A new method for Fourier series expansions: Applications in rotor-seal systems

Complex nonlinearities of rotor-seal systems make it difficult to implement some widely used techniques for nonlinear vibrations. This is partly due to the fact that it is rather cumbersome to expand the nonlinear terms into Fourier series. In this paper, a novel Fourier series expansion method is proposed to circumvent this difficulty. The incremental harmonic balance method is utilized to obtain the solutions of a rotor-seal system, where the complicated nonlinearities are handled by the expansion method. Periodic, double-periodic and triple-periodic solutions are obtained in excellent agreement with numerical results, which shows the validity and efficacy of the proposed solution procedures.     

Limit cycle oscillations of rectangular cantilever wings containing cubic nonlinearity in an incompressible flow

Limit cycle oscillations (LCO) as well as nonlinear aeroelastic analysis of rectangular cantilever wings with a cubic nonlinearity are investigated. Aeroelastic equations of a rectangular cantilever wing with two degrees of freedom in an incompressible potential flow are presented in the time domain. The harmonic balance method is modified to calculate the LCO frequency and amplitude for rectangular wings. In order to verify the derived formulation, flutter boundaries are obtained via a linear analysis of the derived system of equations for five different cases and compared with experimental data. Satisfactory results are gained through this comparison. The problem of finding the LCO frequency and amplitude is solved via applying the two methods discussed for two different cases with hardening cubic nonlinearities. The results from first-, third- and fifth-order harmonic balance methods are compared with the results of an exact numerical solution. A close agreement is obtained between these harmonic balance methods and the exact numerical solution of the governing aeroelastic equations. Finally, the nonlinear aeroelastic analysis of a rectangular cantilever wing with a softening nonlinearity is studied.     

Piecewise constrained optimization harmonic balance method for predicting the limit cycle oscillations of an airfoil with various nonlinear structures

A method is proposed to calculate the periodic solutions of piecewise nonlinear systems. The method is based on analytical derivation of nonlinear multi-harmonic equations of motion. Since periodic variations of nonlinear forces are characterized by different states, the vibration cycle is broken into sequential transition intervals according to the instant sets of state transitions. Analytical formulations of the harmonic coefficients of the nonlinear forces and its derivatives with respect to the harmonic coefficients of displacements are developed. Sensitivities of the harmonic coefficients of periodic solutions are determined for constructing explicit expressions for vibration amplitude levels as a function of structural parameters. Numerical investigations of the limit cycle oscillations and its sensitivities of an airfoil with different piecewise nonlinearities have been performed. The results show that the developed method is capable of determining the periodic solutions and its sensitivities with respect to the structural parameters. In order to guarantee time continuity of the nonlinear force, for the hysteresis model it is not right to track the periodic solutions by using the preload or freeplay as the continuation parameters.     

Construction of approximate analytical solutions to strongly nonlinear damped oscillators

An analytical approximate method for strongly nonlinear damped oscillators is proposed. By introducing phase and amplitude of oscillation as well as a bookkeeping parameter, we rewrite the governing equation into a partial differential equation with solution being a periodic function of the phase. Based on combination of the Newton’s method with the harmonic balance method, the partial differential equation is transformed into a set of linear ordinary differential equations in terms of harmonic coefficients, which can further be converted into systems of linear algebraic equations by using the bookkeeping parameter expansion. Only a few iterations can provide very accurate approximate analytical solutions even if the nonlinearity and damping are significant. The method can be applied to general oscillators with odd nonlinearities as well as even ones even without linear restoring force. Three examples are presented to illustrate the usefulness and effectiveness of the proposed method.     

Constrained parameter-splitting perturbation method for the improved solutions to the nonlinear vibrations of Euler–Bernoulli cantilevers

In this paper, a new method named constrained parameter-splitting perturbation method for improving the solutions obtained from the parameter-splitting perturbation method is proposed for solving the problems in some extremal cases, such as the strongly nonlinear vibration of an Euler–Bernoulli cantilever. The proposed method takes the advantages of both the perturbation method and the harmonic balance method. The idea is that the solution obtained by the parameter-splitting perturbation method is substituted into the equation of motion and then the accumulative error of the equation is minimized for determining the unknown splitting parameters under the constraints constructed under the frame of harmonic balance method. The forced vibration of an oscillator with cubic geometric nonlinearity and inertia nonlinearity and the forced vibration of a planar microcantilever beam with a lumped tip mass are studied as examples to reveal the efficacy of the proposed method. The inspection of the steady-state response including its stability is conducted by means of comparing the frequency-response curves obtained by the proposed method with those obtained by the numerical continuation method and harmonic balance method, respectively, to show the efficacy and the advantages of the proposed method. Meanwhile, the nonlinear ordering effect on the solutions of the proposed method is also studied by comparing the results obtained by using different nonlinear orderings in the systems. In the last, we found through convergence examinations that it is necessary to have corrections to the erroneous solution which are obtained by harmonic balance method and Floquet theory in stability analysis.

     

具任意脱层复合材料梁的非线性谐波响应

研究了具任意脱层复合材料梁的非线性谐波响应问题。基于弹性理论，建立了考虑剪切变形时的复合材料梁脱层的基本方程式。在空间上采用B样条函数和Galerkin积分法，在时间上采用增量谐波平衡法进行计算。通过实例计算，得出了简谐力作用下的非线性动力响应曲线。认为基谐波振动仍是非线性振动的主要部分。     

谐波平衡法与复平均法计算强非线性系统稳态响应的区别

近些年，很多学者致力于利用非线性增强振动响应减少的效果或者能量采集器的效率。因而非线性系统的响应值需要从理论计算方面更准确地预测。另外，根据学者已取得的研究成就，非线性能量汇(NES)中存在的立方刚度非线性可以将结构中宽频域的振动能量传递至非线性振子部分。文章将一种由NES和压电能量采集器组成的NES-piezo装置与两自由度主结构耦合连接，系统受谐和激励作用。文章采用谐波平衡法和复平均法分别推导了系统稳态响应，参照数值结果，对比两种近似解析方法在求解强非线性系统稳态响应时的异同。计算结果表明，系统体现较弱非线性时，二者计算结果差异很小；当系统体现强非线性时，复平均法不能准确地呈现系统高阶响应，提高阶数的谐波平衡法能更准确地表示系统响应值。基于谐波平衡法和数值算法，讨论NES-piezo装置对于系统宽频域减振的影响。与仅加入非线性能量汇情况对比，结果表明NES-piezo装置不会恶化宽频域减振效果，并且在第一阶共振频率附近，可以稍微提高结构减振效率。另外，计算结果也表明，采用恰当的NES-piezo装置可实现宽频域范围的结构减振和压电能量采集一体化。此项研究工作为研究不同情形强非线性系统的响应提供了理论方法的指导。另外，研究结果也为宽频域范围的结构减振和压电能量采集一体化提供了理论依据。     

The jump phenomenon effect on the sound absorption of a nonlinear panel absorber and sound transmission loss of a nonlinear panel backed by a cavity

Theoretical analysis of the nonlinear vibration effects on the sound absorption of a panel absorber and sound transmission loss of a panel backed by a rectangular cavity is herein presented. The harmonic balance method is employed to derive a structural acoustic formulation from two-coupled partial differential equations representing the nonlinear structural forced vibration and induced acoustic pressure; one is the well-known von Karman??s plate equation and the other is the homogeneous wave equation. This method has been used in a previous study of nonlinear structural vibration, in which its results agreed well with the elliptic solution. To date, very few classical solutions for this nonlinear structural-acoustic problem have been developed, although there are many for nonlinear plate or linear structural-acoustic problems. Thus, for verification purposes, an approach based on the numerical integration method is also developed to solve the nonlinear structural-acoustic problem. The solutions obtained with the two methods agree well with each other. In the parametric study, the panel displacement amplitude converges with increases in the number of harmonic terms and acoustic and structural modes. The effects of excitation level, cavity depth, boundary condition, and damping factor are also examined. The main findings include the following: (1)?the well-known ??jump phenomenon?? in nonlinear vibration is seen in the sound absorption and transmission loss curves; (2)?the absorption peak and transmission loss dip due to the nonlinear resonance are significantly wider than those in the linear case because of the wider resonant bandwidth; and (3)?nonlinear vibration has the positive effect of widening the absorption bandwidth, but it also degrades the transmission loss at the resonant frequency.     

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.     

平带驱动系统的振动分析研究进展

综述汽车发动机中带张紧装置的平带驱动系统振动分析的研究进展,在转动振动的力学模型和相应数学模型基础上,总结了用数值方法,复模态分析方法和增量谐波平衡法进行线性振动和非线性振动分析以及打滑预测的结果,在转动－横向耦合振动的力学模型和数学模型基础上,总结了用数值方法,Holzer法,模态分析法和直接多尺度法进行线性振动和非线性振动分析的结果,最后提出需要深入研究的若干问题。     

A finite element study on the large amplitude flexural vibration characteristics of FGM plates under aerodynamic load

The large amplitude flexural vibration characteristics of functionally graded material (FGM) plates are investigated here using a shear flexible finite element approach. Material properties of the plate are assumed to be graded in the thickness direction according to a simple power-law distribution in terms of volume fractions of the constituents. The effective material properties are then evaluated based on the rule of mixture. The FGM plate is modeled using the first-order shear deformation theory based on exact neutral surface position and von Kármán’s assumptions for large displacement. The third-order piston theory is employed to evaluate the aerodynamic pressure. The governing equations of motion are solved by harmonic balance method to study the vibration amplitude of FGM plates under supersonic air flow. Thereafter, the non-linear equations of motion are solved using Newmark’s time integration technique to understand the flexural vibration behavior of FGM plates in time domain (simple harmonic or periodic or quasi-periodic). This work is new in the sense that it deals with the non-linear flutter characteristics of FGM plates under high supersonic airflow accounting for both the geometric and aerodynamic non-linearities. Some parametric study is conducted to understand the influence of these non-linearities on the flutter characteristics of FGM plates.     

1/3 subharmonic solution of elliptical sandwich plates

IntroductionInrecentyears,withtheessentialadvantageoflightweightandhighrigidity ,sandwichplatesandshellshavebeenusedasanimportantpatternofstructuralelementsinaeronautical,astronauticalandnavalengineering .However,nonlinearproblemsforsandwichplatesandshellsareonlyinvestigatedbyafewbecauseofthedifficultiesofnonlinearmathematicalproblems.LiuRen_huaiandXuJia_chu[1,2 ]andothershavemadesomeinvestigationsinthisfield .Bifurcationofnonlinearvibrationforsandwichplateshasnotyetbeeninvestigated .Inthisp…     

Nonlinear vibration of a two-mass system with nonlinear stiffnesses

An analytical approach is developed for nonlinear free vibration of a conservative, two-degree-of-freedom mass–spring system having linear and nonlinear stiffnesses. The main contribution of the proposed approach is twofold. First, it introduces the transformation of two nonlinear differential equations of a two-mass system using suitable intermediate variables into a single nonlinear differential equation and, more significantly, the treatment a nonlinear differential system by linearization coupled with Newton’s method and harmonic balance method. New and accurate higher-order analytical approximate solutions for the nonlinear system are established. After solving the nonlinear differential equation, the displacement of two-mass system can be obtained directly from the governing linear second-order differential equation. Unlike the common perturbation method, this higher-order Newton–harmonic balance (NHB) method is valid for weak as well as strong nonlinear oscillation systems. On the other hand, the new approach yields simple approximate analytical expressions valid for small as well as large amplitudes of oscillation unlike the classical harmonic balance method which results in complicated algebraic equations requiring further numerical analysis. In short, this new approach yields extended scope of applicability, simplicity, flexibility in application, and avoidance of complicated numerical integration as compared to the previous approaches such as the perturbation and the classical harmonic balance methods. Two examples of nonlinear two-degree-of-freedom mass–spring system are analyzed and verified with published result, exact solutions and numerical integration data.     

超声速飞行器壁板非线性颤振响应分析的时域法与频域法对比研究

为考查基于假设模态法在时域中开展壁板非线性颤振分析的可行性,在相同的参数下,分别采用时域方法和频域方法研究了超声速飞行器壁板的非线性颤振响应,并从壁板的颤振幅值、颤振频率和颤振型态三个方面对时域和频域分析结果的一致性作了较详细的比较。首先,基于von Karman应变－位移关系和Mindlin板理论建立考虑几何非线性的壁板力学模型,应用一阶活塞理论分析壁板上单面承受的超声速准定常气动力,基于虚功原理和有限单元法推导壁板的运动微分方程。然后,用壁板的线性固有模态作为假设模态,减缩系统的自由度而得到降阶模型。采用四阶龙格-库塔法对降阶模型作时域数值积分,得到壁板的非线性颤振响应。另一方面,假设壁板的极限环颤振为简谐振荡,可对壁板的非线性刚度作等效线性化处理,进而在频域中直接在有限元（未降阶）模型的基础上分析壁板的颤振幅值、颤振频率和颤振型态。数值分析表明,当极限环颤振为简谐振荡时,时域方法和频域方法的计算结果符合一致。本文最后讨论了时域法和频域法应用在壁板非线性颤振分析中各自的优点和局限性。     

An incremental method for limit cycle oscillations of an airfoil with an external store

This paper proposes an incremental method, which is based on the harmonic balance method, to analyze the nonlinear aeroelastic problem of an airfoil with an external store. The governing equations of limit cycle oscillations (LCOs) of the airfoil are deduced by the harmonic balancing procedure. Different from usual procedures, the harmonic balance equations are not solved directly but instead transformed into an equivalent minimization problem. The minimization problem is solved using the Levenberg–Marquardt method. Numerical examples show that the LCOs obtained by the presented method are in excellent agreement with numerical solutions. The bifurcation of the LCOs is further analyzed using the Floquet theory. It is found that the LCOs exhibit saddle-node, symmetry breaking and period-doubling bifurcations with the wind speed as control parameter. Compared with the harmonic balance method, the presented method has a wider convergence region and hence makes it easier to choose a proper initial guess for iterations.     

Bifurcation and chaos of airfoil with multiple strong nonlinearities

The bifurcation and chaos phenomena of two-dimensional airfoils with multiple strong nonlinearities are investigated.First,the strongly nonlinear square and cubic plunging and pitching stiffness terms are considered in the airfoil motion equations,and the fourth-order Runge-Kutta simulation method is used to obtain the numerical solutions to the equations.Then,a post-processing program is developed to calculate the physical parameters such as the amplitude and the frequency based on the discrete numerical solutions.With these parameters,the transition of the airfoil motion from balance,period,and period-doubling bifurcations to chaos is emphatically analyzed.Finally,the critical points of the period-doubling bifurcations and chaos are predicted using the Feigenbaum constant and the first two bifurcation critical values.It is shown that the numerical simulation method with post-processing and the prediction procedure are capable of simulating and predicting the bifurcation and chaos of airfoils with multiple strong nonlinearities.     

双稳态电磁式振动能量捕获器超谐波响应研究

超谐波响应是非线性振动系统在较大激励下表现的特性,在某种条件下双稳态振动能量捕获系统的超谐波响应可使系统产生优越的输出功率。本文将质量-非线性弹簧-阻尼系统与双稳态振动能量捕获器相结合,提出了附加非线性振子的双稳态电磁式振动能量捕获器,建立系统的力学模型及控制方程。采用两项式谐波平衡法,获得了双稳态系统在简谐激励下产生大幅运动的基谐波和超谐波响应的解析解,借助数值仿真分析了质量比和调频比对双稳态振动能量捕获器产生大幅运动的影响规律,获得了双稳态系统的结构参数的最佳配置范围,且当外部激励频率处于低频段时,系统发电主要表现为超谐波发电,随着激励频率的增大,振动发电系统主要呈现基谐波发电。上述研究,为双稳态能量捕获装置的理论研究提供了参考。     

一种求解局部非线性结构稳态响应的方法

为了降低求解局部非线性结构稳态响应的计算量,基于子结构和阻抗缩聚提出了一种用于求解局部非线性结构稳态响应的计算方法.将局部非线性结构分解为线性子结构和非线性子结构,利用谐波平衡构造各个子结构的阻抗方程,对线性子结构进行缩聚,将局部非线性动力学方程转化为求解一组非线性代数方程组问题,通过迭代求解非线性代数方程组,求解系统的稳态响应.     

高阶谐波平衡方法中非物理解来源分析及改进方法研究

对于周期性非定常问题,高阶谐波平衡（High-order Harmonic Balance, HOHB）方法将非定常方程的解用Fourier 级数展开至一定阶次,从而消除其中的时间导数项,大大降低了计算消耗. 本文以达芬振子方程为例,探讨了HOHB 方法中非物理解的来源,分析结果表明:非物理解出现的原因是在推导过程中非线性项的简化处理导致方程左右两边并不严格相等. 根据非线性项的特点,在其处理过程中扩充子时间层上的时域解,并将非线性项中出现的更高阶谐波截断,使方程左右两边严格相等. 通过对达芬振子方程进行数值模拟发现:改进方法在消除非物理解的同时,也显著减少了计算所需谐波数. 对比参考文献发现,同阶改进方法的精度和原始谐波平衡方法基本相当,证明了本方法的可行性. 最后将本方法应用于具有立方刚度非线性的气动弹性系统中,验证本方法的工程适用性. 但是,当方程中非线性项较多时,本方法所需要的计算消耗会有所增加.      

考虑几何非线性的复合材料机翼气动弹性分析

本文研究结构几何非线性与气动力非平面效应对大展弦比复合材料机翼的气动弹性行为的影响．将非线性有限元法与曲面涡格法结合，计算机翼静气动弹性变形；通过曲面偶极子格网法结合静气动弹性平衡位置处的结构切线刚度，建立气动弹性方程并求解得到机翼颤振速度．针对板模型机翼，分析了迎角对机翼几何非线性气动弹性特性的影响．结果表明：本文复合材料板模型机翼的颤振形式不受水平弯曲模态影响，属于经典弯扭颤振；在几何非线性的影响下，机翼扭转频率随结构变形增大而明显减小，颤振速度随迎角增大而减小．