Instrumentation & Measurement Magazine 26-4 - 50

Fig. 1. Direct and indirect piezoelectric effects.
polarized ferroelectric ceramic, which exhibits the famous
piezoelectric effect. When it is stressed mechanically, whether
in tension, compression or shear, it generates an electric charge
proportional to the force applied to the faces of its poles [21].
In practical accelerometer designs, the piezoelectric element
is arranged in such a way that when the assembly is
vibrated, the mass applies a force to the piezoelectric element
itself proportional to the vibrational acceleration. This can be
understood from the law: Force = Mass x Acceleration (Fig. 2).
For frequencies well below the resonance frequency of the
complete mass-spring system, the acceleration of the ground
will be the same as the acceleration of the base, and the amplitude
of the output signal will therefore be proportional to the
acceleration to which the sensor is subjected.
Structural Health Monitoring
Structural health monitoring is a method of using sensors to
collect data that will be processed and interpreted to create a
control system throughout the life cycle of an asset. Structural
health monitoring is an inspection process that facilitates more
efficient asset management by enabling the owner to know the
operating conditions, implement proper maintenance plans,
reduce costs, increase safety and ensure long-term operation.
Structural health monitoring systems are used to measure the
current performance parameters of a building or infrastructure
and compare them to engineering design specifications
and previous parameters [23]-[24].
Sensor Modeling
Modal analysis is one of the most effective tools for identifying
structural resonance and thus helping to solve noise and vibration
problems. Modal analysis is the study of the dynamic properties
(structural resonance) of a system that is defined independently of
the loads applied to it and the responses it provides. The dynamic
properties (or modal characteristics) of a structure or a system are
made up of what are called eigen (or natural) modes: a mode is
defined by its natural frequency (or its resonance frequency), its
damping and its modal deformation. Structural dynamics studies
the response of structures when loads are applied to them.
In our work, modal analysis is used to choose a suitable
damping rate which reduces the measurement error to a minimum
value, improves accuracy and sensitivity as well as
avoids the resonance phenomenon.
In general, the vibration sensor is considered as a system
made up of a mass, a spring and a shock absorber. In this case,
the displacement is expressed as follows:
z    
t xt y t
 
To model this system, we apply Newton's second law as:
2
  
F m mp z
(1)
(2)
where: p is the Laplace coefficient. We then obtain the following
formula:
mp z cpz kz mp y
22
 
(3)
where: m is mass, c is the friction coefficient, k is the elasticity
coefficient, y is the absolute motion and p2
y: absolute
acceleration.
The use of (4) makes it possible to write the formula of the
seismic mass displacement z:
z=-my p2
/ (mp2
+c p + k)
(4)
To simplify (4), the natural frequency and the damping rate
of piezoelectric sensor are expressed by:
ω0 = (k/m)1/2
ξ = c/ (2mω0
)
The replacement of (5) and (6) in (4) gives:
z= - y p2
/ (p2
+ 2ξ ω0
(5)
(6)
where ω0 is the natural frequency and ξ is the damping rate.
p + ω0
2
)
(7)
Equation (8) is obtained by replacing the p (Laplace coefficient)
with jω in (7) where ω is vibratory movement relative
frequency:
z = ω2
y/ (-ω2
+ 2 jξωω0
+ ω0
2
)
2/ω0
2 results in:
)
(8)
Equation (8) is second-order equation with a complex number.
Multiplying the denominator of (8) by ω0
Fig. 2. The principle of piezoelectric accelerometer.
50
z= ω2
y/ ω0
IEEE Instrumentation & Measurement Magazine
2 (1 - (ω/ ω0)2
+ 2 jξω/ω0
(9)
June 2023
Instrumentation & Measurement Magazine 26-4

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