强非线性多自由度动力系统主共振同伦分析法研究

应用同伦分析方法(HAM)解决强非线性多自由度系统在谐波激振力下的主共振问题．同伦分析方法的有效性独立于所考虑的方程中是否含有的小参数．同伦分析方法提供了一个简单的方法,通过一个辅助参数h-来调节和控制级数解的收敛区域．两个具体算例表明,同伦分析方法得出的结果与修正Linstedt-Poincaré法、增量谐波平衡法的解决方案得出的结果相吻合．     

非线性颤振系统中既是超临界又是亚临界的Hopf分岔点研究

研究了二元机翼非线性颤振系统的Hopf分岔．应用中心流形定理将系统降维,并利用复数正规形方法得到了以气流速度为分岔参数的分岔方程．研究发现,分岔方程中一个系数不含分岔参数的一次幂,故使得分岔具有超临界和亚临界双重性质．用等效线性化法和增量谐波平衡法验证了所得结果．     

夹层椭圆形板的1/3亚谐解

研究了夹层椭圆形板的非线性强迫振动问题。在以5个位移分量表示的夹层椭圆板的运动方程的基础上,导出了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,化为基本解为未知函数的基本微分方程和派生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了椭圆板的1/3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了比较。并且讨论了1/3亚谐解的渐近稳定性。     

基于GMRES算法的弹性结构强耦合小变形流激振动分析

使用混合广义变分原理,将基于Lagrange表述的小位移变形结构振动问题与基于Euler描述的不可压缩粘性流动问题,统一在功率平衡的框架下建立流固系统的耦合控制方程．用有限元格式做空间离散后,再用广义梯形法将有限元控制方程转化为增量型的线性方程组,该方程组的系数矩阵具有非对称性,其中元素含对流效应和时间因子．将GMRES算法与振动分析的Newmark法和流动分析的Hughes预测多修正法结合,发展成一种基于GMRES-Hughes-Newmark的稳定算法,用于计算具有复杂几何边界的强耦合流激振动问题．以混流式水轮机叶道为数值算例的计算表明,模拟结果与试验实测结果吻合较好．     

应用多刚体离散化方法对梁的动力屈曲和后屈曲分析

基于梁的多刚体离散化模型（有限段模型）,建立了梁的链式多刚体-铰链-弹簧系统模型,利用坐标变换方法建立了相应的非线性多自由度系统的参数振动方程,并利用约束参数法对所得到的多度系统的Mathieu-Hill方程进行了梁的动力屈曲分析,得到系统的参数共振域．因为所用的离散化模型与动力方程对梁的变形并无限制,所以可以用所得到的数学模型在其失稳域对梁的动力后屈曲进行数值仿真分析．通过实例的数值仿真,证明了这种梁的参数振动模型与分析方法的正确性．     

谐和与随机噪声联合作用下的粘弹系统

研究了粘弹系统在谐和与随机噪声联合作用下的响应和稳定性问题.用谐波平衡法和随机平均法分析了系统在确定性谐和激励和随机激励联合作用下的响应,讨论了粘弹项、随机扰动项对系统响应的影响.结果表明,在一定条件下,系统具有两个均方响应值和跳跃现象.数值模拟表明,谐波平衡法与随机平均法相结合的研究方法是有效的.     

非线性振动分析中的正交函数法

本文根据谐波平衡法假设周期解的基本思想，提出了一种分析非线性振动特性的正交函数法。将位移展开为谐波的级数形式，根据线性模态和三角级数的正交性导出了一组形式简单的特征方程。有效地解决了平方非线性系统存在漂移项的困难，算例表明：本文方法精度高，收敛快，工作量小。     

基于谐波平衡法的尾流激励的叶片振动降阶模型方法

在航空发动机中下游叶片在上游尾流的作用下易发生受迫振动,严重影响叶片的颤振和疲劳性能.对于这种尾流作用下复杂的流固耦合情况需要一种有效的方法来分析.针对这一问题,提出了基于谐波平衡法的尾流激励的叶片振动降阶模型方法.该方法首先将上游尾流Fourier(傅立叶)分解为若干尾流谐波,并计算各尾流谐波下叶片气动力谐波的振幅,得到尾流引起的叶片气动力;再通过叶片的结构运动方程和气动力降阶模型的耦合分析尾流激励下叶片的振动.算例结果表明,该方法可以快速准确地分析尾流激励下叶片的振动特性.     

线载和弹性支承作用面内运动薄板磁固耦合双重共振

针对磁场环境中具有线载荷和弹性支承作用的面内运动薄板,给出了系统的势能、动能及电磁力表达式,应用Hamilton变分原理,推得面内运动条形板的磁固耦合非线性振动方程.考虑边界为夹支 铰支的约束条件,利用变量分离法和Galerkin积分法,得到了含简谐线载力和电磁阻尼力项的两自由度非线性振动微分方程组.应用多尺度法对主 内联合共振问题进行解析求解,得到了双重联合共振下系统的一阶状态方程和共振响应特征方程.通过算例,得到了面内运动薄板的一阶和二阶共振幅值变化规律曲线图,分析了不同作用量和载荷位置对系统振动特性的影响.结果表明:系统发生主 内双重共振时,解的多值性和跳跃现象明显,弹性支承和线载荷位置对共振现象影响显著;一阶和二阶的共振多值解区域同时出现同时消失,体现了明显的内共振特征.     

非线性振动系统周期解的数值分析

用直接数值积分法求非线性振动系统的周期解,求解时对初始条件进行迭代,使它与终点条件相一致.积分时间区间(即周期)或运动方程中的某些参数,也可在迭代过程中随同变化,积分方法是变步长的. 用这种“打靶”法求周期解,所需计算工作量相对较少.其中误差主要来源于数值积分,故不难估计并控制它足够小.这种方法可处理各种类型的振动问题,如单自由度和多自由度系统的自由无阻尼振动、强迫振动、自激振动和参数振动等等;也能求得不稳定解和那些对参数变动十分敏感的解.解的稳定性根据相关的周期系数微分方程来研究.求共振曲线或其他振动特性曲线时,利用插值方法并自动调节步长来定出迭代始值. 为了阐明这种方法的通用性,计算了若干例子.非线性的描述可用解析函数或任何其他形式,例如分段线性函数.文中还就所得周期解指出了非线性振动的一些值得注意的性质.部分计算结果与已有的近似解或实验结果作了比较.     

Global resonance optimization analysis of nonlinear mechanical systems: Application to the uncertainty quantification problems in rotor dynamics

An efficient method to obtain the worst quasi-periodic vibration response of nonlinear dynamical systems with uncertainties is presented. Based on the multi-dimensional harmonic balance method, a constrained, nonlinear optimization problem with the nonlinear equality constraints is derived. The MultiStart optimization algorithm is then used to optimize the vibration response within the specified range of physical parameters. In order to illustrate the efficiency and ability of the proposed method, several numerical examples are illustrated. The proposed method is then applied to a rotor system with multiple frequency excitations (unbalance and support) under several physical parameters uncertainties. Numerical examples show that the proposed approach is valid and effective for analyzing strongly nonlinear vibration problems with different types of nonlinearities in the presence of uncertainties.     

Nonlinear transversely vibrating beams by the improved energy balance method and the global residue harmonic balance method

Mathematical modeling of many engineering systems such as beam structures often leads to nonlinear ordinary or partial differential equations. Nonlinear vibration analysis of the beam structures is very important in mechanical and industrial applications. This paper presents the high order frequency-amplitude relationship for nonlinear transversely vibrating beams with odd and even nonlinearities using the improved energy balance method and the global residue harmonic balance method. The accuracy of the energy balance method is improved based on combining features of collocation method and Galerkin–Petrov method, and an improved harmonic balance method is proposed which is called the global residue harmonic balance method. Unlike other harmonic balance methods, all the former global residual errors are introduced in the present approximation to improve the accuracy. Finally, preciseness of the present analytic procedures is evaluated in contrast with numerical calculations methods, giving excellent results.     

Validity and accuracy of solutions for nonlinear vibration analyses of suspended cables with one-to-one internal resonance

This study investigates the accuracy of nonlinear vibration analyses of a suspended cable, which possesses quadratic and cubic nonlinearities, with one-to-one internal resonance. To this end, we derive approximate solutions for primary resonance using two different approaches. In the first approach, the method of multiple scales is directly applied to governing equations, which are nonlinear partial differential equations. In the second approach, we first discretize the governing equations by using Galerkin’s procedure and then apply the shooting method. The accuracy of the results obtained by these approaches is confirmed by comparing them with results obtained by the finite difference method.     

Analytical approximations to primary resonance response of harmonically forced oscillators with strongly general nonlinearity

This paper presents an innovative analytical approximate method for constructing the primary resonance response of harmonically forced oscillators with strongly general nonlinearity. A linearization process is introduced prior to harmonic balancing (HB) of the nonlinear system and a linear system is derived by which the accurate analytical approximation procedure is easily and innovatively implemented. The main cutting edge of the proposed method is that complicated and coupled nonlinear algebraic equations obtained by the classical HB method is avoided. With only one iteration, very accurate analytical approximate primary resonance response can be determined, even for significantly nonlinear systems. Another advantage is the direct determination of the maximum oscillation amplitude. This is due to the appropriate form chosen for the approximation with no extra processing required. It is concluded that the result of an initial approximate solution from the two-term (constant plus the first harmonic term) harmonic balance is not reliable especially for strongly nonlinear systems and a correction to the initial approximation is necessary. The proposed method can be applied to general oscillators with mixed nonlinearities, such as the Helmholtz-Duffing oscillator. Two examples are presented to illustrate the applicability and effectiveness of the proposed technique.     

Equivalent linearization method for the flutter system of an airfoil with multiple nonlinearities

The equivalent linearization method (ELM) was extended to analyze the flutter system of an airfoil with multiple nonlinearities. By replacing the cubic plunging and pitching stiffnesses by equivalent quantities, linearized equations for the nonlinear system were deduced. According to the linearized equations, approximate solutions for limit cycle oscillations (LCOs) were obtained in good agreement with numerical results. The influences of the linear and cubic stiffnesses on LCOs were analyzed in detail. Reducing linear pitching stiffness leads to decreasing of the critical flutter speed. For linear plunging stiffness, the opposite is true. Also, it reveals that the bifurcation could be supercritical or subcritical, which is related to the ratio between the coefficient of cubic pitching stiffness and that of plunging one.     

Analysis of flow characteristics of granular material unloaded on nonlinear vibration inclined platform

In this study, the repeated discontinuous friction between granular material and contact platform and structural nonlinearity of inclined vibration platform giving rise to the vibration flow-aiding unloading is a complicated process, which has significant effects on the dynamic behaviors and flow characteristics of granular material. A simplified mathematical model of the inclined vibration platform and granular material is deduced by mechanical properties. Based on the equations of motion and a good degree of accuracy and applicability of the process with calculated data reported in the literature, the approximate analytical solution and flow properties are investigated by using the modified incremental harmonic balance method and numerical integration method. Moreover, the influences of friction coefficient, excitation amplitude, nonlinear stiffness and inclined angle on the complicated dynamic behaviors are explored and discussed. It is shown that the different motion paths of granular material on inclined vibration platform are observed depending on the different parameters. The increasing friction coefficient has complicated effects on the nonlinear dynamic behaviors of the granular material. The excitation amplitude and nonlinear stiffness can effectively control the flow characteristics of granular material at low excitation frequency but the inclined angle presents opposite property. The research may contribute to improve unloading efficiency, predict the motion state of granule and provide a theoretic foundation for further design the unloading system.     

Effects of Cone Angles on Nonlinear Vibration Responses of Functionally Graded Shells北大核心CSCD

The nonlinear vibration responses of functionally graded materials (FGMs) shells with different cone angles under external loads were studied. Firstly, the Voigt model was employed to describe the physical properties along the thickness direction of FGMs conical shells. Then, the motion equations were derived based on the 1st-order shear deformation theory, the von Kármán geometric nonlinearity and Hamilton’s principle. Next, the Galerkin method was applied to discretize the motion equations and the governing equations were simplified into a 1DOF nonlinear vibration differential equation under Volmir’s assumption. Finally, the nonlinear motion equations were solved with the harmonic balance method and the Runge-Kutta method, and the amplitude frequency response characteristic curves of the FGMs conical shells were obtained. The effects of different material distribution functions and different ceramic volume fraction exponents on the amplitude frequency response curves of conical shells were discussed. The bifurcation diagrams of conical shells with different cone angles, as well as time process diagrams and phase diagrams for different excitation amplitudes, were described. The motion characteristics were characterized by Poincaré maps. The results show that, the FGMs conical shells present the nonlinear characteristics of hardening springs. The chaotic motions of the FGMs conical shells are restrained and not prone to motion instability with the increase of the cone angle. The FGMs conical shell present a process from the periodic motion to the multi-periodic motion and then to chaos with the increase of the excitation amplitude. © 2022 Editorial Office of Applied Mathematics and Mechanics. All rights reserved.     

Study of non-linear dynamic behavior of open cracked functionally graded Timoshenko beam under forced excitation using harmonic balance method in conjunction with an iterative technique

The nonlinear modeling and subsequent dynamic analysis of cracked Timoshenko beams with functionally graded material (FGM) properties is investigated for the first time using harmonic balance method followed by an iterative technique. Crack is assumed to be open throughout. During modeling, nonlinear strain–displacement relation is considered. Rotational spring model is adopted in order to model the open cracks. Energy formulations are established using Timoshenko beam theory. Nonlinear governing differential equations of motion are derived using Lagrange's equation. In order to incorporate the influence of higher order harmonics, harmonic balance method is employed. This reduces the governing differential equations into nonlinear set of algebraic equations. These equations are solved using two different iterative techniques. Methodology is computationally easier and efficient as well. This is observed that although assumption of simple harmonic motion (SHM) simplifies the problem, it yields to erroneous results at higher amplitude of motion. However, accuracy of the solution is improved considerably when the contribution of higher order harmonic terms are considered in the analysis. Results are compared with the available results, which confirm the validity of the methodology. Subsequent to that a parametric study on influence of forcing term, material indices and crack parameters on large amplitude vibration of Timoshenko beams is performed for two different boundary conditions.     

Study on frequency response and bifurcation analyses under primary resonance conditions of micro-milling operations

Avoidance of resonance in fluctuation of milling tool is vital for reaching excellence quality and performance of the cutting operation. The cutting tool in resonance condition vibrates with considerable magnitude that causes to increase milling tool wear and manufacturing prices. Analytical study of primary resonances and bifurcation behavior of a micro-milling process, including structural nonlinearities, gyroscopic moment, rotary inertia, velocity-dependent process damping, static and dynamic chip thickness, is chief aim of this article. The milling tool is modeled as a 3-D spinning cantilever beam that is motivated by cutting forces. To get the analytical solution for frequency response function and bifurcation behavior of the system under primary resonances, the method of multiple scales is operated on converted ordinary differential equations that are obtained by applying assumed modes method on nonlinear partial differential equations of tool vibration. The effects of different process parameters and nonlinear terms on the frequency response of the tool tip oscillations are examined. In addition, the effects of detuning parameter and damping ratio on the bifurcation and behavior of the limit cycle under primary resonances are examined. The results shows that these parameters are the bifurcation parameters and Neimark, symmetry breaking, flip, and period-3 bifurcations occur when the detuning parameter is varied.