# Study on the Extraction Simulation of the Cylindrical Features

Authors: Zexiang Zhao; Dongxu Ren; Xinyu Zhao; Bin Li; Ruyi Liu
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

The extraction simulation method and theory of cylindrical components were promoted in this paper. According to the manufacturing principle of the cylindrical components, the main influence factors of their precision manufacturing were analyzed, and their extraction models of cylindrical features were built under the analysis of the errors of the slideway and the spindle as well as the parallelism error between the slideway and the spindle rotary axis. The determination principle of modeling parameters of cylindrical features was studied. According to the extraction strategies of the roundness profiles, generatrix profiles, birdcage profile, and single spiral profile, some cylindrical features were simulated by using the established models, and their corresponding global sizes, calculated sizes, and form errors were evaluated and compared. The simulation experiments showed that the simulation models can meet the needs for the research of the relationship among the size tolerance, the form tolerance, the waviness, and the roughness, and can be taken as the theoretic base for the error budgets of the slideway and the rotation spindle in the design of precision machine tools.


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## Body

## 1. Introduction

With the development of precision and ultraprecision technology, a lot of cylindrical components with high precision grade are needed. High-precision turning and grinding technologies have being developed for hard and brittle materials, such as glass, ceramics, and crystal, nonferrous materials, such as copper alloy and aluminum alloy, and ferrous metals, as well as porous materials. These high-precision cylindrical components are usually indicated by their sizes and tolerances [1], form tolerances [2], such as roundness, straightness, and cylindricity, and surface roughness [3]. Their sizes, form errors, and surface roughness are measured by the length measuring instruments based on the two-point method, the form error measuring instruments, and the roughness measuring instruments, respectively. For a fit of hole and shaft with the same nominal size, their respective comprehensive sizes formed by two-point size and form errors rather than only their respective two-point size will have actual influence on their fit characteristic. Therefore, besides two-point size, the global sizes, such as least-squares size, minimum circumscribed size, maximum inscribed size, and minimax size, and the calculated sizes, such as circumference diameter, area diameter, and volume diameter, for a cylindrical component are stipulated in ISO 14405-1 [4], which can be measured by the cylindrical coordinate measuring machine [5]. Global sizes and calculated sizes are the comprehensive sizes of the actual sizes and form errors of the manufactured cylindrical component. For obtaining the global sizes and calculated sizes of a cylindrical component, its profiles should be first extracted by using the measuring machine above, and then the evaluation of the sizes can be done by using the program developed based on the definitions of global sizes and calculated sizes. The extraction strategies and their sampling parameters of the cylindrical profiles will have influence on the measuring results of the global sizes and calculated sizes. If the sampling number of a cylindrical component is less, the extracted profiles may not characterize the actual size of the components, so that the global sizes and calculated sizes cannot be correctly obtained, and if the sampling number of a cylindrical component is too much, some sampling points may be redundant, so that the measuring and evaluation of the sizes may be long and some cross errors may be introduced in the long measuring process. Therefore, the extraction strategies and their sampling parameters are studied based on the simulation profiles, which are mainly composed of the simulated slideway’s linear error profiles and spindle axis’s rotary radial error profiles of the precision machine tool.


## 2. Modeling of Surface Profiles of Cylindrical Component

### 2.1. Influence Factors of Precision Manufacturing of Cylindrical Component

In the field of precision manufacturing, in order to obtain high-precision surface quality, hard and brittle materials, such as glass, ceramics, and crystal, and nonferrous materials, such as copper alloy and aluminum alloy, are usually turned by single-point diamond turning machines and ferrous metal materials and porous materials are usually ground by the precision grinding machines. In the manufacturing of high-precision cylindrical components, whether or not the high-precision turning or grinding way is used, the influence factors of precision manufacturing of cylindrical components mainly consist of the cutting tool (grinding wheel or turning tool), slideway’s straightness error, rotary accuracy of the spindle, parallelism error between the slideway and the spindle’s rotary axis, external and internal vibration, temperature, etc. Besides, tool abrasion and cutting force also have much influence on the surface quality. The surface roughness is mainly affected by tool’s condition and process parameters, and the surface waviness is generated by vibration and the related factors. For convenience, the influences of slideway straightness, rotary accuracy of the spindle, their parallelism error, and some random errors on the cylindrical component’s surface quality are mainly considered in this paper. The schematic of position relationship of the spindle and the slideway is shown in Figure1.Figure 1
Schematic of position relationship of the spindle and the slideway.


### 2.2. The Error Analysis of the Slideway

In the precision machine tools, the linear slideway, aerostatic bearing slideway, and hydrostatic slideway are usually used. No matter what kind of slideway is adopted in a precision machine tool, there may be some form errors. The slideway’s profile in thez direction may be seen in Figure 2–4, which consists of the form error profile, waviness profile, and roughness profile and can be expressed as follows:(1)ρgpz=ρgfz+ρgwz+ρgrz,where ρgf is the form error profile, which can be expressed as follows:(2)ρgfz=Wz+∑j=1Ngfgfwjzsin2πzjλf+θj,z=0∼L,where W(z) can be calculated by the following equation:(3)Wz=±RA2∓0.5L−z21/2−RA+G,for convex or concave arc profile,±GsineπzL,for convex or concavesineprofile,±0.5GsinekgπzL,kg=2,3,4,5,forsineprofile,where G is the bow altitude of the arc profile or the magnitude of the half sine wave profile. If G is equal to zero, the slideway form error profile characterized by the first item in equation (2) is an ideal straight line. For convex and concave arc profiles, signs “+” and “−” on the left of the brace are selected, respectively, and signs “−” and “+” on the left of the parentheses are selected where z ≤ L/2 and z > L/2, respectively, as shown in Figure 2, where L is the slideway length and RA is the radius of the arc profile. For the convex half sine profile characterized by the first item in equation (2), sign “+” on the left of G is selected, and for the concave half sine profile, sign “−” on the left of G is selected, as shown in Figure 3. For the sine profile, signs “+” and “−” can be selected according to the actual needs. The second item in equation (2) is also a kind of the form error profile, whose wavelength is j times of the cutoff wavelength λf of the form error profile, θj is the initial angle of the jth wave, and Ngf is the maximum times of the upper cutoff wavelength λf of the form error profile, which may be supposed to be five, and the product of times of Ngf and λf is equal to or less than the slideway length L, as shown in Figure 4.Figure 2
Schematic of the arc profile of the slideway characterized by the first item in equation (2). (a) Straight. (b) Convex. (c) Concave.
(a)(b)(c)Figure 3
Schematic of the sine profile of the slideway characterized by the first item in equation (2). (a) Convex (kg = 1). (b) Concave (kg = 1). (c) Sine profile (kg = 2, 3, …).
(a)(b)(c)Figure 4
The form error profile of the slideway characterized by the second item in equation (2).In equation (1), ρgw is the waviness profile with a cutoff wavelength λc along the axis of the cylindrical component, which is expressed as follows:(4)ρgwz=∑j=1Ngwgwwjzsin2πzjλc+φj,z=0∼L,where gwwjz is the magnitude of the jth wave in the waviness profile, whose wavelength is j times of the cutoff wavelength λc of the waviness profile, φj is the initial angle of the jth wave, Ngw is the maximum times of the cutoff wavelength λc, and the product of times of Ngw and λc is equal to or less than λf.In equation (1), ρgr is the roughness profile with a cutoff wavelength λs along the slideway in the z direction, which is expressed as follows:(5)ρgrz=∑j=1Ngrgrwjzsin2πzjλs+ϑj+δz,z=0∼L,where grwjz is the magnitude of the jth wave in the roughness profile, whose wavelength is j times of the cutoff wavelength λs of the roughness profile, ϑj is the initial angle of the jth wave, Ngr is the maximum times of the cutoff wavelength λs of the roughness profile, the product of times of Ngr and λs is equal to or less than λc, and δ(z) are the random numbers between 0 and grw1/5.The waviness and roughness profiles characterized by equations (4) and (5) are similar to the form profile shown in Figure 4, but their cutoff wavelengths and magnitudes are less than those of the form profiles characterized by equation (2).


### 2.3. The Rotary Error Analysis of the Spindle

Through the analysis of the roundness profile’s compositions introduced by the spindle rotary error, tool wear, vibration, and so on, the roundness profile can be expressed as follows:(6)ρrpz,ψz=ρrez,ψz+ρrfz,ψz+ρrwz,ψz+ρrrz,ψz,z=0∼L,ψz=0∼2π,where ρre(z, ψ) is the radial size of the cylinder with some eccentricity and multi‐sine waves to the rotary spindle axis, shown in Figure 5, which can be expressed as follows:(7)ρrez,ψz=ρsk2ψz−e2zsin2ψz−ϕz+ezcosψz−ϕz,where e(z) is the eccentricity of the roundness profile, ϕ(z) is the eccentricity angle of the roundness profile, ψ(z) is the angle between one point in the roundness profile and the x axis in the xoy coordinate system, and ρsk(ψ(z)) is the radial size between the rotary axis of the spindle and the circle with kr UPR at the angle ψ(z), where UPR means undulations per round. ρsk(ψ(z)) can be determined as follows:(8)ρskψz=R+Asin2πψz−msam−1msa,m=1,2,…,kr,ψz=0∼2π,where R is the radius of an ideal cylinder, kr is the UPR number in a roundness profile of the cylinder, A is the magnitude of the sine wave profile, and msa is equal to 2π/kr. The parameters above can be seen in Figure 5.Figure 5
Schematic of the parameters in equations (7) and (8).Based on equation (8), the circle profiles with kr UPR can be simulated, as shown in Figure 6.Figure 6
The circle profiles with differentkr UPR. (a) kr = 1. (b) kr = 2. (c) kr = 3. (d) kr = 4. (e) kr = 5. (f) kr = 6. (g) kr = 7. (h) kr = 8.
(a)(b)(c)(d)(e)(f)(g)(h)In equation (6), ρrf(z, ψ) is the form error profile in the roundness profile, which can be expressed as follows:(9)ρrfz,ψz=∑j=1NrfrfwjzsinπDψzjλf+ζjz,z=0∼L,ψz=0∼2π,where rfwj(z) is the magnitude of the jth wave of the form error profile in the roundness profile, whose wavelength is j times of the cutoff wavelength λf of the form error profile, ζj(z) is the initial angle of the jth wave in the roundness profile, Nrf is the maximum times of the upper cutoff wavelength λf of the waviness profile, which may be supposed to be five, and the product of times of Nrf and λf is equal to or less than the perimeter of the roundness profile.In equation (6), ρrw(z, ψ) is the waviness profile in the roundness profile, which can be expressed as follows:(10)ρrwz,ψz=∑j=1NrwrwwjzsinπDψzjλc+ξjz,z=0∼L,ψz=0∼2π,where rwwj(z) is the magnitude of the jth wave of the waviness profile in the roundness profile, whose wavelength is j times of the cutoff wavelength λc of the waviness profile, ξj(z) is the initial angle of the jth wave in the roundness profile, Nrw is the maximum times of the upper cutoff wavelength λc of the waviness profile, and the product of times of Nrw and λc is equal to or less than λf.In equation (6), ρrr(z, ψ) is the roughness profile in the roundness profile, which can be expressed as follows:(11)ρrrz,ψz=∑j=1NrrrrwjzsinπDψzjλs+ηjz+δψz,z=0∼L,ψz=0∼2π,where rrwj(z) is the magnitude of the jth wave of the roughness profile in the roundness profile, whose wavelength is j times of the cutoff wavelength λs of the roughness profile, ηj(z) is the initial angle of the jth wave, Nrr is the maximum times of the cutoff wavelength λs of the roughness profile, the product of times of Nrr and λs is equal to or less than λc, and δ(ψ(z)) are the random numbers between 0 and rrw1(z)/5.In equations (9)–11, D is the diameter of an ideal cylinder.


### 2.4. The Parallelism Error Analysis of the Slideway to the Spindle Rotary Axis

Owing to that there is the parallelism error of the slideway to the spindle’s rotary axis, the fabricated cylinder may be a cone, and the generatrix profile of which may be expressed as follows:(12)ρsz=S×zL+CD,z=0∼L,where S is the slope of the fabricated generatrix, which represents whether the slideway is parallel to the spindle axis, and CD is the intercept of the generatrix in the zoρs coordinate system. The parameters in equation (12) are shown in Figure 7.Figure 7
Schematic of the parameters in equation (12).


### 2.5. The Profile of the Cylinder Gained by Simulation

According to Sections2.2–2.4, the profile of the cylinder gained by manufacturing simulation can be expressed as follows:(13)ρz,ψz=ρgpz+ρrpz,ψz+ρsz,z=0∼L,ψz=0∼2π.Besides the influence factors mentioned above, the modeling of the cylindrical feature’s profile simulation does not consider the other influence factors, such as heat deflection, tool wear, and vibration.


## 2.1. Influence Factors of Precision Manufacturing of Cylindrical Component

In the field of precision manufacturing, in order to obtain high-precision surface quality, hard and brittle materials, such as glass, ceramics, and crystal, and nonferrous materials, such as copper alloy and aluminum alloy, are usually turned by single-point diamond turning machines and ferrous metal materials and porous materials are usually ground by the precision grinding machines. In the manufacturing of high-precision cylindrical components, whether or not the high-precision turning or grinding way is used, the influence factors of precision manufacturing of cylindrical components mainly consist of the cutting tool (grinding wheel or turning tool), slideway’s straightness error, rotary accuracy of the spindle, parallelism error between the slideway and the spindle’s rotary axis, external and internal vibration, temperature, etc. Besides, tool abrasion and cutting force also have much influence on the surface quality. The surface roughness is mainly affected by tool’s condition and process parameters, and the surface waviness is generated by vibration and the related factors. For convenience, the influences of slideway straightness, rotary accuracy of the spindle, their parallelism error, and some random errors on the cylindrical component’s surface quality are mainly considered in this paper. The schematic of position relationship of the spindle and the slideway is shown in Figure1.Figure 1
Schematic of position relationship of the spindle and the slideway.


## 2.2. The Error Analysis of the Slideway

In the precision machine tools, the linear slideway, aerostatic bearing slideway, and hydrostatic slideway are usually used. No matter what kind of slideway is adopted in a precision machine tool, there may be some form errors. The slideway’s profile in thez direction may be seen in Figure 2–4, which consists of the form error profile, waviness profile, and roughness profile and can be expressed as follows:(1)ρgpz=ρgfz+ρgwz+ρgrz,where ρgf is the form error profile, which can be expressed as follows:(2)ρgfz=Wz+∑j=1Ngfgfwjzsin2πzjλf+θj,z=0∼L,where W(z) can be calculated by the following equation:(3)Wz=±RA2∓0.5L−z21/2−RA+G,for convex or concave arc profile,±GsineπzL,for convex or concavesineprofile,±0.5GsinekgπzL,kg=2,3,4,5,forsineprofile,where G is the bow altitude of the arc profile or the magnitude of the half sine wave profile. If G is equal to zero, the slideway form error profile characterized by the first item in equation (2) is an ideal straight line. For convex and concave arc profiles, signs “+” and “−” on the left of the brace are selected, respectively, and signs “−” and “+” on the left of the parentheses are selected where z ≤ L/2 and z > L/2, respectively, as shown in Figure 2, where L is the slideway length and RA is the radius of the arc profile. For the convex half sine profile characterized by the first item in equation (2), sign “+” on the left of G is selected, and for the concave half sine profile, sign “−” on the left of G is selected, as shown in Figure 3. For the sine profile, signs “+” and “−” can be selected according to the actual needs. The second item in equation (2) is also a kind of the form error profile, whose wavelength is j times of the cutoff wavelength λf of the form error profile, θj is the initial angle of the jth wave, and Ngf is the maximum times of the upper cutoff wavelength λf of the form error profile, which may be supposed to be five, and the product of times of Ngf and λf is equal to or less than the slideway length L, as shown in Figure 4.Figure 2
Schematic of the arc profile of the slideway characterized by the first item in equation (2). (a) Straight. (b) Convex. (c) Concave.
(a)(b)(c)Figure 3
Schematic of the sine profile of the slideway characterized by the first item in equation (2). (a) Convex (kg = 1). (b) Concave (kg = 1). (c) Sine profile (kg = 2, 3, …).
(a)(b)(c)Figure 4
The form error profile of the slideway characterized by the second item in equation (2).In equation (1), ρgw is the waviness profile with a cutoff wavelength λc along the axis of the cylindrical component, which is expressed as follows:(4)ρgwz=∑j=1Ngwgwwjzsin2πzjλc+φj,z=0∼L,where gwwjz is the magnitude of the jth wave in the waviness profile, whose wavelength is j times of the cutoff wavelength λc of the waviness profile, φj is the initial angle of the jth wave, Ngw is the maximum times of the cutoff wavelength λc, and the product of times of Ngw and λc is equal to or less than λf.In equation (1), ρgr is the roughness profile with a cutoff wavelength λs along the slideway in the z direction, which is expressed as follows:(5)ρgrz=∑j=1Ngrgrwjzsin2πzjλs+ϑj+δz,z=0∼L,where grwjz is the magnitude of the jth wave in the roughness profile, whose wavelength is j times of the cutoff wavelength λs of the roughness profile, ϑj is the initial angle of the jth wave, Ngr is the maximum times of the cutoff wavelength λs of the roughness profile, the product of times of Ngr and λs is equal to or less than λc, and δ(z) are the random numbers between 0 and grw1/5.The waviness and roughness profiles characterized by equations (4) and (5) are similar to the form profile shown in Figure 4, but their cutoff wavelengths and magnitudes are less than those of the form profiles characterized by equation (2).


## 2.3. The Rotary Error Analysis of the Spindle

Through the analysis of the roundness profile’s compositions introduced by the spindle rotary error, tool wear, vibration, and so on, the roundness profile can be expressed as follows:(6)ρrpz,ψz=ρrez,ψz+ρrfz,ψz+ρrwz,ψz+ρrrz,ψz,z=0∼L,ψz=0∼2π,where ρre(z, ψ) is the radial size of the cylinder with some eccentricity and multi‐sine waves to the rotary spindle axis, shown in Figure 5, which can be expressed as follows:(7)ρrez,ψz=ρsk2ψz−e2zsin2ψz−ϕz+ezcosψz−ϕz,where e(z) is the eccentricity of the roundness profile, ϕ(z) is the eccentricity angle of the roundness profile, ψ(z) is the angle between one point in the roundness profile and the x axis in the xoy coordinate system, and ρsk(ψ(z)) is the radial size between the rotary axis of the spindle and the circle with kr UPR at the angle ψ(z), where UPR means undulations per round. ρsk(ψ(z)) can be determined as follows:(8)ρskψz=R+Asin2πψz−msam−1msa,m=1,2,…,kr,ψz=0∼2π,where R is the radius of an ideal cylinder, kr is the UPR number in a roundness profile of the cylinder, A is the magnitude of the sine wave profile, and msa is equal to 2π/kr. The parameters above can be seen in Figure 5.Figure 5
Schematic of the parameters in equations (7) and (8).Based on equation (8), the circle profiles with kr UPR can be simulated, as shown in Figure 6.Figure 6
The circle profiles with differentkr UPR. (a) kr = 1. (b) kr = 2. (c) kr = 3. (d) kr = 4. (e) kr = 5. (f) kr = 6. (g) kr = 7. (h) kr = 8.
(a)(b)(c)(d)(e)(f)(g)(h)In equation (6), ρrf(z, ψ) is the form error profile in the roundness profile, which can be expressed as follows:(9)ρrfz,ψz=∑j=1NrfrfwjzsinπDψzjλf+ζjz,z=0∼L,ψz=0∼2π,where rfwj(z) is the magnitude of the jth wave of the form error profile in the roundness profile, whose wavelength is j times of the cutoff wavelength λf of the form error profile, ζj(z) is the initial angle of the jth wave in the roundness profile, Nrf is the maximum times of the upper cutoff wavelength λf of the waviness profile, which may be supposed to be five, and the product of times of Nrf and λf is equal to or less than the perimeter of the roundness profile.In equation (6), ρrw(z, ψ) is the waviness profile in the roundness profile, which can be expressed as follows:(10)ρrwz,ψz=∑j=1NrwrwwjzsinπDψzjλc+ξjz,z=0∼L,ψz=0∼2π,where rwwj(z) is the magnitude of the jth wave of the waviness profile in the roundness profile, whose wavelength is j times of the cutoff wavelength λc of the waviness profile, ξj(z) is the initial angle of the jth wave in the roundness profile, Nrw is the maximum times of the upper cutoff wavelength λc of the waviness profile, and the product of times of Nrw and λc is equal to or less than λf.In equation (6), ρrr(z, ψ) is the roughness profile in the roundness profile, which can be expressed as follows:(11)ρrrz,ψz=∑j=1NrrrrwjzsinπDψzjλs+ηjz+δψz,z=0∼L,ψz=0∼2π,where rrwj(z) is the magnitude of the jth wave of the roughness profile in the roundness profile, whose wavelength is j times of the cutoff wavelength λs of the roughness profile, ηj(z) is the initial angle of the jth wave, Nrr is the maximum times of the cutoff wavelength λs of the roughness profile, the product of times of Nrr and λs is equal to or less than λc, and δ(ψ(z)) are the random numbers between 0 and rrw1(z)/5.In equations (9)–11, D is the diameter of an ideal cylinder.


## 2.4. The Parallelism Error Analysis of the Slideway to the Spindle Rotary Axis

Owing to that there is the parallelism error of the slideway to the spindle’s rotary axis, the fabricated cylinder may be a cone, and the generatrix profile of which may be expressed as follows:(12)ρsz=S×zL+CD,z=0∼L,where S is the slope of the fabricated generatrix, which represents whether the slideway is parallel to the spindle axis, and CD is the intercept of the generatrix in the zoρs coordinate system. The parameters in equation (12) are shown in Figure 7.Figure 7
Schematic of the parameters in equation (12).


## 2.5. The Profile of the Cylinder Gained by Simulation

According to Sections2.2–2.4, the profile of the cylinder gained by manufacturing simulation can be expressed as follows:(13)ρz,ψz=ρgpz+ρrpz,ψz+ρsz,z=0∼L,ψz=0∼2π.Besides the influence factors mentioned above, the modeling of the cylindrical feature’s profile simulation does not consider the other influence factors, such as heat deflection, tool wear, and vibration.


## 3. Determination of Modeling Parameters of Cylindrical Feature

For simulation of precision manufacturing, the equations in Section2 should be discretized. If we use a series of roundness profiles with an equal interval or a single spiral profile to characterize a cylinder’s size feature gained by using precision manufacturing simulation, as shown in Figure 8, there are two characteristic parameters, that is, the axial interval ∆zr of two adjacent roundness profiles and the angular interval ∆ψr of two adjacent sampling points in a roundness profile for the size feature in Figure 8(a). The axial interval ∆zg of two adjacent points in one generatrix profile and the angular interval ∆ψg of two adjacent generatrix profiles are two parameters for the generatrix extraction strategy in Figure 8(b). The birdcage extraction strategy in Figure 8(c) needs four parameters, shown in Figures 8(a) and 8(b), and there are also two characteristic parameters, that is, the pitch ∆zs of a single spiral or the axial interval ∆z1 of two adjacent sampling points or the angular interval ∆ψs of two adjacent sampling points in Figure 8(d). For the need of determining the characteristic parameters, the cylinder’s size feature, the cutoff wavelengths of the profiles determined by equation (13) should be analyzed, which are related to the parameters ∆zr, ∆ψr, ∆zg, ∆ψg, ∆zs(or ∆z1), and ∆ψs. Suppose that the cutoff wavelengths of the form profile, waviness profile, and roughness profile are λf, λc, and λs, respectively, and the selection of the cutoff wavelength λx depends on the compositions of the profiles. If the profile consists of the form error profile, waviness profile, and roughness profile, we select λs as the cutoff wavelength λx, and if the profile includes the form error profile and waviness profile, λc will be selected as λx. Otherwise, λf should be selected as λx.Figure 8
Schematic of parameters of the extraction strategies [6, 7]. (a) Roundness profile (RP). (b) Generatrix profile (GP). (c) Birdcage profile (BP). (d) Single spiral profile (SP).
(a)(b)(c)(d)For the determination of the cutoff wavelengthλx, we can first refer to the determining methods of the sampling length in the measurement of surface roughness to determine the parameter λs. The corresponding sampling length values of the surface roughness values Ra 0.008∼Ra 2.0 or Rz 0.025∼Rz 10.0 can be seen in Table 1.Table 1
The relationship of surface roughness values and sampling length values [8].
Ra (μm)Rz (μm)lr (mm)≥0.008∼0.02≥0.025∼0.100.08>0.02∼0.1>0.10∼0.500.25>0.1∼2.0>0.50∼10.00.8The object of the sampling lengthlr stipulated in reference [8] is to eliminate the influence of waviness on the evaluation of surface roughness, so we may approximately take lr as the cutoff wavelength λc1. On the other hand, the usually adopted parameters of the Gaussian filter [9] in the roundness measurement are shown in Table 2. According to the relationship of the diameter of the measured cylindrical component and the cutoff wavelength of the Gaussian filter, we can gain the cutoff wavelength λc2.Table 2
The relationship of the diameter of the measured component and Gaussian filter’s cutoff wavelength.
Diameter of component (mm)Filter’s cutoff (UPR)Wavelengthλc2 (mm)Diameter of component (mm)Filter’s cutoff (UPR)Wavelengthλc2(mm)≤815≤1.675>80∼250500>1.675∼1.57>8∼2550>1.675∼1.57>2501500>0.523>25∼80150>1.57∼1.675From Tables1 and 2, it is known that λc1 is related to the surface roughness value Ra or Rz, and λc2 depends on the diameter of the measured cylinder. Besides, the cutoff wavelengths of the form error profile, waviness profile, and roughness profile are usually classified as follows: that is, the profile with the wavelengths being equal to or greater than 10 mm belongs to the form error profile, the profile with the wavelengths less than 1 mm is the roughness profile, and the wavelengths of the waviness profile are equal to or greater than 1 mm and less than 10 mm. Therefore, the cutoff wavelength λc3 is 1 mm. We can take one of λc1, λc2, and λc3 as λc based on the need of the evaluation item.According to Nyquist sampling criterion [10], if a longer sampling interval than the Nyquist criterion is specified, the digitized signal suffers from aliasing distortion; therefore, more than 2 points are needed for each wave. For reducing aliasing distortion induced by digitization, the sampled points for each wave with a wavelength of λx are selected as 4, so the sampling interval in each roundness profile is as follows:(14)Δs=λx4,where λx may be λc or λs. The total sampling number of each roundness profile can be gained by the following equation:(15)nr=πDΔs,and the angular interval ∆s = ∆ψr = 2π/nr.Suppose that the cutoff wavelengthλf of a generatrix is equal to 10λc or 10 mm or larger based on the cylindrical component’s size or form tolerance and cylinder’s length L, which does not consider the influence of waviness and roughness profiles, the interval ∆zr of two adjacent roundness profiles may be equal to λf/4∼λf/2.The parameters ∆ψs and ∆zs of a single spiral profile in Figure 8(d) may be equal to the parameters ∆ψr and ∆zr, respectively. The axial interval ∆z1 of two adjacent points in the axial direction is as follows:(16)Δz1=Δzsns.In equations (1)∼(7), z = (i − 1)∆z1 and ψx = (j − 1)∆ψx, where x means r for roundness profiles and s for the single spiral profile. The number of the roundness profiles can be calculated as follows:(17)nz=LΔz1+1.The pitch numberms of the single profile is equal to mr − 1. For setting G in equation (3) and S in equation (12), the size tolerance of component’s diameter and the usual parallelism specification of the slideway and the spindle rotary axis can be considered. Suppose that the size tolerance of the component is Ts, the parallelism index of the slideway and the spindle rotary axis is Tp and Cpk = η, and then size distribution T6σ = Ts/η. We can approximately assign one fourth of T6σ to G and S, respectively. For gfwj(j = 1∼Ngf) in equation (2) and rfwj(j = 1∼Nrf) in equation (9), both gfwj (j = Ngf) and rfwj(j = Nrf) may be set as a quarter of the roundness tolerance or the cylindricity tolerance, and the values of gfwj(j = (Ngf − 1∼1) or rfwj(j = (Nrf − 1∼1) reduce in turn based on a priority number system R10 or others. For grwj(j = 1∼Ngr) in equation (5) and rrwj(j = 1∼Nrr) in equation (11), both grwj(j = Ngr) and rrwj(j = Nrr) may be set as surface roughness Ra, and the values of grwj(j = (Ngr − 1∼1) or rrwj(j = (Nrr − 1∼1) reduce in turn based on a priority number system R10. For gwwj(j = 1∼Ngw) in equation (4) and rwwj(j = 1∼Nrw) in equation (10), both gwwj(j = Ngw) and rwwj(j = Nrw) may be set as two times of the surface roughness Ra, and the values of gwwj(j = (Ngw − 1∼1) or rwwj(j = (Nrw − 1∼1) reduce in turn based on a priority number system R10.For the initial angles in equations (2), (4), (5), and (9)–(11), they may not abide by any rule and may be set according to random probability distribution in a range from zero to 2π.Considering that the radial cutting force of the precision manufacturing is not large, the influence of the bending deformation caused by the radial cutting force on the simulation of cylinder’s size feature is not considered in this paper. On the other hand, even if the roughness profile in equations (1) and (6) is all included, owing to that only cylinder’s size feature needs to be simulated, the simulation of cylinder’s size feature can eliminate the roughness profile.


## 4. Simulation Experiment and Discussion

### 4.1. Simulation Experiment

We took a shaft in Figure9 as an example for the extraction simulation of the cylinder’s size feature based on the models in Sections 2 and 3. The diameter and its tolerance, upper limit deviation, roundness tolerance, cylindricity tolerance, surface roughness Ra, and length of the shaft are shown in Figure 9. Four extraction strategies of cylinder’s feature in Figure 8 were adopted for the cylindrical component in Figure 9. The simulation parameters are shown in Table 3, where X can be replaced by the symbols 0, 1, and 2, respectively. The profiles indicated by the symbol 0 consist of the form, waviness, and roughness profiles, the profiles indicated by the symbol 1 include the form and waviness profiles, and the profiles indicated by the symbol 2 are only the form profiles. In Table 3, tilt_angle is the angle between the axis of the simulated cylinder and the z axis. The cylindrical features shown in Figures 10–17 are simulated according to the parameters in Table 3, and the sampling number of which is 2000, where RP, GP, BP, and SP represent the profiles gained based on the extraction strategies of the roundness profiles, generatrix profiles, birdcage profile, and single spiral profile, respectively.Figure 9
Cylindrical component used for simulation of feature extraction.Table 3
Simulation parameters of cylindrical features of the part in Figure9.
Symbol1N1_X1N3_X3N1_X3N3_X1Y1_X1Y3_X3Y1_X3Y3_XRemarksG (mm)000.0010.001000.0010.001Equation (3)kg00330033Ngf, Ngw, Ngr5Equations (2), (4), (5)gfwj, gwwj, grwjgfw1∼gfw5 = 0.112∼0.560 μm; gww1∼gww5 = 0.066∼0.320 μm; grw1∼grw5 = 0.050∼0.250 μmθj, φj, ϑjRandom values between 0 and 2πe(0) (mm)0.001Equation (7)ϕ(z)Random values from between 0 and 2πR (mm)(50–0.025)/2Equation (8)A (mm)00.000500.000500.000500.0005kr13131313Nrf, Nrw, Nrr5Equations (9)–(11)rfwj, rwwj, rrwjrfw1∼rfw5 = 0.112∼0.560 μm; rww1∼rww5 = 0.066∼0.320 μm; rrw1∼rrw5 = 0.050∼0.250 μmθj, φj, ϑjRandom values between 0 and 2πλf, λc, λsλf = 1.6225 mm; λc = 0.3 mm, λs = 0.06 mmEquations (2), (4), (5), (9)–(11)δ(z), δ(ψ(z)) (mm)Random values between 0 and 0.5 ∗grw1 or between 0 and 0.5 ∗ rrw1Equations (5), (11)S00000.0010.0010.0010.001Equation (13)CD (mm)0L (mm)100tilt_angle (radian)0.001mr,msmr = L/2.5 + 1 = 41, ms = L/2.5 = 40nr,ns500, 625, 1000, 1250, 2000, 2500, 5000, and 10000 in the same extraction sampleng, mgng = 50π/2.5 ≈ 63, mg = mr ∗ nr/ng ≈ 1302Figure 10
Cylindrical features gained based on the simulation parameter symbol “1N1_0.” (a) RP. (b) GP. (c) BP. (d) SP.
(a)(b)(c)(d)Figure 11
Cylindrical features gained based on the simulation parameter symbol “1N3_0.” (a) RP. (b) GP. (c) BP. (d) SP.
(a)(b)(c)(d)Figure 12
Cylindrical features gained based on the simulation parameter symbol “3N1_0.” (a) RP. (b) GP. (c) BP. (d) SP.
(a)(b)(c)(d)Figure 13
Cylindrical features gained based on the simulation parameter symbol “3N3_0.” (a) RP. (b) GP. (c) BP. (d) SP.
(a)(b)(c)(d)Figure 14
Cylindrical features gained based on the simulation parameter symbol “1Y1_0.” (a) RP. (b) GP. (c) BP. (d) SP.
(a)(b)(c)(d)Figure 15
Cylindrical features gained based on the simulation parameter symbol “1Y3_0.” (a) RP. (b) GP. (c) BP. (d) SP.
(a)(b)(c)(d)Figure 16
Cylindrical features gained based on the simulation parameter symbol “3Y1_0.” (a) RP. (b) GP. (c) BP. (d) SP.
(a)(b)(c)(d)Figure 17
Cylindrical features gained based on the simulation parameter symbol “3Y3_0.” (a) RP. (b) GP. (c) BP. (d) SP.
(a)(b)(c)(d)


### 4.2. Evaluation of Cylindrical Features

The evaluation objects of cylindrical features include global sizes, calculated sizes, form errors, waviness, and roughness in this paper. The evaluation results of global sizes and cylindricity errors based on the cylindrical features simulated by using the parameters in Table3 and the evaluation models [5] are shown in Figures 18 and 19, where LSD, MCD, MID, and MMD are the abbreviations of least-squares diameters, minimum circumscribed diameters, maximum inscribed diameters, and minimax diameters, respectively, and LSC, MCC, MIC, and MZC are the abbreviations of cylindricity errors evaluated by least-square criterion, minimum circumscribed criterion, maximum inscribed criterion, and minimum zone criterion [11–14], respectively. The evaluated roundness errors of the cylindrical features based on the roundness profile extraction strategy and the corresponding models [15, 16] are shown in Figure 20, where MEAN, MAX, and MIN represent the mean, maximum, and minimum values of roundness errors for one cylindrical feature, respectively, and their corresponding calculated size is shown in Figure 21, where C_D, A_D, and V_D represent the circumference diameter, area diameter, and volume diameter of the calculated sizes, which were evaluated based on the corresponding roundness profiles above and the models [17], and the meanings of MEAN, MAX, and MIN can be seen in Figure 20.Figure 18
Global sizes and cylindricity errors of the cylindrical features produced under parameter symbols 1N1, 1N3, 3N1, and 3N3. (a) 1N1-global sizes. (b) 1N1-cylindricity errors. (c) 1N3-global sizes. (d) 1N3-cylindricity errors. (e) 3N1-global sizes. (f) 3N1-cylindricity errors. (g) 3N3-global sizes. (h) 3N3-cylindricity errors.
(a)(b)(c)(d)(e)(f)(g)(h)Figure 19
Global sizes and cylindricity errors of the cylindrical features produced under parameter symbols 1Y1, 1Y3, 3Y1, and 3Y3. (a) 1Y1-global sizes. (b) 1Y1-cylindricity errors. (c) 1Y3-global sizes. (d) 1Y3-cylindricity errors. (e) 3Y1-global sizes. (f) 3Y1-cylindricity errors. (g) 3Y3-global sizes. (h) 3Y3-cylindricity errors.
(a)(b)(c)(d)(e)(f)(g)(h)Figure 20
Statistic values of roundness errors of the cylindrical features. (a) Least-square roundness errors. (b) Minimum circumscribed roundness errors. (c) Maximum inscribed roundness errors. (d) Minimum zone roundness errors.
(a)(b)(c)(d)Figure 21
Statistic values of calculated sizes of cylindrical features.For the evaluation of the axis straightness error of a cylinder, the centeroi(ai, bi, zi) of the reference circle of the ith roundness profiles is first determined, as shown in Figure 22, where ρij is the discretized value of ρ(z, ψ(z)) in equation (13), that is, the radial distance from the jth sampling point in the ith roundness profile to the rotary axis, ψij is the angle between the jth sampling point in the ith roundness profile and the x axis, and ei and εi are the eccentricity and eccentric angles of the ith reference circle. The reference circle may be the least-square reference cycle or the minimum circumscribed reference circle or the maximum inscribed reference circle or the minimum zone reference circle, which corresponds to four kinds of evaluation criteria of the roundness error mentioned above.Figure 22
Schematic of center of the reference circle.The actual axis of the simulated cylinder can be expressed as the sequential connection of centers of the reference circles, as shown in Figure23.Figure 23
Schematic of axis of the simulated cylinder.There are two evaluation criterions of the axis straightness error, that is, least-square criterion and minimum zone criterion. Suppose that the ideal axis in Figure23 is characterized by the parameters xa0, ya0, pa, and qa, the evaluation of the axis straightness based on the least-square criterion should meet the requirement as follows:(18)Fa=∑i=1mrdri2=min,where dri is the radial distance from the center oi of the ith reference circle to the ideal axis, as shown in Figure 23, which can be expressed as follows:(19)dri=ai−xa0−pazi2+bi−ya0−qazi2,where the parameters xa0, ya0, pa, and qa of the ideal axis are undetermined ones.Let the partial derivatives ofFa with respect to variables xa0, ya0, pa, and qa be equal to zero, respectively, and a linear system of equations with variables xa0, ya0, pa, and qa can be obtained. After solving the linear system of equations, the parameters xa0, ya0, pa, and qa of the ideal axis in Figure 23 can be gained, and the axis straightness error faLS determined based on the least-square criterion can be expressed as follows:(20)faLS=2max1≤i≤mrdvi,where mr is the number of the roundness profiles and dvi is the vertical distance from the center oi of the ith reference circle to the ideal axis, which can be calculated by the following equation:(21)dvi=ai−xa02+bi−ya02+zi2−pa0ai−xa0+qa0bi−ya0+zi21+pa2+qa2.Based on the definition of the minimum zone criterion, the axis straightness errorfaMZ can be expressed as follows:(22)faMZ=2minXamax1≤i≤mrdvi,Xa=xa0ya0paqaT,where Xa represents the parameters of the ideal axis, which may be obtained through the optimization algorithm, and the parameters of the ideal axis determined based on the least-square criterion are usually taken as their initial values of the parameters of the ideal axis determined by using the minimum zone criterion through optimization.For the evaluation of the generatrix straightness errors of a cylinder, azoρj coordinate system is built for the jth generatrix profile, as shown in Figure 24, where ρji is the radial distance from the ith sampling point in the jth generatrix profile to the z axis and dji is the radial distance from the ith sampling point in the jth generatrix profile to the ideal line, which may be determined by using the least-square criterion or the minimum zone criterion.Figure 24
Schematic of building the ideal line based on the generatrix profile.Suppose that the slope and the intercept of the ideal line areAi and Bj, and then the ideal line in Figure 24 can be expressed as follows:(23)ρj=Ajzi+Bj.The least-square ideal line should satisfy the following condition; that is,(24)Fg=∑i=1mgdji2=∑i=1mgρji−Ajzi−Bj2=min,where mg is the number of the sampling points in the jth generatrix profile.Let the partial derivatives ofFg with respect to variables Aj and Bj be equal to zero, respectively, and a linear system of equations with variables Aj and Bj can be obtained. After solving the linear system of equations, the parameters Aj and Bj of the ideal line in Figure 24 can be gained, and the generatrix straightness error fgLSj of the jth generatrix profile by using the least-square criterion can be expressed as follows:(25)fgLSj=11+Aj2max1≤i≤mgdji−min1≤i≤mgdji.According to the definition of the minimum zone method, the generatrix straightness errorfgMZj can be expressed as follows:(26)fgMZj=11+Aj2minAj,Bjmax1≤i≤mgdji−min1≤i≤mgdji.Through optimizing the parametersAj and Bj of the ideal line and realizing the object function in equation (26), the minimum zone generatrix straightness error can be obtained. The parameters of the least-square ideal axis are usually taken as their initial values of the parameters of the minimum zone ideal axis for optimization.The evaluation results of axis straightness errors can be seen in Figure25(a), where LS-××, MC-××, MI-××, and MZ-×× represent the center coordinates of least-square reference circles, minimum circumscribed reference circles, maximum inscribed reference circles, and minimum zone reference circles for all roundness profiles of one cylindrical feature, respectively, and ××-LS and ××-MZ represent the axis straightness errors evaluated by using the least-square criterion [18] and minimum zone criterion [19], respectively. The evaluation results of generatrix straightness errors [20] of generatrix profiles can be seen in Figure 25(b), and the symbol meanings can be seen in Figures 21 and 25(a).Figure 25
Statistic values of straightness errors of cylindrical features. (a) Axis straightness errors. (b) Generatrix straightness errors.
(a)(b)Besides the statistics of the evaluation results of global sizes, calculated sizes, form errors, such as cylindricity errors, roundness errors, generatrix straightness errors, and axis straightness errors, their waviness and roughness evaluation results were also analyzed, as shown in Figure26.Figure 26
Statistic values of waviness and roughness of cylindrical features.


### 4.3. Discussion

From Figures18(a), 18(c), 18(e), and 18(g), the values of MMD are near those of LSD for all extraction strategies, and the influence of waviness and roughness on the MMD and LSD is less, but there are some differences between values of MZC and LSC for each extraction profile. From Figure 26, the maximum, minimum, and mean values of waviness and roughness have no much difference for generatrix profiles, but the three values above have much difference for the roundness profile.For further researching the influence of sampling number on the evaluation results of global sizes and cylindricity errors, we took the sampling numbernr or ns in Table 3 as 500, 625, 1000, 1250, 2000, 2500, 5000, and 10000, respectively, which are from the same simulated cylindrical profiles under the sampling number nr or ns being 10000. Their corresponding global sizes and cylindricity errors for 8 kinds of simulation parameter symbols and 4 kinds of extraction strategies in Figures 18 and 19 were evaluated. Suppose that the gained global sizes and cylindricity errors are expressed as DESML and CESML, respectively, where subscripts E, S, M, and L represent extraction strategy of profiles, simulation parameter symbol, evaluation method, and sampling number, respectively. Suppose that E from 1 to 4 means RP, GP, BP, and SP, respectively, S from 1 to 8 corresponds to the symbols 1N1_X, 1N3_X, 3N1_X, 3N3_X, 1Y1_X, 1Y3_X, 3Y1_X, and 3Y3_X, respectively, M from 1 to 4 means LSD (LSC), MCD (MCC), MID (MIC), and MMD (MZC), respectively, and L from 1 to 7 means the sampling points 500, 625, 1000, 1250, 2000, 2500, and 5000, respectively. We can obtain the following XAML matrix based on XRSEML values, where XRSEML = DSEML/DSEMI for global sizes and CSEML/CSEM7 for cylindricity errors, M = 1∼4, L = 1∼7 and symbol I means that the sampling number is 10000:(27)XAML=XR11MLXR12MLXR13MLXR14MLXR15MLXR16MLXR17MLXR18MLXR21MLXR22MLXR23MLXR24MLXR25MLXR26MLXR27MLXR28MLXR31MLXR32MLXR33MLXR34MLXR35MLXR36MLXR37MLXR38MLXR41MLXR42MLXR43MLXR44MLXR45MLXR46MLXR47MLXR48ML.The mean, maximum, minimum valuesDsL/DsI or CsL/CsI and the standard deviation values DsL/DsI-std or CsL/CsI-std of XAML for each M and each L can be gained, as shown in Figures 27–29, and the symbols from s1 to s7 correspond to L from 1 to 7, respectively; SG means the group of different sampling numbers simulated under 4 kinds of extraction strategies, and the meaning of 0, 1, and 2 after SG can be seen in the horizontal axis in Figure 28. From Figures 27 and 28, we can see that the sampling points have no much influence on the evaluation results of LSD, MCD, MID, and MMD, and the standard deviation values DsL/DsI-std of LSD is the least and DsL/DsI-std of MCD is the largest among 4 kinds of global sizes. From Figures 27 and 29, we also can see that the sampling number has much influence on the evaluation results of LSC, MCC, MIC, and MZC, and the standard deviation values CsL/CsI-std of LSC is the least. On the other hand, when the extracted profiles for SG2, as shown in Figures 27–29, do not include waviness and roughness profiles, the DsL/DsI-std and CsL/CsI-std of global sizes and cylindricity errors are least among their values of the extracted profiles for SG0, SG1, and SG2, which means that the influence of the sampling number on the measuring results of global sizes and cylindricity sizes is the least when the profiles are extracted under the SG2 condition.Figure 27
DsL/DsI & CsL/CsI of different sampling numbers. (a) LSD and LSC. (b) MCD and MCC. (c) MID and MIC. (d) MMD and MZC.
(a)(b)(c)(d)Figure 28
DsL/DsI-std of different sampling numbers.Figure 29
CsL/CsI-std of different sampling numbers.


## 4.1. Simulation Experiment

We took a shaft in Figure9 as an example for the extraction simulation of the cylinder’s size feature based on the models in Sections 2 and 3. The diameter and its tolerance, upper limit deviation, roundness tolerance, cylindricity tolerance, surface roughness Ra, and length of the shaft are shown in Figure 9. Four extraction strategies of cylinder’s feature in Figure 8 were adopted for the cylindrical component in Figure 9. The simulation parameters are shown in Table 3, where X can be replaced by the symbols 0, 1, and 2, respectively. The profiles indicated by the symbol 0 consist of the form, waviness, and roughness profiles, the profiles indicated by the symbol 1 include the form and waviness profiles, and the profiles indicated by the symbol 2 are only the form profiles. In Table 3, tilt_angle is the angle between the axis of the simulated cylinder and the z axis. The cylindrical features shown in Figures 10–17 are simulated according to the parameters in Table 3, and the sampling number of which is 2000, where RP, GP, BP, and SP represent the profiles gained based on the extraction strategies of the roundness profiles, generatrix profiles, birdcage profile, and single spiral profile, respectively.Figure 9
Cylindrical component used for simulation of feature extraction.Table 3
Simulation parameters of cylindrical features of the part in Figure9.
Symbol1N1_X1N3_X3N1_X3N3_X1Y1_X1Y3_X3Y1_X3Y3_XRemarksG (mm)000.0010.001000.0010.001Equation (3)kg00330033Ngf, Ngw, Ngr5Equations (2), (4), (5)gfwj, gwwj, grwjgfw1∼gfw5 = 0.112∼0.560 μm; gww1∼gww5 = 0.066∼0.320 μm; grw1∼grw5 = 0.050∼0.250 μmθj, φj, ϑjRandom values between 0 and 2πe(0) (mm)0.001Equation (7)ϕ(z)Random values from between 0 and 2πR (mm)(50–0.025)/2Equation (8)A (mm)00.000500.000500.000500.0005kr13131313Nrf, Nrw, Nrr5Equations (9)–(11)rfwj, rwwj, rrwjrfw1∼rfw5 = 0.112∼0.560 μm; rww1∼rww5 = 0.066∼0.320 μm; rrw1∼rrw5 = 0.050∼0.250 μmθj, φj, ϑjRandom values between 0 and 2πλf, λc, λsλf = 1.6225 mm; λc = 0.3 mm, λs = 0.06 mmEquations (2), (4), (5), (9)–(11)δ(z), δ(ψ(z)) (mm)Random values between 0 and 0.5 ∗grw1 or between 0 and 0.5 ∗ rrw1Equations (5), (11)S00000.0010.0010.0010.001Equation (13)CD (mm)0L (mm)100tilt_angle (radian)0.001mr,msmr = L/2.5 + 1 = 41, ms = L/2.5 = 40nr,ns500, 625, 1000, 1250, 2000, 2500, 5000, and 10000 in the same extraction sampleng, mgng = 50π/2.5 ≈ 63, mg = mr ∗ nr/ng ≈ 1302Figure 10
Cylindrical features gained based on the simulation parameter symbol “1N1_0.” (a) RP. (b) GP. (c) BP. (d) SP.
(a)(b)(c)(d)Figure 11
Cylindrical features gained based on the simulation parameter symbol “1N3_0.” (a) RP. (b) GP. (c) BP. (d) SP.
(a)(b)(c)(d)Figure 12
Cylindrical features gained based on the simulation parameter symbol “3N1_0.” (a) RP. (b) GP. (c) BP. (d) SP.
(a)(b)(c)(d)Figure 13
Cylindrical features gained based on the simulation parameter symbol “3N3_0.” (a) RP. (b) GP. (c) BP. (d) SP.
(a)(b)(c)(d)Figure 14
Cylindrical features gained based on the simulation parameter symbol “1Y1_0.” (a) RP. (b) GP. (c) BP. (d) SP.
(a)(b)(c)(d)Figure 15
Cylindrical features gained based on the simulation parameter symbol “1Y3_0.” (a) RP. (b) GP. (c) BP. (d) SP.
(a)(b)(c)(d)Figure 16
Cylindrical features gained based on the simulation parameter symbol “3Y1_0.” (a) RP. (b) GP. (c) BP. (d) SP.
(a)(b)(c)(d)Figure 17
Cylindrical features gained based on the simulation parameter symbol “3Y3_0.” (a) RP. (b) GP. (c) BP. (d) SP.
(a)(b)(c)(d)


## 4.2. Evaluation of Cylindrical Features

The evaluation objects of cylindrical features include global sizes, calculated sizes, form errors, waviness, and roughness in this paper. The evaluation results of global sizes and cylindricity errors based on the cylindrical features simulated by using the parameters in Table3 and the evaluation models [5] are shown in Figures 18 and 19, where LSD, MCD, MID, and MMD are the abbreviations of least-squares diameters, minimum circumscribed diameters, maximum inscribed diameters, and minimax diameters, respectively, and LSC, MCC, MIC, and MZC are the abbreviations of cylindricity errors evaluated by least-square criterion, minimum circumscribed criterion, maximum inscribed criterion, and minimum zone criterion [11–14], respectively. The evaluated roundness errors of the cylindrical features based on the roundness profile extraction strategy and the corresponding models [15, 16] are shown in Figure 20, where MEAN, MAX, and MIN represent the mean, maximum, and minimum values of roundness errors for one cylindrical feature, respectively, and their corresponding calculated size is shown in Figure 21, where C_D, A_D, and V_D represent the circumference diameter, area diameter, and volume diameter of the calculated sizes, which were evaluated based on the corresponding roundness profiles above and the models [17], and the meanings of MEAN, MAX, and MIN can be seen in Figure 20.Figure 18
Global sizes and cylindricity errors of the cylindrical features produced under parameter symbols 1N1, 1N3, 3N1, and 3N3. (a) 1N1-global sizes. (b) 1N1-cylindricity errors. (c) 1N3-global sizes. (d) 1N3-cylindricity errors. (e) 3N1-global sizes. (f) 3N1-cylindricity errors. (g) 3N3-global sizes. (h) 3N3-cylindricity errors.
(a)(b)(c)(d)(e)(f)(g)(h)Figure 19
Global sizes and cylindricity errors of the cylindrical features produced under parameter symbols 1Y1, 1Y3, 3Y1, and 3Y3. (a) 1Y1-global sizes. (b) 1Y1-cylindricity errors. (c) 1Y3-global sizes. (d) 1Y3-cylindricity errors. (e) 3Y1-global sizes. (f) 3Y1-cylindricity errors. (g) 3Y3-global sizes. (h) 3Y3-cylindricity errors.
(a)(b)(c)(d)(e)(f)(g)(h)Figure 20
Statistic values of roundness errors of the cylindrical features. (a) Least-square roundness errors. (b) Minimum circumscribed roundness errors. (c) Maximum inscribed roundness errors. (d) Minimum zone roundness errors.
(a)(b)(c)(d)Figure 21
Statistic values of calculated sizes of cylindrical features.For the evaluation of the axis straightness error of a cylinder, the centeroi(ai, bi, zi) of the reference circle of the ith roundness profiles is first determined, as shown in Figure 22, where ρij is the discretized value of ρ(z, ψ(z)) in equation (13), that is, the radial distance from the jth sampling point in the ith roundness profile to the rotary axis, ψij is the angle between the jth sampling point in the ith roundness profile and the x axis, and ei and εi are the eccentricity and eccentric angles of the ith reference circle. The reference circle may be the least-square reference cycle or the minimum circumscribed reference circle or the maximum inscribed reference circle or the minimum zone reference circle, which corresponds to four kinds of evaluation criteria of the roundness error mentioned above.Figure 22
Schematic of center of the reference circle.The actual axis of the simulated cylinder can be expressed as the sequential connection of centers of the reference circles, as shown in Figure23.Figure 23
Schematic of axis of the simulated cylinder.There are two evaluation criterions of the axis straightness error, that is, least-square criterion and minimum zone criterion. Suppose that the ideal axis in Figure23 is characterized by the parameters xa0, ya0, pa, and qa, the evaluation of the axis straightness based on the least-square criterion should meet the requirement as follows:(18)Fa=∑i=1mrdri2=min,where dri is the radial distance from the center oi of the ith reference circle to the ideal axis, as shown in Figure 23, which can be expressed as follows:(19)dri=ai−xa0−pazi2+bi−ya0−qazi2,where the parameters xa0, ya0, pa, and qa of the ideal axis are undetermined ones.Let the partial derivatives ofFa with respect to variables xa0, ya0, pa, and qa be equal to zero, respectively, and a linear system of equations with variables xa0, ya0, pa, and qa can be obtained. After solving the linear system of equations, the parameters xa0, ya0, pa, and qa of the ideal axis in Figure 23 can be gained, and the axis straightness error faLS determined based on the least-square criterion can be expressed as follows:(20)faLS=2max1≤i≤mrdvi,where mr is the number of the roundness profiles and dvi is the vertical distance from the center oi of the ith reference circle to the ideal axis, which can be calculated by the following equation:(21)dvi=ai−xa02+bi−ya02+zi2−pa0ai−xa0+qa0bi−ya0+zi21+pa2+qa2.Based on the definition of the minimum zone criterion, the axis straightness errorfaMZ can be expressed as follows:(22)faMZ=2minXamax1≤i≤mrdvi,Xa=xa0ya0paqaT,where Xa represents the parameters of the ideal axis, which may be obtained through the optimization algorithm, and the parameters of the ideal axis determined based on the least-square criterion are usually taken as their initial values of the parameters of the ideal axis determined by using the minimum zone criterion through optimization.For the evaluation of the generatrix straightness errors of a cylinder, azoρj coordinate system is built for the jth generatrix profile, as shown in Figure 24, where ρji is the radial distance from the ith sampling point in the jth generatrix profile to the z axis and dji is the radial distance from the ith sampling point in the jth generatrix profile to the ideal line, which may be determined by using the least-square criterion or the minimum zone criterion.Figure 24
Schematic of building the ideal line based on the generatrix profile.Suppose that the slope and the intercept of the ideal line areAi and Bj, and then the ideal line in Figure 24 can be expressed as follows:(23)ρj=Ajzi+Bj.The least-square ideal line should satisfy the following condition; that is,(24)Fg=∑i=1mgdji2=∑i=1mgρji−Ajzi−Bj2=min,where mg is the number of the sampling points in the jth generatrix profile.Let the partial derivatives ofFg with respect to variables Aj and Bj be equal to zero, respectively, and a linear system of equations with variables Aj and Bj can be obtained. After solving the linear system of equations, the parameters Aj and Bj of the ideal line in Figure 24 can be gained, and the generatrix straightness error fgLSj of the jth generatrix profile by using the least-square criterion can be expressed as follows:(25)fgLSj=11+Aj2max1≤i≤mgdji−min1≤i≤mgdji.According to the definition of the minimum zone method, the generatrix straightness errorfgMZj can be expressed as follows:(26)fgMZj=11+Aj2minAj,Bjmax1≤i≤mgdji−min1≤i≤mgdji.Through optimizing the parametersAj and Bj of the ideal line and realizing the object function in equation (26), the minimum zone generatrix straightness error can be obtained. The parameters of the least-square ideal axis are usually taken as their initial values of the parameters of the minimum zone ideal axis for optimization.The evaluation results of axis straightness errors can be seen in Figure25(a), where LS-××, MC-××, MI-××, and MZ-×× represent the center coordinates of least-square reference circles, minimum circumscribed reference circles, maximum inscribed reference circles, and minimum zone reference circles for all roundness profiles of one cylindrical feature, respectively, and ××-LS and ××-MZ represent the axis straightness errors evaluated by using the least-square criterion [18] and minimum zone criterion [19], respectively. The evaluation results of generatrix straightness errors [20] of generatrix profiles can be seen in Figure 25(b), and the symbol meanings can be seen in Figures 21 and 25(a).Figure 25
Statistic values of straightness errors of cylindrical features. (a) Axis straightness errors. (b) Generatrix straightness errors.
(a)(b)Besides the statistics of the evaluation results of global sizes, calculated sizes, form errors, such as cylindricity errors, roundness errors, generatrix straightness errors, and axis straightness errors, their waviness and roughness evaluation results were also analyzed, as shown in Figure26.Figure 26
Statistic values of waviness and roughness of cylindrical features.


## 4.3. Discussion

From Figures18(a), 18(c), 18(e), and 18(g), the values of MMD are near those of LSD for all extraction strategies, and the influence of waviness and roughness on the MMD and LSD is less, but there are some differences between values of MZC and LSC for each extraction profile. From Figure 26, the maximum, minimum, and mean values of waviness and roughness have no much difference for generatrix profiles, but the three values above have much difference for the roundness profile.For further researching the influence of sampling number on the evaluation results of global sizes and cylindricity errors, we took the sampling numbernr or ns in Table 3 as 500, 625, 1000, 1250, 2000, 2500, 5000, and 10000, respectively, which are from the same simulated cylindrical profiles under the sampling number nr or ns being 10000. Their corresponding global sizes and cylindricity errors for 8 kinds of simulation parameter symbols and 4 kinds of extraction strategies in Figures 18 and 19 were evaluated. Suppose that the gained global sizes and cylindricity errors are expressed as DESML and CESML, respectively, where subscripts E, S, M, and L represent extraction strategy of profiles, simulation parameter symbol, evaluation method, and sampling number, respectively. Suppose that E from 1 to 4 means RP, GP, BP, and SP, respectively, S from 1 to 8 corresponds to the symbols 1N1_X, 1N3_X, 3N1_X, 3N3_X, 1Y1_X, 1Y3_X, 3Y1_X, and 3Y3_X, respectively, M from 1 to 4 means LSD (LSC), MCD (MCC), MID (MIC), and MMD (MZC), respectively, and L from 1 to 7 means the sampling points 500, 625, 1000, 1250, 2000, 2500, and 5000, respectively. We can obtain the following XAML matrix based on XRSEML values, where XRSEML = DSEML/DSEMI for global sizes and CSEML/CSEM7 for cylindricity errors, M = 1∼4, L = 1∼7 and symbol I means that the sampling number is 10000:(27)XAML=XR11MLXR12MLXR13MLXR14MLXR15MLXR16MLXR17MLXR18MLXR21MLXR22MLXR23MLXR24MLXR25MLXR26MLXR27MLXR28MLXR31MLXR32MLXR33MLXR34MLXR35MLXR36MLXR37MLXR38MLXR41MLXR42MLXR43MLXR44MLXR45MLXR46MLXR47MLXR48ML.The mean, maximum, minimum valuesDsL/DsI or CsL/CsI and the standard deviation values DsL/DsI-std or CsL/CsI-std of XAML for each M and each L can be gained, as shown in Figures 27–29, and the symbols from s1 to s7 correspond to L from 1 to 7, respectively; SG means the group of different sampling numbers simulated under 4 kinds of extraction strategies, and the meaning of 0, 1, and 2 after SG can be seen in the horizontal axis in Figure 28. From Figures 27 and 28, we can see that the sampling points have no much influence on the evaluation results of LSD, MCD, MID, and MMD, and the standard deviation values DsL/DsI-std of LSD is the least and DsL/DsI-std of MCD is the largest among 4 kinds of global sizes. From Figures 27 and 29, we also can see that the sampling number has much influence on the evaluation results of LSC, MCC, MIC, and MZC, and the standard deviation values CsL/CsI-std of LSC is the least. On the other hand, when the extracted profiles for SG2, as shown in Figures 27–29, do not include waviness and roughness profiles, the DsL/DsI-std and CsL/CsI-std of global sizes and cylindricity errors are least among their values of the extracted profiles for SG0, SG1, and SG2, which means that the influence of the sampling number on the measuring results of global sizes and cylindricity sizes is the least when the profiles are extracted under the SG2 condition.Figure 27
DsL/DsI & CsL/CsI of different sampling numbers. (a) LSD and LSC. (b) MCD and MCC. (c) MID and MIC. (d) MMD and MZC.
(a)(b)(c)(d)Figure 28
DsL/DsI-std of different sampling numbers.Figure 29
CsL/CsI-std of different sampling numbers.


## 5. Conclusions

Based on the analyses of the slideway error, the rotary error of the spindle, and their parallelism error, the simulation models of the slideway profile, the roundness profile, and the parallel profile were built, respectively. The slideway profile consists of the form profile, waviness profile, and roughness profile, and the roundness profile is composed of the eccentric circle, form profile, waviness profile, and roughness profile. According to the built simulation models above and the given values of their corresponding parameters, four kinds of extraction strategies of cylindrical profiles, such as roundness profiles, generatrix profiles, birdcage profile, and single spiral profile, can be simulated, respectively. The determination methods of the modeling parameters of the cylindrical feature were promoted. A cylindrical component with a nominal diameter ofϕ50 mm, the upper limit deviation of 0 mm, the lower limit deviation of −0.025 mm, the roundness tolerance of 0.004 mm, the cylindricity tolerance of 0.007 mm, and the roughness arithmetical mean deviation Ra of 0.4 μm was taken as an example for the simulation of its cylindrical feature. Under the given eight sets of the modeling parameter values, eight groups of cylindrical features of the component were obtained, and each group of cylindrical features is characterized by the roundness profiles, generatrix profiles, birdcage profile, and single spiral profile, respectively. Based on the obtained cylindrical features, their global sizes, calculated sizes, cylindricity errors, roundness errors, axis straightness errors, generatrix straightness errors, waviness values, and values of roughness Ra were evaluated, and their evaluation values were analyzed through statistics. The simulation and analysis results showed that the modeling parameter values have much influence on the simulation of cylindrical features and their evaluation values of global sizes, cylindricity errors, and roundness errors, and for four extraction strategies of cylindrical profiles, the selected sampling number has no much influence on the global sizes, but has much influence on the cylindricity errors. We can use the built extraction strategy models of cylindrical profiles to further research some issues on the relationship between actual size, form errors, and global sizes.



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*Source: 4807535-2020-07-27.xml*