The article deals with the creation of a generalized mathematical model of inclinometer that allows more fully than any known prior mathematical model to study its properties under actual operating conditions. The mathematical model of inclinometer considered non-identical electrical parameters of the primary sensors. Mathematical model of a linear single-axis accelerometer is also built, which is part of the inclinometer

A. N. Krasnov

G. Y. Kolovertnov

V. E. Lyalin

English

Vibroengineering PROCEDIA

 48 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. AUG 2016, VOL. 7. ISSN 2345-0533  
Generalized mathematical model of a linear single-axis 
accelerometer as an integral part of the inclinometer 
A. N. Krasnov1, G. Y. Kolovertnov2, V. E. Lyalin3 
1, 2Ufa State Petroleum Technological University, Ufa, Russia 
3Kalashnikov Izhevsk State Technical University, Izhevsk, Russia 
1Corresponding author 
E-mail: 1ufa-znanie@mail.ru, 3velyalin@mail.ru 
(Received 2 August 2016; accepted 8 August 2016) 
Abstract. The article deals with the creation of a generalized mathematical model of inclinometer 
that allows more fully than any known prior mathematical model to study its properties under 
actual operating conditions. The mathematical model of inclinometer considered non-identical 
electrical parameters of the primary sensors. Mathematical model of a linear single-axis 
accelerometer is also built, which is part of the inclinometer. 
Keywords: inclinometer, information-measuring system, mathematical model, accelerometer. 
1. Introduction and motivation 
Inclinometer is a very complicated electromechanical system functioning under the influence 
of a greater number of influencing factors [1]. Analysis of the requirements for bottom-hole 
information-measuring system (IMS) for a known length of the trajectory of the well showed that 
the basis for the construction of directional navigation sensors must be based on information about 
the spatial rotation of the drilling tool relative to the two non-collinear vectors. The basis of such 
vectors relies vector acceleration of gravity Ԧ݃, magnetic field vector ሬܶԦ and the vector of the Earth 
Earth’s angular velocity ΩሬሬԦ . This proved that the measurement of two collinear vectors are 
sufficient to determine the mobile system with respect to the base. 
To compile mathematical models inclinometer responding to vectors of any physical nature, 
introduced in consideration of the right of the coordinate system (Figs. 1-2). 
  
Fig. 1. The schematic diagram of coordinate systems 
Figs. 1, 2 introduced the notation: ܴ∗ – fixed frame corresponding to the geographic coordinate 
system 0ߦ∗ߟ∗ߞ∗ , it axis 0ߦ∗  tangential to the meridian and the geographic North, axis 0ߟ∗  – 
tangential to the parallels and the East; axis 0ߞ∗ – the vertical space and deep into the earth. Frame 
R corresponds trihedral coordinate system 0ߦߟߞ  in which the axis 0ߦ  is directed along the 
magnetic meridian, the axis 0ߞ – the vertical space and down, axis 0ߟ perpendicular to the axis of 
0ߞ, 0ߦ axes, and thus to get a trihedron was right. Trihedron of coordinates 0ߦߟߞ is rotated relative 
0ߦ∗ߟ∗ߞ∗ to the angle ±݀ of magnetic declination; coordinate system 0ݔଵݕଵݖଵ associated with the 
plane of the borehole inclination, the axis 0ݔଵ of which lies in a horizontal plane and coincides 
GENERALIZED MATHEMATICAL MODEL OF A LINEAR SINGLE-AXIS ACCELEROMETER AS AN INTEGRAL PART OF THE INCLINOMETER.  
A. N. KRASNOV, G. Y. KOLOVERTNOV, V. E. LYALIN 
 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. AUG 2016, VOL. 7. ISSN 2345-0533 49 
with the projection of the hole axis on a horizontal plane. Regulations define downhole plane 
azimuth ߙ true or magnetic, counted, respectively, from the northern areas of geographical or 
magnetic meridian. The plane ݔଵ0ߞ is called the plane of inclination of the well, the well plane 
curvature, or apsidal plane; coordinate system 0ଵݔଶݕଶݖଶ associated with the hole inclination axis, 
which is directed along the tangent to the trajectory of the axis of the axis 0ଵݖଶ at the point 0ଵ, 
then the angle ߠ is the zenith angle – the angle deviation of the borehole axis from the vertical; 
coordinate system 0ଵݔଷݕଷݖଷ associated with the tubular string. Trihedron axes 0ଵݔଷݕଷݖଷ obtained 
from the initial 0ଵݔଶݕଶݖଶ rotation about the axis 0ଵݖଶ by an angle ߮ called reticle (apsidal angle 
whipstock angle drilling tool) [2]. 
 
Fig. 2. The projections of the angular velocity of the Earth rotation axes associated with the wellhead 
2. The model of the system 
The projections of the vectors of the magnetic field of the Earth (MFE) ሬܶԦ, Earth’s angular 
velocity ΩሬሬԦЗ, acceleration of gravity ܩԦ on the axis ܴ∗, ܴ of the frame, will be: 
ΩሬሬԦோ∗(Ωଷcos߶Г, 0 − Ωଷsin߶Г) = (߶Г, 0, ΩВ), Ԧ݃(0,0, ݃),
ሬܶԦ(ܪ, 0, ܼ) = (ܶcos߭, 0, ܶsin߭),
where ߶Г , ߭  – latitude and angle of magnetic inclination; ܪ , ܼ  – horizontal and vertical 
components of the magnetic field of the Earth. Let: 
ܾ = ܼܪ = tg߭,     ܾ
∗ = Ω஻ΩГ = −tg߮Г.
The transition from a fixed reference point ܴ (0ߦߟߞ), associated with the Earth, to the moving frame 
ܴଷ(0ଵݔଷݕଷݖଷ) associated with the column of pipes, by successive rotation through the final ߙ , ߠ , ߮ , 
corresponding to Euler angles (Fig. 3). 
 
a) 
 
b) 
 
c) 
Fig. 3. Consecutive transitions from the fixed bound Earth coordinate system 0ߦߟߦ to the mobile 0ݔଷݕଷݖଷ 
GENERALIZED MATHEMATICAL MODEL OF A LINEAR SINGLE-AXIS ACCELEROMETER AS AN INTEGRAL PART OF THE INCLINOMETER.  
A. N. KRASNOV, G. Y. KOLOVERTNOV, V. E. LYALIN 
50 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. AUG 2016, VOL. 7. ISSN 2345-0533  
Coordinate systems ܴ(0ߦߟߞ) and ܴଷ(0ݔଷݕଷݖଷ) related by the formula: 
൭
ݔଷ
ݕଷ
ݖଷ
൱ = ܣథܣఏܣఈ ൭
ߦ
ߟ
ߞ
൱, 
where ݔଷ, ݕଷ , ݖଷ – the components of a vector Ԧܽ  in the frame ܴଷ; (ߦߟߞ) – components of the 
vector in the frame ܴ, ܣ = ܣఝܣఏܣఈ – the matrix of the direction cosines between the rapper ܴ 
and ܴଷ found as the product of the matrix between the intermediate axes. 
The expressions for the projection of the magnetic field of the Earth ሬܶԦோ, the angular velocity 
of the Earth rotation in inertial space ΩሬሬԦзோ and the vector acceleration of gravity (or gravity) in the 
fixed system of coordinates ܴ(0ߦߟߞ) on the axis Ԧ݃ோ  associated with the downhole projectile 
collinear axis of the frame ܴଷ: 
ሬܶԦோଷ = ܣథܣఏܣఈ ⋅ ሬܶԦோ,    ΩሬሬԦோଷ = ܣథܣఏܣఈ ⋅ ΩሬሬԦзோ, Ԧ݃ோଷ = ܣథܣఏܣఈ ⋅ Ԧ݃ோ.
This chapter offered generalized mathematical models inclinometer with the three still 
enshrined in the well projectile primary sensors, which are obtained by the design of vectors ሬܶԦ, ΩሬሬԦ, 
Ԧ݃  on the sensitivity axis of the magnetic, gyroscopic angular velocity sensors (AVS) and 
accelerometers, which form the basis of the structure of the inclinometer: 
ሬܶԦோଷ = ܣ ⋅ ሬܶԦோ,     ΩሬሬԦோଷ = ܣ ⋅ ΩሬሬԦோబ, Ԧ݃ோଷ = ܣ ⋅ Ԧ݃ோ, (1)
where ܣ = ܣథܣఏܣఈ – the matrix of the direction cosines between the axes of the frames ܴ, ܴ଴, ܴଷ. 
Between these equations there are six ratios: 
ห ሬܶԦோଷห = หሬܶԦோห, หΩሬሬԦோଷห = หΩሬሬԦோబห, | Ԧ݃ோଷ| =| Ԧ݃ோ|, (2)
൫ሬܶԦோଷ ⋅ Ԧ݃ோଷ൯ = ൫ሬܶԦோ ⋅ ݃ோ൯, ൫ΩሬሬԦோଷ ⋅ Ԧ݃ோଷ൯ = ൫ΩሬሬԦோబ ⋅ Ԧ݃ோబ൯, ൫ሬܶԦோଷ ⋅ ΩሬሬԦோଷ൯ = ൫ሬܶԦோ ⋅ ΩሬሬԦோబ൯, (3)
expressing the fact that the orthogonal transformation units projected vectors and their scalar 
products in the original and projected coordinate systems are the same. 
The set of Eqs. (1)-(3) is a generalized mathematical model in matrix-vector form 
gyromagnetic inclinometer with fixedly mounted in a downhole projectile uniaxial magnetic, 
gravity and gyroscopic sensors primary. From it derived the mathematical models only magnetic 
(flux-gate), only the gyroscopic inclinometer or just gravity. By solving the Eq. (1) with respect 
to the desired coordinates ߙ, ߠ, ߮, calculated angular parameters of the spatial orientation of the 
underground mobile device. 
The mathematical model of inclinometer considered non-identical electrical parameters of the 
primary sensors. For uniaxial primary sensors inclinometers (ferroprobes, accelerometer, 
gyroscopic AVS) output signal can be written as [3]: 
௜ܷФ = ܷ଴௜Ф + ܷ௠௜Ф ⋅ ܽ௜,    ௜ܷГ = ܷ଴௜Г + ܷ௠௜Г ⋅ ܿ௜, ௜ܷА = ܷ଴௜А + ܷ௠௜А ⋅ ܾ௜, ݅ = 1,2,3,
where ܷ௠௜Ф , ܷ௠௜Г , ܷ௠௜А  – the highest values ferroprobes output signal AVS gyro, accelerometer, and ܷ଴௜Ф, 
ܷ଴௜Г , ܷ଴௜А  – zero signal sensors, ܽ௜, ܾ௜, ܿ௜, ݅ = 1, 2, 3 – the direction cosines between the vectors ሬܶԦ, ΩሬሬԦ, 
Ԧ݃ and the sensitivity axes of the primary sensor on the assumption that the sensitivity axis of the primary 
sensors collinear axis of the frame ܴଷ. 
Then the calculation of the primary sensor signal with respect to their non-identical electrical 
parameters is carried out by the formulas: 
GENERALIZED MATHEMATICAL MODEL OF A LINEAR SINGLE-AXIS ACCELEROMETER AS AN INTEGRAL PART OF THE INCLINOMETER.  
A. N. KRASNOV, G. Y. KOLOVERTNOV, V. E. LYALIN 
 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. AUG 2016, VOL. 7. ISSN 2345-0533 51 
൫ ଵܷФ − ܷ଴ଵФ ൯
ܷ௠ଵФ
= ܽଵ ,   
൫ܷଶФ − ܷ଴ଶФ ൯
ܷ௠ଶФ
= ܽଶ ,
൫ܷଷФ − ܷ଴ଷФ ൯
ܷ௠ଷФ
= ܽଷ, (4)
( ଵܷГ − ܷ଴ଵГ )
ܷ௠ଵГ
= ܿଵ ,     
(ܷଶГ − ܷ଴ଶГ )
ܷ௠ଶГ
= ܿଶ ,
(ܷଷГ − ܷ଴ଷГ )
ܷ௠ଷГ
= ܿଷ, (5)
( ଵܷА − ܷ଴ଵА )
ܷ௠ଵ௔ = ܾଵ ,    
(ܷଶА − ܷ଴ଶА )
ܷ௠ଶА
= ܾଶ ,
(ܷଷА − ܷ଴ଷА )
ܷ௠ଷА
= ܾଷ. (6)
Eqs. (4)-(6) are a mathematical model of information signals from the transmitters. 
The mathematical model inclinometers also take into account the angular misalignment of the 
primary sensor in the coordinate system associated with the housing as the base surface of the primary 
sensors are generally not coincide with their sensitivity axes, besides the seats beneath them in the 
housing inclinometer fulfilled with limited accuracy due to variation of parameters of technological 
processes of production of components and assembly. Therefore, the model must take into account 
fully the connection between the sensor signals and the components of vector fields ሬܶԦ, Ԧ݃. 
Denoting Ԧ݁௜Ф – guides the unit vectors of the axes of sensitivity ݅-th primary sensor ݅ = 1, 2, 3, 
for example, ferroprobe: 
Ԧ݁௜Ф = cos( Ԧ݁௜Ф, ଓԦ)ଓԦ + cos( Ԧ݁௜Ф, ଔԦ) + cos( Ԧ݁௜Ф, ሬ݇Ԧ),
we obtain the projection of the vector ܣԦ = ܽଵଓԦ + ܽଶଔԦ + ܽଷ ሬ݇Ԧ where ଓԦ, ଔԦ, ሬ݇Ԧ – unit vectors orthogonal 
axes ܴଷ(0ݔଷݕଷݖଷ) as the: 
PrАሬԦ = ܽଵcos( Ԧ݁௜Ф, ଓԦ)ଓԦ + ܽଶcos( Ԧ݁௜Ф, ଔԦ) + ܽଷcos( Ԧ݁௜Ф, ሬ݇Ԧ). (7)
Consequently, the output signals from primary sensors can be represented as: 
௜ܷФ = ܷ଴௜Ф + ܷ௠௜Ф ൣܽଵcos൫ Ԧ݁௜Ф, ଓԦ൯ଓԦ + ܽଶcos൫ Ԧ݁௜Ф, ଔԦ൯ + ܽଷcos൫ Ԧ݁௜Ф, ሬ݇Ԧ൯൧, ݅ = 1,2,3.
Systems of Eq. (7), are the primary scalar equations relating the testimony imperfect 
ferroprobe with an error of inclinometer casing installation. 
Designed inclinometers include accelerometers in its design. They reviewed the working 
principles drawn up its mathematical model investigated the error of the base vibration, and also 
shows the possibility of creating vibration and vibration inclinometers [3]. 
With the help of three accelerometers measured the three projections of the vector acceleration 
of gravity Ԧ݃ on the sensitivity axis accelerometers, which are functionally linked with the desired 
zenith ߠ and reticle ߮ angles. 
Mathematical model of single-axis linear accelerometer includes a mass inertial element ݉௜, its 
movement ݎ௜∗ (݅ = 1, 2, 3). Let inclinometer comprising three uniaxial linear accelerometer sensitive 
axes are mutually orthogonal Ԧ݁ଵ, Ԧ݁ଶ, Ԧ݁ଷ, is moved along a curve ܮ representing the trajectory of the 
well (Fig. 4). At the same time it moves with linear and angular accelerations Ԧܽ, ߝԦ. The accelerometer 
is shown as a mass ݉௜ on a spring. The equation of motion of the inertial element ݅ accelerometer in 
general terms is as follows ݉௜ Ԧܽ௜ = ݉௜ Ԧ݃ + ሬܶԦ௜ + ሬܴԦ௜ : where ሬܶԦ௜ = −݇௜(ݎ௜∗ − ݎ଴௜∗ ) Ԧ݁௜  – the tension 
“electric” spring rate ݇௜ ; ݎ଴௜∗  – the equilibrium position in the absence of acceleration Ԧܽ௜ ; ሬܴԦ௜  – 
communication reaction rail axis Ԧ݁௜. 
The absolute acceleration of the motion ݅th inertial body made up of the acceleration of the 
portable motion Ԧܽ௘௜  (movement along the curve ܮ  and the rotation of the unit vector Ԧ݁௜ ), the 
acceleration of the relative movement Ԧܽ௥௜  (movement along the rail axis Ԧ݁௜ ) and the Coriolis 
acceleration Ԧܽ௞௜: Ԧܽ௜ = Ԧܽ௘௜ + Ԧܽ௥௜ + Ԧܽ௞௜. 
The accelerations and velocities of inertial motion of the body are of the form: 
GENERALIZED MATHEMATICAL MODEL OF A LINEAR SINGLE-AXIS ACCELEROMETER AS AN INTEGRAL PART OF THE INCLINOMETER.  
A. N. KRASNOV, G. Y. KOLOVERTNOV, V. E. LYALIN 
52 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. AUG 2016, VOL. 7. ISSN 2345-0533  
Ԧ߭௥௜ = ݎሶ௜∗ Ԧ݁௜,     Ԧܽ௥௜ = ݎሷ௜∗ Ԧ݁௜, Ԧ߭௘௜ = ݎԦሶଵ + ሬ߱Ԧ × ݎ௜∗ Ԧ݁௜,
Ԧܽ௞௜ = 2 ሬ߱Ԧ × Ԧ߭௥௜∗ ,     Ԧܽ௘௜ = ݎԦሷଵ + ሬ߱Ԧሶ × ݎ௜∗ Ԧ݁௜ + ሬ߱Ԧ × ( ሬ߱Ԧ × ݎ௜∗ Ԧ݁௜),
where ሬ߱Ԧ, ሬ߱Ԧ = ߝԦ – angular velocity and acceleration of the point ݉௜. After the transformation of 
the original equation takes the final form: 
ݎԦሷଵ Ԧ݁௜ − ݎ௜∗ห Ԧ݁ሶ ห
ଶ − Ԧ݃ ⋅ Ԧ݁௜ = −ݎԦሷଵ +
݇௜
݉௜ (ݎ଴௜
∗ − ݎ௜∗),
which is a generalized mathematical model of the accelerator, moving along an arbitrary curve ܮ. 
 
Fig. 4. Explanation of the mathematical model of the accelerometer 
3. Conclusions 
It was found that the measurement of the two non-collinear vectors sufficient to determine the 
orientation of the moving coordinate system relative to the base (fixed) associated with the Earth. 
The basis of the construction of directional sensors on information about the spatial rotation of the 
drilling tool with respect to any two non-collinear vectors ta-cal as vector acceleration of gravity 
vector of the Earth’s magnetic field vector and the Earth’s angular velocity. 
Made a significant contribution to the development of the theory of directional transducers is 
to obtain a generalized mathematical model in matrix-vector form gyromagnetic inclinometer with 
fixedly mounted in a downhole projectile magnetic, gyroscopic, and gravity sensors, which allows 
to obtain mathematical models only magnetic (flux-gate), only gyro, or only gravity (to determine 
a position diverter and zenith angle of inclination) inclinometers. 
References 
[1] Isachenko W. H. Well Inclinometer. Nedra, 1987, p. 216, (in Russian). 
[2] Kovshov G. N., Kolovertnov G. Y. Navigation devices and inclinometer spatial orientation sensors 
control the drilling tool when drilling. Proceedings of the International Conference on “Methods, Tools 
and Technologies of Reception and Processing of the Measuring Information”, 2000, p. 105-106, 
(in Russian). 
[3] Krasnov A. N., Ishinbaev N. A., Kolovertnov G. Y. Measuring channels for use with piezoresistive 
sensors. Materials of the 11th Scientific-Technical Conference “Sensors and Transducers Information 
Systems for Measuring, Monitoring and Control”, 1999, p. 140-141, (in Russian). 
[4] Avery T. E., Burkhart H. E. Height Measurement Principles. Forest Measurements (5th ed.). 
McGraw-Hill, 2002, p. 154. 
[5] Williams M. S., Bechtold M. A., LaBau V. J. Five instruments for measuring tree heights: an 
evaluation. Southern Journal of Applied Forestry, 1994, p. 76-82. 


Generalized mathematical model of a linear single-axis accelerometer as an integral part of the inclinometer

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Generalized mathematical model of a linear single-axis accelerometer as an integral part of the inclinometer

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