Construction of coordinate systems and the shaved gear tooth profile 11

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.