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 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 Methods:A Review and Recent Developments

The harmonic balance(HB)method is one of the most commonly used methods for solving periodic solutions of both weakly and strongly nonlinear dynamical systems.However,it is confined to low-order approximations due to complex symbolic operations.Many variants have been developed to improve the HB method,among which the time domain HB-like methods are regarded as crucial improvements because of their fast computation and simple derivation.So far,there are two problems remaining to be addressed.i)A dozen of different versions of HB-like methods,in frequency domain or time domain or in hybrid,have been developed;unfortunately,misclassification pervades among them due to the unclear borderlines of different methods.ii)The time domain HB-like methods suffer from non-physical solutions,which have been shown to be caused by aliasing(mixture of the high-order into the low-order harmonics).Although a series of dealiasing techniques have been developed over the past two decades,the mechanism of aliasing and the final solution to dealiasing are still not well known to the academic community.This paper aims to provide a comprehensive review of the development of HB-like methods and enunciate their principal differences.In particular,the time domain methods are emphasized with the famous aliasing phenomenon clearly addressed.

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.

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.

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.

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.

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.

混联Ⅱ型惯容非线性能量阱的动力学特性研究

分别使用非线性恢复力、非线性阻尼替代惯容减振系统中的线性恢复力、线性阻尼,并考虑摩擦力的影响,提出了混联Ⅱ型惯容非线性能量阱.建立了主系统的动力学方程,利用谐波平衡法求解系统在简谐激励下的幅频响应曲线.采用弧长算法和数值法相结合的方法研究了系统的惯质比、非线性阻尼、非线性刚度和摩擦力单个参数对其减振性能的影响.发现非线性刚度和非线性阻尼数值的增大会使峰值先减小后增大,不同的是,前者幅频响应曲线逐渐向右上方向弯曲,后者产生峰值的位置向低频段转移.分析了惯质比、非线性阻尼、非线性刚度3种参数两两组合下对系统减振效果的影响.研究表明,在激励幅值为0.005 m时,惯质比和阻尼同时变化减振效果最好:当ε=0.1时,系统主结构位移峰值的最小值约为0.01 m;而在参数ε=0.001时,整体取值范围内其最大值约为0.061 m;当惯质比取得最佳值0.1时,非线性阻尼和非线性刚度κ_(21)的取值范围变大.在摩擦力的作用下,系统的最大幅值都有不同程度的增加.上述研究可为振动系统减振的研究提供参考.

面向光栅刻划机的负刚度结构隔振性能优化

大面积光栅刻划机实验室存在的复杂振动因素,会对它的运行及刻画精度造成影响,针对光栅刻划机设计一种正负刚度并联的被动隔振结构,改善仅有空气弹簧时对低频隔振无效的现象,拓宽隔振频带。为模拟隔振器隔振性能,建立系统的动力学模型和动力学方程,使用仿真软件MATLAB进行仿真,为磁致负刚度弹簧进行尺寸结构设计,对比并联磁致负刚度弹簧前后的隔振效果。仿真结果表明采用空气弹簧和磁致负刚度结构并联后能够有效的隔离低频振动,并且磁吸致负刚度和磁斥致负刚度进行组合能够有效的改善单一磁吸致负刚度或磁斥致负刚度结构的非线性。

直齿锥齿轮系统动态特性参数解域界结构

为研究含间隙直齿锥齿轮系统的幅值跳跃、齿面冲击特性,用谐波平衡法建立系统动力学方程。在频率与间隙、时变刚度、静态综合误差和负载构建的参数平面内用Broyden拟牛顿法和伪弧长延续算法获得系统参数解域界结构,探索系统幅值跳跃、多值解和啮合冲击特性对参数的敏感性。结果表明在系统啮频和轴频主共振区均出现幅值跳跃和多值解现象,在啮频处动态特性更为复杂丰富。小间隙下存在严重的齿面冲击现象,当间隙b>0.98时系统的跳跃与齿面冲击均趋于稳定,系统动态特性对时变啮合刚度不敏感,在高速轻载、较大齿面误差激励下系统的非线性跳跃和齿面冲击均加剧。参数解域界结构为锥齿轮结构设计提供数据支持。

基于非线性能量阱的汽车传动系统扭振抑制研究

对非线性能量阱(nonlinear energy sink, NES)在汽车传动系统扭振抑制中的应用进行了研究。根据传动系统的结构和振动特点,建立了简化的3自由度传动系统-NES耦合动力学模型;基于增量谐波平衡法联合增量弧长法,推导并求解了耦合系统的频率响应,利用Floquet理论对周期解的稳定性进行判断;在频域和时域上对系统的非线性动力学响应及其影响因素进行了分析,并基于能量谱研究了NES的减振性能;最后,基于扩展的5自由度非线性模型对NES进行了参数优化和验证。结果表明,NES的减振性能受其自身刚度、阻尼及发动机激励幅值影响,合理设计NES参数可以高效抑制汽车传动系统的扭转共振,而不恰当的NES参数会促使系统发生高分支周期响应,导致异常振动峰值出现,经优化后的NES可以仅5%的惯量比使传动系统转速波动均方根值降低41.3%,减振效果显著。该研究可为NES在传动系统扭振抑制中的应用及其参数设计提供参考。

大展弦比板弯扭耦合受迫振动分析

对于工程广泛存在的大展弦比结构,首次研究了大展弦比板弯扭耦合受迫振动响应.考虑弦向相对弯曲刚度远大于展向弯曲刚度,将弦向截面等效为刚体,考虑其绕展向轴线的扭转,应用Hamilton原理分别计算大展弦比板的势能与动能,建立两端简支的大展弦比板弯扭耦合受迫振动的动力学控制方程.将动力学控制方程进行Galerkin截断后,通过谐波平衡法和微分求积法/微分求积单元法,计算得到了大展弦比模型弯扭耦合受迫振动稳态响应的解析解及数值解,同时验证了解的截断收敛性及谐波收敛性.最后研究了外激励幅值对结构受迫振动响应的影响.结果将为工程中广泛存在的大展弦比结构设计提供指导.

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.

电力系统摇摆方程的侵入式谐波解及分析

开展了大干扰背景下电力系统摇摆方程的侵入式谐波解研究。首先,针对二次微分代数系统的初值问题,提出了系统化的二次型谐波平衡方法;其次,将电网运动方程映射至解耦空间,扩展了摇摆方程的高阶分块解耦形式,并借助二次型谐波平衡方法,给出了摇摆方程的侵入式谐波解;然后,基于该谐波解分析了电网稳定运动形态特征的数学机理,建立了电力系统暂态稳定的非线性状态-输出方程;最后,单机无穷大系统、IEEE 3机9节点系统的仿真算例进一步验证了所提方法及理论分析的有效性。

基于IHBM的汽车非线性悬架系统定量研究

分析了汽车悬架系统和轮胎的非线性弹簧力和阻尼力,建立了二自由度汽车非线性垂向振动系统的动力学模型.结合增量谐波平衡方法(incremental harmonic balance method,IHBM),对该系统的动力学行为进行定量研究.推导其增量谐波平衡过程,研究增量谐波平衡法的迭代计算过程,采用几个不同的谐波次数,计算系统的近似周期解,确定周期解的稳定性;同时,以路面激励圆频率为参数进行了跟踪计算,得到系统主共振时的幅频响应特性.近似解的计算结果与数值计算结果的对比表明,增量谐波平衡方法的精度可灵活控制,且收敛速度快,结果可靠,是汽车强非线性动力学行为研究的有效方法.

重构谐波平衡法及其求解复杂非线性问题应用

谐波平衡法是求解非线性动力学系统周期解的最常用方法,但对非线性项进行高阶近似需要庞杂的公式推导,限制了该方法的超高精度解算.通过对频域非线性量的时域等价重构,提出了重构谐波平衡法(RHB法),解决了经典谐波平衡法超高阶次计算难题.然而,上述两种方法均要求动力学系统为多项式型非线性,且无法直接用来求解非线性系统的拟周期解.针对上述问题,文章提出一种将RHB法和复杂非线性系统等价重铸法相结合的计算方法,首先将一般非线性问题无损重铸为多项式型非线性系统,然后用RHB法进行高精度求解;针对拟周期响应求解问题,提出基于“补频”思想的RHB方法,通过基频的优化筛选,实现拟周期响应的快速精准捕捉.选取非线性单摆、相对论谐振子和非线性耦合非对称摆等典型系统进行仿真计算,仿真结果表明,所提出的RHB-重铸法在解非多项式型非线性系统的稳态响应时精度保持为10^(-12)量级,达计算机精度,远超现有方法水平.补频RHB法则实现了对拟周期问题的高效解算,拓宽了方法对真实物理响应的求解范围.

有限深度介质上梁非线性能量汇减振的Winkler地基实现

土-结构相互作用对结构动力学特性的影响是一种具有非线性能量汇特征的效应。利用考虑土体质量的Winkler地基梁理论,将有限深度弹性介质等效为非线性能量汇系统的附加质量,建立简谐激励下弹性介质上简支梁系统的非线性动力学模型。采用Galerkin方法和增量谐波平衡法分析了弹性介质上简支梁的非线性动力响应。利用数值计算方法验证了理论结果的正确性,并分析了非线性能量汇的有效性。通过参数优化和分析,揭示了不同参数范围内Winkler地基的减振效果,讨论了其最佳参数范围。研究结果表明:在合理的参数范围内,弹性介质对其支承梁的动力响应有良好的抑制作用,能够快速有效地吸收共振条件下的振动能量,并具有良好的鲁棒性。优化后的非线性能量汇可使梁的共振幅值降低超95%,且具有较宽的减振频带。研究成果从非线性能量汇的角度展现了土-结构相互作用效应的减振机理,为基于弹性地基设计的结构振动抑制提供了理论依据。

一维立方非线性刚度周期结构色散特性研究

研究立方非线性刚度单胞阵列形成一维周期结构后的色散特性,对飞机壁板振动控制的研究具有一定的促进作用。首先构建一维线性刚度周期结构的动力学模型,基于布洛赫理论(Bloch theorem)推导了其色散方程,并对其色散特性和弹性波传播现象进行分析。进而建立含立方非线性刚度单胞的一维周期结构的动力学模型,利用摄动法推导该周期结构的色散方程,分析非线性刚度的软、硬和激励振幅对其色散特性以及弹性波传播产生的影响。最后考虑到飞行器壁板复杂工作环境,避免摄动法仅适用于弱非线性的局限性,给出含立方非线性刚度一维周期结构色散关系的谐波平衡法的求解过程,对比两种方法的求解结果。本文为利用非线性周期结构对飞行器壁板进行振动控制的进一步研究奠定基础,对非线性声子晶体低频减振研究也具有一定的促进作用。

一种新型准零刚度系统结构设计与特性分析

针对传统的线性隔振系统,提出一种由永磁铁与拉伸弹簧组合而成的新型准零刚度隔振系统,设计用于光栅刻划机等精密设备的低频或超低频隔振。首先,建立系统的静力学模型,求出该系统满足准零刚度所需条件,分析各结构参数对隔振性能的影响;其次,通过对准零刚度隔振系统模型的动力学分析,建立该系统在简谐力激励下的非线性动力学方程,通过谐波平衡法求出系统力传递率,分析了系统中阻尼比、立方非线性项、外部干扰幅值对系统力传递率的影响;最后,与传统的线性隔振系统进行对比,验证了该新型准零刚度系统的可行性。