EXPERIMENTAL STUDIES FOR DEVELOPMENT HIGH-POWER AUDIO SPEAKER DEVICES PERFORMANCE USING PERMANENT NdFeB MAGNETS SPECIAL TECHNOLOGY

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Fiabilitate si Durabilitate - Fiability & Durability Supplement no 1/ 2013 Editura “Academica Brâncuşi” , Târgu Jiu, ISSN 1844 – 640X

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METALLOGRAPHIC MICROSTRUCTURE OF THE BONDED WELD... 278 42. Cristina IONICI - BREAKING MODEL OF SINTERED STEELS... 283 43. Cristina IONICI - THE MODEL CRACK GROWTH OF INTER-PORES IN SINTERED STEELS... 287 44. C t linăIANCUă- ABOUT COSMOS/M ANALYSIS CAPABILITIES FOR NONLINEAR MATERIALS 291 45. Stefan IANCU - TRENDS IN PRODUCT DEVELOPMENT: CONCURRENT ENGINEERING AND

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CONSUMPTION IN WATER ELECTROLYSIS REACTION... 369 58. Aurel George POPESCU - AGGRAVATING FACTORS OF AGGING AND WAYS AGAINST THEIR 372 59. Dan DOBROTA, Gheorghe AMZA - DETERMINATION OF THE OPTIMUM LEVEL TO

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AND INTERPRETATION OF A MANUFACTURING SYSTEM………..……… 402 65. Marin NEACSA, George ADÎR, Victor ADÎR, Ancuta ADÎR - ABOUT USING OF THE THEORY

OF THE STOCHASTIC SYSTEMS WITH FINITE STATES IN THE STUDY OF SAFETY OF THE MECHANICAL SYSTEMS………..……… 411 66. Iuliana Carmen B RB CIORUă- A NUMERICAL EXAMPLE TO DETERMINE THE SUPPLIER

PERFORMANCE USING DEA AND AHP PIECEWISE TRAPEZOIDAL FUZZY ... 418 67. Viorica Mariela UNGUREANU - EXISTENCE AND UNIQUENESS OF THE SOLUTIONS FOR A

CLASS OF STOCHASTIC DIFFERENTIAL EQUATIONS WITH INFINITE MARKOVIAN JUMPS 424 68. Valeria Victoria IOVANOV - DEVELOPING A MATHEMATICAL MODEL FOR THE PROCESS

OF PURIFYING INDUSTRIAL RESIDUAL WATERS IN THE SEDIMENTARY TANKS... 431 69. Valeria Victoria IOVANOV - THE ASSESSMENT OF THE AMOUNT OF SOLID PARTICLES IN

THE SEDIMENTARY TANKS AND THEIR EFFLUENT... 433 70. Miodrag IOVANOV - AN EXTREMAL REGION FOR UNIVALENT FUNCTIONS... 436 71. Olimpia-Mioara PECINGINA, Constantin BOGDAN - PSEUDO ALGEBRAS BOOL APPLIED

BL-ALGEBRAS... 440 72. Olimpia-Mioara PECINGINA, Constantin BOGDAN - FILTERS AND IDEAL BOOLE

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73. M d linaă Roxanaă BUNECIă- USING MAPLE TO REPRESENT THE SUBGROUPOIDS OF TRIVIAL GROUPOID X×Z×X... 446 74. Constantin BOGDAN, Olimpia-Mioara PECINGINA - MV-ALGEBRAS AND BL-ALGEBRAS….. 455 75. Constantin BOGDAN, . Olimpia-Mioara PECINGINA - PSEUDO MV-ALGEBRAS SI PSEUDO

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5

CALCULATION OF SURFACE ROUGHNESS PARAMETERS FOR

EXTERNAL CYLINDRICAL GRINDING

Yuri NOVOSELOV1,Sergey BRATAN2, Vladimir BOGUTSKY3,

Yury GUTSALENKO4

1,2,3_{Sevastopol National Technical University, Ukraine,}

4_{National Technical University}_―_{Kharkov Polytechnic Institute}_‖,ă_Ukraine 1,2_Professor,3_{Associate Professor,}4_{Senior Staff Scientist}

1,2,3_{[email protected],}4_{[email protected]}

Abstract: The method of calculating the surface roughness parameters for an external cylindrical

grinding is considered. The offered calculationrelations are taken account of production mode parameters, the grain size of the grinding wheel and the change of state of the tool working surface during machining. This allows to assess the impact of multipass grinding process and to predict the kinetics of changes in surface roughness.

Keywords: cylindrical grinding, machining surface, roughness calculation

1. INTRODUCTION

The main parameters of quality of the processed surface is its roughness and depth of the defective layer. Usually the determination of roughness parameters is added up to tabulation of the profilogram and further calculations in the tables, for example, with the help of computer.

The processes of grinding have a complex stochastic nature, which leads to disorder of indicators of quality of products and does not allow to use all possibilities of finishing methods. Microrelief of grinded surface in the workpiece material is a combination of mappings of the transient surfaces which are formed by the movement of cutting edges in the space of the workpiece. Forms of unit scratches are determined by the forms of cutting edges and the peculiarities of their contact with the material surface.

Analytical relations for definition of the most important parameters of a surface roughness, under the condition that the describing the ordinate random process is stationary and normal, are obtained in works of Yu. Vitenberg, A. Husu, Yu. Linnik and a number of other researchers. Roughness parameters were calculated using the correlation functions. The form of the function was considered well-known, and its coefficients are determined on the basis of experimental studies of grinding process.

Principles of forecasting the most important parameters of a surface roughness depending on technological factors are considered in papers [1, 2]. In [2], where the calculation of roughness parameters is made on the basis of functionals obtained in the theoretical analysis of the processes of forming surfaces, known relations are considerably refined taking into account influence of the processes occurring in a dynamical system.

The developed approach is presented first of all applied to a one-dimensional evaluation of average roughness (arithmetic mean deviation of the profile) Ra which is the main in the

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Organization for Standardization (ISO 4287:1997; in Ukraine it is also DSSU ISO 4287:2012), the American Society of the Mechanical Engineers (ANSI / ASME B46.1-1995), the Russian Federal Agency on Technical Regulating and Metrology (GOST R 25142-82) and other leading national and international subjects of development a supranational technological structure for economic progress of modern civilization. Objects of attention of the fulfilled elaboration are also widely used in the international and national practice such one-dimensional roughness amplitude estimates as profile maximum peak-to-valley height R_max

and profile peak-to-valley deviation by ten points R_z. In accordance with a certain preference

of Ra_{parameter to use for roughness estimate (GOST R 2789-73, etc.) its consideration is the}

main in the work performed.

2. BASIC RELATIONSFOR Ra CALCULATION

Arithmetic mean deviation of the profile Ra is calculated [2] as:



     _n i m g э g и k с u u a r i W D n V V K H V R 0 2 / 3 2 2 / 3 ) ( ) ( 2  

when  r W_m; (1)

0,4 0,6

0,4 0,4 0,4 0,2 0,2

0, 25

( )

u f a

с к и g э g

V t R

K V V n D 



 when  r Wm, (2) where Wm – is the distance from the deepest profile point to the middle line of the profile which is calculated from the condition of the y_m 0, P M( ) 0,5 ,

0

( ) ln 2 0

n

k m

i

G W i r



   

 . At

the value of radial metal removal  r Wm

1 ln 2 m k W G      

  , where ( 1) ( ) ( )

( 3 / 2)

э c b к u g

k

u u

D m K C V V n

G

m V H

  



   



   . (3) With private values m0,5, 1,5 the relation of the (3) takes the form:

1,5

0,598 _g _{э c к}( _u) _g

k

u u

D K V V n G

V H

 

 ; (4)

; ) ( 66 , 13 478 , 1 2 g э g u k c u f f D n V V K V t t r    

 (5)

 

2 13,66

0,739 0,546 ,

( )

u f

c k u g e g

V r

t r r

K V V n D



    

 (6)

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7

overstating along the scratch edges); ng – the number of grain vertices on the unit of the surface of wheel working layer; H_u –the value of the layer of the wheel working surface in depth for calculation of the ng number of abrasive grains; P M( ) – the probability of material removal;m  – indices of the power characteristic; _g – radius of rounding for the top of

abrasive grain; _Vk –speed of grinding wheel; V_u – speed of workpiece; D_e – equivalent diameter; r – radial removal of material from the workpiece surface.

The structure of equations (1) and (2) and the value of indicator of the degree are similar to exponential function existing in the literature, but unlike them, they reflect the physical nature of the process of forming and correspond to the dimensional theory.

3. BASIC RELATIONSFOR Rmax AND Rz CALCULATION

Profile maximum peak-to-valley height R_max_{and profile peak-to-valley deviation by ten}

points Rz are calculated on the depth of the layer in which the surface roughness is distributed (R_max) and mathematical expectations of the distances from the upper boundary of layer up to

five highest points of the profile and the distances from the lower boundary of layer up to the five lowest points of the profile (Rz).For a stationary process, which is close to normal, we can be considered that the distances from the upper boundary of roughness layer to the most protruding tops of the profile are distributed according to the laws similar to the distribution of the distances from the hollows to the lower boundary of roughness layer. In this case the mathematical expectation values of Rmax and Rz parameters are defined [2] as

3/2 max

2

[ ] 2

3 ( )

u f

g k u э

V t M R H

n V V L D

 

 ; (7) 3/2

[ ] 2,95

( )

u f z

g k u э

V t

M R H

n V V L D

 

 . (8) where H t _f  r – value layer of surface roughness (the size of the transition area between the material and the environment).

4. STATEMENT OF THE RESEARCH PROBLEM

One of the main parameters of the tool working surface, which is large extent influence the characteristics of roughness of the workpiece processed surface, is the rounding radius of the grain top _g. According to D. Wakser [3], G. Ippolitov [4] and other researchers [5, 6], radius at the top of the grain depends on the material of abrasive grain, method of production, grain size, mode of tool dressing.

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plane which is perpendicular to the vector of the cutting speed, and there is a blunting of the abrasive grain.

However, according to the above exhibited (1), (2), (5)-(8) relations for the calculation of R_a, R_max and Rz_{roughness parameters does not take into account the transformation} process of the cutting part of the abrasive grain during grinding.

Considering these relationships as a base with reflecting the work of abrasive tools in some initial state, for example, after a pre-dressing, we'll enhance their taking into account changes of the radius of the grain rounding and state of the working surface of the tool during its operation.

5. TO PROVIDE AN IMPROVED RELATIONS FOR Ra, Rz AND Rmax CALCULATING

In the general case it can be write that

0

( ) _g ,

g K g

    (9) where K__g – coefficient acceptant into account change of rounding radius of grain in the

process of work of the abrasive tool; _g₀_– the initial rounding radius of the grain top.

Таbleă1: Initial radius of rounding tops of abrasive grains _g₀

The authors

The granularity according to GOST R 3647-80 and ISO 8486-1,2μ1λλ6(Е)

16 25 32 40 50 63 80 100 125 160 200 F80 F60 F54 F46 F36 F30 F24 F20 F16 F12 F10

The basic size of abrasive grains Bg, µm

160 240 315 400 500 630 800 1000 1250 1600 2000 The initial radius of rounding tops of the grains g₀, µm

A. Baykalov [7] 13 19 – 28 – – – – – 114 – E. Maslov [8] 11 17 25 – 41 – – 76 – – – A. Murdasov [6] – 19 – 30 – – 68 – 97 115 130

D. Wakser [3] 14 21 – 30 – – – – – – –

A.Korolev [1] 12 – – – – 48 – – 93 119 149

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The experimental dependence obtained on the basis of data given in Tab. 1 has the form:

0

0.955

0.0535

g Bg

   , (10)

where B_g – the basic size of abrasive grains to GOST R 3647-80 and ISO 8486-1,2:1996(E), m.

Approximation of a power-law dependence was carried out by the least squares method.

In the Tab. 2 it is shown the comparison of the mean values of the experimental data in Tab. 1 and the values calculated by the formula (10). Graphically this comparison is shown in Fig. 1. Check on the coefficient of correlation and the Fisher criterion showed the adequacy of the proposed dependence (10).

Таbleă2: Comparison of experimental and calculated values of the rounding radius g₀ of the grain tops

Source

The granularity according to GOST R 3647-80 and ISO 8486-1,2μ1λλ6(Е) 16 25 32 40 50 63 80 100 125 160 200 16 F80 F60 F54 F46 F36 F30 F24 F20 F16 F12 F10 F80

TheăbasicăsiгeăВg of abrasive grains, µm

160 240 315 400 500 630 800 1000 1250 1600 2000 160 The rounding radius g₀ of the grain tops, µm

The average value of the experimental data in Tab. 1

12.6 19 26 29 39.5 48 64 76 95 115.3 139.5 12.6

The

calculating by (10)

12.8 19.4 24.5 30.7 38.1 47.6 59.6 74.3 92.4 115.4 143 12.8

With the account of (10) dependence (9) takes the form

0

0,955

( ) _g 0,0535 _g ,

g K g K Bg

       (11) As shown in [2], for any point of the profile of the abrasive grain (Fig. 2) the radius of curvature in the polar coordinate is calculated by the equation:

Figure 1: Comparison of the calculated (1) and experimental (2) dependences between a radius _g₀ at the

(10)

10

3

2 2 2

2 2

( , ) ( , )

( ) .

( , ) 2 ( , ) ( , ) ( , )

g g

g

g g g g

R R

R R R R

                      

  (12)

When combining of the pole of the polar coordinate with the center of curvature of the top of the grain, for angles in neighborhood of _0, the radius-vector of the initial profile is g₀, and its current value is

0

( , ) (1 ( , )) ( , ),

cos

g g

H u

R _ _  _



        

 

  

where _ – polar angle of the points of the profile of the grain top; u - distance from maximum hollows of the

initial profile to the center of curvature of the initial profile of the grain top.

The current radius of the grain top is calculated on the current radius-vector and its first and second derivatives by equation (12) with _ 0:

0 0 0 2 ( ) ( ) , A g A g g g B Be e A BA         

  (13) where _A h V0( k Vu) _; _{B H u} _.

H 

 

  

The coefficient K__g acceptant into account change of rounding radius of grain in the

process of work of the abrasive tool can be represented as

0 ( ) , g g g

K_   

 or after the

conversion:

0( ) 0( )

0.955 2

0,955 0.955

0 0

18,692 (0,0535 ( )(1 ))

,

(0.0535 (1 ( ) ) ( )( ) )

k u k u

g

h V V h V V

H H

g

g k u k u g

H B H u e e

K

B h V V h V V H u B

           _             

     (14)

where h0 – is the relative depreciation of the abrasive material;  – time of work of the

abrasive tool.

Figure 2: Scheme for the calculation of change of the contour of the

(11)

11

In Fig. 3 it is shown the graphics allowing to evaluate the impact time of the work of grinding wheel on the change the radius of rounding the top of the abrasive grain.

Depending on the number ng of grains per the unit of of the grinding wheel included in (1), (2), (4)-(8), also in many respects is defined by the basic size B_g of

abrasive grains. At the same time, the existing experimental data show about a substantial change of the number of cutting edges for the period of the durability of the tool. Some portion of the abrasive grains will be destroyed or to removed from the grinding wheel for each contact with the processed material due to the limited strength of abrasive grains and their fastening in the tool. At the same time new cutting edges lying in the deeper layers of the tool will come into operation. Therefore, in general case, it can be wrote

0

( ) _g

g n g

n  K n (15) where K_n_g– is the coefficient acceptant into account the change in the number of abrasive

grains on the surface of the wheel in the period between dressings; ng₀ – the initial amount of

abrasive grains on the working surface of the wheel.

At [10] the initial quantities of abrasive grains on the surface of the grinding wheels

0

g

n , 1/ 2, were determined with the account of the content V_g% of abrasive grains in the

wheels, the basic size B_g of abrasive grains according to GOST R 3647-80, structure and

hardness (V_g  45% for grinding wheels with the structures of 5...6 and hardness [4]), and implemented by the approximation of the method of least squares that allowed to obtain the dependence of:

0

g

n 0,62·Bg–1,99. (16)

Таble 3: Comparison of the calculating values of the initial amount ng₀ of abrasive grains

Source of calculated values

The granularity according to GOST R 3647-80 and ISO 8486-1,2μ1λλ6(Е) 16 25 32 40 50 63 80 100 125 160 200 F80 F60 F54 F46 F36 F30 F24 F20 F16 F12 F10

The basic size Вg of abrasive grains, µm

160 240 315 400 500 630 800 1000 1250 1600 2000

Figure 3: Impact time  of the work of grinding wheel on the change the radius  _g( ) of rounding the top of the abrasive grain for different values of

(12)

12

The amount ng₀ of abrasive grains, 0 2 1 , g

n

mm

[10] 23.2 9.2 5.7 3.56 2.28 1.44 0.89 0.57 0.366 0.224 0.144 (16) 22.4 9.4 5.6 3.57 2.29 1.44 0.89 0.57 0.369 0.226 0.145

In the Tab. 3 it is given a comparison of the number of grains per

1 mm2 calculated according to [10] and the calculating values by formula (16). Graphically this comparison is shown in Fig. 4. Check on the coefficient of correlation and the Fisher criterion showed the adequacy of the proposed dependence (16).

With the account of (16) the formula (15) takes the form

–1,λλ

0,62· ·

( ) _g .

g n g

n   K B

(17) In work [2] it is obtained the dependency which allows to calculate the change in the number of grain for the period between dressing of the abrasive tools:

0

( ) g g (1 ) k

g g p

p p

z z

n n P

P P

 

  _  _ 

  , (18) where zg – is the number of abrasive grains

that are entering in the work at the contact i

of the tool with the surface; P_p – probability the destruction of grain; v_k – frequency of

rotation of the grinding wheel;  - work time work after dressing.

In the general case, zg depends on the

number ng0 of grains on the surface of the

instrument after dressing, law the distribution of the grain in depth of grinding wheel, radial wear of grinding wheel, durability of fastening of grains and cutting forces arising in the zone of contact, which are random variables. So, if the load on the top of the grains during grinding does not exceed 4 N, then the probability P_p of extraction of grain out off the bond does not exceed 0.01. With the increase of load probability P_p is growing: for

Figure 4: Comparison of the dependences between the size Вg (grit) of abrasive grains and the number of grains per 1 mm2_surface of grinding wheel

0

g

n : 1 – the results of calculations by (16); 2 - according to [10]

Figure 5: The influence of the work time  of the grinding wheel on the change in the number

) (

g n

(13)

13

8 z

P  Í _{the probability}P_p 0.20, at Pz 10Í Pp 0.50. With the further Pz increase Pp

probability is approaching to its maximum value of about 0.87 (P_z 15Í ) [11].

The coefficient Kn_g acceptant into account the change in the number of grains on the surface of the instrument in the process of its work can be represented as

0 ( ) g g n g n K n 

 or after the conversion with the account of the dependencies (16) and (18):

1,99 1,99

(1 (1 ) 0,62(1 ) 1,613

k k

g

g p p

n g

p g

z P P

K B P B                 

. (19)

In Fig. 5 it is shown the curves of the influence the time of work on the change in the number of abrasive grains ng per 1 mm2 of the working surface of the grinding wheel under

its work in the mode of blunting.

b

Figure 6: The influence of the time  the work of the grinding wheel on the a

R _{( ) and}R_z (b) parameters of roughness; B_g320мкм

Equations (1), (2), (5)-(8) for the calculation of the characteristics of surface roughness will take the following form considering the obtained dependences (11) and (17):

1,5 3.025 1,5 0 1,017 ( ) ( ) g g u u a _n

с n k и g э m

i

V H R

K K V V K B_  D W i r



 _  

when  r Wm; (20)

0,4 0,6 0,605

0,4 0,4 0,2 0,4 0,2

0,544

( )

g g

u f g

a

с n к и э

V t B R

K K K_ V V D 

 when r Wm, (21)

where 2 1,51 ; 95, 254 1, 478 ( ) g g f u g f

c n k u ý

t r

V B t

K K V V K D_

    (22) 1.51 2 22.03

0,739 0,546 ;

( )

g g

u g

f

c n k u э

V rB

t r r

K K V V K D_ 

    



(14)

14

1.5 1.99 max

[ ] 2,074

( )

g

u f g

n k u э

V t B

M R H

K V V L D  

 ; (24)

1,5 1,99

[ ] 3,747

( )

g

u f g z

n k u э

V t B

M R H

K V V L D

 

 .

(25)

In Fig. 6 it is shown curves illustrating the influence of time of work of grinding wheel on the parameters of a roughness of the processed surface.

6.CONCLUSION

Feature of the obtained equations (20)-(25) is that the calculations take into account the parameters cutting mode, the grain size of the grinding wheel, as well as operational change of the working surface of the instrument. It allows to estimate influence on the roughness parameters of the large number of passes of abrasive grains on the surface of the workpiece under a multistep grinding process.

The proposed relations allow to predict the kinetics of changes of roughness parameters. In equations (24) and (25) implicitly includes the likelihood of removal of material, which is calculated with taking into account the roughness of the workpiece and it changes with every contact the surface of the workpiece with the instrument.

REFERENCES

[1] KorolevA. V., Novoselov Yu. K. Theoretical-probabilistic foundations of abrasive processing. – In 3 parts. – P. 1: State of the working surface of the tool. – Saratov: Publishing house of the Saratov University, 1987. – 160 p. – In Russian.

[2] Novoselov Yu. K. Dynamics of formation of the surfaces in the abrasive processing. – Sevastopol: Publishing house of the SevNTU, 2013. – 304 p.– In Russian.

[3] Wakser D. B. The influence of the geometry of the abrasive grain on the properties of grinding wheel // The main issues of the high-performance grinding / Ed. by Е. N. М slov. – Moscow: Mechanical Engineering State Publishing House, 1960. – PP.94-126. – In Russian.

[4] Ippolitov G. M. Abrasive-diamond processing. – Moscow: Mechanical Engineering, 1969. – 334 p. – In Russian.

[5] Kremen Z. I., Yuriev V. G., Baboshkin А. F. The technology of grinding in mechanical engineering. – St. Petersburg: Politechnika, 2007. – 424 p. – In Russian. [6] Murdasov A. V., Vulf A. M. Features of operation of grinding wheels with abrasive

grains of different forms // Abrasives and diamonds. – Moscow: Research Institute of the Mechanical Engineering, 1967. – No. 4. – PP. 65-69. – In Russian.

[7] Baykalov A. К. Introduction to the theory of grinding of materials. – Kiev: Scientific Thought, 1978. – 207 p. – In Russian.

[8] Maslov E. N. The theory of grinding of metals. – Moscow: Mechanical Engineering, 1974. – 400 p. – In Russian.

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[10] Abrasive and diamond processing of materials: Reference book / Ed. by А. N. Реznikov. – Moscow: Mechanical Engineering, 1977. – 391 p. – In Russian. [11] Krutikova A. A., Danilenko M. V. Probabilities of wear types for grains of abrasive

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16

KINEMATICS

OF

MATERIAL

REMOVAL

AND

FORMING

OF

SURFACE

AT

GRINDING

Feodor NOVIKOV

Kharkov National University of Economics, Ukraine

Professor

[email protected]

Abstract: The mathematical model of kinematics of material removal and a forming of surfaces is

developed at grinding. Conditions of increase of productivity of processing are defined and new kinematic schemes of high-performance grinding are offered.

Keywords: grinding, wheel speed, processing productivity, thickness of cut

1. INTRODUCTION

Grinding is one of the main methods of machining of the materials, providing high rates of accuracy, quality and productivity of processing [1, 2]. At the same time, its technological capabilities fully aren't used due to the lack scientifically reasonable recommendations about a choice of optimum modes of cutting taking into account strength properties of a working surface of a wheel. In a special measure it belongs to diamond grinding which owing to unique cutting properties of diamond has considerable reserves of increase of productivity of processing when providing requirements for quality and accuracy of processed surfaces [3]. It is important to know first of all regularities material removal and formation of surfaces at grinding at the level of kinematics of microcuts for identification of realization of potential opportunities of process of grinding. It will allow to develop high-performance kinematic schemes of grinding and to define optimum modes of cutting. Therefore the purpose of work is development of mathematical model of kinematics of material removal and a forming of surfaces at grinding.

2. MATHEMATICAL MODEL AND THE SETTLEMENT SCHEME THE GRINDING PROCESS

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17

 

                           



  2 0 2 5 3 2 3 2 1 n i / iT nT pr c n t t y V b V k tg exp y  

 , (1)

where k- superficial concentration of grains of a wheel, piece/ m2; b - the maximum height of a protrusion of tops of grains over level of a linking of a wheel, m; 2 - corner at top of cone-shaped cutting grain; V_с, V_pr - according to the speed of a wheel and preparation, m/s;  1/R_c1/R_pr; R_c, R_pr - respectively radiuses of a wheel and preparation, m; t_iT t_T it - cover coordinate at i contact with a wheel, m;



n



t t

t_n_Т  _T  1  - cover coordinate at ncontact with a wheel, m; t_T - coordinate of the current infinitely thin cover by which the removed allowance is conditionally presented, to m; n - number of passes of a wheel.

The basis of the developed mathematical model of grinding is made by the analytical decision on the description of border of completion of dispergating by cutting grains of the material brought in a zone of cutting, along an arch of contact of a wheel with preparation: 3 2 0 2 0 2 5 2 5 3 6 3 2

2







         n i n i / iT / iT nT max

nT t t

t H t H extr extr

, (2)

where H_max - the maximum thickness of a cut, m; t_nT_extr - coordinate of extreme provision of a cover at which the full profile _n

 

y =0,95 is formed at level H_max, m; t_iT_extr - coordinate of extreme provision of a cover at i_{contact with a wheel, m.}

As shown in fig. 1, this line is drawn on tops of microroughnesses of a processed material, has a difficult configuration, connects a processed surface with processed and by analogy to blade processing determines the provision of a conditional probabilistic surface of cutting at grinding. Characteristic points of border are a basis for calculation of physical and technological parameters of grinding (the maximum thickness of a cut, a roughness of the processed surface, the actual length of contact of a wheel with preparation, etc.). It allows from uniform positions quite unambiguously analytically to describe regularities of process of grinding in all possible range of change of depth of grinding (including ranges of multipass and deep grinding).

c R c V pr R pr V 1 2 3 4 5 H t О 2 О 1 О max H max R b 1 T t

. ..

_.

.

. ..

.

. ..

. . ..

. .

_.

_{. .. .}

..

...

.

..

.

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18

3.MAXIMUM THICKNESS OF A UNIT CUT AND MICROGEOMETRY OF GROUND SURFACE

By calculations it is established that the provision of border is defined by a ratio of two parameters – maximum (given probabilistic) thickness H_max of a cut and grinding depth t. In a case t < H_max (multipass grinding) the border accepts approximately a symmetric form of rather axial plane of grinding, in a t>H_max (deep grinding) – an asymmetric form.

By calculations it is established that the percent of working grains for a case

t>H_max; makes about 50%, and for a case t<H_max – 5ă…ă10%ă(i.e.ăgrainsăpassăalmostă"aă trace in a trace" that as it will be shown above, is an important reserve of increase in productivity of processing).

In a case t>H_max analytical dependence for definition of provision of border assumes a simple air

6 t t H H T max

 , (3)

where t_T– the coordinate of the current elementary (infinitely thin) cylindrical cover by which the removed allowance is conditionally presented in the settlement scheme, m.

Respectively, parameters of border H_max and R_max (parameter of a roughness of processing, m) are described by analytical dependences

33 0 5 0 5 0 3 630 , c , , pr

max _m _V

t V X

H _

           

   ; (4)

4 0 5 0 3 10 , c , pr

max _m _V

V X

R _

          

   , (5)

where X - granularity of a wheel, m; m- volume concentration of grains in a wheel (dimensionless size).

Table 1: Calculated values of thickness of a cut H_max (basic data: R_pr=8010-3_m; c

R =15010-3_m;

X =0,22510-6 _m;_m_=100; c

V =30 m/s; V_pr=1 m/min; t=0,110-3_m)

Authors F. Novikov E. Maslov G. Lurye A. Reznikov Experiment data max

H , к 14,7 0,007 0,12 1,1 10,5

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19

here H_max (to 40%) is connected with that in kinematic model of process of grinding wear of grains of a wheel isn't considered. For the purpose of specification of the received results settlement dependences which contain the new dimensionless parameter

Н

/

х



 definingădegreeăofălinearăаearăofăgrainsăandăchangingăаithină0ă…ă1ă(foră"sharp"ă grain 0, for become blunted 1) are established:

_

_





33 0 2 5 0 5 0 3 1 1 630 , c , , pr

max _m _V

t V X H                       

; (6)



_



_

4 0 5 0 3 2 1 1 10 , c , pr

max _m _V

V X R                     

. (7)

Taking into account parameter  (>0) values also decrease. Therefore, decreases (and even it is eliminated) a divergence between calculated and experimental values

max

H (tab. 1). Comparison of experimental values of the maximum thickness of chip with the corresponding calculated values of parameter H_max showed their approximate coincidence at =0, 2 (tab. 2). It follows from this that the accounting of size of linear wear of grainx (by means of parameter) in settlement dependences allows to bring the theory and practice of grinding into accord.

Table 2: Calculated values H_max and experimental values of the maximum thickness of chips in mm (m=100; X =0,210-6_m;

с

V =30 m/s; R_pr=0,02 m; R_с=0,15 m)

No.

Grinding modes  _Maximum

thickness of chip, mm t, mm Vpr, m/min 0 0,2 0,5

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20

4. STUDY, DISCUSSION AND PROSPECTS FOR THE IMPLEMENTATION OF GRINDING PRODUCTIVE POSSIBILITIES

From all the entering into dependences (6) and (7) parameters, the greatest influence on H_max and R_max  renders. It indicates a prevailing role of size  in formation of the key physical and technological parameters of grinding and confirms the made hypothesis of effective management of grinding process on the basis of size regulation . For its realization the greatest possible productivity of processing caused by strength properties of a working surface of a wheel is determined, i.e. at the fixed (limit) area of cross section of a cut by separate grain



_



_

         1 1 5 0 2 max H ,

S and =0:

                 



  2 0 2 5 3 3 ₂ 450 2 n i / iT nТ max c _t t H t X В V m Q 

 , (8) where B - width of a wheel, m.

Values of the parameter defining percent of working grains, are given in tab. 3.

Table 3: Calculated values  at H_max=1010-6_m

t10-6,ă 1 5 10 50 100

,% 6,65 15,25 43,19 44,0 44,0

It is theoretically established that generally processing productivity Q with increase in depth of grinding t changes on extreme dependence, passing a point of a minimum (fig. 2). Preparation speed V_pr Q/ Bt thus continuously decreases. It is proved that depth of grinding Qis equal in a minimum point t to parameterH_max. It corresponds to transition from the scheme multipass (tH_max) to the scheme of traditional deep grinding (t H_max).

From the physical point of view the minimum of productivity of processing Q

under a condition S const is caused by existence of the shortest on the chip length l

(considering productivity of processing proportional Sl ) as with increase and reduction of depth of grinding t, since value t=H_max, length of chip l increases. In the first case – at the expense of increase in length of an arch of contact of a wheel with preparation, in the second case – at the expense of increase V_pr.

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21

Obviously, for abrasive grinding it is effective since at the expense of action on grains of big loadings the mode of intensive self-sharpening of a wheel is provided and its high cutting ability is maintained. For diamond grinding this condition results in the increased wear of a wheel that, actually, and predetermines low efficiency of application of diamond wheels at high-performance grinding and inexpediency of their use instead of usual abrasive wheels at removal of big allowances.

The received extreme dependence

t

Q defines kinematic conditions of essential increase of productivity the processings consisting in realization of new ratios between parameters t and H_max

(tH_max,tH_max), i.e. in realization of the left and right branches of dependence (fig. 2).

On this basis the new ways of grinding realizing the left branch of dependence are developed Qt. They are based on application of schemes multipass (fig. 3a) and deep (fig. 3c) round external grinding with rather high speed of the preparation close to speed of a wheel; schemes of deep round external grinding by the wheel periphery with rather small speed of preparation and big longitudinal giving (fig. 3e); schemes of deep round external grinding with rather small speed of preparation and additional tangential high -frequency movements of a wheel of big amplitude (fig. 3b); schemes of deep flat face grinding with use of additional high-frequency oscillating motions of a wheel or preparation in the direction, perpendicular to the direction of giving of a wheel (fig. 3d).

It is established that efficiency of grinding in this case is caused by passing of grains almost "a trace in a trace" and possibility of increase H_max at the fixed value S (i.e. the loading operating on grain) that allows message processing with a high speed of a wheel V_с – to 600 m/s and above.

800

600

400

200

0 0,02 0,04 0,06 0,07 0,1 3 2 1 4 G ri n d in g p er fo m a n ce s / m , Q109

Grinding depth t103,m

Figure 2: Dependence grinding perfomance Q on grinding depth t. Basic data: m= 100;

X =0,210-3_m;_B_{= 22,5}_₁₀-3_m;__{=17 m}-1_; c

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22

Application of such conditions will provide increase in productivity of processing by 10 times and more that will well be coordinated with experience of leading machine-tool constructing firms which came for creation of grinders with a speed of wheel up to 300 m/s. Implementation of the offered schemes of grinding assumes development of the new machines providing big speeds of a wheel and preparation. It will allow to change the content of grinding operations cardinally.

REFERENCES

[1] Maslov E. N. Theory of grinding of metals. – Moscow: Mechanical Engineering, 1974. – 319 p. – In Russian.

[2] Lurye G. B. Grinding of metals. – Moscow: Mechanical Engineering, 1969. – 197 p. – In Russian.

[3] Yakimov A. V. Abrasive and diamond processing of shaped surfaces. – Moscow: Mechanical Engineering, 1984. – 212 p. – In Russian.

[4] Novikov F. V. Physical and kinematic bases of high-performance diamond grinding / Tesis for a Doctor's degree. −ăOdessa, 1995. – 36 p. – In Russian.

[5] Ryzhov E. V. Technological methods of increase of wear resistance of machine elements. – Kiev: Scientific Thought, 1984. – 272 p. – In Russian.

1 t .. . . . . . . . . . . .._. . . ._. . . . . . . . . . ._. _. 1 t .. . . . . . . . . . . .. . . . ._. . . . . . . . . . ._. _. 1 2 t t  2 t .. . . . . . . . . . . .. . . . ._. . . . . . . . . . ._. _. 2 t . 1 t . . . . . . . . . . ._. _. . . . . . . . . . . . . . . . _.. . 1 t . . . . . . . . . . ._. _. . . . . . . . . . . . . . . . _.. . 0 V 2 t 1 2 t t  . . . . . . . . . . . . . . . . 0 V 1 t n S c pr V V  1  p S c pr V V  c V V₀

p S pr V pr V pr V pr V pr V c V c V c V c V c V c pr V V  1  p

S V0Vc

с n V S  p S c pr V V  pr p V

S  Sp1

c pr V V  e b d f c V p S p S

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23

SYNTHESIS OF MODEL THE LUENBERGER OBSERVER

FOR EXTERNAL CYLINDRICAL GRINDING PROCESS

Sergey BRATAN1, Denis SIDOROV2, Yury GUTSALENKO3 1,2_{Sevastopol National Technical University, Ukraine}

3_{NationalăTechnicalăUniversitвă―KharkovăPolвtechnicăInst}_i_{tute‖,ăUkraine} 1_Professor,2_{Associate Professor,}3_{Senior Staff Scientist}

1,2_{[email protected],}3_{[email protected]}

Abstract: The problem of diagnosing the actual depth of cut at cylindrical grinding is considered.

A mathematical model of the behavior of the grinding wheel and the workpiece during processing is worked out. According to this model it is produced a synthesis the model of the Luenberger observer with the Kalman filter to control the process of external cylindrical grinding. The developed approach is to improve the accuracy of control and the related with them computational procedures of assessment and management.

Keywords: grinding, control, model, Luenberger observer, Kalman filter

1. INTRODUCTION

The quality of work pieces depends both on the part blank and processing technology, and technological system (TS) properties. Grinding operations research shows that as a rule at the initial time choosing the right characteristics of a tool, cutting mode, optimal grinding cycle assembling provide set-up parameters of accuracy and quality of the detail surface. However, not only the nominal values of these parameters influence the performance properties of the products but their deviations that significantly increase during the production system operation. The presence of abnormalities is the reason of perturbation actions in technological process caused by instability of TS parameters. In addition instability of detail parameters is defined by exposure in the production process of TS of the changing external factors some of which are unknown and aren’tăcontrolledăduringăprocessing.ăThisăproblemăisă particularly acute for finishing operations, where quality parameters of finished products are finally formed and which are the most sensitive to perturbation actions [1].

In the under present-day conditions near 15 ... 20% of finishing operations are carried out by means of external cylindrical grinding. Operations are projected with use of traditional methods not fully accounting the influence of random factors that reduce the stability of manufactured products quality indicators. For quality index stabilization technological modes are assigned on the basis of adverse conditions, for example, the cutting properties renewal of worn grinding wheel is made much earlier than that it requires a real state.

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2. PROBLEM STATEMENT

Thus, for the process of finishing work it is a task of diagnosis as the need to define a number of parameters of the TS in the processing, including those that are not available for direct measurements, what is the purpose of this article. These parameters include, for example, the actual cutting depth, which is largely determined by the laws of material removal and wear of the abrasive tool. Its evaluation task (diagnostics), and therefore a number of other process parameters can be solved using the theory of dynamical Luenberger observers and experience of its application in developments of process control systems of flat grinding [2-4]. According to this experience, besides really existing dynamic object (Fig. 1, the upper part of the scheme), in the scheme of dynamic diagnostic for grinding there are included a dynamic model of the object, the unit of assessments (evaluator), the Kalman filter, the shaping filter, which operates in the mode of the real time.

Interaction between the grinding wheel and work piece is primarily characterized by the parameters of the grinding wheel and work piece form, and their relative position; elastic, damping and other properties of the TS [1]. To develop the grinding wheel and work piece behavior model during machining we are giving the mathematical description of the process that characterizes the interaction between the grinding wheel and work piece.

3.MATHEMATICAL MODEL OF WORKING CONTACT DYNAMICS

As the mathematical model of the grinding wheel can be considered a rotating disk, and circumference in one-dimensional representation. The center of rotation inevitablвă doesn’tă coincide with the center of the shape, which determines the imbalance of the wheel, what is usually explained by the appearance of periodically varying forces generated during grinding. Inăaddition,ătheăsurfaceăofătheăаheelădoesn’tăhaveăanăidealăgeometrв.ăThereăcanăbeădetectedă well-formed and random deviations of form and waviness with amplitude, frequency and phase changing for the tool life period.

Interaction scheme for the external cylindrical grindingprocess has the form shown in Fig. 2.

EXPERIMENTAL STUDIES FOR DEVELOPMENT HIGH-POWER AUDIO SPEAKER DEVICES PERFORMANCE USING PERMANENT NdFeB MAGNETS SPECIAL TECHNOLOGY

CONTENTS

CALCULATION OF SURFACE ROUGHNESS PARAMETERS FOR

EXTERNAL CYLINDRICAL GRINDING



 

KINEMATICS

OF

MATERIAL

REMOVAL

AND

FORMING

OF

SURFACE

AT

GRINDING

 











 

. ..

.

.

. ..

.

. ..

. . ..

. .

.

. .. .

..

...

.

.

..

.

.



























SYNTHESIS OF MODEL THE LUENBERGER OBSERVER

FOR EXTERNAL CYLINDRICAL GRINDING PROCESS

_.

_.

_{. .. .}

_

_

_

_

_

_