30 F. Weber et al. where kh 4 is the pre-sliding stiffness that is selected two orders of magnitude greater than W/Reff 4. The force of the restrainer of concave plate 4 is assumed as linear stiffness force if it is triggered fr 4 D kr 4 .ju4 u3j d4/ sign.u4 u3/ W ju4 u3j d4 0 W ju4 u3j <d4 (4.7) where kh 4 denotes the restrainer stiffness which is also assumed to be two orders of magnitude greater thanW/Reff 4. The equations of motion of concave plate 3 (i D3) and slider mass 2 (i D2) have the same form mi Rui Cfh i C W Reff i .ui ui 1/ Cfr i Dfh .iC1/ C W Reff .iC1/ .uiC1 ui/ Cfr .iC1/ mi Rug (4.8) where the friction and restrainer forces fh i andfr i are formulated analogically with (4.6) and (4.7). The equation of motion of concave plate 1 is given by Eq. (4.8) with i D1 and ui 1 D Pui 1 D0. The equation of motion of concave plate 2 of the double FP is given analogically with (4.5) and the equation of motion of the slider (index 1) of the double FP is given by (4.8)withi D1 and ui 1 D Pui 1 D0. 4.5 Isolation Performance of Mock-Up Triple Friction Pendulum The isolation results of the triple FP that is designed according to the literature [1, 2] and of the double FP with same isolation time period and equal friction coefficients 1 D 2 that are tuned by trial and error are depicted in Figs. 4.2 and 4.3 for the four chosen accelerograms. The following main observation can be made: • The triple FP generates a worse isolation of the structure for most of the PGAs (PGA 2.2 m/s2 for El Centro NS, PGA 0.59m/s2 for Kobe, PGA 0.85m/s2 for Loma Prieta, PGA 1.12m/s2 for Northridge) than the double FP whose equal friction coefficients are tuned by trial and error (suboptimal tuning). • The isolation due to the triple FP deteriorates dramatically when sliding regime V is triggered due to the stiffening behavior at reduced friction of sliding regime V [4]. • The fact that the restrainers of the double FP with same total displacement capacity are triggered at larger PGAs than in case of the triple FP demonstrates that the double FP does not only generate better isolation of the structure but also that the total bearing displacement is smaller. 0 1 2 3 4 7 8 El Centro NS 7.8 0 2 4 6 8 1 3 5 7 restrainers triggered (triple and double FPs) sliding regime V triggered 2.2 a restrainers triggered sliding regime V not triggered double FP, μ1=μ2=6.0% without isolator (max=19.1m/s2) double FP, μ1=μ2=6.5% triple FP, μ1=4%, μ4=13% triple FP, μ1=5%, μ4=15% 5 6 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 00.511.52 2.533.54 Kobe restrainers triggered (triple and double FPs) 0.59 3.82 sliding regime V triggered sliding regime V not triggered without isolator (max=12.22m/s2) double FP, μ1=μ2=5.5% double FP, μ1=μ2=6.5% b triple FP triple FP PGA (m/s2) PGA (m/s2) max( | üs+üg | ) (m/s 2) max( | üs+üg | ) (m/s 2) Fig. 4.2 Isolation performances of triple FP according to literature and conventional double FP tuned by trial and error for El Centro NS (a) and Kobe (b) accelerograms
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