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Fig. 4. Normalized evolution of the first four natural frequencies versus the
liquid level.
Fig. 6. Experimental setup.
Table 2 - Density value of the liquids
used in the experiments
Liquid
Case 1 - water
Case 2 - salt water
Case 3 - cooking oil
Case 4 - motor oil
Density (Kg/dm3
1.010
1.130
0.864
0.860
Fig. 5. Back-propagation trained neural network architecture.
a linear function in the output node. The number of nodes in
the input layer corresponds to the selected number of natural
frequencies; thus, the normalized frequency values of the considered
number of vibration modes corresponding to some
specific liquid level are fed into the input layer of the neural
network model and are mapped into the output value aiming
to provide the corresponding liquid filling level. Fig. 5 shows
the architecture of the utilized neural network when four NF
are used as input data. Numerical information corresponding
to the set of considered frequencies and the corresponding
content level is organized to be used in the NN model using
partitions of 40, 30, and 30% for training, validation, and testing,
respectively.
Experimental Arrangement
As proof of principle, we employed as container a glass tube
of length L = 1,080 mm, with an external diameter of 25.4 mm
and a wall thickness of ≈1 mm. The tube was clamped to the
optical table at the bottom end. Fig. 6 shows a picture of the experimental
setup. A collection of experiments was carried out
using liquids with several density values: deionized water, salt
water, cooking oil, and motor oil, as shown in Table 2. A small
mirror was glued to the tube to reflect the signal light beam.
August 2021
Under static conditions, that is when the tube is not vibrating,
the signal generated by the p-emf sensor is null. To obtain
a signal, the tube is induced to vibrate by hitting the tube with
the gentle and rapid impact of a light wooden stick. A set of
measurements was collected through a series of experiments
in which the content liquid level was increased in steps of 1%,
ranging from empty to the full content capacity. The obtained
data was further organized to be used in the neural network
training stage. Fig. 7a shows an example of the spectral components
corresponding to four natural frequencies obtained
when the tube is empty. Similarly, the vibration spectrum obtained
when the tube is full is presented in Fig. 7b; the shift of
natural frequencies is clearly observed.
Fig. 8 shows, in a spectrogram-like form, an example of the
experimental spectral content evolution of the first four natural
vibration modes, when the content level increases as a
function of the liquid level in steps of 1% of the full content capacity.
These dependences are similar to that described by the
Bernoulli-Euler beam theory. Indeed, the geometrical conditions
of the tube were such that its length was much larger than
its diameter (L/D ≈ 42), so they allow to approximate its behavior
by the Bernoulli-Euler beam theory. In a previous work,
our research team presented a finite-element modelling (FEM)
carried out in COMSOL showing consistency between the obtained
vibration spectral modes and the Bernoulli-Euler beam
theory using several container geometries [32].
IEEE Instrumentation & Measurement Magazine
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