25 Flooring-Systems and Their Interaction with Usage of the Floor 207 Mf is the mass matrix andKf is the stiffness matrix of the empty floor. The damping matrixCf of the empty floor depends on the parameters aandb, and these parameters are tuned such that the minimum damping occurs at the frequency associated with the first mode of the empty floor. A minimum damping equivalent to a logarithmic decrement of 0.1 is assumed. This is considered representative for a reinforced concrete floor and entails that the damping ratio of the empty floor, , assumes a value of approximately 0.016. In the computational model, allowance is made for attaching floor occupancy mass at nodal points of the FE grid. 25.2.2 Usage of the Floor Usage of the floor area will involve adding masses to the floor. For the studies of this paper, added masses are assumed rigidly attached to the floor and modelled as lumped masses, although this is likely to be a simplification of real-life floor occupancy conditions. Attaching each lumped mass to a single node of the model of the floor is also a simplification of matters. However, the approach is considered reasonable for the purposes of the study. The scenarios considered for the floor occupancy are outlined in Fig. 25.1. They are denoted by the integer n taking on values of 1, 4, 9 or 16. The figure represents the number of masses placed on the floor in the particular scenario. Each individual mass is of 75 kg. The scenario n D0 represent the empty floor. Lines are added on the floor area in Fig. 25.1 to provide an understanding of the exact positioning of the masses assumed for calculations. The strategy used for placing masses has been to gradually divide the floor into increasing numbers of squares (1, 4, 9, 16) and to place a mass of 75 kg in the center of each square. As can be seen, the scenarios are arranged such that the floor occupancy mass gradually increases (when moving from scenario n D1 up tonD16). Initially, the occupancy masses are assumed connected to the floor at the midplane of the floor (in the FE model of the floor at z D0 m defining the horizontal plane of the floor model). For this scenario, the masses are fixed to (only influencing) translational degrees of freedom at nodal points of the FE model of the empty floor. In order to consider scenarios in which an added mass is elevated above the floor, study scenarios are arranged assuming values of z above zero, were the value of z defines the vertical distance between the horizontal midplane of the floor and the position of the added masses. In the FE model of the floor, the influence of the added mass is modelled such that its contribution to floor dynamics mimics that of an elevated lumped mass attached to a node in the floor model by an interconnecting massless element with infinite axial and bending stiffness. This corresponds, approximately, to the influence of desks, bookshelves or similar furniture where the center of mass is placed at a finite height above the floor. Fig. 25.1 Layout plan for floor occupancy scenarios. nD1 (open square), nD4 (open diamond), nD9 (open triangle), nD16 (open circle)
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