international journal of non-linear mechanics

International Journal of Non-Linear Mechanics 55 (2013) 170–179

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International Journal of Non-Linear Mechanics

0020-74http://d

n CorrE-m

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Numerical investigating nonlinear dynamic responses to rotatingdeep-hole drilling shaft with multi-span intermediate supports

Kong Lingfei n, Li Yan, Zhao ZhiyuanSchool of Mechanical and Instrumental Engineering, Xi'an University of Technology, Xi'an 710048, Shaanxi, PR China

a r t i c l e i n f o

Article history:Received 31 October 2012Accepted 4 June 2013Available online 14 June 2013

Keywords:Deep hole drillingModal reductionDrilling shaftNonlinear dynamic responseFinite element method

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esponding author. Tel.: +86 18629322811.ail address: [email protected] (K. Lingf

a b s t r a c t

An approach is presented to study the nonlinear dynamic responses to rotating drill shaft with multi-span supports. Based on the finite element method, the drilling shaft is modeled as lots of 2-nodeTimoshenko shaft element model with free-interface that can take the effect of inertia and shear intoconsideration. In these cases, the governing equations of drilling shaft system consist of the coupledlinear and nonlinear components. According to the feature of such systems, a modified transformation isintroduced, by means of which the linear degrees of freedom of the drilling shaft system are reducedsignificantly whereas nonlinear degrees of freedom of the system are retained in the physical space.A modified Newton shooting method is used to obtain the periodic trajectories of the dynamic system.The advantage of this method has reduced much of the computational cost in the past, and thehydrodynamic forces of cutting fluid, cutting forces and unbalance forces can easily be added to the systemequations. Further, the numerical schemes of this study are applied to a large-scale deep-hole drill machinewith two intermediate supports. The periodic dynamic behaviors of the drilling shaft system and the regionof unstable rotation are investigated numerically, whereby revealing some interesting phenomena.

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1. Introduction

Deep-hole drilling has found wide application in such fields asmetallurgical industry, nuclear power and ordnance industry.However, owing to the working environment of drilling shaft inlong and narrow hole, the motion state of observing the drillingshaft during machining is impossible. Therefore, more and moreattention has been paid to how to establish the practical model ofdrilling shaft system to calculate the movement trajectories of thedrilling shaft efficiently without losing dynamic analysis precisiondue to complexity of the problem in the last 20 years [1–5].

At present, much research work has been done on the field of thedrilling shaft dynamic in the world. Chin et al. [6,7] presented theproblem of state monitoring in deep-hole drilling by computersimulation and experiment. Perng and Deng [8,9] established thedifferent models for lateral and longitudinal shaft motion to study thetool eigenproperties of deep-hole drilling, such as numerical Euler–Bernoulli beam model, dynamic flexibility rotating beam model, andflexibility rotating beam mode with static fluid. Such simplifiedmodels do not accurately represent the practical drilling shaft system.Drilling shaft system as a rotating machine is a typical nonlineardynamic system, and hence nonlinear hydrodynamic forces of cuttingfluid do not have analytical formulation in fact. Kong et al. [10]established the concentration mass dynamic model of drilling shaft

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ei).

system with the multi-degrees but without considering the effects ofinertia and shear, and proposed a method to calculate the nonlinearhydrodynamic forces of cutting fluids and their Jacobian matrices ofcompatible accuracy simultaneously, and in addition, it was found thatthe mass eccentricity can inhibit the whirling motion of drilling shaftreported from the experimental results. Hussien et al. [11] studieddynamic drilling shaft modeling with one-span and developed amathematical approach to study the whirling motion of a continuousboring bar-workpiece system by transforming the problem intononhomogeneous equations with homogenous boundary conditions.In reality, the drilling shaft system with one or more intermediatesupports is typically a multi-part continuum. Despite the cutting forcesand the hydrodynamic forces of cutting fluid acting on a few nodalpoints of drilling shaft individually, the effect from the nonlinearity isglobal. Additionally, local nonlinearities and linear components of thedrilling shaft system are coupled.

For chatter detection, Weinert et al. [12] used dynamicalsystems to model the drilling process. They are interested in alocal description of adequate accuracy to predict disturbancessufficiently and to provide insights into how to react in order toprevent them. Therefore, they proposed a phenomenologicalapproach with a special emphasis on the temporal neighborhoodsof instabilities or state transitions from stable drilling to chattervibration and back. Messaoud [13] proposed a phenomenologicalmodel based on the Van Der Pol equation and used nonlinear timeseries modeling to set up an on-line modeling approach of thetime varying dynamics process.

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K. Lingfei et al. / International Journal of Non-Linear Mechanics 55 (2013) 170–179 171

On the other hand, the theories are mentioned in a series ofstudies concerning the dynamic responses to rotating shaft withvarious considerations such as subject to shear deformation,rotatory inertia moment, gyroscopic moment, etc. by usingTimoshenko shaft model [14–16]. Pan et al. [17] proposed amethod for dynamic simulation of multi-body systems includinglarge-scale finite element models of flexibility. Some optimallumped inertia techniques are developed in order to avoid com-putation of the coupling matrics between the rigid-body degrees-of-freedom (DOF) and flexible DOF in the finite element repre-sentation of the flexible bodies [18–20]. Hu et al. [21] establishedthe finite element model coupling the flexible-body dynamics of arotating shaft. The flexible DOF were presented as solid elementswithout modal reduction. However, using the aforementionedmethod, the numerical analysis of dynamic responses to a non-linear model with many degrees of freedom generally needs muchcomputing time and may cause computational problems. In orderto save computing cost and obtain good accuracy, Fey et al. [22]made a research on the long-term behavior of the mechanicalsystem with local nonlinearity by using the component modesynthesis technique. Hu et al. extended their study to formulatethe dynamic responses to a rotating shaft supported by a flexiblesupport structure, and the Craig–Bampton method was employedto reduce the equations of motion [23].

Since drilling shaft system in deep hole machining couple withthe cutting force, the hydrodynamic forces of cutting fluid and themass eccentricity, nonlinear dynamic behaviors of drilling shafthave rarely been discussed in particularly when they are also withone or more intermediate supports. The present work is a majorextension to our previous studies [10], and a system method isdeveloped to satisfy the dynamic design demand of the large-scalecomplex deep hole drilling machine. This paper is organized asfollows. In Section 2, the governing equations, including the effectof inertia and shear, are formulated as lots of 2-node Timoshenkoshaft element model with free-interface. By introducing a mod-ified transformation, the linear degrees of freedom of the drillingshaft system are reduced significantly while the nonlinear degreesof freedom are retained. In Section 3, according to this reducedmodel, a modified Newton shooting method is used to integratethe periodic trajectories of the reduced system. The periodictrajectories of the drilling shaft are used as numerical exampleto demonstrate and validate the proposed method in Section 4. InSection 5, the numerical schemes of this study are applied to alarge-scale deep hole drilling machine with two intermediatesupports. The periodic motion behaviors of the drilling shaftsystem and the region of unstable motion are investigatednumerically, whereby some interesting phenomena are revealed.Section 6 is concerned with the concluding remarks.

Fig. 3. The finite element model of drilling shaft.

Fig. 2. Coordinates of drilling shaft system for calculation.

2. Reduction of the model equations of drilling shaft system

2.1. Model of drilling shaft system with intermediate supports

Fig. 1 displays the boring trepanning association (BTA) drillingmachine. The machine has an external cutting fluid supply andinternal chip transport. The tool head is screwed onto the drilling

Fig. 1. Configuration of deep hole drilling

tube. The high-pressure cutting fluid is supplied through the spacebetween the drilling tube and machine hole, and then removedalong with the chip through the drill tube. The cross-section of thedrilling shaft is round.

The drill shaft is considered as a continuous flexible shaft at theshaft driver while the workpiece is fastened at its end. Hence, atypical drilling shaft system with two intermediate simple sup-ports is shown in Fig. 2. The drilling shaft is modeled as lots of 2-node Timoshenko shaft element model with eight degrees offreedom [24], considering the effects of inertia and shear as shownin Fig. 3. Using the finite element method, flexible drilling shaftequations of lateral motion can be written as

M €Xþ G _Xþ KX¼ Fu þ Fc þ fsðX; _XÞ ð1Þwhere M; G; K∈Rn�n are the mass matrix, gyroscope matrix andstiffness matrix respectively, and their specific forms are listed inAppendix A; XðtÞ∈Rn is the displacement vector. For a drilling shaftwith n nodal points, the displacement vector is of the form

X¼ ½x1; y1;ψ1;φ1;…; xn; yn;ψn;φn�T ð2Þwhere xi, yi and ψ i, φi (i¼1, 2, …, n) are the lateral translations androtation angles of ith nodal point along the horizontal and verticaldirection, respectively.

Unbalance forces: Fu∈Rn is the unbalance forces vector causedby the mass eccentricities of drilling shaft and weight forcesacting on oxy plane, and its form

Fu ¼

meex1ω2 cos ωt þmeey1ω

2 sin ωt

meey1ω2 cos ωt−meex1ω

2 sin ωt þmeg

00⋮

meexiω2 cos ωt þmeeyiω

2 sin ωt

meeyiω2 cos ωt−meexiω

2 sin ωt þmeg

00⋮

8>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>:

9>>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>>;

ð3Þ

machine for non-rotary workpiece.

K. Lingfei et al. / International Journal of Non-Linear Mechanics 55 (2013) 170–179172

where me is the element mass, exi and eyi (i¼1, 2, …, n) aremass eccentricities of the drilling shaft acting on the ith nodalpoint in x and y directions, ω is the rotating speed of the drillingshaft, and g is the acceleration of gravity.Cutting forces: Fc∈Rn is the cutting forces vector. The problemthat we focus on takes the lateral motion of drilling shaft intoconsideration. In Ref. [10], the feed rate is assumed as asmallest constant to establish the cutting forces model, andthis model reported by the experimental results is feasible.Based on above model, the force terms transmitted to thecutting head of drilling shaft equal to a period lateral fluctua-tion of a dynamic origin in terms of Eq. (4). The force equals tothe value of amplitude ðf c0Þ multiplied by the rotationalfrequency

f cx ¼ f c0 sin ωt

f cy ¼ f c0 cos ωt

(ð4Þ

Thus, the cutting forces vector is of the form

Fc ¼ ½0; ⋯; −f cx; −f cy; 0; 0�T ð5Þwhere f cx and f cy are the fluctuation of cutting forces acting onthe cutting head in the x and y directions respectively.Hydrodynamic forces of cutting fluid: fsðX; _XÞ∈Rn is the nonlinearhydrodynamic forces vector caused by cutting fluid. Due to thenonlinear hydrodynamic forces acting on the ith nodal pointinside drilling hole, thus for drilling shaft system with thelength of drilling depth (lc), the nonlinear forces vector is of thespatially localized feature as follows

fsðX; _XÞ ¼ ½0; ⋯;−f sxi xi; yi; _xi; _yi� �

; −f syi ðxi; yi; _xi; _yiÞ; 0; 0; ⋯�T ð6Þ

where f sxi and f syi (i¼1, 2, …, n) are nonlinear hydrodynamicforces of cutting fluid in the x and y directions respectively, andthe computational method of hydrodynamic forces and theirJacobians are considered in Ref. [10].Boundary condition: The boundary condition of clamp side(spindle box side in Fig. 2) of drilling shaft is assumed to befixed(namely, xðz¼ 0Þ ¼ yðz¼ 0Þ ¼ 0 and ψðz¼ 0Þ ¼ φðz¼ 0Þ ¼ 0)and that of intermediate support side is assumed to be simplysupported (namely, xðz¼ lsaÞ ¼ 0, yðz¼ lsaÞ ¼ 0, xðz¼ lsa þ lsbÞ ¼ 0and yðz¼ lsa þ lsbÞ ¼ 0).

2.2. Reduction of the dynamic equations of shaft system

The numerical analysis of the dynamic behaviors of drillingshaft system, consisting of linear components with many degreesof freedom and local nonlinearities, needs much computing timeand may cause computational problem generally. To save comput-ing cost, the Eq. (1) can be partitioned as

Maa Mab

Mba Mbb

" #€Xa

€Xb

( )þ

Gaa Gab

−GTba Gbb

" #_Xa_Xb

( )

þKaa Kab

Kba Kbb

" #Xa

Xb

( )¼

Q a

Q b

( )þ fsa Xa; _Xa

� �0

( )ð7Þ

where Q ¼ Fu þ Fc . The nonlinear degrees of freedom Xa ðXa∈Rna Þare caused by nonlinear equations. The linear degrees of freedomXb ðXb∈R

nb Þ depend on Xa ðnb⪢naÞ.For reducing the degrees of freedom of the linear components

in Eq. (7), X is written as follows

X¼ T1ηn ð8Þ

where

ηn ¼ ½ηn

a; ηn

c �T ð9Þ

T1 ¼ ½Φa; Φb� ð10Þ

In Eqs. (9) and (10), ηn ¼ ½ηn1; ηn2; ⋯; ηnm�T , ηna ¼ ½ηn1; ηn2; ⋯; ηnna �T

and ηnc ¼ ½ηnnaþ1; ηnnaþ2; ⋯; ηnm� T . The columns of matrix Φb∈R

n�nc

with kept elastic eigenmodes are the mass normalized solutionsð−ω2

j Mþ KÞ Φj ¼ 0 (j¼1, 2, …, nc) of the undamped eigenproblem

for ωj∈½0;ωcutÞ[25]. The columns of the matrix Φa∈Rn�na with theresidual flexibility modes are defined in the following

Φa ¼ ½K−1−ΦbΩ−2cc Φ

Tb �

Iaa0ba

" #ð11Þ

where Iaa∈Rna�na , 0ba∈Rnb�na are the identity matrix and the zero

matrix respectively. The matrix Ωcc∈Rnc�nc is a diagonal matrixwith the kept angular eigenfrequencies lower than or equal to ωcut

[26]. Then

X¼ T1ηn⇒Xa

Xb

( )¼

Φaa Φac

Φba Φbc

" #ηna

ηnc

( )ð12Þ

where

Φaa∈Rna�na ; Φac∈Rna�nc ; Φba∈Rnb�na ; andΦbc∈R

nb�nc :

Starting from Eq. (12), ηna and ηn

c can be written as

ηna

ηnc

( )¼ Φ−1

aa −Φ−1aaΦac

0 Icc

" #Xa

ηnc

( )⇒ηn ¼ T2η ð13Þ

This results in the following total transformation

X¼ T1T2η¼ Tη ð14Þ

X¼ Tη⇒Xa

Xb

( )¼

Iaa 0ΦbaΦ−1

aa Φbað−Φ−1aaΦacÞ þΦbc

" #Xa

ηnc

( )ð15Þ

where η∈Rm. It is evident in Eq. (15) that the number of nonlineardegrees of freedom Xa is not changed, and so the influences arewholly retained.

Applying the transformation Eq. (15), the reduced componentequations become

TTMT €ηþ TTGT _ηþ TTKTη¼ TTQ þ TT fs ð16Þ

After reduction, the n (¼na+nb) order equations of the drillingshaft system are reduced to m (¼na+nc) order equations. Thehydrodynamic forces of cutting fluid, cutting forces and unbalanceforce can be easily added to the reduced governing equations.From Eqs. (12)–(16), it is evident that the external force vectors ofthe drilling shaft and nonlinear effects definitely remain in thereduced Eq. (16).

The equation of motion of the reduced shaft system is given by

Mn €ηþ Gn _ηþ Knη¼ F ð17Þ

where

Mn ¼ TTMT; Gn ¼ TTGT; Kn ¼ TTKT; and F¼ TTQ þ TT fs:

When state variables q¼ ½ηT ; _ηT �T ðq¼ ½q1; q2;…q2m�T Þ areintroduced, the corresponding system equations in state spaceare written as Eq. (18), making integration of the system responseeasy to be done by modified traditional shooting method

_q¼_η

Mn−1ðF−Gn _η−KnηÞ

( )ð18Þ


3. Integration of the reduced model equations by modified thetraditional shooting method

The transient responses to the reduced system should beintegrated step by step numerically. In order to guard the generalityof the proposed method, Eq. (18) can be written as follows

_q¼_η

Mn−1ðF−Gn _η−KnηÞ

( )⇒ _q¼ fðq; t; αÞ ð19Þ

where t∈R is time. α∈R is the design parameter of drilling shaft.It is supposed that a periodic trajectory of the drilling shaft

system in Eq. (19) exists and its period is T . In order to show theperiod T explicitly in the equation, the system in Eq. (19) istransformed into Eq. (20) by using t ¼ Tτ and then

dqdτ

¼ T fðq; Tτ; αÞ ð20Þ

Obviously, the period of the periodic trajectories of the system inEq. (20) is changed to 1.

The initial values of q0i ¼ βi and T0 (i¼1, 2, …, 2m) are selected,

where q0(q0 ¼ ½q01; q02;…q02m�T ) denotes the state vector at time

τ¼ 0. Using the initial values q0i and T0, we integrate the systemEq. (20) from τ¼ 0 to τ¼ 1. Then, the value of q1 is obtained, inwhich q1ðq1 ¼ ½q11; q12;…q12m�T Þ denotes the state vector at τ¼ 1.Obviously, the value of q1 depends on the initial values, and q1 is a

function of βðβ¼ ½β1; β2;…β2m�T Þ and T0, namely q1ðβ; T0Þ.A periodic trajectory is found if the following criterion is

satisfied.

riðβ; T0Þ ¼ q1i ðβ; T0Þ−q0i ¼ q1i ðβ; T0Þ−βi ¼ 0 ð21Þ

where ri∈R (i¼1, 2, …, 2m) is the value error. If the chosen initialvalues of q0i and T0 just satisfy the above criterion, the periodictrajectory of the system is found. In other words, the selectedinitial values of q0i and T0 are the exact solutions. However, it isalmost impossible in practice. To obtain the initial values of q0i andT0, the traditional shooting method is modified by the iterationprocedure in detail as follows.

Step 1: the value error riðβ; T0Þ is calculated based on Eq. (22)after integrating from τ¼ 0 to τ¼ 1.

r1ðβ; T0Þ ¼ q11ðβ; T0Þ−β1r2ðβ; T0Þ ¼ q12ðβ; T0Þ−β2

⋮r2mðβ; T0Þ ¼ q12mðβ; T0Þ−β2m

8>>>><>>>>:

ð22Þ

Step 2: In order to obtain the iteration increment during theiteration process, riðβ; T0Þ is expanded to a Taylor series near toβi and T0, and retain the linear terms as

r1 þ ∂r1∂β1

Δβ1 þ ∂r1∂β2

Δβ2⋯þ ∂r1∂β2m

Δβ2m þ ∂r1∂T0 ΔT0 ¼ 0

r2 þ ∂r2∂β1

Δβ1 þ ∂r2∂β2

Δβ2⋯þ ∂r2∂β2m

Δβ2m þ ∂r2∂T0 ΔT0 ¼ 0

⋮r2m þ ∂r2m

∂β1Δβ1 þ ∂r2m

∂β2Δβ2⋯þ ∂r2m

∂β2mΔβ2m þ ∂r2m

∂T0 ΔT0 ¼ 0

8>>>>><>>>>>:

ð23Þ

Step 3: In Eq. (22), the partial derivative of riðβ; T0Þ with respectto βj and T0 is found. The following formulas can then beobtained.

∂ri∂βj

¼ ∂q1i∂βj

−∂βi∂βj

¼ ∂q1i∂βj

−ξiji; j¼ 1; 2;…; 2m ð24Þ

∂ri∂T0 ¼ ∂q1i

∂T0 i¼ 1; 2;…; 2m ð25Þ

where ξij is the Kronecker symbol, namely

ξij ¼1 i¼ j

0 i≠j

(ð26Þ

Step 4: To obtain ð∂q1i =∂βjÞ and ð∂q1i =∂T0Þ in Eqs. (24) and (25),the derivative of Eq. (20) with respect to βj and T0 is derived,then the following formulas can be obtained

ddβj

dqdτ

� �¼ T0 dfðq; T0τ; αÞ

dβjð27Þ

d

dT0

dqdτ

� �¼ dðT0fðq; T0τ; αÞÞ

dT0 ð28Þ

By changing the derivation orders of Eqs. (27) and (28), theequations become

ddτ

dqidβj

!¼ T0 ∑

2m

s ¼ 1

∂f iðq; T0τ; αÞ∂qs

dqsdβj

" #; i; j¼ 1; 2; …; 2m ð29Þ

ddτ

dqidT0

� �¼ f iðq; Tτ; αÞ þ T0 ∂f iðq; T0τ; αÞ

∂T0

þT0 ∑2m

s ¼ 1


dqsdT0

!i¼ 1; 2;…; 2m ð30Þ

Let ðdqi=dβjÞ ¼ ϑij and ðdqi=dT0Þ ¼ ϑiT , then the initial valueproblem of an ordinary differential equation can be formed asfollows:

dϑijdτ ¼ T0 ∂f iðq;T0τ;αÞ

∂q1ϑ1j þ ∂f iðq;T0τ;αÞ

∂q2ϑ2j þ⋯þ ∂f iðq;T0τ;αÞ

∂q2mϑ2mj

h idϑiTdτ ¼ f iðq; T0τ;αÞ þ T0 ∂f iðq;T0τ;αÞ

∂T0 þ T0 ∑2m

s ¼ 1


ϑiT

!8>>><>>>:

i; j¼ 1; 2; …; 2m ð31Þ

By selecting the initial values of q0i ¼ βi and T0, the initial condition

in Eq. (31) can be discovered that ϑ0ij ¼ ξij ( the Kronecker symbol)

and ϑ0iT ¼ 0. Consequently, the values of ϑ1ij ¼ ðdq1i =dβjÞ and

ϑ1iT ¼ ðdq1i =dT0Þ can be obtained in Eq. (31) integrating fromτ¼ 0 to τ¼ 1. At the same time, the system in Eq. (20) should beintegrated during the iteration procedure to calculate the value of

ð∂f iðq; T0τ; αÞ=∂qjÞ (i, j¼1, 2, …, 2m) in Eq. (31).

Step 5: Next, the values of ð∂q1i =∂βjÞ and ð∂q1i =∂T0Þ calculatedabove are substituted into Eqs. (24) and (25) in Step 3. Then, the

values of ð∂ri=∂βjÞ and ð∂ri=∂T0Þ can be obtained.

Step 6: Then, the values of ð∂ri=∂βjÞ and ð∂ri=∂T0Þ are substitutedinto Eq. (23) in Step 2. It is noticed that a set of the 2m-orderlinear equations with 2m+1 variables will be formed

(Δβ1; Δβ2; … Δβ2m; ΔT0). In order to solving the values of

Δβ1; Δβ2; …; Δβ2m and ΔT0, one variable must be fixed. Hence,the optimal method is depicted in detail as follows. In ri, if thevalue of rλ is the least, the initial value of βλ will be selected as afixed value because it is the closest to that of the actual periodictrajectory of the system. In other words, the column corre-sponding to the least rλ should be removed from the coefficient

matrix of Eq. (23) at next iterative process. However, if ΔT0 is

the least in ri, the column corresponding to ΔT0 cannot be


removed because the period T0 of the periodic trajectories ofthe system is certain.Step 7: Finally, if riðβ; T0Þ satisfies the criterion in term ofEq. (21), the contrary transform τ¼ t=T can be made. Thus,the periodic trajectory of the reduction system in Eq. (19) canbe obtained. Otherwise, let q0i ¼ βi þ Δβi (i¼1, 2, …, 2m) andT0 ¼ T0 þ ΔT0, and repeat these procedures from Step 1 untilthe precision required is satisfied.

4. Validation of the proposed method

To validate the proposed method, a computer program hasbeen developed to compute the nonlinear dynamic behaviors of a

Fig. 4. The trajectories of the drilling shaft center at #9 station for ω¼905 r/min: (a) comand full DOF model; (b) comparsion of the trajectories of the drilling shaft center by th



drilling shaft system with two intermediate supports. The drillingshaft is discretized into 8 finite elements with 9 nodal points(shown in Fig. 2). The corresponding parameters are as follows:the length of the drilling shaft is l¼4000 mm, the inner diameterof the drilling shaft is 19 mm, the external diameter of the drillingshaft is 26 mm, drilling depth is lc¼427 mm, the diameter ofdrilling hole is 28 mm, mass eccentricities of the drilling shaftacting on the ith nodal point inside drilling hole (ex¼ey¼11.2 μm)and other points outside drilling hole (ex¼ey¼10 μm) have thesame rotating phase angle. Based on definition of cutting forces inEq. (4), the value of fluctuation amplitude is f c0¼0.067 KN and thedisplacements of intermediate supports are lsa ¼ lsb ¼ ðl−lcÞ=3 inthis paper. The specifications of other parameters are given inAppendix B.

parsion of the trajectories of the drilling shaft center by the 12-eigenmodes modele modified shooting method and Runge–Kutta method.




By using the presented method, the stable periodic trajectoriesof the drilling shaft system are shown in Figs. 4–7. Figs. 4–7(a) showthe comparsion of the trajectories of the drilling shaft center by thereduction model and full degrees of freedommodel for ω¼905 r/min.Simultaneously, Figs. 4–7(b) are presented in contrast with the4th order Runge–Kutta method. It is shown that the periodicresponse to the drilling shaft system has enough accuracy when the12-eigenmodes model is used, and the influence of the solutionmethod on the accuracy of results is validated. Additionally, the stableperiodic of drilling shaft indicates that the drilling tool is suppliedlowly by energy from whirling vibration, and the high-precision holecan be obtained in these parameters [27].

5. Numerical examples and discussion

The drilling shaft system with intermediate supports is typicallya multi-part continuum with local nonlinearity. The nonlinearforces act only on the ith nodal point inside drilling hole. In orderto test the method mentioned above more stably, the nonlinear

Fig.7. The trajectories of the drilling shaft center at #6 station for ω¼905 r/min: (a) comand full DOF model; (b) comparsion of the trajectories of the drilling shaft center by th

Fig. 8. The period-doubling trajectories of the drilling shaft center for ω¼1010 r/m

dynamic responses to drilling shaft system with two intermediatesupports are analyzed. Taking the drilling shaft length asl¼3200 mm, inner diameter as 19 mm, external diameter as35 mm, drilling depth as lc¼400 mm, the diameter of drilling holeas 28 mm, rotating speed as ω¼1010 r/min, mass eccentricities asex¼ey¼7.2 μm (acting on all nodal point), the leading Floquetmultiplier −1.0943 is calculated. Based on Floquet theory [28](a real eigenvalue on the negative real axis), the periodic trajectoriesof the drilling shaft is a period-doubling motion, as shown in Fig. 8.The period-doubling motion of cutting tool in drilling processimplies that the work-piece surface gives a regeneration nature tothe multi-lobe formation. Furthermore, because of the period-doubling motion of the drilling tool, the edge of tool repeats cuttingwork-piece twice, and makes the roundness error of the holeincrease under the fixed direction of surface, due to the circle-land cutting more [27,29].

Taking rotating speed ω as bifurcation parameter, the move-ment performance of drilling shaft system is a quasi-periodicmotion, as shown in Fig. 9. Thus, the movement trajectories ofthe drilling shaft presented obviously the asymmetrical beat


in: (a) at #9 station; (b) at #7 station; (c) schematic diagram of lobing hole.

Fig. 9. The quasi-periodic trajectories of the drilling shaft center for ω¼1152 r/min: (a) at #9 station; (b) at #7 station; (c) poincaré map on x–y plane of the drilling shaftcenter at #9 station (torus attractors); (d) poincaré map in the x–y plane of the drilling shaft center at #7 station (torus attractors); (e) schematic diagram of waviness hole.

Fig. 10. The unstable trajectories of the drilling shaft center for ω¼1531 r/min, lc¼600 mm, and ex¼ey¼7.2 μm: (a) at #9 station; (b) at #7 station.


phenomenon and the sub-harmonic fractional frequency vibrationfor ω¼1152 r/min. This occurs because the effect of drill wander-ing with the larger energy of whirling vibration reduces themovement stability of drilling shaft during the workpiece drilling,due to the fluctuation of cutting force, the disturbance of thehydrodynamic forces of cutting fluid and unbalanced forces.The quasi-periodic motion of drilling tool directly generates the

instantaneous changes of cutting thickness under the hole surface,and forces the circle-land to cut more or less. So the hole profile isalmost wave formation [27,30].

Taking rotating speed as ω¼1531 r/min and drilling depth aslc¼600 mm, the unstable motion of the center of the drilling shaftsystem is shown in Fig. 10. In comparing the movement trajec-tories of the drilling shaft center at #9 stations with that at #7

Fig. 11. The chaotic trajectories of the drilling shaft center for ω¼1531 r/min, lc¼600 mm and ex¼ey¼5.1 μm: (a) at #9 station; (b) at #7 station; (c) the time series at #9station in the y direction; (d) the power spectrum at #9 station in the y direction.

Fig. 12. The region of the rotating state of drilling tool.


stations, the difference of movement trajectories of the drillingshaft center is clearly visible. Considering the same rotating speedand drilling depth, only while the mass eccentricities of drillingshaft are ex¼ey¼5.1 μm, the chaotic motion of the center of thedrilling tool appears as shown in Fig. 11. Fig. 11(d) shows that thepower spectrum of y of shaft center at #9 station reveals the maincharacteristics of chaotic spectrumwith low energy, wideband andunrepeatable. The occurrence of these phenomena above caneasily lead to the unstable motion of the drilling shaft coupledbending and torsional vibration, because of the unsynchronizationmovement of the drilling shaft. Consequently, the drilling tool iseasily subjected to be broken, worn and fractured, as well as thehole surface is scratched.

Moreover, it is seen from Figs. 8 and 9(a) that the higherrotating speed yields the bigger whirling magnitude. So the biggerroundness error will be generated. This trend is in agreement withthe experimental results [29]. In addition, the regularity of thewhirling motion at #9 station in Fig. 10 is better than in Fig. 11.This is because the mass eccentricity can inhibit the whirlingmotion of drilling shaft [10].

In the actual deep hole drilling process, the biggest concern ofthe machine operator is: the working state of the drill shaft isstable or unstable, and how to select the rotating speed para-meters under the different drilling depths for improving thequality of hole surface. Hence, in order to guard the methodmentioned above more usefully, the stable or unstable mode of therotating state of drilling shaft are analyzed. Taking the drillingshaft length as l¼2000 mm, inner diameter as 26 mm, externaldiameter as 35 mm, the diameter of drilling hole as 37 mm, masseccentricities as ex¼ey¼13 μm (acting on all nodal point) and thesame rotating phase angle, the stable region of the drilling tool indrilling hole process is obtained, as shown in Fig. 12. If the cuttingparameters are selected in A region, the nonlinear vibration ofdrilling tool is the stable periodic motion. Thus, the higher-precision hole can be obtained and the working life of drilling

tool is extended in this parameter area. However, if selecting thehigher rotating speeds under the different drilling depth, such asin the B or C region, the motion states of drilling shaft are variedfrom stable periodic motion to period-doubling motion or quasi-periodic motion. Consequently, the roundness error of drilling holebecomes larger and makes the forms of hole as multi-lobe orwaviness. The reason for this is the shaft whirling at higher speeds.Further, if selecting the rotational speeds in D region, the move-ment motion of the drilling tool is unstable motion. These caneasily lead to the drilling tool broken, worn or even to the machinedamaged. In the actual drilling, the parameters in D region arealways not allowed to be selected. Additionally, the stable area ofdrilling shaft is decreasing with the drilling depths increasing. Thisis because the longer shaft inside hole is less rigid, and thus, asmaller disturbance from excitation forces, such as the cuttingforces or the hydrodynamic forces, will easily lead to the unstablemotion of drilling tool.


6. Conclusions

A new mathematical approach is developed to calculate thenonlinear dynamic behaviors of drilling shaft with multi-spansupports. The variational problem of lateral vibration of drillingshaft is deduced and the finite element model of lateral vibrationof drilling shaft is established by using Timoshenko elementmodel. The shear deformation and rotary inertia are taken intoaccount in the model of drilling system. Mode synthesis techniquewith free-interface is modified to represent a reduced method ofdegrees-of-freedom. The practical flexible drilling shaft systemwith multi-supports is used as subject to test the presentedreduced method. The results show that the presented reducedmethod can reduce a large number of degrees-of-freedom offlexible drilling shaft system, effectively saving the computationalcosts under the condition of maintaining nonlinear analysis pre-cision for the system.

The numerical schemes of this study are applied to a large-scaledeep-hole drill machine with two intermediate supports. Theperiodic dynamic behaviors of the drilling shaft system and thestable region of rotation were detected with the present method.The results not only reveal many interesting phenomena of drillingshaft system with intermediate supports, but also show thatalgorithm is very stable and effective.

Acknowledgments

This work is supported by National Natural Science Foundationof China (Grant nos. 51105305 and 51075327), the Major ResearchProgram of Shaanxi Province of China (13115 Project, Grant no.2009ZDKG-25), Natural Science Foundation of Shaanxi Provinceof China (Grant no. 2011JQ7012) and Natural Science Foundation ofDepartment of Education of Shaanxi Province of China (Grant nos.2010JK695 and 12JK680).

Appendix A

The lateral translations and rotation angles of a typical pointwithin the element can be related to the nodal displacementvector Xe which is of the form

Xe ¼X1

X2

( )¼ ½x1; y1;ψ1;φ1; x2; y2;ψ2;φ2�T

By using Timoshenko shaft element model, the translationaland rotational function can be written respectively as

xðz; tÞyðz; tÞ

( )¼

N1 0 0 N2 N3 0 0 N4

0 N1 N2 0 0 N3 N4 0

" #Xe ¼ ½Nt �Xe

ψðz; tÞφðz; tÞ

( )¼

0 Γ1 Γ2 0 0 Γ3 Γ4 0Γ1 0 0 Γ2 Γ3 0 0 Γ4

" #Xe ¼ ½Γr �Xe

where ½Nt �, ½Γr �, N1, N2, N3, N4, Γ1, Γ2, Γ3, Γ4 are writtenrespectively as

Nt½ � ¼Ntx

Nty

" #; Γr½ � ¼

Γrψ

Γrφ

" #;

N1 ¼1

1þ ϕð1þ ϕ−ϕζ−3ζ2 þ 2ζ3Þ;

N2 ¼l

1þ ϕ1þ ϕ

2

� �ζ−

4þ ϕ

2ζ2 þ ζ3

�

N3 ¼1

1þ ϕðϕζ þ 3ζ2−2ζ3Þ; N4 ¼

l1þ ϕ

−ϕ

2ζ þ ϕ−2

2ζ2 þ ζ3

�

Γ1 ¼ −6

ð1þ ϕÞ lðζ−ζ2Þ; Γ2 ¼

11þ ϕ

1þ ϕ−ð4þ ϕÞζ þ 3ζ2 �

Γ3 ¼6

ð1þ ϕÞ lðζ−ζ2Þ; Γ4 ¼

11þ ϕ

ðϕ−2Þζ þ 3ζ2 �

where ϕ¼ ð12EI=κ AGl2Þ, ζ¼ ðz=lÞ. E is the Young's modulus, I is themoments of inertia, κ is the shear coefficient, G is the shearmodulus and A is the cross-sectional area of the drilling shaft.

The kinetic energy Te of the shaft element can be given by

Te ¼ 12_XeTMe

t_Xe þ 1

2_XeTMe

r_Xe þ 1

2Ieω2− _X

eTSe _X

e

where

Met ¼

Z l

0ρA½Nt �T ½Nt �dz; Me

r ¼Z l

0Jd½Γr �T ½Γr�dz; Ie

¼Z l

0Jpdz; S

e ¼ ω

Z l

0Jp½Γrφ�T ½Γrφ�dz:

ρ is the mass density of the shaft material, Jd and Jp are thediametral and polar mass moments of inertia of shaft per unitlength.

The potential energy Ve of the shaft element can be given by

Ve ¼ 12XeT Ke �

Xe

where ½Ke� ¼ ½Kbe� þ ½Ksh�, ½Kbe� and ½Ksh� are the elastic stiffnessmatrix and shear stiffness matrix respectively, which are of theforms

Kbe½ � ¼Z l

0EI

∂½Γr �∂z

� �T ∂½Γr�∂z

� �dz

½Ksh� ¼Z l

0κAG

−ð∂½Nty�=∂zÞ−½Γrψ �−ð∂½Ntx�=∂zÞ þ ½Γrφ�

" #T −ð∂½Nty�=∂zÞ−½Γrψ �−ð∂½Ntx�=∂zÞ þ ½Γrφ�

" #dz

The equations of motion for the complete drilling shaft systemcan be obtained by assembling the contribution of each compo-nent equation of motion, namely T¼ ΣNseg

i ¼ 1Tei and V¼ ΣNseg

i ¼ 1Vei , and

the final equations can be expressed in the form as

M €Xþ G _Xþ K X¼ Fu þ Fc þ fsðX; _XÞ

Appendix B

The relevant calculation parameters are used:

Young's modulus (E): 206�109 PaDensity ðρÞ: 7.87�103 kg/m3

Shear modulus (G): 81�109 PaDynamic viscosity of cutting fluid: 0.026 Pa sPressure of cutting fluid: 2�106 Pa.

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