Despite the well known benefits of physical units, matrices, and matrix algebra in engineering computations, most engineering analysis packages are essentially dimensionless. They simply hold the engineer responsible for selecting a set of engineering units and making sure their use is consistent. While this practice may be satisfactory for the solution of self-contained and well established problem solving procedures, where the structure of the output is well known and understood, identifying and correcting unintentional errors in the solution of new and innovative computations can be significantly easier when units are an integral part of the computation procedure. This report begins with a description of the data structures and algorithms needed to represent and manipulate physical quantity variables, and matrices of physical quantities. The second half of this report focuses on the implementation of Aladdin, a new computational environment for matrix and finite element calculatio...

ABSTRACT: Despite the well known benefits of physical units, matrices, and matrix

This work explores symbolic and numeric solutions to the Lyapunov matrix equation as it applies to performance-based assessment of base-isolated structures supplemented by modified bang-bang control. Traditional studies of this type rely on numeric simulations alone. This study is the first to use symbolic analysis as a means of identifying key “cause and effect ” relationships existing between parameters of the active control problem and the underlying differential equations of motion. We show that symbolic representations are very lengthy, even for structures having a small number of degrees of freedom. However, under certain simplifying assumptions, symbolic solutions to the Lyapunov matrix equation assume a greatly simplified form (thereby avoiding the need for computational solutions).Regarding the behavior of the bang-bang control strategy, further analysis shows: (1) for a 1-DOF system, the actuator force acts very nearly in phase, but in opposite direction to the velocity (90 ◦ out of phase and in opposite direction to the displacement), and (2) for a wide range of 2-DOF nonlinear base-isolated models, bang-bang control is insensitive to nonlinear deformations in the isolator devices. Through nonlinear time-history analysis, we see that one- and two-DOF models are