Derivation of Equation of Motion for the Pillow-Shape Seismic Base Isolation System

Abstract

Rolling-based seismic isolation systems, in which rollers of circular or non-circular section are used, are less expensive and easier to manufacture. However, this type of isolation suffers from either lack of restoring or re-centering capability, or weakness against uplift forces. To resolve the first shortcoming the use of elliptical as well as pillow-shape rolling parts has been suggested, and for the second one, using some uplift restrainers has been proposed. In this paper a kind of base isolating device, called Pillow-Shape Seismic Base Isolation System (PSBIS) is introduced which has both re-centering and uplift-restrained capabilities. The paper is concentrated on derivation of equation governing the motion of PSBIS device. To derive this equation, since the behavior of the device is highly nonlinear the use of Lagrange equation of motion is the most appropriate approach. For this purpose, considering the rotation angle of the pillow-shape roller as the generalized coordinate, the horizontal and vertical components of the pillow-shape roller acceleration have been formulated and the kinetic and potential energy terms as well as the virtual work of the non-conservative, resulted from rolling resistance and seismic forces, have been developed to be substituted in the basic Lagrange equation. To verify the derived equation of motion a simplified sample of the PSBIS device was built and used experimentally, and also the motion of the pillow-shape parts was traced by drawing them in Auto-CAD software. Results show the validity of the derived equation of motion.