1 Using Laser Vibrometry for Precise FRF Measurements in Experimental Substructuring 5 uk =( Ie k)T(qv t +q v θ ×I r k) =( Ie k)T ⎡ ⎢⎣ 1 0 0 0 Ir k z −Ir k y 0 1 0 −Ir k z 0 Ir k x 0 0 1 Ir k y −Ir k x 0 ⎤ ⎥⎦ ⎡ ⎢⎢ ⎢⎢ ⎢⎢ ⎢⎣ qv x qv y qv z qv θx qv θy qv θz ⎤ ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎦ =Rukq v (1.9) The reduction operator Ruk is a 3 ×6 matrix relating the sensor displacement uk to the generalized coordinates qv. The relation can be extended to all sensor channels u2 and all virtual points q: u2 =Ruq (1.10) This spatial reduction onto the IDM is based on the assumption of an almost rigid behaviour of the interface around the virtual point. Since in reality this condition is not always fully satisfied, a residual termμ, which represents the unprojected flexible motion, must be added: u2 =Ruq +μ (1.11) As the number of IDMs is typically lower than the number of interface DoFs, the problem is overdetermined and is handled with a least square procedure. To find the q that best approximates the measured response u2, a residual cost function, namely the squared error μTμ, has to be minimized. The least square projection is performed by applying the Moore-Penrose pseudo-inverse of Ru: q = RT uRu −1 RT uu2 =Tuu2 (1.12) In general, a weighting matrix can be used to gain more control over the error minimization by adjusting the importance of certain DoFs in the transformation [2, 11]. 1.2.2.2 Virtual Point Forces The derivation of the force reduction matrix is similar. The full set of input forces f2 has to be related to the generalized forces mv t =[mv x,mv y,mv z] and moments mv θ =[mv θx ,mv θy ,mv θz] . For one single impact f h with orientation Ie h at a distance Irh from the virtual point, the relation can be written as follows: mv = ⎡ ⎢⎢ ⎢⎢ ⎢⎢ ⎢⎣ mv x mv y mv z mv θx mv θy mv θz ⎤ ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎦ = ⎡ ⎢⎢ ⎢⎢ ⎢⎢ ⎢⎣ 1 0 0 0 1 0 0 0 1 0 −Ir h z Ir h y Ir h z 0 −Ir h x −Ir h y Ir h x 0 ⎤ ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎦ Ie hf h =RT fh f h (1.13) The operator RT fh is the6×1 matrix describing the reduction of input forces f h to the generalized forces mv. The extended formulation for all virtual points and the full set of forces can be written: m=RT ff2 (1.14) Note that Rf assumes the same exact formulation as Ru if a ‘collocated’ setup (sensor channels and forces in the same position and direction) is chosen. Unlike the displacements transformation, the problem is underdetermined and therefore a standard least square is not applicable. The goal is to find the forces ˜f2 that realize the generalized forces mwith a minimal ‘effort’, or in mathematical
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