弹簧静变形和重力加速度从固有频率表达式中消失的条件

教学经常使用固有频率表达式不显含弹簧静变形和重力加速度的振子;但也能找到相反的情形.本研究探讨前者的充要条件.分析发现如果弹簧轴线在质点通过静平衡位置时沿其轨迹的切向,则弹簧静变形从固有频率表达式中消失.在满足这一前提下,进一步要求质点轨迹在静平衡位置的曲率为0,或沿铅垂方向,则重力也无回复力的效果,从而重力加速度在固有频率表达式中不出现.该结论不仅有助于理解重力与固有频率之间的关系,也可指导振动例题编制和选用.     

关于弹簧振子固有频率的进一步讨论

近年来,国内外的一些研究生入学试题中,常见的一个题目是,在不能忽略弹簧质量的前题下计算弹簧振子的固有频率。这个问题本质上牵涉到弹性介质中波的传播问题,比较复杂.但是,在某些简化假设下,可以用计算固有频率的能量法──雷利法[1]来近似求解,并证明在此情况下弹簧振子的固有(圆)频率为[2]:式中k为弹簧的倔强系数,m为弹簧的质量,M为振动物体的质量. 现在,我们证明(1)式只是精确解在一定程度下的近似,同时讨论(1)式的应用条件和因而引起的误差,以便对(1)式有个较深入的理解. 我们仍然需要作一些简化处理,即把弹簧简化为一根均匀的弹性杆…     

无负荷质量弹簧振动固有频率的研究

通过弹簧和弹性细杆之间的类比,推导出无负荷质量弹簧在几种常见边界条件下纯纵向振动的频率方程及固有频率的公式并进行了讨论。     

弹簧振子系统振动的周期

本文分析了弹簧质量不可忽略时，对振子周期的影响．引用瑞利法讨论系统振动的基频，最后按多自由度理论对精确求解进行了讨论；通过比较，阐明瑞利法的适用范围及精度．文中还提到计算振动周期的简捷方法──能量法．     

重离子碰撞中单粒子自由度的平衡

从占有数的主方程出发,在近似的类跃迁几率下,分别变一体碰撞和二体碰撞过程,定量估计了单粒子自由度的局域弛豫时间大小和特征,特别是对局域温度的依赖.并在二体碰撞情况下,同其它方法的结果做了比较.     

变形核振动态的微观处理

本文在处理对力的粒子数守恒方法的基础上,提出了处理长程力的一个方法,利用此方法分析了重变形核的振动态的微观机构．变形核的振动态被看成是在长程力（四极力、八极力等）的影响下,Ｋ~π相同的若干“对拆散态”和“对激发态”的某种相干混合．给出了长程力在各“对拆散态”和“对激发态”之间的矩阵元公式．在此方法中,避免了通常在ＢＣＳ方法基础上处理振动时所碰到的粒子数不守恒、假态、波函数不正交等困难．也没有采用ＲＰＡ方法中那些近似假定．而且基态或激发态的不同的堵塞效应已自动考虑在内。ｎｐ力对导致振动态的出现起了重要作用．作为例子,我们分析了稀土偶偶核的γ振动和锕系偶偶核的ｂ振动态的内部结构．具体说明了以下事实：（ｌ）振动带位置随不同核的变化规律（图１,３）;（２）从基态到振动态的跃迁几率有显著加强;（３）讨论了重变形核中什么区域可以出现低能级的振动带．本文还分析了奇Ａ核的γ振动态,它们被看成是Ｋ~π相同的三粒子态和单粒子态的相干混合．以Ｈｏ～（１６５）为例,分析了奇Ａ核γ振动的各种特点．在这里堵塞效应起了很重要的作用．     

弹簧振子实验不确定度的分析研究

采用静力伸长法和振动周期法测量并计算弹簧振子的劲度系数,通过分析不确定度,比较了用静伸长法求弹簧劲度系数中逐差法和用最小二乘法作Excel直线拟合这两种数据处理方法的精确性差异,并通过计算得出测量弹簧劲度系数时静伸长精确度高于振动法的结论。实验结果具有一定的理论意义,并对于改进弹簧振子实验过程有重要的指导作用。     

转动弹簧的微变形分析及其应用

教材中经典的弹簧振子的弹簧不会转动,但工程上弹簧轴线会转动的情形比比皆是。弹簧轴线转动会造成几何非线性,但是围绕平衡位置的微幅振动仍是振动学习的最重要任务。本文先用解析法导出了弹簧微变形的近似表达式,再利用直观的几何关系验证这一表达式,而后者易于理解和记忆。利用3个例题对表达式的威力进行了验证。     

外电场对弹簧振子系统运动特性的影响

设计了一个受周期驱动带电弹簧振子的实验装置，并建立系统的动力学方程，用线性稳定性分析的方法确定平衡点并分析平衡点的稳定性，利用数值计算并结合多种分析方法求解和判断非线性方程解的各种性质，得到存在电场力作用情况下系统的一些新特性。     

基于功率流方法的多自由度系统吸振控制

针对附加动力吸振器的多自由度系统,利用结构导纳理论,引入附加动力吸振器后振动系统的等价结构导纳矩阵,推导了振源输入系统的总功率流、主振系的净功率流以及吸振器吸收的功率流。并且以功率流为控制量,探讨了通过附加单个或多个动力吸振器对多自由度系统进行单模态或多模态控制的问题,揭示了吸振器的控制机理和效果。     

Camera calibration from vanishing points in a vision system

Camera calibration has been studied for many years and there are many methods available to find the parameters precisely. However, most existing methods require information of the known scene points in general three-dimensional positions for the calibration. A simple, geometrically intuitive method is proposed. The intrinsic parameters of the camera are determined by using the vanishing points in each image. The rotation matrix of the projection matrix is computed from the vanishing and image edges and the translation matrix are obtained with additional translation motion between the viewpoints. Our approach does not need any a priori information about the cameras being used. Computer simulations and real data experiments are carried out to validate our method.     

Kubo formula for frequency dispersion of dielectric permittivity and static conductivity of the Coulomb system

It is proved that the Kubo formula for the conductivity σ(ω) is valid at real frequencies ω. On this basis, an exact relation is derived for the static conductivity σst of the Coulomb system. It is shown that the static conductivity is determined by the time correlation function in the limit t→∞. It is proved that the permittivity ε(ω) satisfies the Kramers-Kronig relations which take into account a singularity associated with static conductivity.     

The vanishing viscosity limit for a dyadic model

A dyadic shell model for the Navier-Stokes equations is studied in the context of turbulence. The model is an infinite nonlinearly coupled system of ODEs. It is proved that the unique fixed point is a global attractor, which converges to the global attractor of the inviscid system as viscosity goes to zero. This implies that the average dissipation rate for the viscous system converges to the anomalous dissipation rate for the inviscid system (which is positive) as viscosity goes to zero. This phenomenon is called the dissipation anomaly predicted by Kolmogorov’s theory for the actual Navier-Stokes equations.     

Resonance response of a single-degree-of-freedom nonlinear vibro-impact system to a narrow-band random parametric excitation

The resonant response of a single-degree-of-freedom nonlinear vibro-impact oscillator with a one-sided barrier to a narrow-band random parametric excitation is investigated. The narrow-band random excitation used here is a bounded random noise. The analysis is based on a special Zhuravlev transformation, which reduces the system to one without impacts, thereby permitting the applications of random averaging over "fast" variables. The averaged equations are solved exactly and an algebraic equation of the amplitude of the response is obtained for the case without random disorder. The methods of linearization and moment are used to obtain the formula of the mean-square amplitude approximately for the case with random disorder. The effects of damping, detuning, restitution factor, nonlinear intensity, frequency and magnitude of random excitations are analysed. The theoretical analyses are verified by numerical results. Theoretical analyses and numerical simulations show that the peak response amplitudes will reduce at large damping or large nonlinear intensity and will increase with large amplitude or frequency of the random excitations. The phenomenon of stochastic jump is observed, that is, the steady-state response of the system will jump from a trivial solution to a large non-trivial one when the amplitude of the random excitation exceeds some threshold value, or will jump from a large non-trivial solution to a trivial one when the intensity of the random disorder of the random excitation exceeds some threshold value.     

The Knizhnik-Zamolodchikov system as a deformation of the isomonodromy problem

It is shown that the Knizhnik-Zamolodchikov (KZ) equation (and corresponding vector bundle) can be viewed as a quantization of the isomonodromy problem for differential equations with several singular points.     

An approximate method for determining the static deflection and natural frequency of a cracked beam

This paper provides an approximate method to determine the stiffness and the fundamental frequency of a cracked beam. The cracked beam is first represented as an un-cracked beam with equivalent reduced sections around the cracks. The effect of the cracks is explained, visualised and quantified using the equivalence concept developed for stepped beams with periodically variable cross-sections. Then an alternative expression of the improved Rayleigh method is provided to calculate the natural frequencies of a beam with a variable stiffness distribution along its length. As the method is insensitive to the assumed mode shapes, it avoids the difficulty in choosing appropriate mode shapes and yields accurate results. This is shown using several examples to compare the results determined using the proposed method and the Finite Element method (FEM). The method greatly simplifies the calculation of cracked beams with complicated configurations, such as a beam with several cracks, a cracked beam with concentrated masses, a beam with cracks close to each other, and a beam with periodically distributed cracks.     

Definition of the actinide region of static quadrupole-octupole deformation

Ground state spins of odd-A nuclei in the region just beyond ²⁰⁸Pb are compiled. They are compared with the theoretically predicted spins of an axially symmetric, reflection asymmetric nucleus with octupole deformation ϵ₃ = 0.08 to determine the region of static quadrupole-octupole deformation. Coexistence of different shapes and the corresponding spectra in the same nucleus are predicted in the transition regions.