Redundancy in Residual Vectors for Missing Mass Effect in Coupled Modal Synthesis

It has been illustrated in the past that a modal synthesis gives incorrect modal properties and seismic response of nonclassically damped coupled systems when all the modes of uncoupled primary and secondary systems are not included. The effect of missing mass contained in the truncated high frequency modes can be represented using residual modal vectors. This paper illustrates two possibilities in which the residual modal matrix may contain redundant vectors leading to a breakdown of the originally proposed method. First, secondary systems with a support inclined to the primary system orthogonal connecting degrees of freedom can have residual vectors that are scalar multiple of each other (parallel). Second, the number of residual and nonrigid modal vectors can be more than the total number of secondary system modes (equal to the number of secondary system degrees of freedom) in multiply connected simple secondary systems. A solution is presented to identify and eliminate both these types of redundant vectors using an alternative form of Gram-Schmidt’s orthogonalization.