轴向运动大挠度板的非线性动力学行为

该文研究了受周期激励轴向运动大挠度板横向振动的稳定性及分岔现象。在von Kàrmàn非线性大挠度板理论基础上,利用达朗贝尔原理建立系统的动力学模型。通过Galerkin截断,将时间变量和空间变量、位移函数和应力函数耦合在一起的偏微分方程离散,得到系统运动常微分方程。利用数值方法分析板随轴向运动速度、外激励力幅值、长宽比和轴向拉力变化时的运动分岔行为。利用最大Lyapunov指数和Poincaré映射图识别系统的动力学行为。结果发现,当板的某些参数变化时,系统出现分岔现象。不同参数时,系统呈现周期运动、倍周期运动、概周期运动,甚至混沌运动。     

非线性弹簧支承悬臂输液管道的分岔与混沌分析

研究悬臂输液管道系统在自激励、参数激励和外激励联合作用下的非线性动力学行为,揭示系统运动的规律。建立了非线性弹簧支承悬臂输液管道的运动微分方程,以线性弹簧支承条件下悬臂梁的固有频率和振型函数作为近似,采用李兹-伽辽金方法对非线性运动微分方程进行离散化,经过数值计算,利用分岔图、相图和功率谱图分析系统的非线性动力学响应,得到了流体平均流速和流体与管道质量比对系统周期运动和混沌运动的影响规律。研究结果表明,当流体平均流速较小时,系统的响应首先表现为周期运动,随着流体平均流速的增大,系统的响应通过系列倍周期分岔而进入混沌运动,又经由系列倍周期倒分岔转化为周期运动。随着流体与管道质量比的减小,系统出现混沌运动的临界流体平均流速值减小,这说明通过改变流体与管道质量比参数可以控制系统的振动形态。     

分段非线性轧机辊系系统的分岔行为研究

考虑轧机液压压下缸和平衡缸对辊系的约束影响,建立轧机辊系的分段非线性动力学模型。分别采用奇点稳定性理论和奇异性理论分析了该分段非线性系统在自治情况和非自治情况下的分岔特性,得到不同系统参数下的分岔形态。最后根据某轧机结构参数,模拟了轧机分段非线性辊系系统在不同的外部扰动下的局部分岔行为,发现随着外扰参数的变化该系统是周期运动、倍周期运动以及混沌等多种运动形态相互交替的复杂动力学系统,外部扰动参数的变化影响系统的运动形态,这为研究和抑制轧机辊系振动问题提供了理论参考。     

分析弹性地基一般支承输流管道的动力学特性

研究Pasternak双参数地基一般支承输流管道的线性固有频率及非线性动力学特性。综合考虑管道黏弹性系数、地基的剪切效应、线性刚度的影响,建立了系统运动微分方程。根据两端一般支承的边界条件推导出线性系统固有频率方程,分析了基础激励与脉动流作用下,流速对系统非线性动力学特性的影响。数值结果表明,管道一阶临界流速随弹性系数的增大呈现先增大后减小的趋势,当弹性系数足够大时,管道随流速的增加发生一阶、二阶模态耦合现象;系统响应随流速变化呈现由倍周期分岔过渡到混沌运动的特性;当管内流体流速足够大时,系统响应保持混沌运动状态。     

参数激励驱动微陀螺系统的非线性振动特性研究

对于一类典型的切向梳齿驱动型微陀螺,建立两自由度、具有刚度立方非线性和参数激励驱动的微陀螺系统动力学模型。考虑主参数共振和1∶1内共振的情况,利用多尺度法获得周期解的解析形式,并利用分岔理论,得到Hopf分岔条件,结合数值模拟系统的动力学响应,揭示系统参数对驱动和检测模态振幅和分岔行为的影响机制。研究结果表明,在1∶1内共振和较大的载体角速度下,激励频率的变化容易引起微陀螺振动系统的多稳态解、振幅跳跃现象和概周期响应等复杂动力学行为。     

含磨损故障的齿轮传动系统非线性动力学特性

为揭示磨损故障对于齿轮传动系统非线性动态特性的影响,利用Archard和Weber-Banaschek公式分别计算了齿面动态累积磨损量和磨损齿轮对的时变啮合刚度。建立含有非线性齿侧间隙、内部误差激励和含磨损故障的时变啮合刚度的三自由度齿轮传动系统平移-扭转耦合动力学方程。采用变步长Gill积分方法对动力学模型进行了数值仿真分析,以系统的激励频率为分岔参数,计算系统的对应的分岔图;引入GRAM-SCHMIDT方法对系统的Jacobi矩阵进行正交化处理,计算系统的李雅普诺夫指数谱,同时结合Poincaré映射图和功率谱验证了李雅普诺夫指数谱和分岔图计算结果的正确性。通过研究发现了系统内部存在的丰富非线性现象,包括倍周期分岔途径、阵发性途径和多种拟周期通过锁相进入混沌的现象;在系统经由拟周期进入混沌的过程中发现了交替出现的拟周期与锁相现象以及拟周期运动时功率谱分量存在的Farey序列现象。研究结果表明含有磨损故障的齿轮传动系统具有非常复杂的动力学特性,而系统由周期运动进入混沌运动的途径也是丰富多样的。     

一类非线性相对转动系统的动力学行为及其机理分析

研究了一类具有非线性刚度的相对转动系统的动力学行为。应用Routh-Hurwitz稳定性理论判断了相对转动系统平衡点的稳定性。应用分岔理论研究了平衡点失稳时的分岔行为,推导了平衡点产生fold分岔的条件,进而通过仿真得到了平衡点在双参数平面上的分岔集及单参数分岔曲线,研究了不同参数区域内平衡点的个数以及稳定性问题。应用分岔图研究了相对转动系统随平方非线性刚度系数及激励角频率变化的全局动力学行为,获得了周期三以及混沌等动力学行为。通过调整平方非线性刚度系数得到了慢变外激励下相对转动系统中的对称式和不对称式fold/fold簇发行为。     

一类冲击钻进机械系统的动力学特性

建立了一类冲击钻进机械系统的动力学模型,分析了系统周期冲击运动的类型,给出了判定系统发生钻进运动的条件,并采用数值计算的方法分析了系统的动力学行为和系统参数对系统钻进效果的影响。结果表明：擦边分岔处是系统两种截然不同运动的“分界线”,“擦边”运动导致 运动转迁为 运动或形成复杂的长周期运动或混沌;当激振频率 在 周期运动的冲击速度峰值附近时,系统具有最大冲击速度和最佳钻进量     

两端弹性支承裂纹管道的非线性动力学特性

研究两端弹性支承输流管道含圆周方向裂纹时的非线性动力学特性。首先,推导出裂纹管道的模态函数与局部柔度系数,然后运用Galerkin离散技术将管道运动方程在模态空间中展开,采用非线性动力学仿真方法得到管道系统响应随各参数变化的分岔图和最大Lyapunov指数图。数值结果表明,这种两端弹性支承的特殊边界裂纹管道在参数激励、自激励和外激励联合作用下,表现出丰富的非线性动力学特性,分别出现周期运动、概周期运动、阵发性混沌和混沌等多种响应形式。     

陀螺系统的受迫振动及其时滞反馈控制

对简谐激励下陀螺系统的受迫振动及其在含时滞的位移和速度反馈控制下的动力学行为进行研究。利用拉格朗日方程,建立了两自由度陀螺系统的运动微分方程。考虑主共振和1:1内共振的情况,采用平均法得到了平均方程。通过对平均方程进行化简,得到了关于系统振幅的分岔方程,分别讨论了各个参数对系统振幅的影响。根据奇异性理论,分析了参数变化对系统分岔行为的影响。对受迫陀螺系统施加含时滞的位移和速度反馈控制,讨论了反馈增益和时滞对系统振幅的影响。     

黏弹性轴向运动变张力梁非线性动力学

在黏弹性轴向运动梁横向参数振动的非线性动力学行为研究中,首次计入因速度变化引起的、沿梁的径向变化的、轴向变张力的影响。给出描述变张力轴向运动梁横向非线性振动的偏微分—积分控制方程。基于微分求积法给出轴向运动梁横向非线性参数振动的数值解,通过观察梁中点的位移、速度随时间变化的历程,识别轴向运动系统的非线性动力学行为。同时,通过从数值解中提取的相图、Poincaré映射图和频谱分析,考察轴向运动梁横向振动的分岔与混沌特性,揭示了工程应用中的非线性轴向运动系统的混沌动力学行为。     

Dynamic analysis of an axially translating viscoelastic beam with an arbitrarily varying length

The nonlinear free vibration of an axially translating viscoelastic beam with an arbitrarily varying length and axial velocity is investigated. Based on the linear viscoelastic differential constitutive law, the extended Hamilton’s principle is utilized to derive the generalized third-order equations of motion for the axially translating viscoelastic Bernoulli–Euler beam. The coupling effects between the axial motion and transverse vibration are assessed under various prescribed time-varying velocity fields. The inertia force arising from the longitudinal acceleration emerges, rendering the coupling terms between the axial beam acceleration and the beam flexure. Semi-analytical solutions for the governing PDE are obtained through the separation of variables and the assumed modes method. The modified Galerkin’s method and the fourth-order Runge–Kutta method are employed to numerically analyze the resulting equations. Further, dynamic stabilization is examined from the system energy standpoint for beam extension and retraction. Extensive numerical simulations are presented to illustrate the influences of varying translating velocities and viscoelastic parameters on the underlying dynamic responses. The material viscosity always dissipates energy and helps stabilize the transverse vibration.     

Asymptotic analysis on nonlinear vibration of axially accelerating viscoelastic strings with the standard linear solid model

Nonlinear parametric vibration of axially accelerating viscoelastic strings is investigated via an approximate analytical approach. The standard linear solid model using the material time derivative is employed to describe the string viscoelastic behaviors. A coordinate transformation is introduced to derive Mote’s model of transverse motion from the governing equation of the stationary string. Mote’s model leads to Kirchhoff’s model by replacing the tension with the averaged tension over the string. An asymptotic perturbation approach is proposed to study principal parametric resonance based on the two models. The amplitude and the existence conditions of the steady-state responses are determined by locating the nonzero fixed points in the modulation equations resulting from the solvability condition. Numerical results are presented to highlight the effects of the material parameters, the axial-speed fluctuation amplitude, and the initial stress on steady-state responses.     

Hamiltonian dynamic analysis of an axially translating beam featuring time-variant velocity

A dynamic analysis is presented for an axially translating cantilever beam simulating the spacecraft antenna featuring time-variant velocity. The extended Hamilton’s principle is employed to formulate the governing partial differential equations of motion for an axially translating Bernoulli–Euler beam. Further, the assumed modes method and the separation of variables are utilized to solve the resulting equation of motion. Attention is focused on assessing the coupling effects between the axial translation motion and the flexural deformation during the beam extension or retraction operations upon the vibratory motion of a beam with an arbitrarily varying length under a prescribed time-variant velocity field. A number of numerical simulations are also presented to illustrate the qualitative features of the underlying mechanical vibration of an axially extending or contracting flexible beam. In general, the transverse beam vibration is stabilized during extension and unstabilized during retraction. The axial acceleration of a translating beam does not affect the transverse vibratory system stabilization.     

The nonlinear vibration of an initially stressed laminated plate

Nonlinear partial differential equations of motion for a laminated plate in a general state of non-uniform initial stress are presented in various plate theories. This study uses Lo’s displacement field to derive the governing equations. The higher-order terms in Lo’s theory can be disregarded, to obtain the equations of simpler forms and even other theories for laminated plate. These nonlinear partial equations are transformed to ordinary nonlinear differential equations using the Galerkin method. The Runge–Kutta method is used to obtain the ratio of nonlinear frequency to linear frequency. The numerical solutions of an initially stressed laminate plate based on various plate theories obtained by the Galerkin and Runge–Kutta method are presented herein. Using these equations with various theories, the nonlinear vibration behavior of laminated plate is studied. The results show that apparent discrepancies exist among the various displacement fields, which indicates the transverse shear strain, normal strain and initial stress state have great effect on the vibration behavior of laminate plate under nonlinear vibration.     

轴向运动导电薄板的磁弹性振动

针对磁场环境中轴向运动导电薄板的磁弹性振动问题进行研究。在给出薄板运动的动能、应变能以及电磁力虚功表达式的基础上,应用哈密顿变分原理,推得磁场中轴向运动矩形薄板的磁弹性振动方程。基于麦克斯威尔电磁场方程并考虑相应的电磁关系式,得到薄板所受电磁力的表达式。针对横向磁场中矩形板的自由振动问题,通过位移函数的设定并应用伽辽金积分法,得到三种边界约束条件下轴向运动薄板的磁弹性振动微分方程。通过数值算例,给出了不同边界条件下矩形板的磁弹性振动特性曲线图,分析了轴向运动速度和磁感应强度等参量对薄板固有振动频率的影响,讨论了临界速度的变化规律。     

轴向变速黏弹性Rayleigh梁非线性参数振动稳态响应

研究非线性轴向运动黏弹性Rayleigh梁因速度周期变化产生的亚谐波共振。轴向运动速度在平均速度附近做简谐周期性脉动。通过取物质导数的Kelvin本构关系描述Rayleigh梁的黏弹性。运用多尺度近似解析方法,构建轴向运动Rayleigh梁的非线性偏微分方程的可解性条件,分析参数振动稳态响应的振幅与扰动速度频率关系。并运用微分求积方法直接离散非线性Rayleigh梁的控制方程,以验证近似解析方法分析。通过数值算例,分析了系统参数对稳态响应曲线的影响。     

基于微分求积法的轴向流作用下二维板复杂响应研究

研究轴向流作用下两端简支二维板线性稳定性及非线性复杂响应。用微分求积法对流场方程及二维板连续型运动方程统一离散,通过流固耦合边界条件将流场势函数用板的横向振动位移变量表示,获得二维板横向振动位移变量控制方程。通过特征值分析获得复频率及临界流速随流道高度变化。通过对壁板非线性动力响应数值模拟,采用分岔、相平面及庞加莱截面图等揭示壁板将发生的周期运动、拟周期、混沌等多种运动形式表明,壁板由周期倍化分岔或拟周期运动通向混沌。     

Nonlinear transversal vibration of an axially moving viscoelastic string on a viscoelastic guide subjected to mono-frequency excitation

In this paper, the nonlinear transversal vibration of an axially moving viscoelastic string on a viscoelastic guide subjected to a mono-frequency excitation is considered. The model of the viscoelastic guide is a parallel combination of springs and viscous dampers. The governing equation of motion is developed using Hamilton’s principle. Applying the method of multiple scales to the governing partial differential equation, the solvability condition and approximate solutions are derived. Three cases, namely primary, subharmonic and superharmonic resonances are studied and appropriate analytical solutions are obtained. The effect of mean value velocity, force amplitude, guide stiffness and viscosity coefficient of the string on the frequency-response and bifurcation points is investigated. Findings are in good agreement with results extracted from numerical modeling.