American Journal of Applied Science and Technology
151
https://theusajournals.com/index.php/ajast
VOLUME
Vol.05 Issue 05 2025
PAGE NO.
151-155
10.37547/ajast/Volume05Issue05-32
Features of The Mathematical Description of Typical
Links of Electromechanical Systems
X.U. Sarimsakov
Doctor of economic sciences, docent, Andijan machine-building university, Uzbekistan
Received:
31 March 2025;
Accepted:
29 April 2025;
Published:
31 May 2025
Abstract:
The variety of methods for mathematical description of electromechanical sys-tems, the feasibility of
combining different approaches to modeling different types of elements - all this leads to the need to take into
account the specifics of the mathematical description of each element of the system. Structural and dynamic
properties of typical elements of electromechanical objects are analyzed and sys-tematized.
Keywords:
Electromechanical systems, mechatronic systems, modeling, actuators, mathemat-ical description.
Introduction:
Modern electromechanical systems are an integral
part of technical means used in industry, mechanical
engineering, energy and transport. Due to the
widespread use of such objects in various areas of
human activity, intensive work has been carried out
for many years to improve them. Simple objects have
been replaced by controlled electromechanical
objects with a wide range of functions and improved
performance characteristics.
Since significant energy consumption is observed
during the operation of electromechanical systems,
intensive work has been carried out for many years to
improve their energy characteristics. The tasks of
improving dynamic characteristics by using automatic
control systems also remain relevant. Depending on
the task, the automated control system must provide
the
necessary
operating
modes
of
the
electromechanical system with control of speed,
power, mechanical torque, frequency, etc. Currently,
for modern electromechanical systems, there is a
steady trend towards increasing requirements for the
accuracy of complex movements of actuators,
provided that their speed of movement increases.
This became possible with the advent of powerful
semiconductor power converters and high-precision
digital control systems, which laid the foundation for
the development of mechatronic systems [1-3].
Therefore, when considering the structural and
dynamic features of modern electromechanical
systems, it is advisable to focus on electromechanical
systems of the mechatronic type. Taking into account
their structural and dynamic features will help
determine ways to improve the efficiency of their
operation.
A feature of mechatronic systems is that they cover a
wide range of tasks due to the versatility of
computerized control, monitoring and diagnostic
systems. Depending on the features of the actuator
(control object), a mechatronic system can use a
different set of sensor devices and different
algorithms for generating control actions. To improve
the efficiency of controlling complex multichannel
and multi-connected objects, their reference and
predictive models are used. Additional requirements
are put forward for such models, in particular, the
ability to function in real and accelerated time modes,
high reliability, the ability to synchronize the current
state of the model relative to a real operating object,
etc. [4-6].
The use of computerized control systems in
mechatronic devices has made it possible not only to
improve control quality indicators, but also to solve a
set of problems associated with controlling complex
systems with a large number of electromechanical
converters [7]. The specificity of these problems is
that when generating control signals, it is necessary
to take into account their interconnectedness
through the commonality of the control object, which
can have a complex structure. To solve such
problems, it is necessary to create mathematical
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American Journal of Applied Science and Technology (ISSN: 2771-2745)
models of multi-connected control objects. Note that
for such control objects, the construction of
parametric models causes significant difficulties, so it
is necessary to find other ways of their mathematical
description. An effective approach to the
construction of mathematical models of multi-
connected dynamic objects is the use of a
mathematical description in the form of integral
equations and their systems using identification
methods.
METHODOLOGY
A distinctive feature of modern mechatronic systems
is the use of the principles of unification, aggregation
and typification [1]. This allows for the design of a
mechatronic system based on unified blocks, which, if
necessary, can be replaced with similar standard
blocks. Such flexibility of the structure of a
mechatronic system allows for prompt changes in it
in order to optimize the structure for a specific class
of problems. In addition to structural flexibility,
mechatronic systems also have software flexibility
due to the use of programmable microcontrollers and
microcomputers
in
control, monitoring
and
diagnostic systems. Changing control algorithms in
mechatronic systems, in the vast majority of cases, is
carried out by changing software modules. Another
positive moment for mechatronic systems is a
significant increase in the efficiency of control,
monitoring and diagnostic algorithms when using
mathematical models of control objects. At the same
time, at the current stage of development of
mechatronic systems, there is a need to create
effective high-speed algorithms for modeling the
dynamics of various actuators as control objects.
When describing the dynamics of complex
mechanical elements (long kinematic transmissions,
spatial frame structures and mechanisms), which
consist of both homogeneous and heterogeneous
elements (beams, rods, plates, shells, etc.) with
different types of connections between themselves
(connections through elastic and damping elements,
rigid and movable-hinge connections, etc.), the use of
a universal method of mathematical description
causes significant difficulties. In addition, the
presence of different types of connections and
different types of motion between elements
(longitudinal, transverse, torsional vibrations) also
complicates the task of mathematical description.
Such a variety of interaction methods can be
reproduced using structural models, which can
consist of different types of links combined into a
single block-structural scheme.
Let us consider the dynamic properties of typical
objects with distributed parameters, which are
present in modern controlled electromechanical
systems.
Remote rods of industrial robots. When industrial
robots with long remote rods designed to carry loads
operate, the elastic compliance of the robot links has
a significant effect on the trajectory of the gripping
unit. Due to the presence of distributed masses and
elastic links, elastic oscillations occur, the amplitude
of which can be unacceptably large, which causes
inaccurate operation of the robot's actuators.
Therefore, it is necessary to have a mathematical
model of the robot links taking into account the
distribution of parameters, which will allow us to
study the effect of elastic deformations on the
accuracy of processing program movements.
Let us consider a mechanical model of an industrial
robot (Fig. 1), which consists of two rectilinear links 1
and 2 and a gripping unit [3, 8]. The first link (OO
1
) is
a rod of rectangular or circular cross-section and
moves along the axis
O
x
3
in the vertical direction.
Fig. 1. Kinematic diagram of an industrial robot.
O
x
1
x
2
x
3
0
O
1
2
O
2
EJ
,
P
,
s
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American Journal of Applied Science and Technology (ISSN: 2771-2745)
At point O there is a cylindrical hinge connecting links
OO
1
and OO
2
. Link 2 rotates around an axis parallel to
the plane x
1
x
2
. At point O
2
there is a wrist hinge by
means of which the gripping unit is connected to link
OO
2
. The moment of inertia of the load m is
sufficiently small relative to the axis passing through
its center of gravity perpendicular to the plane of
oscillation. Then the equation of oscillation of link OO
of the industrial robot has the form
(
)
( )
4
2
1
1
4
2
,
y
y
EJ
s
m
x
x
f x t
x
t
+
+
−
=
where EJ,
ρ, s
are respectively the bending rigidity,
material density and cross-sectional area of the link;
y is the deflection of the x-section of link 2; f
1
is the
linear density of external forces.
Introducing a system of relative units
u = y/l, ξ = x/l, τ
= t/t
0
, f(ξ, t) = l
3
f
1
(x, t)/(EJ), t
0
= l/ν
0
,
v
0
= √EJ/(ρs/l)
,
where t
0
and
ν
0
are the model time and speed, we
obtain the equation of motion of the second link of
the industrial robot
(
)
( )
4
2
1
1
1
,
u
u
f
+ +
−
=
with boundary conditions
2
2
3
0
0
1
0
2
2
3
0
u
u
u
u
d
d
d
=
=
=
=
=
=
=
=
where
μ = m/(ρls)
is the ratio of the mass of the load
to the mass of the OO
2
link.
Drill pipe columns of drilling rigs. When constructing
wells, an important problem is to ensure high
technical and economic indicators of the drilling
process. This problem is especially relevant when
constructing deep and super-deep wells. The main
process during drilling is the operation of the bit to
deepen the wellbore. A distinctive feature of
automatic bit feed control systems is the presence of
a drill pipe column, through which the bottomhole
parameters are measured (axial load and bit
movement speed), as well as the transmission of the
control action from the surface to the bottomhole.
Fig. 2. Simplified diagram of the drilling rig bit feed mechanism.
The influence of the drill column is manifested in a
significant distortion and delay of information
received from the bottomhole, and the control
action, which is transmitted in the opposite direction.
Therefore, the organization of the bit feed control
process is associated with significant difficulties. The
productivity of drilling rigs can be increased by taking
into account the dynamic characteristics of the drill
string when transmitting mechanical forces from the
wellhead to the bottomhole. [3, 9]
A simplified structural diagram of the drill bit feed
regulator is shown in Fig. 2.
The axial reaction of the face and the reaction of the
bit are applied to the lower end of the column, and
the forces of gravity, viscous friction, and inertia are
distributed along the length. The drill pipe column,
taking into account a number of assumptions, can be
considered an elastic homogeneous rod with
distributed mass, elasticity, and viscous friction. The
displacement of the sections of the column elements
is described by a differential equation in partial
derivatives
2
2
2
2
2
2
u
u
u
a
c
t
t
x
+
=
,
(1)
where
( )
/ 2
a
h
m
=
,
/
c
Es m
=
; m is the
mass of a unit length of the column; u is the
displacement of the column cross-section relative to
1
2
3
4
5
6
7
8
9
10
Д
6 - a winch drum
7 - a gearbox;
8 - a bit feed electric
motor;
9 - a power converter;
10 - a control device
1 - a drill pipe string
2 - a downhole motor
3 - a drill bit
4 - a fixed end of the
wireline
5 - a load cell
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American Journal of Applied Science and Technology (ISSN: 2771-2745)
the equilibrium position; E is the modulus of
elasticity; s is the cross-sectional area of the column;
x is the coordinate of the column cross-section; a is
the damping coefficient; h is the damping coefficient
per unit length of the column; c is the velocity of
propagation of the displacement along the column.
The speed of movement of the drill pipe section
ν
and
the increase in force P are determined by the
expressions
( )
,
u x t
v
t
=
,
( )
,
u x t
P
Es
x
= −
.
When using a surface bit feed controller, the control
action for the drill string is the displacement rate of
the upper end of the string
( )
0
v t
. The boundary
condition for this end of the string is
( )
( )
0
0
,
x
u x t
v t
t
=
=
.
(2)
The boundary conditions for the lower end of the drill pipe string are
( )
( )
,
x l
l
u
P l t
Es
zv t
x
=
= −
=
,
(3)
where l is the length of the column;
ν
l
is the bit speed;
z is the parameter determining the interaction of the
bit with the rock (at an absolutely hard bottomhole z
→ ∞
). Equations (1)
—
(3) define the dynamic
characteristics of the drill string as an object with
distributed parameters. However, the presented
mathematical model does not take into account the
fact that real drill strings consist of parts with
different physical properties and also belong to
developing systems. To ensure the necessary
adequacy of the mathematical model of the drill pipe
column, it is necessary to involve new approaches to
its construction, in particular, the structure-oriented
approach, which involves representing the model of a
complex dynamic object in the form of a set of
mathematical descriptions that allows taking into
account the features of each structural element in its
mathematical
description
and
numerical
implementation. As part of the application of the
structural approach to the construction and
numerical implementation of models of dynamic
distributed objects, it is advisable to determine, by
analogy with the structural modeling of objects with
lumped parameters, typical structural blocks.
RESULTS AND DISCUSSION
For this purpose, we will find out what types of
actuators with distributed parameters are present in
electromechanical systems. In general, they can be
divided into two large groups: spatially one-
dimensional
and
multidimensional.
Spatially
multidimensional ones include actuators of industrial
manipulators with complex frame spatial structures,
drilling rigs, etc. Spatially one-dimensional objects are
divided into rod and ring-shaped. Objects of ring
nature include drives of conveyors and escalators,
drives for feeding manipulator platforms for
reinforcement and winding, drives for positioning the
heads of printing devices, etc. Rod objects are cables
of lifting installations, elevators, towing systems, long
kinematic transmissions, drill pipe columns, long
shafts, remote rods of industrial robots. The
classification scheme for objects with distributed
parameters of electromechanical systems is shown in
Fig. 3.
Fig. 3. Classification of actuators with distributed parameters of electromechanical systems.
Rod
Ring-shaped
Mixed
SPATIALLY ONE-DIMENSIONAL
SPATIALLY
MULTIDIMENSIONAL
With longitudinal
deformation
With torsional
deformation
With bending
deformation
Drives for feeding
platforms of
manipulators for
reinforcement.
Drives for
conveyors.
Drives for
escalators.
Actuators of industrial
manipulators with
complex frame spatial
structures.
Drilling rigs.
Cables of lifting
installations.
Towing systems.
Long kinematic
transmissions.
Remote rods of
industrial
robots.
Booms of lifting
installations.
Long shafts
Drill pipe
columns
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American Journal of Applied Science and Technology (ISSN: 2771-2745)
Analyzing the dynamic characteristics of the objects
shown in the diagram, we can conclude that when
describing them mathematically, their initial models
can be reduced (by decomposition) to a structure
whose elements are mathematical descriptions of
distributed links with three types of deformation
(longitudinal, torsion and bending). Therefore, when
describing the dynamic characteristics of such
objects, the typical element is a linearly extended
element, which, without taking into account the
dissipation of energy and resistance forces, is
described by a system of partial differential equations
( )
( )
( )
( )
( )
( )
2
2
2
2
2
2
2
2
4
2
4
2
,
,
0;
,
,
0;
,
,
0,
p
u x t
u x t
E
x
t
x t
x t
GJ
I
x
t
y x t
y x t
EJ
s
x
t
−
=
−
=
−
=
(4)
where
u(х, t)
is the longitudinal shear of the section;
(х, t)
is the angle of twist of the section around the
longitudinal axis;
y(х, t)
is the displacement of the
section from the centerline in the transverse
direction; G is the shear modulus; J
p
is the polar
moment of inertia of the cross section; I is the
moment of inertia of a unit of length;
is the density
of the substance; E is the modulus of elasticity.
CONCLUSION
Thus, the features of modern electromechanical
systems should be sought in the plane of analysis of
the properties of actuators (control objects) in order
to obtain for them the corresponding mathematical
dependencies in a form that would make it possible
to effectively solve the problems of their modeling.
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