On the Optimal Designing of Parameters of Working Platforms of Vulcanizer by the Method of Random Search

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International Journal of Emerging Technology and Advanced Engineering

Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 9, Issue 2, February 2019)

21 On the Optimal Designing of Parameters of Working Platforms

of Vulcanizer by the Method of Random Search

George Filatov

Professor, Doctor of Techn. Sciences, Ukrainian State University of Chemical Technology, Ukraine

Abstract – the article is presented an application of method of random search for optimal designing of parameters of the working platforms of vulcanizer for manufacturing of tires of complete profile. adduced: the calculated scheme of the support platforms of vulcanizer, the formulation of problem of mathematical programming, the results of optimization. Is noted, that the optimal vulcanizer is lighter than serial more than on 25 percent.

Keywords-- Optimal Designing, Random Search Method, Working Platforms of Vulcanizer.

I. INTRODUCTION

As one of the applications of random search method is the optimal designing of the platforms vulcanizing tires, restored at full profile 1-170 MCM GOST 13530-81.

The vulcanizer is designed to restore of casings and tubeless tires of diagonal and radial construction with the outer diameter from 785 to 1100 mm, with internal / landing / diameter from 381 to 508 mm and the width of the profile from 170 to 310 mm by means of the method of new protector using a sectoral and not sectoral molds. Press force on one mold is 1.7 MN. The mold consists of two mold halves mounted on the upper and lower platforms, which perceive the main load occurring during vulcanization. The platforms are the ribbed welded construction. One of the requirements for platforms vulcanizer is their stiffness. Therefore, vulcanizer [1] have an increased mass and require more energy consumption during operation.

As noted above, the mold half is mounted on a platform which perceives the arched effort

P

. The pressure on the platform is transmitted via the support ring, the mean diameter of which is according to design considerations is taken equal mm. The mold may be considered as a beam lying on two supports (Fig. 1).

II. THE CALCULATED SCHEME AND THE FORMULATION

THE PROBLEM OF OPTIMAL DESIGNING

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International Journal of Emerging Technology and Advanced Engineering

22

Fig. 2. Fragment of the calculated scheme of As can be seen from the Fig.2, arc length: support ring of mold

0 0 1

360

1

1 arccos

2

2 ;

360

1 arccos

2

2 











_

















 







R

Z

R

l

R

Z

R

l

_n _n



.

Concentrated forces acting on the beam axis:



























 















_











_ 0 1

360

1 arccos

1

1 arccos

2

2 R

Z

R

Z

Rq

l

q

P

_n _n _n



,

where:

l

P

q



 the value of the distributed load,

P



spacer force;

R



circumference of the support ring of mold;

Z

 ent height.

By setting the values of

Z

, we find concentrated forces n

P

and build the diagram of the bending moments. Calculation on the strength of the supporting platforms of vulcanizer is performed with account the maximum normal stresses. The normal stresses were determined in each of 12 cross-sections, the most typical of which are shown on Fig.3. For each of these cross sections were determined the areas, the static moments of areas, the moments of inertia and the resisting moments.

The condition of strength for i-th cross-section takes the form:



1 ,

2 ,...,

12 

perm







i

W

M

i z i P i



, (1)

where

M

_Pi



M

_maxi  the calculated bending

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International Journal of Emerging Technology and Advanced Engineering

23

The condition of stiffness:

perm 2

2 2 2 1 2 1 1 max

12

12 EI

f

L

M

EI

L

M

f







, (2)

where

f

_perm − the permissible technological

2 1 2 1

,

M

,

I

,

I

M

 respectively maximum bending moments and moments of inertia in two mutually perpendicular cross sections.

 



1 ,

2 ,

3 ,

4 ;

1 ,

2 ,...,

11 

min

1







j

i

F

n

i

j



.

(3)

III. THE FORMULATION OF THE PROBLEM OF

MATHEMATICAL PROGRAMMING AND THE RESULTS OF

CALCULATIONS

As a criterion of rationality was adopted a minimum total area of four characteristic cross-sections of the upper and lower platforms of vulcanizer are shown in Fig.3:

Thus, the problem is formulated as follows [2]: to find such dimensions of thickness of plates and edges of platforms of vulcanizer at which the weight of platforms of

vulcanizer would be minimal, at the performance of the restrictions of strength (1) and stiffness (2). The analog of mass in this formulation is the total area of cross-section of platforms (3).

We introduce the notation

x

_j





_j and we formulate the mathematical programming problem: find a minimum of the function:

 



1 ,

2 ,

3 ,

4 ;

1 ,

2 ,...,

11 

1







j

i

x

F

n

i

j

i

(4)

at the performance of restrictions:

 

x



 



 

x



0 ,



i



1 ,

2 ,....,

12 ;

j



1 ,

2 ,....

11 

g

_i _j



_i _j (5)

0

12

12 )

(

2 2 2 2 1 2 1 1

1

_























EI

L

M

EI

L

M

f

x

g

_i _j (6)

 

x



x



x





0 ,



j



1 ,

2 ,....,

11 

g

_j _j _j _j (7)

 

x



x





x



0 ,



k



1 ,

2 ,....,

11 

(4)

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Fig.3. The main cross-sections of support platforms of vulcanizer The condition of stiffness:

perm 2

2 2 2 1 2 1 1 max

12

12 EI

f

L

M

EI

L

M

f







, (2)

perm

f

− the permissible technological

2 1 2 1

,

M

,

I

,

I

M

 respectively maximum bending

moments and moments of inertia in two mutually perpendicular cross sections.

IV. THE FORMULATION OF THE PROBLEM OF

MATHEMATICAL PROGRAMMING AND THE RESULTS OF

CALCULATIONS

As a criterion of rationality was adopted a minimum total area of four characteristic cross-sections of the upper and lower platforms of vulcanizer shown in Fig.3:

 



1 ,

2 ,

3 ,

4 ;

1 ,

2 ,...,

11 

min

1 







j

i

F

n

i

j

(5)

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25

Thus, the problem is formulated as follows [2]: to find such dimensions of thickness of plates and edges of platforms of vulcanizer at which the weight of platforms of vulcanizer would be minimal, at the performance of the restrictions of strength (1) and stiffness (2). The analog of mass in this formulation is the total area of cross-section of platforms (3).

We introduce the notation

x

_j





_j and we formulate the mathematical programming problem: find a minimum of the function:

 



1 ,

2 ,

3 ,

4 ;

1 ,

2 ,...,

11 

1







j

i

x

F

n

i

j

i (4)

At the performance of restrictions:

 

x



 



 

x



0 ,



i



1 ,

2 ,....,

12 ;

j



1 ,

2 ,....

11 

g

_i _j



_i _j (5)

0

12

12 )

(

2 2 2 2 1 2 1 1

1

_























EI

L

M

EI

L

M

f

x

g

_i _j (6)

 

x



x



x





0 ,



j



1 ,

2 ,....,

11 

g

_j _j _j _j (7)

 

x



x





x



0 ,



k



1 ,

2 ,....,

11 

g

_k _j _j _j (8)

The conditions (7) - (8) represent the geometric restrictions on the area of the search. Thus, the mathematical model of load-bearing structures of vulcanizer (4) - (8), is described in the space of optimized parameters by the closed area of the order of 11, formed by the 35 restrictions of the physical and geometric character, with the non-linear objective function (4), minimum of which is required to find.

For solving of described problem is used algorithm of global random search SGEF [3].

Algorithm is implemented in the form of a computer program, the optimization of parameters of the upper and lower platform of vulcanizer 1-170 MCM was produced

under the following initial data: the bending moments

41 ,

573

1



M

kNm,

M

₂



251 ,

49

kNm: modulus of

elasticity

E



2 

10

5MPa; allowable stress

120

perm





MPa, the permissible deflection

2 perm

0 ,

0758

10







f

m. The boundaries of change of

optimized parameters were adopted the following: 2

2

₅

₁₀

10

[image:5.612.56.556.609.727.2]

1 







_j





 m. The calculation results are given in Table 1.

Table 1

Estimated values of the thickness of plates and ribs of vulcanizer 1-170 GMU

№

The type of

project perm

f

mm

F

cm

2

Dimensions of the elements of cross-sections, mm

1

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26

V. DISCUSSION OF OPTIMIZING RESULTS

Comparison of the results is adopted for the optimal and serial projects. This comparison shows that the optimal design on 26.6% lighter than serial. The stresses in any of the cross sections do not reach the permissible values. The decisive restriction is the restriction on stiffness. In the serial vulcanizer as in optimal, the permissible deflection

was

0 ,

0758



10

2m. Thus, the optimal vulcanizer was as hard as a serial, but less material capacious.

In the Table 1 also are listed the value of the objective function and the size of the thickness of the slabs and ribs of optimal platform vulcanizer at the tighter tolerances on

the deflection

f

_perm. Thus, when

f

_perm=

0 ,

07 

10

2m, i.e. when the stiffness increased on 7.6%, the weight of the optimal platform was on 7.7% less than of the serial

project, and at

f

_perm=

0 ,

06 

10

2m, i.e. at the increasing of stiffness on 20.84% , the mass of the optimal design was on 3.3% less than of serial project.

And only when permitted deflection

perm

f

=

0 ,

05 

10

2m, the optimal project became heavier on 7.3%.

Consequently, in the framework of the proposed mathematical model can obtain more rigid and at the same time less metal capacious bearing structures of platforms of vulcanizer without disrupting their strength.

REFERENCES

[1] Tsyganok I.P. Vulcanizing equipment of factories of rubber industry . − M: Mechanical engineering, 1967. − 324 p.

[2] Filatov G.V. On the question of the rational designing by method of random search the parameters bearing structures vulcanizing tires, restored at full profile // Questions of chemistry and chemical technology. − Dnepropetrovsk. UGHTU. – 2002. − №6. P.136-140. [3] Filatov G.V. The Global Method of Random Search with

Controlled Boundaries of the Interval Parameters to be Optimized. // − International Journal of Emerging Technology & Advanced Engineering, Volume 6, Issue 8, September, 2016, p.p.231-247.