=Paper=
{{Paper
|id=Vol-2899/paper005
|storemode=property
|title=Mathematical modeling of drive transitive process with linear change in control input
|pdfUrl=https://ceur-ws.org/Vol-2899/paper005.pdf
|volume=Vol-2899
|authors=Dmitriy Ershov,Irina Lukyanenko,Evgeniy Zlotnikov
}}
==Mathematical modeling of drive transitive process with linear change in control input==
https://ceur-ws.org/Vol-2899/paper005.pdf
Mathematical modeling of drive transitive process with linear
change in control input
Dmitriy Ershov 1,2, Irina Lukyanenko 1 and Evgeniy Zlotnikov 2
1
Saint-Petersburg University of Aerospace Instrumentation (SUAI), 67, BolshayaMorskaia str., St Petersburg,
190000, Russia
2
Saint-Petersburg Mining University, 2, 21st Line, St Petersburg 199106, Russia
Abstract
The paper researches into dynamic parameters of transitive processes in electrical drive of
technological equipment in acceleration mode given changes in control input under law of
linearity. Analysis of two-mass dynamic model allows for solutions of differential equations
describing three stages in the drive acceleration mode. Dynamic and static components of drive
errors are discussed in relation to its velocity as being conditioned on the moment of load, the
total moment of the drive inertia, stiffness ratio of static parameter and pre-defined
programmed acceleration.
Keywords 1
Technological equipment, transitive process, drive acceleration, motor angular velocity
1. Introduction
Conditions under which manufacturing and industrial equipment operates require setting a process
of transition either to produce a pre-defined principle of changes in actuator angular velocity, or as a
requirement for pre-determined speed of equipment operation, or to produce limited and controlled
dynamic loads. The required parameters determining transitive process can be achieved by setting the
law for changes in control input –voltage Uc(t), which is transferred to the armature. This law is made
operational by a specialized transducer.
2. Mathematical Modeling
The control input that is changed linearly can be described by the following equation:
Uс
t 0 t pt (1)
kF
where F is magnetic flux, k is designed parameter of motor, εp is the pre-determined value of
programmed acceleration provided by transducer operation.
The moment of motor can be given as:
Mm = βmεpt - βmωm (2)
The dynamic model is given in Figure 1:.
Differential equations for drive motion are done as:
𝐽 𝜔 𝑀 𝑀 ,
𝐽𝜔 𝑀 𝑀 ,
M m m p t mm
III International Workshop on Modeling, Information Processing and Computing (MIP: Computing-2021), May 28, 2021, Krasnoyarsk,
Russia
EMAIL: fetcat@mail.ru (Dmitriy Ershov); irina.n.lukyanenko@gmail.com (Irina Lukyanenko); zlotnik_evg@mail.ru (Evgeniy Zlotnikov)
ORCID: 0000-0001-9384-7590 (Dmitriy Ershov); 0000-0002-9647-0781 (Evgeniy Zlotnikov)
© 2021 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
31
Analysis and description of the drive in acceleration mode requires determining the static mechanical
parameter of the motor [1,2,7], with the moment of load considered a constant. The above system can
be reduced to one equation in relation to ωm.
Figure 1: Dynamic model of technological drive
𝐽 𝐽 𝜔 𝛽 𝜔 𝛽 𝜀 𝑡 𝑀 (3)
At the initial point in acceleration given t=0 ωm=0, the control input given (1) equals zero ω0(0) = 0
and consequently given (2) the moment of motor also equals zero. Hence, (3) conditions the drive
motion to be possible only if the moment of motor Mm has values higher than the moment of load M i0
. The mode of acceleration is characterized by a stage when the moment of motor is continuously
increasing with time while mechanical system remains static. This stage can be described by the
following equation:
m p t M i0 0
This first stage finishes with t=t1, for which the condition of M m t1 m p t1 M i0 holds. It produces
M i0
t1 .
m p
At the second stage in acceleration mode, given t>t1, the mechanical system is set in motion and the
moment of motor is described as:
M m t m p t t1 M i0 mm .
The equation describing the system motion in the second stage can be done as:
𝑇 𝜔 𝜔 𝜀 𝑡 𝑡 ,
where Tm J m J i m .
1
Solution to equation (3) produces equation which determines changes in angular velocity at the
second stage of acceleration mode:
1t t
(4)
m t m t t1 Tm p 1 e T
m
.
0
M
Given dependence (1), relation of p t1 i , produces:
m
1
t t
(5)
m i p
J J 1 m
e T
M i0
m t 0 t
m m .
M i0
The value of m0 determines reductions in the angular velocity value in relation to the pre-
m
set value ω0(t), determined at the drive input and conditioned on the moment of load M i0 - this produces
drive velocity static error. The value of M md J m J i p determines the dynamic component of
moment Mm generated by the motor and providing conditions for the drive acceleration with the pre-
32
determined acceleration rate εp. The third addend in (5) is given as reduction in the angular velocity in
relation to pre-determined value ω0(t), conditioned on the dynamic load and considered drive velocity
J Ji p
dynamic error. The value of md m Tm p is regarded the pre-determined value of dynamic
m
error. The resulting function (5) can be given as:
1
t t
m t 0 t 1 e Tm
0 d
(6)
m m
.
t t1
Given t=t2=3Tm+t1 the function e 0,05 and then given t>t2 the angular velocity ωm(t) can be
Tm
described as:
m t 0 t m0 md (7)
The second stage of the drive acceleration mode is completed in time t=t3, in which the angular
velocity in idle running ω0 reaches a certain pre-determined value 0* . Time moment t3 is determined
0*
in the following manner: t3 .
p
The third stage of the drive acceleration mode, when t>t3, can be described with differential equation
(3) done as:
𝐽 𝐽 𝜔 𝛽 𝜔 𝛽 𝜔* 𝑀
Dividing all components of equation by βm produces the following equation:
𝑇 𝜔 𝜔 𝜔* 𝑀 𝛽 .
The difference 0 M i m determines the values of angular velocity m0 under the steady-state
* 0 1
mode of drive motion. Differential equation describing the third stage of the drive acceleration mode
can be done as:
𝑇 𝜔 𝜔 𝜔 .
Solution to the last equation produces condition according to which the value of angular velocity at
this stage is determined by the function:
t t3 t t3
m t m0 md e T m0 Tm p e T
m m (8)
Figure 2: demonstrates the diagrams for dependencies 0 t p t , 0 t p t t1 ,
0 t m t . The curve abc is plotted based on (4). When the point of the curve in which t=t1 reached,
the drive mechanical system is set in motion, transiting from static to dynamic mode. Given this
condition, velocity time error t equals m0 . It is determined by the value of the moment of load M i0 and
stiffness ratio of static parameter βm. Given changes in time value t in the range of t1t2 the drive acceleration mode is determined by region bc of the curve, in which the value of angular
velocity is described by function (7). The steady-state value of dynamic error in velocity md is the
higher the higher the total moment of the drive inertia Jm+Ji and the pre-determined programmed
acceleration εp, and is the lower the higher βm.
The diagram of dependence of equation (8) is shown with the curve cd. The value of angular velocity
m t asymptotically approaches the value of m0 , which determines the drive angular velocity in
steady-state mode of drive motion. Given t>t4=3Tm+t3 angular velocity m t is different from the
value m0 by less than 5%, and transitive process is considered completed. Under the steady-state mode
of the drive operation drive velocity static error m0 preserves.
33
Figure 2: Diagram of the motor angular velocity dependence
The t4 time range of transitive process of the third stage in drive acceleration is then calculated as:
*
t4 3Tm t3 3Tm 0 (9)
p
3. Conclusion
It follows from expression (9) that reductions in time intervals required for transitive process or
increases in performance speeds of the drive can be gained by lower values of mechanical constant Tm
and higher values of programmed acceleration εp. Observance of the first condition requires either
reduced values of total moment of inertia or increased values of stiffness ratio of static parameter βm.
Observance of the second condition requires increases in values of the motor starting torque which is
subjected to a number of limitations conditioned on the parameters of the motor operation [3-6, 8-15].
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