Matlab code for damping identification using energy ... - CFD4Aircraft

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under study. Starting from eq. (4), in the case of viscous damping and considering the initial instant t=0, it becomes ∫ T ∫ T ẋ T Cẋ dt = ẋ T g(t) dt (5) 0 0 where C is the viscous damping matrix, which can be written as C = p∑ c i L i (6) i=1 where L i ∈ R n×n is a matrix which indicates the location of the i th of p different viscous damping sources of amplitude c i . Figure 1: Absolute dashpot connecting DOF 2 to the ground In the case of an absolute dashpot connecting one degree of freedom (e.g. degree-of-freedom 2, see figure 1) of the structure to the ground, L i takes the form ⎛ ⎞ 0 0 0 ··· 0 0 1 0 ··· 0 L i = 0 0 0 ··· 0 ⎜ ⎝ . . . . .. ⎟ 0 ⎠ 0 0 0 0 0 (7) Figure 2: Relative dashpot connecting DOF 1 to DOF 2 In this case the pattern approach does not help the reduction of the number of unknowns, but helps a systematic procedure to define the damping sources in an automated way. 2
In the case of a relative dashpots connecting two degrees of freedom together (e.g degree-offreedom 1 and 2, see figure 2), L i takes the form ⎛ ⎞ 1 −1 0 ··· 0 −1 1 0 ··· 0 L i = 0 0 0 ··· 0 (8) ⎜ ⎝ . . . . .. ⎟ 0 ⎠ 0 0 0 0 0 which allows the reduction of the number of parameters to identify from 4 to 1. If the damping between two consecutive degrees of freedom is assumed to be the same for all the different couples (figure 3) representing, for example, the material damping between identical Figure 3: Identical relative dashpots connecting consecutive DOFs elements or similar connections or joints between parts of the structure, L i can take the form ⎛ ⎞ 1 −1 0 ··· 0 0 −1 2 −1 ··· 0 0 0 −1 2 ··· 0 0 L i = . ⎜ . . . .. (9) −1 0 ⎟ ⎝ 0 0 0 −1 2 −1 ⎠ 0 0 0 0 −1 1 reducing the number of non-zero unknowns, in a 10 degrees of freedom example, from 28 to 1. Assuming p different possible configurations for the damping sources, the energy equation (5) can be arranged as c 1 ∫T 0 ẋ T L 1 ẋ dt + c 2 ∫T 0 ẋ T L 2 ẋ dt + ...+ c p ∫T 0 ẋ T L p ẋ dt = ∫ T 0 ẋ T g(t) dt (10) By exciting the structure with q excitations at different frequencies, different versions of eq. (10) are obtained and arranged in a matrix form ⎡ ⎤ ⎧ ⎫ ∫T 1 ∫T 1 ∫T 1 ẋ T L 1 ẋ dt ... ẋ T L p ẋ dt ⎧ ⎫ ẋ T g 1 (t) dt 0 0 ⎪⎨ c 1 ⎪⎬ ⎪⎨ 0 ⎪⎬ . . . . = ⎢ T ⎣ ∫ q ∫T q ⎥ ⎪⎩ ⎪ . (11) ⎭ ⎦ c p ∫T q ẋ T L 1 ẋ dt ... ẋ T L p ẋ dt ⎪⎩ ẋ T g q (t) dt ⎪⎭ 0 0 3 0
Page 1: Matlab code for damping identificat
Page 5 and 6: Analytical integration of experimen
Page 7 and 8: The integrals present in eq. (11) b
Page 9: Using the code When the file “En

Matlab code for damping identification using energy ... - CFD4Aircraft

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