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.

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.

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.

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.

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.

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.

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.

The Reduced Space Method for Calculating the Periodic Solution of Nonlinear Systems

A hybrid method combined the reduced Sequential Quadratic Programming(SQP)method with the harmonic balance method has been developed to analyze the characteristics of mode localization and internal resonance of nonlinear bladed disks.With the aid of harmonic balance method,the nonlinear equality constraints for the constrained optimization problem are constructed.The reduced SQP method is then utilized to deal with the original constrained optimization problem.Applying the null space decomposition technique to the harmonic balance algebraic equations results in the vanishing of the nonlinear equality constraints and a simple optimization problem involving only upper and lower bound constraints on the optimization variables is formed and solved.Finally,numerical results are given for several test examples to validity the proposed method.The efficiency of the solution method to trace the family of energy dependent nonlinear modes is illustrated.The localization nonlinear normal modes of bladed disks related to various types of internal resonances are explored.

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。

基于非线性谐波法的多级轴流压缩机叶顶间隙流动分析

为了探究叶顶间隙对多级轴流压缩机的内部流动和气动性能的影响,对某12级多级轴流压缩机进行研究。首先,基于数值模拟预测出叶顶间隙为近零间隙(C_(n0))、设计间隙(C_(d))、1.5倍设计间隙(1.5C_(d))、2倍设计间隙(2C_(d))的压缩机性能曲线;然后,在此基础上采用非线性谐波法(NLH)对不同间隙模型进行非定常数值模拟计算;最后,对模拟结果的内部流场状况进行分析。研究结果表明:叶顶间隙大小不仅影响压缩机的流量特性曲线,而且在某个间隙时压缩机的性能最佳,同时在稳定的工作范围内存在一个最佳间隙,即失速裕度最佳值。通过对压缩机叶顶区域流场的分析,从非定常流场的角度阐明了叶顶泄漏流形成的涡对压缩机性能影响的原因,在近零间隙模型中,叶顶间隙涡不明显,由间隙过小导致堵塞引起叶片吸力面出现较大范围的高速流动区域。当叶尖间隙在最优值左右时,叶尖泄漏流动抑制了流动分离。而随着叶顶间隙的不断增大,叶顶间隙引起涡的尺度和强度也相应增大,使压缩机性能降低。

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。

轻微弯曲输流管道的非线性强迫振动特性

为了掌握轻微弯曲输流管道在外激励下的非线性振动特性,基于哈密顿原理建立了轻微弯曲输流管道的非线性振动方程,并使用伽辽金截断法将振动方程离散为非线性常微分方程组。利用增量谐波平衡法求解系统的非线性动力学响应,并使用Floquet理论研究系统响应的稳定性和分岔行为。研究结果表明,轻微弯曲输流管道在一些参数下同时具有软弹簧特性和硬弹簧特性。分析系统参数对响应的影响表明,随着管道弯曲程度增加,系统的响应特性由硬弹簧特性转变为软弹簧特性。可见,管道的初始弯曲是影响输流管道系统振动特性的重要因素,因此在输流管道的设计中必须考虑管道弯曲构型的影响。

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。

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.

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.