234 F.L.M. dos Santos et al. Strain gauges have been commonly used for static load testing of mechanical products in the aeronautic, automotive and mechanical industry. Moreover, fatigue testing [5], durability analysis and lifetime prediction [6] has also been a common application where strain gauges are used. This sort of testing is a common part of the product development process, and additional information on product durability and dynamic performance can be assessed by obtaining the modal parameters of the system, while still using the same instrumentation. A very important contribution on the field of strain measurement are the fiber optic sensors, or Fiber Bragg Grating (FBG) sensors [7, 8]. Their robustness to magnetic interference, added to the easiness of creating sensor arrays with multiple sensors, plus the possibility of embedding these sensors in composite structures, makes for an attractive solution for use in SHM systems. The availability of such an array of sensors, ready to be used and adequate for modal testing, is another incentive to carrying out a strain modal analysis, saving up on time and instrumentation. Another application of dynamic strain measurements is related to the strain displacement relations [9]. In many systems, strain gauges are used as the standard vibration sensor, especially when size or sensor location is an issue. Such is the case in aerospace applications, like gas turbines, wind turbines and helicopters [10], where size and weight are very restricted, and any sensor place on a blade should affect its aerodynamic properties as little as possible. One particular use of the strain measurements and strain to displacement relations is the strain pattern analysis (SPA), where strain measurements are used to predict blade displacements. 23.2 Theoretical Background To obtain the strain modal formulation, one can start with the fundamental theory of modal analysis. Modal theory states that the displacement on a given coordinate can be approximated by the summation of a nnumber of modes: u.t/ D n X iD1 i qi .t/ (23.1) where u is the displacement response in x direction, i is the ith (displacement) vibration mode, and qi is the generalized modal coordinate andt is time. For small displacements, given the theory of elasticity, the strain/displacement relation is: "x D @ @x u (23.2) And similarly, the same relationship exists between the strain vibration modes and the displacement modes: i D @ @x i (23.3) This way, by the relations on Eqs. (23.2) and (23.3), the expression on (23.1) can be rewritten as: ".t/ D n X iD1 i qi .t/ (23.4) Moreover, the relationship between the generalized modal coordinateq and an input force F is: qi Dƒ 1 i i F; withƒi D. ! 2mi Cj!ci Cki / (23.5) where mi , ci andki are the ith modal mass, modal damping and modal stiffness, and!is the excitation frequency. Substituting (23.5) into (23.4), the relation between a force input and a strain output, in terms of displacement and strain modes is represented as: "i D n X iD1 i ƒ 1 i i F (23.6) And finally, the strain frequency response function (SFRF) can be obtained, in matrix form: ŒH" D n X iD1 ƒ 1 i f i gf i gDŒ Œƒ 1 Œ T (23.7)
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