Modal mass of clamped beams and clamped plates

The calculation of the forced vibration response of a clamped beam or clamped plate often involves the calculation of the modal (or generalized) mass. Calculation of this term for simply-supported structures is relatively easy and results in a value that is half the mass of the structure for beams, and a quarter of the mass of the structure for plates and cylinders. However, calculation of this term for clamped beams and clamped plates is algebraically more complicated and has led to the presentation of several formula in research literature that appear vastly different. This paper contains a derivation of the modal mass of a clamped beam and clamped plate and shows that several of the formula for the modal mass in the research literature are equivalent.