# Dynamics of a Flexible Roof Test Model under Ambient Vibrations Measurements
## 2 The Case Study: A Tensile Structure Saddle Roof  
The cable net with hyperbolic paraboloid shape [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15] has two types of cables: upward (i.e., load bearing for gravitational loads) and downward cables (i.e., stabilizing for gravitational loads) under gravitational loads (Figure 1a). The former type increases its internal traction, while the latter decreases its internal traction. Under suction induced by wind (Figure 1b), it is the opposite. When the suction is non-uniformly distributed on the roof (Figure 1c), some problems may occur because some parts of the upward cable can be almost horizontal. This means that it loses its internal traction. This is very dangerous for the roof's global stability. At the same time, some parts of the downward cable exceed the limit of traction, which can result in a breaking collapse because cables are made of harmonic steel, resulting in non-ductile elements. Finally, under extreme conditions of wind (Figure 1d), upward cables can invert their curvature, causing roof instability. To investigate the aeroelastic behavior of these roofs, a wind tunnel experimental campaign was conducted at the Wind Engineering Laboratory of Cracow University of Technology. A geometrical sample was selected, and six different geometries were studied. Finite Element Method geometric nonlinear analyses were performed to predict the prototype dynamics, modal frequencies and displacements under wind action [8]. Modal analyses have confirmed, as well known, that these roofs are characterized by several close natural frequencies, which makes the construction and calibration of the test model very difficult. Test models were constructed in a 1:200 geometrical scale using 39 steel ropes along \(L_{1}\) and 39 steel ropes along \(L_{2}\) (Figure 2a), to reproduce the prototype structural setup. In Figure 2a, \(f_{1}\) and \(f_{2}\) are the upward and downward cable sags, respectively, \(L_{1}\) and \(L_{2}\) are the upward and downward cable spans, respectively and, finally, \(H_{1}\) and \(H_{2}\) are the minimum and maximum roof distance from the ground. The cable net was covered by a silk fabric to reproduce the aerodynamics. The dynamic characteristics of the model were identified by the acquisition of free (ambient) vibration at four accelerations (Figure 2b). Very small and light accelerometers were used in this research (with a weight below 1 g). In total, 49 nodes were identified for the square roof plan. Figure 1: Structural behavior: gravitational load (**a**), suction (**b**), ununiformed suction (**c**), extreme suction (**d**). Several methods were used to identify and compare the results, such as the Welch power spectral density, the random decrement technique, empirical mode decomposition and the Frequency Domain Decomposition (FDD) [32; 33]. FDD is a common output-only technique used in different structural engineering fields for dynamic identification purposes, where it can be useful for structural health monitoring due to its applicability when the input data are unknown. The FDD technique is based on approximating a complicated system response by subdividing it into independent SDOF ones that correspond to each significant mode. Further related details can be found in [15] and the next chapter. The FDD technique involves the main steps listed below:  
* Spectral density matrix estimations starting from the registered raw data. * Performing SVD of the spectral density matrices. * If different setups of the test are available, then averages of the first singular values of all setups should be evaluated, as well as the averages of the second, etc. * Finally, the average singular value peaks should be captured. For well-distinguished modes, the main singular value (the 1 st value) should always be identified. * Otherwise, as in the case of close or repeated modes, all singular values should be captured (the 2 nd, the 3 rd, etc., in addition to the 1 st one). Natural frequencies range from about 10 = to 30 Hz for each test model.