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

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