Int J Adv Manuf Technol (2017) 91:25–37 DOI 10.1007/s00170-016-9727-5
ORIGINAL ARTICLE
A methodology for helical mill-grinding of tiny internal threads made of hard brittle materials Hang Gao 1 & Shouxiang Lu 1 & Anqiang Yang 1 & Yongjie Bao 1
Received: 7 June 2016 / Accepted: 7 November 2016 / Published online: 16 November 2016 # Springer-Verlag London 2016
Abstract This study proposes a methodology for helical millgrinding of tiny internal threads made of hard brittle materials such as SiCp/Al composites. The methodology uses the helical mill-grinding method incorporating with a diamond formgrinding wheel. A mathematical model is established to predict thread form errors and provide a rational range of wheel parameters, such as variation of tool profile angle Δα and ratio of the wheel diameter to the thread major diameter η. Based on the methodology, a grinding wheel is developed for processing the M2 internal threads in a validation experiment. The study demonstrates that an M2 internal thread made of the SiCp/Al composite of 45% SiC volume fraction is successfully machined in 5 min with pitch error <0.08% and angle error <0.3%. The thread profile on the pitch diameter is within the axial equivalent tolerance zone (0–0.016 mm), which indicates that the thread precision reaches the H4 level. Keywords Internal thread machining . Helical mill-grinding . Principle error . SiCp/Al composite
* Hang Gao
[email protected] Shouxiang Lu
[email protected] Anqiang Yang
[email protected] Yongjie Bao
[email protected] 1
Key Laboratory for Precision and Non-traditional Machining Technology of Ministry of Education, Dalian University of Technology, No.2 Linggong Road, Ganjingzi District, Dalian 116024, China
1 Introduction Thread processing methods such as tapping, turning, and rolling are usually used on materials, such as metals and plastics. However, it is difficult, sometimes impossible, to make M2 internal threads with machining precision of H5 on metal matrix composites, silicon carbide ceramics, and other hard and brittle materials. Because of the hard-to-machine characteristics and the structural features of Φ1–2 mm tiny holes, the processing efficiency with cemented carbide or CVD diamond coating taps is extremely low(1–3 h per hole). Moreover, tap stall or even breakage in a hole frequently occurs. With the development of the aerospace, national defense, automobile, and microelectronics industries, demand for making high precision tiny internal threads on hard and brittle materials is increasing. Exploring and developing precision and high efficiency processing methods of tiny internal threads on hard and brittle materials has become one of the challenges in engineering. To a great extent, the choice of a thread processing method depends on workpiece material, the part structure and the size of the thread. Tapping is a common method of internal thread processing, and there have been in-depth studies on mechanical model, processing technology, and surface integrity of the tapping processes [1, 2]. Vibration-assisted tapping has been applied to hard-to-machine metals such as titanium alloys [3]. However, in the process of tapping tiny threads on hard and brittle materials, it is difficult to completely avoid tap breakage. Thread rolling is a cold forming and chipless thread processing method with compressive residual stress left in the finished surface, which is beneficial to improving the fatigue strength of a screw thread and is playing a dominant role in thread processing of large quantities [4, 5]. Thread rolling technology is appropriate for both internal threads and external threads. The rolling process has been optimized and the
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rolling tool has been designed [6, 7] based on the optimization. Nevertheless, the rolling method is limited primarily to metals with good plasticity and is not used in processing materials with poor flow property. Turning is another traditional thread processing method. In recent years, the research of multi-point thread turning has improved the productivity of thread processing [8]. Limited by the relative motion of the cutting tool and workpiece, the turning method is mainly used in the axis-symmetrical part thread processing [9, 10]. Thread whirling is an efficient and precise machining process for manufacturing of screws and worms, which is applied more and more extensively [11, 12]. With the development of the numerical control technology, thread milling method has been widely used. The method is not subject to the restrictions of workpiece structure and has high technological flexibility [13–15]. It allows elimination of tool change when holes with different diameters are to be threaded. It also provides higher cutting speed, greater thread accuracy and better chip evacuation conditions than tapping [16]. Araujo et al. [16, 17] studied thread milling process in depth and established a mechanical model that is considered more realistic. Wan et al. [18] modeled the mechanics and dynamics of thread milling operations and predicted the stability of the operation as a function of spindle speed, axial depth of cut, cutter path, and tool geometry. Lee et al. [19] built a cutting force prediction model composed of surface and edge force components, which can be used in optimization of process parameters so as to raise productivity. Skoczylas et al. [20] used tapered end mill to machine worm thread and the possibility of concave thread making by tool of straight line was substantiated. It was found that a high compatibility of the shape of the machined profile with theoretical one took place. As everyone knows, it is more likely to get a high precision machining surface through grinding method relative to cutting. Form grinding is widely used to produce precision screws, worms, and gears [21]. Chiang et al. [21, 22] proposed a simplified two-dimensional numerical simulation method for form grinding of the thread on cylindrical workpieces and a geometric approach to determining the grinding wheel profile. Thread processing belongs to the category of form machining. The actual profile of a thread is determined by tool shape and trajectory. If the tool shape is designed according to the shape of the theoretical thread, the machined profile and the theoretical profile may appear inconsistent; such inconsistency means principle error. To solve this problem, the principle error should be calculated and then eliminated or suppressed through tilting the tool axis or modifying the tool shape. Wu et al. [23] established a mathematical model for machining of screw rotors with worm-shaped tools and computed the normal errors of the generated cutting lines. Chiang et al. [22] avoided undercutting and interference by means of tilting the grinding wheel axis in the process of thread form grinding and
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calculated the minimum tilt angle that avoids undercutting. Mohan et al. [12] made an attempt to simulate the whirling process and proposed a rational design of tool profile, and the validation of tool profile was carried out by simulation and worms machining experiment. Lee et al. [24] provided a novel methodology to design the tooth profile of the thread mill by comparing the “to-be” thread profile and the “as-is” thread profile which was analyzed by means of NC cutting simulation. The thread mill was modified until the thread profile and the “to-be” thread profile became congruent with each other. According to the literatures of thread processing, form grinding is a feasible method for precision machining internal threads of hard and brittle materials. The study on how to eliminate the interference phenomenon and reduce the thread processing principle error is carried out mainly by means of numerical simulation and modification of tool profile so as to make the actual profile close to the theoretical profile. This paper presents mathematical modeling of principle error of internal threads based on helical mill-grinding. First, the characteristics of the helical mill-grinding process are described. Then, the details of principle error modeling are presented and an analytical expression is provided. Error reduction strategies are also proposed based on the principle error model. Finally, a form-grinding wheel is developed for machining tiny internal threads on hard and brittle materials. A machining experiment of M2 internal threads on SiCp/Al composites is carried out to validate the principle error model.
2 Helical mill-grinding method of internal threads A schematic diagram of the helical mill-grinding of internal threads is shown in Fig. 1. The wheel axis and the hole axis are parallel and the eccentric distance is e. The wheel rotates around its own axis at a high speed to complete the main machining. The feed motion of the wheel is along a spiral line: on one hand, the wheel axis rotates around the hole axis for uniform circular motion. On the other hand, the wheel axis goes along the axial direction for uniform linear motion. The
Fig. 1 Schematic diagram of helical mill-grinding of internal threads
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process parameters of the helical mill-grinding are as follows: the rotational speed of the wheel is n (r/min), the circular feed speed of the wheel is f (mm/min), and the axial feed speed of the wheel is g (mm/min). The relationship of thread pitch P, circular feed speed f, and axial feed speed g is given as P = 2πe g/f. When a form-grinding wheel whose thread profile is consistent with metric thread profile is employed to process internal threads, there can be interference between the actual enveloping surface generated from the wheel motion and the hypothetical metric thread profile. Such interference results from the principle error. Due to the existence of the principle error, the thread dimension may be out of tolerance, which reduces both the yield strength and tensile strength of threaded part [25]. To explore the analytical expressions for principle error, to analyze the principle error influence factors, and to reduce the principle error by rational design of wheel shape are the key to the feasible helical mill-grinding method. On the condition that the relative motion between the wheel and workpiece is invariable, changing tool geometric angle may be a feasible way to reduce the principle error. In this study, the modeling of principle error is based on the change of the tool geometric angle. The metric wheel angle is reduced by Δα, as is shown in Fig. 2. The metric thread profile is shown in Fig. 3. P stands for thread pitch; H is fundamental triangle height; pffiffiffi H ¼ 3=2P; D is major diameter; D2 is pitch diameter, D2 = D−2 × 3/8H = D−0.6495P; D1 is minor diameter, D1 = D−2 × 5/8H = D−1.0825P; h0 = 5/8H is the height of thread; α = 60° is thread angle.
3 Modeling of principle error of helical mill-grinding A geometric sketch is presented to help in establishing the principle error mathematical model of helical mill-grinding internal threads, as shown in Fig. 4. A rectangular coordinate system is set up to describe the helical mill-grinding and is shown in Fig. 4a. Diameter of the form-grinding wheel is Ds, eccentricity between the wheel axis and the basic hole axis is e, diameter of the basic hole cylinder is D 1 , and diameter of the major
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Fig. 3 Metric thread profile
diameter cylinder is D. Assuming that there is an arbitrary cylinder between the basic hole cylinder and the major diameter cylinder and the radius of the arbitrary cylinder is R, a space curve Γ is formed as a result of the intersection of the arbitrary cylinder and the contour surface of the wheel. Due to the structural symmetry of the wheel, the shape of the space curve Γ remains unchangeable in the process of wheel rotation. When the wheel is in a feed motion along a spiral line, the space curve Γ is in a helical motion on the arbitrary cylinder simultaneously; each point of Γ will form a spiral line. Among the numerous spiral lines, the outermost spiral line is defined as the profile spiral line. Changing the value of R continuously between D1/2 and D/2; a series of profile spiral lines will form thread contour surface. If we define the intersection of the metric thread contour surface and the arbitrary cylinder as the metric spiral line, the principle error should be the axial difference between the profile spiral line and the metric spiral line on the arbitrary cylinder. The space curve Γ consists of four sections: two space curves Γ1 and Γ2 formed by the intersection of the arbitrary cylinder and the conical surface of the wheel which is symmetrical about the XOY plane, and two straight line segments formed by the intersection of the arbitrary cylinder and wheel cylinder, respectively. Obviously, the profile spiral line can only be formed by Γ1 and Γ2, so it is enough to investigate space curve Γ1. The wheel profile is projected on the YOZ plane as is shown in Fig. 4b. The pitch is set as P, and the coordinate of Q is (D/2, P/16). If the metric thread angle is set as α and the variation of the tool profile angle is set as Δα, an equation of the straight line PQ is given as: α−Δα D P z ¼ −tan ⋅ y− þ ð1Þ 2 2 16
Fig. 2 Schematic diagram of grinding wheel profile
The coordinate of conical tip P is as follows: D−DS DS a−Δa P ; ⋅tan þ 2 2 2 16
ð2Þ
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Fig. 4 Geometric sketch of principle error mathematical model. a Rectangular coordinate system for helical mill-grinding method. b The projection of wheel profile on the YOZ plane. c The projection of wheel profile on the XOY plane
If we rotate PQ around wheel axis POS 360°, the upper conical surface equation of wheel is obtained according to the equation of PQ and the P coordinate: s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi DS a−Δa P D−DS 2 a−Δa ⋅tan y− z¼ þ x2 ⋅tan þ − 2 2 2 16 2
ð3Þ
8 > > > > > > > <
The cylindrical coordinate parameter equation of the arbitrary cylinder is as follows: x ¼ RcosðθÞ ð4Þ y ¼ RsinðθÞ The parameter equation of space curve Γ1 is obtained from Eqs. (3) and (4) as follows:
x ¼ RcosðθÞ y ¼ RsinðθÞ ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi DS a−Δa P D−DS 2 a−Δa 2 ⋅tan RsinðθÞ− þ − þ ðRcosðθÞÞ ⋅tan z¼ > 2 2 2 16 2 > > > > D D π π > 1 > : ≤ R ≤ ; −θS ≤ θ ≤ þ θS 2 2 2 2
The metric thread profile equation on YOZ plane is as follows: a D P z ¼ −tan ð6Þ þ ⋅ y− 2 2 16 If the radius of arbitrary cylinder is R, the intersection point of metric spiral line Γ0 and YOZ plane is (R, −tan(α/2)(R−D/2) + P/ 16), and the parameter equation of Γ0 which crosses through the intersection point is as follows: 8 x0 ¼ RcosðθÞ > > > > y0 ¼ RsinðθÞ > > a < P π D P z0 ¼ þ ⋅ θ− −tan ⋅ R− ð7Þ 2π 2 2 2 16 > > > > D1 D > > ≤ R ≤ ; −∞≤ θ ≤ þ ∞ : 2 2
ð5Þ
Known from Eq. (5), Γ1 is a smooth and continuous curve. At arbitrary point where R is decided, the direction vector of the tangent of Γ1 is s ¼ xðθÞ̇ ; yðθÞ;̇ zðθÞ̇ . Known from Eq. (7), the direc tion vector of the tangent of Γ0 is s0 ¼ x0 ðθÞ̇ ; y0 ðθÞ;̇ z0 ðθÞ̇ . If the profile spiral line is formed by point of tangency on Γ1, then s ¼ s0 , which means zðθÞ̇ ¼ z0 ðθÞ̇ . If we set Ds/D = η, D−Ds = (1−η)D = a, tan(α/2−Δα/2) = b, (P/ πab)2 = A, sin(θ) = B, we get the following equation: s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi! 1 Aa Aa2 B¼ −4 ⋅ðA−1Þ ð8Þ þ 2 R R2 So, the coordinate of point of tangent M is (R, θM, zM), in which θM = arcsin(B). In the XOY plane, as is shown in
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Fig. 4c, set the projection point of the intersection line of the arbitrary cylinder and wheel cylinder as C, the angle between segment OC and Y axis as θs. Obviously, the value of θM is between (π/2−θs) and (π/2 + θs). Substituting θM into Eq. (5), we get the following equation: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! η a2 P ZM ¼ ⋅b þ ⋅D− R2 −RaB þ ð9Þ 4 2 16 Then the expression of θs will be solved. As is shown in Fig. 4c, there is geometrical relationship in triangle OCD and triangle OsCD which is OsD + OOs = OD. According to the Pythagorean theorem, the abovementioned equation is equivalent to the following equation: ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s ffi DS 2 2 D−DS pffiffiffiffiffiffiffiffiffiffiffiffiffi ð10Þ ¼ R2 −H 2 −H þ 2 2 Obviously, −1 H θS ¼ sin R
ð11Þ
We get the following equation according to Eqs. (10) and (11): ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 0v u 2 u 1 ð1−2ηÞ⋅D2 A ð12Þ θS ¼ sin−1 @t1− R þ 4R ð1−ηÞ2 ⋅D2 Obviously, both θ s and θ M are functions of R. Set D = 2 mm, D1 = 1.567 mm, P = 0.4 mm, α = 60°, Δα = 4°, and η = 0.75, on the condition that the value of R is between D1/2 and D/2; the functional image of θM、(π/2−θs) and (π/2 + θs) are plotted as is shown in Fig. 5 with the help of Matlab. It is easy to know that there exists a critical value Rs. On the condition that the value of R is between D1/2 and Rs, the value of θM is between (π/2−θs) and (π/2 + θs) and the
profile spiral line is formed by point of tangency on Γ1. On the condition that the value of R is between Rs and D/2, the profile spiral line is formed by the endpoint of Γ1. Because equation θM = π/2−θs is transcendental equation and there is no analytical solution, we can only get the approximate solution of Rs through numerical method. The parameter equation of profile spiral line ΓM which crosses through point of tangent M on Γ1 is as follows: 8 x ¼ R⋅cosðθÞ > < y ¼ R⋅sinðθÞ ð13Þ > : z ¼ P ðθ−θM Þ þ Z M 2π The axial difference value of ΓM and Γ0 on the YOZ plane is obtained according to Eqs. (7) and (13): δ1 ¼ z
π π P P D a π ⋅tan −z0 ¼ − ⋅θM þ Z M þ R− − 2 2 4 2π 2 2 16
ð14Þ
Substituting Eqs. (8) and (9) into Eq. (14), we get the following equation: P P α−Δα D α þ R− ⋅tan þ δ1 ¼ − sin−1 ðBÞ⋅tan 2 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 4 s2π 1 2 2 2 @ ηD − R2 −ð1−ηÞDRB þ D ⋅ð1−ηÞ A D1 ≤ R < RS 4 2 2 ð15Þ It is easy to know from Fig. 4b, c that the cylindrical coordinate of the endpoint of Γ1 is (R, π/2-θs, P/16). Therefore, the equation of the profile spiral line ΓE is as follows: 8 xE ¼ R⋅cosðθÞ > > > < yE ¼ R⋅sinðθÞ π ð16Þ θ− −θS > > P > 2 : zE ¼ ⋅P þ 2π 16 The axial difference value of ΓE and Γ0 on the YOZ plane is obtained according to Eqs. (7) and (16): π π P D α δ 2 ¼ zE ⋅tan ⋅θS þ R− −z0 ¼ ð17Þ 2 2 2π 2 2 Substituting Eq. (12) into Eq. (17), we get following equation: ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 0v u 2 u P 1 ð1−2ηÞ⋅D2 A ⋅sin−1 @t1− R þ δ2 ¼ 4R 2π ð1−ηÞ2 ⋅D2 D α D ⋅tan þ R− RS ≤ R ≤ 2 2 2
Fig. 5 Relative position of θM, (π/2−θs), and (π/2 + θs)
ð18Þ
It is known from Eqs. (15) and (18) that on the condition that the major diameter D, the thread pitch P, and the thread
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angle α are decided, there is a certain functional relationship between the principle error and the following three parameters: the arbitrary cylinder radium R, the ratio of the wheel diameter to the thread major diameter η, and the variation of tool profile angle Δα. Summing up the above, an analytical expression of the principle error is as follows: 8 D1 > > ≤ R < R ð R; η; Δα Þ δ < 1 S 2 ð19Þ δðR; η; ΔαÞ ¼ D > > : δ2 ðR; ηÞ RS ≤ R ≤ 2 Obviously the value of η is between 0 and 1. But the abovementioned scope is too broad in practice. On one hand, the wheel diameter must be less than the base hole diameter so as to retract the wheel smoothly in practice, that means Ds = ηD < D1 = D−1.0825P, so η < 1–1.0825P/D. On the other hand, it must be sure that the integrated thread form is obtained. Hence, if we do not consider the wheel rod diameter, the minimum wheel diameter should be greater than the double height of thread h0, which means Ds = ηD > 2 h0, so η > 2 h0/D = 1.0825P/D. Known from Eqs. (8), (9), and (12), the value of η must meet the following conditions: 2 Aa −4 ðA−1Þ ≥ 0 ð20Þ R2 a2 ≥0 4 2 1 ð1−2ηÞ⋅D2 1− ⋅ Rþ ≥0 4R ð1−ηÞ2 ⋅D2
R2 −RaB þ
ð21Þ ð22Þ
Substituting 1.0825P/D < η < 1–1.0825P/D into Eqs. (20)–(22), respectively, we can get that on the condition Δα < 27.228°, Eqs. (20) and (21) work and Eq. (22) always stands up. Generally speaking, the variation of tool profile angle Δα is controlled within 10°, so Eqs. (20), (21), and (22) always stand up. To sum up, the scope of η is as follows: 1:0825
P P < η < 1−1:0825 D D
radial direction; however, the principle error investigated in this paper is an axial error. Consequently, the principle error is analyzed according to the axial equivalent tolerance, which is illustrated in Fig. 6. The thread pitch diameter error is an important standard to judge whether the thread is qualified or not. Therefore, if the principle error is controlled within 1/3 of the thread pitch diameter axial equivalent tolerance zone, it will be considered that the principle error being in a reasonable range. The geometric relationship of the thread pitch diameter axial equivalent tolerance and the metric thread pitch tolerance is as follows: T D0 ¼ T D2 tanðα=2Þ. 2
Taking the M2 internal thread for example, the wheel parameters, such as η、ξ and Δα, are optimized through discussing the effect of R, η, and Δα on the principle error shown below. 4.1 Effect of R on principle error (a) The effect of R on principle error δ is directly reflected on the thread profile error. When Δα = 0°, the R-δ curves with different η are shown in Fig. 7a. As can be seen from the diagram, there are dividing points Rs1–Rs6 on the curves. With regard to a single curve, on the condition that R < Rs, the principle error curve is determined by Eq. (15) and on the condition that R > Rs, the principle error curve is determined by Eq. (18). With the increase of η, the value of Rs decreases, the principle error curve gets higher position, and the interval of the curves is larger. (b) When η = 0.5, the R-δ curves with different Δα are shown in Fig. 7b. The curves are divided into two parts, which are the misalignment part and overlap part, respectively. In the misalignment part of the curves, on the condition that Δα increases from −8°to 8°, the tilt direction of curves change from declivitous to acclivitous. Known from the local amplification figure, the value of
ð23Þ
4 Analysis of effect of R, η, and Δα on principle error When processing metric internal thread of a certain size with the helical mill-grinding method, if we want to decide the geometrical dimensions of the wheel, the ratio of wheel diameter to thread major diameter η, the ratio of wheel rod diameter to wheel diameter ξ, and the variation of tool profile angle Δα must be determined. The metric thread tolerance zone is in the
Fig. 6 Thread equivalent tolerance zone
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Fig. 7 Effect of R on principle error. a The R-δ curves when Δα = 0°. b The R-δ curves when η = 0.5
dividing point Rs decreases with the increase of Δα. On the condition of R > Rsmax, the curves are overlapped.
On the basis of the above analysis, the principle error generated when machining M2 internal thread using helical millgrinding method is not evenly distributed along the thread profile. According to the influence length of the principle error, the error determined by Eq. (15) accounts for most of the length while the error determined by Eq. (18) accounts for only a little length.
tolerance zone are shown in the diagram, from which it is easy to find the reasonable value of η and Δα. This study aims to solve the machining problem of tiny internal thread, thus the stiffness of the wheel rod must be considered. If the wheel rod diameter is set as d0, d0/Ds = ξ, then the condition that there is no interference between the wheel rod and the base hole wall is d0 + 2 h0 < Ds in the process of machining, which means 0 < ξ < 1–1.0825P/ηD. With regard to a small wheel used to make M2 internal threads, the wheel rod stiffness is sensitive to its diameter size.
4.2 Effect of η and Δα on principle error To explore the effect of η and Δα on principle error, the value of R is set as D2/2 and the principle error at thread pitch diameter is investigated. Figure 8 shows the η-δ relation curve on different value of Δα. It is known from Eq. (23) that for M2 internal thread, the range of η is 0.2165–0.7835. Thus, the minimum and the maximum values of η in the diagram are 0.22 and 0.78, respectively. It can be seen that with the increase of η, the principle error at thread pitch diameter increases with an acceleration. If the value of η is fixed, the principle error at thread pitch diameter decreases with the increase of Δα. The H4 level unilateral axial equivalent tolerance zone of thread pitch diameter and 1/3 range of such
Fig. 8 Effect of η and Δα on principle error (M2)
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Fig. 9 Effect of η and Δα on principle error under different thread major diameter. a Effect of η on principle error. b Effect of Δα on principle error
The wheel rod will deform or even break in the machining process if ξ is too small. Consequently, on the premise of guaranteeing the principle error to be within a certain range, both the values of ξ and η should be as close to the upper limit value as possible. Considering the demands of chip evacuation, η should be slightly smaller than the upper limit. Based on the analysis of the above, for the M2 internal thread processing, the value of η and Δα are determined to be 0.75 and 4°, respectively. Under the above conditions, it is found that ξmax = 0.7113; by considering the manufacturing error of the wheel and the base hole, ξ = 0.6 is chosen. It is worth mentioning that when processing larger internal thread, the strategy is different to choosing the value of η and Δα. Figure 9a shows the effect of η on principle error at thread pitch diameter under different thread major diameter. It is easy to know that the larger the thread major diameter is, the upper the position of η-δ curve is. And with the increase of η, the principle error at thread pitch diameter increases with an acceleration. Therefore, on the premise that the wheel rod stiffness is enough, choosing smaller η is helpful to reduce the principle error when machining larger internal thread. Figure 9b shows the effect of Δα on principle error at thread pitch diameter under different thread major diameter. As is
illustrated in the diagram, with the increase of the thread major diameter, the Δα-δ curve tilts more seriously. And the value of δ is more concentrated and close to zero when Δα is close to 0°. In consequence, when machining internal thread with larger major diameter, it is helpful to reduce the principle error with smaller |Δα|. Taking M10 internal thread for example, on the condition that η = 0.5 and Δα = 0°, the principle error curve is shown in Fig. 10. It is clear that the curve is in the 1/3 range of unilateral axial equivalent tolerance zone of thread pitch diameter. Therefore, on the condition that a small value of η can be chosen, there is no need to change the wheel angle and the principle error is in the allowable range.
5 Validation experiment of helical mill-grinding method 5.1 Preparation of grinding wheel High SiC volume fraction SiCp/Al composite is hard and brittle material with poor machinability [26, 27]. Moh’s hardness of the reinforcement phase SiC is 9.5, which is higher than that of most abrasive materials. As Moh’s hardness of the synthetic diamond is 10, harder than SiC, diamond was employed to prepare a grinding wheel through sintering the diamond powders with bronze bond on a cemented carbide rod. The sintered wheel was modified through the microEDM process and the required wheel shape was obtained, and is shown in Fig. 11.
5.2 Experimental material and equipment
Fig. 10 The principle error curve for M10 internal threads
The experimental material was SiCp/Al composite with a 45% SiC volume fraction. The thickness of the workpiece was 3 mm. The workpiece and the material metallograph are shown in Fig. 12.
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5.4 Experimental results and analysis 5.4.1 The stiffness analysis of the wheel rod
Fig. 11 Actual profile of sintered diamond wheel
The experiment was carried out on a three axis high-speed vertical machining center (MikronHSM500). The spindle maximum power was 16 KW, maximum speed was 54,000 rpm, and repetitive positioning accuracy was 5 μm. The wheel and the machined thread were observed through a VHX-600E microscope which was produced by Keyence Company. The grinding force was measured by Kistler 9257B dynamometer; the Kistler 5080 amplifier and 5697A data acquisition were employed to read and save data from the force sensor.
5.3 Experimental method A developed wheel was employed in this experiment. The machining parameters were n = 20,000 r/min and f = 3 mm/ min. The workpiece was cut in half along the axial direction by the wire-cut electric discharge machining (WEDM) after the thread was machined for the sake of observation. Thread pitch and thread angle were measured with the following method: taking advantage of the segment and angle measurement functions of the microscope, three thread teeth in the middle of the workpiece were measured, and the average value was taken as the result.
Fig. 12 Workpiece and the material metallograph
With regard to helical mill-grinding of tiny internal thread, the stiffness of the wheel rod should be considered. Excessive wheel rod deformation under grinding force will lead to outof-tolerance of thread profile. The force analysis of the wheel rod is shown in Fig. 13a. The load position is simplified to a point on the edge of the wheel disk. At some point in machining process, the grinding force consists of three force components, which are normal force Fn, tangential force Ft, and axial force Fa, respectively. The wheel diameter is far less than the length of the rod, so the effect of Fa on the bending deformation of the wheel rod is ignored. Only forces in X-Y plane are taken into consideration, as depicted in Fig. 13b. The normal force is decomposed into Fnx and Fny, and the tangential force is decomposed into Ftx and Fty. The following equations are obtained: F x ¼ F nx þ F tx ¼ F n cosγ þ F t sinγ ð24Þ F y ¼ F ny − F ty ¼ F n sinγ− F t cosγ From Eq. (24), the following equations are obtained:
F n ¼ F x cosγ þ F y sinγ F t ¼ F x sinγ− F y cosγ
ð25Þ
The normal force Fn is the most important factor which gives rise to the rod bending deformation, as is shown in Fig. 13c. The wheel rod is similar to a cantilever beam and will bend under the action of Fn. The maximum deflection is y and the rotation angle of wheel is β. From Eq. (25), it is easy to qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi know that the maximum value of Fn is F 2x þ F 2y . Small grinding force is obtained under high rotate speed (20,000 r/ min) and low feed speed (3 mm/min), as shown in Fig. 14. Known from the figure, the value of Fx and Fy vary with time like sinusoid approximately, and the maximum value of Fn is 1.46 N. The wheel rod deformation is simulated through finite element analysis. The elasticity modulus of the cemented carbide rod is 630GPa and the Poisson’s ratio is 0.22 [28, 29]. As is shown in Fig. 13d, the maximum deflection y is 2.89 μm, which is far less than the rod length (l = 10 mm). According to the national standard, the pitch diameter allowable deviation of H4 level is 28 μm, and the axial equivalent allowable deviation is 16 μm. Taking error copying effect into consideration, the position error of wheel disk is in the allowable range. Consequently, on the condition of appropriate rod length and grinding parameters, the effect of rod deformation on machining accuracy is within the acceptable range.
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Fig. 13 The stiffness analysis of the wheel rod. a The force analysis of the wheel rod. b The force analysis in the X-Y plane. c The schematic diagram of the wheel rod deformation. d The finite element analysis result of the deformation
Fig. 14 The grinding force curve (n = 20,000 r/min, f = 3 mm/min)
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Fig. 15 The processing quality of M2 thread. a The tap blockage. b The incomplete thread by tapping. c The complete thread by helical millgrinding. d The surface quality of thread by tapping. e The surface
quality of thread by helical mill-grinding (n = 8000 r/min, f = 10 mm/ min). f The surface quality of thread by helical mill-grinding (n = 20,000 r/min, f = 3 mm/min)
5.4.2 Surface quality and machining accuracy of M2 internal thread
parameters, the surface quality of helical mill-grinding method is better than that of tapping method. An M2 internal thread is completed with processing time 4.7 min. Figure 15c shows the cross section of a thread and demonstrates the feasibility of the helical millgrinding method on M2 internal threads of hard and brittle materials. The measurement results of the thread pitch and thread angle of Fig. 15c were as follows: the average thread pitch was 400.31 μm and the average thread angle was 60.18°. The thread pitch and thread angle of the metric M2 internal thread were 400 μm and 60°, respectively. Consequently, the thread pitch error was 0.08% and the thread angle error was 0.3%. A single internal thread profile was measured and is shown in Fig. 16. It is clear that the thread profile was in the thread pitch diameter axial equivalent tolerance zone (0–0.016 mm), which indicates that the thread pitch diameter precision reaches the H4 level.
Compared with tapping, there are obvious advantages of helical mill-grinding method when machining tiny internal thread on difficult-to-machine material such as high volume fraction SiCp/Al composites. As is shown in Fig. 15a, b, the tap is easily broken and blocked in the thread hole and can only machine an incomplete thread while a completed thread is obtained through helical mill-grinding method (Fig. 15c). Comparing the thread surface quality under different machining methods, burrs are found on the surface of tapping (Fig. 15d) and tiny grooves are found on the surface of helical mill-grinding (Fig. 15e, f). Comparing the thread surface quality under different grinding parameters, it is found that the surface is much smoother under higher rotate speed and lower feed speed. Therefore, under reasonable grinding
5.4.3 The effect of grinding wheel wear on the thread profile
Fig. 16 Measurement of single thread profile of M2 internal thread
The grinding wheel wear condition is monitored during the processing as shown in Fig. 17. With the increase of machined thread number, grinding wheel wear more and more seriously and the main abrasion position is the grinding wheel tip. From Fig. 17a, b, we can see that there is little change of the thread profile when tiny tip abrasion happens. While there is growing influence on the thread profile with the increase of the tip abrasion, as is shown in Fig. 17c, d. After machining 10 threads, the wheel wear and the thread profile error are all serious.
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Fig. 17 The influence of grinding wheel wear condition on thread profile
6 Conclusions This study presents a methodology for helical mill-grinding of tiny internal threads made of hard brittle materials. In this study, a mathematical model of the principle error is established and an analytical expression of the principle error is obtained. The optimized wheel parameters are determined based on the analysis of the principle error model. The helical mill-grinding method is experimentally validated. The main conclusions are as follows: 1. On the conditions that the major diameter D, thread pitch P, and thread angle α are fixed, there is a certain functional relationship between the principle error and the following three parameters: the arbitrary cylinder radius R, the ratio of the wheel diameter to the thread major diameter η, and the variation of the tool profile angle Δα. 2. As to the machining of the metric internal thread, there are theoretical value ranges of η and ξ, which are 1.0825P/ D < η < 1–1.0825P/D and 0 < ξ < 1–1.0825P/ηD, respectively. In order to ensure the wheel rod stiffness, the value of ξ should be close to the maximum value as much as possible. The value of η should be close to the upper limit when the thread major diameter is small. On the contrary, when the thread diameter is large, there is more advantage to reducing the principle error through choosing a small value of η. 3. Aiming at the machining of the M2 internal thread on SiCp/Al composites with a SiC volume fraction of 45%, the optimized wheel parameters are as follows: wheel diameter Ds = 1.5 mm, wheel rod diameter d0 = 0.9 mm, and wheel angle α = 56°. The stiffness of the wheel rod is proved enough through finite element analysis. On reasonable process parameters, the M2 internal thread is completed with a processing time of 4.7 min.
The testing results indicate that the thread pitch error is 0.08% and the thread angle error is 0.3%. The thread profile is in the thread pitch diameter axial equivalent tolerance zone (0–0.016 mm), which indicates that the thread pitch diameter precision reaches the H4 level. The grinding wheel wear condition is monitored during the processing and the main abrasion position is the grinding wheel tip. Acknowledgements The authors would like to thank Professor Bi Zhang for proofreading the manuscript and are grateful to the financial supports of the Science Fund for Creative Research Groups (Grant No. 51321004), the National Natural Science Foundation of China (NSFC) (Projects No: 51475073), and the Fundamental Research Funds for the Central Universities (DUT16QY01).
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