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 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) domain 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.

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.

Revisiting the space-time gradient method:A time-clocking perspective, high order difference time discretization and comparison with the harmonic balance method

This paper revisits the Space-Time Gradient(STG) method which was developed for efficient analysis of unsteady flows due to rotor–stator interaction and presents the method from an alternative time-clocking perspective. The STG method requires reordering of blade passages according to their relative clocking positions with respect to blades of an adjacent blade row. As the space-clocking is linked to an equivalent time-clocking, the passage reordering can be performed according to the alternative time-clocking. With the time-clocking perspective, unsteady flow solutions from different passages of the same blade row are mapped to flow solutions of the same passage at different time instants or phase angles. Accordingly, the time derivative of the unsteady flow equation is discretized in time directly, which is more natural than transforming the time derivative to a spatial one as with the original STG method. To improve the solution accuracy, a ninth order difference scheme has been investigated for discretizing the time derivative. To achieve a stable solution for the high order scheme, the implicit solution method of Lower-Upper Symmetric GaussSeidel/Gauss-Seidel(LU-SGS/GS) has been employed. The NASA Stage 35 and its blade-countreduced variant are used to demonstrate the validity of the time-clocking based passage reordering and the advantages of the high order difference scheme for the STG method. Results from an existing harmonic balance flow solver are also provided to contrast the two methods in terms of solution stability and computational cost.

Almost Periodic Solutions by the Harmonic Balance Method

We consider non-autonomous ordinary differential equations in two cases.One is the one-dimensional case that admits a condition of hyperbolicity,and the other one is the higher-dimensional case that admits an exponential dichotomy.For differential equations of this kind,we give a rigorous treatment of the Harmonic Balance Method to look for almost periodic solutions.

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.

Subharmonic resonance of a clamped-clamped buckled beam with 1:1 internal resonance under base harmonic excitations

The subharmonic resonance and bifurcations of a clamped-clamped buckled beam under base harmonic excitations are investigated.The nonlinear partial integrodifferential equation of the motion of the buckled beam with both quadratic and cubic nonlinearities is given by using Hamilton’s principle.A set of second-order nonlinear ordinary differential equations are obtained by spatial discretization with the Galerkin method.A high-dimensional model of the buckled beam is derived,concerning nonlinear coupling.The incremental harmonic balance(IHB)method is used to achieve the periodic solutions of the high-dimensional model of the buckled beam to observe the nonlinear frequency response curve and the nonlinear amplitude response curve,and the Floquet theory is used to analyze the stability of the periodic solutions.Attention is focused on the subharmonic resonance caused by the internal resonance as the excitation frequency near twice of the first natural frequency of the buckled beam with/without the antisymmetric modes being excited.Bifurcations including the saddle-node,Hopf,perioddoubling,and symmetry-breaking bifurcations are observed.Furthermore,quasi-periodic motion is observed by using the fourth-order Runge-Kutta method,which results from the Hopf bifurcation of the response of the buckled beam with the anti-symmetric modes being excited.

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.

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.

NES cell

A broadband adaptive vibration control strategy with high reliability and flexible versatility is proposed.The high vibration damping performance of nonlinear energy sink(NES)has attracted attention.However,targeted energy transfer may cause severe vibration of NES.Besides,it is difficult to realize pure nonlinear stiffness without the linear part.As a result,the reliability of NES is not high.The low reliability of NES has hindered its application in engineering.In addition,the performance of NES depends on its mass ratio of the primary system,and NES lacks versatility for different vibration systems.Therefore,this paper proposes the concept of NES cell.The advantages of the adaptive vibration control of NES are applied to cellular NES.By applying a large number of NES cells in parallel,the reliability of NES and its versatility to complex vibration structures are improved.An elastic beam is used as the primary vibration structure,and a limited NES is used as the cell.The relationship between the vibration suppression effect of NES cells and the number of NES cell is studied.In addition,the effect of the distribution of NES cells on the multi-mode resonance suppression of the beam is also studied.In summary,the mode of the primary structure can be efficiently controlled by a large number of lightweight NES cell.Therefore,the lightweight NES cell is flexible for vibration control of complex structures.In addition,it improves the reliability of NES applications.Therefore,the distributed application of NES cells proposed in this paper is a valuable vibration suppression strategy.

Harmonic balance-based approach for optimal time delay to control unstable periodic orbits of chaotic systems

As a classical technique for chaos suppression,the time-delayed feedback controlling strategy has been widely developed by stabilizing unstable periodic orbits(UPOs)embedded in chaotic systems.A critical issue for achieving high controlling precision is to search for an appropriate time delay.This paper proposes a simple yet effective approach,based on incremental harmonic balance method,to determine the optimal time delay in the delayed feedback controller.The time delay is adjusted within the iterative scheme provided by the proposed method,and finally converges to the period of the target UPO.As long as the optimal time delay is fixed,moreover,the attained solution makes it quite convenient to analyze its stability according to the Floquet theory,which further provides the effective interval of the feedback gain.

Research on linear/nonlinear viscous damping and hysteretic damping in nonlinear vibration isolation systems

A nonlinear vibration isolation system is promising to provide a high-efficient broadband isolation performance.In this paper,a generalized vibration isolation system is established with nonlinear stiffness,nonlinear viscous damping,and Bouc-Wen(BW)hysteretic damping.An approximate analytical analysis is performed based on a harmonic balance method(HBM)and an alternating frequency/time(AFT)domain technique.To evaluate the damping effect,a generalized equivalent damping ratio is defined with the stiffness-varying characteristics.A comprehensive comparison of different kinds of damping is made through numerical simulations.It is found that the damping ratio of the linear damping is related to the stiffness-varying characteristics while the damping ratios of two kinds of nonlinear damping are related to the responding amplitudes.The linear damping,hysteretic damping,and nonlinear viscous damping are suitable for the small-amplitude,medium-amplitude,and large-amplitude conditions,respectively.The hysteretic damping has an extra advantage of broadband isolation.

Forced response of quadratic nonlinear oscillator: comparison of various approaches

The primary resonances of a quadratic nonlinear system under weak and strong external excitations are investigated with the emphasis on the comparison of different analytical approximate approaches.The forced vibration of snap-through mechanism is treated as a quadratic nonlinear oscillator.The Lindstedt-Poincar′e method, the multiple-scale method, the averaging method, and the harmonic balance method are used to determine the amplitude-frequency response relationships of the steady-state responses.It is demonstrated that the zeroth-order harmonic components should be accounted in the application of the harmonic balance method.The analytical approximations are compared with the numerical integrations in terms of the frequency response curves and the phase portraits.Supported by the numerical results, the harmonic balance method predicts that the quadratic nonlinearity bends the frequency response curves to the left.If the excitation amplitude is a second-order small quantity of the bookkeeping parameter,the steady-state responses predicted by the second-order approximation of the LindstedtPoincar′e method and the multiple-scale method agree qualitatively with the numerical results.It is demonstrated that the quadratic nonlinear system implies softening type nonlinearity for any quadratic nonlinear coefficients.

Nonlinear vibration analysis of pipeline considering the effects of soft nonlinear clamp

Soft nonlinear support is a major engineering project,but there are few relevant studies.In this paper,a dynamic pipeline model with soft nonlinear supports at both ends is established.By considering the influence of the Coriolis force and centrifugal force,the dynamical coupling equation of fluid-structure interaction is derived with extended Hamilton’s principle.Then,the approximate analytical solutions are sought via the harmonic balance method.The amplitude-frequency response curves show that different effects can be determined by approximate analysis.It is demonstrated that the increase in the fluid velocity can increase the amplitude of the pipeline system.The frequency range of unstable response increases when the fluid pressure raises.The combination of the soft nonlinear clamp and the large geometrical deformation of the pipeline affects the nonlinear vibration characteristic of the system,and the external excitation force and damping have significant effects on the stability.

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.

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.

接地限幅型非线性能量汇

非线性能量汇(NES)以能量定向传输和共振自动捕获的特性在被动振动控制方面表现突出.为了提高实用性,本文提出了一种接地限幅型NES.该设计可以封装在一个盒子里.然而,由于NES的振动区域有限,需要考虑其控制效率.因此,通过近似解析方法、数值仿真方法,特别是强迫振动实验,研究接地限幅型NES的动力学特性.本研究中涉及的所有参数均通过基于实验平台的参数辨识得到.通过双曲正切函数将分段非线性拟合成连续非线性.然后在分析处理过程中使用谐波平衡法(HBM)来提高解的准确性.直接数值方法产生的结果与实验记录一起验证了解析方法.对控制参数的分析表明,适当的限位弹簧可以有效地降低NES的振动,同时控制效率的代价很小.总之,这项工作为NES的约束提供了一种简单可靠的方法,有利于NES的设计,拓宽了NES在工程中的应用.

Flutter analysis of an airfoil with multiple nonlinearities and uncertainties

An original method for calculating the limit cycle oscillations of nonlinear aeroelastic system is presented.The problem of detemining the maximum vibration amplitude of limit cycle is transfomed into a nonlinear optimization problem.The hamonic balance method and the Floquet theory are selected to construct the general nonlinear equality and inequality constraints.The resulting constrained maximization problem is then solved by using the MultiStart algorithm.Finally,the proposed approach is validated and used to analyse the limit cycle oscillations of an airfoil with multiple nonlinearities and uncertainties.Numerical examples show that the coexistence of multiple nonlinearities may lead to low amplitude limit cycle oscillation.