Model of Surface Texture for Honed Gear Considering Motion Path and Geometrical Shape of Abrasive Particle

The relationship between workpiece angular velocity ω_w and honing tool angular velocity ω_h is expressed as:where φ_w(t) and φ_h(t) are rotation angles of workpiece and honing tool, which are equal to the dot product of workpiece angular velocity ω_w and honing tool angular velocity ω_h over time, respectively; n_w and n_h are the tooth number of workpiece and honing tool, respectively; i is a transmission ratio.

As shown in Figure 1, spatial coordinate systems of workpiece and honing tool are established, including a workpiece fixed coordinate system S_wf (O_wf-x_wf-y_wf-z_wf), a honing tool fixed coordinate system S_hf (O_hf-x_hf-y_hf-z_hf), a workpiece rotation coordinate system S_w (O_w-x_w-y_w-z_w) and a honing tool rotation coordinate system S_h (O_h-x_h-y_h-z_h). The origins of coordinate systems S_wf and S_w are located at the centroid of workpiece. The origins of coordinate system S_hf and S_h are fixed at the centroid of honing tool. Any two axes in a coordinate system are perpendicular to each other. The axes x_wf, x_w, x_hf, and x_h coincide and pass through the centroids of the workpiece and honing wheel. The axes z_w and z_h are perpendicular to the end faces of workpiece and honing tool, respectively. d is the distance between the workpiece centroid and honing tool centroid. γ is a cross axis angle which is equal to the difference between helix angle of honing tool (β_h) and helix angle of workpiece (β_w), i.e., γ = |β_h − β_w|.

In coordinate system S_w, the tooth flank of workpiece is an involute helicoid. The tooth flank equation r_w(ζ, η) of workpiece is [29]:where r_b is the basic circle radius of workpiece; η is an angle parameter of the intersection point between involute and base circle, which varies from η_s to η_f; ζ is an angle parameter of involute, which varies from ζ_s to ζ_f; p_w is the workpiece lead.

The relative velocity v between workpiece and honing tool can be calculated by:where v_wf,w and v_wf,h are velocities on the tooth flank of workpiece and honing tool, respectively; ω_wf,w and ω_wf,h are angular velocities of workpiece and honing tool, respectively; r_wf,w(ζ, η, φ_w(t)) and r_wf,h(ζ, η, φ_w(t)) are the tooth flank equations of workpiece and honing tool in coordinate system S_wf, respectively.

According to the meshing theory, the meshing equation is expressed as follows:where the normal vector n of workpiece tooth flank in coordinate system S_wf is obtained by Eq. (5):where n_x, n_y, and n_z are the components of normal vector n along x_wf, y_wf, and z_wf coordinate axes, respectively.

Substituting Eqs. (3) and (5) into Eq. (4), the meshing equation can be changed as:where U, V, and W are expressed as:

The honing tool tooth profile equation is obtained, as shown in Eq. (8)where M_h,w is the transformation matrice for coordinate systems S_w-S_h.

During analyzing the contact between workpiece and honing tool, the tooth flank of honing tool is enveloped by the tooth flank of workpiece based on the envelope theory. In gear power-honing, the tooth flank of workpiece is enveloped by the tooth flank of honing tool. To ensure that the tooth flank of honed gear is an involute helicoid, the contact lines produced by above two enveloping processes should be consistent. Consequently, the coordinates of all points on the tooth flank of workpiece need to satisfy Eq. (9).

The derivation of Eq. (9) is shown in Eqs. (10)–(16). Firstly, Eq. (6) can be changed as:where δ = arctan(V/U). Eq. (10) has a real solution only when −1 ≤ cos(δ + φ_w(t))  ≤ 1, i.e., U² + V² ≥ W². Those points on conjugate surfaces corresponding to all real solution form the effective region. Those points satisfying U² + V² = W² construct the meshing boundary line.

Secondly, Eq. (6) can be also written as:

Thirdly, by deriving Eq. (9) from φ_w, it can be obtained as:

By squaring Eqs. (6) and (9), it has

Finally, it can be obtained as:

On the meshing boundary line, U² + V² = W², then

Because ∂f/∂φ_w = 0 is derived from the conjugate meshing equation f = 0, those points on the workpiece tooth flank satisfying ∂f/∂φ_w = 0 and f = 0 form the meshing boundary line.

The distinctive structure of honed gear surface contains numerous curved machining marks which result from the scratching of abrasive particles with the rotation of honing tool and workpiece.

In the process of gear power-honing, any micro-area of the tooth surface is engaged in time sequence, and the dynamic meshing contact point is marked as B_e, and its position vector expression (B_e) must meet the requirements of Eq. (17).

When one of ζ_e1 or η_e2 is fixed, another is solved by carrying out numerical iterative search algorithms like Newton-Raphson method.

In this section, the dynamic meshing contact point B_e is assumed to be the center of the cross section where an abrasive particle of honing tool penetrates workpiece. By the numerical calculation for motion paths of abrasive particles, machining marks are obtained. The surface model of honed gear is established in coordinate system S_wf for honed gear as following procedures (Table 1).

Step 1: A point B_e is obtained under a certain rotation angle φ_h(T), as shown in Figure 2a. The point B_e⁽⁰⁾ (e = 1,2, …, k) is taken as an endpoint of each machining mark Γ. The generation process of the eth (1 ≤ e ≤ k) machining mark Γ_e is taken as an example.

Step 2: By rotating the honing tool with a small discrete rotation angle ∆φ_h,e, point B_e⁽⁰⁾ moves to point B_e⁽¹⁾ with relative velocity v_e⁽⁰⁾, as shown in Figure 2b. The radius vector B_e⁽¹⁾ of point B_e⁽¹⁾ is judged whether it meets Eqs. (6) and (9) simultaneously. If it meets, the equation Γ_e⁽¹⁾ of machining mark Γ_e⁽¹⁾ between two points B_e⁽⁰⁾ and B_e⁽¹⁾ is obtained; if it does not, the calculation stops. The radius vector B_e⁽¹⁾ is expressed as:where v_e⁽⁰⁾ satisfies to the Eq (3).

The equation Γ_e⁽¹⁾ of machining mark Γ_e⁽¹⁾ is obtained by:

Step 3: By continuously rotating the honing tool with m small discrete rotation angles ∆φ_h,e, point B_e⁽¹⁾ eventually moves to point B_e^(m), as shown in Figure 2c. It is assumed that radius vector B_e^(m+1) of point B_e^(m+1) does not meet Eq. (6) or Eq. (9). The radius vector B_e^(j) of point B_e^(j), and the equation Γ_e^(j) of machining mark Γ_e^(j) between two points B_e^(j−1) and B_e^(j) (j = 1, 2, 3, …, m) are respectively expressed as:where v_e^(j−1) is the relative velocity at point B_e^(j−1).

Step 4: As shown in Figure 2d, the equation Γ_e of eth machining mark Γ_e is expressed as:

Step 5: All machining marks Γ₁, Γ₂, …, Γ_k are calculated one by one, which constitute a complete honed tooth flank (S_h) with a distinctive structure. The S_h can be expressed as:

The machining marks produced by gear power-honing are some three-dimensional space curves. In this paper, the orientation of machining mark is characterized through angle between machining mark direction and tooth profile direction, as shown in Figure 2d, which is expressed as:

If the calculated angle λ is larger than 90° by Eq. (24), angle λ is calculated by Eq. (25):

The material removal mechanism of cylindrical gear power-honing is similar to that of cylindrical grinding. The abrasive particles move along the machining marks and remove gear material, forming a series of micro-grooves. Roughness R_a is selected as a quantitative index to evaluate the distinctive surface texture. In this section, a roughness model is developed considering the effects of abrasive particle geometrical shape and material plastic pile-up. The following assumptions are made firstly:

Based on the above assumptions, Ono et al. [21] proposed that the maximum groove height H caused by the scratching of abrasive particles on ground surface:where ∆ is an average spacing between adjacent abrasive particles; θ is a half apex angle of the abrasive particle; p is a parameter determined by grinding conditions; q is a parameter related to the shape and distribution of abrasive particles (q = 0.4 and 0.5 for conical and spherical abrasive particles, respectively).

However, the gear power-honing has more complex contact between workpiece and tool, and lower cutting speed, which differs from cylindrical grinding process. The complex contact mainly affects the orientation of machining marks. The equivalent curvature radius of workpiece is different from the directions on a tooth flank. Therefore, the influence of machining mark orientation and cutting speed needs to be taken into account in the calculation of parameter p, which can be expressed as:where c is the grinding times during a period of spark-out honing, which is related to honing tool speed and spark-out time; ρ_w and ρ_h are the curvature radii of workpiece and honing tool at a discrete point of the contact line, respectively, which are related with the corresponding normal curvatures κ_w and κ_h along the orientation of machining mark:

Considering the width of geometric contact between workpiece and honing tool, the normal curvatures κ_w and κ_h are replaced by average curvatures κ_w,Γ and κ_h,Γ.

The speed v_w,Γ of workpiece along the direction of machining mark is calculated by:

The speed v_h,Γ of honing tool along the direction of machining mark is calculated by:

The material plastic pile-up is an inevitable phenomenon in abrasive machining. The degree of material pile-up is closely related to the hardness of honing tool and the shape/size of abrasive particles. For different shaped abrasive particles, the average spacing ∆ of abrasive particles is [30, 31]where M_od is a granularity of abrasive particles; S_tr is an organization number; N_e is a coefficient on the number of effective abrasive particles [8]. By substituting Eq. (32) into Eq. (26), the maximum groove heights H_c and H_s corresponding to conical and spherical abrasive particles can be obtained.

Figure 3a and b shows the schematics of cross section of grooves caused by scratching of a conical abrasive and spherical abrasive, respectively. The height of material pile-up ridges h_c and h_s corresponding to conical and spherical abrasive particles is calculated by Eq. (33).where τ is the material removal rate as shown in Eq. (34) [32].where the A, A₁ and A₂ are shown in Figure 3 and can be obtained from basic geometric operations.

Therefore, considering the material plastic pile-up, the maximum groove height H becomes

Based on the relationship between the maximum groove height H and average roughness R_a [21], the surface roughness R_a can be established by:

According to the theoretical model, if workpiece parameters, honing tool parameters, and honing process parameters are known, the surface texture of honed gear will be calculated and the distribution of roughness R_a will be obtained for honed gear.

Figure 5a and b shows the surface texture of honed gear and local surface textures of seven small areas selected from one of single teeth. From the surface texture, it has many curved machining marks like a fish skeleton. These machining marks asymptotically approach the pitch line and form a distinctive surface texture. The surface texture allows a nice adhesion of thin oil layer on a whole tooth flank, resulting in a low noise behavior of honed gears [6]. Due to the low cutting speed which usually varies from 0.5 m/s to 15 m/s in gear power-honing, thermal structural damages do not occur for honed gears. Three lines approximately vertical to the orientation of machining marks are selected randomly in each small area. By averaging the roughness R_a measured from the position of three lines for each area, the roughness R_a for each area is obtained, as shown in Table 3. According to the workpiece parameters, honing tool parameters, and process parameters, the roughness R_a is calculated by the proposed model, also shown in Table 3. The maximum relative error between the measured and calculated values of R_a is less than 10% when the shape of abrasive particle is assumed to be spherical. On the contrary, the maximum relative error is close to 40% when the shape of abrasive particle is assumed to be conical. This indicates that the abrasive shape has a great influence on roughness and the spherical abrasive particle has more excellent prediction accuracy. The roughness results for the second and third groups (No. 2 and No. 3 in Table 3) show that roughness R_a increases as the speed of honing tool decreases. By comparing the roughness results for the first and second groups (No. 1 and No. 2 in Table 3), it shows that roughness R_a for the second group does not increase basically when the speed of honing tool decreases, which may be attributed to the decrease of tooth number and helix angle of honing tool.

The model is verified by experimental results, and applied to investigate the effects of process parameters on the distribution of speeds on tooth flank, orientations of machining marks, and roughness theoretically. The process parameters include cross axis angle γ, tooth number of honing tool n_h, and rotational speed of honing tool u_h. The speeds include v_wf,w, v_wf,h and v. The values of v_wf,w, v_wf,h, v, angle λ, and roughness R_a are calculated under different process parameters (Table 4). Then, the distribution characteristics of the orientations of machining marks and roughness are discussed, which are introduced in detail in the following sections.

Figure 6a, c and e shows three contour maps for speeds v_wf,w, v_wf,h and v calculated under γ = 5°, n_h = 113, and u_h = 860 r/min, respectively. It has been found that the X values of v_wf,w and v_wf,h increase linearly, while the value of v first decreases and then increases gradually from tooth root to tooth top at a fixed X value. Meanwhile, the values of v_wf,w, v_wf,h, and v always remain unchanged from the left end to the right end at a fixed Y value. This is because the values of v_wf,w and v_wf,h are proportional to the rotation radius of each contact point. According to the meshing theory of gear, the minimum value of relative speed v is located at pitch line of honed gear and increases gradually toward both sides of pitch line. Figure 6b, d and f shows the calculated values of v_wf,w, v_wf,h, and v at X = 0 mm under different process parameters (Table 4). It can be observed from Figure 6b that the values of v_wf,w do not change with cross axis angle γ, while it increases with tooth number n_h or rotational speed u_h of honing tool. This is because when workpiece parameters are determined, the change of cross axis angle γ is only caused by the change of helix angle of honing tool, which will not affect workpiece speed. Moreover, according to Eq. (1), the change of the value of v_wf,w with tooth number n_h and rotational speed u_h can be explained effectively. In Figure 6d, the value of v_wf,h is negatively correlated with angle γ, while positively correlated with tooth number n_h and rotational speed u_h. The change of the values of v_wf,h with angle γ can be explained by Eqs. (2) and (3). The diameter of honing tool increases with tooth number n_h, which introduces the increasing rotation radius of each contact point, thereby leading to a rising v_wf,h. In Figure 6f, the curve of relative speed v changes from a V-shaped curve to an approximately straight line with an increase of angle γ. This is because that pitch radius of workpiece becomes small with the increase of angle γ, so the abscissa of the minimum relative speed becomes smaller and the corresponding contact point on workpiece tooth flank moves towards the direction of tooth root. When the point exceeds the range of active flank, a phenomenon appears that the relative speed v increases monotonically from tooth root to tooth top. In Figure 6f, the relative speeds v increase with tooth number n_h or rotational speed u_h. This is because the relative speed v is positively related to the magnitude of angular velocity ω_w, which increases with tooth number n_h and rotational speed u_h according to Eq. (1).

Figure 7 shows the complete surface texture of honed gear calculated under γ = 5°, n_h = 113, and u_h = 860 r/min. It can be clearly seen that there are numerous uniquely curved machining marks like a fish skeleton on the tooth flank. Figure 8a shows the contour map for angle λ calculated under γ = 5°, n_h = 113, and u_h = 860 r/min. It can be observed that the angle λ first increases and then decreases gradually from tooth root to tooth top and the maximum value is 90° just located on the pitch line of workpiece. This is consistent with the change rule of machining marks orientation seen in Figure 7.

Figure 8b shows the values of calculated angle λ at X = 0 mm under different process parameters (seen in Table 4). It can be observed that the values of angle λ are always zero when γ = 0°. However, with the increasing angle γ among a range of γ > 0°, the curve of angle λ changes from a convex curve to a drop-down curve. This is because the spindles of honing tool and workpiece are parallel when γ = 0°. In this case, the motion between the workpiece and the tool is equivalent to the motion of a gear pair with parallel axes. Without considering the axial displacement of workpiece, there is no velocity along the longitudinal direction for the workpiece, introducing an unchanged angle λ on the whole tooth flank of workpiece. When 0° < γ ≤ 10°, the component of relative velocity v along the tooth profile direction has a sine function relationship with angle γ according to Eq. (3). What’s more, the abscissa of the maximum relative speed becomes smaller and the corresponding contact point on workpiece tooth flank moves towards the tooth root direction, which has the same reason as that of minimal relative speed v. Thus, as shown in Figure 8b, when the angle γ is 15° or 20°, the angle λ decreases monotonically from tooth root to tooth top. If the angle γ increases to 90°, the axes of honing tool and workpiece will be perpendicular. In this case, the motion between a workpiece and a tool is equivalent to the motion of a gear pair with vertical axes. The machining marks with a longitudinal direction will be generated.

It can be seen from Figure 8b that the angle λ changes slightly with tooth number n_h at a fixed Y value. This is mainly because a small amount of change in tooth number has little effect on the change of transmission ratio i. Thus, the orientation of machining mark changes slightly according to Eqs. (23) and (24). There is no change for angle λ with honing tool rotational speed u_h at a fixed Y value. This is because the honing tool rotational speed u_h has an impact on machining efficiency, which is independent of the orientation of machining mark.

Figure 9a shows the contour map for tooth flank roughness R_a calculated under γ = 5°, n_h = 113, and u_h = 860 r/min based on the hypothesis of spherical abrasive particles. It can be observed that the value of R_a first increases and then decreases gradually from tooth root toward tooth top. The value of R_a always remains unchanged from the left end to the right end at fixed Y value. According to Eqs. (25) and (26), the value of roughness R_a is negatively related to the ratio of v_wf,w to v_wf,h which is a fixed value shown in Figure 6. Besides, it is positively related to curvature κ_w,Γ while negatively related to curvature κ_h,Γ. And the variation range of curvature κ_h,Γ is far less than that of curvature κ_w,Γ. Therefore, the change of curvature κ_w,Γ is a major factor for the change of roughness R_a. Figure 10 displays the calculated results for curvature κ_w,Γ under γ = 5°, n_h = 113, and u_h = 860 r/min.

Figure 9b shows the calculated roughness R_a under different process parameters (seen in Table 4) at X = 0 mm. It can be observed that with the increase of angle γ, roughness R_a decreases obviously. This is mainly because the ratio of v_wf,w to v_wf,h increases obviously with angle γ. There are some odd large values of roughness R_a near the tooth root when γ = 0°. This is because when γ = 0°, the ratio of v_wf,w to v_wf,h is near 1 at the position of pitch line so Eqs. (25) and (26) are not suitable for this condition. The roughness R_a decreases slightly with the increasing tooth number n_h. This is because the transmission ratio i increases slightly with tooth number n_h, which leads to a slight increase in the ratio of v_wf,w to v_wf,h. With the increasing honing tool rotational speed u_h, the roughness R_a decreases accordingly. This is because more abrasive particles are involved in removing workpiece material with the increasing honing rotational speed u_h. Therefore, according to Eqs. (25) and (26), the roughness R_a is reduced correspondingly, while the distribution characteristics of roughness R_a remain unchanged.