the calculation and optimal control by the technological...

American Journal of Science and Technology

2017; 4(1): 5-12

http://www.aascit.org/journal/ajst

ISSN: 2375-3846

Keywords Technological System,

Functioning,

In-Process Workpiece,

Mathematical Modeling,

Optimal Control

Received: November 9, 2016

Accepted: December 5, 2016

Published: March 2, 2017

The Calculation and Optimal Control by the Technological Processes of a Puller Shaft Turning Processing

Azimov B. M., Sulyukova L. F.

Centre for Development Hardware and Software Products, IT University, Tashkent, Uzbekistan

Email address [email protected] (Azimov B. M.), [email protected] (Sulyukova L. F.)

Citation Azimov B. M., Sulyukova L. F. The Calculation and Optimal Control by the Technological

Processes of a Puller Shaft Turning Processing. American Journal of Science and Technology.

Vol. 4, No. 1, 2017, pp. 5-12.

Abstract The questions of application of modern methods and algorithms for the optimal control

on technological processes for a cotton-harvester machines’ harvester device puller shaft

turning processing are considered in the article. Components of the cutting force and

meanings of axis stretching force by turning are determined. The necessary conditions of

the technological system optimal control are investigated through the use of the

Pontryagin maximum principle. The optimal values of geometrical, constructive and

functional parameters of the processed workpiece are obtained.

1. Introduction

The puller of cotton-harvesting machine device consists of the shaft with length

lv=0.95m, diameter d=0.026m and taking off brushes mounted into its section which

have the function to take off reeled by spindle raw cotton for throwing into the entrance

camera to transport into the machine bunker. Mainly, the puller work efficiency demands

on its details, especially the shaft producing accuracy. One of the main ways of puller

shaft mechanical processing is the turning [1−4].

The determinative research phase is the control object formalization to determine the

capacity to apply computing methods in the research process. When the task is

formalized completely, i.e. the completed mathematic model is available; it may be

solved by one of numerical methods using the computing device.

2. Construction of the Kinematic Scheme and Dynamic

Models

The kinematic scheme and the dynamic model of the technological system (TS) were

built to develop mechanical processing ways (figure 1, 2). Using the second kind

Lagrange equation the math model of TS puller shaft turning processing was formed [5],

(1)

−−+−=−−−−=

cvv

vv

Mcbj

cbMj

)()(

)()(

212122

2121111

ϕϕϕϕϕϕϕϕϕϕ

ɺɺɺɺ

ɺɺɺɺ

6 Azimov B. M. and Sulyukova L. F.: The Calculation and Optimal Control by the Technological

Processes of a Puller Shaft Turning Processing

where j1, j2 – TS rotating masses inertion moments, N⋅m⋅s2

;

1 2,φ φɺɺ ɺɺ − TS rotating masses corner accelerations in the

processing, s-2

;1 2,φ φɺ ɺ − TS rotating masses corner speeds in

the processing, s-1

;1 2,φ φ − TS rotating masses corner shifts in

the processing, rad; b – the coefficient of sticky resistance of

in-processing shaft, N⋅m⋅s/rad; с − the coefficient of rigidity

of in-processing shaft, N⋅m/rad; Мd, Мr –TS driving and

resistance moment, N⋅m.

Figure 1. The kinematic elements and cutting characteristics: 1–speed

direction of the cutting resulting moving; 2−speed direction of the cutting

main moving; 3–workplate; 4–the examining point of cutting edge; 5−speed

direction of the feed moving.

Figure 2. TS dynamic model.

3. Calculation of Cutting Force

Components

The cutting force components meanings Px, Py, Pz are

determined by the technological conditions that means the

cutting geometrical regimes [6, 7].

The cutting force components by the turning are calculated

by the analytical formula

Pz(x,y)=10·Cp· x

pt ·Sy·V

n·Kp= 10 300

0.11·0.082

0.75·0.102835

-0.15·0.5394=34.822N,

where Cp=300–the coefficient accounting processing

conditions; x=1.0, y=0.75, n=-0.15 – degree indexes;

tр=0.1mm – cutting deep; S=0.082 mm/rot − feed; V=102.835

mm/min=0.102835 m/min –cutting speed; Кр–total

correction coefficient accounting conditions changing with

the reference to table ones:

0.62 1.0 1.0 1.0 0.87 0.5394p p p p zp p

K K K K K Kµ φ λ γ= = ⋅ ⋅ ⋅ ⋅ =

wherep

Kµ − correction coefficient accounting the in-

processing material properties; , , ,p p p zp

K K K Kγ φ λ −

coefficients accounting corresponding geometrical chisel

parameters.

By elastic small rigidity details lines model building and by

their processing in the elastic deformed state bending moments

on X axis are taken in to account as more essential qualities, so

as elastic deformations on this axis exert dominated influence

to form an error in the longitudinal direction.

Relative longitudinal deformations of in-processing shaft

by the stretching [8]

.

Absolute shaft extension by the stretching

,

then the permissible meaning of stretching force is

,

where p

σ −the permissible stress by stretching; F – the area

of shaft cross-section.

Taking into account axis stretching force and the absolute

extension of in-processing shaft by the stretching relative

longitudinal deformation scan be determined by the Hook

slaw [8]

.

Then the calculation meaning of axis stretching force will

be equal in our case

Preliminary necessary shaft rigidity in the corresponding

moment of resistance is determined [8]

s

l

l∆=ε

EF

lPl

p ⋅=∆

per рР Fσ = ⋅

[ ]E

рσε =

. c 1 lPx ∆⋅=

,00000856.0)013.0(14.3101.2

95.006.13

101.2

95.006.13

,0001617.0)026.0(1.0101.8

95.0822.34013.0

311311

410

=⋅⋅

⋅=⋅⋅⋅

⋅=⋅⋅

=

=⋅⋅⋅

⋅⋅=⋅

⋅=

вв

vy

p

p

vzвк

rrEF

lP

IG

lРr

πϕ

ϕ

American Journal of Science and Technology 2017; 4(1): 5-12 7

Rigidity and sticky resistance coefficient are determined

by rolling and stretching

where к

φ ир

φ − shaft rolling corners by rolling and stretching

rigidity; cк, cр и cv– coefficient of rigidity by rolling, transfor-

mated rigidity by rolling and total rigidity of in-processing

shaft; bv – coefficient of shaft sticky resistance; l–in-

processing shaft length; G, E–shaft material displacement

and elastic modules; Jp– polar and axis inertion moments; F–

shaft cross-section area; rв – in-processing shaft diameter; ω− process frequency.

4. Solution of Optimal Control

Problem

According to above-mentioned, the task of TS optimal

control can be choose [9, 10].

In the beginning TS is instate

(2)

It is required to choose such control u(t), which transfer in-

processing shaft moving into the beforehand set final state

(3)

By the way it is required, the transition process time was

the least. So the control aim reduces to functional

minimization.

. (4)

By conditions (2), (3)

(5)

(6)

where f(…)– persistent differential function with its

derivations; u(t)–piece wise persistent function on the

segment [t0, T].

To investigate the TS optimal control necessary conditions

we use Pontryagin maximum principle [9, 10].

To formulate the maximum principle we enter the

Hamilton-Pontryagin function for TS

(7)

and adjoin system

(8)

with control limit 1.u ≤

The necessary condition must be to the task decision

. (9)

Moving to the optimal control determining on the base of

(7), we form the function

, (10)

and get the math model characterizing the control 1 1

1

1u M

j=

by in-processing shaft driving and the control 2

1с cu M

j= ,

2 0sin

cu u u tω= +

(u0–its fluctuation amplitude relatively of

the middle index), characterizing the force in the instrument

putting point.

So, if 0 1,f ≡ then

0 0( , ( ), ( ))J u t t T tφ φ = − , in this case the

task (2)−(6) is named as the rapid operation task.

The object is the steady-state system and the task (4)

means, that and f U don`t depend on the time clearly that

means

. (11)

If the steady-state task (4), (11) has the optimal control u(t)

and the optimal trajectory 0( )tφ , so there is non-zero vector

of conjugate variables 1 2

( ( ), ( )), ( ) nt t t Rψ ψ ψ ∈ , meeting

the conditions (9), that means the maximum condition was

performed (7)

. (12)

0.013 34.8222799.63 ,

0.0001617

0.013 34.82252885.514 ,

0.00000856

с в zк

к к

ср

р

М r P Nmс

rad

М Nmс

rad

φ φ

φ

⋅ ⋅= = = =

⋅= = =

1

52885.514 0.0000085634.822 .

0.013

p p

x

в

cP N

r

φ⋅ ⋅= = =

2799.63 52885.514 55685.144 ,v k р

Nmc c с

rad= + = + =

0.64 0.64 2799.632.18 ,

2 2 3.14 130.83

vк

c Nmsb

radπω⋅ ⋅= = =

⋅ ⋅

0.64 0.64 55685.14443.376

2 2 3.14 130.83

vv

c Nmsb

radπω⋅ ⋅= = =

⋅ ⋅

).0()0( ),0()0( 0i0 ϕϕϕϕ ɺɺ ==i

,),1( 0 ),()( ),()( 0i0 niTttttti =≤≤== ϕϕϕϕ ɺɺ

∫=T

t

dtttutfttuJ

0

))),(),(())(),(,( 0

0 ϕϕϕ

),),(),(()( ttutft ϕϕ =ɺ

,, 0 TttUu ≤≤∈

⟩⟨+−== utuftuH i ,),,(),,,,( 0

0 ψϕψψϕ

+−=∂∂

−=−=∂∂

−=

+−=∂∂

−=−=∂∂

−=

−−

−−

2

1

21

4

222

1

2

3

21

2

1

11

2

122

1

1

1

11

,

,

ψψψψψ

ψψψψψ

vv

vv

bjy

H

dt

dcj

y

H

dt

d

bjy

H

dt

dcj

y

H

dt

d

)),(,,),((max),,),(),(( 00 ψψϕψψϕ ttutHttutH iiUu

ii ∈=

[ ]

[ ]

−−+−===

−+−−===

23142

2

442,32

3142

1

1221,11

)()(1

,

)()(1

,

uyycyybj

yyy

yycyybj

uyyy

vv

vv

ϕϕ

ϕϕ

ɺ

ɺɺ

( , , ) ( , ), ( )f t y u f y u U t U= =

0)(0 ≤= consttψ



So as the adjoin system (8) is homogeneous relatively of

iψ , the equation constant can be choosen arbitrarily (12) so

that

. (13)

From the conditions 1

maxu

H≺

is 2

u signψ= by

20.ψ ≠

Then the boundary value problem of the maximum principle

will be written as

. (14)

In these cases the boundary value problem of the

maximum principle will consist of system (14), boundary

conditions (2) and (3), are from (9), and condition (13).

Form Hamilton-Pontryagin function

(15)

It is clear, that condition (9) will give off the function

2 2, 0.u signψ ψ= ≠

In this case the boundary value

problem (10), (14) consists of [10]

. (16)

Then

, k=2,4,…,2n, (17)

that means the control uk(t) can have only one switching point.

5. Discussion of Experimental

Results

The adjoin system with variations of constructive

parameters was investigated to determine the auxiliary

function (8) by the numerical method bi, сi, ji.

Systems (1), (8), (14) were decided by the Runge-Kutta

numerical methods application. The control uk(t), giving the

function maximum (9), was determined in the field (17).

System decisions results processing (8) were, that inertion

moments and elastic and dissipative forces changing change

the variables 1

ψ ,1

ψɺ ,2

ψ ,2

ψɺ function, that means it changes

in-processing shaft driving. So it is necessary to determine

the adjoin system variables, providing TS normal functioning

to increase the accuracy in-processing shafts form and

dimensions [10].

On the base of received parameters bi, сi, ji for decision of

adjoin system and boundary volume problem of the

maximum principle graphic depending of in-processing shaft

speeds and accelerations in transition process were got

(figure 3).

The graph shown in figure 3 gives the results of the

solution of the control algorithm at ( ) 1u t = ± .

Figure 3. Graphics of auxiliary functions 1,3−1

ψ , 2

ψ ; 2,4−1

ψɺ , 2

ψɺchanging in transition process getting by (14) the maximum principle

boundary value problem solving: 5,7 – corner speeds 1 2,φ φɺ ɺ

and 6,8 –

corner accelerations 1 2,φ φɺɺ ɺɺ

of the shaft: 1,2,5,6 − by u(t)=+1; 3,4,7,8− by

u(t)=-1.

The results of system numerical decisions (1), represented

in tables 1, 2 and on figure 4, take possibility to determine

the geometrical, constructive and functional parameters of in-

processing shaft.

Table 1. Meanings and indexes of in-processing puller shaft geometrical, constructive and functional parameters.

№ Parameters Meaning Index

1 2 3 4 1. Geometric dimensions of the in-processing shaft

1.1. Length of the in-processing shaft 950 mm

1.2. Diameter of the in-processing shaft 26 mm 2. Processing conditions:

2.1. Cutting speed −1000

DnV

π= 102.835 mm/min

2.2. Feed −V

sn

= 0.082 mm/rot

2.3. Processing period − Т 9.27 min

2.4. Cutting deep − tр 0.1 mm 2.5. The shaft number of rotations – n rot/min

3. Constructive characteristics of the in-processing shaft

3.1. Material − steel 35

3.2. Permissible stress by stretching − σр 900 kgs/mm2

3.3. Solidity limit −σ 0.35⋅HB kgs/mm2

0( ) 1 0 t Ttψ = − ≤ ≤

[ ][ ]

−−+−===−+−−===

−

−

23142

1

244232

3142

1

1222111

)()( , ,

)()( , ,

ψϕϕψϕϕ

signyycyybjyyy

yycyybjsignyyy

vv

vv

ɺɺ

ɺɺ

. 424102

222101

++=++=

yyH

yyH

ɺ

ɺ

ψψψψψψ

0

2 ( )i mH f u t uψ= − +

2

2

2

1, ( ) 1( )

1, ( ) 1k

tu sign t

t

ψψ

ψ>

= = − <


№ Parameters Meaning Index

3.4. Total coefficient of the in-processing shaft rigidity – сv 55685.144 N⋅m/rad

3.5. Coefficient of the in-processing shaft rolling rigidity – ск 2799.63 N⋅m/ rad

3.6 Coefficient of the in-processing shaft rigidity by stretching – ср 52885.514 N⋅m/ rad

3.7. Total coefficient of the in-processing shaft sticky resistance − bv 43.376 N⋅m-s/rad

3.8. Inertion moment of machine cartridge - j1 0.56755 N⋅m⋅s2

3.9. Inertion moment of stretching mechanism-j2 0.002938 N⋅m.s2

3.10. Eccentricity−е 1.0 mm

4. Parameters of TS functioning

4.1. Driving moment − М1 73.597828 N⋅m

4.2. Force in direction of cutting speed− Pz 34.822 N

4.3. Resistance moment М2= rsh ·Pz 0.4527 N⋅m 4.4. Calculation index of axis stretching force − Px1 34.822 N

4.5. Absolute shaft extension by stretching − l∆ 0.00000011 m

4.6. Coefficient of storage fixing 3.28474

4.7. The shaft turn corner by bend α : 00.5 /180α π= ⋅ 0.5 rad

Figure 4. The manner of parameters changing of technological process functioning by the puller of cotton-harvester machine device turning.

Figure 4 allows graphically display the results of the solution of a mathematical model of the technological system of the

puller shaft turning processing (1).

Table 2. Parameters meanings of functioning of the shaft processing technological process.

Т,s 1,ɺφφφφ s-1

1ɺɺφφφφ , s-2

М1,Nm 2 ,ɺφφφφ s-1 2ɺɺφφφφ , s-2

М2, Nm Рz, N Рy, N Рx, N nsh,min-1

0 0 132.17 73.6 0 -207.71 -0.4527 -34.82 -13.05 -6.1 0

0.1 13.08 130.54 72.7 13.08 206.78 0.4528 34.83 13.06 6.1 125

0.2 26.168 130.54 72.7 26.168 206.95 0.453 34.86 13.07 6.1 250

0.3 39.25 130.54 72.7 39.25 206.43 0.452 34.877 13.04 6.1 375

0.4 52.33 130.54 72.7 52.33 207.09 0.453 34.88 13.08 6.1 500

0.5 65.42 130.54 72.7 65.42 206.71 0.4527 34.82 13.06 6.1 625

0.6 78.5 130.54 72.7 78.5 206.91 0.453 34.85 13.07 6.1 750

0.7 91.59 130.54 72.7 91.59 206.97 0.453 34.86 13.07 6.1 875

0.8 104.67 130.54 72.7 104.67 206.91 0.453 34.85 13.07 6.1 1000

0.9 117.75 130.54 72.7 117.74 206.22 0.451 34.74 13.03 6.1 1125

1 130.84 130.54 72.7 130.83 205.77 0.45 34.66 13 6.1 1250

There are analogue empirical formulas to determine the

forces Py and Px. But to simplify and accelerate the forces

index calculation it is recommended to decide Py and Px

accordingly the following correlations [4]:

Pz=34.822H,Py=(0.25−0.5)·Pz,Px=(0.1−0.25)·Pz,

0.013 34.822 0.4527 Nm,с sh z

М r Р= ⋅ = ⋅ =0.375 0.375 34.822 13.058 N,

y zР Р= ⋅ = ⋅ =

0.175 0.175 34.822 6.09 Nх z

Р Р= ⋅ = ⋅ = ,

where rsh – radius of the in-processing shaft.

6. Establishing of Theoretical

Conformities of a Detail Conduct

by Longitudinal-Diametrical Bend

The equation of unrigged shaft elastic line was decided on

the base of the calculation scheme to evaluate possibilities of

methods and establish theoretical conformities of a detail



conduct by longitudinal−diametrical bend (figure 5).

Figure 5. Calculation scheme of stresses and elastic line of the shaft by

stretching: е=0.001– eccentricity of stretching force putting; Мv–cutting forces

moment; Mx=Px1·e=34.822·0.001=0.034822 Nm –stretching effort moment.

Description of unrigged shaft elastic line by longitudinal

and diametrical bend can be represented in form of the fourth

order differential equations with the constant coefficients

[11].

. (18)

This equation decision gives the total elastic line equation

for stretching and performing arbitrary diametrical leading

beam

where 0

48

уP ly

EI

⋅= − ,

0 0 0, , y y y′ ′′ ′′′ − accordingly deflection, turn

corner, the second and the third derivates in the coordinates

beginning [8]; 3...102k = − coefficient determining the

storage fixing method [12]; f(x)–function of diametrical

loading influence [11].

Elastic line equations on segments I and II (calculation

scheme figure 5) are

. (19)

The initial parameters are determined by the following:

,

where l a

lα −= ; l–detail length; a–coordinate of diametrical

loading putting.

Taking into account that the stretching moment was putat

the coordinates beginning it is necessary to determine the

initial parameter 0

y′′ . We find it by differentiation of (19).

. (20)

After the multiplication of (20) to bending and stretching

rigidity we will get the equation of bending moment on

segment I, taking into account

; .

Then

. (21)

If by х=0, МI(0)=M, so from (21) is [8]

,

where2 sh k

у r φ= ⋅ .

The function of diametrical loading influence

.

Finally, deflections equations on segments will be

024 =′′− ii yky

)()sin()cos1( 0000 xfkxkxykxykxyyy +−′′′+−′′+′+=

+−′′′+−′′+′=−′′′+−′′+′=

)()sin()cos1(

)sin()cos1(

000

000

xfkxkxykxykxyy

kxkxykxykxyy

II

I

1

0

1

[ ( 1 cos ) sin ](1 cos )(1 cos )

cos sin

( sin cos 1)(1 cos ) sin ,

cos sin

y

x

х

P kl kl kl kly kl

kP kl kl kl

М kl kl kl klkl

Р kl kl kl

α α α− + − − ′ = − − − + ⋅ −

+ − − + + −

1

0

1

( 1 cos ) sin sin cos 1

cos sin cos sin

y

x х

P kl kl kl М kl kl kly

kP kl kl kl Р kl kl kl

α α α− + − + − ′′′= − ⋅ − −

kxkykxkyyII

sincos 2

0

2''

0

'' ′′′−′−=

''

2

bx

MEJ

y=

''

2

cosbMEF

y

α=

'' ''

2 2( ) ; ( ) cosb bEI y M x EF y M x α⋅ = ⋅ = ⋅

EF

eP

EI

yPy

xy ⋅+

⋅−=′′ 12

0

)(sin)(

)(11

axkkP

P

P

axPxf

x

y

x

y −+−

−=

[ ]1 1

1 1 1

1 1

( ) (1 cos ) (1 cos ) [ (1 cos ) sin ]

(1 cos ) ( sin ),

( ) ( ) ( ) sin ( ).

у

I

x x

у

x x x

у у

II I

x x

P My x A kl kl x B kl kl kx

P P

P AM M Bkx kx kx

P P P

P Py x y x x a k x a

P k P

α

= − − − − + − − −⋅ ⋅ − − + − −

= − − + −

⋅


where ( 1 cos ) sin

cos sin

kl kl klA

kl kl kl

α α α− + −=−

; sin cos 1

cos sin

kl kl klB

kl kl kl

+ −=−

.

On the base of the system numerical decisions (1) and calculation scheme deflections and puller shaft processing accuracy

were calculated, the results were represented in table 3 and on figures 5, 6.

Figure 6. The manner for accuracy changing of technological process functioning of the puller turning of cotton-harvester machine device.

Table 3. The results of calculation of deflections and puller shaft processing accuracy.

Т, s 0y , µm Iy , µm II

y , µm defy , µm

0 -97.22 32300 32300 31300

0.1 0.018 -6.38 -6.38 -6.36

0.2 0.0585 -7.71 -7.71 -7.655

0.3 -0.06295 -3.67 -3.67 -3.7335

0.4 0.09134 -8.8 -8.8 -8.714

0.5 0.00263 -5.853 -5.853 -5.85

0.6 0.4888 -7.392 -7.392 -7.3436

0.7 0.0628 -7.856 -7.856 -7.7935

0.8 0.04888 -7.392 -7.392 -7.3436

0.9 -0.1133 -1.994 -1.994 -2.108

1 -0.21978 1.5483 1.5483 1.32854

7. Conclusion

The influence of inertion moments and elastic and

dissipative forces to changing of in-processing shaft moving

was investigated. Changing of TS front and back beams

inertion moment influences essentially to in-processing shaft

corner speed sand accelerations. The variation of in-

processing shaft coefficients of rigidity, sticky resistance and

stretching forces was performed to reduce the range of corner

speeds and accelerations changing. Rigidity increase at the

expense of stretching lead to reducing of in-processing shaft

deformation and reducing of transition process. The

amplitude of corner speeds fluctuations is reduced

considerably by the increase of in-processing shaft sticky

resistance coefficient. It confirms that the amplitude and

frequency of in-processing shaft corner speeds and

accelerations fluctuations depend on the inertion moment and

elastic and dissipative forces. Thus, corresponding meanings

of driving moment, stretching force and cutting forces

moments were determined for set meaning of in-processing

shaft inertion moments of rotating masses and rigidity and

sticky resistance coefficients.

References

[1] Rigidity and vibration by turning. http://tehnoline.ru/files/ theory/ turning/1-1-1.htm.

[2] Processing by cutting. GOST25762-83/ http://www.tehlit.ru/ Pages/43670.htm.

[3] Soloviyov V. V. Providing of the accurate processing by using of technological processes math models “Engineering investigations”, 2004. №2. P.100−104.



[4] Turning processing. http://www.bibliotekar.ru/spravochnik-54/13.htm

[5] Azimov B. M., Sulyukova L. F. Math modeling and optimal control of small rigidity details technologies processing//The reference book. Engineering journal. М.: Mechanical engineering. 2012. №3. P.45−51.

[6] http://osntm.ru/cp_xp_yp_np.html.

[7] http://osntm.ru/kfip_kgp_krp.html.

[8] Kobakova M. Yu. Applied mechanics: a text book for students. SAFU, 2014. −130 pages.

[9] Kravchenko S. A., Nabikin A. U., Birukov V. P. Improvement of efficiency of the automated control system of non grid shafts surface contour in lathing. №3(34), 2012. Samara aerospace university. 2012. P.339−348.

[10] Vasiliev F. P. Numerical methods of extreme tasks solutions. М.: Science,1988. Pages 421−485.

[11] Azimov B. M., Sulyukova L. F. Calculation of the metal cutting and longitudinal tensile forces components and optimal control of grinding processing of a small rigidity shafts// Uzbek journal of the problems of informatics and energetic. T.: Science and technology. 2015. №3-4. P.64−73.

the calculation and optimal control by the technological...

Documents