Mathematical modeling of drive transitive process with linear change in control input

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

• Keywords 1 
• Technological equipment
• transitive process
• drive acceleration
• motor angular velocity

1. Introduction 

The control input that is changed linearly can be described by the following equation: 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: The dynamic model is given in Figure 1:.

Differential equations for drive motion are done as:

U с t   0 t    pt

m m Russia

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 Mi0 . 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 pt  M i0  0  

This first stage finishes with t=t1, for which the condition of M m t1    m pt1  M i0 holds. It produces t1 

M 0   m p

i . moment of motor is described as: where Tm   Jm  Ji  m1 . second stage of acceleration mode:

At the second stage in acceleration mode, given t>t1, the mechanical system is set in motion and the The equation describing the system motion in the second stage can be done as: 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.

     m

,  .  tt1   

.  m t   0 t  

M 0

i   m

 Jm  Ji  p 1  e Tm  M 0 

m The value of m0 

i determines reductions in the angular velocity value in relation to the preset value ω0(t), determined at the drive input and conditioned on the moment of load Mi0 - this produces drive velocity static error. The value of M md   Jm  Ji  p determines the dynamic component of moment Mm generated by the motor and providing conditions for the drive acceleration with the predynamic error. The value of md  error. The resulting function ( 5 ) can be given as: 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  Jm  Ji  p  Tm p is regarded the pre-determined value of dynamic

Given t=t2=3Tm+t1 the function e Tm  0,05 and then given t>t2 the angular velocity ωm(t) can be described as:

m t   0 t   m0  md  

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 *  p tt1 

m m t   0 t   m0  md 1  e Tm 

  

.   * *

. .

Solution to the last equation produces condition according to which the value of angular velocity at this stage is determined by the function: m t   m0  md etTmt3  m0  Tm pe Tm tt3   in the following manner: t3  0 . ( 3 ) done as: 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  0 . It is determined by the value of the moment of load Mi0 and m stiffness ratio of static parameter βm. Given changes in time value t in the range of t1<t<t2 acceleration of the drive mechanical system is determined by region ab of the curve in which the total error in velocity is increased by the dynamic components rising in the range of 0 to the steady-state value of md . Within this time range the angular velocity value is determined by dependencies ( 4 ) - ( 6 ). Given t>t2 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  0 , which determines the drive angular velocity in m 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. 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   p0   ( 9 ) 

3. Conclusion 

4. References 

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]. [10] L. B. Alekseeva, Use of mechanical models for the analysis of antivibration mounting, IOP Conf.

Series: Earth and Environmental Science 194( 2 ) (2018) 22001. [11] L. B. Alekseeva, Determination of transient behavior characteristics in drive control of machines,

Mining Informational and Analytical Bulletin 4 (2016) 18–25. [12] O. M. Bolshunova, A. A. Korzhev, A. M. Kamyshyan, Adaptive control system of dump truck traction electric drive, IOP Conference Series: Materials Science and Engineering, 327( 5 ) (2018) 052007. [13] A. A. Korzhev, O. M. Bolshunova, I. N. Voytyuk, A. Vatlina, Mathematical simulation of transient operation modes of an electric drive of a centrifugal pump for a slurry pipeline, E3S Web of Conferences 140 (2019). DOI: 10.1051/e3sconf/201914004012. [14] E. K. Eshchin, Calculations of dynamic operating modes of electric drives of self-propelled mining machines, Journal of Mining Institute 233 (2018) 534–538. [15] D. I. Shishlyannikov, A. A. Rybin, Assessment of load of beam-balanced pumping units by electric motor power indicators, Journal of Mining Institute 227 (2017) 582.

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