1 Semi-Active Base Isolation of Civil Engineering Structures Based on Optimal Viscous Damping and Zero Dynamic Stiffness 5 0.10 0.15 0.2 0.25 0.3 0.35 stiffness coefficient (kN/m) -2000 -1000 1000 2000 3000 ktotal = kR-eff-nominal 0 0.05 kcontrol ktotal = kcontrol + kR-eff ktotal = 0 kcontrol =kR-eff control law #1 (CL #1) -3000 kcontrol= kR-eff (kR-eff = 0.5kR-eff-nominal) 0 - 0.10 0.15 0.2 0.25 0.3 0.35 stiffness coefficient (kN/m) -2000 -1000 1000 2000 3000 0 0.05 kcontrol ktotal = kcontrol + kR-eff ktotal = 0 -3000 kcontrol = kR-eff (kR-eff = 0.5kR-eff-nominal) 0 kcontrol =kR-eff control law #2 (CL #2) - ktotal = kR-eff-nominal a b bearing displacement amplitude U (m) bearing displacement amplitude U (m) Fig. 1.2 Controlled stiffness and total bearing stiffness due to (a) control law #1 and (b) control law #2 0.10 0.15 0.2 0.25 0.3 0.35 stiffness coefficient (kN/m) -3000 -2000 -1000 1000 2000 3000 ktotal = kR-eff-nominal 0 0 0.05 kcontrol = kR-eff kcontrol ktotal = kcontrol + kR-eff ktotal = 0 kcontrol = 0 (kR-eff = kR-eff-nominal) control law #3 (CL #3) - 0.10 0.15 0.2 0.25 0.3 0.35 stiffness coefficient (kN/m) -3000 -2000 -1000 1000 2000 3000 0 0 0.05 kcontrol ktotal = kcontrol + kR-eff kcontrol = 0 kcontrol = - kR-eff ktotal = 0 (kR-eff = kR-eff-nominal) control law #4 (CL #4) ktotal = kR-eff-nominal a b bearing displacement amplitude U (m) bearing displacement amplitude U (m) Fig. 1.3 Controlled stiffness and total bearing stiffness due to (a) control law #3 and (b) control law #4 • Control law #4 (CL #4, Fig. 1.3b): The effective radius Reff of the curved surface is equal to Reff nominal. The controlled stiffness is formulated to produce zero dynamic stiffness at UD0andktotal DW/Reff nom at U Umax D0.25 m. Between UD0 and UDUmax the controlled stiffness is a linear function of U. The main difference between CL #1 and CL #3 (and between CL #2 and CL #4) is that the maximum (positive) and minimum (negative) controlled stiffness coefficients due to CL #1 (and CL #2) are only 50% of the maximum negative controlled stiffness of CL #3 (and CL #4) due to the different designs of Reff for CL #1 (and CL #2) and CL #3 (and CL #4). The control law leading to smaller controlled stiffness is more suitable for controllable dampers since the emulation of large stiffness with semi-active dampers is inherently combined with the generation of damping that is larger than the desired viscous damping given in (1.4) whereby the actual stiffness and damping of the actual semi-active force are far from their desired counterparts. Detailed information on the emulation errors of desired stiffness and damping with controllable dampers is beyond the scope of this paper but can found in [10]. The main difference between CL #1 (and CL #3) and CL #2 (and CL #4) is that CL #1 (and CL #3) results in zero dynamic stiffness at UDUmax which improves the isolation of the structure at large PGAs due to earthquakes between DBE and MCE while CL #2 (and CL #4) generate zero dynamic stiffness at UD0 which improves the isolation of the structure due to earthquakes up to DBE.
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