关于含参变量特征值轨迹偏转现象的探讨

含参变量的特征值轨迹的偏转现象出现在许多力学和理论物理问题中。本文通过两个实例,对特征值轨迹偏转现象进行了深入的探讨。先讨论了一个两维平板振动问题,通过求特征值的上、下界的方法,确定了该问题的精确特征值呈现了曲线(含一个参变量)偏转和曲面(含两个参变量)偏转现象;然后讨论了一个双震荡器的振动问题,发现当某个参数发生变化时的特征值曲线是由相交变为偏转的。且当一对特征值曲线偏转前后,两个特征向量几乎互相改变了方向;而当一对特征值曲面偏转前后,对应的特征向量的分量也发生偏转。

Loci Veering Phenomenon in Parameter Dependent Eigenvalue Problems

Loci veering phenomenon arises naturally in a wide variety of eigenvalue problems in mechanics and theoretical physics, which is further investigated and the previous work is extended here via two examples. The eigenvalues of a vibrating plate is disscussed, evaluating the upper and lower bounds of the eigenvalues, it is shows that the exact eigenvalues and endowed with curve veering(one parameter) and surface veering(two parameters). Then a problem of free vibration of a coupled oscillator is discussed, where the eigenvalue loci of this problem can turn from crossing to veering by changing the value of a parameter, after two eigenvalue curves veering, the corresponding eigenvectors almost interchange their directions, after a couple of eigenvalue surfaces veering, the components of the corresponding eigenvectors also veer.