Micro Cad 2003 Paper

ABOUT TOOTH FLANK LINE DEVIATION BY CUTTING USING FELLOW’S CUTTER Márton Máté1, Dénes Hollanda2 1-PhD.Dipl.-Eng, lecturer, Sapientia University of Transsylvania, P-ta Trandafirilor 61, 4300 Targu-Mures, Romania. E-mail: [email protected] 2-PhD.Dipl.-Eng, professor, Sapientia University of Transsylvania, P-ta Trandafirilor 61, 4300 Targu-Mures, Romania. E-mail: [email protected] ABSTRACT The paper presents a mathematical model of the real enveloping process by cutting using Fellow’s cutter. As well as known, the classical theory considers that the tooth flank of the cut gear results as the envelope of the surfaces generated by the cutter’s edges through its helical motion [1]. The classical theory doesn’t take into consideration that the circular motion overlaps the helical motion, and as a consequence, the generating surface family will be deformed. The model of the real relative motion of the tool emphasizes some interesting properties of the generating surfaces. The magnitude of the deformations is estimated on a particular numerical example. 1.THE GEOMETRICAL MODEL OF THE CUTTER-GEAR MOTION The model of the relative motion is presented in the figure 1. Here are used 4 coordinate systems, with the significance given below. z G(0) xG(0)

OG(0) zG q (u)

L

z1t x1t λ

i21λ z2t

x2t

O1t ψ 1(u)

y1t

yG x1

OG O1

xG L-s(u)

yG(0)

y1

i21ψ1(u) y2t

O2t

Figure 1. The model of the relative cutter-workpiece motion

The system Ot2xt2yt2zt2 is fastened to the workpiece. It executes a rotation of a constant angular velocity around the axis Ot2zt2. The system OGxGyGzG is related to the cutter. It executes a helical movement characterised by the parameter pE of the helical slider. System O1x1y1z1 belongs to this slider. Finally, system O1tx1ty1tz1t is the coordinate system of the generating surface described by the cutter’s edge. The raison of considering of this last system stays in the fact that helical movement of the cutter related to the slider, rotation of the slider and rotation of the cu t gear happen simultaneously. A generating surface is described by the tool’s edge during one stroke. The next stroke will repeat the same motion, but in other position relative to the workpiece. That leads to the necessity of considering a system that executes the gearing motion, transporting the generating surfaces. This system can be named the system of the equivalent gear. A reference position of the systems O2tx2ty2tz2t and O1tx1ty1tz1t is when axis O2tx2t and O1tx1t are opposite. They are oriented in this case along the centerline O1tO2t. When gearing, the arbitrary position of these are given by angles λ and i21λ, where i21 is the gearing ratio. Let’s consider now the beginning of the stroke. Till the cutter executes a helical motion relative to the slider, the slider will effectuate a planetary motion relative to the staying system O2tx2ty2tz2t. In this case, system O1tx1ty1tz1t is fixed too. The generating surface is obtained relative to both systems. The generating surface’s equations related to the system Ot2xt2yt2zt2 are described in [2]. This paper proposes to deduce the equations relative to the system O1tx1ty1tz1t. 2. THE MATHEMATICAL MODEL Let’s consider the hypotheses that the vertical component of the helical motion is realised by a classical flywheel-rod mechanism. The vertical velocity can be modeled as a parabolic function. Using the limit conditions of zero velocity at the endpoints of the stroke, and imposing that the maximum value appears at the middle of the stroke length, it results: v( t ) = −

6L 2 6L t + 2 t t 3act t act

(1)

tact is the time of the active stroke, and L the stroke length. When integrating the velocity function, the motion law is obtained: t

 t s( t ) = ∫ v( t ) dt = − 2L  t act 0

3

  t  + 3L    t act

  

2

(2)

Introducing the parameter u as the quotient t/tact, the relations become simpler. The values of the parameter u are between [0,1], where u=0 corresponds to the beginning, and u=1 to the end of the active stroke. Due to the helical slider the arbitrary stroke length s(u) of the cutter is accompanied by a rotation of angle q(u) around axis O1z1, where

2π s( u ) (3) pE The planetary motion of system O1x1y1z1 related to the workpiece is determined by angle function Ψ(u), that depends on the circular feed st, stroke number ncd, the frontal modulus mt, and teeth number z1 of the cutter: q( u ) =

ψ( t ) =

s t n cd ⋅t 30 m t z1

(4)

Considering that active stroke time tact represents the f-th part of the whole stroke time, function (4) can be written as follows: ψ( t ) =

s t n cd s n t 60 f t ⋅ ⋅ t act = t cd ⋅ ⋅ 30 n cd z s t act 30 n cd z s n cd t act

As result, the expression of the angle function depending on parameter u is: ψ( u ) =

2 st f ⋅u m t z1

(5)

Using the functions defined above, the transformation matrix from the cutter’s system to the generating surface system can be written as cos a ( u ) M1t ,G =

sin a ( u )

0 A w cos λ − A w cos( i 21 ψ( u ) − λ )

− sin a ( u ) cos a ( u ) 0

A w sin λ + A w sin ( i 21 ψ( u ) − λ )

0

0

1

L − s( u )

0

0

0

1

(6)

where a ( u ) = (1 + i 21 ) ψ( u ) + q( u ) . The parametric expressions of the cutting edge relative to the cutter’s system [3] are: x G ( ϕ) = R bA [ cos( ϕ − η) + v A ( ϕ) sin ( ϕ − η) ] y G ( ϕ) = R bA [ sin ( ϕ − η) − v A ( ϕ) cos( ϕ − η) ]

z G ( ϕ) = p A ( ϕ − v A ( ϕ) )

(7)

Using (6) and (7), the parametric expressions of the generating surface result as follows:

x G ( ϕ) = R bA [ cos e( u ) + v A ( ϕ) sin e( u ) ] + A w ( cos λ − cos( i 21ψ( u ) − λ ) ) y G ( ϕ) = R bA [ sin e( u ) − v A ( ϕ) cos e( u ) ] + A w ( sin λ + sin ( i 21ψ ( u ) − λ ) )

z G ( ϕ ) = p A ( ϕ − v A ( ϕ ) ) + L − s( u )

(8)

e( u ) = ϕ − (1 + i 21 ) ψ( u ) − q ( u ) − η Analyzing the expressions of the generating surfaces, it can be concluded: -the generating surface is depending on the parameter λ that signifies the position of the generating system relative to the workpiece system; -the generating surfaces are not helical surfaces with constant helix parameter; -the enveloped tooth flank is not a helical surface with constant helix parameter. 3.MODEL ANALYSIS. The mathematical model was tested for a cutter having normal modulus mn=5mm, teeth number z1=25, specific frontal correction ξ1=+0.56,rake angle γ=6°, top relief angle α=5°, pitch helix angle 28.33°, using a helical slider with pE=829mm. The cut gear has z2 =35 teeth, and a specific profile correction ξ 2=+0.25. Axis distance Aw depends on the teeth numbers and the specific corrections. It was determined using a numerical method required by the involute function. After that, the radius of the rolling cylinder Rw1 was calculated. If the generating surface is a helical surface, the intersection between itself and the rolling cylinder results a helix. Mathematically it signifies that the same point of the cutting edge moves on the rolling cylinder, describing the helix. Solving the system formed by the parametric equations of the generating surface, and the implicit equation of the rolling cylinder, a dependence between parameters ϕ, u and λ can be established. This can be written as Φ( ϕ, u , λ ) = [ x G ( u , λ ) ] 2 + [ y G ( u , λ ) ] 2 − R 2w1 = 0

(9)

The function can be analyzed only using numerical approaches. The proposed method consists in the fixing of the position parameter λ and, for a range of u∈ (0,1), corresponding to arbitrary edge positions during the stroke, ϕ edge parameter will be determined. Here the secant method was used. The variation of the ϕ edge parameter is presented in figure 2. Figure 3 presents different slices of this surface, corresponding to different λ values. For every λ value, the parameter pair (ϕ,u) determines a point of the intersection curve between the rolling cylinder and the generating surface. The deviation from the theoretical helix can be best illustrated on the layed out rolling cylinder. Three types of differences were calculated here: the angulary difference between the theoretical helix line and the real intersection curve’s regression line (figure4), the distance on the axis direction between the same lines (figure5), and the distance between the real curve and the theoretical helix (figure 6).

ϕ [°] 29.2

29

28.8 u [%]

λ [°] 20

0

-20

80

60

40

20

Figure 2. The ϕ parameter’s surface ϕ

λ=23.6°

0.512

λ=11.8° λ=0

0.51

λ=-11.8°

0.508

λ=-23.6° 0.506 0.504 0.502 0.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Figure 3 Slices in ϕ parameter’s surface min] ∆ β w[ 0

1

2

-23.6

-11.8

0

11.8

Figure 4 Angular deviation

λ [°]

u

It can be concluded that the real generating surface differs from the theoretical helical surface. As a consequence, the helical deviation of this determines a deviation of the tooth direction, which leads to the deterioration of the quality of coupling. λ= 23.6°

0.04

λ= 14.2°

[mm] ∆

λ= 4.7°

0.02 0 -0.02

λ= -4.7° -0.04

λ= -14.2° λ= -23.6°

-0.06

-40

-30

-20

-10

x[mm]

Figure 5 Distance between the theoretical helix and the regression line of the intersection curve, on the direction of axis ∆[mm]

λ=-14.2° λ=-23.6° λ=-4.7°

0.05

λ=4.7°

0 λ=14.2°

-0.05

-0.1

λ=23.6 °

40

30

20

10

x[mm]

Figure 6. Distance between the theoretical helix and the intersection curve, on the direction of axis REFERENCES 1.Litvin F.L., A fogaskerékkapcsolás elmélete, Műszaki Könyvkiadó, Budapest, 1972. 2. Hollanda D., Máté M., A generálófelület meghatározása metszőkerekes fogaskeréklefejtésnél, X. Országos Gépész Találkozó, Székelyudvarhely, 2002. 3. Máté M., Contribuţii la optimizarea parametrilor constructivi-functionali ai cuţitelor roată cu dinţi înclinaţi, 1999. (PhD Dissertation)