CHANGE IN THE CUTTING DEPTH OF TEETH GRINDING WHEELS IN THE WAY OF THE COPYING AND STABILITY Текст научной статьи по специальности «Медицинские технологии»

CHANGE IN THE CUTTING DEPTH OF TEETH GRINDING WHEELS IN THE WAY OF

THE COPYING AND STABILITY

Rasulov N.M.

doctor of science, Professor, Azerbaijan Technical University Mammadov A.S.

candidate of science associate Professor, Azerbaijan Technical University

Shabiyev E.T.

Azerbaijan Technical University

When the grinding gear teeth by copying the depth of cut on the profile of the tooth turns uneven. It reduces the efficiency of the process. The influence of the technological factors on the cutting depth during the grinding of teeth by the method of copying. A system of equations for determining changes in the depth of cut at any participant evolution tooth profile. Shows the direction of providing a relatively uniform distribution of the depth of cut on the profile of the tooth.

Keywords: cutting depth, profile, copy, error, abrasive wheel

ИЗМЕНЕНИЕ ГЛУБИНЫ РЕЗАНИЯ ПРИ ШЛИФОВАНИИ ЗУБЬЕВ КОЛЕС КОПИРОВАНИЕМ И ПУТИ ЕЕ С СТАБИЛИЗАЦИИ

Расулов Нариман Могбил оглы

доктор технических наук, профессор кафедры Технология машиностроения, Азербайджанский технический университет

Маммадов Арастун Салман оглы кандидат технических наук, доцент кафедры Технология машиностроения, Азербайджанский технический университет

Шабиев Елгюн Тагы оглы ассистент кафедры Метрологии и стандартизация, Азербайджанский технический университет

При шлифовании зубьев зубчатых колес путем копирования глубина резания по профилю зуба получается неравномерной. Снижается эффективность процесса. Анализируется влияние технологических факторов на глубину резания при шлифования зубьев методом копирования. Получена система уравнений для определения изменения глубины резания на любой участике эволвентного профиля зуба. Выявлено направление обеспечения относительно равномерного распределения глубины резания по профилю зуба.

Ключевые слова: глубина резания, профиль, копирование, погрешность, абразивный круг.

introduction. Gears are widely used in a variety of machines, assemblies, devices, etc. [1-4]. They often have a high performance. A performance of any technical device, including gears, formed workmanship of its components. An effective way of the providing such indicators is the grinding gears of involute working surfaces of gear wheels and in their manufacture. One of the main methods for grinding involute profile is a copy method [4, 5].

Grinding method is copying a disc abrasive wheel. Profile grinding wheel simulates troughs between the adjacent teeth. When the module of the teeth of 10 mm simultaneously grind both sides of a cavity at least 9 mm module opposite sides of two neighboring cavities. Grinding allowance is the 0.2-0.3 mm thickness of the tooth. The treatment is carried out in 3-4 passes. Achieved machining accuracy is 0.01-0.015 mm, a surface roughness of Ra = 0,32 microns [4, 5].

In the grinding of the teeth on the method of copying the profile of the grinding wheel wear unevenly, leading to a noticeable error form of the gear of the teeth, particularly the first and the last. Abrasive circle periodically ruled by a diamond tool [3-5].

It is known that a change in processing depth of cut surfaces is the one of the causes of error processing. If the cylindrical surface of the workpiece under the grinding has an error of shape in the form of taper surface after grinding,it has no cylindrical and conical shape [5].

Despite the fact that the shaped surface provides a certain allowance, the constant for the entire profile, the actual allowance in the feed direction of the grinding wheel on the depth varies

according to the profile. So, when gear grinding by copying the actual differences between the cutting depth for involute tooth profile (the thickness of the material removal tool in the feed direction on the depth of cut), decreasing, copied to the working surface of the treated tooth. Error processing appears related to the technological process of inheritance. Based on the above,the one of the way of increasing the quality of grinded teeth may be ensure the stability of the actual depth of cut in the grinding, by controlling the influencing factors.

The aim is to identify functional relationships between the actual depth of cut and its influencing factors in the grinding of teeth by copying and determining the direction of these relationships management.

Technological factors causing change the cutting depth in the grinding of teeth. Parameters that cause uneven depth of cut across the grinding surface of the teeth, depending on the stage of their formation may be divided into two groups:

- Factors inherent in the process of sanding related to the organization and holding of the grinding operation. These include mainly: installation error of the workpiece; error profiling of the grinding wheel; errors associated with non-uniform wear of the abrasive wheel; geometric accuracy of the machine; errors associated with thermal and elastic deformations of elements of technological system.

- Errors in the wheel-workpiece-inherent in the process of gear cutting and formed by cutting teeth blade tool, ie, grinded surfaces a manufacturing error in the previous grinding operations. These include: the beating of the ring gear relative to the bore axis; mixing teeth plane symmetry with respect to the

hole axis; involute profile error; pitch error and tooth thickness; etc.

Among these factors, in our view, the most noteworthy subjective factors related to the organization and the conduct of the formation of the teeth having a relatively large value. The analysis of the influence of the accuracy of the grinding of some of them, which are the most significant.

Regulation (slope) of the ground surface (Fig. 1). The most significant changes affecting the depth of cut in the grinding factor is the positioning mechanism being ground involute tooth profile with respect to the direction of tool feed on the cutting depth [6-7]. When the grinding teeth by copying the actual depth of cut in the feed direction of the tool unstable, getting different values involute profile and very different from the allowance

processing the Z, provided stable throughout the processing of the surface (Figure 1, a, t1 # t2 # t3 # Z;. Where t1, t2 and t3 -cutting depths at different portions of the profile). Such a change in the depth of cut is peculiar gear grinding process by copying and related provisions of the tooth being ground and certain parts of its involute profile in the cutting zone. The position of the tooth as a factor influencing the depth of cut, briefly called us "verticality (or tilt)" schlifuemogo tooth.

Installation preform. Uneven depth of cut in the grinding of the teeth is caused by errors and even home when installing the wheel - workpiece in four directions:

- Offset planes of symmetry of two adjacent teeth and the profile of the abrasive wheel (5) (Figure 1b, t2 # t4 # t5 # Z; where t2, t4 and t5 - cutting depths at different portions of the profile.)

Fig. 1. Scheme changes the cutting depth: the ideal organization process (a), when bias (b) and in rotation (a) of the plane of symmetry with respect to the adjacent teeth of the tool

- Rotate the plane of symmetry of two adjacent teeth with respect to the profile of a circle by an angle ip, error of the angular position of being grinded teeth (Fig 1, in the, t11 # t1 # t12 # t2113 # t3 # Z; where t11, t12 and t13 - cutting depth at different. Profile sections).

- Toothing runout relative to an axis of the base hole (e),

- Deviation of the direction of the teeth on the direction of the double table moves.

The greatest value of the difference of cutting depth is achieved on profiles, opposite to the direction of eccentricity of the teeth (Fig. 2). Changing the cutting depth range only because of the heartbeat of the ring gear relative to the wheel-workpiece axis of rotation during grinding At, = 2e. The thickness of the outermost surface layer in the direction of eccentricity removes the tooth material is ~ (t + e), and opposite to it of the tooth is ~ (t-e). At the same time, Ate - formed four types of primary errors leading to eccentricity: a deviation from uniaxial hole, which is the technological base, and rotating workpiece when gear-cutting - e1, beating grooving tool - e11, a deviation from the uniaxial base hole and spin the wheel-workpiece when gear grinding - e2, beating of abrasive circle - e21. If we consider that the value of the beating the formation of the teeth tools insignificant (e11^0; e21^0), we obtain:

e = Ve2 + e2 and = ZyR+l

It is known that the value of the eccentricities e1 and e2 are determined by the values guaranteed by the gaps between the mounting surfaces of the mandrel and the technological bases of the workpiece. So, for the leveling of values (e^min; e2^min)

if and when the cutting teeth and grinding of the workpiece on the mandrel expander design.

The functional relationship between the changes in the depth of cut and its influencing factors can be expressed in a generalized form:

At = f(Ah, AZ, Ae, Ay, Al, Ad, Au, AT, Ay, A5, Ai-1) where Ah - change in the depth of cut (hereinafter error), which is a consequence of verticality sanding profile

AZ - error due to differences on the left of the actual margins of grinding for different teeth,

A5 - the error associated with the displacement of the symmetry planes of adjacent teeth (or tooth) and the profile of the tool (Figure 2b.)

Ay - error associated with angular deviation of the symmetry planes of the adjacent teeth (or tooth) and the profile of the tool (see Figure 2 in.)

Al - error due to deviation of the direction of the tooth, Ad - error resulting from geometric inaccuracies of the grinding machine,

Au - error due to inaccurate profiling abrasive wheel and uneven wear,

AT - error resulting from the instability of thermal deformation elements of technological system,

-pogreshnost Ay, which is a consequence of the elastic deformation of the elements of the technological system,

Ae - displacement of the ring gear axis relative to the axis of the hole,

Ai-1 - error grinded other teeth formed at the operation processing tool blade teeth.

Fig. 2. Scheme of the influence of the eccentricity of the ring gearthe position of abrasive teeth we get:

Error Ah direction and have certain specific values for toothed surfaces with the given parameters (module, number of teeth) depending on the organization operation. Other errors are random values and directions in space. Therefore, the largest value changes the cutting depth can be expressed by the formula:

At = Ah +

+A2 Z + A1e + A> + A1S + A2/ + A2d + A2u + A1T + A2 y + A2_1

Uneven depth of cut on the profile of the teeth, the vertical profile of the teeth and the uneven depth of cut along the tooth being ground produce a number of negative consequences, such as the amount of material removal width of a single grinding wheel is relatively large, the intensity of the tool wear when cutting high and quickly distorted its operating profile; the working surface of the abrasive wheel, and its volume, taking the heat originating in the cutting zone, is relatively small; formed a high heat treatment on the treated surface, creates a condition of "burning" the surface being treated, etc.

Based on the above, the identification of patterns of distribution of the depth of cut on the involute profile, depending on the position of the tooth is being the ground of paramount importance.

The pattern of changes in the depth of cut for involute profile. Analysis of the mechanism of formation when the grinding involute profile and the above drawbacks shows that the main causes of these shortcomings is almost complete agreement with the direction of the vertical feed the tool to the cutting depth and involute profile (almost vertical) [9-10]. As a result, the unevenness is formed on the profile cutting depth under all operating passages circle.

We derive a mathematical model of the actual depth of cut (allowance in the feed direction of the tool) with each working stroke when the grinding teeth copying method (Fig. 3).

Assume that, for grinding the tooth profile is provided across the uniform machining allowance profile depth Z (Fig. 3). Point G is the start of the involute. The radius of the base circle R. Somehow involute angle 0 formed point A Profile. The actual depth of cut at this point is equal to the t, wherein t = AB # Z.

Analysis of the selected grinding scheme shows that the maximum depth of cut (t ) involute profile at the beginning (point G) and the minimum (tmin) at its end.

The technological challenge is to determine the dependence tx = f(0 ) and management to identify ways to improve the efficiency of grinding teeth copying method.

Fig. 3. Scheme to change the cutting depth

Obviously, when Z stability allowance across the tooth surfaces, the tooth surface of the workpiece to be equidistant involute working surface. The wires form MB from point B, owned by the involute profile of the workpiece. To output t =f(0 ) is extending forming points A and B to their intersection with the axis of symmetry of neighboring teeth (simultaneously with the symmetry axis of the abrasive wheel) at the points С and E. Assume that the rotation angle of the involute start point G relative to the symmetry axis is 0. From point B, BD conducts parallel to the generator of the NE. Then, according to the construction of the conditions of the parallelogram BDEA, triangle BCD and size of chains:

AB||CE; BD||AE; AB=tx=DE; FB=Z; BC1MO и BD1NO; BC=MC-MF-FB=MC-MF-Z, (1)

CD=OC-OE-ED; (2)

zBCE=9O°-(0+0 +0 )=0„ 0 =0-0

4 x r 1 t fx

Here, 0 - involute angle desired point A profile

0t - involute angle formed by the cutting depth at point A,

0f - involute angle points B and F.

We find the lengths of the sides ABCD. For simplicity, we denote tx = X. From AONE:

OE =

NO

R

cosZNOE cos(0x + в)

(3)

NE=NOtgZNOE=Rtg(0x+0)

NA = pA = NG =

2nR6x nR6x

360

180

BD = AE = NE -NA = Rtg(6x + 6)--

nR6x 180

(4)

where pA- involute radius of A profile, From geometric constructions is obvious that

OC = ■ R

NA = NG

cos

(01+0X + 0 )

(5)

MC=MOtgZMOC=Rtg(0 +0 +0)

MF = Dp =-

HF 180

Using (6) and (7) (1), we obtain:

nR(et + ex) ВС = Rtg(0t + 6x + 6)--V— -Z

(6) (7)

180

According to the theorem of cosines, CD2=BC2+BD2-2BC^BD^cos

CD

2 _

nR(Gt + вх) Rtgtft + ex + e)--v__ ^ - z

180

+

+

Rtg(ex + e)

nRdx

180

nR(6t + вх)

Rtgtft + ex + e)--- z

Rtg(ex + e)-

nRdx

180

180 ■cos6t

(8)

We take into account (3), (5) and (8) (2) and transforming, we obtain:

X =

cos(0; + e, + e) œs(et + в)

[fleece + + й)

180

zp + + fl) - - 2 [fitpce, + вх + в)

жД(е, + вг) 180

^Rtg(_ex + e)

uR 6,

1

180

■cosdï

Accordingto the theoremofsinesofABCD:

nR6x Rtg(ex + e)--

180

R

sin0t = R

cos(0t + вх + в) cos (0x + 0)

X

cos (Gt + вх + в) (11)

If in equations (9) and (11) take 0x = 0, we obtain the equations corresponding to the maximum depth of cut, X = tmax:

cos(et + e) cose r nRet f

-J[fitg(et + e)—jäör-z] + (Rtge)2 -

T nR6t l '

-2Ätg e ■ [r tg(et + e)cosdt

RR

Rtge sin et = (—;-T---x) cos(et + e)

s t ^cos(et + e) cos 0 y t J

A system of equations consisting of (9) and (11). Decision regarding itsXcan obtainamathematical model of the cutting depth. In particular - the local values 0x, equation (9) and (11) contain two unknowns (X and 0t). Not presents difficulties of its decision with respect to X = tx using modern software (eg, Matlab, etc.).

By entering into the formulas (9) and (11) the values of R and 0, expressed modulus m and the number of teeth z of any wheel you can determine the maximum depth of cut, and compare it

with an allowance Z.

Analysis of the equations (9) and (11) reveals that with increasing angle 0 is reduced depth of cut at a given value the Z allowance, improved grinding conditions. Hence, an increase in the slope involute profile causes a decrease in the depth of cut, it provides a more uniform distribution. When 0 = 900 provided tmax = Z. However, during the grinding of teeth by copying software 0 = 900 is not possible, as this grinding wheel touches the head of the adjacent tooth. Fig. 4 shows a graphical representation of a generalized tx = f(0x) with a minimum of 0^min.

Application of the equations (9) and (11) allowed to determine the extreme (limit) values in grinding depth of cut of the 1-st and 5-th tooth (Fig. 4) with parameters m = 4 mm, number of teeth z = 40. It is revealed that, for a given value of Z = 0,3 mm allowance around the involute profile of the maximum cutting depth for the first tooth at the beginning of the profile is t = 2,04 Z = 0,612 mm, its minimum value at the end of involute

max

profile is t = 1,75Z mm; 5-th tooth profile is in the beginning of t = 1,47Z, its minimum value at the end of the profile is

max

t . = 1,36Z. Thus, with increasing inclination of the tooth being ground, depth of cut decreases unevenness on the profile.

Г TtR8, и

_ R R

1 _(cos(0t + e)" cose"

Fig. 4. Dependence of cutting depth of the involute profile angle

Conclusions. A

- For the leveling influence of installation errors on the uneven depth of cut if and when cutting teeth and grinding of i the workpiece on the mandrel expander design,

- When grinding teeth of the gears by copying cutting depth i varies for involute tooth profile; maximum depth of cut involute profile corresponds to the beginning and end of the minimum-

value profile,

- With increasing inclination of involute profile and removing it from the starting of the involute sections of the value of cutting depth decreases and approaches the value of the allowance; to improve the efficiency of teeth grinding process by copying necessary to grind the teeth having a relatively high inclination.

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