求解弱非比例阻尼系统实模态解的阻尼矩阵摄动法

提出了一种求解弱非比例阻尼振动系统实模态解的摄动方法和将非比拟阻尼矩阵分解为比例阻尼矩阵和余项阻尼矩阵的方法.对于弱非比例阻尼振动系统,通过同时对阻尼矩阵和响应矢量进行小参数摄动,将原非比例阻尼系统分解为一系列的比例阻尼振动系统,在此基础上用正则模态变换将各阶比例阻尼的摄动方程解耦,从而求得原非比例阻尼振动系统的近似解析解.计算实例表明,此方法的结果与数值计算结果十分吻合.     

旋转粘弹性夹层梁非线性自由振动特性研究

对旋转粘弹性夹层梁的非线性自由振动特性进行了分析.基于Kelvin-Voigt粘弹性本构关系和大挠度理论,建立了旋转粘弹性夹层梁的非线性自由振动方程,并使用Galerkin法将偏微分形式振动方程化为常微分振动方程.采用多重尺度法对非线性常微分振动方程进行求解,通过小参数同次幂系数相等获得微分方程组,并通过求解方程组及消除久期项来获得旋转粘弹性夹层梁非线性自由振动的一次近似解.用数值方法讨论了粘弹性夹层厚度、转速和轮毂半径对梁固有频率的影响.结果表明:固有频率随转速增大而增大,随夹层厚度增大而减小,随轮毂半径的增大而增大.     

轴向运动薄板非线性振动及其稳定性研究

应用增量谐波平衡法(IHB法)研究轴向运动薄板横向非线性振动特性及其稳定性.通过Hamilton原理推导出了非惯性参考系下四边简支轴向运动薄板的横向振动微分方程,然后利用Galerkin方法离散运动方程.对离散后的非线性方程组应用IHB法进行非线性振动分析,研究了在固有频率之比ω20/ω10接近于3:1情况下,外激励频率ω在ω10附近的具有内部共振的基谐波响应.最后用多元Floquet理论分析了系统周期解的稳定性,其中采用Hsu方法来计算转移矩阵.通过对具体例子的数值计算,分别得到了自由振动和不同外激励下的频幅相应曲线,通过对比运动梁模型和运动薄板模型的计算结果,分析了各种模型的适用范围.     

发动机总成悬置系统解耦方法研究

目前汽车发动机动力总成悬置系统设计的主要任务是选择悬置元件的刚度、位置和角度,使悬置系统自由振动模态频率避开发动机怠速激励力频率与车身自振频率,并尽量提高各模态振型的解耦程度,从而提高悬置系统隔振效果.悬置系统按预定频率严格解耦设计是使设计出的悬置系统模态频率完全等于按汽车设计频率规划预定的频率,并使各模态的振型严格解耦,即各向振动能量的解耦度等于1.本文从悬置系统的自由振动方程出发给出了对悬置系统按预定频率严格解耦设计的方程组,可以利用广义逆矩阵的理论求该方程组的解,亦可通过方程组构造函数进而求出该方程组的解,从而提供比当前的悬置系统模态优化设计更为简便高效的优化设计方法.相应的算例验证了本文提出的按预定频率严格解耦设计方程和求解方法的正确性.     

一般阻尼动力系统非奇异摄动法

借助矩阵摄动理论,将模态叠加法运用于一般阻尼矩阵的动力学方程求解结构的动响应是一种较为理想的方法.但当系统的外荷载激振频率接近于系统的固有频率时,直接将阻尼矩阵作为摄动矩阵,会使解产生奇异,并导致求解失败或误差过大,这是因为模态坐标下的动力学方程是无阻尼方程.为了解决这一问题,本文考虑在模态坐标的动力学方程中保留一定的阻尼.即将阻尼做分解,代入振动方程,得到不同阶次摄动方程,再将摄动方程变换到模态坐标,即采用非奇异摄动方法.最后通过数值算例,得到一阶、二阶摄动,将其与精确解进行比较.精度明显得到改善,基本趋于精确解.从而验证了本方法的精确性和有效性.     

非对称强非线性振动特征分析

提出了构造一类非线性振子解析逼近周期解的的初值变换法.用Ritz-Galerkin法,将描述动力系统的二阶常微分方程,化为以振幅、角频率和偏心距为独立变量的不完备非线性代数方程组;关键是考虑初值变换,增加补充方程,构成了以角频率、振幅和偏心距为变量的完备非线性代数方程组.作为例子利用初值变换法求解了相对论修正轨道方程的六种分岔周期解.给出了非对称振动的幅频曲线和偏频(偏心距与角频率的关系)曲线.发现了固有角频率漂移现象.     

基于LMI 的线性时滞系统输出动态反馈镇定

考虑线性时滞系统的输出动态反馈镇定问题.利用自由参数矩阵对闭环系统进行适当变换,并结合相应的Lyapunov-Krasovskii泛函得到了时滞相关的控制器存在性判据.利用控制器参数化方法,将控制器参数与泛函参数的求解归结为线性矩阵不等式解的形式,从而克服了时滞无关性及求解非凸优化问题所导致的保守性.仿真算例验证了结论的有效性.     

哈密顿体系下矩形薄板自由振动的一般解

根据弹性薄板自由振动问题的基本方程，把问题引入到哈密顿对偶体系中．x方向模拟为时间，选取弯矩，等效剪力，转角和挠度为对偶向量，得到了在不同边界条件时关于x轴对称和反对称时的解析解．算例研究了四边固支薄板的自由振动情形，从而推广了哈密顿体系的应用范围，验证了哈密顿体系求解方法在自由振动问题中的有效性．     

基于降阶迭代的振动系统部分特征结构配置北大核心CSCD

采用速度-位移主动控制,提出了一种解决振动系统部分特征结构配置问题的新方法。首先,利用部分特征结构配置的无溢出特性,给出了振动系统速度-位移反馈控制器的参数化表达式。然后,从闭环系统特征关系出发,将部分特征结构配置问题转化为西尔维斯特矩阵方程的求解问题。接着,对西尔维斯特矩阵方程进行降阶处理,并将其转化为等价的低阶矩阵方程组。之后,通过构造新的迭代格式求解矩阵方程组,并由此得到控制器的数值解。最后,进一步给出了迭代格式的收敛性证明及部分特征结构配置问题的求解算法。数值算例验证了所提算法的有效性。     

位移反馈分数阶PID控制对单自由度线性振子的影响

研究了基于位移反馈分数阶PID控制的单自由度线性振子的自由振动,通过平均法得到了系统的一阶近似解析解.发现分数阶PID控制器的比例环节以等效线性刚度的形式影响系统的动力学特性,积分环节以等效线性负阻尼和等效线性刚度的形式影响系统的动力学特性,微分环节以等效线性阻尼和等效线性刚度的形式影响系统的动力学特性.对近似解析解和数值解进行了比较,二者吻合良好,验证了近似解析解的正确性.从近似解析解和分数阶系统的特征方程两个角度对系统的稳定性进行了分析.最后利用系统的时间响应性能指标分析了位移反馈分数阶PID控制器的系数和分数阶阶次变化时,对单自由度线性振子控制性能的影响.     

Exact dynamic and static element stiffness matrices of nonsymmetric thin-walled beam-columns

An improved numerical method to exactly evaluate 14 × 14 dynamic and static element stiffness matrices is proposed for the spatial free vibration and stability analysis of nonsymmetric thin-walled straight beams subjected to eccentrically axial loads. Firstly equations of motion and force-deformation relations are rigorously derived from the total potential energy for a uniform beam element with nonsymmetric thin-walled cross-section. Next a system of linear algebraic equations with nonsymmetric matrices is constructed by introducing 14 displacement parameters and transforming the higher order simultaneous differential equation into the first order simultaneous equation. And then explicit expressions for displacement parameters are exactly evaluated by solving a generalized eigenproblem with complex eigenvalues. Finally exact element stiffness matrices are determined using force-deformation relations. Particularly straightforward application of the present method may not give the exact static stiffness because of existence of multiple zero eigenvalues in case of static buckling problems. Accordingly, a modified numerical method to resolve this difficulty is developed for two cases depending on the initial state of stress resultants. In order to demonstrate the validity and the accuracy of this method, the natural frequencies and buckling loads of nonsymmetric thin-walled beam-columns having bending-torsional deformation modes are evaluated and compared with analytical and F.E. solutions or results analyzed by ABAQUS’s shell element.     

消隔组合X型减振器设计与动力学分析

本文将一种X型结构与线性弹簧阻尼减振器相结合构成一种兼具消振和隔振性能的新型X型减振器.基于拉格朗日方法建立单自由度线性振子耦合消隔组合X型减振器系统的动力学方程,应用谐波平衡法得到系统稳态响应的近似解析解,并通过Runge Kutta法得到系统的数值解验证解析解的正确性.讨论了不同的基础激励下新型X型减振器对系统的响应幅值以及位移传递率的减振效果.此外,分析了不同参数对系统的幅频响应曲线以及位移传递率的影响.研究结果表明,新型X型减振器不仅可以将线性刚度转化为非线性刚度,还可以为系统提供负刚度,拥有优秀的消振和超低频隔振性能.     

A method for solving systems of linear interval equations applied to the Leontief input–output model of economics

A new approach to solving systems of linear interval equations based on the generalized procedure of interval extension is proposed. This procedure is based on the treatment of interval zero as an interval centered around zero, and for this reason it is called the “interval extended zero” method. Since the “interval extended zero” method provides a fuzzy solution to interval equations, its interval representations are proposed. It is shown that they may be naturally treated as modified operations of interval division. These operations are used for the modified interval extensions of known numerical methods for solving systems of linear equations and finally for solving systems of linear interval equations. Using a well known example, it is shown that the solution obtained by the proposed method can be treated as an inner interval approximation of the united solution and an outer interval approximation of the tolerable solution, and lies within the range of possible AE-solutions between the extreme tolerable and united solutions. Generally, we can say that the proposed method provides the results which can be treated as approximate formal solutions sometimes referred to as algebraic solutions. Seven known examples are used to illustrate the method’s efficacy and advantages in comparison with known methods providing formal (algebraic) solutions to systems of linear interval equations. It is shown that a new method provides results which are close to the so-called maximal inner solutions (the corresponding method was developed by Kupriyanova, Zyuzin and Markov) and the algebraic solutions obtained by the subdifferential Newton method proposed by Shary. It is important that the proposed method makes it possible to avoid inverted interval solutions. The influence of the system’s size and number of zero entries on the results is analyzed by applying the proposed method to the Leontief input–output model of economics.     

Hamilton体系下中厚板静力弯曲和自由弯曲振动问题的一类模型及通解

对于中厚板的静力弯曲和自由弯曲振动问题,引入两个辅助函数,采用胡海昌在Reissner板理论基础上提出的中厚板微分方程及边界条件,将两类问题的控制方程引入Hamilton体系,分别得到Hamilton体系下中厚板静力弯曲和自由振动问题的微分方程组模型. 比较后得到了Hamilton体系下中厚板静力和振动问题的统一模型,其特点是: 微分方程组模型的统一形式中Hamilton矩阵在对角线位置有2个零子块矩阵. 对于中厚板静力和振动问题,比较了所得齐次微分方程组的特征根,给出齐次微分方程组的通解并进行了比较,从而使问题的求解更理性化和合理化,求解过程遵循一套统一的方法论,便于把这类解法推广到其它问题.     

集中谐载力作用下三边固定一边自由板的受迫振动

应用混合变量最小作用量原理求解了三边固定一边自由矩形板在任意集中谐载作用下的受迫振动问题，建立了求解这类问题受迫振动稳态解的新方法，对于结构工程抗震问题中的振动分析给出了一个新的解决途径，所获得的计算结果，跟现有的文献进行比较证明了其正确性，因此该方法不仅具有重要的理论价值，而且可以为工程实际直接采用。     

Flexibility analysis of a tracking control method for air-breathing hypersonic vehicles

This paper studies the flexibility of a tracking control method originally proposed by the authors for air-breathing hypersonic vehicles （AHVs）. The main feature of this method is to design the tracking controller without canceling but using aero-propulsive, as well as elevator-to-lift couplings. By introducing a virtual input, the tracking controller and external reference trajectories are simultaneously obtained by solving a system of linear algebraic equations. This system of linear algebraic equations is always solvable and the solution space of the corresponding homogeneous system is of dimension 3, which leads to much freedom in choosing or defining the free variables. The flexibility is reflected by the fact that the flight requirements of AHVs are involved in the definition of the free variables. Three case studies on different maneuvers, i.e., flight at constant dynamic pressure, flight at variant dynamic pressure and flight with fast climb rate are considered to verify the flexibility of this method. Simulation results show its effectiveness and flexibility.     

Taylor polynomial solutions of systems of linear differential equations with variable coefficients

A Taylor collocation method has been presented for numerically solving systems of high-order linear ordinary, differential equations with variable coefficients. Using the Taylor collocation points, this method transforms the ODE system and the given conditions to matrix equations with unknown Taylor coefficients. By means of the obtained matrix equation, a new system of equations corresponding to the system of linear algebraic equations is gained. Hence by finding the Taylor coefficients, the Taylor polynomial approach is obtained. Also, the method can be used for the linear systems in the normal form. To illustrate the pertinent features of the method, examples are presented and results are compared.     

Numerical solution of an integral equations system of the first kind by using an operational matrix with block pulse functions

This article proposes a simple efficient method for solving a Volterra integral equations system of the first kind. By using block pulse functions and their operational matrix of integration, a first kind integral equations system can be reduced to a linear system of algebraic equations. The coefficient matrix of this system is a block matrix with lower triangular blocks. Numerical examples show that the approximate solutions have a good degree of accuracy.     

An expansion–iterative method for numerically solving Volterra integral equation of the first kind

Most integral equations of the first kind are ill-posed, and obtaining their numerical solution often leads to solving a linear system of algebraic equations of a large condition number. So, solving this system is difficult or impossible. For numerically solving Volterra integral equation of the first kind an efficient expansion–iterative method based on the block-pulse functions is proposed. Using this method, solving the first kind integral equation reduces to solving a recurrence relation. The approximate solution is most easily produced iteratively via the recurrence relation. Therefore, computing the numerical solution does not need to solve any linear system of algebraic equations. To show the convergence and stability of the method, some computable error bounds are obtained. Numerical examples are provided to illustrate that the method is practical and has good accuracy.