2.2 Modelling of system damping For lightly damped structures, appropriate results may be obtained by imposing the real damping assumption (real modeshapes). The real damping assumption is imposed by adding a viscous term in the equation of motion: (15) where the damping matrix C is: 1 (2 ) − − = ⋅ ⋅ C Φ diag Φ T ζω , (16) and (17) where Φis the matrix of the eigen-modes Φi obtained by solving the eigen-problem: ( ) 2 −K M Φ =0 i i ω , (18) and 2 iω is the i-th eigen-value of Eq.(18). Modal damping ratios iζcan be evaluated from: ( ) ( ) 2 = = ⋅ i i i f ζ ζ ζ π ω , (19) where the damping ( ) f ζ is defined by means of control coefficients zγ and B-spline functions zB defined on a uniformly spaced knot vector: ( ) ( ) ( ) ( ) [ ] 1 ( ) ; ; 0,1 , = = = ⋅ = + ⋅ − ∈ ∑ zn z z ST FI ST z f f u B u f f u f f u ζ ζ γ (20) where fST and fFI are, respectively, the lower and upper bound of the frequency interval in which the spline based damping model is defined. In order to take into account the contribution to the FRF of modes outside the input data frequency interval, ( ) 1.2 = ⋅ ST MAX f f is considered, where MAX f is the maximum frequency adopted in the input data. 3. Updating The parametrization adopted for the elastic constraints and for the damping model is employed in an updating procedure based on Frequency Response Functions experimental measurements. The measured FRFs ( ) X bH ω, with b=1,…, , are collected in a vector ( ) h X ω : (21) The dynamic equilibrium equation in the frequency domain, for the spline-based finite element model, can be defined by Fourier transforming Eq.(15), where ( ) ~ ( ) = F : (22) where ( ) Z ωis the dynamic impedance matrix and ( ) ( ) ( ) 1− = H Z ω ω is the receptance matrix. Since the vector contains non-physical displacements and rotations, the elements of the matrix ( ) H ω cannot be directly 1 1 2 2 2 0 0 0 2 (2 ) 0 0 0 2 = ⋯ ⋮ ⋮ ⋱ … diag ζ ω ζ ω ζω ζ ω , ( ) ⋅ + ⋅ + ⋅ = + ɺɺ ɺ f M δ C δ δ F K ∆K , ( ) ( ) ( ) 1 = ℓ ⋮ h X X X H H ω ω ω . ( ) ( ) ( ) 2 1− − + + + ⋅ = ⋅ = ⋅ = ɶ ɶ ɶ ɶ f M C K ∆K δ Z δ H δ F j ω ω ω ω , ℓ ℓ ɶδ 130
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