Abstract: 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.

Key words: vibration of a flexible rod, resistance, separation of variables, complex eigenproblem, orthogonality and completness