This paper presents a time-domain optimal design procedure for adaptive negative stiffness devices (ANSDs) installed in inelastic building frames. The structure is modeled as a nonlinear shear-type system with hysteretic restoring forces, classical viscous damping with approximately uniform modal damping ratios, and story-wise ANSDs composed of a negative stiffness device and a nonlinear fluid viscous damper. A state-space formulation suitable for gradient-based optimization is derived, and an objective functional is defined that combines a shear-energy measure based on squared story-shear histories over an ensemble of ground motions with penalties on changes in ANSD stiffness and damping coefficients. Sensitivities of this functional with respect to the device parameters are obtained via an adjoint (Lagrange multiplier) formulation, evaluated in discrete time, which enables an efficient iterative tuning scheme. The methodology is demonstrated using an eight-story reinforced-concrete frame, idealized as a planar shear building, and subjected to 15 recorded earthquakes. The tuned devices reduce the shear-energy measure for most records, yield modest decreases in peak base shear and floor accelerations for critical motions, and prevent collapse under one previously critical record with only moderate adjustments of the ANSD parameters.
Assaf Shmerling (Sat,) studied this question.