Minimum zone fitting and error evaluation for arc correction type convex contour of bearing roller

Abstract

The minimum zone fitting and error evaluation for the arc-modified convex contour of a bearing roller have important applications for consistency detection and quantificational research of the elastohydrodynamic lubrication of a bearing roller. Based on the definition of the shape error and the geometric characteristics of the arc corrected roller convexity line of the bearing, a new fitting and error evaluation method for the total convexity contour of a bearing roller is presented. First, the reference cutoff points of the arc segment and straight line are determined based on the curvature difference of each measurement point. Then, the measuring points on both sides of the two reference cutoff points are selected as auxiliary cutoff points for arc fitting. The fitting error is obtained based on the minimum area method. Finally, a series of tangent equations are obtained based on the tangent principle between a line and two arcs, and the straightness error is determined by calculating the distances between the measuring points and the tangents. The example results show that an arc-modified convex contour can be fitted, and its global error can be evaluated effectively and precisely using the presented method. This study also provides a new idea for the minimum zone fitting of multi-segment curves along a plane.

Keywords

Convexity of bearing roller contour error minimum zone method piecewise curve fitting form error evaluation

Introduction

A rolling bearing is a basic part used in the mechanical industry, which is widely applied in all types of mechanical equipment.¹ A cylindrical (tapered) roller is an important part of a rolling bearing.² To ensure a good performance of the bearing in engineering projects, the generatrix of the bearing roller is usually designed into a convex shape.³ The usual modified generatrix of a bearing roller includes a full arc type, arc correction type and logarithmic curve type. Among these, an arc modified convex roller can make contact along the entire roller length without an edge effect.^4,5 Thus, when the stability of bearing operation is taken as the design criterion, the arc-modified convex structure is widely used in roller crown designs.⁶

The current superfinishing process can achieve a highly efficient and precise modification for the roller crown, and although many enterprises can produce better convex curved rollers, the consistency of the crown is not very high.⁷ The results indicate that a significantly different contact stress and elastohydrodynamic lubrication film can be produced owing to a minor profile error of the convexity line of a modified roller (convexity size error, convexity centre offset error and convexity shape error),^8,9 which affects the performance of the bearing.^10,11

The convexity curve is composed of a middle straight line and two arcs (Figure 1). The fitting method and error evaluation method for an arc and straight line can be found in various studies,^12,13 and many successful fitting methods have been developed.^14–16 However, there have only been a few reports on the overall minimum zone fitting method of multi-segment curves in a plane. Benko¹⁷ and Karniel¹⁸ studied the constraints of various curves and surfaces used in reverse engineering projects, and provided a two-dimensional curve characteristic expression and a curve constraint expression; however, the global constraint fitting problem of a two-dimensional curve has not been solved.^19,20 Therefore, a study on the algorithm of the overall minimum region fitting and an error evaluation for the crown curve of an arc-modified bearing roller is of significant value for a quantitative study of the convexity profile, the formation mechanism of roller crown superfinishing, and the effective fitting of the plane cross-section data applied in reverse engineering projects.

Figure 1.

Arc-modified type convex curve.

The convexity curve of an arc-modified bearing roller is composed of a middle straight line and two arcs, and has obvious geometric characteristics. If a single quadratic function is used to fit a convex curve, it becomes difficult to obtain a precise fitting and good overall effect. To overcome this problem, in this study, a method for piecewise fitting and multi-value optimisation is adopted to realise the curing fitting for the crown curve of a bearing roller. The proposed method overcomes the limitations of existing methods that can only fit a single class of curves, providing a new method to fit a multi-segment curve on a plane. In addition, the proposed method introduces the minimum zone method into the fitting and error evaluation of the multi-segment curve, ensuring that the obtained reference curve better reflects the curve represented by the measurement points and providing more accurate guidance for the correction of the roller machining process.

Overview and scope

Convexity curve equation of arc-modified roller

As shown in Figure 1, the curve of an arc-modified roller refers to a case in which a straight generatrix of the bearing roller is modified into a convex curve, the middle of which is an approximately straight line, and the two ends are indented toward the entity contraction with a variable curvature. For an ideal curve type, the middle straight line is tangential to the two curves at either end. Without a loss of generality, the modified generatrix of a cylindrical (conical) roller can be described through equation (1) (note that in a design, $R_{1} = R_{2}$ is generally required).

{\begin{matrix} {(x - a_{1})}^{2} + {(y - b_{1})}^{2} = R_{1}^{2} (x_{0} \leq x \leq x_{1}) \\ y = Ax + B (x_{1} \leq x \leq x_{2}) \\ {(x - a_{2})}^{2} + {(x - b_{2})}^{2} = R_{2}^{2} (x_{2} \leq x \leq x_{3}) \end{matrix}

(1)

where

{\begin{matrix} A = \frac{b_{1} - b_{2}}{a_{1} - a_{2}} \\ B = \frac{(a_{2} b_{1} - a_{1} b_{2}) \pm R_{1} \sqrt{{(a_{1} - a_{2})}^{2} + {(b_{1} - b_{2})}^{2}}}{a_{2} - a_{1}} \end{matrix}

(2)

Definition and assumption of profile error of arc-modified roller

As shown in Figure 2, a series of measured points are distributed on both sides of the ideal profile. According to the definitions of profile error in ISO, the profile error of the arc-modified roller can be defined as the area between two envelope lines that can contain the actual measured profile. In this study, in order to facilitate the rapid and accurate evaluation of the profile error of the arc-modified roller, first, it is assumed that the number of measurement points is limited but sufficient to describe the profile of the arc-modified roller. Second, it is assumed that the measurement points and reliable. In other words, all measurement points are obtained through error separation techniques (the error separation techniques are not the focus of this study), and there is no random error. Therefore, these measurement points can be used to describe the reality of a roller profile.

Figure 2.

Profile error of arc-modified roller.

The flow of the proposed profile error evaluation method

Figure 3 shows the overview of the proposed method in this study. First, input the measuring points (Step (1)) and according to these points determine the reference dividing points (two points) between the arcs and the straight line (Step (2)). Next, select a certain number of measuring points on both sides of each reference dividing point to form the corresponding auxiliary dividing point group (Step (3)). The actual dividing points of arcs and straight line are included in the range of the point groups. Then, for each arc segment, gradually combine other measuring points of the arc with the corresponding auxiliary dividing points to form a series of data groups (Step (4)). For each data group, the arc centre and equation that realise the arc have the minimum profile error can be calculated by the proposed minimum zone fitting algorithm in this study (Step (5)). And based on the obtained arcs, the fitting equation and fitting error of straight line segment are calculated (Step (6)). By comparing the fitting errors between the arcs and straight line, the overall error of convex curve is determined (Step (7)). Finally, compared with all the overall errors, the minimum value is the minimum profile error of the convex curve, and the corresponding arc and line equations are the minimum zone fitting equations of the convex curve (Step (8)).

Figure 3.

Overall of the profile error evaluation method.

Minimum zone fitting and error evaluation for arc-modified roller

Determine the reference dividing points

Assume that the measured points are expressed by $P_{i} (x_{i}, y_{i}) (i = 1, 2, . . ., N)$ .

A convexity curve is composed of a middle straight line and two arcs. When measuring a crown curve, only a series of measuring point coordinates can be obtained; however, the position of the dividing point between the straight line and the arcs at both ends cannot be obtained. It can be determined from geometric knowledge that although the measuring points are not on the ideal crown curve, the curvature of the measuring points on the straight-line segment, and the curvature of the measuring points on the circular segment, are still significantly different. Therefore, the curvature difference of each measuring point can be used to determine the reference cutoff point between the straight-line segment and the circular segment.

The curvature $K_{i} (i = 2, 3, . . ., N - 1)$ of the measuring points $P_{i} (x_{i}, y_{i}) (i = 2, 3, . . ., N - 1)$ can be gained using the three-point arc method, the calculation principle and process of which are shown in Figure 4 and equation (3),²¹ respectively.

K_{i} = \frac{4 S}{L_{i} Q_{i} L_{i + 1}}

(3)

where $L_{i}$ , $L_{i + 1}$ , $Q_{i}$ are the side length of the triangle constructed by the three points, $S$ is the area of the constructed triangle, these parameters can be calculated by equation (4).

{\begin{matrix} L_{i} = | | P_{i} - P_{i - 1} | | = \sqrt{{(x_{i} - x_{i - 1})}^{2} + {(y_{i} - y_{i - 1})}^{2}} \\ Q_{i} = | | P_{i + 1} - P_{i - 1} | | = \sqrt{{(x_{i + 1} - x_{i - 1})}^{2} + {(y_{i + 1} - y_{i - 1})}^{2}} \\ L_{i + 1} = | | P_{i + 1} - P_{i} | | = \sqrt{{(x_{i + 1} - x_{i})}^{2} + {(y_{i + 1} - y_{i})}^{2}} \\ S = \pm \frac{1}{4} \\ \sqrt{(L_{i} + L_{i + 1} + Q_{i}) (L_{i + 1} + Q_{i} - L_{i})} \sqrt{(L_{i} - L_{i + 1} + Q_{i}) (L_{i + 1} + L_{i} - Q_{i})} \end{matrix}

(4)

where $S$ attains a positive value when $P_{i - 1} P_{i} \times P_{i - 1} P_{i + 1} > 0$ ; otherwise, $S$ has a negative value.

Figure 4.

Calculation of the curvature of the measured points.

The curvature difference $E_{i}$ of a measuring point refers to the difference between the curvature $K_{i}$ of the measuring point $P_{i} (x_{i}, y_{i})$ and the curvature $K_{i - 1}$ of the previous measuring point $P_{i - 1} (x_{i - 1}, y_{i - 1})$ , which can be calculated by equation (5).

E_{i} = K_{i} - K_{i - 1}

(5)

Because the dividing point between a straight-line segment and arc segment is the point where the curvature is discontinuous, we can take the measuring point corresponding to the maximum curvature difference $\max (E_{i})$ as the reference dividing point between the left arc and the straight line. This point is expressed by $P_{g} (x_{g}, y_{g})$ , and the measuring point corresponding to the minimum curvature difference $\min (E_{i})$ as the reference dividing point between the right arc and the straight line, which is expressed by $P_{h} (x_{h}, y_{h})$ , as shown in Figure 5.

Figure 5.

Dividing point and auxiliary dividing points.

Determine the auxiliary dividing points

As shown in Figure 5, taking $t$ measuring points on both sides of the reference dividing points $P_{g}$ and $P_{h}$ as auxiliary dividing points, the left auxiliary dividing points are expressed by $P_{g - t}, . . ., P_{g + t}$ , and the right auxiliary dividing points are expressed by $P_{h - t}, . . ., P_{h + t}$ .

Minimum zone fitting for circular arcs

As shown in Figure 5, based on the established process of the auxiliary dividing points, it can be seen that the measuring points $P_{1} ~ P_{g - t - 1}$ must be the measuring points on the left arc, and the measuring points $P_{h + t + 1} ~ P_{N}$ must be the measuring points on the right arc. Further, when the auxiliary dividing points can be on the left arc or on the straight line (such as $P_{g}$ and $P_{h}$ ), the position of the auxiliary dividing points can only be determined by comparing the size of fitting error. The minimum zone fitting method for the left and right circular arcs are described in the following paragraphs.

Determining the reference centre of the left circular arc

From the measured points $P_{1} ~ P_{g - t - 1}$ , three points of a roughly uniform distribution are selected, which are expressed as $Q_{1} (u_{1}, v_{1})$ , $Q_{2} (u_{2}, v_{2})$ and $Q_{3} (u_{3}, v_{3})$ , respectively. Using the principle of determining a circle by three points, the coordinate of the reference centre of the left circular arc $G_{0} (x_{0}, y_{0})$ is calculated by equation (6).

{\begin{matrix} x_{0} = \frac{u_{s}}{u_{x}} \\ y_{0} = \frac{v_{s}}{v_{x}} \end{matrix}

(6)

where $u_{s}, u_{x}, v_{s}, v_{x}$ can be calculated by equation (7).

{\begin{matrix} u_{s} = (v_{1} - v_{2}) (v_{1} v_{2} + u_{3}^{2}) + (v_{2} - v_{3}) (v_{2} v_{3} + u_{1}^{2}) \\ - (v_{1} - v_{3}) (v_{1} v_{3} + u_{2}^{2}) \\ u_{x} = 2 v_{1} (u_{3} - u_{2}) - 2 v_{2} (u_{1} - u_{3}) + 2 v_{3} (u_{2} - u_{1}) \\ v_{s} = (u_{1} - u_{2}) (u_{1} u_{2} + v_{3}^{2}) + (u_{2} - u_{3}) (u_{2} u_{3} + v_{1}^{2}) \\ - (u_{1} - u_{3}) (u_{1} u_{3} + v_{2}^{2}) \\ v_{x} = 2 u_{1} (v_{3} - v_{2}) - 2 u_{2} (v_{1} - v_{3}) + 2 u_{3} (v_{2} - v_{1}) \end{matrix}

(7)

Construct auxiliary centres of the left circular arc

As shown in Figure 6, using the centre point $G_{0} (x_{0}, y_{0})$ as a control point, a regular square is constructed using side length d. The size of d can be set according to the processing requirements. In this way, the four auxiliary centres (i.e. the four vertexes of a square, expressed as $G_{0 q} (x_{0 q}, y_{0 q}) (q = 1, 2, 3, 4)$ ) can be determined. The coordinates of the auxiliary centres are calculated by equation (8).

{\begin{matrix} x_{01} = x_{02} = x_{0} + 0.5 d \\ x_{03} = x_{04} = x_{0} - 0.5 d \\ y_{01} = y_{04} = y_{0} + 0.5 d \\ y_{02} = y_{03} = y_{0} - 0.5 d \end{matrix}

(8)

Figure 6.

Establishment of auxiliary centres.

Minimum zone fitting of the left arc

Supposing the dividing points $P_{g - t}, . . ., P_{g + t}$ on the left side, $2 t + 1$ data groups $P_{1} (x_{1}, y_{1}) ~ P_{g - t - 1 + j} (x_{g - t - 1 + j}, y_{g - t - 1 + j}) (j = 1, 2, 3, \dots, 2 t + 1)$ can be constructed by combining the measuring points $P_{1} ~ P_{g - t - 1}$ with the dividing points $P_{g - t}, . . ., P_{g + t}$ . Thus, for each data group, with the reference centre $G_{0} (x_{0}, y_{0})$ , the radius of the measuring points $P_{1} ~ P_{g - t - 1 + j} (x_{g - t - 1 + j}, y_{g - t - 1 + j})$ is gained by equation (9), and the radial extreme difference of the left circular arc $Δ R_{0 j}$ can be calculated by equation (10).

R_{0 i} = \sqrt{{(x_{i} - x_{0})}^{2} + {(y_{i} - y_{0})}^{2}} (i = 1, 2, \dots, g - t - 1 + j)

(9)

Δ R_{0 j} = \max (R_{0 i}) - \min (R_{0 i})

(10)

As shown in Section ‘Construct auxiliary centres of the left circular arc’, four auxiliary centres $G_{0 q} (x_{0 q}, y_{0 q}) (q = 1, 2, 3, 4)$ have been constructed as the centre of the left circular arc. Thus, for each data group, the four extreme radial difference $Δ R_{qj} (q = 1, 2, 3, 4; j = 1, 2, 3, \dots, 2 t + 1)$ can be obtained by calculating the distances between the measured points $P_{1} ~ P_{g - t - 1 + j} (j = 1, 2, 3, \dots, 2 t + 1)$ and the auxiliary centre $G_{0 q} (q = 1, 2, 3, 4)$ . Among the four $Δ R_{qj}$ , the minimum value is saved as $Δ R_{gj}$ , and the auxiliary centre corresponding with the $Δ R_{gj}$ is expressed as $G_{gj} (x_{gj}, y_{gj})$ .

For the left circular arc, there are $2 t + 1$ datasets: $P_{1} ~ P_{g - t - 1 + j} (j = 1, 2, 3, \dots, 2 t + 1)$ . When $j = 1$ , the measured points are $P_{1} ~ P_{g - t}$ . With the reference centre $G_{0} (x_{0}, y_{0})$ , the minimum radial extreme difference is expressed by $Δ R_{01}$ . With the auxiliary centres $G_{0 q} (x_{0 q}, y_{0 q}) (q = 1, 2, 3, 4)$ , the minimum radial extreme difference is expressed by $Δ R_{g 1}$ , and the auxiliary centre corresponding with the $Δ R_{g 1}$ is expressed by $G_{g 1} (x_{g 1}, y_{g 1})$ . In addition, when $j = 2$ , the measured points are $P_{1} ~ P_{g - t + 1}$ . With the reference centre $G_{0} (x_{0}, y_{0})$ , the minimum radial extreme difference is expressed by $Δ R_{02}$ . With the auxiliary centres $G_{0 q} (x_{0 q}, y_{0 q}) (q = 1, 2, 3, 4)$ , the minimum radial extreme difference is expressed by $Δ R_{g 2}$ , and the auxiliary centre corresponding with the $Δ R_{g 2}$ is expressed by $G_{g 2} (x_{g 2}, y_{g 2})$ , and so on. Based on the definition above, the detailed processes of the minimum zone fitting of the left arc are as follows:

Step 1: Calculate the extreme radial difference

When $j = 1$ , with the auxiliary centres $G_{0 q} (x_{0 q}, y_{0 q}) (q = 1, 2, 3, 4)$ as the centre of the left circular arc, the maximum radius of the measured points $P_{1} ~ P_{g - t}$ can be obtained by calculating the distances between the measured points $P_{1} ~ P_{g - t}$ and the auxiliary centre $G_{0 q}$ , as defined by equation (11).

\begin{matrix} R_{q 1 i} = \sqrt{{(x_{i} - x_{0 q})}^{2} + {(y_{i} - y_{0 q})}^{2}} \\ (i = 1, 2, \dots, g - t; q = 1, 2, 3, 4) \end{matrix}

(11)

For each auxiliary centre and the measured points $P_{1} ~ P_{g - t}$ , there is an extreme radial difference. Because there are four auxiliary centres, there are four extreme radial differences, as shown by equation (12). Among the four extreme radial differences, the minimum one is the minimum zone containing the measured points $P_{1} ~ P_{g - t}$ , which is expressed as $Δ R_{g 1}$ (as shown by equation (13)), whereas the centre (one of the auxiliary centres) corresponding to $Δ R_{g 1}$ is expressed as $G_{a} (x_{a}, y_{a})$ .

Δ R_{q 1} = \max (R_{q 1 i}) - \min (R_{q 1 i}) (q = 1, 2, 3, 4)

(12)

Δ R_{g 1} = \min (Δ R_{q 1})

(13)

Step 2: Optimisation fitting

Comparing $Δ R_{g 1}$ and $Δ R_{01}$ , if $Δ R_{01} \geq Δ R_{g 1}$ , it shows that $Δ R_{g 1}$ is closer to the ideal minimum zone containing the measured points $P_{1} ~ P_{g - t}$ , and the centre $G_{a} (x_{a}, y_{a})$ is also closer to the centre of the minimum zone circular arc; hence, the reference centre changes to point $G_{a} (x_{a}, y_{a})$ (as the red square in Figure 7, here, assume $G_{a}$ is $G_{02}$ ). Subsequently, a new regular square is re-established using the side length d, as described in Section ‘Construct auxiliary centres of the left circular arc’, and steps 1 and 2 are repeated. Otherwise, if $Δ R_{01} < Δ R_{g 1}$ , the reference centre is unchanged, but $d = 0.618 d$ (as the blue square in Figure 8), and as described in Section ‘Construct auxiliary centres of the left circular arc’, and steps 1 and 2 are repeated. When the side length of the regular square is less than the preset value $δ$ (in this study, $δ = 0.00001 mm$ ), the search terminates, and it can be considered that the search centre is reaching close to the actual centre of the minimum zone circular arc. At this moment, the minimum values of $Δ R_{01}$ and $Δ R_{g 1}$ are the contour error of the left circular arc, and the minimum of $Δ R_{01}$ and $Δ R_{g 1}$ is expressed as $F_{g 1}$ (as shown in equation (14)), whereas the centre corresponding to $F_{g 1}$ is expressed as $G_{g 1} (x_{g 1}, y_{g 1})$ .

F_{g 1} = \min (Δ R_{01}, Δ R_{g 1})

(14)

Figure 7.

Reference centre change.

Figure 8.

Auxiliary point reconstruction.

Step 3: Fitting equation of left circular arc and its fitting error

According to steps 1 and 2, the minimum zone fitting errors $F_{gj} (j = 1, 2, 3, \dots, 2 t + 1)$ and their corresponding centre $G_{gj} (x_{gj}, y_{gj}) (j = 1, 2, 3, \dots, 2 t + 1)$ can be obtained. Thus, the minimum zone fitting equation of the left circular arc can be obtained by equation (15).

(x - x_{gj})^{2} + (y - y_{gj})^{2} = R_{gj}^{2} (j = 1, 2, 3, \dots, 2 t + 1)

(15)

Fitting equation of right circular arc and its fitting error

From the measured points $P_{h + t + 1} ~ P_{N}$ , three points of roughly uniform distribution are selected. And using the method described in Section ‘Determining the reference centre of the left circular arc’, the coordinate of the reference centre $G_{0} (x_{0}, y_{0})$ of the right circular arc can be obtained. Then, combining the measuring points $P_{h + t + 1} ~ P_{N}$ and the dividing points $P_{h - t}, \dots, P_{h + t}$ in turn, $2 t + 1$ data groups $P_{h - t - 1 + k} ~ P_{N} (k = 1, 2, 3, \dots, 2 t + 1)$ of the right circular arc can be constructed. Finally, using the same method described in Sections ‘Construct auxiliary centres of the left circular arc’ and ‘Minimum zone fitting of the left arc’, the radial extreme difference under the reference centre of each data group can be calculated, the auxiliary centres can be constructed, and the minimum zone fitting errors $F_{hk}$ , the corresponding centre $G_{hk} (x_{hk}, y_{hk})$ , the minimum zone fitting equation of the right circular arc (shown by equation (16)) can be calculated step by step.

(x - x_{hk})^{2} + (y - y_{hk})^{2} = R_{hk}^{2} (k = 1, 2, 3, \dots, 2 t + 1)

(16)

Fitting of straight-line segment and its fitting error

Assuming that the general equation of a straight-line segment of the crown curve is expressed as $y = Ax + B$ , its constraint is tangential to the circular arcs at both ends.

Each of the fitted left circular arcs and each of the fitted right circular arcs are combined sequentially, and the $(2 t + 1)^{2}$ common tangent line corresponding to the actual condition of the measured contour can be obtained. The tangent points on the left arc are expressed as $T_{gjk} (X_{gjk}, Y_{gjk}) (j = 1, 2, 3, \dots, 2 t + 1) (k = 1, 2, 3, \dots, 2 t + 1)$ , and the tangent points on the right arc are expressed as $S_{hjk} (X_{hjk}, Y_{hjk}) (j = 1, 2, 3, \dots, 2 t + 1) (k = 1, 2, 3, \dots, 2 t + 1)$ ; in addition, based on the constraint condition, the parameters of the common tangents and the tangent points are achieved by equations (17) and (18).

{\begin{matrix} (A_{jk}^{2} + 1) [x_{gj}^{2} - R_{gj}^{2} + {(B_{jk} - y_{gj})}^{2}] + {[A_{jk} (B_{jk} - y_{gj}) - x_{gj}]}^{2} = 0 \\ (A_{jk}^{2} + 1) [x_{hk}^{2} - R_{hk}^{2} + {(B_{jk} - y_{hk})}^{2}] + {[A_{jk} (B_{jk} - y_{hk}) - x_{hk}]}^{2} = 0 \end{matrix}

(17)

{\begin{matrix} X_{gjk} = [x_{gj} - A_{jk} (B_{jk} - y_{gj})] / (A_{jk}^{2} + 1) \\ Y_{gjk} = A_{jk} X_{gjk} + B_{jk} \\ X_{hjk} = [x_{hk} - A_{jk} (B_{jk} - y_{hk})] / (A_{jk}^{2} + 1) \\ Y_{hjk} = A_{jk} X_{hjk} + B_{jk} \end{matrix}

(18)

For the type of combination between one of the left circular arcs and one of the right circular arcs, there are four common tangents and four pairs of tangent points. According to the actual known curve, the y-coordinate of the tangent point is greater than the y-coordinate of the centre of the circular arcs, that is, the tangent points corresponding to $Y_{gjk} > y_{gj}$ and $Y_{hjk} > y_{hk}$ are the sachgemaess tangent points, and the straight-line equation determined by the left tangent point $T_{gjk} (X_{gjk}, Y_{gjk})$ (where, $Y_{gjk} > y_{gj}$ ) and right tangent point $S_{hjk} (X_{hjk}, Y_{hjk})$ (where, $Y_{hjk} > y_{hk}$ ) are common sachgemaess tangent equations.

The straightness error of the common tangent line can be calculated by equation (19).

D_{jki} = \pm \frac{A_{jk} x_{i} + B_{jk} - y_{i}}{\sqrt{A_{jk}^{2} + 1}} (X_{gjk} < x_{i} < X_{hjk})

(19)

where $D_{jki}$ attains a positive value when the measured points $P_{i} (x_{i}, y_{i})$ (where, $X_{gjk} < x_{i} < X_{hjk}$ ) are located outside of the common tangent line; otherwise, $D_{jki}$ has a negative value.

According to the definition of a straightness error, the straightness error of the common tangent is shown by equation (20).

\begin{matrix} Δ D_{jk} = \max D_{jki} - \min D_{jki} (j = 1, 2, 3, \dots, 2 t + 1) \\ (k = 1, 2, 3, \dots, 2 t + 1) \end{matrix}

(20)

Overall error of the convex curve

It can be seen from the above fitting process of the left and right arcs and the common tangent that, for the measured data of the convexity profile, the ${(2 t + 1)}^{2}$ curve of convexity profile can be fitted. For each convexity curve (left circular arc segment + common tangent line + right circular arc), only the region formed by the maximum fitting error of the three curves (left circular arc, common tangent line and right circular arc) can contain the entire convexity curve.

According to the definition of a curve contour error, it can be determined that this maximum fitting error is the contour error of the entire convexity curve. Because the ${(2 t + 1)}^{2}$ convexity curve is fitted by the above fitting process, the ${(2 t + 1)}^{2}$ overall contour error of the convexity curve (marked as $E_{jk}$ ) can be obtained, as described by equation (21).

\begin{matrix} E_{jk} = \max (F_{gj}, Δ D_{jk}, F_{hk}) (j = 1, 2, 3, \dots, 2 t + 1) \\ (k = 1, 2, 3, \dots, 2 t + 1) \end{matrix}

(21)

Among the ${(2 t + 1)}^{2}$ overall contour errors $E_{jk}$ , the minimum $E_{jk}$ (marked as $E$ ) is the minimum zone error containing all measured points, as described by equation (22).

E = \min (E_{jk})

(22)

Recording the values of $j$ and $k$ corresponding to $E$ , the parameters of the left circular arc, the right circular arc and the common tangent line can be gained using equations (15)–(17), respectively. The minimum zone fitting equation of the entire convexity curve is obtained. In other words, the curve formed by the combination corresponding to $E$ is the entire curve to be fitted.

Example certification

Data construction

It is assumed that the standard equation of the convexity curve is equation (23). Taking 44 points on the curve and recording the points as $S_{i} (x_{i}, y_{i}) (i = 1, 2, 3, \dots, 44)$ , one point is taken on the normal lines of the convexity curve passing the points $S_{i} (x_{i}, y_{i})$ , which are expressed as $T_{i} (x_{i}, y_{i})$ , and when $\bar{T_{i} S_{i}} = Δ E_{i}$ and $Δ E_{i} \leq \pm 0.01$ , $Δ E_{i}$ is given a positive value when the points $T_{i} (x_{i}, y_{i})$ are located outside of the convexity curve; otherwise, $Δ E_{i}$ has a negative value. To make the simulation data more general, the data points $T_{i} (x_{i}, y_{i})$ are contra-rotated by 10°, and translated as 3 and 5 mm along the X- and Y-axes respectively, to obtain the analogue measurement data $P_{i} (x_{i}, y_{i})$ at any location along the plane (Table 1).

Table 1.

Simulation data (mm).

No.	$S_{i} (x_{i}, y_{i})$		$Δ E_{i}$	$T_{i} (x_{i}, y_{i})$		$P_{i} (x_{i}, y_{i})$
	x	y		x	y	x	y
1	0	0	0.0080	−0.0080	0.0000	2.9921	4.9986
2	0.0759	0.8682	−0.0070	0.0829	0.8670	2.9311	5.8682
3	0.3015	1.7101	0.0090	0.2931	1.7132	2.9912	6.7381
4	0.6698	2.5000	−0.0060	0.6751	2.4970	3.2312	7.5763
5	1.1698	3.2139	0.0090	1.1621	3.2204	3.5852	8.3733
6	1.7861	3.8302	−0.0050	1.7893	3.8264	4.0977	9.0790
7	2.5000	4.3301	0.0020	2.5010	4.3284	4.7114	9.6970
8	3.2899	4.6985	−0.0090	3.2933	4.6891	5.4290	10.1897
9	4.1317	4.9240	0.0050	4.1309	4.9290	6.2122	10.5714
10	5.0000	5.0000	−0.0012	5.0000	5.0012	7.0551	10.7962
11	6.0000	5.0000	0.0060	6.0000	5.0060	8.0396	10.9718
12	7.0000	5.0000	−0.0050	7.0000	4.9950	9.0263	11.1347
13	8.0000	5.0000	0.0080	8.0000	5.0080	10.0088	11.3211
14	9.0000	5.0000	−0.0050	9.0000	4.9950	10.9959	11.4819
15	10.0000	5.0000	0.0030	10.0000	5.0030	11.9793	11.6635
16	11.0000	5.0000	−0.0040	11.0000	4.9960	12.9653	11.8302
17	12.0000	5.0000	0.0060	12.0000	5.0060	13.9484	12.0137
18	13.0000	5.0000	−0.0030	13.0000	4.9970	14.9348	12.1785
19	14.0000	5.0000	0.0050	14.0000	5.0050	15.9182	12.3600
20	15.0000	5.0000	−0.0100	15.0000	4.9900	16.9056	12.5189
21	16.0000	5.0000	0.0100	16.0000	5.0100	17.8869	12.7123
22	17.0000	5.0000	−0.0020	17.0000	4.9980	18.8738	12.8741
23	18.0000	5.0000	0.0040	18.0000	5.0040	19.8576	13.0536
24	19.0000	5.0000	−0.0050	19.0000	4.9950	20.8440	13.2184
25	20.0000	5.0000	0.0070	20.0000	5.0070	21.8267	13.4039
26	21.0000	5.0000	−0.0030	21.0000	4.9970	22.8133	13.5677
27	22.0000	5.0000	0.0040	22.0000	5.0040	23.7969	13.7482
28	23.0000	5.0000	−0.0090	23.0000	4.9910	24.7840	13.9091
29	24.0000	5.0000	0.0020	24.0000	5.0020	25.7668	14.0936
30	25.0000	5.0000	−0.0080	25.0000	4.9920	26.7534	14.2574
31	26.0000	5.0000	0.0020	26.0000	5.0020	27.7365	14.4409
32	27.0000	5.0000	−0.0040	27.0000	4.9960	28.7223	14.6086
33	28.0000	5.0000	0.0020	28.0000	5.0020	29.7060	14.7882
34	29.0000	5.0000	−0.0050	29.0000	4.9950	30.6921	14.9549
35	30.0000	5.0000	0.0080	30.0000	4.9920	31.6774	15.1256
36	30.8683	4.9240	−0.0070	30.8695	4.9309	32.5443	15.2164
37	31.7101	4.6985	0.0090	31.7070	4.6900	33.4109	15.1246
38	32.5000	4.3301	−0.0060	32.5030	4.3353	34.2564	14.9135
39	33.2139	4.8302	0.0090	33.2075	3.8226	35.0392	14.5309
40	33.8302	3.2139	−0.0050	33.8341	3.2172	35.7614	14.0436
41	34.3301	2.5000	0.0020	34.3284	2.4990	36.3729	13.4221
42	34.6985	1.7101	−0.0090	34.7079	1.7135	36.8831	12.7144
43	34.9240	0.8682	0.0050	34.9191	0.8674	37.2380	11.9179
44	35.0000	0.0000	−0.0040	35.0040	0.0000	37.4722	11.0784

From the established process of the simulation data, it can be seen that the simulated measured data represent the transformed convexity curve. Here, the radius of the circular arc at both ends is 5 mm, and the centre of the circular arc at both ends is $O_{1} (7.9240, 5.8682)$ mm and $O_{2} (32.5442, 10.2094)$ mm, and the slope and intercept of the common tangent are 0.1763 and 9.5482 mm, respectively.

{\begin{matrix} {(x - 5)}^{2} + y^{2} = 25 (0 \leq x < 5) \\ y = 5 (5 \leq x < 30) \\ {(x - 30)}^{2} + y^{2} = 25 (30 \leq x < 35) \end{matrix}

(23)

Determine the dividing points

The curvature and curvature difference of the simulated data points can be obtained using equations (3) and (5). As shown in Figure 9, the minimum and maximum values of the curvature difference are $- 0.0843$ and $0.0655$ , respectively, corresponding to the simulated data points, namely, the 9th and 35th points. Therefore, the 9th and 35th measuring points are selected as the left and right reference dividing points of the convexity curve, respectively. For an easier calculation, one point is taken before and after the reference dividing points $P_{9}$ and $P_{35}$ as the auxiliary dividing points, that is, $P_{g} = P_{9}$ , $P_{h} = P_{35}$ and $t = 1$ .

Figure 9.

Curvature and curvature difference of data points.

Fitting the circular arcs

Selecting the measured points $P_{2}$ , $P_{5}$ and $P_{8}$ on the left circular arc, and selecting the measured points $P_{37}$ , $P_{40}$ and $P_{43}$ on the right circular arc, the reference centre coordinates are obtained using equation (6), that is, $O_{1} (7.8058, 5.9333)$ mm and $O_{2} (32.6042, 10.2748)$ mm.

To prove the universality of the value of side length $d$ , in this study, 0.1, 0.15 and 0.2 mm are set as the foursquare side lengths for different datasets. The stopping criteria for the iterations is $δ = 0.00001$ mm, and the fitting results of the left and right circular arcs are as shown in Tables 2 and 3.

Table 2.

Minimum zone fitting results of the left circular arcs (mm).

Data points	Side length of square	Stop condition	Centre coordinates		Radius	Error
Data points	d	δ	x _gj	y _gj	R _gj	F _gj
P ₁–P₈	0.1	0.00001	7.8558	5.8833	4.9368	0.0188
P ₁–P₉	0.15	0.00001	7.9105	5.8730	4.9875	0.0168
P ₁–P₁₀	0.2	0.00001	7.9121	5.8729	4.9896	0.0173

Table 3.

Minimum zone fitting results of right circular arcs (mm).

Data points	Side length of square	Stop condition	Centre coordinates		Radius	Error
Data points	d	δ	x _hk	y _hk	R _hk	F _hk
P ₃₆–P₄₄	0.1	0.00001	32.5458	10.2061	5.0020	0.0182
P ₃₅–P₄₄	0.15	0.00001	32.5415	10.1985	5.0091	0.0170
P ₃₄–P₄₄	0.2	0.00001	32.4349	10.0947	5.1403	0.0756

As shown in Tables 2 and 3, although the search area has different side lengths, but is under the same search termination conditions, the fitted results are basically consistent with the set of circular arc parameters, which shows that the proposed method for arc fitting is correct and effective. In the third row of Table 3, the fitting error is far greater than the set value, indicating that point $P_{34}$ is not on the right circular arc, and may be a point on a straight line.

Common tangent parameters and errors

Using the parameter of the circular arcs in Tables 2 and 3 as well as the method described in Section ‘Fitting of straight-line segment and its fitting error’, the nine common tangent lines and their errors can be obtained, the results of which are listed in Table 4.

Table 4.

Fitting results of the common tangent.

No.	Left tangent point/mm	Right tangent point/mm	Data points	Slope	Intercept	Error/mm
No.	T_gjk(X_gjk,Y_gjk)	S_hjk(X_hjk,Y_hjk)		A _jk	B _jk	ΔD _jk
1	6.9917, 10.7439	31.6704, 15.1308	P ₉–P₃₅	0.1778	9.5010	0.0405
2	6.9916, 10.7439	31.6646, 15.1303	P ₉–P₃₄	0.1778	9.5009	0.0408
3	6.9822, 10.7422	31.5252, 15.1548	P ₉–P₃₃	0.1798	9.4869	0.0459
4	7.0437, 10.7846	31.6765, 15.1319	P ₁₀–P₃₅	0.1765	9.5415	0.0138
5	7.0436, 10.7846	31.6708, 15.1314	P ₁₀–P₃₄	0.1765	9.5414	0.0135
6	7.0341, 10.7829	31.5315, 15.1559	P ₁₀–P₃₃	0.1785	9.5273	0.0482
7	7.0452, 10.7867	31.6768, 15.1319	P ₁₁–P₃₅	0.1764	9.5438	0.0126
8	7.0451, 10.7867	31.6711, 15.1315	P ₁₁–P₃₄	0.1764	9.5437	0.0124
9	7.0356, 10.7850	31.5318, 15.1559	P ₁₁–P₃₃	0.1784	9.5296	0.0468

Overall results of convex curve

For the convenience of analysis and comparison, all fitting results are listed in Table 5.

Table 5.

Fitting results of convex curve (mm).

No.	Left arc parameters and errors				Right arc parameters and errors				Line parameters and errors			Max error
No.	x _gj	y _gj	R _gj	F _gj	x _hk	y _hk	R _hk	F _hk	A _jk	B _jk	ΔD _jk	E _jk
1	7.8558	5.8833	4.9368	0.0188	32.5458	10.2061	5.0020	0.0182	0.1778	9.5010	0.0405	0.0405
2	7.8558	5.8833	4.9368	0.0188	32.5415	10.1985	5.0091	0.0170	0.1778	9.5009	0.0408	0.0408
3	7.8558	5.8833	4.9368	0.0188	32.4349	10.0947	5.1403	0.0756	0.1798	9.4869	0.0459	0.0459
4	7.9105	5.8730	4.9875	0.0168	32.5458	10.2061	5.0020	0.0182	0.1765	9.5415	0.0138	0.0182
5	7.9105	5.8730	4.9875	0.0168	32.5415	10.1985	5.0091	0.0170	0.1765	9.5414	0.0135	0.0170
6	7.9105	5.8730	4.9875	0.0168	32.4349	10.0947	5.1403	0.0756	0.1785	9.5273	0.0482	0.0756
7	7.9121	5.8729	4.9896	0.0173	32.5458	10.2061	5.0020	0.0182	0.1764	9.5438	0.0126	0.0182
8	7.9121	5.8729	4.9896	0.0173	32.5415	10.1985	5.0091	0.0170	0.1764	9.5437	0.0124	0.0173
9	7.9121	5.8729	4.9896	0.0173	32.4349	10.0947	5.1403	0.0756	0.1784	9.5296	0.0468	0.0468

As can be seen from Table 5, the overall error of the fifth convexity curve is the smallest. At the same time, the corresponding minimum zone curve equation can be obtained. To demonstrate the accuracy and effectiveness of the proposed method, the data processing results obtained based on the proposed method are compared with the preset used in this study and the results obtained using the traditional least square method,²² as shown in Table 6.

Table 6.

Comparison of data processing results.

Crowned curve	Parameters of left arc		Parameters of right arc		Straight line		error
Crowned curve	Centre	Radius	Centre	Radius	Slope	Intercept	error
Preset	7.9240, 5.8682	5	32.5442, 10.2094	5	0.1763	9.5482	≤0.02
The least square method	7.9330, 5.8579	5.0124	32.5526, 10.2109	4.9962	0.1761	9.5501	0.0209
The proposed method	7.9105, 5.8730	4.9875	32.5415, 10.1985	5.0091	0.1765	9.5414	0.0170

It can be seen from Table 6 that the parameters and errors of the convexity curve obtained through the proposed method are consistent with the parameters and errors of the predetermined convexity curve, which verifies the effectiveness of the method proposed herein. Moreover, compared with the convexity curve obtained by the least square method (with a contour error of 0.0209), the convexity curve obtained by the proposed method (with a contour error of 0.0170) has a smaller contour error, proving that the convexity curve obtained by the proposed method better reflects the curve represented by the measurement points.

Conclusion

In this paper, a method for the overall minimum zone fitting and error evaluation of the convex contour of an arc-corrected bearing roller was presented. The proposed method has the following advantages. First, the proposed method overcomes the limitations of existing methods that can only fit a single class of curves, providing a new method to fit a multi-segment curve on a plane. It uses data segmentation and the curvature analysis of the discrete points to obtain the demarcation points between the straight lines and circular arcs. And on this basis, the multi-segment curve fitting on the plane is realised by piecewise curve fitting and global optimisation technique based on the tangent constraints. Second, the minimum zone fitting method is introduced into the curve fitting and error evaluation of the multi-segment curve. Compared with the traditional least squares method, when all measurement points are reliable, the proposed method can fit the contour curve of measuring points with less error and higher accuracy. However, in practice, there may be random errors in the measurement points, which can have an impact on profile fitting and error evaluation. We hope to conduct research in this area in the future.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by National Natural Science Foundation of China (No. 5222780130), Key Scientific Research Project of Colleges and Universities in Henan Province (No. 23A580002), Henan Postdoctoral Science Foundation (HN2022167) and Open Foud of State Key Laboratory of Tribology in Advanced Equipment (No. SKLTKF20A03).

ORCID iD

Zhenyu Ma

Data availability statement

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

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