Aerospace and Electronic Systems - August 2018 - 41

Sarego, Zaccariotto, and Galvanetto

Figure 6.

Neural Network diagram for the reconstruction of the time history of the
impact force.

Figure 5.

Positions for the impacts for training the Neural Network for evaluating
impact locations: the four stars are the sensors and the black squares the
25 impact locations.

net is made by Levenberg-Marquardt back-propagation algorithm
[26]. The adopted network architecture is the simplest architecture
[30], resulting from an analysis carried out with different numbers
of neurons per hidden layer, able to be trained fast and giving reasonable mean squared errors. For this study, no more hidden layers
were considered, since a one-hidden layer ANN can approximate
any continuous map [30], [31].
The training has been done with 90% (48) of the samples, while
the validation and test sets are both made of 5% (3) of the samples.
The performance of the ANN training is evaluated by using the
mean squared error and the regression analysis, which shows the
correlation between output and targets.
This ANN has been employed for reconstructing the force of
three different impacts:
C

Additional impacts, applied on a finer grid of points, were simulated to train the Neural Network for reconstructing the impact
location. For this reason, 25 positions were defined as shown in
Figure 5, for each position 18 impacts with different velocities,
those shown in Table 2, and different masses (2.41, 1.2, 0.6 kg)
were carried out. Therefore, a total number of 450 impacts were
simulated and for each of them the four sensor signals were collected with a sampling frequency of 500 kHz. For this network, the
sampling frequency has been increased for a more precise definition of the maximum displacement recorded by the sensors, while
the number of impact locations have been increased for a more
accurate prediction of unknown impact locations.

impact n° 1 is located at the position identified by the blue
circle in Figure 7a and the velocity is 1.27 m/s (the results
are shown in Figure 7b);
impact n° 2 is located at the position identified by the blue
cross in Figure 7a and its velocity is 1.49 m/s (the results are
shown in Figure 7c);
impact n° 3 is located at the position identified by the blue
cross in Figure 7a and its velocity is 1.4 m/s (the results of
force reconstruction are shown in Figure 7d).

The normalized mean squared error (NMSE) is computed as
1
N
NMSE ( % ) =

NUMERICAL RESULTS
Two Neural Networks are used in this study. The first neural network
is built for the reconstruction of the time history of the force, while
the second used is for the identification of the impact location.
As for the force reconstruction algorithm, following the procedure suggested in [16], the inputs of the ANN are the real and
imaginary frequency components of the discrete Fourier transform
of the time history of the displacement signals from the four sensors located as shown in Figure 4; the targets are the real and imaginary frequency components of the discrete Fourier transform of the
time history of the impact force. As mentioned in [16], since in the
frequency domain the components of the force and sensor signals
are complex numbers, complex-valued ANNs may provide better
results for the reconstruction of the time history of the force. The
feed-forward network is made of one hidden layer of 20 neurons
and one linear output layer of 402 neurons (see Figure 6). Defined
in [29] as the total number of weights and biases, the complexity
of this feedforward neural network is 40,601. The training of the
AUGUST 2018

 (F
N

i =1

S, i

FS FR

− FR,i

)

× 100

where FS and FR are respectively the simulated and reconstructed
force, FS and FR their mean values, and N the total number of time
steps of the simulation.
For each impact, the NMSE of the reconstructed time histories
of forces with respect to the originally simulated values are reported in Table 3. The NMSE is computed in the [0, 10] ms time interval and it depends on the time duration of the impact: as shown in
Figures 7b-7d, the simulated impact force is zero when there is no
more contact between the plate and the impactor, while the inverse
Fourier transform of the results of the ANN is not. In the evaluation of the NMSE, the reconstructed force is assumed to be equal
to zero after the first instant in which it assumes a negative value.
These results are in agreement with the result (NMSE = 17%)
of the best performance ANN (out of 100-cycles-trained ANNs)
obtained by [16], while the average error for the 100-cycles-trained
networks reported in [16] is 58%. The most important parameter,
which determines the occurrence of damage, is the peak force,

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Aerospace and Electronic Systems - August 2018

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