Design optimization of the cone-grinding wheel

CHAPTER 4 PLUNGE GEAR SHAVING WITH TOOTH MODIFICATIONS

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

(R2 coordinate) 3 4 4 5 5 5

(Z2 coordinate) 3 3 3 3 3 3

(R2 coordinate) 3 3 4 3 4 4

(Z2 coordinate) 3 3 3 3 3 4

(10 3 )

emean ⁻ mm 9.805 1.721 1.193 0.645 0.185 0.185

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

h

h h

h

α

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.

Y2 Z² R2

θ

Figure 4.4 Cylindrical coordinate used for sampling data points.

To p

Root

X Y Z

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

2, _f Z Z

Yf g, h

Z Z

h, k

Y Y Xk

, 1

Z Zk

2 20

φ φ

−

φ

γ

Figure 4.8 Coordinate systems of gear shaving machine

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

X Y Z

Figure 4.10 Topographic errors between theoretical and ground shaving cutter tooth surfaces (∑ vs. ^T_S₁ ∑ ). ^G_S₁

Unit: 10^-3mm

12.3

14.3

14.5

12.7

14.5

12.7 12.3

14.3

Z=-10mm Z=10mm

X Y Z

Figure 4.11 Topographic errors between theoretical and ground shaving cutter tooth surfaces (∑ vs. ^T_D₁ ∑ ).^G_D₁

Figure 4.12 Flowchart of design optimizations of the cone grinding wheel.

S.A.P.

E.A.P.

Unit: 10^-3mm

1.9

2.1

2.2

1.9

2.2

1.9 1.9

2.1

Z=-10mm Z=10mm

X Y Z

Figure 4.13 Topographic errors between theoretical and ground shaving cutter tooth surfaces optimized by θ_c (∑ vs. ^T_S₁ ∑ ). ^G_S₁

X Y Z

Figure 4.14 Topographic errors between theoretical and ground shaving cutter tooth surfaces optimized by θ_c (∑ vs. ^T_D₁ ∑ ). ^G_D₁

Y ′g

Z ′g

θc

Figure 4.15 Parameterized profile of grinding wheel for optimization.

X Y Z

Figure 4.16 Topographic errors between theoretical and ground shaving cutter tooth surfaces by profile optimization (∑ vs. ^T_D₁ ∑ ). ^G_D₁

(a)

(b)

Figure 4.17 Plunge shaving cutter used in the experiment (a) overview (b) scaled view.

(a)

(b)

Figure 4.18 Pre-shaved gear used in the experiment (a) overview (b) detailed view.

Figure 4.19 Measured data of the pre-shaved gear used in the experiment.

Figure 4.20 Plunge shaving cutter ground by the grinding wheel of the re-sharpening machine.

(a)

(b)

Figure 4.21 NACHI shaving machine used in the experiment (a) machine overview (b) cutter and work gear setup.

Figure 4.22 Measured data of the shaved gear after the experiment.

CHAPTER 5 DESIGN OF SERRATIONS ON GEAR PLUNGE SHAVING CUTTER

The plunge shaving cutter infeed movement towards the gear is radial to the gear axis without transverse movement. The stroke of the cutter is very short and the shaving time using this method is the shortest. The shaving cutter tooth surfaces have serrations extending from the top land to the root fillet of the teeth whose sharp edges exert a cutting action on the work gear due to the relatively lengthwise sliding motion. Serrations are an important feature for all shaving cutters, but for plunge cutters their importance is vital. The serrations related to such type of cutter must have a helical pattern and they have to be manufactured with a very high precision CNC slotting machine in order to obtain the best results. Thus, the surface roughness of the work gear after shaving is affected primarily by the arrangement of these serrations. In this chapter, an method for optimizing the serration displacement is proposed.

5.1 Design parameters for the shaving cutter serrations

As described by Hsu[10] and shown in Fig. 5.1, plunge shaving is characterized by a radial feed stroke without transverse feed. The serrations in consecutive shaving cutter teeth must be shifted longitudinally so that the cutting marks on the work gear move in a lengthwise direction. The plunge shaving cutter serrations are staggered in a helix as shown in Fig. 5.2. Thus, the design parameters for the serrations of the shaving cutter are the serration pitch p , the helix direction of the serrations, and the number of starts _s z , which indicates _os the number of serration pitches moved in the lengthwise direction when the shaving cutter is rotated once fully.

When the work gear meshes with the shaving cutter, the same tooth of the work gear meshes with the shaving cutter teeth every N teeth. Since the number of teeth ₂ N of the ₁

shaving cutter might not be an integer multiple of the number of teeth N of the work gear, ₂ the same work gear tooth will mesh with different shaving cutter teeth. For example, if the number of work gear teeth is 15 and the number of shaving cutter teeth is 157 (and the shaving cutter’s initial contact tooth is labeled tooth #1), the same tooth of the work gear is meshed with the shaving cutter at tooth #1, #16, #31, #46, …, #151, #9, #24, …, in sequence.

That is, the consecutive cutting marks on the same tooth of the work gear shift in a lengthwise direction due to the helix arrangement of the serrations. The lengthwise distance between the consecutive cutting marks is noted by the symbold , defined as serration displacement, which _s can be expressed as follows:

2 2

p fraction part when fraction part

N N

d z N z N

p fraction part when fraction part

N N

Fig. 5.1 shows a diagram of a shaving cutter and a work gear. If the helix of the serrations moves in a right-hand (left-hand) direction and the shaving cutter rotates in a counterclockwise direction, the directional shift of the cutting mark on the work gear surface is from left to right (right to left).

However, the above is the viewpoint from shaving cutter. If the surface roughness of gear as well as cutting efficiency need to be addressed, the 3-D mathematical model needs to be constructed.

The shaving cutter is provided with a plurality of serrations with sharp cutting edges, which are produced on a slotting machine. In Hsu’s [10] model, the serrations in 3-D space were obtained by calculating the intersections of slotting cutter and the shaving cutter tooth surface and then interpolated by cubic splines to obtain the cutting edges. After these, the cutting paths were calculated for analyses. However, to achieve design optimization, the calculation cycle needs to be integrated and automated. It’s too inefficient to calculate serrations by intersection method when every time the design parameters are changed. To

simplify the process, the cutting edges are directly obtained by sampling and interpolating points from the shaving cutter tooth surface obtained in the previous chapter so that tooth crowning can be considered simultaneously., as shown in Fig. 5.3(a). Moreover, the serration shift SF was calculated by Eq.5.2 at the first place:

/ 1 s s

SF =n p N (5.2)

, which indicated SF was controlled by defining parameters p and _s z . In this way, the _os design space was relatively small, and it was also unnecessary to use two parameters. In this chapter, SF is directly defined as a continuous design parameter.

By specifying the values of p and _s SF , the cutting edges are defined in the 3-D space:

[ 1]

ce ce ce ce

r = x y z (5.3)

According to the coordinate system of gear shaving machine shown in Fig.4.8, the cutting paths can be calculated by:

2 21 2

( ) ( )

cp ce

r φ =M φ r (5.4)

In the next section, a numerical example is provided for design optimization of shaving cutter serration.

5.2 Numerical examples and discussion

The trace of the plunge shaving cutter’s cutting edge on the work gear during the cutting process is illustrated schematically in Fig. 5.4 (Hsu, [10]). At the beginning of a shaving cycle, the cutting edge presses into the work gear surface, moves on its tooth surface due to the relative motion, and then removes chips when contact stress reaches local ultimate stress.

Whereas the front cutting edge removes chips as it moves, the top land of the serration (i.e.

the tooth surface of the shaving cutter), rather than removing chips, plows into the work gear tooth surface without a cutting action and extrudes material in the direction opposite to the cutting direction. Therefore, the front cutting edge produces the desired cutting action, while the top land of the serration produces unwanted friction and material flow. The following

example illustrates the design optimization of cutter serrations.

Example 5.1

The basic data for the work gear and shaving cutter are listed in Table 5.1, while the data for the serrations are listed in Table 5.2. Fig. 5.4 shows the simulations of the first and the subsequent cutting marks in the mid-tooth height of the work gear. The distance between the cutting marks between succeeding cuts, termed the “cutting mark displacement”, is directly proportional to the serration displacement. The abscissa in Fig. 5.4 represents the cutting depth (unit:μm), while the ordinate represents the cutting mark displacement in the lengthwise direction of the work gear (unit:mm) where the cutting marks are repeated within a certain range.

Fig. 5.4 (b) and (c) show the end of the first and second cutting cycle simulation, whose higher peak are termed the “first peak” and “second peak”. The ratio between the height of the second peak and that of the first peak in the first cutting cycle is known as the “cutting-down ratio”, which is used as an index for cutting efficiency. In this example, the height of the second peak is 1.590 mμ and the cutting-down ratio is 0.318.

According to the basic data of work gear and shaving cutter, the cutting-down ratio is optimized. The optimum design problem is formulated:

find serration shift SF

that minimizes cutting-down ration r _c subject to SF N× ₁≥ p_s

, in which SF =n p N_s _s / ₁ originally, and it is directly selected as the design variable rather than n or _s p for larger design space. The minimum and maximum searching steps are _s 0.001 and 0.1 mm. The cutting-down ratio has been improved from 0.318 to 0.242 (1^st peak=3.1 mμ , 2^nd peak=0.7 mμ ) by 23.9%. The scaled views of 1^st peak and 2^nd peak are shown in Fig. 5.5.

5.3 Concluding remarks

In this chapter, the design concepts of shaving cutter serrations are presented, and design optimization is also achieved. The advantages of the proposed model are listed:

1. The proposed mathematical model considers the tooth modification of shaving cutter tooth derived in chapter 4.

2. The proposed mathematical model can be adopted for design optimizations because the design variables are changed to be independent of each other.

3. The cutting-down ratio has been reduced by 23.9%, which means the shaving time can be reduced by almost 1/4 per piece.

Table 5.1 Basic data for the work gear and corresponding shaving cutter.

Work gear

Number of teeth (N2) 13 Normal module in pitch circle (mpn) 1.750 mm Normal circular tooth thickness (spn2) 3.270 mm Normal pressure angle in pitch circle (αpn) 20∘

Outside radius (ro2) 27.600 mm Form radius (rf2) 21.490 mm

Face width 24 mm

Helix angle in pitch circle (βp2) 5∘R.H.

Plunge gear shaving cutter

Number of teeth (N1) 137 Helix angle in pitch circle (βp1) 10∘R.H.

Face width 30 mm

Normal circular tooth thickness (spn1) 0.529 mm Operating data of shaving procedure

Operating center distance (Eo) 130.612 mm Operating crossed angle (γo) 14.722∘

Cone grinding wheel

Cone angle (θc) 2.961∘

Pressure angle (α) 13.882∘

Rc 350 mm

Table 5.2 Basic data for the serrations.

Serration pitch 1.85 mm R.H.

Number of starts 62 Serration displacement -0.216 mm

Figure 5.1 Diagram of a shaving cutter and a work gear (Hsu, 2006).

Figure 5.2 Diagram of a shaving cutter with helix-staggered serrations (Hsu, 2006).

Serration pitch Radial infeed only

Figure 5.3. Design parameters of plunge shaving cutter serration (a) serration pitch (b) serration displacement.

Unit: μm

Depth

Repeating range Unit: mm Unit: μm

Depth

Repeating range Unit: mm cutter

cutter cutter work gear work gear work gear

(a)

(b)

(c)

First peak

Second peak

Figure 5.4 Cutting path calculations (Hsu, 2006) (a) path of the cutting edge on the work gear (b) end of the first cutting cycle simulation (c) end of the second cutting cycle simulation.

Figure 5.5 Optimized cutting-down ratio of Example 5.2 (a)1^st cut (b) 1^st peak (c) 2^nd peak.

(a)

(b)

(c)

First peak

Second peak

CHAPTER 6 CONCLUSIONS AND FUTURE WORKS

6.1 Conclusions

This dissertation has investigated gear tooth crowning induced by transverse and plunge gear shaving. In the past, these are very time consuming because the machine setting and the shaving cutter need to be modified back and forth by trial and error. In this dissertation, mathematical models for analyses and design optimization are proposed to solve this problem.

For transverse shaving, influences of machine setting parameters and cutter assembly errors have been observed. Design optimization for robustness of gear transmission error has also been accomplished. For plunge shaving, the analytical descriptions of crowned gear and hence plunge shaving cutter have been constructed so that the grinding wheel can be optimized to minimized the topographic error. The cutting trace of plunge shaving cutter has also been analyzed so that the final real tooth forms can be predicted. Besides, the shaving efficiency has also been improved. Based on the results of the numerical examples, the following conclusions have been drawn:

1. 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 _h pivot and pin. The horizontal and the center distance errors Δh and ΔE₀ are also proved significant to gear tooth crowning.

2. 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. Among the 11 selected parameters, the coefficient

a concerning the modification of shaving cutter and the angle c θ between the guide

way and horizontal on the shaving machine contribute the most to transmission error.

3. To manufacture the double crowned gear by transverse shaving, the machine setting parameter and cutter profile coefficient can be calculated and optimized beforehand without trial and error, which reduces the required time for development.

4. 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 crowning can be compensated by adjusting the cone angle, and the profile crowning can be implemented by modifying the profile of the grinding wheel.

5. The proposed mathematical model of shaving cutter serration considers the tooth modification of shaving cutter tooth.

6. The proposed mathematical model of shaving cutter serration can be adopted for design

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