On Vibration Problems With Discretely Distributed Loads-A Rigorous Formalism

Three problems in simple vibration theory are treated, the longitudinal vibration of a prismatic bar, the lateral vibration of a beam, and the vibration of a circular plate, each with arbitrarily placed discrete loads. It is shown that with the appropriate definition of an inner product on a Hilbert space the spatial differential operator may be made selfadjoint with all of the formal advantages of that reduction. The key to the solution of the problem is in the redefinition of the inner product from the normal one to one which is slightly more complicated but obvious once it is seen.