design for manufacturability,it is necessary to understand how each electronic component affects the amplifier.In this paper,we use the harmonic balance method to conduct a comprehensive study on the impact of nonuniformity and fabrication yield of the electronic components on the performance of a JTWPA.The results provide insightful and scientific guidance for device design and fabrication processes.

A vertical track nonlinear energy sink

Eliminating the effects of gravity and designing nonlinear energy sinks(NESs)that suppress vibration in the vertical direction is a challenging task with numerous damping requirements.In this paper,the dynamic design of a vertical track nonlinear energy sink(VTNES)with zero linear stiffness in the vertical direction is proposed and realized for the first time.The motion differential equations of the VTNES coupled with a linear oscillator(LO)are established.With the strong nonlinearity considered of the VTNES,the steady-state response of the system is analyzed with the harmonic balance method(HBM),and the accuracy of the HBM is verified numerically.On this basis,the VTNES prototype is manufactured,and its nonlinear stiffness is identified.The damping effect and dynamic characteristics of the VTNES are studied theoretically and experimentally.The results show that the VTNES has better damping effects when strong modulation responses(SMRs)occur.Moreover,even for small-amplitude vibration,the VTNES also has a good vibration suppression effect.To sum up,in order to suppress the vertical vibration,an NES is designed and developed,which can suppress the vertical vibration within certain ranges of the resonance frequency and the vibration intensity.

时域谐波平衡求解器中非物理解出现根源探析

The time domain harmonic balance method is an attractive reduced order method of analyzing unsteady flow for turbomachines. However, the method can admit non-physical solutions. Non-physical solutions were encountered from a three-blade-row compressor configuration in a time domain harmonic balance analysis. This paper aims to investigate the root cause of the non-physical solutions. The investigation involves several strategies, which include increasing the number of harmonics, increasing the number of time instants, including scattered modes,including the rotor-rotor interaction, and the use of a new method-the approximate time domain nonlinear harmonic method. Numerical analyses pertinent to each strategy are presented to reveal the root cause of the non-physical solution. It is found that the nonlinear interaction of unsteady flow components with different fundamental frequencies is the cause of the non-physical solution. The non-physical solution can be eliminated by incorporating extra scattered modes or using the approximate time domain nonlinear harmonic method.

Harmonic balance method with alternating frequency/time domain technique for nonlinear dynamical system with fractional exponential

Comparisons of the common methods for obtaining the periodic responses show that the harmonic balance method with alternating frequency/time （HB-AFT） do- main technique has some advantages in dealing with nonlinear problems of fractional exponential models. By the HB-AFT method, a rigid rotor supported by ball bearings with nonlinearity of Hertz contact and ball passage vibrations is considered. With the aid of the Floquet theory, the movement characteristics of interval stability are deeply studied. Besides, a simple strategy to determine the monodromy matrix is proposed for the stability analysis.

Improving convergence of incremental harmonic balance method using homotopy analysis method

We have deduced incremental harmonic balance an iteration scheme in the （IHB） method using the harmonic balance plus the Newton-Raphson method. Since the convergence of the iteration is dependent upon the initial values in the iteration, the convergent region is greatly restricted for some cases. In this contribution, in order to enlarge the convergent region of the IHB method, we constructed the zeroth-order deformation equation using the homotopy analysis method, in which the IHB method is employed to solve the deformation equation with an embedding parameter as the active increment. Taking the Duffing and the van der Pol equations as examples, we obtained the highly accurate solutions. Importantly, the presented approach renders a convenient way to control and adjust the convergence.

Spectral Method for Three-Dimensional Nonlinear Klein-Gordon Equation by Using Generalized Laguerre and Spherical Harmonic Functions

In this paper,a generalized Laguerre-spherical harmonic spectral method is proposed for the Cauchy problem of three-dimensional nonlinear Klein-Gordon equation. The goal is to make the numerical solutions to preserve the same conservation as that for the exact solution.The stability and convergence of the proposed scheme are proved.Numerical results demonstrate the efficiency of this approach.We also establish some basic results on the generalized Laguerre-spherical harmonic orthogonal approximation,which play an important role in spectral methods for various problems defined on the whole space and unbounded domains with spherical geometry.

Incremental harmonic balance method for periodic forced oscillation of a dielectric elastomer balloon

Dielectric elastomer(DE) is suitable in soft transducers for broad applications,among which many are subjected to dynamic loadings, either mechanical or electrical or both. The tuning behaviors of these DE devices call for an efficient and reliable method to analyze the dynamic response of DE. This remains to be a challenge since the resultant vibration equation of DE, for example, the vibration of a DE balloon considered here is highly nonlinear with higher-order power terms and time-dependent coefficients. Previous efforts toward this goal use largely the numerical integration method with the simple harmonic balance method as a supplement. The numerical integration and the simple harmonic balance method are inefficient for large parametric analysis or with difficulty in improving the solution accuracy. To overcome the weakness of these two methods,we describe formulations of the incremental harmonic balance(IHB) method for periodic forced solutions of such a unique system. Combined with an arc-length continuation technique, the proposed strategy can capture the whole solution branches, both stable and unstable, automatically with any desired accuracy.

Primary resonance of fractional-order Duffing–van der Pol oscillator by harmonic balance method

The dynamical properties of fractional-order Duffing–van der Pol oscillator are studied, and the amplitude–frequency response equation of primary resonance is obtained by the harmonic balance method. The stability condition for steady-state solution is obtained based on Lyapunov theory. The comparison of the approximate analytical results with the numerical results is fulfilled, and the approximations obtained are in good agreement with the numerical solutions. The bifurcations of primary resonance for system parameters are analyzed. The results show that the harmonic balance method is effective and convenient for solving this problem, and it provides a reference for the dynamical analysis of similar nonlinear systems.

INCREMENTAL HARMONIC BALANCE METHOD FOR AIRFOIL FLUTTER WITH MULTIPLE STRONG NONLINEARITIES

The incremental harmonic balance method was extended to analyze the flutter of systems with multiple structural strong nonlinearities. The strongly nonlinear cubic plunging and pitching stiffness terms were considered in the flutter equations of two-dimensional airfoil. First, the equations were transferred into matrix form, then the vibration process was divided into the persistent incremental processes of vibration moments. And the expression of their solutions could be obtained by using a certain amplitude as control parameter in the harmonic balance process, and then the bifurcation, limit cycle flutter phenomena and the number of harmonic terms were analyzed. Finally, numerical results calculated by the Runge-Kutta method were given to verify the results obtained by the proposed procedure. It has been shown that the incremental harmonic method is effective and precise in the analysis of strongly nonlinear flutter with multiple structural nonlinearities.

Fully discrete Jacobi-spherical harmonic spectral method for Navier-Stokes equations

A fully discrete Jacobi-spherical harmonic spectral method is provided for the Navier-Stokes equations in a ball. Its stability and convergence are proved. Numerical results show efficiency of this approach. The proposed method is also applicable to other problems in spherical geometry.

APPLICATION OF HARMONIC ANALYSIS METHOD TO RESEARCH ON ROTOR AIRLOADS

According to the rotor vortex theory,the rotor circulation and the rotor induced velocity are developed into Fourier series.The circulation distribution along blade spanwise is expressed in terms of segment-by-segment linear functions.In consequence the induced velocity equations and the circulation equations are derived.The engineering application of the rotor vortex theory is provided.Then the induced velocity and its harmonic components are obtained to provide a quantitative basis for the vortex model.For calculating each harmonic component of the induced velocity a simplified method is put forward which considers the effects of each order circulation with neglecting those of higher order.The method saves the computer time and is of significant benefit.

Harmonics analysis of the ITER poloidal field converter based on a piecewise method

Poloidal field（PF） converters provide controlled DC voltage and current to PF coils. The many harmonics generated by the PF converter flow into the power grid and seriously affect power systems and electric equipment. Due to the complexity of the system, the traditional integral operation in Fourier analysis is complicated and inaccurate. This paper presents a piecewise method to calculate the harmonics of the ITER PF converter. The relationship between the grid input current and the DC output current of the ITER PF converter is deduced. The grid current is decomposed into the sum of some simple functions. By calculating simple function harmonics based on the piecewise method, the harmonics of the PF converter under different operation modes are obtained.In order to examine the validity of the method, a simulation model is established based on Matlab/Simulink and a relevant experiment is implemented in the ITER PF integration test platform.Comparative results are given. The calculated results are found to be consistent with simulation and experiment. The piecewise method is proved correct and valid for calculating the system harmonics.

Professor Ruixia Pei's Clinical Experience in Treating Globus Hystericus with"Harmonizing Method"

To summarize professor Pei9s experience in treating globus hystericus.Methods:Learn from your teacher.Results:Professor Pei had been suffering from the disease for more than 30 years and had her unique views on the treatment of globus hystericus,which are mostly for emotional dysfunction,liver Qi is not comfortable,functioning of Qi is not adjusted,Yin and Yang imbalance.The basic principle of treatment is to grasp the core pathogenesis,take harmony as the method and balance as the duration,harmonize Qi,regulate Yin and Yang,use drugs to disperse the liver and rectify Qi,as well as auxiliary treatment with products to promote blood circulation,remove blood stasis,dryness and dampness,and clear heat.

Clinical Experience of"'Harmonizing MethodM based on Usage of Minor Bupleurum Decoction

Minor Bupleurum Decoction is a common TCM compound and a classic prescription that can best reflect the"harmonizing method',of TCM.But there are many differences on the usage and dosage of Minor Bupleurum Decoction.Combined with the clinical treatment experience,this paper summarizes the experience of Minor Bupleurum Decoction in the clinical treatment of diseases.It is hoped that relevant suggestions can be provided for clinicians to improve the clinical therapeutic effect of Minor Bupleurum Decoction.

Adjacent mode resonance of a hydraulic pipe system consisting of parallel pipes coupled at middle points

The coupling vibration of a hydraulic pipe system consisting of two pipes is studied.The pipes are installed in parallel and fixed at their ends,and are restrained by clips to one bracket at their middle points.The pipe subjected to the basement excitation at the left end is named as the active pipe,while the pipe without excitation is called the passive pipe.The clips between the two pipes are the bridge for the vibration energy.The adjacent natural frequencies will enhance the vibration coupling.The governing equation of the coupled system is deduced by the generalized Hamilton principle,and is discretized to the modal space.The modal correction is used during the discretization.The investigation on the natural characters indicates that the adjacent natural frequencies can be adjusted by the stiffness of the two clips and bracket.The harmonic balance method(HBM)is used to study the responses in the adjacent natural frequency region.The results show that the vibration energy transmits from the active pipe to the passive pipe swimmingly via the clips together with a flexible bracket,while the locations of them are not node points.The adjacent natural frequencies may arouse wide resonance curves with two peaks for both pipes.The stiffness of the clip and bracket can release the vibration coupling.It is suggested that the stiffness of the clip on the passive pipe should be weak and the bracket should be strong enough.In this way,the vibration energy is reflected by the almost rigid bracket,and is hard to transfer to the passive pipe via a soft clip.The best choice is to set the clips at the pipe node points.The current work gives some suggestions for weakening the coupled vibration during the dynamic design of a coupled hydraulic pipe system.

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 harmonic-wave bio-thermal method for continuous monitoring skin thermal conductivity and capillary perfusion rate

The revelation of thermal energy exchange mechanism of human body is challenging yet worthwhile,because it can clearly explain the changes in human symptoms and health status.Understanding,the heat transfer of the skin is significant because the skin is the foremost organ for the energy exchange between the human body and the environment.In order to diagnose the physiological conditions of human skin without causing any damage,it is necessary to use a non-invasive measurement technique by means of a conformal flexible sensor.The harmonic method can minimize the thermal-induced injury to the skin due to its low heat generating properties.A novel type of computational theory assessing skin thermal conductivity,blood perfusion rate of capillaries in the dermis,and superficial subcutaneous tissues was formed by combining the multi-medium thermal diffusion model and the bio-thermal model(Pennes equation).The skins of the hand back of six healthy subjects were measured.It was found that the results revealed no consistent changes in thermal conductivity were observed across genders and ages.The measured blood perfusion rates were within the range of human capillary flow.It was found that female subjects had a higher perfusion rate range(0.0058-0.0061 s^(-1))than male subjects(0.0032-0.0049 s^(-1)),which is consistent with invasive medical studies about the gender difference in blood flow rates and stimulated effects in relaxation situations.

A Methodology for Assessing Axial Compressor Stability with Inlet Temperature Ramp Distortion

Based on a small perturbation stability model for periodic flow,the effects of inlet total temperature ramp distortion on the axial compressor are investigated and the compressor stability is quantitatively evaluated.In the beginning,a small perturbation stability model for the periodic flow in compressors is proposed,referring to the governing equations of the Harmonic Balance Method.This stability model is validated on a single-stage low-speed compressor TA36 with uniform inlet flow.Then,the unsteady flow of TA36 with different inlet total temperature ramps and constant back pressure is simulated based on the Harmonic Balance Method.Based on these simulations,the compressor stability is analyzed using the proposed small perturbation model.Further,the Dynamic Mode Decomposition method is employed to accurately extract pressure oscillations.The two parameters of the temperature ramp,ramp rate and Strouhal number,are discussed in this paper.The results indicate the occurrence and extension of hysteresis loops in the rows,and a decrease in compressor stability with increasing ramp rate.Compressor performance is divided into two phases,stable and limit,based on the ramp rate.Furthermore,the model predictions suggest that a decrease in period length and an increase in Strouhal number lead to improved compressor stability.The DMD results imply that for compressors with inlet temperature ramp distortion,the increase of high-order modes and oscillations at the rotor tip is always the signal of decreasing stability.

Nonlinear dynamic analysis of cantilevered piezoelectric energy harvesters under simultaneous parametric and external excitations

The nonlinear dynamics of cantilevered piezoelectric beams is investigated under simultaneous parametric and external excitations. The beam is composed of a substrate and two piezoelectric layers and assumed as an Euler-Bernoulli model with inextensible deformation. A nonlinear distributed parameter model of cantilevered piezoelectric energy harvesters is proposed using the generalized Hamilton＇s principle. The proposed model includes geometric and inertia nonlinearity, but neglects the material nonlinearity. Using the Galerkin decomposition method and harmonic balance method, analytical expressions of the frequency-response curves are presented when the first bending mode of the beam plays a dominant role. Using these expressions, we investigate the effects of the damping, load resistance, electromechanical coupling, and excitation amplitude on the frequency-response curves. We also study the difference between the nonlinear lumped-parameter and distributed- parameter model for predicting the performance of the energy harvesting system. Only in the case of parametric excitation, we demonstrate that the energy harvesting system has an initiation excitation threshold below which no energy can be harvested. We also illustrate that the damping and load resistance affect the initiation excitation threshold.