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- Milatz J.M.W., van Zolingen J.J. The Brownian Motion of Electrometers. Physica, 1953, vol. 19, issue 1, pр. 181–194.

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- DOI: 10.21122/2220-9506-2016-7-1-16-23 16 Introduction Designing a supersensitive gravi-inertial 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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- . 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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- Forward R.L. Measurement of static field gradient: US Patent no. 3273397, Filed 05.06.1964.

Total in-text references: 1- In-text reference with the coordinate start=3118
- Prefix
- DOI: 10.21122/2220-9506-2016-7-1-16-23 16 Introduction Designing a supersensitive gravi-inertial 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
- Exact
- [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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- Bell C.C. Torsionally resonant gravity gradient sensor: US Patent no. 3564921, Filed 23.02.1971.

Total in-text references: 1- In-text reference with the coordinate start=3118
- Prefix
- DOI: 10.21122/2220-9506-2016-7-1-16-23 16 Introduction Designing a supersensitive gravi-inertial 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
- Exact
- [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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- Dias R.A., Cretu E., Wolffenbuttel R., Rocha L.A. Pull-in-based μg-resolution accelerometer: Characterization and noise analysis. Sensors and Actuators A: Physical, 2011, vol. 172(1), рр. 47–53.

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- Trusov A.A. Ultra high quality factor and wide dynamic range inertial MEMS for north-finding and tracking applications. Whitepaper Available at: http://www.alexandertrusov.com/uploads/pdf/2013-UCI-Trusov-whitepaper-FM-IMU.pdf (accessed 01.12.2015).

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- Forward R., Bell C., Morris J. Rotating gravitational sensors. Gravity Research Foundation Essay. Available at: http://www.gravityresearchfoundation.org/pdf/awarded/ 1965/ forward_bell_morris.pdf (accessed 19.12.2015).

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- Bernstein J. An Overview of MEMS Inertial Sensing Technology. Sensors online. Available at: http:// www.sensorsmag.com/sensors/acceleration-vibration/ an-overview-mems-inertial-sensing-technology-970 (accessed: 19.02.2015).

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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 read-out system
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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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- Zhang W.M., Yan H., Peng Z.K. Electrostatic pull-in instability in MEMS/NEMS: A review. Sensors and Actuators A: Physical, 2014, vol. 214, рр. 187–218.

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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 pull-in 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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- Fuligni F., Lorenzini E., Bordoni E., Lazarewicz A.R., Iafolla V. Development of a High-Sensitivity, NonCryogenic Astrophysical Gravity Gradiometer for Space21 borne Use Observatory. Proceedings of 14th Annual Gravity Gradiometry Conference, United States Air Force Academy, Colorado Springs, Colorado, 11–12 February 1986. USA, 1986, рр. 374–392.

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- Typically the electric field forces in measuring devices with capacitive readout are too small to achieve the pull-in 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 gravi-inertial 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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- Mekid S., Gilavdary I. Differential capacitance torque sensor: US Patent no. 8893563, Nov. 25, 2014.

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- Typically the electric field forces in measuring devices with capacitive readout are too small to achieve the pull-in effect. But there is possibility to adjust the stiffness in a narrow interval [9]. In
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- [10]
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- the gravi-inertial 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.

- In-text reference with the coordinate start=4664
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- McNeil A., Lin Y., Miller T. Differential capacitive sensor and method of making same: US Patent no. 7610809, Nov. 3, 2009.

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- Capacitive sensors are non-linear 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 quasi-static displacement of PM in the gravi-inertial 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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- Gilavdary I., Mekid S., Riznookaya N. [Controlling of sensitivity of the sensor with differential electrostatic transducers]. Pribory i metody izmerenij [Devices and methods of measurements], 2015, vol. 6, no. 2, рр. 163–172 (in Russian).

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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 quasi-static displacement of PM in the gravi-inertial 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 quasi-static mode and i

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- Investigation of the stability of quasi-static 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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- [12]
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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

- In-text reference with the coordinate start=6700
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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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- Korn G.A., Korn T.M. Mathematical handbook for scientists and engineers: definitions, theorems, and formulas for reference and review. New York, Dover Publications, 2000, 1130 p. 22

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- static equilibrium of PM In this table, the parameter f – natural frequency of the PM when the electric field is switch off (U = 0), f0 – the natural frequency of the PM if the electric field is turned on. Figure 3 – The relations of g5(v) (continuous line) and g3(v) (dashed line) 18 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: are the real roots of an algebraic equation of the fourth order: (7) This equation can be solved analytically.

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