Missing mass effect in coupled analysis. I: Complex modal properties

New formulations are developed to account for the effect of uncalculated high frequency rigid modes in the seismic analysis of coupled primary-secondary systems by modal synthesis approach. The effect of uncalculated rigid modes is included in terms of residual modal vectors for both the uncoupled primary and the multiply supported secondary systems. The reduced eigenvalue problem, obtained by transforming the original equation of motion in which the displacements at the secondary system degrees of freedom (DOF) are expressed relative to the base of the primary system, gives incorrect results when all the modes of the secondary system are not included and the effect of high frequency modes is included through residual modal vectors. An alternate formulation is presented in which the original equation of motion is transformed such that the displacements at the secondary system DOF are expressed relative to the primary system connecting DOE The reduced eigenvalue problem is solved using the transformed equation of motion for the nonrigid and the residual modal vectors of the two uncoupled systems.