Abstract: The vibration of a string or a rod with their ends connected to particles（concentrated mass）,orsimilar problems,result in a class of unconventional Sturm-Liouville eigenvalue problem after separation of variables.The completeness of the eigenfunctions of such eigenvalue problems is proved.It is also proved that the n-th eigenfunction has n-1 nodes（zeros）.

Key words: unconventional Sturm-Liouville eigenvalue problem, eigenfunctions, completeness, nodes（zeros）