Research objectives

Although the previous studies have significantly improved understanding of the characteristics of the shaving process, they largely focused on theoretical analyses and manufacture of shaving cutter. However, a complete mathematical model of gear shaving has not been established, which can facilitate the control of gear tooth crowning and predict the real finished tooth forms. The main objectives of this dissertation including:

1. Investigating the important influences of shaving machine setting parameters and cutter assembly errors on gear tooth crowning.

2. Investigating the transmission error of the gear pair shaved by traditional shaving machine, and also carrying out robust design.

3. Improving the efficiency of design and manufacture of plunge shaving cutter for shaving gears with crowning.

4. Constructing mathematical model of gear plunge shaving to improve the cutting efficiency.

CHAPTER 2

GEAR SHAVING: FUNDAMENTALS

2.1 Gear shaving cutter

During all shaving processes, the work-piece and the cutter rotate together like a pair of helical gears with crossed axes. The shaving cutter receives the motion from the motor and drives the work-piece which is coupled onto a freely-moving shaft. The meshing is therefore free, which is the main reason why indexing errors are not easy to improve.

The reason why axes are crossed is to create an action of relative sliding between the cutter tooth flank and the gear tooth. As shown in Fig. 2.1. the cutter prolonged along its axis in such a way as to form a cylinder if the b axis were tree to move, the wheel coupled on it would have a movement around the cutter axis in the V direction linked to the cross angle γ . Speed V is the result of two components, precisely V and ₁ V . _s V is compensated by _s the natural cutter rotation, while the V component would move the workpiece along the ₁ cutter axis. In fact, wheel b cannot move in that direction because it is clamped on its axis and the axis, in turn, is fixed. It is quite clear that in order to compensate for this missing movement, a sliding action will occur between the two tooth flanks.

The shaving cutter teeth are manufactured with slots, normally called serrations as shown in Fig. 2.2, which run along the tooth profile. These serrations form a series of cuffing edges.

These are the edges that generate the removal of metal in the form of tiny chips, as a result of the sliding effect between the gear tooth flank and the cutter tooth flank. The shape, dimensions and manufacturing methods of these serrations will be deaft wfth at a later stage.

The crossed-axes angle is the difference between the cutter helix angle and the gear helix angle. The range of the values of such angle is normally between 10^D and 15^D. If the value is higher than 15^D, cutting is easier, cutting capacity increases and the sliding speed increases,

thereby shortening the life of the cutting edges. However, the control of the profile and the lead direction is lost because of the reduced guiding action. On the other hand, if the angle decreases from 10^D onwards, a progressive upsetting effect is obtained that tends to reproduce the conditions of parallel axes. In cases where gears do not permit a regular crossed-axes angle due to their shape, it can go as down to 3° thus obtaining a shiny surface.

2.2 Methods of gear shaving

There are various shaving methods which substantially differ according to the direction of the movement given to the cutter. The choice of one method or another depends on the work- piece shape, on the machine characteristics and on the type of production of that has to be performed, i.e. small batches or big series. For some methods the cutter has to be studied carefully as regards both serrations and the tooth surface geometry.

2.2.1 Transverse shaving

The relative feed motion between the cutter and the gear takes place in the direction of the gear axis as shown in Fig. 2.3. The gear to be shaved reciprocates in the direction of its own axis while the gear and the tool are in mesh. With each reciprocation, a small amount of radial feeding of the shaving cutter occurs. The theoretical table stroke is as long as the face width of the gear to be shaved and it is recommended to calculate 1 extra stroke per module in order to guarantee clean shaving of the edges. This method is unsuitable for shaving shoulder gears.

2.2.2 Diagonal Shaving

As shown in Fig. 2.4, the gear to be shaved reciprocates obliquely in relation to its own axis while the gear and the tool are in mesh. The diagonal angle is achieved either by positioning the workpiece table obliquely or by interpolating two machine axes. With each reciprocation, radial feeding of the shaving cutter occurs. In general the diagonal angle can be

between 0 and 40 degrees but should not be above 25 degrees for reasons of wear.

2.2.3 Underpass shaving

As shown in Fig. 2.5, underpass shaving is basically the same as diagonal shaving but with a diagonal angle of 90 degrees. With underpass shaving, there is no axial table reciprocation. Instead the workpiece reciprocates perpendicularly to its own axis. The shaving cutter must be wider than the gear to be shaved and its serrations must be placed on a helix.

All tooth corrections must be made to the shaving cutter as it will not be possible to realize them through axial movements on the machine.

2.2.4 Plunge shaving

As shown in Fig. 2.6, with this method there is no worktable translation but only a radial feed of the workpiece against the shaving cutter. The shaving cutter must be wider than the gear to be shaved and the serrations of the shaving cutter must be in the form of a helix in order to produce the relative tooth flank feed. Plunge shaving is particularly suited to shaving shoulder gears. In this case, however, all tooth modifications also must be made to the shaving cutter as it will not be possible to realize them through axial movements on the machine.

Figure 2.1 Sliding on the tooth flanks of the crossed-axes meshing.

Figure 2.2 Shaving cutter serrations.

Figure 2.3 Transverse gear shaving.

Figure 2.4 Diagonal gear shaving.

Gear at end of stroke Gear at start of stroke

Shaving cutter

γ 90

^。

Figure 2.5 Underpass gear shaving.

Shaving cutter

γ Gear

Figure 2.6 Plunge gear shaving.

CHAPTER 3

TRANSVERSE GEAR SHAVING WITH TOOTH CROWNING

Transverse gear shaving has been widely adopted for shaving helical/spur gears with large face width because the whole gear can be finished by the axial infeed of shaving cutter.

If plunge shaving is to be adopted under this condition, without axial infeed, the face width of shaving cutter must be large enough to shave the whole gear tooth surface, which inevitably causes much higher cutter cost. To induce lead crowning of the shaved gear by transverse shaving on a traditional shaving machine, the crowning mechanism is needed as shown in Fig.

3.1, in which the work table will rock in the shaving process. The pivot can be fed horizontally only, and the pin will move along the guideway. Once the angle θ between the guideway and the horizontal is specified (≠0) in the shaving process, the rocking motion of the work table can be achieved. When θ = °0 , the work table will move horizontally without rocking and will therefore not produce any crowning effect. This can also be achieved by a modern CNC shaving machine by simulating the rocking motion, however, with a much higher cost. On the other hand, to induce gear tooth crowning in profile direction, the profile of shaving cutter itself should be modified at the first place. This chapter proposes the mathematical model of the traditional shaving machine. Then, both gear tooth lead and profile crownings are investigated by analyses of parameters. Finally, design optimization for robustness of gear transmission error has also been conducted.

3.1 Transverse shaving with lead crowning

3.1.1 Construction of coordinate systems and the shaved gear tooth profile

The crowning mechanism can be further parameterized as shown in Fig. 3.2, where d _v and d are the vertical and horizontal distances between the pin and pivot at the initial _h position. While the pivot (work table) moves z horizontally in shaving from position I to _t

position II, the pin will move a distance d_p along the guideway. The rotating angle of the

The coordinate system of the shaving process can be simplified and illustrated as shown in Fig. 3.3, where the cutter assembly errors including horizontal, vertical, and center distance errors, are considered. The coordinate systems S_s and S′ are connected to the shaving ₂ cutter and the work gear, respectively, while S_d is the fixed coordinate system; S′_h and S′_v are auxiliary coordinate systems for importing assembly errors into the horizontal and vertical directions; the angle hΔ denotes the horizontal assembly error, the angle vΔ denotes the vertical assembly error, and ΔE₀ indicates the error in the center distance. Other parameters in Fig. 3.3 are also described as follows: z denotes the traveling distance of the shaving _t cutter along the axial direction of the work gear; Cdenotes the distance between the pivot and center of the work gear; γ denotes the angle between the two crossed axes; E₀ represents the center distance; φ_s and φ represent the angles of rotation of the cutter and ₂ the gear, respectively, which are related to each other in the shaving operation.

If the shaving cutter is assumed to be a helical involute gear, the surface profile and its unit normal can be represented by Eqs. 3.2 and 3.3 derived by Litvin [3], where u_s and v _s

Tooth profile r_s and surface unit normal n_s of the shaving cutter represented in the coordinate system S can be transformed into the coordinate system of work gear _s S′ ₂ constructed in Fig.3.3 by Eq. 4:

' ' '

There are two major kinematical parameters φ_s and z in transverse shaving gear with _t lead crowning so that two meshing equations (Eqs. 3.5 and 3.6) are required to calculate the enveloping surface of the work gear, where n₂^' is derived from n_s through the same coordinate transformation mentioned above.

2 respectively, which are related to each other in the shaving operation by Eq. 3.7 when the gear ratio is assumed fixed value, where T_s and T₂ denote the numbers of shaving cutter and gear, respectively. Eq. 3.7 needs to be modified as shown in Eq. 3.8 if there exists feeding of the shaving cutter along the rotation axis of the work gear, where φ_2s denotes the additional rotation angle of the work gear in transverse shaving and its representation is shown in Eq. 3.9 (Lin, 2006), in which r_p2 and β_p2 denote the radius and the helical angle on the pitch circle,

2 2

Considering Eq. 3.4 to Eq. 3.9 simultaneously, the tooth surface of the shaved gear with lead crowning can be obtained under the ideal conditions (Δ = Δ =h v 0^D and Δ = mm). E₀ 0 Simulation of gear shaving is carried out with properties of the work gear, the shaving cutter and machine settings listed in Tables 3.1 and 3.2. Comparing with standard gear tooth surface (θ = °0 ), the deviations are calculated according to Eq.3.10:

2 2 on gear tooth surfaces with and without lead crowning, respectively. As illustrated in Fig. 3.4, the dashed line represents the standard tooth surface, while the solid line represents the gear tooth with lead crowning. The effect of lead crowning is obvious that deviations are large at

10

z= mm and z= −10mm, and no crowning is produced at z=0mm.

3.1.2 Effects of machine setting parameters on the tooth lead crowning

From the developed mathematical model, in this section, four important machine setting parameters are investigated including the angle between the guideway and the horizontal θ, the vertical and horizontal distances between the pivot and the pin d and _v d , and the _h distance between the pivot and the center of the work gear C.The effects of these parameters are illustrated through the following examples.

Example 3.1

The fundamental properties of the shaving cutter and the work gear are listed in Table 3.1 and parameters d and _v d are changed to investigate the variations of lead crowning of the _h left tooth surface on the operating pitch circle within the tooth face width (-10mm~10mm) as shown in Figs. 3.5 and 3.6.

Case1~Case6: d_v =188mm,C=385mm,θ =2 50 '^D , d_hvaries from 350mm to 600mm in steps of 50mm.

Case7~Case12:d_h =545mm,C=385mm, θ =2 50 '^D , d_vvaries from 50mm to 300mm in steps of 50mm.

It is found that the effects of parameterd_von crowning are extremely small compared to that induced by d_h, and the variation of the parameter d_h is in inverse proportion to the amount of lead crowning. It is thought to be more effective to control this parameter in the design or assembly of the crowning mechanism.

Example 3.2

The properties of the shaving cutter and the work gear are listed in Table 3.1, and the machine parameters d , _v d and _h θ are listed in Table 3.2. With variations of C (Case a:C=200mm; Case b:C=300 mm; Case c: C=400mm), as shown in Fig. 3.7, the result shows that the amount of lead crowning is in inverse proportion to the value of the variation of C but with a low sensitivity.

Example 3.3

The properties of the shaving cutter and the work gear are listed in Table 3.1, while the machine parameters d , _v d and _h C are listed in Table 3.2. With variations of θ (Case A:θ =1 50^{D '}; Case B:θ =2 50^{D '}; Case C: θ =3 50^{D '}; Case D: θ =4 50^{D '}), as shown in Fig. 3.8, it is observed that the parameter θ indeed has great influences on the amount of tooth lead crowning in positive proportion manner.

From Examples 3.1 to 3.3, it can be summarized that:

(1) The amount of gear tooth lead crowning is moderately sensitive to machine setting parameter d in the inverse proportion sense. _h

(2) The amount of gear tooth lead crowning is very sensitive and proportional to the machine setting parameter θ .

In gear shaving, parameters θ and d can be adjusted to approach the desired amount _h of lead crowning. Results of this section can be further utilized with the technique of optimization to determine the best setting of the shaving machine for different work gears and shaving cutters.

3.1.3 Simulation of gear shaving with cutter assembly errors

Two more examples are provided in this section to show the influences of cutter assembly errors on the quality of shaved gear. In Example 3.5, cutter assembly errors including horizontal (Δh), vertical (Δv) and center distance (Δ ) are investigated by E₀ comparing them with the perfect work gear (shaved without assembly errors). However, the cutter assembly errors take effects not only on lead crowning, but also on circular tooth thickness. Consequently, the theoretical work gear must be derived beforehand (without lead crowning, but still with cutter assembly errors). This is demonstrated in Example 3.4. The influences of cutter assembly errors on lead crowning can then be studied by subtracting the tooth surface derived in Example 3.4 from that obtained considering lead crowning and assembly errors.

Example 3.4

The properties of the shaving cutter and the work gear are listed in Table 3.1, and the machine setting parameters are: θ =0^D (no lead crowning of gear tooth), d_v =188mm,

h 545

d = mm and C=385mm. Cutter assembly errors are considered in the following cases.

(i) vertical error Δ = ±v 0.02^D (ii) horizontal error Δ = ±h 0.02^D

(iii) error of center distance Δ = ± mm E₀ 1

Taking condition (i) as an example, as shown in Figure 3.9, the pressure angle of the right tooth surface is increased while it is decreased on the left tooth surface. Simulations of the other two conditions are completed in the same way, and demonstrated in the next

example.

Example 3.5

The properties of the shaving cutter and the work gear are listed in Table 3.1, and the machine setting parameters are listed in Table 3.2. The same cutter assembly errors as those in Example 3.4 are considered. Variations of lead crowning are shown in Figs. 3.10, 3.11 and 3.12, in which the values of ideal condition are included for comparison. It can be seen that the horizontal and the center distance errors Δh and Δ have more significant effects on E₀ lead crowning than those caused by the vertical errorΔv.From the results shown in examples 3.4 and 3.5, it can be concluded that the cutter assembly errors have significant effects on both the circular tooth thickness and tooth lead crowning.

By simulating transverse shaving with the crowning mechanism, the effects of machine setting parameters and cutter assembly errors on the tooth lead crowning of the work gear have been investigated through numerical examples. The crowning effect has been shown sensitive to the angle θ between guideway and horizontal as well as the horizontal distance

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

3.2 Transverse shaving with double crowning

The double crowned gear (gear teeth crowned both in lead and profile directions) is an excellent example of significant improvements in transmission by gear tooth crowning (Litvin, 2001), and it can be manufactured efficiently by gear shaving with CNC shaving machine (Litvin, 2001). However, it is indeed costly for gear manufacturer to replace a traditional shaving machine by a CNC one. The coordinated motions of traditional shaving machine is driven by mechanism instead of controller, and, nevertheless, only lead crowning can be conducted. In this section, gear tooth surface with double crowning manufactured by traditional shaving machine is derived and then examined.

To manufacture a double crowned gear by this machine, the shaving cutter needs to be modified in the tooth profile direction, and it can be generated by the parabolic rack cutter shown in Fig. 3.13. a is the parabolic coefficient used to control the profile of the rack _c cutter. Involute profile of shaving cutter can also be generated by simply setting a_c = , 0 represented as Eq. 3.11.

2 0 1 ^T

a =⎡⎣uc −a uc c ⎤⎦

r (3.11)

Through the coordinate transformations from rack cutter to shaving cutter, which is shown in Figs. 3.14 and 3.15, the locus can be obtained shown as Eqs. 3.12 and 3.13.

( ) ( ) [ ( , ) ( , ) ( , )]^T

By deriving the normal vector Eq. 3.14 and the meshing equation Eq. 3.15, the tooth surface of shaving cutter r can be obtained. _s

s s

The transverse shaved gear tooth surface can then be derived through the process similar to those in the previous section. The locus equation of shaving cutter:

2(u ,θ ,ψ_{c c s}, , ) 1]ϕs zt ^T = 2_s( , )[ (ϕs zt s u ,θ ,ψc c s) 1]^T

[r M r (3.16)

, and the meshing equations:

2

2 = 2_s( , ) (ϕs zt _s u ,θ ,ψc c s)

n L n (3.19)

Considering Eq. 3.11 to Eq. 3.19 simultaneously, the tooth surface of the shaved gear with double crowning can be obtained. Simulation of gear shaving is carried out with data of the work gear, the shaving cutter and machine settings listed in Tables 3.1 and 3.2 with

1.4 10 3

ac = × ⁻ . As shown in Fig. 3.16, the effect of crowning especially in the profile direction can be clearly observed at z=0mm.

3.3 Design optimization for robustness of gear transmission error 3.3.1 Transmission error analysis of the shaved gear

Transmission error of the shaved gears can be calculated by simulation of gear meshing.

Considering a gear pair composed of a double crowned gear (gear 2) and an involute gear (gear 4, directly generated by rack cutter with a_c = ), the coordinate systems can be 0 illustrated as Fig. 3.17. Coordinate system S X Y Z₂( ₂, ,_{2 2}) is fixed on gear 2, and operating. Transforming the vectors and unit normal vectors of the gear tooth surfaces to coordinate system S X Y Z_q( _q, ,_{q q}), the two meshing surfaces ∑ and ₂ ∑ must satisfy ₄ respective tooth surfaces illustrated in Figs. 3.18.

In the 3-D space, Eqs. 3.20 and 3.21 include six scalar equations. By the given equation

that for unit normal vectors n⁽_q²⁾ =n⁽_q⁴⁾ =1, only five independent scalar equations need to be

Considering meshing equations Eqs. 3.15 (gears 2 and 4), 3.17 and 3.18 simultaneously, 9 independent equations are used to solve 10 variables including

u

_c,

θ

_c,

ψ

_s,

φ

_s,

z

_t,

φ

₂

′

,

u

4,

θ

₄,

ψ

_s

′

and

φ

₄

′

.

u

_c,

θ

_c,

ψ

_s,

φ

_s and

z

_tare variables for deriving shaved gear (gear 2);

u

₄,

θ

₄ and

ψ

_s

′

are variables for deriving involute gear (gear 4). By inputting values of

φ

₂

′

directly, the relation between real rotating angles

φ φ

₄′ ′( )₂ can be calculated.

Transmission error is defined as the differences between the real and the theoretical rotation

Transmission error is defined as the differences between the real and the theoretical rotation

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