21 Evaluation of Mass-Spring-Damper Models for Dynamic Interaction Between Walking Humans and Civil Structures 173 where t is time (s), F(t) is the walking force (N), Wis the static weight of the subject (kg), n is the harmonic number, N is the total number of harmonics, ˛n is the Fourier coefficient of the nth harmonic, which is also known as dynamic load factor (DLF), fp is pacing frequency and ¿n is the phase shift of the nth harmonic. Based on the above mentioned simulation groups, nine cases of simulations are considered in this study: • Deterministic force and non-interactive model (DF-N). • Measured force and non-interactive model (MF-N). • Deterministic force and Silva’s model [8] of walking people (DF-Si). • Measured force and Alonso’s model [6] of walking people (MF-A). • Measured force and Shahabpoor’s model [7] of walking people (MF-Sh). • Measured force and Silva’s model [8] of walking people (MF-Si). • Measured force and Toso’s model [9] of walking people (MF-T). • Measured force and Van Nimmen’s model [10] of walking people (MF-V). • Measured force and Zhang’s model [11] of walking people (MF-Z). It is worth mentioning that the simulated vibration responses are repeated six times to replicate the experimentally measured vibration responses. 21.4 Results and Discussion 21.4.1 Results In this section, the simulated vibration responses described in Sect. 21.3.3 are compared with their experimental counterparts. Two approaches are utilised for comparison, and as follows: • Maximum 1-second Root Mean Square (RMS). • Statistical comparison of vibration responses. The process of comparing the simulations is schematically illustrated in Fig. 21.4 and the description of each procedure is presented in this section. The maximum 1-second RMS is calculated for each measured and simulated vibration response. There are six walking tests/simulations for each case (i.e. six maximum 1-second RMS), and the average value is calculated and considered for comparison. This average value is normalised by its experimental counterpart and the results are presented in Fig. 21.5. The maximum 1-second RMS does not provide a clear description of the overall vibration responses, as it is based on the maximum 1-second of the vibration response only. The cumulative distribution of the vibration responses can provide more informative description. Hence, it is used to compare the measured and simulated vibration responses. Figure 21.6 presents the cumulative distribution function (CDF) of the experimental and simulated vibration responses for a test subject at two pacing frequencies. The concept of fractiles is utilised to compare the CDF of the measured and simulated vibration responses. In this study, a certain fractile of a vibration signal refers to the proportion of vibration signal with magnitude up to that value. The fractile related to any probability of non-exceedance (0–100%) can be extracted from the CDF plots. Figure 21.7 compares the fractiles of two simulations with their experimental counterparts. Interestingly, the fractiles of up to 90% probability of non-exceedance of all simulation cases follow almost a linear line which does not necessarily slope at 45ı (Fig. 21.7). This means there is a trend of over- or under-estimation of vibration responses for vibration magnitudes with probability of non-exceedance up to 90%. Hence, the ratio of simulated/measured fractiles for this range is utilised to estimate the slope of the best linear line that passes through them (Fig. 21.7). This slope is the ratio of the vertical/horizontal components of the line. A slope of 1.1, for example, indicates an overestimation for vibration response for up to 90% of the vibration signal. This feature is used to compare the measured and simulated responses for all simulation cases, and the values of the slope for all simulation cases are presented in Table 21.2. The overall performance of each model is demonstrated by calculating the average absolute difference (from 1.0) of each fractile ratio as shown in the last row in Table 21.2.
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