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

Abstract

Gear power-honing is mainly applied to finish small and medium-sized automotive gears, especially in new energy vehicles. The distinctive curved surface texture greatly improves the noise emission and service life of honed gears. The surface texture for honed gear considering the motion path and geometrical shape of abrasive particles has not been investigated. In this paper, the kinematics of the gear honing process is analyzed, and the machining marks produced by the abrasive particles of honing wheel scratching abrasive particles against the workpiece gear are calculated. The tooth surface roughness is modeled considering abrasive particle shapes and material plastic pile-ups. This results in a mathematical model that characterizes the structure of the tooth surface and the orientation of the machining marks. Experiments were used to verify the model, with a maximum relative error of less than 10% when abrasive particles are spherical. Based on this model, the effects of process parameters on the speeds of discrete points on the tooth flank, orientations of machining marks and roughness are discussed. The results show that the shaft angle between the workpiece gear and the honing wheel and the speed of the honing wheel is the main process parameters affecting the surface texture. This research proposes a surface texture model for honed gear, which can provide a theoretical basis for optimizing process parameters for gear power-honing.

1 Introduction

Hardened gear is a key component of the transmission system of new energy vehicles (NEVs), which is one of the main noise sources of NEVs [1]. Finishing processes for hardened gears have received increased attention as NEVs place higher demands on reducing gear meshing noise [2]. Gear grinding and gear power honing are the two main methods for finishing hardened gears [3]. Compared with gear grinding, gear power honing can avoid tooth surface burn due to its low grinding speed [4]. On the other hand, power honing will produce a unique texture on the tooth surface, resulting in low noise performance [4,5,6,7]. This is one of the most important advantages of power honing. However, the mechanism for forming features such as texture distribution and roughness on power-honed tooth surfaces has not been adequately investigated in published sources. Therefore, this study aims to discuss the honed gear surface texture formation mechanism and propose a modeling method to provide a theoretical basis for optimizing process parameters.

Some scholars have conducted in-depth research on the processing mechanism, forming characteristics of gear honing and the optimization and modification of honed gear [8,9,10,11,12,13]. For example, Bergs [14] calculated the velocity at the contact point between the tool and the workpiece and proposed a gear-honing process force model to improve the robustness of the honing process. Liang et al. [15] established the vector expression of cutting speed at the contact point for gear external honing and introduced a relationship between cutting speed at contact points and the surface quality of gears. Furthermore, it confirmed that the surface texture of the honed gear has a visible difference from the top to the root of the tooth. Talu et al. [16] compared the tooth surface textures of straight bevel gears using pulsed electrochemical honing and pulsed electrochemical finishing processes to provide support for process parameter optimization. Yuan et al. [17] proposed a tooth surface roughness modeling method based on the conical abrasive grain assumption, established a roughness model along the tooth surface and tooth orientation direction, and investigated the effect of the axis intersection angle on the roughness. Han et al. [18] developed a roughness model based on experimental data using the response surface method and similarly discussed the effect of process parameters on roughness. The above models can reflect the complex motion and contact states in gear honing. However, there are some shortcomings in these models regarding surface textures. For example, the modeling process may ignore the influence of different abrasive shapes and the variation of the roughness distribution at different gradients of the tooth surface due to specific textures. Yang et al. [19] have mentioned that the irregular texture produced by power honing can reduce gear meshing noise. Therefore, it is necessary to study the surface texture by obtaining the machining marks on the honed gear first. Then the process parameters are guided to reduce the NVH of NEVs by using the surface texture characteristics of honing gears.

The gear-honing tool is essentially a grinding wheel in the shape of an internal helical gear [20]. The workpiece material is removed and the tooth flank is finished by rubbing, ploughing, and cutting abrasive particles. In the process, machining marks are generated. Ono et al. [21] built the classical theory of grinding mechanism based on the concept of subsequent grinding edge. Based on the assumption of uniformly distributed abrasive particles, Wang et al. [22] obtained the surface roughness of an aero-engine blade with different curvature variations by superimposing a simulation model of contact deformation at the grinding interface with the equations of the trajectory of the abrasive grains. Kang et al. [23] constructed a model of the distribution, size, and shape of active and broken abrasive grains based on image processing data and established a three-dimensional morphological model of the grinding wheel that agrees with the actual. Elwasli et al. [24] discussed the effect of initial surface topography on wear and friction behavior and accurately predicted the wear behavior of smooth and rough surfaces. Xiong et al. [25] studied the mechanism of cutting-edge burr formation by considering the material build-up effect in the micro drill grinding process. Moreover, much concrete research discussed the relationship between the surface roughness and the number of effective abrasive particles according to the mass of grinding test data [26, 27]. These researches establish many roughness models considering the abrasive particle shape, material plastic pile-up and effective abrasive particle number for ground surface topography/structure. However, research on gear has focused on optimizing grinding process parameters by investigating the mechanism of surface texture generation on ground teeth [8, 28]. The model of surface texture for honed gear considering these characteristics is still scarce. Therefore, it is important to study the honed surface texture by calculating its roughness considering the above characteristics.

The literature review shows that modeling the surface texture of honed gears suffers from a lack of consideration of different abrasive shapes and the distribution of different gradients of roughness on tooth surfaces. In conjunction with grinding theory, a theoretical model of the surface texture of honed gears is proposed in this study to address these gaps. The machining marks on the tooth flank were obtained by numerical calculation. The roughness was modeled considering the effects of abrasive particle geometrical shape and material plastic pile-up to quantify the honed surface texture. A series of experiments on gear power-honing were carried out to verify the effectiveness and robustness of the roughness model. Moreover, the speeds of discrete points on the tooth flank, the orientations of machining marks and the roughness of honed gear were discussed under different processing parameters.

2 Theoretical Model

The common application of gear power-honing is the hard finishing of external gears. In this case, the contact between a workpiece and a honing tool with geometrically undefined cutting edges is equivalent to the contact of a gear pair with intersecting axes. In the modeling, a series of machining marks are obtained by numerical calculation, generated by scratching of abrasive particles, and superimposed to build the surface texture of honed gear.

2.1 Kinematics of Gear Honing Process

The relationship between workpiece angular velocity ωw and honing tool angular velocity ωh is expressed as:

$$\frac{{\left| {{\varvec{\omega}}_{{\text{w}}} } \right|}}{{\left| {{\varvec{\omega}}_{h} } \right|}} = \frac{{\varphi_{{\text{w}}} (t)}}{{\varphi_{{\text{h}}} (t)}} = \frac{{n_{{\text{h}}} }}{{n_{{\text{w}}} }} = i,\quad(t \ge 0),$$
(1)

where φw(t) and φh(t) are rotation angles of workpiece and honing tool, which are equal to the dot product of workpiece angular velocity ωw and honing tool angular velocity ωh over time, respectively; nw and nh 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 Swf (Owf-xwf-ywf-zwf), a honing tool fixed coordinate system Shf (Ohf-xhf-yhf-zhf), a workpiece rotation coordinate system Sw (Ow-xw-yw-zw) and a honing tool rotation coordinate system Sh (Oh-xh-yh-zh). The origins of coordinate systems Swf and Sw are located at the centroid of workpiece. The origins of coordinate system Shf and Sh are fixed at the centroid of honing tool. Any two axes in a coordinate system are perpendicular to each other. The axes xwf, xw, xhf, and xh coincide and pass through the centroids of the workpiece and honing wheel. The axes zw and zh 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 (βh) and helix angle of workpiece (βw), i.e., γ = |βhβw|.

Figure 1
figure 1

Spatial coordinate systems of workpiece and honing tool and schematic of the contact between workpiece and honing tool

In coordinate system Sw, the tooth flank of workpiece is an involute helicoid. The tooth flank equation rw(ζ, η) of workpiece is [29]:

$${\varvec{r}}_{\text{w}} (\zeta ,\eta ) = \left[ {\begin{array}{*{20}l} {x_{\text{w}}} \hfill \\ {y_{\text{w}} } \hfill \\ {z_{\text{w}} } \hfill \\ \end{array} } \right]\text{ = }\left[ {\begin{array}{*{20}l} {r_{\text{b}} \cos (\eta { + }\zeta ) + r_{\text{b}} \zeta \text{sin}(\eta { + }\zeta )} \\ {r_{\text{b}} \sin (\eta { + }\zeta ) - r_{\text{b}} \zeta cos(\eta + \zeta )} \\ {\eta p_{{\text{w}}} } \\ \end{array} } \right],$$
(2)

where rb is the basic circle radius of workpiece; η is an angle parameter of the intersection point between involute and base circle, which varies from ηs to ηf; ζ is an angle parameter of involute, which varies from ζs to ζf; pw is the workpiece lead.

The relative velocity v between workpiece and honing tool can be calculated by:

$$\begin{aligned} \varvec{v} &= {\varvec{v}_{\text{wf},\text{w}}} - {\varvec{v}_{\text{wf},\text{h}}} \\ &= {\varvec{\omega}_{\text{wf},\text{w}}} \times {\varvec{r}_{\text{wf},\text{w}}}(\zeta ,\eta ,{\varphi_{\text{w}}}(t)) - {\varvec{\omega}_{\text{wf,h}}} \times {\varvec{r}_{\text{wf,h}}}(\zeta ,\eta ,{\varphi_{\text{w}}}(t)) \\ &= {\omega_{\text{w}}}\left[ \begin{array}{l} \begin{aligned} &- {y_{\text{w}}}(1 - \cos \gamma /i) + {z_{\text{w}}}\cos {\varphi_{\text{w}}}(t) \cdot \sin \gamma /i \\ &- d\sin {\varphi_{\text{w}}}(t) \cdot \cos \gamma /i \end{aligned} \hfill \\ \begin{aligned} &{x_{\text{w}}}(1 - \cos \gamma /i) - {z_{\text{w}}}\sin {\varphi_{\text{w}}}(t) \cdot \sin \gamma /i \\ &- d\cos {\varphi_{\text{w}}}(t) \cdot \cos \gamma /i \end{aligned} \\ - ({x_{\text{w}}}\cos {\varphi_{\text{w}}}(t) - {y_{\text{w}}}\sin {\varphi_{\text{w}}}(t) + d) \cdot \sin \gamma /i \end{array} \right], \end{aligned}$$
(3)

where vwf,w and vwf,h are velocities on the tooth flank of workpiece and honing tool, respectively; ωwf,w and ωwf,h are angular velocities of workpiece and honing tool, respectively; rwf,w(ζ, η, φw(t)) and rwf,h(ζ, η, φw(t)) are the tooth flank equations of workpiece and honing tool in coordinate system Swf, respectively.

According to the meshing theory, the meshing equation is expressed as follows:

$${\varvec{v}}\left( {\zeta_{1} ,\eta_{1} ,\varphi_{{\text{w}}} (t)} \right) \cdot {\varvec{n}} = 0,$$
(4)

where the normal vector n of workpiece tooth flank in coordinate system Swf is obtained by Eq. (5):

$$\begin{aligned} {\varvec{n}} &= \left[ {\begin{array}{*{20}l} {n_{x} } \hfill & {n_{y} } \hfill & {n_{z} } \hfill \\ \end{array} } \right]^{{\text{T}}} \\ &{ = }\left[ {\begin{array}{*{20}l} {\zeta \cdot r_{{\text{b}}} \cdot p_{{\text{w}}} \cdot \sin (\eta + \zeta + \varphi_{{\text{w}}} (t))} \\ { - \zeta \cdot r_{{\text{b}}} \cdot p_{{\text{w}}} \cdot \cos (\eta + \zeta + \varphi_{{\text{w}}} (t))} \\ {\zeta \cdot r_{{\text{b}}}^{2} } \\ \end{array} } \right], \\ \end{aligned}$$
(5)

where nx, ny, and nz are the components of normal vector n along xwf, ywf, and zwf coordinate axes, respectively.

Substituting Eqs. (3) and (5) into Eq. (4), the meshing equation can be changed as:

$$U\cos \varphi_{{\text{w}}} (t) - V\sin \varphi_{{\text{w}}} (t) = W,$$
(6)

where U, V, and W are expressed as:

$$\left\{ {\begin{array}{*{20}l} {U = - (n_{{{\text{w,}}z}} \cdot x_{{\text{w}}} - n_{{{\text{w,}}x}} \cdot z_{{\text{w}}} ) \cdot \sin \gamma - n_{{{\text{w,}}y}} \cdot d \cdot \cos \gamma ,} \hfill \\ {V = - (n_{{{\text{w,}}z}} \cdot y_{{\text{w}}} - n_{{{\text{w,}}y}} \cdot z_{{\text{w}}} ) \cdot \sin \gamma + n_{{{\text{w,}}x}} \cdot d \cdot \cos \gamma ,} \hfill \\ {W = (\cos \gamma - 1/i) \cdot (n_{{{\text{w,}}y}} \cdot x_{{\text{w}}} - n_{{{\text{w,}}x}} \cdot y_{{\text{w}}} ) + n_{{{\text{w,}}z}} \cdot d \cdot \sin \gamma .} \hfill \\ \end{array} } \right.$$
(7)

The honing tool tooth profile equation is obtained, as shown in Eq. (8)

$$\left\{ {\begin{array}{*{20}{l}} {{\varvec{r}_w} = {{\varvec{r}_w}}(\zeta ,\eta ),} \\ {{\varvec{r}_h} = {\varvec{M}_{\text{h,w}}}{\varvec{r}_w},} \\ {U\cos {\varphi_{\text{w}}}(t) - V\sin {\varphi_{\text{w}}}(t) = W,} \end{array}} \right.$$
(8)

where Mh,w is the transformation matrice for coordinate systems Sw-Sh.

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).

$$U^{2} + V^{2} \ge W^{2} .$$
(9)

The derivation of Eq. (9) is shown in Eqs. (10)–(16). Firstly, Eq. (6) can be changed as:

$$\cos (\delta + \varphi_{{\text{w}}} (t)) = \frac{W}{{\sqrt {U^{2} + V^{2} } }},$$
(10)

where δ = arctan(V/U). Eq. (10) has a real solution only when −1 ≤ cos(δ + φw(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:

$$f(\zeta ,\eta ,\varphi_{{\text{w}}} (t)) = U\cos \varphi_{{\text{w}}} (t) - V\sin \varphi_{{\text{w}}} (t) - W = 0.$$
(11)

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

$$\frac{\partial f}{{\partial \varphi_{{\text{w}}} (t)}}(\zeta ,\eta ,\varphi_{{\text{w}}} (t)) = - U\sin \varphi_{{\text{w}}} (t) - V\cos \varphi_{{\text{w}}} (t).$$
(12)

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

$$\left\{ \begin{aligned} &U^{2} \cos^{2} \varphi_{{\text{w}}} (t) + V^{2} \sin^{2} \varphi_{{\text{w}}} (t) \hfill \\& - 2UV\sin \varphi_{{\text{w}}} (t)\cos \varphi_{{\text{w}}} (t) = W^{2} , \hfill \\ &U^{2} \sin^{2} \varphi_{{\text{w}}} (t) + V^{2} \cos^{2} \varphi_{{\text{w}}} (t) \hfill \\& + 2UV\sin \varphi_{{\text{w}}} (t)\cos \varphi_{{\text{w}}} (t) = (\partial f/\partial \varphi_{{\text{w}}} (t))^{2} . \hfill \\ \end{aligned} \right.$$
(13)

Finally, it can be obtained as:

$$U^{2} + V^{2} = W^{2} + (\partial f/\partial \varphi_{{\text{w}}} (t))^{2} .$$
(14)

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

$$\frac{\partial f}{{\partial \varphi_{{\text{w}}} (t)}}(\zeta ,\eta ,\varphi_{{\text{w}}} (t)) = 0.$$
(15)

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

$$\left\{ {\begin{array}{*{20}l} {{\varvec{r}}_{{\text{wf,w}}} = {\varvec{r}}_{{\text{wf,w}}} (\zeta ,\eta ,\varphi_{{\text{w}}} (t)),} \hfill \\ {f(\zeta ,\eta ,\varphi_{{\text{w}}} (t)) = 0,} \hfill \\ {\partial f(\zeta ,\eta ,\varphi_{{\text{w}}} (t))/\partial \varphi_{{\text{w}}} (t) = 0.} \hfill \\ \end{array} } \right.$$
(16)

2.2 Surface Modeling of Honed Gear

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 Be, and its position vector expression (Be) must meet the requirements of Eq. (17).

$$\left\{ {\begin{array}{*{20}{l}} {{\varvec{B}_e}(T) - {\varvec{r}_{\text{w}}}({\zeta_{e1}},{\eta_{e2}}) = 0 \quad (e = 1,2, \ldots ,k; \, e1 \leq m; \, e2 \leq n), \, } \\ {U({\zeta_{e1}},{\eta_{e2}}) \cdot \cos {\varphi_{\text{w}}}(T) - V({\zeta_{e1}},{\eta_{e2}}) \cdot \sin {\varphi_{\text{w}}}(T) - W({\zeta_{e1}},{\eta_{e2}}) = 0.} \end{array}} \right.$$
(17)

When one of ζe1 or ηe2 is fixed, another is solved by carrying out numerical iterative search algorithms like Newton-Raphson method.

In this section, the dynamic meshing contact point Be 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 Swf for honed gear as following procedures (Table 1).

Table 1 Surface modelling process for honed gears

Step 1: A point Be is obtained under a certain rotation angle φh(T), as shown in Figure 2a. The point Be(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 Γe is taken as an example.

Figure 2
figure 2

Modeling process of surface texture of honed gear

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

$${\varvec{B}}_{e}^{(1)} = {\varvec{B}}_{e}^{(0)} + \Delta \varphi_{{{\text{h}},e}} \cdot {\varvec{v}}_{e}^{(0)} ,$$
(18)

where ve(0) satisfies to the Eq (3).

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

$${\varvec{\varGamma}}_{e}^{(1)} = {\varvec{B}}_{e}^{(1)} - {\varvec{B}}_{e}^{(0)} .$$
(19)

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

$$\begin{aligned} {\varvec{B}}_{e}^{(j)}& = {\varvec{B}}_{e}^{(j - 1)} + \Delta \varphi_{{{\text{h}},e}} \cdot {\varvec{v}}_{e}^{(j - 1)} \\ &= {\varvec{B}}_{e}^{(0)} + \Delta \varphi_{{{\text{h}},e}} \cdot {\varvec{v}}_{e}^{(0)} + \Delta \varphi_{{{\text{h}},e}} \cdot {\varvec{v}}_{e}^{(1)} + ... + \Delta \varphi_{{{\text{h}},e}} \cdot {\varvec{v}}_{e}^{(j - 1)} \\ &= {\varvec{B}}_{e}^{(0)} + \Delta \varphi_{{{\text{w}},e}} /i \cdot {\varvec{v}}_{e}^{(0)} + \Delta \varphi_{{{\text{w}},e}} /i \cdot {\varvec{v}}_{e}^{(1)} + ... + \Delta \varphi_{{{\text{w}},e}} /i \cdot {\varvec{v}}_{e}^{(j - 1)} , \\ \end{aligned}$$
(20)
$$\begin{aligned}{\varvec{\varGamma}}_{e}^{(j)} &= {\varvec{B}}_{e}^{(j)} - {\varvec{B}}_{e}^{(j - 1)} = \Delta \varphi_{{{\text{h}},e}} \cdot {\varvec{v}}_{e}^{(j - 1)} \\ &= \Delta \varphi_{{{\text{w}},e}} \cdot {\varvec{v}}_{e}^{(j - 1)} /i, \\ \end{aligned}$$
(21)

where ve(j−1) is the relative velocity at point Be(j−1).

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

$${\varvec{\varGamma}}_{e} = \left\{ {\left. {{\varvec{\varGamma}}_{e}^{(0)} ,{\varvec{\varGamma}}_{e}^{(1)} ,{\varvec{\varGamma}}_{e}^{(2)} ,...,{\varvec{\varGamma}}_{e}^{(m)} } \right\}} \right..$$
(22)

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

$$S_{{\text{h}}} = \left\{ {{\varvec{\varGamma}}_{1} ,{\varvec{\varGamma}}_{2} ,{\varvec{\varGamma}}_{3} ,...,{\varvec{\varGamma}}_{e} ,...,{\varvec{\varGamma}}_{k} } \right\}.$$
(23)

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:

$$\lambda = \arccos \left( {\frac{{({\varvec{B}}_{e}^{(j)} - {\varvec{B}}_{e}^{(j - 1)} ) \cdot \frac{{\partial {\varvec{r}}_{{\text{w}}} ({\varvec{B}}_{e}^{(j)} )}}{\partial \zeta }}}{{\left| {{\varvec{B}}_{e}^{(j)} - {\varvec{B}}_{e}^{(j - 1)} } \right| \cdot \left| {\frac{{\partial {\varvec{r}}_{{\text{w}}} ({\varvec{B}}_{e}^{(j)} )}}{\partial \zeta }} \right|}}} \right) \cdot \frac{180^\circ }{{\uppi }}.$$
(24)

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

$$\lambda = 180^\circ - \arccos \left( {\frac{{({\varvec{B}}_{e}^{(j)} - {\varvec{B}}_{e}^{(j - 1)} ) \cdot \frac{{\partial {\varvec{r}}_{{\text{w}}} ({\varvec{B}}_{e}^{(j)} )}}{\partial \zeta }}}{{\left| {{\varvec{B}}_{e}^{(j)} - {\varvec{B}}_{e}^{(j - 1)} } \right| \cdot \left| {\frac{{\partial {\varvec{r}}_{{\text{w}}} ({\varvec{B}}_{e}^{(j)} )}}{\partial \zeta }} \right|}}} \right) \cdot \frac{180^\circ }{{\uppi }}.$$
(25)

2.3 Roughness Modeling Considering Shape of Abrasive Particle

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 Ra 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:

  1. (1)

    Ignore an effect of vibration from gear honing machine;

  2. (2)

    ignore the influence of thermal deformation on gear honing machine;

  3. (3)

    and the abrasive particles are distributed uniformly on honing tool surface.

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:

$$H = \left( {\frac{15}{{16}} \cdot\Delta ^{3} \cdot \cot \theta \cdot p} \right)^{q} ,$$
(26)

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:

$$p = \frac{1}{c} \cdot \frac{{v_{{{\text{w},\Gamma }}} }}{{v_{{{\text{w},\Gamma }}} - v_{{{\text{h},\Gamma }}} }}\sqrt {\frac{1}{2}\left(\frac{1}{{\rho_{{\text{w}}} }} - \frac{1}{{\rho_{{\text{h}}} }}\right)} ,$$
(27)

where c is the grinding times during a period of spark-out honing, which is related to honing tool speed and spark-out time; ρw and ρh 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 κw and κh along the orientation of machining mark:

$$\kappa_{{\text{w}}} = {1 \mathord{\left/ {\vphantom {1 {\rho_{{\text{w}}} }}} \right. \kern-0pt} {\rho_{{\text{w}}} }},$$
(28)
$$\kappa_{{\text{h}}} = {1 \mathord{\left/ {\vphantom {1 {\rho_{{\text{h}}} }}} \right. \kern-0pt} {\rho_{{\text{h}}} }}.$$
(29)

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

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

$$v_{{{\text{w},\Gamma }}} = {\varvec{v}}_{{\text{wf,w}}} \cdot {\varvec{v}} \cdot \frac{1}{{\left\| {\varvec{v}} \right\|}}.$$
(30)

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

$$v_{{{\text{h},\Gamma }}} = {\varvec{v}}_{{\text{wf,h}}} \cdot {\varvec{v}} \cdot \frac{1}{{\left\| {\varvec{v}} \right\|}}.$$
(31)

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]

$$\Delta = \left\{ \begin{aligned} & 65.3M_{\text{od}}^{ - 1.4} \sqrt[3]{\frac{{\uppi }}{{32 - S_{\text{tr}} } \cdot \frac{1}{N_{\text{e}} }}}\quad{\text{for conical abrasive particle,}} \hfill \\& 137.9M_{\text{od}}^{ - 1.4} \sqrt[3]{\frac{\uppi }{32 - S_{\text{tr} } \cdot \frac{1}{N_{\text{e} }}}}\quad{\text{for spherical abrasive particle,}} \hfill \\ \end{aligned} \right.$$
(32)

where Mod is a granularity of abrasive particles; Str is an organization number; Ne is a coefficient on the number of effective abrasive particles [8]. By substituting Eq. (32) into Eq. (26), the maximum groove heights Hc and Hs 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 hc and hs corresponding to conical and spherical abrasive particles is calculated by Eq. (33).

$$\left\{ {\begin{array}{*{20}{l}} {{h_{\text{c}}} = \sqrt {\frac{1 - \tau }{2}} {H_{\text{c}}} }&{ {\text{for conical abrasive particle,}}} \\ {{h_{\text{s}}} = \frac{{\sqrt {1 - \tau } }}{2}{H_{\text{s}}}}&{\text{for spherical abrasive particle,}} \end{array}} \right.$$
(33)

where τ is the material removal rate as shown in Eq. (34) [32].

$$\tau = \frac{{A - \left( {A_{1} + A_{2} } \right)}}{A},$$
(34)

where the A, A1 and A2 are shown in Figure 3 and can be obtained from basic geometric operations.

Figure 3
figure 3

Cross section of scratch grooves including material pile-up ridges caused by scratching of a conical abrasive particle and b spherical abrasive particle. The cross section is vertical to movement direction of abrasive particle

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

$$H = \left\{ {\begin{array}{*{20}l} {H_{{\text{c}}} + h_{{\text{c}}} \quad{\text{for conical abrasive particle,}}} \hfill \\ {H_{{\text{s}}} + h_{{\text{s}}}\quad {\text{for spherical abrasive particle}}{.}} \hfill \\ \end{array} } \right.$$
(35)

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

$$R_{{\text{a}}} = 0.256H.$$
(36)

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 Ra will be obtained for honed gear.

3 Experimental Design

To validate the theoretical model, a series of honed gears were prepared under different process parameters which were selected from an effective range of gear-honing machines [17]. When the tooth number of honing tool is set to the maximum value of 113, the value of the rotation speed of honing tool, 876.11 r/min, is approximately at the midpoint of effective parameter range (720–1000 r/min); when the tooth number of honing tool is set to the minimum value of 105, the value of the rotation speed of honing tool, 900.95 r/min, is approximately at the golden section point of effective parameter range. The gears were first machined by a gear grinding machine (YW7232 CNC, CHMTI) under the same process parameters. The gear material was 20CrMnTiH. Next, the gears were finished by a gear honing machine (HMX-400, Fassler), as shown in Figure 4a. The parameters of workpiece and honing tool were shown in Table 2. The binder material of honing tool was ceramic, and the material of abrasive particles was microcrystalline corundum, as shown in Figure 4b. The infeed rate of honing tool, oscillation time and spark-out time are 0.408 mm/min, 45 s and 3.6 s, respectively.

Figure 4
figure 4

Experimental design for honing tests: a Gear honing machine and honed gears, b Honing tool and its surface abrasive particles, c Single tooth and its measurement areas

Table 2 Workpiece and honing tool parameters

The tooth flank structure and its roughness were measured. As shown in Figure 4c, the honed gears were cut by a diamond wire cutting machine to obtain many single teeth. The single teeth were cleaned by an ultrasonic cleaning machine with alcohol for at least 10 min. Seven small areas were selected from each tooth flank along the tooth longitudinal and tooth profile directions. The surface texture for every single tooth was obtained by an ultra-depth 3D microscope (VHX-1000C, Keyence). The local surface texture and roughness for each area were obtained by laser scanning confocal microscope (OLS4000, Olympus Corporation).

4 Results and Discussions

4.1 Experimental Results and Model Verification

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 Ra measured from the position of three lines for each area, the roughness Ra for each area is obtained, as shown in Table 3. According to the workpiece parameters, honing tool parameters, and process parameters, the roughness Ra is calculated by the proposed model, also shown in Table 3. The maximum relative error between the measured and calculated values of Ra 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 Ra 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 Ra 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.

Figure 5
figure 5

Measurement results of a complete surface texture obtained by an ultra-depth 3D microscope, b local surface textures of seven measured areas obtained by a laser scanning confocal microscope

Table 3 Process parameters for theoretical investigation

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 nh, and rotational speed of honing tool uh. The speeds include vwf,w, vwf,h and v. The values of vwf,w, vwf,h, v, angle λ, and roughness Ra 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.

Table 4 Process parameters for theoretical investigation

4.2 Effects of Process Parameters on Speed

Figure 6a, c and e shows three contour maps for speeds vwf,w, vwf,h and v calculated under γ = 5°, nh = 113, and uh = 860 r/min, respectively. It has been found that the X values of vwf,w and vwf,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 vwf,w, vwf,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 vwf,w and vwf,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 vwf,w, vwf,h, and v at X = 0 mm under different process parameters (Table 4). It can be observed from Figure 6b that the values of vwf,w do not change with cross axis angle γ, while it increases with tooth number nh or rotational speed uh 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 vwf,w with tooth number nh and rotational speed uh can be explained effectively. In Figure 6d, the value of vwf,h is negatively correlated with angle γ, while positively correlated with tooth number nh and rotational speed uh. The change of the values of vwf,h with angle γ can be explained by Eqs. (2) and (3). The diameter of honing tool increases with tooth number nh, which introduces the increasing rotation radius of each contact point, thereby leading to a rising vwf,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 nh or rotational speed uh. This is because the relative speed v is positively related to the magnitude of angular velocity ωw, which increases with tooth number nh and rotational speed uh according to Eq. (1).

Figure 6
figure 6

Contour maps for a speed vwf,w (mm/s) of workpiece along the direction of machining marks, c speed vwf,h (mm/s) of honing tool along the direction of machining mark, and e relative speed v (mm/s) calculated with γ = 5°, nh = 113, and uh = 860 r/min; b vwf,w (mm/s), d vwf,h (mm/s) and f v (mm/s) under different process parameters at X = 0 mm

4.3 Effects of Process Parameters on Orientation of Machining Mark

Figure 7 shows the complete surface texture of honed gear calculated under γ = 5°, nh = 113, and uh = 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°, nh = 113, and uh = 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 7
figure 7

Calculated complete surface texture of honed gear under γ = 5°, nh = 113, and uh = 860 r/min

Figure 8
figure 8

Calculated angle λ (degree) for machining marks: a the contour map under γ = 5°, nh = 113, and uh = 860 r/min, b the line chart under different process parameters at X = 0 mm

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 nh 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 uh at a fixed Y value. This is because the honing tool rotational speed uh has an impact on machining efficiency, which is independent of the orientation of machining mark.

4.4 Effects of Process Parameters on Roughness

Figure 9a shows the contour map for tooth flank roughness Ra calculated under γ = 5°, nh = 113, and uh = 860 r/min based on the hypothesis of spherical abrasive particles. It can be observed that the value of Ra first increases and then decreases gradually from tooth root toward tooth top. The value of Ra 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 Ra is negatively related to the ratio of vwf,w to vwf,h which is a fixed value shown in Figure 6. Besides, it is positively related to curvature κw,Γ while negatively related to curvature κh,Γ. And the variation range of curvature κh,Γ is far less than that of curvature κw,Γ. Therefore, the change of curvature κw,Γ is a major factor for the change of roughness Ra. Figure 10 displays the calculated results for curvature κw,Γ under γ = 5°, nh = 113, and uh = 860 r/min.

Figure 9
figure 9

Calculated roughness Ra (μm) for honed gear: a the contour map under γ = 5°, nh = 113, and uh = 860 r/min, b the line chart under different process parameters at X = 0 mm

Figure 10
figure 10

Curvature κw,Γ (mm−1) under γ = 5°, nh = 113, and uh = 860 r/min

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

5 Conclusions

In this paper, a theoretical model of surface texture for honed gear is developed. The machining marks on tooth flank are obtained by numerical calculation. Considering the geometrical shape of abrasive particles and the plastic pile-up of gear material, a roughness model is established and then verified by experiments. The effects of process parameters on the speeds of discrete points on the tooth flank, orientations of machining marks and roughness are discussed.

The detailed conclusions are described as follows:

  1. (1)

    The calculated surface texture and machining marks orientations show similar laws to the experimental results. The orientation angle of machining mark first increases and then decreases gradually from the tooth root toward tooth top, while it always remains unchanged from the left end to the right end. The angle has a complex change with the increase of cross axis angle. Additionally, the angle changes little with the increase of tooth number, and has no change with a change of rotational speed of honing tool.

  2. (2)

    The model of roughness Ra has a high calculation accuracy with a maximum error of less than 10% when the abrasive particles are assumed to be spherical. The distribution of roughness is non-uniform from tooth root to tooth top. With the increase of angle cross axis angle, the roughness decreases obviously, and the non-uniform degree of tooth flank roughness is improved effectively. The roughness decreases with the increasing tooth number or speed of honing tool.

  3. (3)

    The speed of contact point along the direction of machining mark is one of main factors affecting the formation of surface texture. The relative speed first decreases and then increases gradually from the tooth root toward tooth top, while it always remains unchanged from the left end to right end. Besides, the relative speed changes from V-shaped curve to approximate straight-line with the increase of cross axis angle, and positively relates to cross axis angle and honing tool speed, respectively.

The research can provide a theoretical basis for optimization of process parameters for gear power-honing. In future, parameters optimization of honing process will be studied. The surface texture model will also be further optimized, for example by incorporating the influence of the height of the grinding grains.

Data availability

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

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Acknowledgements

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Funding

Supported by National Key Research and Development Plan (Grant No. 2020YFE0201000); Chongqing Municipal Special Postdoctoral Science Foundation (Grant No. XmT20200021); Liuzhou Municipal Science and Technology project (Grant No. 2021AAB0101).

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Contributions

HC and HX were in charge of the whole trial; YL wrote the manuscript; XH revised the manuscript; JW assisted with sampling and laboratory analyses. All authors read and approved the final manuscript.

Authors' Information

Yuhu Liu, born in 1997, is currently a master candidate at State Key Laboratory of Mechanical Transmissions, Chongqing University, China.

Xiaohui Huang, born in 1997, is currently a PhD candidate at State Key Laboratory of Mechanical Transmissions, Chongqing University, China.

Huajun Cao, born in 1978, is currently a professor at Chongqing University, China. He received his PhD degree from Chongqing University, China, in 2004. His research interests include advanced manufacturing technology, green manufacturing and equipment, manufacturing systems engineering.

Jiacheng Wang, born in 1995, is currently a master candidate at State Key Laboratory of Mechanical Transmissions, Chongqing University, China.

Huapan Xiao, born in 1988, is currently an assistant professor at Chongqing University, China. He received his PhD degree from Xi’an Jiaotong University, China, in 2020. His research interests include precision and ultra-precision abrasive processing technology and detection.

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Correspondence to Huajun Cao.

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Liu, Y., Huang, X., Cao, H. et al. Model of Surface Texture for Honed Gear Considering Motion Path and Geometrical Shape of Abrasive Particle. Chin. J. Mech. Eng. 36, 96 (2023). https://doi.org/10.1186/s10033-023-00910-9

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