“端面受到空气阻力的弹性杆的振动”续谈

Complement to"Vibration of a flexible rod with one end subject to air resistance"

采用传统的分离变量法,讨论了端面受阻尼作用的弹性杆振动问题.分离变量得到了一种新类型的本征值问题:本征值为复数,相应的本征函数也是复函数.尽管本征函数本身不具有正交性,但是重新组合之后能够构造出一组正交完备函数组,从而为求解这一定解问题奠定了良好的理论基础.笔者分析了这一解法的理论含义,提出了一系列基本的理论问题,并回答了其中的部分问题.

The vibration of a flexible rod with one end subjected to air damping is discussed in the framework of the usual method of separation of variables.A new type of eigenproblem is solved and complex eigenvalues and eigenfunctions are obtained.The eigenfunctions themselves are not orthogonal mutually,but they can be combined to construct a complete and orthogonal system of functions.Therefore,the vibration of the rod can be determined by the initial conditions.The validity of this approach is analysed to probe its theoretical meaning.Some questions are suggested and attempt is pursued to find a partial answers of them.