AUGMENTED MODAL DIMENSIONS FOR SEISMIC EVALUATION OF BUILDINGS

An N-degree-of-freedom structure can be represented by using N single-degree-of-freedom modal sticks/oscillators, which form an N-dimensional modal space. Because each modal stick represents a modal equation of motion, we always take it for granted that a modal stick is one-dimensional. This paper shows that a modal equation of motion could be multi-dimensional. Accordingly, the corresponding modal stick/oscillator is also multi-dimensional. In other words, we can augment the dimensions of a modal space. It is worth recalling that with the imaginary unit, the one-dimensional real number axis was augmented into the two-dimensional complex plane, which is influential in mathematics and science. Likewise, augmented modal dimensions are influential in seismic evaluation of structures. For instance, a three-degree-of-freedom modal stick can simultaneously reflects the three force-displacement relationships of a two-way asymmetrical building subjected to its modal inertia force. Moreover, a multi-degree-of-freedom modal stick allows the modal responses in the three directions to be different from each other. This feature is appealing for modal analysis of non-proportionally damped buildings, such as buildings with supplemental damping, or buildings with soil-structure interaction. On the other hand, it is common to use multiple tuned mass dampers to control a translation-rotation coupled vibration mode. Nevertheless, based on the concept of augmented modal dimensions, a translation-rotation coupled tuned mass damper was developed to control a translation-rotation coupled vibration mode. That novel tuned mass damper is a straightforward and simple solution to the modal control of asymmetrical buildings. Moreover, through conducting modal fusion, a self-mass damper, designated as top-story mass damper, was developed for the seismic control of the first triplet of vibration modes of two-way asymmetrical buildings. In short, augmented dimensions of modal space push the boundaries of seismic evaluation of structures.