Published: 01 December 2017

Transient vibration analysis of BTA deep-hole drilling shaft system

Guohong Ma1
Xingquan Shen2
1School of Mechanical Engineering, North University of China, Taiyuan, People’s Republic of China
2Taiyuan University of Technology, Taiyuan, People’s Republic of China
Corresponding Author:
Guohong Ma
Views 72
Reads 26
Downloads 1476

Abstract

Dynamics of deep-hole drilling shaft system is closely related to hole processing quality. From the viewpoint of rotor dynamics and fluid-structure interaction, the governing equation of the drilling shaft system for lateral vibration is obtained taking into account of fluid-structure interaction, rotational inertia, gyroscopic effect, the effect of motion constraints and frictional damping generated by surrounding fluid. The influence of rotational angular velocity and compressive axial force on transient vibration of drilling shaft is mainly examined. It has been found that rotational angular velocity has an obvious effect on the lateral vibration of drilling shaft, whereas the lateral vibration of drilling shaft does not change significantly with the increase of compressive axial force.

1. Introduction

The boring trepanning association (BTA) deep-hole drilling shaft system is a complex system. The dynamic behavior of drilling shaft exerts an unfavorable influence on cutting quality [1-3]. Kong [4] constructed the dynamic model to investigate nonlinear dynamic responses taking into account of the influence of cutting force fluctuation, auxiliary support and mass eccentricity. Perng [5] analyzed the eigenproperties of spinning deep-hole drilling shaft containing flowing fluid and subject to compressive axial force. Al-Wedyan [6] considered the interaction between the workpiece and drilling shaft to investigate whirling vibration of deep-hole drilling shaft. Kenichiro [7] investigated chatter vibration considering support position of drilling shaft in detail.

In this paper, the governing equation is obtained taking into account of fluid-structure interaction, rotational inertia and gyroscopic effect. The influence of rotational angular velocity and compressive axial force on vibration is mainly examined.

2. The equation of motion

As shown in Fig. 1, the drilling shaft rotating at angular velocity ω and conveying cutting fluid with flow velocity U is subjected to torque on drill head, compressive axial force on drill head and support constraints. The drilling shaft is modeled as a Rayleigh beam which is clamping at one end and hinging at the other end. The equations of motion [6-8] are given by:

1
ρpAp+ρfAf2wxt2-ρpIp+ρfIf4wxt2z2+2ω3wytz2+Cfwxt -T3wyz3+P+pfAf+ρfAfU22wxz2+2ρfAfU2wxtz+EIp4wxz4 +Kawx+Cawxtδ(z-za)+Kbwx+Cbwxtδ(z-zb)=0,
2
ρpAp+ρfAf2wyt2-ρpIp+ρfIf4wyt2z2-2ω3wxtz2+Cfwyt +T3wxz3+P+pfAf+ρfAfU22wyz2+2ρfAfU2wytz+EIp4wyz4 +Kawy+Cawytδ(z-za)+Kbwy+Cbwytδ(z-zb)=0,

where ρf, ρp is density of cutting fluid and drilling shaft; di, de is internal and external diameter of drilling shaft; Ap, Af is cross sectional area of drilling shaft wall and bore; Ip, If is moment of inertia of drilling shaft and bore; P is compressive axial force; T is torque; Ka, Kb and Ca, Cb are support stiffness and damping. wx, wy is transverse displacement.

Fig. 1Schematic diagram of deep-hole drilling system

Schematic diagram of deep-hole drilling system

The dimensionless quantities are introduced as follows:

ηx=wxl, ηy=wyl, ξ=zl, ξa=al, ξb=bl, Π=lTEIp, u=ρfAfEIp1/2Ul,
τ=tEIpρpAp+ρfAf1/2/l2, β=ρfAfρfAf+ρpAp, γ=(ρfIf+ρpIp)l2ρpAp+ρfAf,
Ω=ρpAp+ρfAfEIp1/2l2ω, ka=Kal3EIp, kb=Kbl3EIp, cf=l2Cf[EIp(ρpAp+ρfAf)]1/2,
ca=lCa[EIpρpAp+ρfAf]1/2, cb=lCb[EIpρpAp+ρfAf]1/2, Γ=l2P+pfAfEIp,

and the equation is rearranged as:

3
2ηxτ2-γ4ηxτ2ξ2+2Ω3ηyτξ2+cfηxτ-Π3ηyξ3+Γ+u22ηxξ2 +2βu2ηxτξ+4ηxξ4+kaηx+caηxτδ(ξ-ξa)+kbηx+cbηxτδ(ξ-ξb)=0,
4
2ηyτ2-γ4ηyτ2ξ2-2Ω3ηxτξ2+cfηyτ+Π3ηxξ3+Γ+u22ηyξ2 +2βu2ηyτξ+4ηyξ4+kaηy+caηyτδ(ξ-ξa)+kbηy+cbηyτδ(ξ-ξb)=0.

3. Method of solution

The Eqs. (9) and (10) are discretized using Galerkin method. The displacement at any point ξ can be expressed as:

5
ηxξ,τ=i=1mϕiξpiτ,
6
ηyξ,τ=j=1nψjξqjτ,

where pi(τ) and qj(τ) represents the unknown time-dependent generalized coordinates and ϕi(ξ) and ψj(ξ) is the corresponding orthogonal eigenfunction of the beam, given by:

7
ϕiξ=cosλiξ-coshλiξ-σisinλiξ-sinhλiξ, σi=cosλi+coshλisinλi+sinhλi,
8
ψjξ=cosλjξ-coshλjξ-σjsinλjξ-sinhλjξ, σj=cosλj+coshλjsinλj+sinhλj,
9
tanλi=tanhλi,
10
tanλj=tanhλj,

where λi and λj are the solution of Eqs. (9) and (10).

Substituting Eqs. (5) and (6) into Eqs. (3) and (4), multiplying both sides of Eqs. (3) and (4) by the jth eigenfuction ϕj(ξ) and integrating from 0 to 1, Eqs. (3) and (4) can be rewritten as:

11
Mp¨+C+Fp˙+K+Hp-Rq˙-Sq=0,
12
Mq¨+(C+F)q˙+(K+H)q+Rp˙+Sp=0

where:

M=B0-γB2, C=cfB0+2βuB1, F=caB5+cbB6, R=2γΩB2,
K=Γ+u2B2+B4, S=ΠB3, H=kaB5+kbB6,

and the elements of B0 through B6 are shown as follows:

B0i,j=01ϕiξϕjξdξ=01ψiξψjξdξ,
B1i,j=01ϕiξϕ'jξdξ=01ψiξϕj'ξdξ,
B2i,j=01ϕiξϕj''ξdξ=01ψiξϕj''ξdξ
=01ϕiξϕj''ξdξ=01ψiξϕj''ξdξ,
B3i,j=01ϕiξψj3ξdξ=01ψiξϕj3ξdξ,
B4i,j=01ϕiξϕj4ξdξ=01ψiξψj4ξdξ,
B5i,j=ϕiξaϕjξa=ψiξaψjξa,
B6i,j=ϕiξbϕjξb=ψiξbψjξb.

To solve it simply, the state vector is introduced as follows:

13
Z=pp˙qq˙T.

Thus, Eqs. (11) and (12) are transformed into its first-order form:

14
Z˙=AZ,

where:

A=OIOO-M-1(K+H)-M-1(C+F)M-1SM-1ROOOI-M-1S-M-1R-M-1(K+H)-M-1(C+F).

The matrix I is the identity matrix.

4. Numerical results and discussion

The parameters of the drilling shaft system are listed in Table 1. In the study, letting the dimensionless quantity u= 0.5, cf= 1.5, ca= 7.32×102, cb= 3.81×102, ka= 9.21×104, kb= 4.80×104, Π= 0.05, ξa= 0.4 and ξb= 0.8, respectively.

Table 1The parameters of the drilling shaft system

Density of drilling shaft
ρz (kg/m3)
Density of cutting fluid
ρf (kg/m3)
Internal diameter
d1 (mm)
External diameter
d2 (mm)
Length
l (m)
7.8×103
0.865×103
20
26
5

4.1. Effect of rotational angular velocity on vibration

There is some fluctuation in the amplitude of lateral vibration displacement of drilling shaft in Fig. 2(a), 3(a) and 4(a). However, with the passage of time, the overall trend is decreasing. Eventually, the motion of drilling shaft is stable with equal amplitude vibration.

Fig. 2Lateral vibration displacement and radial displacement trajectory

Lateral vibration displacement and radial displacement trajectory

a) (Ω= 10, Γ= 10)

Lateral vibration displacement and radial displacement trajectory

b) (Ω= 10, Γ= 10)

By comparing Fig. 2(b), 3(b) and 4(b), the radial displacement trajectory changes from linear to explosive trajectory, which shows that the movement of the drilling shaft is out of control. This will seriously affect the processing quality of the hole.

Fig. 3Lateral vibration displacement and radial displacement trajectory

Lateral vibration displacement and radial displacement trajectory

a) (Ω= 50, Γ= 10)

Lateral vibration displacement and radial displacement trajectory

b) (Ω= 50, Γ= 10)

Fig. 4Lateral vibration displacement and radial displacement trajectory

Lateral vibration displacement and radial displacement trajectory

a) (Ω= 100, Γ= 10)

Lateral vibration displacement and radial displacement trajectory

b) (Ω= 100, Γ= 10)

4.2. Effect of axial compressive force on vibration

With the increase of axial compressive force, there is no obvious change in radial displacement trajectory of drilling shaft, as shown in Fig. 5. In a certain range, axial compressive force has little effect on the lateral vibration of drilling shaft.

Fig. 5Radial displacement trajectory of drilling shaft for different levels of axial compressive force Ω= 50: a) Γ= 5, b) Γ= 10, c) Γ= 15

Radial displacement trajectory of drilling shaft for different levels of axial compressive force  Ω= 50: a) Γ= 5, b) Γ= 10, c) Γ= 15

a)

Radial displacement trajectory of drilling shaft for different levels of axial compressive force  Ω= 50: a) Γ= 5, b) Γ= 10, c) Γ= 15

b)

Radial displacement trajectory of drilling shaft for different levels of axial compressive force  Ω= 50: a) Γ= 5, b) Γ= 10, c) Γ= 15

c)

5. Conclusions

A systematic mathematical model for drilling shaft system is established. The results indicate that lateral vibration of drilling shaft is becoming more and more fierce with the increase of rotational angular velocity and does not change significantly with the increase of axial compressive force.

References

  • Raabe N. Dynamic Disturbances in BTA Deep-Hole Drilling: Modelling Chatter and Spiraling as Regenerative of the Effects. Advances in Data Analysis, Data Handling and Business Intelligence. Studies in Classification, Data Analysis, and Knowledge Organization, Springer, Berlin, Heidelberg, 2009, p. 745-754.
  • Maleki M. I., Nouri M., Madoliat R. Investigating chatter vibration in deep drilling, including process damping and the gyroscopic effect. International Journal of Machine Tools and Manufacture, Vol. 49, Issues 12-13, 2009, p. 939-946.
  • Philip B., Lamar V., Michael T. Low-frequency regenerative vibration and formation of lobed holes in drilling. Journal of Manufacturing Science and Engineering Transactions of the ASME, Vol. 124, Issue 2, 2002, p. 275-285.
  • Kong L. F., Li Y., Zhao Z. Y. Numerical investigating nonlinear dynamic responses to rotating deep-hole drilling shaft with multi-span intermediate supports. International Journal of Non-Linear Mechanics, Vol. 55, 2013, p. 170-179.
  • Perng Y. L., Chin J. H. Theoretical and experimental investigations on the spinning BTA deep-hole drill shafts containing fluids and subject to axial forces. International Journal of Mechanical Sciences, Vol. 41, Issue 11, 2001, p. 1301-1322.
  • Al Wedyan H.-M., Bhat R. B., Demirli K. Whirling vibrations in boring trepanning association deep hole boring process: analytical and experimental investigations. Journal of Manufacturing Science and Engineering-Transactions of the ASME, Vol. 129, Issue 1, 2007, p. 48-62.
  • Matsuzaki K., Ryu T., Sueoka A. Theoretical and experimental study on rifling mark generating phenomena in BTA deep hole drilling process. International Journal of Machine Tools and Manufacture, Vol. 88, 2015, p. 194-205.
  • Raffa F. A., Vatta F. Gyroscopic effects analysis in the Lagrangian formulation of rotating beams. Meccanica, Vol. 34, 1999, p. 357-366.

About this article

Received
03 November 2017
Accepted
15 November 2017
Published
01 December 2017
SUBJECTS
Mechanical vibrations and applications
Keywords
deep hole drilling
dynamic behavior
fluid-structure interaction
Acknowledgements

This work was financially supported by the National Natural Science Foundation of China (51175482).