182 M. Vanali et al. 1. each PGRF (i.e., one for each person on the structure) is considered as a moving excitement. Hence, the equivalent matrixGH(!) (see Eq. 22.4) changes in time; 2. a fraction of the apparent mass m fr(!) is calculated: m fr .¨/ D m nd M eq .!/ (22.7) where mis the number of people on the structure and nd is the number of points in which the structure is discretised. Then, m fr .¨/ is applied to each of the nd points. Thus, the PGRF in each point can be expressed as: f GR i .!/ Dm fr .!/ Rxi .!/ D m fr .!/! 2xi .!/ (22.8) In terms of the full displacement vector x(!), fGR becomes: fGR .!/ DWnHWT nx.!/ D ! 2m fr .!/Wnx.!/ (22.9) where Wn is a nd x nd identity matrix, H(!) is a nd x nd diagonal matrix containing the fractions of the equivalent apparent mass (i.e. H.!/ D Wn!2m fr .!/). Substituting Eq. (22.9) into Eq. (22.1), we obtain (neglecting f): G 1 .!/ C!2m fr .!/Wn x.!/ DG 1 H .!/x.!/ D f ACTIVE .!/ (22.10) where GH(!) is the nd nd matrix representing the equivalent set of FRFs describing the dynamic behaviour of the joint system composed by the structure and the people. Obviously, the behaviour of this joint system is an average behaviour because m fr is employed. The second approach of the previous list assumes a fixed form of GH(!) in time. Therefore, this assumption makes the simulation of the structure response fast and easy under the computational point of view. Moreover, when the number of people on the structure is increases, the accuracy of this easy-to-apply approach is expected to increase as well. Finally, the response of the structure to the movement of people can be calculated as the convolution between the AGRFs and the unit impulse response functions of the joint system. These unit impulse response functions can be achieved by applying the inverse Fourier transform to the FRFs composing GH(!). As for the AGRFs to be applied, Fig. 22.3 provides an example of how to build them by using the database of AGRFs mentioned previously. Such an example refers to a person ascending a staircase and the forces depicted are related to the force exerted by this person on three consequent steps. Fig. 22.3 Simulation of the AGRFs exerted by a person ascending a staircase on three consequent steps. The portion of the time-histories where the forces are not null are recorded AGRFs like those of Fig. 22.2. The force signals are in Newton
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