基于速度协调法的齿轮碰撞动力学分析

建立了基于速度协调法的齿轮副碰撞动力学模型,提出了针对该模型的"碰撞"数值算法,讨论了系统在不同参数激励下的动力学特性。采用经典龙格库塔法进行了Adams/Matlab联合仿真,证明了该算法的精确性和有效性,并利用该算法计算系统周期解对应的离散状态转移矩阵,进而求得了Floquet乘子,并借此判断了系统周期解的稳定性。研究结果表明:在相同速度波动量的条件下,齿轮系统动态误差、相对速度和加速度与主动轮的转速成正比;系统输入转速决定了系统周期解的稳定性,在不同转速下系统出现不同周期的运动,甚至出现混沌;拖力矩的出现打破了系统运动的对称性,使得系统由双边碰撞转变为单边碰撞,从而在某种程度上降低了系统的振动和噪声;另外,系统周期解的稳定性对拖力矩非常敏感,在拖力矩较小时,系统不受速度波动量和拖力矩波动量的影响,始终处于不稳定状态。     

分数阶单自由度线性振子双侧对称碰撞振动分析

基于含分数阶微分的单自由度线性双侧刚性碰撞模型,研究了双侧对称碰撞振动系统在简谐激励下的稳定性和分岔行为。利用平均法得到分数阶线性系统的等效刚度和等效阻尼,获得碰撞振动的稳态解;利用迭代法得到更精确的瞬态固有频率,从而获得碰撞振动的瞬态解。在此基础上,得到了双侧对称碰撞振动系统的近似解析解。根据近似解析解,分析了对称n-1-1周期运动的存在条件,并利用Poincaré映射研究了n-1-1周期运动的稳定性。详细分析了当外界激励频率、分数阶阶次和间隙变化时系统的分岔行为。分析结果表明,在双侧对称分数阶振动碰撞系统中,存在着擦边分岔、音叉分岔、倍周期分岔和混沌运动。     

一类具有时滞的Volterra微分系统周期解的存在唯一性

探讨了一类具有时滞的Volterra微分系统的周期解问题。利用基本解矩阵、Floquet理论等工具，对Volterra系统进行积分变换，构造其系统解新的表达式；利用Krasnoselskii不动点方法，获得了所研究系统周期解的存在性；利用压缩映射原理获得了系统周期解唯一性的充分性条件。最后，通过实例验证了结论的有效性。     

内圈故障滚动轴承系统周期运动的倍化分岔

针对轴承内圈破损故障,建立轴承三自由度分段非光滑的故障模型,研究内圈故障滚动轴承系统周期运动的倍化分岔现象和混沌行为;求出系统的切换矩阵后,将得到的切换矩阵结合光滑系统的Floquet理论来分析轴承非光滑系统周期运动发生倍化分岔的条件。通过在碰撞面处建立Poincaré映射,用数值方法进一步揭示轴承系统的周期运动经倍化分岔通向混沌的现象;结果表明,当旋转频率接近临界分岔点时,系统有1个Floquet特征乘子接近-1,系统发生周期倍化分岔,随着旋转频率的增加,系统经历了周期二解的Nermark-Sacker分岔,随后又经历了多周期、混沌等复杂的非线性行为。对该故障轴承系统分岔和混沌的研究,可为大型高速旋转机械的安全稳定运行提供可靠的设计与故障诊断依据,也为实际设计时提供理论指导和技术支持。     

一类对称约束碰振系统的余维二擦边分岔的存在条件EI北大核心CSCD

类似光滑系统的余维二分岔的分类方法,余维二擦边分岔被划分为三种类型,分别是擦边点退化、退化环擦边(非双曲)以及两个擦边事件同时发生。分析了一个二自由度对称约束的碰撞振动系统,得到了该系统第二类余维二擦边分岔的存在条件。考虑双侧擦边周期运动,理论推导出双侧擦边周期运动的存在性条件;利用不连续映射方法,得出1/1/n碰撞周期运动发生鞍结分岔和倍周期分岔的解析表达式;结合双擦边周期运动的存在性条件和1/1/n碰撞周期运动的分岔条件,推导出发生余维二擦边分岔时满足的解析表达式,并以周期1运动为例,给出了余维二擦边分岔点的分布。     

一类两自由度分段线性非光滑系统的分岔与混沌

研究了一类两自由度分段线性非光滑系统周期运动的分岔现象和混沌行为.求出系统在各分界面处的切换矩阵,应用Floquet理论分析了该系统周期运动发生Neimark-Sacker分岔和倍化分岔的条件,然后建立Poincare映射,通过数值方法进一步揭示了系统发生的Neimark-Sacker分岔,倍化分岔和亚谐分岔现象.对该系统分岔和混沌的研究,有助于工程中此类弹性碰撞系统的优化设计.     

碰撞阻尼器系统的分岔、混沌与控制

对碰撞阻尼器振动系统推导了周期解存在的条件，并利用Poincare映射和数字仿真进行了分岔与混沌运动的研究。计算结果表明，这种非线性碰撞振动系统在特定的参数条件下，除了稳定的周期运动形态外，还会沿着倍周期分岔、HOPF分岔及拟周期环面破裂等分岔进入混沌运动。因此，为了有效地利用碰撞阻尼器特性控制振动，在设计和使用碰撞阻尼器时应考虑参数满足周期运动的条件，避免由于自身的非线性特性而产生的混沌运动。     

三自由度含间隙碰撞振动系统Poincaré映射Hopf-Hopf交互分岔的反控制EI北大核心CSCD

对一类含间隙碰撞振动系统Poincaré映射的Hopf-Hopf交互分岔进行了反控制研究。首先,基于碰撞振动系统建立了六维Poincaré映射,由于六维映射相应雅克比矩阵的特征值没有解析的表达式,这使得由特征值特性描述的传统临界分岔准则在确定控制增益中具有很大的局限性。针对这个局限性,建立了包含特征值分布条件、横截条件和非共振条件的显式临界准则。所建立的准则与传统的分岔准则等价,但并不依赖雅克比矩阵特征值的直接计算。然后,针对碰撞的不连续特性导致的隐式Poincaré映射在闭环系统控制设计中的困难,发展了一种基于原碰撞系统的线性反馈控制方法。最后,数值分析给出了在指定的参数点所设计的映射Hopf-Hopf交互分岔的环面解,进一步验证了理论分析的正确性。     

三自由度含间隙碰撞振动系统Poincaré映射Hopf-Hopf交互分岔的反控制

对一类含间隙碰撞振动系统Poincaré映射的Hopf-Hopf交互分岔进行了反控制研究。首先,基于碰撞振动系统建立了六维Poincaré映射,由于六维映射相应雅克比矩阵的特征值没有解析的表达式,这使得由特征值特性描述的传统临界分岔准则在确定控制增益中具有很大的局限性。针对这个局限性,建立了包含特征值分布条件、横截条件和非共振条件的显式临界准则。所建立的准则与传统的分岔准则等价,但并不依赖雅克比矩阵特征值的直接计算。然后,针对碰撞的不连续特性导致的隐式Poincare映射在闭环系统控制设计中的困难,发展了一种基于原碰撞系统的线性反馈控制方法。最后,数值分析给出了在指定的参数点所设计的映射Hopf-Hopf交互分岔的环面解,进一步验证了理论分析的正确性。     

双组份Degasperis-Procesi方程的有界行波解?

本文研究双组分Degasperis-Procesi方程的有界行波解。利用平面动力系统的分岔理论,分析了双组分Degasperis-Procesi方程对应行波系统在参数平面不同区域的分岔相图。进而,依据动力系统相轨中的同宿轨、周期轨与非线性波动方程的孤立波解、周期波解之间的关系,在一定的参数条件下,获得了双组份Degasperis-Procesi方程的孤立波解和周期波解,并借助数值模拟给出了部分解的图像。     

非线性能量阱振动抑制效果理论分析与试验研究

Aiming at eliminating the broadband line spectrum generated by the periodic rotation of submarine mechanical equipment, the vibration suppression effect was investigation and the structural parameter optimization of nonlinear energy sink (NES) were carried out under harmonic excitation. The dynamic model of mechanical equipment coupled with the nonlinear energy sink was established, and the periodic motion and stability of the coupled system were analyzed by incremental harmonic method, arc-length continuation method and Floquet theory. The influence of damping, mass ratio and stiffness on the vibration suppression effect of the nonlinear energy sink was discussed by taking the system vibration energy as evaluation criterion. Furthermore, the optimized damping and stiffness were obtained through local optimization algorithm, also the robustness of the vibration suppression effect was studied. The results show that the proposed method is in good agreement with Runge-Kutta numerical method, which can effectively construct the complete image of the coupled system periodic solution. Besides, weak damping is a prerequisite for a nonlinear energy sink to have good vibration suppression effect and robustness within a range of 20% of excitation frequency and of 50% amplitude after parameter optimization. A NES, parallel with vertical linear springs and constituted by flexible hinges under preloaded state, was proposed and related test research was also carried out to verify the theoretical findings.     

考虑摩擦效应的颗粒阻尼器力学模型研究及参数分析

颗粒阻尼具有高度非线性特性且影响因素复杂,在土木工程减振控制的研究和应用处于简单理论探究与实验阶段,尚未形成成熟可靠的理论对实际设计和应用进行指导。在阻尼颗粒未发生堆积时,考虑阻尼颗粒与阻尼器腔体之间的碰撞过程和摩擦效应,构建一种颗粒阻尼器-单自由度结构系统力学模型,并求得该力学模型在简谐激励下位移响应的解析解。该模型能够充分反映颗粒阻尼器的碰撞和非碰撞过程,其相轨迹可以体现颗粒阻尼器的复杂非线性特征。通过单层钢框架在简谐激励下的电磁振台试验对颗粒阻尼器的理论及运动特性进行验证,证明该理论模型的合理性和解析结果的正确性。最后对颗粒阻尼器进行颗粒质量、激振幅值、激振频率和阻尼颗粒运动间隙的参数分析,并与已有冲击阻尼器模型进行对比,结果表明所构建的力学模型能更加合理地评价颗粒阻尼器的减振性能。     

Dynamical responses for a vertical vibration model of a four-roller cold rolling mill under combined stochastic and harmonic excitations

Considering that random fluctuations can affect the rolling process of cold rolling mills, leading to abnormal cold mill performance and defects in the rolled product. Therefore, it is crucial to study the influence of random fluctuations on the system of cold rolling mills. In this paper, taking a model of the vertical vibration of a four-roller cold rolling mill under harmonic excitation as a prototype class for a real system, the effects of random fluctuations on the system response are analyzed. Firstly, based on the deterministic vertical vibration model of a four-roller cold rolling mill under harmonic excitation, a stochastic vertical vibration model of a four-roller cold rolling mill with random fluctuation as Gaussian white noise is introduced. Subsequently, the vertical vibration model of a four-roller rolling mill is theoretically analyzed by the averaging method and the stochastic averaging method, respectively. The effectiveness of the proposed theoretical method is verified by numerical simulation results, and the influences of harmonic excitation and random excitation on the system response are also investigated in detail. Finally, the results show that the noise induces the occurrence of stochastic transitions and bifurcations, and the steady-state probability density function and time history diagrams are given to further explain the existence of these dynamic phenomena. The related research can provide theoretical guidance for the realization of vibration control and reliability design in the four-roller cold rolling mill system.     

常数激励与简谐激励联合作用下Duffing系统的非线性振动

以Duffing系统为研究对象,探讨常数激励与简谐激励联合作用下系统的骨架曲线及幅频响应特性,重点考察常数激励的影响。采用谐波平衡法求解该系统的振动方程,得到幅频响应关系,并给出了骨架曲线以及周期解的稳定性分析。采用幅频响应曲线和骨架曲线表征系统的基本动力学性质,讨论了常数激励和简谐激励幅值对系统幅频曲线性态和骨架曲线形态的影响。研究发现,系统振动响应中直流分量与谐波分量的振幅同步变化,但变化趋势相反。此外,两者骨架曲线的形态均是先向左微偏后转为向右弯曲,因此,在某些参数条件下,对应一个激励频率的周期解可能有5组,其中3组为稳定性,2组为不稳定解。进一步通过增大常数激励发现其能够对该系统造成的“刚度增强”效应,但同时也会伴随着愈加显著的“刚度渐软”特性。相应地,对于特定的简谐激励幅值,随着常数激励的增大,系统的幅频曲线能够由纯硬特性转变为软硬特性共存,甚至纯软特性。但是,在大尺度下观察,常数激励对系统骨架曲线的影响主要表现在骨架曲线根部的形态上,即随着简谐激励幅值的增大,常数激励对系统共振频率的影响变弱,不同常数激励下的骨架曲线趋于一致。     

具有分数导数本构关系的粘弹性浅拱的非线性动力学行为

利用分数导数本构模型描述材料的粘弹性特性,建立了粘弹性浅拱在横向荷载作用下的动力学方程。利用Galerkin截断法并结合边界条件分别得到了一阶和二阶Galerkin系统的控制微分方程。通过数值计算,分析了简谐激励下一阶Galerkin系统的非线动力学行为。研究表明:随着外激励幅值的变化,粘弹性浅拱系统可以通过倍周期分岔或阵发性两条路径进入混沌;固定外激励幅值、频率以及阻尼系数等状态参数,不同初始条件下,系统可以出现多周期解共存、周期解与混沌解共存的现象。     

振动磨的周期运动稳定性与分岔

由于碰撞的存在,振动磨系统的响应呈现出复杂的周期运动或混沌运动。本文将振动磨的模型进行简化,建立一维、两自由度、受简谐激振力作用的振动磨碰撞振动力学模型。基于Poincare映射原理,根据映射Jacobi矩阵分析振动磨周期运动的稳定性,并通过理论分析和数值仿真,研究振动磨周期运动的稳定性与分岔,以及由倍周期分岔通向混沌的过程。     

一类Hopfield型神经网络的概周期解

研究一类具概周期输入的Hopfield型神经网络模型,通过应用常微分方程定性理论证明了模型系统概周期解、有界解的存在唯一性及概周期吸引子的存在性。     

复合固体层中兰姆波的积累二次谐波发生与传播研究

文章提出一种分析复合固体层中兰姆波的积累二次谐波发生与传播的方法。基于导波的部分平面波分析理论，建立了复合固体层中非线性兰姆的理论模型。在分析过程中，将非线性作为线性情形的二阶微扰进行处理的方法。尽管复合固体层中兰姆波具有频散性质，分析结果表明，对于一些特定条件，二次谐波的振幅可以随传播距离积累增长。根据边界条件和初始激发条件，得到积累二次谐波的形式解，在此基础上进行了数值分析，给出一些有趣的声场分布。     

On the existence of periodic solution for equation of motion of thick beams having arbitrary cross section with tip mass under harmonic support motion

A cantilever beam having arbitrary cross section with a lumped mass attached to its free end while being excited harmonically at the base is fully investigated. The derived equation of vibrating motion is found to be a non-linear parametric ordinary differential equation, having no closed form solution for it. We have, therefore, established the sufficient conditions for the existence of periodic oscillatory behavior of the beam using Green’s function and employing Schauder’s fixed point theorem. The derived equation of vibration motion is found to be a non-linear parametric ordinary differential equation, having no closed form solution for it. To formulate a simple, physically correct dynamic model for stability and periodicity analysis, the general governing equations are truncated to only the first mode of vibration. Using Green’s function and Schauder’s fixed point theorem, the necessary and sufficient conditions for periodic oscillatory behavior of the beam are established. Consequently, the phase domain of periodicity and stability for various values of physical characteristics of the beam-mass system and harmonic base excitation are presented.     

Some aspects of chaotic and stochastic dynamics for structural systems

In this paper, the bifurcation behaviour of an externally excited four-dimensional nonlinear system is examined. Throughout this paper, a two-degree-of-freedom shallow arch structure under either a periodic or a stochastic excitation will be considered. For the case when the excitation is periodic, the local and global behaviour is examined in the presence of principalsubharmonic resonance and1:2 internal resonance. The method of averaging is used to obtain the first order approximation of the response of the system under resonant conditions. A standard Melnikov type perturbation method is used to show analytically that the system may exhibit chaotic dynamics in the sense of Smale horseshoe for the 1:2 internal resonance case in the absence of dissipation. In the case of stochastic excitation, the stability of the stationary solution is examined by determining themaximal Lyapunov exponent andmoment Lyapunov exponent in terms of system parameters. An asymptotic method is used to obtain explicit expressions for various exponents in the presence of weak dissipation and noise intensity. These quantities provide almost-sure stability boundaries in parameter space. When the system parameters lie outside these boundaries, it is essential to understand the nonlinear behaviour. The method of stochastic averaging is applied to obtain a set of approximate Itô equations which are then examined to describe the local bifurcation behaviour.