whirling vibrations in boring trepanning association deep

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Hussien M. Al-Wedyan1

e-mail: [email protected]

Rama B. Bhate-mail: [email protected]

Kudret Demirlie-mail: [email protected]

Department of Mechanical and IndustrialEngineering,

Concordia University,1455 De Maisonneuve Boulevard,

Montreal, Quebec H3G 1M8, Canada

Whirling Vibrations in BoringTrepanning Association DeepHole Boring Process: Analyticaland Experimental InvestigationsAn approach to study the whirling motion of the BTA (boring trepanning association)deep hole boring system is presented by introducing the system excitation in the form ofinternal forces between the boring bar and the workpiece. This involves nonhomogeneousboundary conditions with homogeneous equations. The mathematical approach with theboring bar-workpiece internal cutting forces and external suppression forces will trans-form the problem into nonhomogeneous equations with homogenous boundary condi-tions. Using this approach the whirling motion of the boring bar is obtained at differentpoints on the boring bar-workpiece assembly. External suppression forces will reduce thewhirl amplitude at the same locations. Further, an experimental investigation is carriedout on the BTA deep hole boring process while drilling at four cutting speeds usingproximity pickups for displacement measurements. �DOI: 10.1115/1.2280610�

ntroductionChatter vibrations are self-excited vibrations encountered inachining operations. A portion of the power supplied to the ma-

hine is converted into vibratory power during the machining op-ration, which sets up sustained vibrations in the tool and theorkpiece. In deep hole boring, the load transmitted from theoring-bar to the workpiece while drilling consists of a meanomponent corresponding to the transmitted power and a fluctu-ting increment of a dynamic origin. The latter originates fromithin the complete system of the boring bar-workpiece assembly

nd the nature of the drilling process. Over the last twenty years,here has been an increased research effort to investigate chatteribration in deep hole boring process.

Bayly et al. investigated regenerative vibrations in differentachining processes �1�. They studied the cutting and rubbing

orces in a chisel drilling edge in addition to tool vibration, whichntroduces an error in the hole size or “roundness error” of therilled piece. A mechanism of torsional chatter was investigatedxperimentally by Bayly et al. �2� and the analysis was carried outn the frequency domain to find the chatter frequencies and bound-ries of stability. The engagement and disengagement is highlyonlinear during the drilling process. The roundness error ineaming, due to regenerative vibration, was investigated by Baylyt al. �3� using a quasi static model. It was shown that the toolith N teeth caused holes with N+1 or N−1 “lobes” which are

elated directly to the forward and backward whirl motion. Statis-ical process planning was used to describe the relationship be-ween the machining parameters and the quality of the bored sur-ace by Weinert et al. �4�. During the drilling process of deep holeoring, undesirable vibrations are initiated due to slender BTArills having larger length to diameter ratio with low torsional andending stiffnesses. Keraita et al. �5� theoretically correlated thecoustic emission during cutting to the workpiece-tool geometrynd the cutting conditions. They showed that the instability ofutting or chatter is due to combination of structure, cutting con-itions, and tool geometry. Litak et al. �6� theoretically investi-

1Author to whom correspondence should be addressed.Contributed by the Manufacturing Engineering Division of ASME for publication

n the JOURNAL OF MANUFACTURING SCIENCE AND ENGINEERING. Manuscript receivedune 21, 2004; final manuscript received March 13, 2006. Review conducted by
. C. Shin.
8 / Vol. 129, FEBRUARY 2007 Copyright ©

aded 15 Nov 2007 to 133.5.72.206. Redistribution subject to ASME

gated the chaotic harmful chatter vibrations, which caused insta-bilities during the cutting process.

Kovacic �7� used the tool rake angle, shear angle, and the feedrate to propose a nonlinear model to investigate chatter vibrations.The mechanism of chatter vibration was studied experimentallyby Marui et al. �8� using six different spindle-workpiece combi-nations having different properties. They found that the phase lagbetween the forces and chatter displacement is related to the en-ergy induced by chatter.

Whirling vibration in the chisel drill was experimentally mea-sured by Fuji et al. �9� in order to investigate how the whirlingvibrations developed in the chisel drill. They used three differentchisel drills with different web thicknesses. Fuji et al. �10� studiedthe interactions among the effect of drill geometry and drill flank,in starting and developing of the whirling motion, where they alsofound that the flank surface of the cutting edge was responsiblefor damping the vibration. Ema et al. �11� investigated the whirl-ing vibrations in a workpiece having a pilot hole, where they alsofind that the whirling motion is a regenerative vibration caused bycutting forces and friction while drilling. A mathematical modelwas presented by Batzer et al. �12� for a chisel drill with a zerohelix angle to determine the displacement of the assumed rigidtool and rigid workpiece, where they consider only the axial vi-brations and ignore the transverse motion. They used a singledegree of freedom model that was solved numerically to find thechip thickness and the time lag for the chip formation. Cuttingtests were done to verify the results.

The deep hole boring process is used to bore holes with usuallyhigh length to diameter ratios seeking better surface finish, goodroundness, and straightness. The process usually depends upon thefollowing hole requirements: Diameter of the bored hole, thedepth of the bored hole, the quality of the hole surface, the dimen-sion, parallelism, and straightness. Due to the fact that the boringbar-cutting head combination is slender which is highly needed toproduce holes with different length to diameter ratios, this kind ofdrilling is subjected to disturbances such as chattering vibrations.Despite the abundance of studies done in this field, chatter vibra-tions are still not fully understood. Whirling motion is vibration inthree dimensions, which affects the accuracy of the bored piece. Itis well known now that the deep hole boring process is usedextensively to drill expensive workpieces and hence process pre-
cision is of prime importance. To achieve the best process plan
2007 by ASME Transactions of the ASME

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ith the aim of minimizing the risk of the workpiece damage, aomprehensive investigation of the dynamics involved in the pro-ess, both analytical and experimental, is highly important.

athematical Modeling of the Boring Bar-WorkpiecessemblyThe boring bar has an intermediate simple support as shown in

ig. 1, where the interaction point between the cutting tool headnd the workpiece is also shown. Continuity conditions are as-umed for the deflection, slope, and moment at the interactionoint, and shear force increases with the addition of the cuttingorce.

The boring bar is considered as a continuous beam clamped athe bar driver while the workpiece is clamped at its end. Hence,e can consider the boring bar-workpiece system as a multispaneam and the transverse vibration of this beam in the Y-Z planeas the following governing partial differential equations in the Ynd Z directions as:

EI�4Wy

�X4 �X,t� + M�2Wy

�t2 �X,t� = Fy�t� �1�

nd

EI�4Wz

�X4 �X,t� + M�2Wz

�t2 �X,t� = Fz�t� �2�

here I, M, Wy, Wz, Fy, and Fz, respectively, are

I1, M1, W1y, W1z, Py and Pz, in 0 � X � L1

I2, M2, W2y, W2z, 0 and 0, in L1 � X � L2

I3, M3, W3y, W3z, 0 and 0, in L2 � X � L

ll the constants and variables used in the mathematical modelre given in Appendix A. The boundary conditions are

W1y�0,t� = 0.0 �3a�

W1y� �0,t� = 0.0 �3b�

W1y�L1,t� = 0.0 �3c�

W2y�L1,t� = 0.0 �3d�

W1y� �L1,t� = W2y� �L1,t� �3e�

W1y� �L1,t� = − W2y� �L1,t� �3f�

ig. 1 Boring bar-workpiece assembly with the cutting forcesnd control forces

W2y�L2,t� = W3y�L2,t� �3g�

ournal of Manufacturing Science and Engineering


W2y� �L2,t� = W3y� �L2,t� �3h�

W2y� �L2,t� = − W3y� �L2,t� �3i�

EI2W2y� �L2,t� = − EI3W3y� �L2,t� − Fcy�t� �3j�

W3y�L,t� = 0.0 �3k�

W3y� �L,t� = 0.0 �3l�

W1z�0,t� = 0.0 �4a�

W1z� �0,t� = 0.0 �4b�

W1z�L1,t� = 0.0 �4c�

W2z�L1,t� = 0.0 �4d�

W1z� �L1,t� = W2z� �L1,t� �4e�

W1z� �L1,t� = − W2z� �L1,t� �4f�

W2z�L2,t� = W3z�L2,t� �4g�

W2z� �L2,t� = W3z� �L2,t� �4h�

W2z� �L2,t� = − W3z� �L2,t� �4i�

EI2W2z� �L2,t� = − EI3W3z� �L2,t� − Fcz�t� �4j�

W3z�L,t� = 0.0 �4k�

W3z� �L,t� = 0.0 �4l�

The system is subjected to nonhomogeneous boundary conditionsor time-dependent boundary conditions as given in Eqs. �3j� and�4j�. A modified approach will be used to transform this probleminto a problem consisting of nonhomogeneous differential equa-tions with homogeneous boundary conditions which will besolved by modal analysis. To this end let us assume a solution ofthe boundary-value problem described by the equations of motionabove in the Y and Z directions in the form

W1y�x,t� = v1y�x,t� 0 � X � L1

W2y�x,t� = v2y�x,t� + g2�x�Fcy�t� L1 � X � L2 �5�

W3y�x,t� = v3y�x,t� + g3�X�Fcy�t� L2 � X � L

W1z�x,t� = v1z�x,t� 0 � X � L1

W2z�x,t� = v2z�x,t� + g2�x�Fcz�t� L1 � X � L2 �6�

W3z�x,t� = v3z�x,t� + g3�x�Fcz�t� L2 � X � L

The functions g2�x� and g3�x� are chosen to render the boundaryconditions for the variables v1y�x , t� ,v2y�x , t� ,v3y�x , t� andv1z�x , t� ,v2z�x , t� ,v3z�x , t� homogeneous.

The functions g2�x� and g3�x� are not unique and severalchoices may be acceptable �13�. The second step is to introducethe boundary conditions of the current problem into Eqs. �5� and�6�. Consider Eq. �3j�

�2v2y� �L2,t� + v3y� �L2,t� = − Fcy�t��2g2��L2� + g3��L2� +1

EI3�

�7�

For the right hand side of Eq. �7� to be zero

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�2g2��L2� + g3��L2� +1

EI3= 0 �8�

ollowing similar procedure, the boundary conditions from Eq.3� result in

g2�L1� = 0 �3d��

g2��L1� = 0 �3e��

g2��L1� = 0 �3f��

g3�L2� − g2�L2� = 0 �3g��9�

g3��L2� − g2��L2� = 0 �3h��

g3��L2� + g2��L2� = 0 �3i��

g3�L� = 0 �3k��

g3��L� = 0 �3l��here the corresponding equation number is also specified.Assuming that

g2��X� = D1X + D2 �10�ubstitute Eq. �10� into Eq. �8�, the following equation is obtained:

g3��X� = − � 1

EI3+ �2�D1X + D2�� 11�

ombining Eqs. �10� and �11� gives

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g2�X� =X4

24D1 +

X3

6D2 +

X2

2D3 + XD4 + D5 �12�

g3�X� =− X3

6EI3− �2�X4

24D1 +

X3

6D2� +

X2

2D6 + XD7 + D8 �13�

where the constants D1–D8 are evaluated by applying the set ofboundary conditions in Eq. �9�. The values of these constants arepresented in Table 1.

Now introducing Eqs. �5� and �6� into Eqs. �1� and �2�, respec-tively, we obtain a set of nonhomogeneous differential equations.In the Y direction

EIvy��x,t� + Mvy�x,t� = Fy�t� �14�where

Table 1 Values for the constants of integration for g1„x… andg2„x…

Constant Value

D10.1723

D2 −0.4032D3

0.4551D4 −0.3339D5

0.1803D6

0.0113D7

0.0374D8

0.0571

Fy�t� = Py�t��x − a� and I = I1, M = M1 0 � X � L1

= − EIg2��x�Fcy�t� − Mg2�x�Fcy�t� and I = I2, M = M2 L1 � X � L2

= − EIg3��x�Fcy�t� − Mg3�x�Fcy�t� and I = I3, M = M3 L2 � X � L

nd in Z direction

EIvz��x,t� + Mvz�x,t� = Fz�t� �15�here

Fz�t� = Pz�t��x − a� and I = I1, M = M1 0 � X � L1

= − EIg2��x�Fcz�t� − Mg2�x�Fcz�t� and I = I2, M = M2 L1 � X � L2

= − EIg3��x�Fcz�t� − Mg3�x�Fcz�t� and I = I3, M = M3 L2 � X � L

t this stage the homogeneous equations will be solved to find theatural frequencies with the corresponding eigenfunctions for theollowing system of equations:

EIvy��x,t� + Mvy�x,t� = 0 �16�

EIvz��x,t� + Mvz�x,t� = 0 �17�

nder free vibration conditions, we assume

vy�x,t� = Vy�x�ei�t �18�

vz�x,t� = Vz�x�ei�t �19�

onsider X=X /L, then d /dX= �d /dX��1/L�, and accordingly weave

v��X� − �4vy�X� = 0 �20�

vz��X� − �4vz�X� = 0 �21�

where

�4 =�2ML4

EI, becomes

�14 =

�2M1L14

EI1, 0 � X � L1

�24 =

�2M2L24

EI2, L1 � X � L2

�34 =

�2M3L4

, L2 � X � 1
EI3
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L1 =L1

L, L2 =

L2

L, L =

L

L= 1

he solutions in the three regions are

v1y�x� = A1 cos �1X + A2 sin �1X + A3 cosh �1X

+ A4 sinh �1X, 0 � X � L1


+ A8 sinh �2X, L1 � X � L2 �22�


+ A12 sinh �3X, L2 � X � 1

nd in the Z direction

v1z�x� = B1 cos �1X + B2 sin �1X + B3 cosh �1X

+ B4 sinh �1X, 0 � X � L1


+ B8 sinh �2X, L1 � X � L2 �23�


+ B12 sinh �3X, L2 � X � 1

1y�X�, v2y�X� 3y�X� in the Y direction and v1z�X�, v2z�X�, and

3z�X� in the Z direction have to satisfy the conditions that theirespective fourth derivatives are equal to a constant multiplied byhe functions. All the constants Ai and Bi, i=1, . . . ,12 are evalu-ted using the following boundary conditions in the Y direction:

v1y�0,t� = 0.0 �24a�

v1y� �0,t� = 0.0 �24b�

v1y�L1,t� = 0.0 �24c�

v2y�L1,t� = 0.0 �24d�

v1y� �L1,t� = v2y� �L1,t� �24e�

v1y� �L1,t� = − v2y� �L1,t� �24f�

v2y�L2,t� = v3y�L2,t� �24g�

v2y� �L2,t� = v3y� �L2,t� �24h�

v2y� �L2,t� = − v3y� �L2,t� �24i�

�2v2y� �L2,t� = − v3y� �L2,t� �24j�

v3y�L,t� = 0.0 �24k�

v3y� �L,t� = 0.0 �24l�

v1z�0,t� = 0.0 �25a�

v1z� �0,t� = 0.0 �25b�

v1z�L1,t� = 0.0 �25c�

¯
v2z�L1,t� = 0.0 �25d�


v1z� �L1,t� = v2z� �L1,t� �25e�

v1z� �L1,t� = − v2z� �L1,t� �25f�

v2z�L2,t� = v3z�L2,t� �25g�

v2z� �L2,t� = v3z� �L2,t� �25h�

v2z� �L2,t� = − v3z� �L2,t� �25i�

�2v2z� �L2,t� = − v3z� �L2,t� �25j�

v3z�L,t� = 0.0 �25k�

v3z� �L,t� = 0.0 �25l�

From Eqs. �24a� and �24b�, it is found that A1=−A3 and A2=−A4, respectively. The same for Eqs. �25a� and �25b�, it is foundthat B1=−B3 and B2=−B4, respectively.

These equations are written in the form

�G�10�10A10�1 = 0 and �H�10�10B10�1 = 0 �26�

For a nontrivial solutions �G�=0, �H�=0. Anyone will give thecharacteristic equation, where �G� and �H� are the determinant ofthe coefficient matrices �G� and �H�, respectively. The matrix withthe coefficients is shown in Appendix A.

Plotting the frequency equation against a=��n yields the rootsof the frequency equation for the first five natural frequencies ofthe cutting tool-boring bar-workpiece assembly. Table 2 presentsthe first five natural frequencies.

The normal modes corresponding to these natural frequenciesare:

v1yi�x� = A3�cosh �1iX − cos �1iX� − sin �1iX + sinh �1iX

v2yi�x� = A5 cos �2iX + A6 sin �2iX + A7 cosh �2iX + A8 sinh �2iX

�27�

v3yi�x� = A9 cos �2iX + A10 sin �2iX + A11 cosh �2iX + A12 sinh �2iX

v1zi�x� = B3�cosh �1iX − cos �1iX� − sin �1iX + sinh �1iX

v2zi�x� = B5 cos �2iX + B6 sin �2iX + B7 cosh �2X + B8 sinh �2iX

�28�

v3zi�x� = B9 cos �3iX + B10 sin �3iX + B11 cosh �3iX + B12 sinh �3iX

The values for Ai and Bi, i=3,5 , . . . ,12 are presented in Table 3.Going back to the main equations in the Y direction:

EIvy��x,t� + Mvy�x,t� = Fy�t� �29�

Table 2 The first five natural frequencies of the boring bar-workpiece system

Natural frequencynumber �n �Hz�

1 11.2352 25.2803 53.8634 88.6125 93.254

where

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Fy�t� = Py�t��x − a� and I = I1, M = M1 0 � X � L1


= − EIg3��x�Fcy�t� − Mg3�x�Fcy�t� and I = I3, M = M3 L2 � X � 1

nd in Z direction we have

EIvz��x,t� + Mvz�x,t� = Fz�t� �30�here

Fz�t� = Pz�t��x − a� and I = I1, M = M1 0 � X � L1


= − EIg3��x�Fcz�t� − Mg3�x�Fcz�t� and I = I3, M = M3 L2 � X � 1

Table 3 The arbitrary constants for the normal modes at the first five natural frequencies

�1 �2 �3 �4 �5

A3 , B3−0.0908 −0.2037 −0.4286 −0.6893 −0.7374

A5 , B51.5703 1.1701 −0.8996 0.0313 −0.2971

A6 , B6−0.9189 1.1507 −0.3758 0.5280 0.4780

A7 , B7−2.2355 −8.1567 −67.9999 −662.3000 −710.7308

A8 , B82.8708 8.3050 68.0048 662.3000 710.7308

A9 , B9−0.0125 −0.1034 −0.4449 −0.4875 −0.4726

A10 , B101.8644 1.8197 1.2021 0.3080 0.2349

A11 , B11−0.0054 −0.0723 −0.3149 0.4626 0.7224

A12 , B12−1.7543 −1.3410 −0.2369 −0.5921 −0.8355

n order to solve the nonhomogeneous differential Eqs. �29� and30�, assume the solution in terms of normal modes

vy�x,t� = n=1

�

vyn�x��yn�t�

�31�

here



vz�x,t� = n=1

�

vzn�x��zn�t�

Introducing Eq. �31� into Eq. �29� in the Y direction we obtain

n=1

� ��yn�t�vyn��x� + �yn�t�M

EIvyn�x�� = Fy�t� �32�

where

Fy�t� =Py�t��x − a�

EIand I = I1, M = M1 0 � X � L1


= − EIg3��x�Fcy�t� − Mg3�x�Fcy�t� and I = I3, M = M3 L2 � X � 1

nd introduce Eq. �31� into Eq. �30� in the Z direction, we obtain

n=1

� ��zn�t�vzn��x� + �zn�t�M

EIvzn�x�� = Fz�t� �33�



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Fz�t� =Pz�t��x − a�

EIand I = I1, M = M1 0 � X � L1


= − EIg3��x�Fcz�t� − Mg3�x�Fcz�t� and I = I3, M = M3 L2 � X � 1

nd since v1yn�x�, v2yn�x�, v3yn�x� and v1zn�x�, v2zn�x�, v3zn�x� and �n satisfies Eqs. �20� and �21�, Eqs. �32� and �33� can be written as,rst in the Y direction:

n=1

� ��yn�t� + �n2�yn�t��

M

EIvyn�x�� = Fy�t� �34�

here

Fy�t� =Py�t��x − a�

EIand I = I1, M = M1 0 � X � L1

= − �g2��x�Fcy�t� +M

EIg2�x�Fcy�t�� and I = I2, M = M2 L1 � X � L2

= − �g3��x�Fcy�t� +M

EIg3�x�Fcy�t�� and I = I3, M = M3 L2 � X � 1


n=1

� ��zn�t� + �n2�zn�t��

M

EIvzn�x�� = Fz�t� �35�

here

Fz�t� =Pz�t��x − a�

EIand I = I1, M = M1 0 � X � L1

= − �g2��x�Fcz�t� +M

EIg2�x�Fcz�t�� and I = I2, M = M2 L1 � X � L2

= − �g3��x�Fcz�t� +M

EIg3�x�Fcz�t�� and I = I3, M = M3 L2 � X � 1

o find the solution in the three regions, we have to uncouplehese equations using the orthogonal property of the eigenfunction

nd integrate with respect to X over the domain. The orthogonalroperty in Appendix A is used. Hence, we obtain an infinite setf uncoupled ordinary differential equations in Y direction for thehree regions of the beam

�yi�t� + 2�1i2 �yi�t� + �1i

4 �yi�t� = Nyi�t�

�yi�t� + 2�2i2 �yi�t� + �2i

4 �yi�t� = Nyi�t� �36�

�yi�t� + 2�3i2 �yi�t� + �3i

4 �yi�t� = Nyi�t�

here is a modal damping ratio.



Similarly in Z direction

�zi�t� + 2�1i2 �zi�t� + �1i

4 �zi�t� = Nzi�t�

�zi�t� + 2�2i2 �zi�t� + �2i

4 �zi�t� = Nzi�t� �37�

�zi�t� + 2�3i2 �zi�t� + �3i

4 �zi�t� = Nzi�t�where

Nyi�t� = Fy�t��0

L1

v1yi�x�dx +�L1

L2

v2yi�x�dx +�L2

1

v3yi�x�dx�Nzi�t� = Fz�t��L1

v1zi�x�dx +�L2

v2zi�x�dx +�1

v3zi�x�dx�
0 L1 L2
Fy�t� =1

EIv1yi�a�Py�t� and I = I1, M = M1 0 � X � L1

= − �G2yi* Fcy�t� + G2yiFcy�t�� and I = I2, M = M2 L1 � X � L2

= − �G3yi* Fcy�t� + G3yiFcy�t�� and I = I3, M = M3 L2 � X � 1

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Fz�t� =1

EIv1zi�a�Pz�t� and I = I1, M = M1 0 � X � L1

= − �G2zi* Fcy�t� + G2ziFcz�t�� and I = I2, M = M2 L1 � X � L2

= − �G3zi* Fcz�t� + G3ziFcz�t�� and I = I3, M = M3 L2 � X � 1

ll other constants are shown in Table 7 in Appendix A. Thehirling motions at different sections of the boring bar-workpiece

ystem are plotted at a speed of 60 rad/s and with the external andnternal forces in the form of

Py�t� = P0y cos�60t + �

Pz�t� = P0z sin�60t + �

Fcy�t� = Fc + Fc0 cos��t�

Fcz�t� = Fc + Fc0 sin��t�

ith

a = 0.1789

he solution for the set of equations in Y direction Eqs. �36� and37� are obtained by the convolution integral or superposition in-egral. It is based on the superposition of the responses of theystem to a sequence of impulses. Let the variable of integratione �� between the limits of integration �0� and �t� and the elemen-al impulse is Nn��d�. So the complete solution for these equa-ions with zero initial conditions is

�yi�t� = � 1

�i�

0

t

Nyi��e−wi�t−�� sin �i�t − ��d��38�

�zi�t� = � 1

�i�

0

t

Nzi��e−wi�t−�� sin �i�t − ��d��he complete solution for the current problem is as follows:

W1y�x,t� = i=1

�

v1yi�x�� 1

�i�

0

t

Nyi��e−wi�t−�� sin �i�t − ��d��0 � X � L1

W2y�x,t� = i=1

�

v2yi�x�� 1

�i�

0

t

Nyi��e−wi�t−�� sin �i�t − ��d��+ g2�x�Fcy�t� L1 � X � L2 �39�

W3y�x,t� = n=1

�

v3yi�x�� 1

�i�

0

t

Nyi��e−wi�t−�� sin �i�t − ��d��+ g3�x�Fcy�t� L2 � X � 1


W1z�x,t� = i=1

�

v1zi�x�� 1

�i�

0

t

Nzi��e−wi�t−�� sin �i�t − ��d��¯ ¯
0 � X � L1


W2z�x,t� = i=1

�

v2zi�x�� 1

�i�

0

t

Nzi��e−wi�t−�� sin �i�t − ��d��+ g2�x�Fcz�t� L1 � X � L2 �40�

W3z�x,t� = n=1

�

v3zi�x�� 1

�i�

0

t

Nzi��e−wi�t−�� sin �i�t − ��d��+ g3�x�Fcz�t� L2 � X � 1

Substituting the forces in the solution in the Y directions

W1y�x,t� = i=1

�

vy1i�x�� 1

�i�

0

t � 1

EI1v1yi�a�Py��

− �G2yi* Fcy�� + G2yiFcy��


�e−wi�t−�� sin �i�t − ��d��,

in the range of 0 � X � L1

W2y�x,t� = i=1

�

vy2i�x�� 1

�i�

0

t � 1




�e−wi�t−�� sin �i�t − ��d�� + g2�x�Fcy�t�,

in the range of L1 � X � L2 �41�

W3y�x,t� = i=1

�

vy3i�x�� 1

�i�

0

t � 1




�e−wi�t−�� sin �i�t − ��d�� + g3�x�Fcy�t�,

in the range of L2 � X � 1

And in the Z direction



Iahwteffs

wiittd

mnb

qb

a

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Downlo

W1z�x,t� = i=1

�

vz1i�x�� 1

�i�

0

t � 1

EI1v1zi�a�Pz��

− �G2zi* Fcz�� + G2ziFcz��


�e−wi�t−�� sin �i�t − ��d��,

in the range of 0 � X � L1

W2z�x,t� = i=1

�

vz2i�x�� 1

�i�

0

t � 1




�e−wi�t−�� sin �i�t − ��d�� + g2�x�Fcz�t�,

in the range of L1 � X � L2 �42�

W3z�x,t� = i=1

�

vz3i�x�� 1

�i�

0

t � 1




�e−wi�t−�� sin �i�t − ��d�� + g3�x�Fcz�t�,

in the range of L2 � X � 1

n order to arrive at the final solution, the integrals in Appendix Are used. It is seen from those integrals that the final solution willave a transient and steady state solution. The transient solutionill die out after a while, and interest is in the steady state solu-

ion of the current problem. The steady state solution for the self-xcited motion at �=60 rad/s which is below the first naturalrequency, is plotted in Fig. 2 showing both the Z and Y signalsor the boring bar-workpiece system with and without the suppres-ion forces. Figure 3 is a one-figure plot of the system with and

ithout the suppression forces at L=0.41, L=0.81, and L=0.91. Its obvious from Figs. 2 and 3 that the whirling motion is oscillat-ng around the mean value of the force. This is due to the assump-ion of the force terms in the Y and Z direction that, in our case,he forces transmitted from the boring-bar to the workpiece whilerilling consists of a component corresponding to the mean trans-

itted power �Fc� plus a fluctuating component �Fco� of a dy-amic origin in terms of Eq. �43�. The power transmitted to theoring bar equals to the torque multiplied by the rotational fre-

uency. The torque equals to �Fc� multiplied by the radius of theoring bar �boring bar diameter is 1 in�. The value of �Fco� is

ssumed to be 10% of �Fc�

¯
Fcy�t� = Fc + Fc0 cos��t�


Fcz�t� = Fc + Fc0 sin��t� �43�This excitation that produces the latter originates from within thecomplete system of the boring bar-workpiece assembly and thenature of the drilling process.

Experimental Setup, Description, and InstrumentsThe whirling motion of deep hole boring-cutting head system is

investigated experimentally. None of the previous studies con-cerning vibrations and chatter in the deep hole boring system dealtwith the whirling vibrations of the boring bar-cutting head system,at the initial stage of drilling operations and during the machiningprocess. Noncontacting type proximity pickups were used to mea-sure the whirling motion of the boring bar. An electronic packagewith LABVIEW™ software was used to obtain the whirl orbit of theboring bar as shown in Figs. 4 and 5. The experiments were con-ducted to validate the analytical predictions of whirling vibrationsof the rotating boring bar-cutting head system described in theprevious section. The boring bar is a flexible circular, hollow

Fig. 2 The Y „—… and Z „¯… signals with and without suppres-sion forces for the boring bar-workpiece system at �

=60 rad/s in 0< X< L1 at L=0.41

Fig. 3 The Z and Y signals for the boring bar-workpiece sys-tem in 0< X< L1, L1< X< L2, and L2< X<1 at L=0.41, L=0.81, and¯
L=0.91.
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haft, clamped at one end and simply supported by the stuffingox in the middle and the other end is coupled with the workpieceo be bored. It is driven by a variable speed motor to control thepeed of cut. Two noncontacting type displacement transducers,Master for vertical displacement �Y� and Slave for the horizontalisplacement �Z�� which operate using eddy current principle, aresed to measure whirl amplitudes. The sensors were named Mas-er and slave due to the synchronization between them. The whirlrbit was detected using data acquisition system �DAQ� alongith LABVIEW computer software.The proximity pick ups are used to measure the transverse mo-

ion of the boring bar �displacement along Y and Z directions� athe onset of the boring bar-workpiece engagement. They were00 mm from the driving unit. This was approximately in an un-upported portion of the boring bar. By reducing the vibrations inhis region, the vibrations of the boring bar in general were ex-ected to be reduced as confirmed by the measurements done onhe workpiece subsequently. An actual implementation wouldave to measure the whirling motion close to the cutting tool andompensate for the whirling with exciters mounted close to it. The

ig. 4 Schematic presentation of experimental setup, theroximity pickups were located at 1.25 m from the boring barutting head

Fig. 5 The frame and the two measuring sensors



present study establishes the effectiveness of the active controlscheme in suppressing the excessive whirling motion of the bor-ing bar in general.

As the boring bar rotates relative to the sensors during the bor-ing process, the eddy current developed starts modulating the os-cillator voltage signal which is demodulated to provide an outputsignal proportional to the displacement. This displacement signalcan be recorded and analyzed. The output of this transducer is atime varying continuous voltage analogous to the quantity beingmeasured and a scale factor or a calibration constant will deter-mine the value of the measured quantity.

Kaman Instrumentation Displacement Measuring SystemModel KD-2310 was used to make the precision noncontact dis-placement measurements in the experiment. The system includes asensor, a 3 m coaxial cable, and a signal conditioning electronicspackage. This system uses the principle of impedance variationcaused by eddy currents induced in a conductive metal target.

The coupling between a coil in the sensor and a target is de-pendent upon their displacement or the gap between them. Theoutput voltage of the system is proportional to the distance be-tween the face of the sensor and any metallic target.

KD-2310 systems are most stable when the target is near theface of the sensor. Also, the sensitivity to the cable movement,dielectric constant, magnetic field, etc., is the greatest when thetarget is at full-scale displacement. Parallelism between the targetand the sensor is acceptable as long as it does not exceed 15 deg.The specifications of the DAQ card used in the experiment arefound in �14�. The devices can scan multiple channels at the samemaximum rate as their single-channel rate.

Experimental Results and DiscussionThe following experiments were carried out to measure the

whirl orbits of the boring bar-cutting head-workpiece interactionat different speeds of rotation, at the beginning of drilling opera-tion. A frequency response analysis is followed.

Two proximity sensors were mounted at a point where the bor-ing bar is moving at a constant velocity. Once the proximity pick-ups were installed they were calibrated for the machining opera-tion. The bipolar output calibration procedure �KamanInstrumentation manual� was carried out. The output voltageranges from a negative voltage for the first half of the measuringrange to a positive output for the second half of the range. Thesignals sensed by the sensors are routed into the data acquisitionpackage, which consists of demodulators, data acquisition card,and a computer with LABVIEW™ software to capture the signalsand analyze them. The calibration figures are shown in Figs. 6 and7. The cutting head is supported by the work piece and a smallfeed �around 30 mm/min� is provided because rotating the boring

Fig. 6 The master calibration with y=1.3787x+0.0003

bar without giving some feed will damage the cutting head. A flow



rhwa1os8

cae1i19

Ff

Fs

the slave „continuous line… signals and, „b… the whirl orbit

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ate of 40 G/min was used. The experiments are carried out atigh speeds of rotation in order to overcome the resistance of theork piece. The investigations were at the lowest speed of 1200,

nd the highest speed provided by the boring machine at440 rpm. The other speeds were recommended by the machineperator at 1280 and 1359 rpm. The experimental setup and achematic of the boring bar-workpiece system are shown in Figs.and 9.Figure 10�a� shows the master and slave signals for the lowest

utting speed of 1200 rpm. The signals from the master and slavere plotted in Fig. 10�a�. As shown in Fig. 10�b� the whirl diam-ter was approximately 1000 �m. Increasing the speed to280 rpm, the corresponding master and slave signals are shownn Fig. 11�a�. The corresponding whirl mode is shown in Fig.1�b� and seems to be more stable with a diameter equal to00 �m. At a speed equal to 1359 rpm the two signals are shown

Fig. 7 The slave calibration with y=1.3379x+0.0002

ig. 8 Part 2: Schematic representation the experiment at dif-erent speeds of rotation

ig. 9 Part 2: Picture of the experiment done at different
peeds of rotation


Fig. 11 At a speed of 1280 rpm „a… the master „dot line… and

Fig. 10 At a speed of 1200 rpm „a… the master „dot line… and

the slave „continuous line… signals, „b… The whirl orbit

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Fthe slave „continuous line… signals, „b… The whirl orbit

58 / Vol. 129, FEBRUARY 2007

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Table 4 The bar rotational speed associated with the majorwhirl ellipse amplitude for experiment part 2

Rotational speed�rpm�

Major amplitude of the whirling ellipse��m�

1200 10001280 9801359 8951440 710

ig. 12 At a speed of 1359 rpm „a… the master „dot line… andhe slave „continuous line… signals, „b… The whirl orbit

ig. 13 At a speed of 1440 rpm „a… the master „dot line… and
Fig. 14 Amplitude spectral density for a speed of „a… 1200 rpm for Z and Y, and „b… 1280 rpm for Z and Y


i1s

Ff

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n Fig. 12�a�. The whirl diameter decreased to 895 �m as in Fig.2�b�. For the maximum speed of 1440 rpm the two signals arehown in Fig. 13�a�. The whirl mode seems to be more and more

Table 6 The spectral

Spindle speed �rpm� First Second

1200 11.06 20.251280 8.43 21.921359 12.76 22.221440 11.41 24.51

ig. 15 Amplitude spectral density for a speed of „a… 1359 rpmor Z and Y, and „b… 1440 rpm for Z and Y



stable with a diameter equal to 710 �m as in Fig. 13�b�. Thereason for the reduced whirl diameter may be due to more stabil-ity when the speed is increased while cutting. Also a small feedwas given during the experiments, and hence the whirl mode wasnot measured at the same point for the four speeds. The first andsecond computed natural frequencies for the boring bar-workpiecesystem are 11.23 and 25.28 Hz, respectively. The first and secondnatural frequencies obtained experimentally were in the range of8.43–11.41 and 20–24.51 Hz, respectively. The 1200 rpm was faraway from the first natural frequency and close to the secondnatural frequency. The boring speed of 1280, 1359, and 1440 rpmwere below the second computed value and in the range of theexperimentally obtained natural frequency. Table 4 summarizesthe rotational speed with the major amplitudes of the ellipticalwhirl orbit of the boring bar.

The measured time-domain data is also converted to frequencydomain data via Fourier transform. A frequency analysis of thesignals will show the natural frequencies of the boring-bar cuttinghead assembly. Under steady state conditions, the response will bea periodic or nonperiodic process and hence can be represented bya Fourier series. If one of those frequencies is close to the reso-nant frequency of the vibrating system then the correspondingoscillations may be the dominant part of the response.

The power spectral density, �PSD�, is the amount of power perunit �density� of frequency �spectral� as a function of the fre-quency. The power spectral density, PSD, describes how thepower �or variance� of a process is distributed over the frequencyrange. Mathematically, it is defined as the Fourier transform of theautocorrelation of the process. An equivalent definition of PSD isthe squared modulus of the Fourier transform of the process,scaled by a proper constant term. The nature of the representationof the Fourier series in this way becomes clear if we think of it asa superposition of sinusoidal oscillations of all possible frequen-cies.

Frequency response at the beginning of the boring process isshown in Fig. 14�a� where there are five spectral peaks, at11.06 Hz, at 20.25 Hz, at 29.67 Hz, at 58.82 Hz and at 78.34 Hz.In Fig. 14�b� there are six spectral peaks at 8.43, 21.92, 31.12,35.34, 42, and 63.70 Hz. In Fig. 15�a� there are seven spectralpeaks at 12.76, 22.22, 32.13, 44.17, 65.76, and 307.94 Hz. In Fig.15�b� there are six spectral peaks at 11.41, 24.51, 32.91, 47.077,70.08, and 93.33 Hz. From these figures we notice that the maxi-mum spectral peak is at the first natural frequency of the system.Table 5 lists the first and second natural frequencies, both com-puted and experimentally obtained. Table 6 summarizes the spec-tral peaks for part-2. The rotational speeds were far beyond thefirst natural frequency of the boring bar-workpiece system as seen

Table 5 The first and second computed and experimentallyobtained natural frequencies for part 2

Natural frequencies ��n�Computed

�Hz�Experimentally obtained

�Hz�

�111.23 �8.43–11.41�

�225.28 �20.25–24.51�

ks for Figs. 14 and 15

Spectral peaks �Hz�

Third Fourth Fifth ¯

29.67 58.82 78.34 ¯

31.12 35.34 42 63.732.13 44.17 65.76 307.4432.91 47.077 70.08 93.33

pea

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n Tables 4 and 5. The whirl amplitude is decreasing while theotational speeds are increasing as seen in Table 4. This meanshat the rotational speeds are far beyond the second natural fre-uency of the boring bar-workpiece system. The experimentallyound natural frequency is in the range of �20.25–24.51� Hz aseen in Table 5. But because the whirling amplitude was decreas-ng this means that the second natural frequency should be around0 Hz. Other spectral peaks beyond the 20 Hz were due to somerrors in the machine such as gear noise, coolant flow rate, chipemoval rate, and errors in the measuring instruments, etc.

onclusions

A mathematical approach was developed to study the whirlingotion of a continuous boring bar-workpiece system in a BTA

eep hole boring process. This model has been used to simulatehe whirling motion at different locations of the boring bar-orkpiece system. The mathematical model to study the whirlingotion of the boring bar-workpiece assembly transformed the ho-ogenous equations with nonhomogeneous boundary condition

nto a problem with nonhomogeneous equations with homogenousoundary conditions. The whirling motion was reduced with theddition of external forces on the boring bar in the two directions.he computed fundamental natural frequency of the boring bar-orkpiece model was 11.23 Hz, which was validated experimen-

ally in the range of 8.43–11.41 Hz. The whirl amplitude was theighest at 1200 rpm and started decreasing while the speedncreased, which means that the second natural frequency isround 20 Hz.

Table 7 The values of the different

Function Description Function

G2yi*

�L1

L2v2yi�x�g2��x�dx M1

G2yi �L1

L2v2yi�x�M2

EI2g2�x�dx

M2

G3yi* �

L2

1v3yi�x�g3��x�dx M3

G3yi �L2

1v3yi�x�

M3

EI3g3�x�dx

I1

G2zi*

�L1

L2v2zi�x�g2��x�dx A1

G2zi �L1

L2v2zi�x�M2

EI2g2�x�dx

I2

G3zi* �

L2

1v3zi�x�g3��x�dx A2

G3zi �L2

1v3zi�x�

M3

EI3g3�x�dx

I3

� j4

�2Lj4Mj

�EI� j

A3

�2 ,�2 ,�2 M2

M3,L2

4

L34

,�EI�2

�EI�3

�



Nomenclature

A1 ,A2 cross section area of the boring bar and work-piece �m2�

do ,di outside and inside diameter of the boring bar�m�

dw diameter of the workpiece �m�E Young’s modulus �N/m2�

I1 , I2 area moment of cross section of the boring barand workpiece �m4�

M1 mass per unit length of the boring bar assem-bly �Kg/m�

M2 mass per unit length of the workpiece assem-bly �Kg/m�

t time �sec�W1�x , t� displacement of the boring bar to the left of

the stuffing box �m�W2�x , t� displacement of the boring bar to the right of

the stuffing box �m�W3�x , t� displacement of the workpiece �m�

� weight density of the boring bar �N/m3��t� generalized coordinate

�n natural frequency of the system �Hz�

Appendix A

A.1 Constants and Functions. See Table 7.

nstants in the mathematical model

Description Function Description

�

gA1

�1i4

�i4�1�1

�1

M2L24

�EI�2

�

gA2

�2i4

�i2 M2L2

4

�EI�2

�

gA3

�3i4

�i2�3�1�1

�3�2�1

M2L24

�EI�2

64�do

4−di4�

�1 �EI�1

�EI�2

4�do

2−di2�

�3 �EI�1

�EI�3

64�do

4−di4�

�1 A1

A2

4�do

2−di2�

�3 A1

A3

64dw

4 �1 L14

L24

4dw

2 �3 L14

L34

�� j

Lj�2� �EI� jg

�Aj �1/2

Fc

Fco

0.8 KN0.08 KN

co



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A.2 The Natural Frequency of the Boring Bar-Workpiece System

�Y� = �g11 g12 0 0 0 0 0 0 0 0

g21 g22 − g23 − g24 g25 g26 0 0 0 0

g31 g32 g33 − g34 − g35 − g36 0 0 0 0

0 0 g43 g44 g45 g46 − g47 − g48 − g49 − g410

0 0 − g53 g54 g55 g56 g57 − g58 − g59 − g510

0 0 g63 g64 g65 g66 − g67 − g68 g69 g610

0 0 g73 − g74 g75 g76 g77 − g78 g79 g710

0 0 g83 g84 g85 g86 0 0 0 0

0 0 0 0 0 0 g97 g98 g99 g910

0 0 0 0 0 0 − g107 g108 g109 g1010

�
here
g11 = cosh�C1L1a� − cos�C1L1a�, g12 = sinh�C1L1a� − sin�C1L1a�

g21 = cosh�C1L1a� + cos�C1L1a�, g22 = sin�C1L1a� + sinh�C1L1a�

g23 = cos�C2L1a�, g24 = sin�C2L1a�, g25 = cosh�C2L1a�

g26 = sinh�C2L1a�, g31 = sinh�C1L1a� + sin�C1L1a�

g32 = cosh�C1L1a� − cos�C1L1a�

g33 = sin�C2L1a�, g34 = cos�C2L1a�, g35 = sinh�C2L1a�

g36 = cosh�C2L1a�, g43 = cos�C2L2a�, g44 = sin�C2L2a�

g45 = cosh�C2L2a�, g46 = sinh�C2L2a�, g47 = cos�C3L2a�

g48 = sin�C3L2a�, g49 = cosh�C3L2a�, g410 = sinh�C3L2a�

g53 = sin�C2L2a�, g54 = cos�C2L2a�, g55 = sinh�C2L2a�

g56 = cosh�C2L2a�, g57 = sin�C3L2a�, g58 = cos�C3L2a�

g59 = sinh�C3L2a�, g510 = cosh�C3L2a�, g63 = cos�C2L2a�

g64 = sin�C2L2a�, g65 = cosh�C2L2a�, g66 = sinh�C2L2a�


g610 = sinh�C3L2a�, g73 = sin�C2L2a�, g74 = cos�C2L2a�

g75 = sinh�C2L2a�, g76 = cosh�C2L2a�, g77 = sin�C3L2a�

g78 = cos�C3L2a�, g79 = sinh�C3L2a�, g710 = cosh�C3L2a�


g86 = sinh�C2L1a�, g97 = cos�C3La�, g98 = sin�C3Lr�

g99 = cosh�C3La�, g910 = sinh�C3La�, g107 = sin�C3La�

g108 = cos�C3La�, a109 = sinh�C3La�, a1010 = cosh�C3La�

¯
n the above, the quantities Ci are


C1 = �C1M2L2

4

�EI�2�1/4

, C2 = �M2L24

�EI�2�1/4

C3 = �C2M2L2

4

�EI�2�1/4

, a = ��n

A.3 Integrals

�0

t

cos�� + ��e−��t−�� sin ��t − ��d�

= −1

2��2 + 4��− 4 sin��t + �� − 2 cos��t + ��

+ 2e−�t sin��t + �� + 4e−�t sin��t + ��

+ 2e−�t cos��t − �� + 2e−�t sin��t − ��

�0

t

e−��t−�� sin ��t − ��d�

= −1

��2 + 1��− 1 + e−�t cos��t� + e−�t sin��t��

�0

t

cos��e−��t−�� sin ��t − ��d�

= −1

��2 + 4��− 2 sin��t� − cos��t� + 2e−�t sin��t�

+ 2e−�t sin��t� + e−�t cos��t��

�0

t

sin�� + ��e−��t−�� sin ��t − ��d�

= −1

2��2 + 4��− 2 sin��t + �� + 4 cos��t + ��

+ 2e−�t cos��t − �� − 2e−�t sin��t − ��

− 2e−�t cos��t + �� − 4e−�t cos��t + ��

�0

t

sin��e−��t−�� sin ��t − ��d�

=1

��2 + 4�� sin��t� − 2 cos��t�

−�t −�t
+ e sin��t� + 2e cos��t��
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A.4 The Orthogonal Property of the Eigenfunction

�0

L1

v1yn�x�v1ym�x�dx +�L1

L2

v2yn�x�v2ym�x�dx

+�L2

1

v3yn�x�v3ym�x�dx = �nm, if m = n

�0

L1

v1yn�x�v1ym�x�dx +�L1

L2

v2yn�x�v2ym�x�dx

+�L2

1

v3yn�x�v3ym�x�dx = 0, if m � n

�0

L1

v1zn�x�v1zm�x�dx +�L1

L2

v2zn�x�v2zm�x�dx

+�L2

1

v3zn�x�v3zm�x�dx = �nm, if m = n

�0

L1

v1zn�x�v1zm�x�dx +�L1

L2

v2zn�x�v2zm�x�dx

+�L2

1

v3zn�x�v3zm�x�dx = 0, if m � n

¯ ¯ ¯
here L1=0.559, L2=0.823, and L=1.


References�1� Bayly, P., Lamar, M., and Calvert, S., 2002, “Low-Frequency Regenerative

Vibration and the Formation of Lobed Holes in Drilling,” J. Manuf. Sci. Eng.,124, pp. 275–285.

�2� Bayly, P., Metzler, S., Schaut, A., and Young, K., 2001, “Theory of TensionalChatter in Twist Drills: Model, Stability Analysis and Composition to Test,” J.Manuf. Sci. Eng., 123, pp. 552–561.

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