自制金割效应物理摆的研究性实验

金割效应物理摆是将数学中的优选法与物理实验中的复摆结合起来并用计算机处理数据的自制教学仪器.把金割效应物理摆作为研究性实验进行开设,更有利于培养学生的开拓精神和创新能力.     

BRST invariance and the conical pendulum

The Hamiltonian formulation of the BRST method for quantizing constrained systems developed recently by Nemeschanskyet al is applied to the well-known problem of the conical pendulum in classical mechanics. The similarity of the system to a gauge theory wherein the two constraints serve as generators of local Abelian gauge transformations is also pointed out. The definition of the physical states of the system as a gauge theory and also as a BRST invariant theory is then discussed in some detail.     

用线性插值法求单摆运动周期的近似解

利用线性插值的方法,提出了一个任意摆角条件下的单摆运动周期的近似公式.并利用Mathematica软件将该公式与精确解进行了比较,还与其他近似公式进行了对比.结果表明,该公式的近似程度相当好.     

Stability analysis of structurally unstable man–machine system involving time delays

In this paper, a structurally unstable man–machine system is considered and a rigorous mathematical analysis is performed to study the influence of time delays in the control force on stability of the equilibrium solution of this system. Results on stability, instability and conditions for the preservation of stability (instability), besides the existence of bifurcation are presented.     

测量圆盘转动惯量产生误差的曲线研究

结合三线摆实验的基本过程,进行实验数据的采集。利用投影法来测量圆盘的转角大小,借助于matlab的cftool工具箱,对数据进行曲线拟合,对测出来的二组数据拟合,得到转角与百分差曲线图。通过误差的大小比较,得出圆盘转动惯量测量值与理论值的百分差随转角变化最接近于三次曲线。     

Optimal feedback gains of a delayed proportional-derivative (PD) control for balancing an inverted pendulum

In the dynamics analysis and synthesis of a con-trolled system, it is important to know for what feedback gains can the controlled system decay to the demanded steady state as fast as possible. This article presents a sys-tematic method for finding the optimal feedback gains by taking the stability of an inverted pendulum system with a delayed proportional-derivative controller as an example. First, the condition for the existence and uniqueness of the stable region in the gain plane is obtained by using the D-subdivision method and the method of stability switch. Then the same procedure is used repeatedly to shrink the stable region by decreasing the real part of the rightmost charac-teristic root. Finally, the optimal feedback gains within the stable region that minimizes the real part of the rightmost root are expressed by an explicit formula. With the optimal feedback gains, the controlled inverted pendulum decays to its trivial equilibrium at the fastest speed when the initial val-ues around the origin are fixed. The main results are checked by numerical simulation.     

钟摆模型极限环的存在性研究

钟摆系统是一类典型的分段光滑系统,结合Filippov系统刻画语言,解释了当钟摆无能量补充时,钟摆最终会停止在滑动集上的原因.利用数值模拟的方法,给出钟摆系统在有能量补充时,存在极限环的条件.最后,结合环域定理证明了一般的钟摆模型存在唯一稳定的极限环.     

三线摆实验中摆角偏大产生的误差及修正方法

研究了三线摆实验中摆角偏大(5°θ20°)时对测量转动惯量产生的误差和修正办法。实验发现随着摆角的增大,测得的圆盘转动惯量也会相应偏大,从而导致测量误差增大。文章通过进一步的理论分析解释了这一现象,并对测量结果进行了系统误差修正。     

Tunneling of trapped-atom Bose condensates

We obtain the dynamics in number and phase difference, for Bose condensates that tunnel between two wells of a double-well atomic trap, using the (nonlinear) Gross-Pitaevskii equation. The dynamical equations are of the canonical form for the two conjugate variables, and the Hamiltonian corresponds to that of a momentum-shortened pendulum, supporting a richer set of tunneling oscillation modes than for a superconductor Josephson junction, that has a fixed-length pendulum as a mechanical model. Novel modes include ‘inverted pendulum’ oscillations with an average angle of π; and oscillations about a self-maintained population imbalance that we term ‘macroscopic quantum self-trapping’. Other systems with this phase-number nonlinear dynamics include two-component (interconverting) condensates in a single harmonic trap, and He³B superfluids in two containers connected by micropores.     

从机械能守恒的角度推导单摆简谐振动的周期

本文给出了利用基本的物理原理结合高等数学公式的方法推导单摆做简谐运动的周期公式,可以简明扼要地理解单摆简谐运动的过程.