Nonlinear dynamics of a spur gear pair with time-varying stiffness and backlash based on incremental harmonic balance method

In this paper the incremental harmonic balance method (IHBM) is extended to analyze the nonlinear dynamics of a spur gear pair and some new results are obtained. At first the dynamical model of a spur gear pair is established, where the backlash, time-varying stiffness and static transmission error are all included. Here the time-varying stiffness and static transmission error are represented by the multi-order harmonic series through Fourier expansion. Based on the IHBM, the general forms of the periodic solutions for this system are founded, which is useful to obtain the solutions with arbitrary precision. And the difference between the frequency-response to the multi-order and single-order harmonic is analyzed. Then the effects of the multi-order harmonic on the kinds of the periodic solutions are also investigated by IHBM, and the comparison with the numerical solutions shows the validity of the proposed method. At last the influence of the damping ratio and the excitation amplitude on frequency-response curves is researched, which presents some useful information to analyze and/or control the dynamics of gear system.     

二次非线性圆板的1/2亚谐解

计及材料的非线性弹性和粘性性质，研究圆板在简谐载荷作用下的1／2亚谐解，导出相应的非线性动力方程。提出一类强非线性动力系统的叠加－叠代谐波平衡法。将描述动力系统的二阶常微分方程化为在本解为未知函数的基本微分方程和分岔解为未知函数的增量微分方程。通过叠加－叠代谐波平衡法得出圆板的1／2亚谐解。同时，对叠加－叠代谐波平衡法和数值积分法的精度进行比较。并且讨论了1／2亚谐解的渐近稳定性。     

软式非线性同步振动沉桩系统的动力学分析

对软式非线性同步振动沉桩系统进行动力学特性研究。首先,建立同步振动沉桩系统的软式非线性振动模型,采用一次近似解的幅频特性方程判定系统周期解稳定性问题;然后,利用选取的参数分析系统幅频特性关系,并且根据幅频特性曲线确定系统多解处的稳定解问题,以及讨论沉桩系统参数(激振频率、土的刚度和阻尼、激振器的偏心距等)对系统动力学特性的影响;最后,基于Matlab/Simulink采用四阶龙格-库塔法运算程序进行数值仿真确定系统周期解稳定性。通过理论和仿真系统地分析了系统周期解的稳定性特性,以及系统各参数对系统周期解的影响。     

基于多重打靶技术的连续法求解非线性自治系统的周期解

提出用基于多重打靶技术的地法注解非线性自治系统的周期解,论述了求解的基本原理,基中对基于多重打靶技术Ｎｅｗｔｏｎ迭代格式进行了特殊处理,将自治系统的待定周期作为初始向量的一个分量,通过多重打靶迭代求出。对Ｖａｎｄｅｒｐｏｌ方程和二自由度耦合激振动进行了计算,计算结果与Ｌａｕ等人用量增量谐波平衡法所得的结果完全吻合。     

Parametric vibrations of a horizontal beam with a concentrated mass at one end

This investigation treats the steady state response of parametric vibration of a simply supported horizontal beam, carrying a concentrated mass at one end and subjected to a periodic axial displacement excitation at the other end under the influence of gravity. Non-linear terms arising from longitudinal inertia of a concentrated end mass and beam elements are included in the equation of motion. By using the one mode approximation and applying Galerkin's method, the governing equation of motion is reduced to a non-linear ordinary differential equation with periodic coefficient. The harmonic balance method is applied to solve the equation and the dynamic response is derived. Experimentally determined amplitude-frequency curves are presented, and are found to be in good agreement with the theory.     

Dynamic analysis of harmonically excited non-linear structure system using harmonic balance method

An analytical method is presented for evaluation of the steady state periodic behavior of nonlinear structural systems. This method is based on the substructure synthesis formulation and a harmonic balance procedure, which is applied to the analysis of nonlinear responses. A complex nonlinear system is divided into substructures, of which equations are approximately transformed to modal coordinates including nonlinear term under the reasonable procedure. Then, the equations are synthesized into the overall system and the nonlinear solution for the system is obtained. Based on the harmonic balance method, the proposed procedure reduces the size of large degrees-of-freedom problem in the solving nonlinear equations. Feasibility and advantages of the proposed method are illustrated using the study of the nonlinear rotating machine system as a large mechanical structure system. Results obtained are reported to be an efficient approach with respect to nonlinear response prediction when compared with other conventional methods.     

A new method for blade forced response analysis with dry friction dampers

An original method which is based on the harmonic balance method and the OptQuest NonLinear Programs MultiStart algorithm is proposed to study the maximum vibration amplitude of periodic motion in bladed disks with dry friction dampers. In the proposed approach, the nonlinear algebraic equations obtained from the application of harmonic balance method are considered as general nonlinear equality constraints and the objective for this analysis is the maximization of the vibration amplitude. Then, the MultiStart algorithm is used to locate the global optimal solution. Finally, numerical examples show that this optimal strategy presented is reliable and effective.     

Higher order explicit solutions for nonlinear dynamic model of column buckling using variational approach and variational iteration algorithm-II

This paper deals with the determination of approximate solutions for a model of column buckling using two efficient and powerful methods called He’s variational approach and variational iteration algorithm-II. These methods are used to find analytical approximate solution of nonlinear dynamic equation of a model for the column buckling. First and second order approximate solutions of the equation of the system are achieved. To validate the solutions, the analytical results have been compared with those resulted from Runge-Kutta 4th order method. A good agreement of the approximate frequencies and periodic solutions with the numerical results and the exact solution shows that the present methods can be easily extended to other nonlinear oscillation problems in engineering. The accuracy and convenience of the proposed methods are also revealed in comparisons with the other solution techniques.     

Accurate higher-order analytical approximate solutions to nonconservative nonlinear oscillators and application to van der Pol damped oscillators

In general, this paper deals with general nonlinear oscillations of a nonconservative and single degree-of-freedom system with odd nonlinearity and, in particular, it presents accurate higher-order analytical approximate solutions to van der Pol damped nonlinear oscillators having odd nonlinearity and the Rayleigh equation. By combining the linearization of the governing equation with harmonic balancing and the method of averaging, we establish accurate analytical approximate solutions for the general weakly damped nonlinear systems. Unlike the classical harmonic balance method, simple linear algebraic equations instead of nonlinear algebraic equations are obtained upon linearization prior to harmonic balancing. The combination of these two methods results in very accurate transient response of the periodic solution. In addition and for the first time, this paper also presents a method for deducing fourth-, fifth- and higher-order linearized governing equations from the lower-order equations without the requirement of formulating the problem from the first principle. Three examples including the van der Pol damped nonlinear oscillator are presented to illustrate the excellent agreement with approximate solution using the exact frequency.     

Self-excited oscillations of a string on an elastic foundation subject to a nonlinear feed-forward force

We investigate the self-excited oscillations of a string on an elastic foundation that is subject to a nonlinear feed-forward force. The feed-forward follows that of a model first proposed by Steele and Baker [1] for an active cochlear, and is due to the gain factor profile which depends on the string displacement. In order to determine the bifurcation structure induced by the nonlinear feed-forward mechanism, we formulate a taut string initial-boundary-value problem with periodic boundary conditions which is reduced to a finite order modal dynamical system. We employ an asymptotic multiple-scales method to obtain slowly varying evolution equations that enable an analytical derivation of the periodic system response and analysis of its orbital stability. The resulting bifurcation structure includes multiple regions of both stable and unstable coexisting periodic solutions defined by primary and secondary Hopf stability thresholds. Numerical verification of the bifurcation structure determines the accuracy of the analytically predicted periodic self-excited response and reveals the existence of quasiperiodic combination-tone solutions and complex nonstationary solutions that emerge in a range of the asymptotically predicted unstable solutions. This analysis enables construction of a comprehensive analytical bifurcation structure and may shed light on mechanisms governing complex multi-component spectra that have been documented for spontaneous otoacoustic emissions in the mammalian inner ear.     

The continuous Poincare-like cell mapping method and its application to nonlinear dynamics analysis of a bearing–rotor system

Based on the idea of continuous cell mapping, an improved Poincare-like cell mapping method is developed. This method can be used for global analysis of nonlinear dynamic systems without missing the periodic solutions and the domains of attraction. A bearing–rotor system is analyzed by using this method, and the influences of initial conditions on transient movement are discussed in detail.     

椭圆齿轮驱动的结晶器非正弦振动传动系统动力学及运动稳定性分析

建立了椭圆齿轮驱动的结晶器非正弦振动传动系统动力学模型,推导了动力学方程。结果表明,椭圆齿轮驱动的结晶器非正弦振动传动系统的质量矩阵、刚度矩阵和阻尼矩阵都随曲柄位置变化,为一周期时变参数系统。采用谐波平衡法求解系统的周期解。基于单特征值假设和福洛开（Flo-quet）理论,推导了特征值求解公式,利用所求特征值可判断系统周期解的稳定性。本文的工作为解决结晶器振动平稳性问题,更好地应用该振动装置打下了基础。     

液固混合介质隔振系统的主共振分析

研究一类具有分段线性—非线性非光滑特性液固混合介质隔振器的主共振解析解。采用摄动法求解非线性段的瞬态响应,由常微分方程理论给出了线性段的瞬态响应。根据响应的连续性与周期性条件,联合接缝法与摄动法分析周期激励作用下系统的主共振响应。而后基于弹性恢复力的傅里叶展开,给出谐波平衡法求解非光滑隔振系统响应的一般步骤,并得到了一阶和二阶近似解。采用龙格—库塔算法对解析方法的有效性进行验证,结果表明:接缝法和数值计算的结果较为吻合,而谐波平衡法的二阶近似比一阶近似解更为精确。     

齿轮系统周期运动稳定性研究

针对含间隙的强非线性齿轮系统动力学模型，用数值方法研究了当系统参数和初始条件变化时周期运动的稳定性。基于Floquet分岔理论将预测一校正算法用于讨论参数变化时周期解的稳定性，得到精确的分岔点参数值；通过胞映射法求得周期吸引子的吸引域，引入稳定性品质因子用以定量分析初始条件变化时周期运动的稳定性。该研究结果可为非线性动力学行为的分析和齿轮系统的设计提供参考。     

Nonlinear dynamics of a planetary gear system with multiple clearances

Presented in this paper is on the nonlinear dynamics of a planetary gear system with multiple clearances taken into account. A lateral–torsional coupled model is established with multiple backlashes, time-varying mesh stiffness, error excitation and sun-gear shaft compliance considered. The solutions are determined by using harmonic balance method from the equations in matrix form. The theoretical results from HBM are verified by using the numerical integration. Finally, effects of parameters are discussed.     

粘性阻尼双线性滞迟振子简谐激励响应的Krylov Bogoliubov计算方法

运用KrylovBogoliubov慢变参数法,研究了含有立方非线性粘性阻尼双线性滞迟振子简谐激励响应计算问题,并根据具有周期系数的常微分方程Floquet理论分析了定常响应的稳定性,指出了由于立方非线性因素的存在,响应的幅频曲线可能出现鞍结分叉,即跳跃现象。     

Model updating of locally non-linear systems based on multi-harmonic extended constitutive relation error

Improving the fidelity of numerical simulations using available test data is an important activity in the overall process of model verification and validation. While model updating or calibration of linear elastodynamic behaviors has been extensively studied for both academic and industrial applications over the past three decades, methodologies capable of treating non-linear dynamics remain relatively immature. The authors propose a novel strategy for updating an important subclass of non-linear models characterized by globally linear stiffness and damping behaviors in the presence of local non-linear effects. The approach combines two well-known methods for structural dynamic analysis. The first is the multi-harmonic balance (MHB) method for solving the non-linear equations of motion of a mechanical system under periodic excitation. This approach has the advantage of being much faster than time domain integration procedures while allowing a wide range of non-linear effects to be taken into account. The second method is the extended constitutive relation error (ECRE) that has been used in the past for error localization and updating of linear elastodynamic models. The proposed updating strategy will be illustrated using academic examples.     

一种改进的增量谐波平衡法及其在非线性振动中的应用

对于一般的增量谐波平衡法而言,在求解分段线性系统周期响应时存在收敛速度慢的缺点。针对这一缺点,本文根据最小二乘法原理和增量过程提出了一种改进的增量谐波平衡算法,通过和原有算法进行对比发现二者之间存在着统一的算法形式,因此只要对原有算法作简单的改进即可方便地使用此方法。利用此方法成功计算了齿轮传动分段线性系统的周期解,通过对计算结果比较,发现迭代次数要比一般的增量谐波平衡法少30%左右。从而可以看出这种算法具有收敛速度快的优点。     

Analytical approximation of nonlinear vibration of string with large amplitudes

Study of nonlinear problems in strings with large amplitude is a very important research area in many fields of physics and engineering. variational approach method (VAM) is in particular selected because the method is appropriate to solve nonlinear vibration of a constanttension string. VAM is an explicit method with high capability for resolving strong nonlinear oscillation system problems. It has been found that VAM is well suited for a range of parameters and the approximate frequencies and periodic solutions show a good agreement with the exact ones. This paper compares the various aspects of VAM in relative to exact approaches and higher-order approximate solutions for the constant-tension string. The comparison indicates that VAM is very fast, effective and convenient. The method does not require any linearization or small perturbation, and it leads to high accuracy of the solutions in a single iteration.     

Numerical simulation of a binary gas flow inside a rotating cylinder

A new approach to calculate the axially symmetric binary gas flow is proposed. Dalton’s law for partial pressures contributed by each species of a binary gas mixture (argon and helium) is incorporated into numerical simulation of rarefied axially symmetric flow inside a rotating cylinder by using the time relaxed Monte-Carlo (TRMC) scheme and the direct simulation Monte-Carlo (DSMC) method. The results of flow simulations are compared with the analytical solution and results obtained by Bird [1]. The results of the flow simulations show better agreement than the results obtained by Bird [1] in comparison with the analytical solutions. However, the results of the flow simulations using the TRMC scheme show better agreement than those obtained using the DSMC method in comparison with the analytical solutions.