23 Strain-Based Dynamic Measurements and Modal Testing 235 The expansion of (23.7) is: 2 6 6 6 4 H" 11 H" 12 H" 1Ni H" 21 H" 22 H" 2Ni : : : : : : : : : : : : H" No1 H" No2 H" NoNi 3 7 7 7 5 D n X iD1 ƒ 1 i 2 6 6 6 4 1i 1i 1i 2i 1i Ni i 2i 1i 2i 2i 2i Ni i : : : : : : : : : : : : Noi 1i Noi 2i Noi Ni i 3 7 7 7 5 (23.8) where No represents the number of strain gauge measurement stations (or the number of output measurements) and Ni represents the number of excitation points (or the number of inputs). The columns of the matrix correspond to the strain responses due to the excitation points along the rows of the matrix. Some important characteristics can be inferred from Eq. (23.8). First of all, differently from displacement FRFs, the SFRF matrix is not symmetric, that is, for instance, H" 12 ¤H" 21. This means that reciprocity is not guaranteed for strain modal analysis—exciting point a and measuring point b will not yield the same FRF as if exciting point b and measuring point a. Moreover, any column of the SFRF matrix contains all the information regarding the strain modes ( ), while any row of the SFRF matrix contains information about the displacement modes ( ). This particular property leads to practical applications—to obtain the strain mode shapes, one must use a fixed excitation point and measure the strain responses. On the other hand, by using a strain gauge as a fixed reference sensor and moving the excitation point (as with impact testing), the displacement mode shapes can be obtained. Due to the similarity of the strain modal formulation and the displacement modal formulation, the same modal identification methods can be used in both cases, as long as the appropriate caution is taken. In this article, the PolyMAX identification method [11] was used without any modifications. Moreover, there is the possibility of obtaining reciprocal FRFs in strain modal testing, if the excitation input is not a force that acts on displacement, but is actually a direct strain input. This case is achievable if, for instance, a piezo patch actuator is used, but it will not be covered in this study, as the most common methods of carrying out modal analysis still use displacement based excitation sources. The following section on the experimental analysis will show some experimental examples of the characteristics mentioned above. 23.3 Experimental Analysis Three analysis cases were chosen to illustrate some of the characteristics of strain modal analysis seen in the previous section. The first test subject, a small composite wind turbine blade, was tested with strain gauges and an impact hammer, and the experimental results were compared with a finite element (FEM) simulation model. Then, a composite T-shaped beam was tested using piezo-based strain sensors, accelerometers and shaker excitation. Finally, a large composite helicopter main rotor blade was tested with shaker and hammer tests. 23.3.1 Wind Turbine Blade A small composite wind turbine blade was used on the first strain modal test [12]. For this purpose, 20 strain gauges were glued to the surface of the blade and an impact hammer with an impedance head was used to excite the structure at several locations. The blade was fixed at its root, to impose a cantilevered condition. Of the 20 strain gauges, one of them consisted of a strain gauge rosette to measure purely shear strain, while the other 19 strain gauges were aligned with the radius of the blade and were measuring normal strain. Figure 23.1 shows the wind turbine blade, its sensor locations represented in the acquisition software and a finite element model of the blade. The first step of the experimental procedure is to measure the strain frequency response functions (SFRFs), that are used later on the modal analysis procedure. Figure 23.2a shows the SFRF of an arbitrary strain gauge, where the resonance peaks are clearly visible. Moreover, the phase shift due to the resonances is the same for the SFRF, where the phase shifts in 180 degrees whenever there is a resonance peak. Additionally for this experiment, a reciprocity check was carried out to verify if the theory for strain modal analysis was correct—for this purpose, two measurement points were picked and the impact hammer was used to excite those
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