Roundness modeling in BTA deep hole drilling
Do Hun China, Moon Chul Yoonb, Sung Bo Simb
aGraduate School of Mechanical Engineering, Pukyong National University, Busan, Republic of Korea
bSchool of Mechanical Engineering, Pukyong National University, Busan, Republic of Korea
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Source of information: http://www.elsevier.com/wps/find/journaldescription.cws_home/525017/description#description
ABSTRACT
The modeling of deep hole geometry which was generated for boring and trepanning association (BTA) drilling was performed and its characteristics was discussed, also the effects of internally BTA drilled round pro?le are analyzed and its geometric modeling reliability was veri?ed by the experiments of roundness testing, especially in BTA drilling operation. In this study, a harmonic geometric round model with the parameter of harmonic function was established. This relationship is also used to provide physical meanings to harmonic lobes generated by the proposed roundness model for a pro?le of BTA drilling, especially those caused by the spindle error motions of the BTA tool.In general, the theoretical roundness pro?le of a hole with an arbitrary multi-lobe can be calculated. But in real experiments, the two- to six-lobe hole pro?le was frequently measured. The most frequently measured one is three- and ?ve-lobe pro?le in experiments. With these results, It was predicted that the reliability of proposed harmonic model has been veri?ed theoretically and experimentally by a large number of real pro?le estimation of the hole and the experimental results that was produced by BTA drilling operation in deep hole making operation. This new modeling method is expected to provide desirable insights into the advanced tolerance analysis of circular hole making in BTA drilling process.
Keywords: BTA (boring and trepanning association) drill; Fractional frequency; Lobe profile; Roundness; Spindle error
1.Introduction
The modeling application in BTA (boring and trepanning association) hole drilling is frequently found in the wide range of industries, such as a hole drilling of machinery part, nuclear power, oil gas, and aerospace one. The round shape modeling of BTA drilling, considering spindle error motion is essential for the analysis of the lobe shape and its tolerance for circular and cylindrical profile. So, roundness modeling has received substantial attention in literature. This kind of roundness modeling is given a substantial attention in literature and some papers are reviewed below. Some products require a high demand on quality as well as precise dimension and shape tolerances. So, the experimental result about this quality was discussed in making a hole in SK3 alloy tool steel with BTA tool under the several cutting conditions, and compared with the theoretical modeling. Generally, the deep hole drilling is defined as the length to diameter ratio is bigger than 20. The BTA drilling and gun drilling are clas sified in this type. Many researchers have studied about the BTA drilling. Sakuma et al. [1] studied about the profile of deep hole and its generating mechanism experimentally. This result is focused on the fields of the optimum cutting edge design of BTA drill that generates minimum cutting force. Also, Sawabe et al. [2] established a relationship between the radial error motion of a spindle and the resultant part profile for turning processes after machining. Damir [3] developed an approximate harmonic model to determine the amplitude spectra of harmonic roundness lobe profiles in different machining processes. Cho and Tu [4] also developed the roundness lobe modeling of machined parts for tolerance analysis. They showed that the spindle error motion of a spindle could result in a similar roundness part profile. For example, a three-lobe spindle error motion frequency can result in a three-lobe circular profile. Their analysis, however, was also focused on the error motions with integer multiples of spindle rotational frequencies and also considers the importance of fractional frequency error motions and tool vibration in turning operation. Furthermore, there are many researchers who are interested in the roundness behaviors of the hole [5–11]. In this study, a new roundness modeling of BTA drilling and its characteristics were investigated. By comparing with the theoretical and experimental lobe results, the reliability of the model was proven. A more complete characteristics of BTA drilling for the alloy tool steel and the accuracy of hole were investigated by theoretical calculation and experimental measurements by performing CNC solid type BTA drilling.The BTA drill head used for drilling was depicted in Fig. 1.
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2. Roundness modeling of BTA drilling
A roundness profile model in BTA drilling is developed to describe the effect of spindle error motion on the roundness profile of drilled hole. Fig. 2 is a hole-generating model of BTA drilling developed by considering the revolution of spindle error of BTA tool center. A roundness lobe profile is the inner line of an object drilled in a given plane, and this circular profile is defined at a plane perpendicular to the axis of a cylinder shape. For a cylinder, the profile can be defined at the surface intersected by any plane passing through a common center. If there is a spindle error in BTA tool shaft of deep hole drilling it soon becomes stable in continuous rotating state. And the center of the BTA tool revolves in a circular or oval path of stationary state at last. But irregular crossing and wandering through the center at initial rotating state may also appear.
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By assuming a BTA tool motion in a stable revolution that the trace of tool center follows circular movements, the idealized harmonic model may be shown in Fig. 1.In Fig. 2, the point Oc is the center of absolute coordinate and Oi is the instantaneous center of BTA tool with spindle error motion that rotates in a harmonic circle. The radius, Ra (t,
), that is the radius of the locus rotates at a revolution frequency of 
a , and its radius is Oc Oi. The vector representation from the point Oc to Oi may be written as follows:
where j is imaginary unit. But,
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The point Tp is the fixed edge position of the BTA tool and it is caused by the revolution of the BTA tool center and workpiece vibration with the amplitude of Rw(t), into the x direction because of its poor fixing. In Fig. 2, the distance 
, from Oi to T, may be Rl(t) and the distance between Oi and Tp may be represented as Rp(t). Then, the BTA tool rotates with radius Rp(t) and it has rotating frequency of 
locating its center at Oi. So the right point trace of radius Rp(t) matches the real profile of radial distance R1(t), considering the spindle error motion. Also, the movement of the position Tp, considering in workpiece is also the movement of spindle error into x direction with its amplitude of 
and a frequency of 
. So it is defiined that the final profile of the roundness may be caused by the combination of the spindle axis error motion that is the rotating with a radius around the absolute center and the low stiffness workpiecevibration. These final traces of the edge, Tp, are the profiles of the roundness geometry BTA drilled. So, the equation that represents this trace may be summarized as follows [7]:
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The condition that some arbitrary position of BTA tool cutting edge coincides with the same position in same plane after one rotation may be defined as the closed lobe profile and its condition may be summarized as follows:
but,
If the position does not coincide with the preposition in plane after one revolution it is not a closed lobe. It caused a non-continuous profile. But if the values 
and 
be a integers, then it reveals a closed lobe profile. Furthermore, the amplitude of trace locus of lobe profile R1(t) may be rewritten as follows:
The mode may be calculated in frequency domain by using the discrete fourier transform as follows:
where N: sampling number, fk: number of lobe (0, 1, ...,n-1), 
: amplitude of i-th sampled radius vector, R1(fk): complex amplitude of k-lobe component.
By using discrete form of Eq. 7, it generates frequency of lobe profile by the natural modes of 
and 
or their combination in frequency domain that are its intrinsic natural modes.
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