180 M. Vanali et al. Fig. 22.1 Human-structure interaction: PGRFs (fGR) and AGRFs (fACTIVE) person and are only due to the active movement of the person. The AGRF can be described as the force exerted by a moving person on a structure with an infinite stiffness. To sum up, the PGRF depends on the dynamic features of the person and on the motion of the structure. Conversely, the AGRF is produced by an active movement which is not related to the dynamics of the structure. According to Fig. 22.1, the following equation describes the dynamics of the structure occupied by people: x.!/ DG.!/ f ACTIVE .!/ f GR .!/ Cf .!/ (22.1) where x is the vector of the degrees-of-freedom in which the structure has been discretised, Gis the matrix containing the frequency response functions (FRF) of the empty structure, f is the vector of the external forces, fACTIVE is the vector of the AGRFs and fGR is the vector of the PGRFs. The expression of Gis [27, 28]: G.!/ D n XjD1 ¥j¥ Tj !2 j !2 C2i j!!j (22.2) where !j is the jth eigenfrequency, j is the associated non-dimensional damping ratio and ¥j is the jth mode shape vector (scaled to the unit modal mass) measured/evaluated at discrete points (the superscript T indicates transposition). Finally, nis the (arbitrary) number of considered modes, ! is the circular frequency and i is the imaginary unit. Since the eigenvectors are known at discrete (m) points, the matrixG(!) is the m mmatrix containing the FRFs for these points. We can now take into account two different situations: 1. passive people (i.e., people who is not moving) on the structure; 2. active people (i.e., people who is moving) on the structure. The two following subsections will show how to use the model of Eq. (22.1) to describe the dynamics of the whole system in the two listed cases. 22.2.1 The Case of Passive People In the case of passive people the AGRFs are null. As for the PGRFs, they can be described by the apparent mass of each person. The apparent mass M (!) is the ratio between the force at the interconnection with the structure and the acceleration of the connection point Rxi: f GR i .!/ DM .!/ Rxi .!/ D M .!/! 2xi .!/ DH.!/xi .!/ (22.3) If we consider more than one person, all the f GR i functions can be expressed in matrix formHand Eq. (22.1) becomes: G 1 .!/ CWHWT x.!/ DG 1 H .!/x.!/ Df .!/ (22.4)
RkJQdWJsaXNoZXIy MTMzNzEzMQ==