18 Nonlinear Damping in Floor Vibrations Serviceability: Verification on a Laboratory Structure 143 Time System Response ap Output Resonance build-up Time Input Excitation p0 Input Fig. 18.3 Input excitation and system response at resonance Table 18.4 Effective mass for the first bending mode Stage Effective mass calculations per MEScope mode shapes 1 0.470M 2 0.502M Therefore, Ru.t/ p0 D k=m 2 k D 1 2 m (18.23) Equation (18.22) can also be written as; Ru.t/ D p0 2 m (18.24) When both sides are divided by gravitational acceleration; Ru.t/ g D p0 2 mg D p0 2 We (18.25) ap D p0 2 Me (18.26) For Eq. (18.26), ap is the peak acceleration; p0 is the excitation amplitude of the sinusoidal forcing function (Fig. 18.3); is the modal damping ratio and Me is the effective mass for the first bending mode. Equation (18.26) will be used for determining the modal damping ratios of the laboratory structure when the structure was in resonant condition. While effective mass is basically an indication of mass sensitivity to a dynamic excitation, it can be calculated if the mode shapes are available. The mode shapes for the laboratory structure are calculated experimentally by using the modal data collected with accelerometers. When each accelerometer location is assigned as a lumped mass, data for each lumped mass location can then be loaded into a commercial vibration analysis software (MEScope) and the mode shapes can be formed. Since the mode shape of a structure is known, the effective mass for that specific mode can be calculated. This is how the mode shape vectors for the first bending mode are determined for both stages of the structure. Each mode shape vector was normalized and effective mass for the first bending mode was calculated (Table 18.4). As shown in Table 18.4, Stage 1 resulted in an effective mass of 0.470 M for the first bending mode. Likewise, Stage 2 resulted in 0.502 M for the effective mass of the first bending mode. The effective mass value is about 7% higher for the Stage 2 than the Stage 1. This make sense since Stage 1 is structurally stiffer than Stage 2. The amount of vibrating effective mass would be less for a stiffer structure.
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