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3119
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DOI: 10.21122/222095062016711623
Introduction
Designing a supersensitive graviinertial sensors
measuring linear and angular accelerations of moving objects with second derivatives of gravitational
potential, on the Earth surface and in circumplanetary space is a problem that stands in front of science and developers since the late 50th century until
currently
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[1–6]
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.
Typically such sensors comprise a sensing mass
(often called as a movable mass, or proof mass (PM))
retained relative to the housing by an elastic mechanical coupling. This elastic coupling is characterized
by a natural frequency of free oscillations of PM
along the axis of the sensor sensitivity.
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3580
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This elastic coupling is characterized
by a natural frequency of free oscillations of PM
along the axis of the sensor sensitivity. In order to
increase the sensitivity of the sensor, it is required
to reduce this frequency, the internal noise and the
noise of a readout system
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[7]
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.
Actually the capacitive microelectromechanical
(MEM) – accelerometers are broadly known where
electrical capacitors are used for reading of the desired signal, and MEM capacitive actuators, where
electrical capacitors and electrical fields are used to
control the movement of the elastically suspended
PM and to drive its resonant frequency.
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where
electrical capacitors are used for reading of the desired signal, and MEM capacitive actuators, where
electrical capacitors and electrical fields are used to
control the movement of the elastically suspended
PM and to drive its resonant frequency.
Actuators usually establish the limits of motion
control while the change of the resonant frequencies
are limited by the pullin effect
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. This effect is due
to the fact that, if PM deflects from their equilibrium
position, the electrostatic forces will grow faster than
the elastic force holding the PM near the equilibrium
position.
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Typically the electric field forces in measuring devices with capacitive readout are too small to
achieve the pullin effect. But there is possibility to
adjust the stiffness in a narrow interval
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[9]
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. In [10]
the graviinertial sensor was proposed in which the
function of the capacitive sensor and actuator are
combined into a single differential capacitive system. In this sensor, it was assumed that the electric
field forces are enough to compensate elastic forces
in the direction of the sensitive axis.
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4665
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Typically the electric field forces in measuring devices with capacitive readout are too small to
achieve the pullin effect. But there is possibility to
adjust the stiffness in a narrow interval [9]. In
 Exact

[10]
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the graviinertial sensor was proposed in which the
function of the capacitive sensor and actuator are
combined into a single differential capacitive system. In this sensor, it was assumed that the electric
field forces are enough to compensate elastic forces
in the direction of the sensitive axis.
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5364
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Capacitive sensors are nonlinear due to the
physical properties of electrical capacitors. Therefore, differential electrostatic systems are often used
in measuring instruments, because nonlinearity may
be partly compensated there
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. However, the effect of asymmetry of the differential capacitive systems is still not fully explored. Such study was carried out for a quasistatic displacement of PM in the
graviinertial sensors [12], where it was shown that
it is the asymmetry of the differential electrostatic
system that sets a limit to reduce the torsion stiffness
of the suspension of PM using the electrostatic field.
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5569
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However, the effect of asymmetry of the differential capacitive systems is still not fully explored. Such study was carried out for a quasistatic displacement of PM in the
graviinertial sensors
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[12]
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, where it was shown that
it is the asymmetry of the differential electrostatic
system that sets a limit to reduce the torsion stiffness
of the suspension of PM using the electrostatic field.
The purpose of this work is within the framework of a unified approach to investigate the effect of
asymmetry of the nonlinear differential electrostatic
system on PM movement in quasistatic mode and
i
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Investigation of the stability of quasistatic mode
of the PM in an electric field
A simplified scheme of a gravi inertial sensor
chosen for the calculations is shown in Figure 1. The
description details of this scheme and original calculations of the capacitor with the inclined plate are
given in
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.
The elastic Mm torque and the electrical torque
Me affect the PM in this sensor. Dependence of the
total torque acting on the PM angle φ deviations
from the equilibrium position can be written as [12]:
(1)
where: v
m
=
φ
φ
; parameter φm
h
L
a
a
=
01
2
lnrelated
to the geometry of the system; h0 – the gap between
the capacitor’s plates when φ=0;
a
h
r
L
a
h
r
L
1
0
2
0
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6701
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The elastic Mm torque and the electrical torque
Me affect the PM in this sensor. Dependence of the
total torque acting on the PM angle φ deviations
from the equilibrium position can be written as
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[12]
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:
(1)
where: v
m
=
φ
φ
; parameter φm
h
L
a
a
=
01
2
lnrelated
to the geometry of the system; h0 – the gap between
the capacitor’s plates when φ=0;
a
h
r
L
a
h
r
L
1
0
2
0
1
2
1
2
=+
=−
ln,ln;
k – the mechanical torsion stiffness;
k
k
B
B
mCU
m
1
0
2
1
2
==
φ
φ
;.
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Figure 3 – The relations of g5(v) (continuous line) and
g3(v) (dashed line)
γvv
vk
v
()=
()−−
()+
22
1
2
18
1
.
gkkk528151321()()().νννγνγνγ=−++++−+
gkk32281321()().ννγνγνγ=+++−+
Eq.(4) has an analytical solution. This solution
can be found using the trigonometric Vieta formulas
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. So that, if the polynomial of third order has the
form av3 + bv2+ cv+ d = 0, the parameters Q, R, S and
y have to be determined using the formulas:
(5)
If S > 0, the equation g3(v) = 0 has three real
roots, and the position of the equilibrium vst and quasistatic PM deviations between the points v1 and v2
are stable.
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