22 Numerical Model for Human Induced Vibrations 181 where Wis a matrix representing the connection of msubjects to the structure (i.e., it is a matrix composed by 0 and 1 elements [6]). According to the Woodbury matrix identity [29], the new matrix GH of the FRFs of the new system composed by the structure and the people is: GH D G 1 CWHWT 1 DG GW H 1 CWTGW 1WTG (22.5) To find the expression of GH, the Hfunctions must be known for all the people. It is possible to use measuredHfunctions as well as approximated models available in literature [30, 31]. 22.2.2 The Case of Active People In the case of active people the AGRFs are non-null. Equation (22.1) can be used to estimate the structural response due to people movement on the structure. To this purpose, we need an estimation of the AGRFs and PGRFs. As for the AGRFs, it is possible to build a database of forces for the considered kind of movement (e.g., as in [7] for people ascending and descending staircases). Figure 22.2 shows some examples of the collected AGRF time-histories for people ascending and descending staircases. As for PGRFs, we can still use Eq. (22.3), but we have to keep in mind that in this case the functionHchanges with time because people change posture during the movement. Furthermore, people move over the structure so that the PGRFs change point of application in time. As for the former point, an equivalent apparent mass M eq is defined taking into account many different postures of the subject during the motion. Basically, each step is split into different positions (Ppositions) and the apparent mass M a;i is estimated (e.g., with experimental tests as in [7]) for each of these postures. Actually, each M a;i is the average of the apparent mass of different people in the given posture. Then, M eq is calculated as: M eq .!/ D P XiD1 ˛iM a;i .!/ (22.6) where the ˛i coefficients are the weights. They are chosen in order to describe properly the amount of time spent by the people in the corresponding posture within the cycle time T (i.e., the time of a single step). In other words, we freeze P postures within the movement cycle and we treat each of them as a static posture. In this scenario, each person produces a PGRF related to the apparent mass M eq. Then, the PGRFs can be applied to the structure in two different ways: Fig. 22.2 Two examples of recorded AGRF time-histories for people ascending a staircase (up) and two examples of recorded AGRF timehistories for people descending a staircase (down)
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