Spectral modal combination of linearly related variables

Modal spectral superposition for single-direction excitation is reviewed, with special attention to the intervening cross-directional estimator. A method for calculating this cross-estimator in terms of three standard estimators, the x-, y-, and 45°-direction excitation estimators, is presented. Addressing the question of the signs of cross-estimators, it is shown that the problem dwells in double-sum superposition formulas rather than in the cross-estimators themselves. In that respect, it is shown that in a double superposition formula, the modal coordinates of strongly coupled modes must enter with the same sign, so that there is no loss of information in the suppression of signs in spectral ordinates. This discussion is in itself a deterministic justification of double-sum spectral superposition formulas, showing that for closely spaced modal frequencies, the formula is an approximation rather than estimation. A formula is derived for the estimator of a linear combination of two variables, equivalent to calculating the estimator by directly applying the spectral superposition formula to the linear combination of the modal components of the two basic variables. It only requires the cross-variable estimator, in addition to the standard estimators of the two basic variables. It is stressed that contrary to common belief, statical or kinematical linear combinations can be checked or otherwise used, if the appropriate cross-estimators are retrieved from analysis. Cross-estimators are interpreted as the product of the standard estimator of one variable times the pseudo-synchronous value of the other variable. The appropriateness of the synchronous value concept is verified, and it is used to propose a member single or double curvature criterion.