18 Numerical and Experimental Studies on Scale Models of Lightweight Building Structures 175 The scaling is introduced by defining the following dimensionless parameters: D As Af , D Is If , D xs xf D ls lf . (18.7) By inserting the dimensionless parameters in the energy expressions for the scaled and the full models, it can be shown that Ts D Tf (18.8) and Vs D 3 Vf . (18.9) Scaling parameters could be introduced also for the material properties (Eand ) and the time. It is, however, assumed here that the same material is used for both models. Not scaling the time results in the eigenfrequencies being preserved. Inserting Eqs. (18.8) and (18.9) in the first equation in Eq. (18.3) results in Ls D Tf 3 Vf D Tf 4 Vf . (18.10) By comparing the result to the second equation in Eq. (18.3), it is found that the scaling condition b D1 results in 4 D1 , ls lf 2 D hs hf . (18.11) When using this condition for creating a scale model of a beam, the eigenfrequency of the bending modes are preserved. It can be seen that bending in the z-direction is unaffected by the width. Hence, if a length scaling is assumed, the scaled height can be determined or vice versa. A corresponding expression can be determined for the width by considering bending in the y-direction. The procedure presented above can be employed for other types of deformations, such as the torsion of beams or the bending of plates. The type of deformation to consider in the scaling depends on the type that is expected to dominate the dynamics of the structure. 18.3.1 Example: Wooden Building Structure The structure studied here was designed to represent the physics involved in low-frequency (below 100 Hz) vibration transmission in multi-storey wood buildings. Specifically, a type of construction called timber volume element (TVE) buildings was used as reference for designing the experimental model. A main feature of such buildings, from a dynamical point-of-view, is the use of elastomer layers for vibration isolation. The load-bearing structure in TVE buildings consists of wood frames covered by plasterboards and particleboards. The buildings are constructed by stacking box-like volume elements (the TVEs) with layers of elastomers in-between. The experimental model considered here, illustrated in Fig. 18.2, consists of parts of two TVEs, one comprising a floor with walls on top and the other comprising a ceiling with walls below. Only half the height of the walls, compared to complete TVEs, is included in the model. The floor and the ceiling consist of a number of primary beams (seven and ten, respectively) attached to edge beams and have surfaces of particleboard and plasterboard, respectively. The walls consist of seven primary beams each, attached to edge beams on one side, and have surfaces of plasterboard. Two sides of the model have one type of walls, apartment separating walls, and the other two sides have another type of walls, facade walls. In Table 18.1, the dimensions are shown for the floor, the ceiling and the two types of walls. Elastomer blocks of the material Sylodyn NB [3] are placed in-between the two TVE structures. A total of 28 elastomer blocks, each being 100 95 25mm3 large, are placed with a centre-to-centre distance (cc) of 600 mm along the walls. The outer dimensions of the full experimental model are 4000 3600 2800mm3. The scaling of the beams was made using Eq. (18.11) for bending in the two directions, resulting in relations between the length and the height, and between the length and the width,
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