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

The same procedure is used for the measurement from the ten accelerometers as
⎧
⎪⎨
ẍ i (t) =
⎪⎩
ẍ 1i (t)
ẍ 2i (t)
ẍ 3i (t)
ẍ 4i (t)
ẍ 5i (t)
ẍ 6i (t)
ẍ 7i (t)
ẍ 8i (t)
ẍ 9i (t)
ẍ 10i (t)
⎫
⎪⎬ ⎪⎨
=
⎪⎭
⎧
⎪⎩
u 1i
u 2i
u 3i
u 4i
u 5i
u 6i
u 7i
u 8i
u 9i
u 10i
⎧
⎫⎪ ⎬ ⎪⎨
sin(ω i t)+
⎪ ⎭
⎪⎩
v 1i
v 2i
v 3i
v 4i
v 5i
v 6i
v 7i
v 8i
v 9i
v 10i
⎫⎪ ⎬
⎪ ⎭
cos(ω i t) (15)
or
ẍ i (t) =u i sin(ω i t)+v i cos(ω i t) (16)
so that velocities and displacements can be calculated by analytical integration as
ẋ i (t) = 1 ω i
(−u i cos(ω i t)+v i sin(ω i t)) (17)
x i (t) =− 1 ω 2 i
(u i sin(ω i t)+v i cos(ω i t)) (18)
400
300
Measured
Sine-fit
200
Acceleration ( m/s 2)
100
0
−100
−200
−300
0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01
Time (s)
Figure 7: Typical measured and sine-fit acceleration, DOF 1
Using these expressions for forces and velocities allows the analytical calculation of the integrals
in eq. (11) avoiding problems due to numerical integration and with the precise measurement
of the period T too. For the case where the single frequency excitation is at frequency
ω i and all the measurements are fitted to harmonic functions at the same frequency only, the
period T i is simply
T i = 2π
(19)
ω i
6