18 Nonlinear Damping in Floor Vibrations Serviceability: Verification on a Laboratory Structure 141 Table 18.2 Test and FE model results for Stage 1 Sine excitation amplitude at 8 Hz frequency Acceleration response amplitude (Test) Acceleration response amplitude SAP2000 (N) (lbs) (%g) Calculated damping ratio ( ) using Me D0.470M (%g) Acceleration response amplitude ratio: SAP2000/Test 56.5 12.7 0.14 0.00690 0.135 0.964 118.8 26.7 0.26 0.00781 0.252 0.969 198.8 44.7 0.36 0.00944 0.348 0.967 Table 18.3 Test and FE model results for Stage 2 Sine excitation amplitude at 6.95 Hz frequency Acceleration response amplitude (Test) Acceleration response amplitude SAP2000 (N) (lbs) (%g) Calculated damping ratio ( ) using Me D0.502M (%g) Acceleration response amplitude ratio: SAP2000/Test 33.8 7.6 0.12 0.00451 0.1226 1.022 53.4 12.0 0.17 0.00503 0.1736 1.021 120.1 27.0 0.24 0.00801 0.2453 1.022 195.7 44.0 0.32 0.00979 0.3271 1.022 18.3.2 Stage 2 Configuration For Stage 2, the structure was put in resonance again by a shaker placed at the midspan, at four levels of excitation amplitudes. The results are shown in Table 18.3. 18.4 Damping Ratio Back-Calculations Using Effective Mass When the structure is in resonance, the resulting peak acceleration amplitude can be used to calculate the effective mass of the specific mode. The relationship between the effective mass, peak acceleration amplitude, the input excitation amplitude and the modal damping ratio can be derived. It is known that for harmonic vibration with viscous damping: mRuCcPuCku Dp0 sin!t (18.1) u.t/ Duc.t/ Cup.t/ (18.2) The transient decay component is: uc.t/ De !nt ŒAcos !Dt CBsin!Dt (18.3) The steady-state response component is: up.t/ DC sin!t CDcos !t (18.4) For ! D!nI CD0 and DD .ust/0 2 (18.5) For ! D!n and zero initial conditionsI AD .ust/0 2 and BD .ust/0 2p1 2 (18.6)
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