A new method for the construction of integrable hamiltonian systems

A new method for the construction of integrable Hamiltonian system is proposed. For a given Poisson manifold, the present paper constructs new Poisson brackets on it by making use of the Dirac-Poisson structure^[1], and obtains further new integrable Hamiltonian systems. The constructed Poisson bracket is usual non-linear, and this new method is also different from usual ones^[2–4]. Two examples are given.     

广义Hamilton系统的研究概况

用广义Ｐｏｉｓｓｏｎ括号直接定义的广义Ｈａｍｉｌｔｏｎ系统是经典Ｈａｍｉｌｔｏｎ系统的一种推广。由于它的维数没有限制，同时它又保持了经典Ｈａｍｉｌｔｏｎ系统的主要性质，因此具有更大的普遍性和广泛的应用前景。本文简要介绍了广义Ｈａｍｉｌｔｏｎ系统理论的发展历史、基本概念和研究现状。     

The Hamiltonian Structures of 3D ODE with Time-Independent Invariants

ＴＨＥＨＡＭＩＬＴＯＮＩＡＮＳＴＢＵＣＴＵＲＥＳＯＦ３ＤＯＤＥＷＩＴＨＴＩＭＥ－ＩＮＤＥＰＥＮＤＥＮＴＩＮＶＡＲＩＡＮＴＳＧｕｏＺｈｏｎｇ－ｈｅｎｇ（郭仲衡）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＰｅｋｉｎｇＵｎｉｖｅｒｓｉｔｙ，Ｂｅｉｊ...     

STOCHASTIC OPTIMAL CONTROL FOR THE RESPONSE OF QUASI NON-INTEGRABLE HAMILTONIAN SYSTEMS~

A strategy is proposed based on the stochastic averaging method for quasi nonintegrable Hamiltonian systems and the stochastic dynamical programming principle. The proposed strategy can be used to design nonlinear stochastic optimal control to minimize the response of quasi non-integrable Hamiltonian systems subject to Gaussian white noise excitation. By using the stochastic averaging method for quasi non-integrable Hamiltonian systems the equations of motion of a controlled quasi non-integrable Hamiltonian system is reduced to a one-dimensional averaged Ito stochastic differential equation. By using the stochastic dynamical programming principle the dynamical programming equation for minimizing the response of the system is formulated.The optimal control law is derived from the dynamical programming equation and the bounded control constraints. The response of optimally controlled systems is predicted through solving the FPK equation associated with It5 stochastic differential equation. An example is worked out in detail to illustrate the application of the control strategy proposed.     

拟不可积Hamilton系统响应的随机最优控制

提出了一种基于拟不可积Hamilton系统随机平均法和随机动态规划原理的控制策略.它可以对受高斯白噪声激励的拟不可积Hamilton系统进行非线性随机最优控制,以使系统的响应最小化.利用拟不可积Hamilton系统随机平均法可以将受控的拟不可积Hamilton系统降维成一维的Ito随机微分方程.利用随机动态规划原理可以为系统响应最小化问题建立动态规划方程.在控制力为有界的条件下,从动态规划方程中可以确定出最优控制规律.受控系统的响应是通过求解与Ito随机微分方程相联系的FPK方程得到的,用一个例子阐述了这一随机最优控制策略的实施过程.     

Vibration characteristics and flutter analysis of a composite laminated plate with a store

The effects of an external store on the flutter characteristics of a composite laminated plate in a supersonic flow are investigated. The Dirac function is used to formulate the interaction between the plate and the store. The first-order piston theory is used to describe the aerodynamic load. The governing equation of the composite laminated plate with an external store is established based on the Hamilton principle. The mode shapes are constructed by the admissible functions which are a set of characteristic orthogonal polynomials generated directly by the Gram-Schmidt process, and the boundary constraint is modeled as the artificial springs. The frequency and mode shapes of the plate under different boundaries are determined by the Rayleigh-Ritz method. The validity of the proposed approach is confirmed by comparing the results with those obtained from the finite element method (FEM). The effects of the mounting position, the center of gravity position and the mounting points spacing of the external store on the flutter boundary are discussed for both the simply supported and cantilever plates, respectively, which correspond to the two installation sites of the external store, i.e., the belly and wings of the aircraft.     

受约束多体系统一种新的违约校正方法

本文针对受约束多体系统的违约问题提出了一种新的违约校正方法．同以往的违约校正方法相比，本方法具有物理意义明确，计算工作量小，校正效果明显的优点．本文的方法对动力学方程的破坏较小，能同任何积分方法配合使用，可有效地将约束方程的违约控制在给定的精度范围内．数值仿真的结果表明了本方法的有效性．     

随机激励的耗散的哈密尔顿系统的平稳解

本文首先为一般的随机激励的耗散的哈密尔顿系统得到精确的平稳解,然后在此基础上为类似而更为一般的系统发展了等效非线性系统法。     

随机激励的耗散的哈密尔顿系统的平稳解

本文首先为一般的随机激励的耗散的哈密尔顿系统得到精确的平稳解,然后在此基础上为类似而更为一般的系统发展了等效非线性系统法     

离散时间系统Ляпунов函数的构造与稳定性判定

本文旨在以矩阵的相似变换理论为基础,以电子计算机为工具,提供一种无需求解离散?矩阵方程,而直接由系统矩阵来构造离散时间系统?函数和判定系统稳定性的方法.     

复杂构造系统的可控性设计计算方法

本文将反分析技术应用到复杂构造系统的设计计算中去，形成一种可控性设计计算方法，即可按人们指定的系统力学性能直接推算系统的设计参数。文中采用超级元方法，对具有复杂细观构造的纤维增强复杂材料层合板进行单参数反分析及可控性设计。     

强非线性振动系统的渐近解法

本文推广文[1]中方法于求解广泛一类强非线性振动系统,导出了适于近似定性分析和定量计算的简便公式。作为例子研究了修正的Vander Pol振子,最后给出数字Poincaré映射结果,从近似定性分析和定量计算两方面证明了本文求得的方法的有效性。     

随机和区间非齐次线性哈密顿系统的比较研究及其应用

随着计算机技术的飞速发展,更高效、更稳定和长时间模拟能力更强的数值算法需求迫切.哈密顿系统辛算法与传统算法相比在稳定性和长期模拟方面具有显著优越性.但动力系统中不可避免地存在大量不同程度的不确定性,动力学分析中需要考虑这些不确定性的影响以确保合理有效性.然而,目前考虑参数不确定性的哈密顿系统响应分析的研究基础还比较薄弱.为此,本文考虑随机和区间参数不确定性,对两种不确定性非齐次线性哈密顿系统分析计算结果进行了比较研究,从而突破了传统哈密顿系统的局限性,并应用于结构动力响应评估中.首先,针对确定性非齐次线性哈密顿系统,提出了考虑确定性扰动的参数摄动法;在此基础上,分别提出了随机、区间非齐次线性哈密顿系统的参数摄动法,得到了它们响应界限的数学表达;随后,用数学理论推导得到了区间响应范围包含随机响应范围的相容性结论;最后,两个数值算例在较小时间步长下验证了所提方法在结构动力响应中的可行性和有效性,体现了随机、区间哈密顿系统响应结果之间的包络关系,并在较大时间步长下与传统方法相比较凸显了哈密顿系统辛算法的数值计算优势、与蒙特卡洛模拟方法相比较验证了所提方法的精度.     

陶瓷阻力曲线测定的新方法——断裂栅法

将零厚度单向试件栅刻蚀工艺应用于晶须增韧氮化硅陶瓷基复合材料表面，通过记录试件所在桥路的输出电压值，可得到随载荷不断增大裂纹逐渐向前扩展过程中每一时刻的裂尖位置和裂纹长度，从而得到该种材料的阻力曲线。     

极坐标哈密顿体系约当型与弹性楔的佯谬解

讨论了极坐标弹性平面哈密顿体系的当型,并通过约当型的求解,直接给出了相关弹性楔体佯谬问题的解,从理论上阐明了经典弹性力学中某些佯谬问题的出现是由于其对应的是哈密顿体系中特殊的约当型解,同时也很自然地为该类问题提供了一个通用,有效的求解方法。     

材料力学考核评价方式的改革与探索 1)

现有考核评价方式侧重期末考试而忽略过程考核,结合材料力学课程,探索一套注重过程考核的评价体系,综合评价学生对本课程的掌握程度,即将期末考试成绩在总评中的占比由70%降低至50%,平时作业、随堂测验、模块测验、实验考核和案例分析成绩各占比10%。新评价方式更加注重对学生学习过程的考核,使学生在整个课程学习过程中保持高度紧张感,提高学习的时效性和目的性,进而提高学生学习材料力学的热情和材料力学课堂教学效果。     

近哈密顿系统的Hopf分岔

简化了Wiggins提出的关于近哈密顿系统的Hopf分岔条件,并结合硬弹簧Duffing系统,研究了该类系统的Hopf分岔行为,并用数值积分的方法验证了结果的正确性。     

NONLINEAR INTERNAL RESONANCE OF FUNCTIONALLY GRADED CYLINDRICAL SHELLS USING THE HAMILTONIAN DYNAMICS

Internal resonance in nonlinear vibration of functionally graded （FG） circular cylin- drical shells in thermal environment is studied using the Hamiltonian dynamics formulation. The material properties are considered to be temperature-dependent. Based on the Karman-Donnell＇s nonlinear shell theory, the kinetic and potential energy of FG cylindrical thin shells are formu- lated. The primary target is to investigate the two-mode internal resonance, which is triggered by geometric and material parameters of shells. Following a secular perturbation procedure, the underlying dynamic characteristics of the two-mode interactions in both exact and near resonance cases are fully discussed. It is revealed that the system will undergo a bifurcation in near resonance case, which induces the dynamic response at high energy level being distinct from the motion at low energy level. The effects of temperature and volume fractions of composition on the exact resonance condition and bifurcation characteristics of FG cylindrical shells are also investigated.     

亏损系统模态优化控制Riccati方程的求解方法

讨论亏损重复特征值的Ｒｉｃｃａｔｉ方程的模态控制算法．由于亏损特征值的Ｊｏｒｄａｎ块矩阵的阶数ｍ小于状态方程的阶数ｎ,即ｍ＜＜ｎ;所提出的模态控制算法可极大地减少计算工作量．其中的数值例子说明了该方法的有效性．     

固体力学中的Hamilton正则表示

主要给出一系列关于固体力学中偏微分方程（组）的Ｈａｍｉｌｔｏｎ正则表示．其中包括２，４，６阶线性对称偏微分方程、柱壳问题及弹性力学中混合动力问题等的Ｈａｍｉｌｔｏｎ正则表示．