4 F. Weber et al. in order to produce: 1. the controlled stiffness kcontrol that is controlled as function of the bearing displacement amplitude Uto compensate for the passive stiffness of the curved surface given by W/Reff and thereby produce zero dynamic stiffness by kcontrol <0 for maximum decoupling of the structure from the ground, and 2. the controlled damping force copt c Pu that dissipates the same amount of damping as resulting from optimal linear viscous damping. The desired optimal viscous damping coefficient copt is reduced by the viscous damping coefficient c that is energy equivalent to the friction damping of the lubricated curved surface [2] c 4 W !iso U (1.5) in order to dissipate the cycle energy of optimal viscous damping. Since c is inversely proportional to the displacement amplitude U of the isolator, i.e. c U 1, c may become greater than c opt at small U which necessitates the distinction of cases in (1.4). Notice that (1.5) represents an approximation because c according to Eq. (1.5) is derived based on the constant isolation radial frequency !iso Dr g Reff (1.6) but the actual frequency of the bearing displacement due to earthquake excitation is time-variant and therefore not detectable in real-time. However, the approximation (1.5) represents a good engineer’s solution as the actual frequency is in the vicinity of !iso. The actual force of the semi-active oil damper can only produce the dissipative forces of the desired active control force f desired active , that is f actual semi active D f desired active W Pu f desired active 0 0 W Pu f desired active <0 (1.7) The formulation (1.7) assumes that control force constraints such as minimum and maximum forces of the semi-active oil damper and control force tracking errors do not exist. Hence, the formulation (1.7) of the semi-active force represents the ideal behavior of a controllable damper. 1.4.2 Adaptive Controlled Stiffness The maximum decoupling of the structure from the shaking ground and therefore minimum absolute structural acceleration Rus C Rug is obtained fromzero dynamic stiffness of the isolator [9]. Since the passive (and positive) stiffness of the isolator is given by W/Reff , the controlled stiffness kcontrol must be negative to reduce the total stiffness ktotal of the isolator to zero under dynamic operation. However, ktotal D0 for the entire bearing displacement range could not re-center the structure sufficiently. Hence, four adaptive stiffness control laws are suggested that produce zero dynamic stiffness either at UD0 or at U Umax due to the MCE: • Control law #1 (CL #1, Fig. 1.2a): The effective radius Reff of the curved surface is 50% of the nominal effective radius Reff nominal generating the targeted isolation time period Tiso D3.5 s. The controlled stiffness is formulated to produce ktotal Dkcontrol CW/Reff DkR eff nominal DW/Reff nominal at UD0 and zero dynamic stiffness, i.e. ktotal D0, at U Umax D0.25 m. Between UD0 and UDUmax the controlled stiffness is a linear function of U. • Control law #2 (CL #2, Fig. 1.2b): The effective radius Reff of the curved surface is 50% of Reff nominal. The controlled stiffness is formulated to produce zero total stiffness at UD0andktotal DW/Reff nominal at U Umax D0.25 m. Between UD0 and UDUmax the controlled stiffness is a linear function of U. • Control law #3 (CL #3, Fig. 1.3a): The effective radius Reff of the curved surface is equal to Reff nominal. The controlled stiffness is formulated to produce ktotal DW/Reff nominal at U D0 and zero dynamic stiffness at U Umax D0.25 m. Between UD0 and UDUmax the controlled stiffness is a linear function of U.
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