Int J Adv Manuf Technol (2011) 53:241–245 DOI 10.1007/s00170-010-2815-z

ORIGINAL ARTICLE

Calculation of the virtual pitch thread diameter using the cloud of points from CMM Igor Alexeevich Shchurov

Received: 13 January 2010 / Accepted: 23 June 2010 / Published online: 11 July 2010 # Springer-Verlag London Limited 2010

Abstract This paper presents a new calculation procedure of the virtual pitch thread diameter using the cloud of points from coordinate measuring machines (CMM). The procedure involves scanning of thread profiles using a CMM, calculation of points sets belonging to the left and right thread flanks, next determination of the thread profile which belongs to the adjoining helical surface. Its parameters guarantee the required diameter calculation. Estimation of this diameter comparing it with the standard one measured by CMM “Lapik KIM1000” and the theoretical maximum pitch thread diameter shows that this approach gives a reliable result. Keywords Thread . Virtual pitch thread diameter . Coordinate measuring machine (CMM) . Cloud of points

1 Introduction Coordinate measuring machines (CMM) let to get clouds of points, which discretely describe components surfaces. Often CMM are used to control the measured profile. CMM compare the profile feature with the preprogrammed perfect profile and establishes the deviation between them [1, 2]. Such parameters as size, position, or form are commonly measured and then the conclusions about the accordance of dimensions and tolerances features to the international and national

standards are made. But it is difficult to estimate the required uncertainty correctly [3]. Many researches present methods of surfaces estimation based on the CAD system parametric technique and sets of measurement points [4]. But one of the most complicated surfaces deprived of sufficient attention is thread. Modern CMM used for the thread measurement usually have programs which determine only such parameters as: major (minor) diameter, pitch diameter, pitch, and flank angles [5, 6]. Furthermore, there is another very important parameter of the thread accuracy. It is the virtual pitch thread diameter. As it is known, the virtual pitch thread diameter is the pitch diameter of an imaginary thread of perfect pitch and angle, cleared at the crests and roots but having the full depth of straight flanks, which would just assemble with the actual thread over a specified length of engagement. This diameter includes the cumulative effect of variations in lead (pitch), flank angle, taper, straightness, and roundness [7–9]. This parameter can be controlled by measuring instruments which are used either with thread measuring jaws or thread measuring rollers [10]. However, modern CMM have no programs for calculation of the virtual pitch thread diameter based on the set of measured points [3]. The following sections describe the calculation of the virtual pitch thread diameter using the cloud of points from CMM.

2 Calculation of the virtual pitch thread diameter I. A. Shchurov (*) Department of Machine and Cutting Tools, South Ural State University, Chelyabinsk 454080, Russia e-mail: [email protected] URL: http://www.instr.susu.ru

2.1 Source data The source data for this task is a set of the thread surface point coordinates. R(1) is the set of measured

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points Mi , belonging to the thread surface (see Fig. 1). ð1Þ ð1Þ ð1Þ Each point coordinates are Xi ; Yi ; Zi . Nominal parameters of the thread surface are: d and d1 are, respectively, the major and minor thread diameters (here in equations, we will use these symbols both for external and internal thread). P is the thread pitch; β and γ are, respectively, right and left flank angles; G—the number of thread starts. The thread type is given so: t = 1 corresponds for external thread and t=–1 for internal one. Parameter V=1 defines that the thread is right-hand and V=–1 left-hand one. The maximum errors of the major and minor diameters are marked as: 2Δ1 and 2Δ2. For the first approximation step, it is possible to assume Δ1 ¼ Δ2 ¼ P=10: 2.2 Calculation procedure The calculation of the virtual pitch thread diameter includes the following stages.

The number of thread grooves (n) is necessary to calculate the required parameter for the full thread length. It is determined by the following equations: ð1Þ

ð1Þ

n ¼ ½ðZk  Zs Þ=P þ 1; ð1Þ ð1Þ Zs ¼ min fZi g; ð1Þ ð1Þ Zk ¼ max fZi g;

ð1Þ

ð1Þ

where fZi g is the set of the points coordinates Z. It is important to put the standard thread profile in Y0Z place from the origin of the coordinate system. But usually, the thread is located accidentally. Therefore, all the points must be rotated about Z axis and shifted along it. For that reason, the point on the thread crest needs to be determined. Thus, from ð1Þ all of the points Mi from the set R(1) belonging to the ð1Þ interval Zi Zð1Þ s þ P the point with the largest radius vector is determined: ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r  ð1Þ2 ð1Þ2 rmax ¼ max Xi þ Yi ;

ð2Þ

ð1Þ

where the point coordinates are: Xmax ¼ Xi ;Ymax ¼ ð1Þ ð1Þ Yi ; Zmax ¼ Zi and its polar angle is 8max ¼ arctgðYmax = Xmax Þ. It is known that the required parameter is defined only on the thread flanks. Therefore, it is necessary to use these flanks only. The measured points set R(1) will be transformed with the consequent extracting from it the set R(2) of ð2Þ ð2Þ ð2Þ ð2Þ points Mi ðXi ; Yi ; Zi Þ belonging to the flanks (see (2) Fig. 2). New sets R are obtained solving the following equation for all the points from the set R(1): 0

1 0 0 ð1Þ 1 1 ð2Þ Xi X 0 B ð2Þ C @ B ið1Þ C A 0 þ Mz ð8max Þ@ Yi A @ Yi A ¼ ð2Þ ð1Þ Z max Zi Zi

ð3Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1Þ2 ð1Þ2 simultaneously for ðXi þ Yi Þ>d=2  P=Δ1 and for qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1Þ2 ð1Þ2 ðXi þ Yi Þ>d1 =2 þ P=Δ2 , where Mz (8 ) is the max

Fig. 1 Sets of measured points (examples): a tap thread surface, right three-start thread, right single-start thread, and left single-start thread. The program screenshot

elemental rotational matrix about Z by the 8max angle. After that it is necessary to examine left and right thread flanks separately and to find their adjoining (3) screw R (4) of points  surfaces. The  setsð4ÞR ð4Þ and ð4Þ ð4Þ ð3Þ ð3Þ ð3Þ ð3Þ and Mi ðXi ; Yi ; Zi Þ belonging, Mi Xi ; Yi ; Zi respectively, to the left and right flanks of the thread (see Fig. 3) are determined below.

Int J Adv Manuf Technol (2011) 53:241–245

243

Fig. 2 Points set of the flank surface R(2)

Fig. 4 CMM “Lapik KIM-1000”

All the points are examined in Y0Z section. For every ð2Þ point from the set R (2) its polar angle is 8i ¼ ð2Þ ð2Þ arctgðYi =Xi Þ. Modified z-coordinate of the point is calculated as

Similar to it, points from the set R(2) for which the condition:

»

Zi ¼

ð2Þ V8i GP=2p:

ð4Þ (2)

If z-coordinate of the point from the set R satisfies the following condition then this point belongs to the set R(3): »

ðj  1ÞP < Zi < jP  P=2; where j = -G...n.

Fig. 3 Point sets R(3) and R(4) of the left and right thread flanks

ð5Þ

»

jP  P=2 < Zi < jP;

ð6Þ (4)

is true form the set R . The point from the left thread profile belonging to the adjoining helical surface can now be calculated. We determine the adjoining helical surface with the given axis as the linear helical surface which has nominal parameters (the flank angles—β and γ, the pitch—P) and contacts the real flank so that all the points of the later lie on the one side from the ð3Þ adjoining surface. For each point Mi from the set R(3) its

Fig. 5 Setup for the thread measurement with the CMM

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are some other exceptions: (a) the calculation of the modified z-coordinate, here before solving the Eq. 7 it is 0 0 necessary to add: Zi ¼ Zi þ P; (b) the calculation of the parameter brt i is executed as brti ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 ð4Þ2 ð4Þ2 ðXi þ Yi Þ þ tgðgÞZi :

ð9Þ

Now, the virtual pitch thread diameter can be calculated. The distance h is determined from the top of the fundamental triangle to the pitch thread diameter line. This thread is an ideal parallel thread, which is used for the determination of the virtual pitch diameter (see Fig. 2): h ¼ ðPtgðbÞtgðgÞÞ=ðtgðbÞ þ tgðgÞÞ:

ð10Þ

Fig. 6 Cloud of points in the axial plane (from the CMM)

ð3Þ ð3Þ coordinate are polar angle 8ð3Þ i ¼ arctgðYi =Xi Þ and zdetermined by the equation 0

Zi ¼

ð3Þ Zi



ð3Þ V8i GP=2p

P

ð3Þ Int½ðZi



0

»

»

dvp 2 ¼ ðblf þ brt Þ  2h:

ð11Þ

ð3Þ V8i GP=2pÞ=P;

ð7Þ 0

After that, the virtual pitch diameter dvp 2 is determined by two lines as

3 Experimental investigation and discussion

0

but if Zi < 0 then Zi ¼ Zi þ P: ð3Þ For the every point Mi , the parameter blf i blf i (parameter from the line equation—Y=kX+b) is determined as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 ð3Þ2 ð3Þ2 blfi ¼ ðXi þ Yi Þ  tgðbÞZi : ð8Þ »

Among all the points from the set R(3) blf ¼ maxfblf i g if » t=–1 or blf ¼ minfblf i g if t=1 are calculated. The points of the right thread profile belonging to the adjoining helical surface also can be calculated. The calculation can be made by analogy with previous item. Thus, instead of the set R(3) the set R(4) is used, and instead of the upper indexes (3), it is necessary to use (4). Also, there Fig. 7 Transformed points from four axial sections and the adjoining ideal thread profile

3.1 Experimental investigation This calculation method was empirically checked. The CMM “Lapik KIM-1000” was used for the measurement of the screw with nonstandard thread M32×2.5 and the tap with standard thread Tr 32×6 (see Figs. 4 and 5). At first, we have measured three parts of the screw. Measurements were made by the standard method specified for the CMM in four axial planes, located through 60° from the vertical plane (see Fig. 5). The cloud of points describes the length of six full ridges. The step along thread axis (see Fig. 6) was 0.07 mm. We have got about 1,000 points for each part.

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The program of the CMM “Lapik KIM-1000” has given us such average results for three screw parts: the measured pitch thread diameter is d2 =30.361 mm, the major diameter is d=32.074 mm, the pitch is P=2.498 mm, the angles are β=31°44′25″ and γ=30°50′16″. These parameters were calculated using the average lines of the points set for each thread flank.

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diameter is d2 =29.202 mm. This diameter defined at the measurement laboratory is d2 =29.193 mm. For this reason, the proposed calculation method of the virtual pitch thread diameter using the cloud of points from CMM gives us a reliable result. Author supplies this article with computer program so that everybody can check his own data using this method.

3.2 Calculation of the virtual pitch thread diameter 4 Conclusions According to the described procedure, the computer program has been created. As it is shown in Fig. 7, the points of all profiles were transformed into the one plane. Moreover, the lines of the adjoining ideal thread profile were plotted through these points. The results obtained with the new vpð1Þ program for the profiles mentioned above are d2 ¼ vpð2Þ vpð3Þ 30:708 mm, d2 ¼ 30:686 mm, d2 ¼ 30:661 mm, vpð4Þ d2 ¼ 30:608 mm. Simultaneous calculation of all these profiles has given the following result: d2vp ¼ 31:253 mm. Comparing this value with the measured value with the pictures (see Fig. 7) of ideal profile, we can conclude that there is a qualitative agreement between them. 3.3 Estimation of the virtual pitch thread diameter and discussion Pitch diameter of metric screw thread is determined as d2 ¼ d  0:6495 P (ISO 724:1993). So in the described example, the standard pitch thread diameter is d2 ¼ 32  0:6495 2:5 ¼ 30:37625 mm. Therefore, the virtual pitch thread diameter is larger than the measured one for more than 0.892 mm and is larger than the standard pitch thread diameter for more than 0.877 mm. There is the standard equation for the calculation of the maximum pitch thread diameter according to the thread tolerance zone: dvp 2 ¼ d2 þ 1:732ΔP þ 0:18PðΔb þ ΔgÞ; where ΔP, Δβ, and Δγ are, respectively, the tolerance of the pitch and left and right flank angles. For the described example: d2vp ¼ 30:376 þ 1:732  0:046 þ 0:18  2:5ð1:73 þ 0:83Þ ¼ 31:608: As it is shown above, the value of the virtual pitch thread diameter lies between the standard pitch thread diameter and the maximum pitch thread diameter. The measurement of the tap with standard thread Tr 32× 6 gave us such average results: the virtual pitch thread

This issue presents a new calculation procedure of the virtual pitch thread diameter using the cloud of points from CMM. The procedure involves scanning of thread profiles using CMM, calculation of the sets of points belonging to the left and right thread flanks and after all determination of the thread profile belonging to the adjoining helical surface. Its parameters let us to compute the required diameter. The estimation of this diameter comparing it with the standard one measured by CMM and with the theoretical maximum pitch thread diameter confirms that this approach has given us a reliable result.

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