The vibratory movement is a displacement characterized by a direction and amplitude. The direction of motion is described in terms of some arbitrary Cartesian or orthogonal coordinate systems.
The majority of vibration sensors can be considered as a system with a single degree of freedom which is characterized by a mass (m), a spring (coefficient of elasticity k), and a damping device (damping c) as shown in figure (Fig.2):
The piezoelectric sensor does not measure the absolute motion y (t) of the vibrating structure, but it measures the relative motion z (t) that must be interpreted to extract the information on the absolute motion [30-46].
The relative vibratory motion is given by the following relation:
z(t) = x(t) – y(t) (1)
Applying the first law of motion, we can write the following equation:
∑F = mγ = m d2z/dt2 (2)
The second-order differential equation showing the movement of the mass-spring-damper system can be written as follows:
m(d2z/dt2) + c (d2z/dt2) + k z(t) = - m(d2y/dt2) (3)
With: m is the mass, c is the coefficient of friction, k is the elasticity coefficient, y is the absolute motion and d2y/dt2 is the absolute acceleration.
When we replace (d/dt) by the Laplace coefficient (s), equation (3) is expressed as follows:
m s2 z +c s z + k z = -ms2 y (4)
The natural frequency of the piezoelectric sensor must be greater than the frequency of the vibratory movement (ω0 ˂˂ ω) to avoid the effect of the resonance phenomenon, it can express as a function of the seismic mass of the sensor and the coefficient of elasticity, its expression illustrated by the following relation:
ω0 = (k/m)1/2 (5)
The damping rate is the ratio between the damping coefficient and the seismic mass and the natural frequency, then; we can write:
ξ = c/ (2mω0) (6)
ω0 is the natural frequency, ω is the frequency of the vibratory motion and ξ is the damping rate.
By substituting equations (5) and (6) in equation (4), the relative movement z is expressed as follows:
z = - y s2/ (s2 + 2ξ ω0 s + ω02) (7)
We replace the Laplace coefficient (s) by (jω), and equation (7) becomes:
ω2y/ (-ω2 + 2 jξωω0+ ω02) (8)
By multiplying the denominator of equation (8) by ω02/ω02, equation (9) is obtained:
z = ω2y/ ω02 (1 – (ω/ ω0)2+ 2 jξω/ω0) (9)
Equation (9) is a complex function; its module is extracted as follows:
z = ω2 y / [ω02 [(1–(ω/ω0)2)2 + (2ζω/ω0)2]1/2] (10)
This equation represents the vibrations relative movement measured by the piezoelectric sensor.
It must extract the equation of the sensor measurement accuracy to improve it; it is done by the multiplication of the equation (10) by ω02:
ω02 z = ω2 y / [(1 – (ω/ ω0)2)2 + (2ζω/ω0)2]1/2 (11)
According to equation (11), the absolute acceleration of the structure is expressed by:
d2y/dt2 = ω2y (12)
The relative acceleration of the vibratory movement is given as follows:
d2z/dt2 = ω02z (13)
To obtain optimum measurement accuracy, he must choose the sensor according to the condition ω/ω0<< 1, while equation (10) becomes:
ω02 z ≈ ω2 y (14)
So:
d2z/dt2 ≈ d2y/dt2 (15)
The measurement accuracy of the piezoelectric sensor equals the absolute acceleration divided over the relative acceleration, the expression is as follows:
P= (d2y/dt2) / (d2z/dt2) = [(z ω02)/(y ω2)] (16)
P: the piezoelectric sensor accuracy
Then, from equation (12), accuracy is defined by the following formula:
P= [(1 – (ω/ω0)2)2 + (2ζω/ω0)2]1/2 (17)
The new relation who relates the relative movement as a function of accuracy is extracted by the use of equation (10), we obtain:
z = (ω2y P)/ ω02 (18)
This equation makes it possible to improve the parameters of the piezoelectric sensor and to propose a new conception of the latter.