Concluding remarks

In this chapter, the transverse shaving of the traditional gear shaving machine has been simulated, and the gear tooth surface of shaved gear has also been constructed. The effects of machine setting parameters and cutter assembly errors on the work gear surface and the tooth

lead crowning have been investigated through numerical examples. For the crowning mechanism, the crowning effect is shown to be sensitive to the angle θ between guideway and horizontal as well as the horizontal distance d between pivot and pin. The horizontal _h and the center distance errors Δh and Δ are also proved significant to gear tooth E₀ crowning. The results can be used as instructions for design, assembly and calibration of gear shaving machine and shaving cutter.

To have a double crowned gear shaved by traditional gear shaving machine for better performance in transmission error, four categories of parameter need to be considered:

modification of shaving cutter, assembly errors of shaving cutter, assembly errors of gears and machine setting parameters. It has been proved that the gear pair with optimized parameters indeed possesses better quality in transmission. Besides, among the eleven selected parameters, the coefficient a_c concerning the modification of shaving cutter and the angle θ

between the guide way and horizontal on the shaving machine contribute the most to product quality (transmission error).

Table 3.1 Parameters of work gear and shaving cutter.

Parameter description Parameter value Work gear

Number of teeth (T₂) 36

Normal module on pitch circle (m_pn2) 2.65 Normal circular tooth thickness on pitch circle (s_pn2) 4.858mm Normal pressure angle on pitch circle (α_pn2) 20^D Helical angle on pitch circle (β_p2) 10^DL.H.

Outer diameter 105.9mm

Face width 28.4mm

Shaving cutter

Number of teeth (T₂) 73

Helical angle on pitch circle (β_ps) 22^DR.H.

Face width 25.4mm

Normal circular tooth thickness on pitch circle (s_pns) 3.348mm

Table 3.2 Machine setting parameters in gear shaving.

Machine setting parameters

Angle between guideway and horizontal (θ ) 2 50^{D '} Vertical distance between pivot and pin (d ) _v 188mm Horizontal distance between pivot and pin (d )_h 545mm Distance between pivot and center of work (C) 385mm

Table 3.3 Basic data of the gear pair (Example 3.6).

Basic data Gear 2 Gear 4

Module (m ) _n 2.65 2.65

Tooth number (T ) _i 36 36

Helix angle (β_i) 10^D L.H. 10^D R.H.

Pressure angle (α_n) 20^D 20^D

Table 3.4 Factors and their levels in transmission error analysis Level

Factor Unit

1 2 3 A θ degree 1°50′ 2°50′ 3°50′

B d _v mm 500 545 590

C d _h mm 350 385 420

D C mm 150 188 226

E Δ γv degree -0.02 0 0.02

F Δ γh degree -0.02 0 0.02

G Δ E mm -1 0 1

H Δv degree -0.02 0 0.02

I Δh degree -0.02 0 0.02

J Δ Eo mm -1 0 1

K a _c - 0.0001 0.0005 0.001

Table 3.5 The S/N ratios of the transmission error analyses.

Table 3.6 Factor response table for the transmission error analyses.

A B C D E F G H I J K Level 1 -3.5233 -5.4453 -6.8664 -6.7792 -5.9369 -7.5113 -6.6523 -6.9550 -8.3829 -8.8092 1.6553 Level 2 -8.9130 -6.0746 -6.1161 -5.0610 -7.2417 -6.7247 -6.6952 -7.6961 -4.6305 -5.0177 -7.2805 Level 3 -7.0376 -7.9540 -6.4914 -7.6337 -6.2953 -5.2378 -6.1264 -4.8227 -6.4605 -5.6470 -13.8486

Effect 5.3897 2.5086 0.7503 2.5726 1.3048 2.2735 0.5688 2.8734 3.7524 3.7915 15.5039

Ranking 2 7 10 6 9 8 11 5 4 3 1

θ

Figure 3.1 Crowning mechanism of the traditional shaving machine.

dp

dv

dh

ψt

θ

II pin

I pin

II pivot I

pivot

zt

Figure 3.2 Parametric representation of the crowning mechanism.

f

Figure 3.3 Coordinate system of the gear shaving machine with considerations of cutter assembly errors.

Figure 3.4 Lead crowning of the shaved gear.

-7 -6 -5 -4 -3 -2 -1 0

-10 -8 -6 -4 -2 0 2 4 6 8 10

Case 1 Case 2 Case 3 Case 4 Case 5 Case 6

Figure 3.5 Lead crowning under different conditions (dv=188mm,C=385mm, θ=2°50´, dh=350~600mm, Δdh =50mm).

-5 -4 -3 -2 -1 0

-10 -8 -6 -4 -2 0 2 4 6 8 10

Case 7 Case 8 Case 9 Case 10 Case 11 Case 12

Figure 3.6 Lead crowning under different conditions (dh=545mm, C=385mm, θ=2°50´, dv=50~300mm, Δdv=50mm).

-5 -4 -3 -2 -1 0

-10 -8 -6 -4 -2 0 2 4 6 8 10

Case a Case b Case c

Lead crowning of left gear surface (μm)

Face width (mm)

Figure 3.7 Lead crowning under different conditions (dv=188mm, dh=545mm, θ=2°50´, C=200~400mm, ΔC= 100mm).

-9 -7 -5 -3 -1

-10 -8 -6 -4 -2 0 2 4 6 8 10

Case A Case B Case C Case D

Figure 3.8 Lead crowning under different conditions (dv=188mm, dh=545mm, C=385mm, θ=1°50´~4°50´, Δθ=1°).

164.1

Figure 3.9 Variations of circular tooth thickness from vertical error (a) Δ =v 0.02^D (b) 0.02

Δ = −v ^D. (a)

(b)

-2.85

Ideal condition Positive vertical error Negative vertical error

Face width (mm)

Figure 3.10 Tooth lead crowning affected by vertical cutter assembly error.

-2.85

Ideal condition Positive horizontal error Negative horizontal error

Amount of lead crowning (μm)

Face width (mm)

Figure 3.11 Tooth lead crowning affected by horizontal cutter assembly error.

-2.13

Ideal condition Positive center distance error Negative center distance error

Amount of lead crowning (μm)

Face width (mm)

Figure 3.12 Tooth lead crowning affected by cutter assembly error of the center distance.

α

n

Figure 3.13 Normal section of the parabolic rack cutter.

θ c

Figure 3.14 Coordinate systems of the rack cutter from normal to axial section.

X c

Figure 3.15 Coordinate systems of the generating motion between the rack cutter and the shaving cutter.

mm

Figure 3.16 Double crowning of the saved gear.

Y4 Figure 3.17 Coordinate systems of the meshing gear pair.

q(2)

Figure 3.18 Meshing of the two gear tooth surfaces.

°

Figure 3.19 Transmission error of the gear pair (gear 2 lead crowned only).

°

Figure 3.20 Transmission error of the gear pair (gear 2 double crowned).

Transmission error 3. Assembly errors of shaving cutter

center distance 2. Assembly errors of gears

center distance vertical distance between the pin

and pivot at the initial position horizontal distance between the pin

and pivot at the initial position distance between the pivot and center of the work gear

θ dv

dh

C

Figure 3.21 Fishbone diagram of the factors.

-20 -15 -10 -5 0 5

A1A2A3 B1 B2B3 C1C2 C3 D1D2D3 E1 E2 E3 F1 F2 F3 G1G2G3 H1H2H3 I1 I2 I3 J1 J2 J3 K1K2K3

Figure 3.22 Factor response graph for the transmission error analyses.

Figure 3.23 Transmission errors caused by original and optimized parameters.

CHAPTER 4

PLUNGE GEAR SHAVING WITH TOOTH CROWNING

Gears are the most important components in transmission systems. Modifications of gear teeth (gear tooth crowning) can accommodate errors and deformations encountered in the manufacture, assembly, and operation of gear pairs. Among the four basic shaving methods, plunge shaving is the most advanced gear finishing technique which only needs radial infeed.

Its advantages include increased productivity, accuracy, long tool life, and a simple machine structure (Bianco, 2000). The basic meshing condition of 3D crossed-axis helical gear pair was first derived by Litvin (1994), and it has been widely adopted as the fundamental assumption for simulation of gear shaving. For shaving methods other than plunge shaving, the gear tooth crowning is accomplished by tooth modifications of the shaving cutter and the coordinated motions between cutter and gear. For the plunge shaving method, however, it only depends on the surface geometry of the plunge shaving cutter. Traditionally, the cutter surface geometry results from a cutter re-sharpening machine by trial and error. Focusing on the surface geometry of shaving cutter, what’s really significant is precision of the region between SAP (start of active profile) and EAP (end of active profile) as shown in Fig. 3.1. In gear shaving, SAP of cutter tooth shaves the gear tooth root while EAP shaves the gear tooth tip. In this chapter, the analytical description of the gear with tooth crowning is first constructed by B-spline surface fitting. Then, the grinding wheel profile is parameterized and optimized for minimizing the surface deviations of theoretical and ground (from re-sharpening machine) tooth surfaces of the plunge shaving cutter. Efficiency is greatly improved by avoiding the traditional trial and error method, and the constructed mathematical model of plunge shaving cutter can be further utilized as the base for extending researches.

4.1 Surface interpolation of modified gear tooth

To integrate the modified gear tooth surfaces into the analytical process, especially for those with both lead and profile crowning, B-spline surface interpolation is selected for its ease of manipulation. In practice, the sampling points of the modified surface can be obtained by CMM (Coordinate Measuring Machine) or from other sources. In this paper, for studying purposes, the most commonly used numerical model is adopted for generation of interpolating points: the tooth flank is modified in profile (root to tip) and lead (side to side) directions independently as shown in Fig. 4.2 (Wagaj, 2002). The magnitude at the tip a and the gear _t roll angle at the start α_t define the boundaries of the tip relief. Between these two points, the profile follows a linear trajectory. Similarly, the magnitude a and the starting roll angle _r α_r define the starting point of a root modification. The amount of lead crowning is denoted by

h.

A B-spline representation enables the simulation of surface irregularities and the control of small tooth geometric modifications, such as rounding and reliving. Given a grid of sampling points D_k,A (0≤ ≤k m and 0≤ ≤A n) and orders p and q (degrees p− and 1 points. Once the numbers of sampling points m and n are selected, then the B-spline orders p and q are limited by Eq. 4.2:

Solving Eq. 4.1, P_{i j,} ’s can be obtained, and the interpolated B-spline surface ∑ can be ₂^I

where u and v denote the surface parameters; 2 denotes the surface of gear tooth; and I denotes the surface obtained by interpolation.

B-spline interpolation can be implemented using the MATLAB Spline Toolbox^○R by specifying a set of data points, and either knot sequences or orders of the interpolated surface.

The process of obtaining the B-spline surface is illustrated in Fig. 4.3. For example, to interpolate the gear tooth surfaces described in Tables 4.1 and 4.2, including three target gear tooth surfaces (standard ∑ , lead crowned _S2 ∑ , and double crowned _L2 ∑ ), three sets of _D2 sampling points must be acquired from the model shown in Fig. 4.2. In order to obtain gridded data points for interpolations, the target surfaces are sampled in the cylindrical coordinate system (represented by R , ₂ θ₂ and Z ) for the nature of the cylindrical gear, as ₂ shown in Fig. 4.4.

By specifying the number of sampling points m and n in the R and ₂ Z coordinates as ₂ well as the respective orders p and q, three interpolated tooth surfaces can be expressed as

2( ,2 2)

I

S R Z

∑ , ∑^I_L2( ,R Z_{2 2}), and ∑^I_D2( ,R Z_{2 2}). Three sets of data points (30 30× in R and ₂

Z coordinates for each set) are also sampled from the three target surfaces for error analysis. 2

The interpolation error E is defined as a 30×30 matrix composed of values of arc length ^I between the respective points (with the same values of radius and at the same axial cross-section) uniformly sampled from the target surface ∑ and the interpolated₂ surface

2

where r denotes the value of radius; _{i j,} x_{i j,} ,y and _{i j,} x_{i j}^I_, ,y_{i j}^I_, denote the points on ∑ and ₂

2

∑ , respectively; and the maximum I e_max^I as well as the mean e_mean^I can be obtained.

Based on the error analysis in this section, E is not sensitive at all to the parameters in ^I the Z2 coordinates, including n and q. Even with lead crowning, the non-linearity remains quite small compared with that caused by profile crowning. Table 4.3 presents six conditions of ∑ . Increasing m and p improve the values of ^I_S2 e_mean^I and e_max^I , which are commonly required to be less than 10^-3 mm. Although the errors are the same in conditions 5 and 6, the parameters in condition 6 are preferable for higher differentiability. Errors of surface ∑ ^I_L2

are similar to that of ∑ . Nevertheless, for surface ^I_S2 ∑ , under conditions similar to ^I_D2 condition 6, m must increase to 14, as shown in Fig. 4.5, because of higher non-linearity due to profile crowning.

The interpolated surfaces ∑ and ^I_L2 ∑ are compared with ^I_D2 ∑ for validations as ^I_S2 shown in the topographic charts Figs. 4.6 and 4.7. The topographic errors between top and root on the left and right tooth flanks are calculated for points of specific radiuses and Z cross-sections. ∑ is presented in straight solid lines as the base of comparison, while ^I_S2 ∑ ^I_L2 and ∑ are presented in dashed lines. In Fig. 4.6, ^I_D2 ∑ is shown with lead crowning only ^I_L2

and in Fig. 4.7, ∑ is shown with both lead and profile crownings. Note that the values of ^I_D2 the topographic differences are all represented in arc length.

4.2 Topographic error analysis of the plunge shaving cutter

The tooth surface of the shaving cutter is usually finished last using a cone grinding wheel on the shaving cutter re-sharpening machine. Because the topographic accuracy of the plunge shaving cutter maps directly onto the work gear, it is important to identify the

topographic error of the ground tooth surfaces ∑^G_S1, ∑^G_L1 and ∑^G_D1 in comparison to the

theoretical ∑^T_S1, ∑^T_L1 and ∑^T_D1where 1 denotes the surface of shaving cutter tooth; G and T denote the surfaces derived from the re-sharpening machine and the interpolated surfaces, respectively.

The basic meshing condition for the crossed helical gear set (Litvin, [3]) is used to calculate the basic geometric data for the shaving cutter. It needs the following eight basic items: the tooth number Z , the normal circular tooth thickness ₁ s_pn1, the helix angle β_p1 of the shaving cutter; the tooth number Z , the normal circular tooth thickness ₂ s_pn2, the helix

angle β_p2 of the work gear, and the normal module m and pressure angle _pn α_pn, of the shaving cutter and work gear. Fig. 4.8 shows the coordinate system of the CNC shaving machine (Hsu, 2006). Considering coordinate transformation

1 1 1 1 12 2 2 2 2 velocity on auxiliary coordinate system Sh.

Likewise, as shown in Fig. 4.9, the ground surfaces of shaving cutters ∑^G_S1, ∑^G_L1 and

G1

∑D can be obtained by coordinate transformations from Sg (grinding wheel) to Ss (shaving

cutter)

and the meshing equation ( , , )_{g g m}

g u θ φ = n ‧v_m^{( )sg} =0 (4.8)

An example is provided for illustration. Note that all the topographic errors are all calculated in the axial cross-section rather than normal one.

Example 4.1

For interpolated gear tooth surfaces ∑^I_S2 and ∑^I_D2 (see Tables 4.1 and 4.2), the corresponding data of plunge shaving cutter and grinding wheel are presented in Tables 4.4.

Obtaining theoretical tooth surfaces ∑^T_S1 and ∑^T_D1 as well as ground tooth surfaces ∑^G_S1

and illustrated in Figs. 4.10 and 4.11 by 5×9 grids of specified radiuses and Z cross-sections, in which theoretical surfaces are presented in straight solid lines, while the corresponding ground ones are shown in dashed lines. From SAP to EAP. of the cutter tooth, errors between

1 T

∑D and ∑^G_D1 (Fig. 4.11) are less than those between ∑^T_S1 and ∑^G_S1 (Fig. 4.10). This is because of the nature of a grinding wheel with cone angle; that is, it presents a parabolic shape on the pitch line of a shaving cutter tooth, so that a specific amount of lead crowning can be compensated.

4.3 Design optimization of the cone-grinding wheel

Compared with the setting parameters of the shaving cutter re-sharpening machine, the

cone angle θ_c of the grinding wheel results in more influences on topographic errors of the final product. Traditionally, θ_c is modified back and forth for the desired accuracy, which is time-consuming. In this section, two examples are provided for illustrating design optimization by adjusting θ_c (Example 4.2) and the profile of the grinding wheel (Example 4.3).

Example 4.2

The objective of this example is to minimize the topographic errors between the theoretical tooth surface and the ground one. The flowchart is shown in Fig. 4.12, and the process is integrated by the MATLAB Optimization Toolbox. The problem is formulated as:

find θ_c

Considering 45 inequality constraints and one boundary constraint, the design variable θ_c is modified iteratively to obtain the optimum of topographic error. The process of calculating

, Topo

ei j based on θ_c is considered as e_{i j}^Topo_, ( )θ_c , and the Sequential Quadratic Programming (SQP) algorithm is adopted, where information of finite difference is used instead of gradients.

The minimum and maximum searching steps are 0.001 and 0.1 degree. For standard surfaces

1 T

∑S and ∑^G_S1 shown in Fig. 4.13, the topographic errors are minimized when θ_c is modified from 10° (initial value) to 1° (optimum value), which is active for the boundary constraint 1° ≤θ_c ≤ °30 . In practice, it’s very difficult to make a grinding wheel with cone angle less than 1° so that the topographic error can’t be improved any more by only adjusting

θc. For double crowned tooth surfaces ∑^T_D1 and ∑^G_D1, the errors are sill large at SAP and EAP (Fig. 4.14) due to profile crowning. To eliminate the errors, the profile of the grinding

wheel also needs to be modified.

Example 4.3

Based on Fig. 4.9, the profile of grinding wheel is parameterized as shown in Fig. 4.15.

Coordinate system S ′_g is attached to the unmodified profile. Within the effective length

A B

L +L (LA=4.2mm, LB=5.5mm), the profile with four sections are defined by w , _A h , _A w _B and h . Represented in _B S ′_g , the four points A2, A1, B1and B2 are fitted by a B-spline curve

with order 4. To improve the topographic errors between ∑^T_D1 and ∑^G_D1, the process of optimization is divided into two levels. The problem formulation for level 1:

find θ_c

The minimum and maximum searching steps are 0.001 and 0.1 degree, and the first level converges efficiently to θ_c =2.382^D, and the problem formulation of level 2:

find [x= w h w h_A, ,_{A B}, ]_B

Following the similar concepts of programming in Example 4.2, the optimum design is presented in Table 4.5, in which the minimum and maximum searching steps are 0.0001 and 0.1 mm. The profile of the grinding wheel is considered straight sided initially. When it reaches optimum, the profile is modified for conjugation to the shaving cutter. The

topographic errors between ∑^T_D1 and ∑^G_D1 are shown in Fig. 4.16, where the errors are all controlled below 10^-3mm.

Experiment of Example 4.3 has been conducted for validation of the proposed method.

Fig. 4.17 shows the plunge shaving cutter with an enlarged view of the cutting edges. Fig.

4.18 shows the pre-shaved gear with an enlarged view of the obviously scalloped tooth surfaces measured as shown in Fig. 4.19. Firstly, the shaving cutter is ground by the grinding wheel on the re-sharpening machine (Fig. 4.20), on which the grinding wheel are modified first by the dresser according to the calculated cone angle and profile parameters. Then, the gear is plunge shaved on NACHI shaving machine with the setup shown in Fig. 4.21.

Materials of gear and cutter are SCM435 and M2, and the operating speed as well as plunge infeed are set as 150 RPM and 1mm/min, respectively. The shaved gear is measured as shown in Fig. 4.22 and the mean values of tooth crowning are recorded in Table 4.6. It is found that:

1. most of the scallops are eliminated and the surface roughness is greatly improved, especially in the lead direction;

2. the amounts of modifications, though with little deviations, are close to the original design values;

3. larger modifications are induced on left flank compared with right flank; this is because the left flank is the driving one in shaving with larger cutting force;

4. efficiency is greatly improved by adopting the proposed method instead of trial and error.

4.4 Concluding remarks

Design and manufacture of the plunge-typed gear shaving cutters has always been a challenge, especially for those used to manufacture gears with tooth crowning. This chapter proposes a method for design and manufacture of the plunge shaving cutter for gears with tooth crowning analytically, rather than trial and error. By integrating B-spline interpolation, differential geometry, and design optimization, the goal is achieved. To interpolate gears with

both lead and profile crownings (double crowned), more sampling points are needed in the radial direction. In manufacturing a shaving cutter, the lead modification can be compensated by adjusting the cone angle, and the profile modification can be implemented by modifying the profile of the grinding wheel. Efficiency is greatly improved by avoiding traditional trial and error method through the proposed one. Besides, an analytical description of the modified gear tooth surface is also constructed, which can be utilized for extending research on serrations and shaving process.

Table 4.1 Basic data of the target surfaces.

Table 4.2 Data of gear tooth modification (crowning) of the target surfaces.

Parameter ∑S2 ∑ _L2 ∑ _D2

a t 0 0 6e-3mm

αt _{N/A N/A 31.8°}

a r 0 0 6e-3mm

αr _{N/A N/A 28.2°}

h 0 6e-3mm 6e-3mm

Gear data

Diameter of base circle d_b2 119.618mm Diameter of addendum circle d_add2 126.71mm

Diameter of root circle d_r2 119.33mm Diameter of pitch circle d_p2 123.915mm Normal pressure angle in pitch circle α_{pn2 14.5°}

Face width fw₂ 18mm Gear tooth number Z₂ 79 Helix angle in pitch circle β_{p2 17°}

Normal circular tooth thickness s_pn2 2.32mm

Table 4.3 Conditions and errors of B-spline surface interpolation (∑ ). ^I_S2 Condition

Parameter

1 2 3 4 5 6

m

(R2 coordinate) 3 4 4 5 5 5

n

(Z2 coordinate) 3 3 3 3 3 3

p

(R2 coordinate) 3 3 4 3 4 4

q

(Z2 coordinate) 3 3 3 3 3 4

(10 3 )

I

emean ⁻ mm 9.805 1.721 1.193 0.645 0.185 0.185

3

max^I (10 )

e ⁻ mm 15.6745 4.55 1.795 1.8539 0.0679 0.0679

Table 4.4 Data of cutter and grinding wheel for Example 4.1.

Table 4.5 Results of the second level optimization in Example 4.2.

Initial design Optimum design

wA 0.1 mm 0.561 mm

hA 0 mm 0.056 mm

wB 0.1 mm 0.538 mm

hB 0 mm 0.235 mm

Plunge shaving cutter

Normal circular tooth thickness s_pn1 2.464mm Tooth number Z₁ 139 Helix angle in pitch circle β_{p1 20°}

Face width fw₁ 20mm Diameter of start of active profile (SAP.) 225.922mm

Diameter of end of active profile (EAP.) 219.537mm Operating center distance E_o 173.04mm

Operating crossed angle γ_{o 3.002°}

Grinding wheel and grinding machine Operating cone pitch radius Rc 350mm

Cone angle θ_{c 10°}

Pressure angle α 4.675°

Operating radius r_o 111.03mm

Table 4.6 Achieved tooth modifications of gear after the experiment for validating Example 4.3.

Left flank (10^-3mm) Right flank (10^-3mm) Design Real Design Real Profile

Tip 6 6.9 6 5.2 Root 6 6.1 6 5.1

Lead 6 5.8 6 5.4

SAP.

EAP.

Figure 4.1 SAP and EAP of a shaving cutter tooth.

a

_t

a

_t

a

_r

a

_r

a

_t

a

_r

h

h h

h

α

_t

α

_r

Unmodified Modified

Involute

Lead Tip

Root

Figure 4.2 Model of gear tooth crowning (Wagaj, 2002).

B-spline surface

Figure 4.3 Flowchart of interpolation error analysis.

X2

Y2 Z² R2

θ

2

Figure 4.4 Cylindrical coordinate used for sampling data points.

0

To p

Root

X Y Z

Figure 4.7 Validations of interpolated surfaces (∑ vs. ^I_S2 ∑ ). ^I_D2

X2

Y2

2, _f Z Z

Xf

Yf g, h

Z Z

Xg

Yg

Xh

h, k

Y Y Xk

, 1

Z Zk

X1

Y1

2 20

φ φ

−

φ

1

γ

o

Eo

Figure 4.8 Coordinate systems of gear shaving machine

X1

Figure 4.9 Coordinate systems of shaving cutter re-sharpening machine.

在文檔中 具齒面修整之齒輪刮削研究 (頁 31-0)