Research Article Dynamics Analysis of Unbalanced Motorized Spindles Supported on Ball Bearings

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1 Shock and Vibation Volume 216, Aticle ID , 1 pages Reseach Aticle Dynamics Analysis of Unbalanced Motoized Spindles Suppoted on Ball Beaings Junfeng Liu, 1 Tao Lai, 1 and Xiaoan Chen 2 1 College of Mechatonic Engineeing and Automation, National Univesity of Defense Technology, Changsha 4173, China 2 The State Key Laboatoy of Mechanical Tansmission, Chongqing Univesity, Chongqing 444, China Coespondence should be addessed to Junfeng Liu; ljf @163.com Received 21 Januay 216; Revised 1 August 216; Accepted 16 Octobe 216 Academic Edito: Gianluca Gatti Copyight 216 Junfeng Liu et al. This is an open access aticle distibuted unde the Ceative Commons Attibution License, which pemits unesticted use, distibution, and epoduction in any medium, povided the oiginal wok is popely cited. This pape pesents an impoved dynamic model fo unbalanced high speed motoized spindles. The poposed model includes a Hetz contact foce model which takes into the intenal cleaance and an unbalanced electomagnetic foce model based on the enegy of the ai magnetic field. The nonlinea chaacteistic of the model is analysed by Lyapunov stability theoy and numeical analysis to study the dynamic popeties of the spindle system. Finally, a dynamic opeating test is caied out on a DX1A- 24/2-type motoized spindle. The good ageement between the numeical solutions and the expeimental data indicates that the poposed model is capable of accuately pedicting the dynamic popeties of motoized spindles. The influence of the unbalanced magnetic foce on the system is studied, and the sensitivities of the system paametes to the citical speed of the system ae obtained. These conclusions ae useful fo the dynamic design of high speed motoized spindles. 1. Intoduction Duetothecomplexityoftheeliabilityandpefomance poblems caused by pofound high speed effects [1 3], the machine tools capable of achieving the high speed cutting wee fist intoduced commecially in 198s, which lags behind the theoy of high speed metal cutting by 5 yeas [4]. Motoized spindles ae the coe components of CNC machine tools and geatly impove the cutting speed, while the inevitable unbalance of the spindle system caused by unexpected eos duing manufactuing and installation pocess stongly influences the machining poductivity and finish quality of wokpieces in high speed opeating conditions [5 11]. Theefoe, it is necessay to deeply study the dynamic popeties of unbalanced machine tool motoized spindle systems. With the low-fiction and the convenience in commodity standadization, ball beaings ae widely used in high speed motoized spindles [12]. Beaing cleaance concens the contact state between the balls and the inne and oute ings and affects the vibation esponse of the oto system [13]. Hasha pesented a model fo studying the stuctual vibations of ball beaing-oto systems due to adial intenal cleaance [14]. Sopanen and Mikkola poposed a dynamic model of a ball beaing-oto system with six degees of feedom to study the effects of the beaing cleaance on the natual fequencies and the vibation esponses ofthesystem[15,16].basedonit,gaofoundthatappopiate negative intenal cleaance helps to educe the vibation esponses of the spindle and impove the stability of the system [17, 18]. Beaing peload could keep the beaing in a negative intenal cleaance state by eliminating the cleaance, but too lage peload will poduce much fictional heat, leading to the ise of tempeatue and the decline of beaing life [19]. With the use of the Tansfe Matix Method, Jiang and Mao investigated the vaiable optimum peload fo a machine tool spindle [2]. Alfaes and Elshakawy constucted a mathematical model based on five degees of dynamic system to study the influences of axial peload of angula ball beaings on the vibation behavio of a ginding machine spindle and calculated the optimum initial axial peload fo the system [21]. Seies of eseaches on the dynamic pefomance and the optimal design method of the unbalanced oto-ball beaing system of motoized spindles wee deeply discussed in above woks. Motoized spindles ae equipped with built-in motos and thei output ends ae diectly connected with tools, and

2 2 Shock and Vibation Font beaings Stato Spindle y x 1 Roto Cutting tool Rea beaings Figue 1: A typical high speed motoized spindle system. o c e Φ o 1 x m m m k 1 c 1 k 2 c 2 Figue 3: Ai gap eccenticity of the oto in motoized spindles. k bx1 c bx1 k by1 c by1 e k bx2 c bx2 k by2 c by2 Figue 2: The simplified model of a motoized spindle system. the oto of the system is acted upon by an additional unbalanced magnetic foce except the unbalanced centifugal foce. The unbalanced magnetic foce is caused by the nonunifom ai gap between the stato and the oto in opeation and affectsthedynamicchaacteisticsofthesystem.inviewof this fact, this pape poposes the matix expession of the magnetic stiffness based on the gap magnetic field enegy. Then a dynamic model fo high speed motoized spindles is established with the consideation of unbalanced magnetic foce, centifugal foce, and nonlinea Hetz contact foce. The vibation esponse and citical speed of the system ae systematically calculated and expeimentally validated. Finally, the sensitivities of the magnetic, mechanical, and beaing paametes to the citical speed ae pesented to instuct the dynamic design of unbalanced high speed motoized spindles. 2. Dynamic Model of Unbalanced Motoized Spindle System Figue 1 shows a typical motoized spindle system which includesastato,aspindleshaft(oto),acuttingtool,a tool holde, and beaing components. The system can be simplified to thee disc elements. Consideing the motions of the discs along the vetical and hoizontal diections, the spindle-beaing system is egaded as a system with six degees of feedom, as shown in Figue 2. Specifically, the font and ea beaings ae teated as spings and dampes in paallel connection, and the magnetic foce is teated as spings between the stato and the oto Model of Unbalance Magnetic Foce. The gap eccenticity of the spindle is shown as in Figue 3, o is the inne cicle cente of stato (the oigin of coodinates), o 1 is the oute cicle cente of oto, and oo 1 =(x 2 +y 2 ).5. c is the centoid of oto, e is the mass eccenticity of oto, α is the angle between the x-axis and the cicumfeential position whee the ai gap width is equal to δ and β is the angle between the xaxis and the cicumfeential position whee the ai gap width is the least. The built-in moto in high speed machine tool spindle system is geneally a thee-phase AC induction moto, so the enegy of the gap magnetic field of the moto is [22] W= R gl g 2 Λ ε n cos n (α β) n= [F sm cos (ω f t pα) +F m cos (ω f t pα φ 1 φ 2 )] 2 dα. Specifically, the lagest component, which is n =, plays a dominant ole; othe components, which ae n =, take diffeent popotion, espectively. Because the effective elative eccenticity ε is fa less than 1, expand the high ode temsin(1)andtakethefisttheetems,and(1)canbe conveted into W= R gl g Λ 2 + x2 y 2 (1+ x2 +y 2 + x σ cos α+y sin α σ cos 2α + xy sin 2α) σ2 [F sm cos (ω f t pα) +F m cos (ω f t pα φ 1 φ 2 )] 2 dα, whee R g is the adius of the inne cicle of stato; L g is effective length of oto; Λ is the magnetic conduction of unifom ai gap, and Λ =μ /k μ δ, μ is the ai pemeability, k μ is the satuation, and k μ = 1 + δ Fe /k δ, k is the calculation coefficient of unifom ai gap, δ Fe is the equivalent ai gap of feomagnetic mateials, δ is the width of unifom ai gap, σ=k μ δ, ω f is the speed of otating magnetic field, and ω f =f/p, f is the cuent fequency, p is the numbe of pole-pais, and F sm and F m ae the amplitudes of theephase esultant magnetomotive foces of the stato and the oto, espectively; φ 1 +φ 2 isthephaseangleinwhichtheoto (1) (2)

3 Shock and Vibation 3 cuent lags behind the stato cuent, and φ 1 =x 1s / 1, φ 2 = x 2s / 2, x 1s and x 2s aetheleakageeactanceofpephase winding of the stato and the oto, espectively; 1 and 2 aetheesistancesofpephasewindingofthestatoandthe oto, espectively. Equation (2) can be expessed by W = 1 2 VT K e V + V T K e, (3) whee V ={x,y} T and K e is the magnetic stiffness matix of the gap magnetic field: K 11 = R gl g Λ [F sm cos (ω f t pα) {[1+cos (2α)] +F m cos (ω f t pα φ 1 φ 2 )] 2 }dα K 12 =K 21 = R gl g Λ [F sm cos (ω f t pα) {[sin (2α)] +F m cos (ω f t pα φ 1 φ 2 )] 2 }dα K 22 = R gl g Λ [F sm cos (ω f t pα) {[1 cos (2α)] +F m cos (ω f t pα φ 1 φ 2 )] 2 }dα. K e is the vecto of the gap magnetic field enegy: K 1 = R gl g Λ {cos α [F sm cos (ω f t pα) +F m cos (ω f t pα φ 1 φ 2 )] 2 }dα K 2 = R gl g Λ {sin α [F sm cos (ω f t pα) +F m cos (ω f t pα φ 1 φ 2 )] 2 }dα Hetz Contact Foce Model of Ball Beaings. In the otoball beaing system of a high speed motoized spindle, the inne and oute aces of the beaings ae fixed to the otating shaft and the beaing housing, espectively. Balls can be consideed to make pue olling movement between the inne ace and the oute ace with equal spacing. Theefoe fo the ball beaing in any time, the elastic defomation associated with the contact point at the jth ball depends on the displacement of oto and the initial beaing cleaance, given by (4) (5) Δ j =x b sin θ j +y b cos θ j γ, (6) whee x b and y b ae the displacements at the beaing along x-andy-diections, espectively; θ j is the azimuth of the jth ball, and θ j =(/N b )(j 1) + (d i /(d i +d o ))ωt, N b is the numbe of balls, d i is the inne diamete of inne ace, and d o is the oute diamete of oute ace; γ is the initial beaing cleaance, which can be detemined accoding to the initial beaing peload [15]. Consideing that the contact foce is nonnegative, the local Hetz contact foce and deflection elationship fo beaing can be witten as [18] F θj = { k b Δ 2/3 j ; Δ j > { ; Δ { j. k b is contact stiffness which is given as 1 k b ={ (ki ) 2/3 +(k } o )2/3 3/2 (7), (8) whee k i =k o =(2 2/3) (E/(1 μ))(σρ) /2 (δ ) 3/2, E is the elastic modulus, and μ is the Poisson atio. The cuvatue sum Σρ and defomation coefficient δ ae obtained fom Hais [23]. Similato[24],thecontactdampingcanbewittenas c b =1 5 k b. (9) By summing the local contact foces, the total estoing foce components of the ball beaing along the x- andy- diections ae given by F bx = c b x b F by = c b y b N b j=1 N b j=1 F θj sin θ j ; F θj cos θ j. (1) 2.3. Motional Diffeential Equations and Stability Analysis of the Spindle System. As the middle pat of the spindle system is simplified to a disc element, the gap eccenticity of the moto is consideed as unifom. Without egad fo the influence of gyoscopic moment, the diffeential equations govening motions of the motoized spindle systems can be obtained with Newton second law: m m m m x +c 1 ( x x )+k 1 (x x )= N 1 F bx1 y +c 1 ( y c )+k 1 (y y )= m g N 1 F by1 x +c 1 ( x x )+c 2 ( x x )+k 1 (x x ) +k 2 (x x )+K 11 x +K 12 y =F cx K 1 y +c 1 ( y )+c 2 ( y )+k 1 (y y ) +k 2 (y y )+K 21 x +K 22 y =F cy m g K 2

4 4 Shock and Vibation m m x +c 2 ( x x )+k 2 (x x )= N 2 F bx2 y +c 2 ( y )+k 2 (y y )= m g N 2 F by2, (11) whee x and y, x and y,andx and y ae the displacements of mass elements distibuted at the font beaing, oto, and the ea beaing along x- and y-diections, espectively.itshouldbenoticedthatthevaiabley is defined along the opposite diection of the gavity. m, m,andm ae the masses of the spindle distibuted at the font beaing, middle between the two sets of beaings, and the ea beaing, espectively, k 1 and k 2 ae the bending stiffness of the spindle, c 1 and c 2 ae the mateial damping of the spindle, espectively, and c q =.1/ω k q, q = 1,2; N 1 and N 2 ae the numbe of the two sets of beaings, espectively; F cx =m eω 2 cos φ; F cy =m eω 2 sin φ, wheeω is the angula velocity of oto, and ω = (1 s)ω f, s is the slip of moto, φ is the eccentic angle of oto mass, and φ=ωt. Without the influence of the unbalanced magnetic foce, the values of the magnetic stiffness matix and vecto of the gap magnetic field in (11) ae zeo. Equations(1),(2),(3),(6),(7),(8),(9),(1),and(11) constitute the dynamic model of the unbalanced motoized spindle system. Thee ae coupling items in the model due to the magnetic stiffness; meanwhile the total estoing foces F bx, F by ae nonlinea functions of vibation amplitudes in x- and y-diections, so the model has stong nonlinea chaacteistic and the nonlinea vibation theoy must be used [25]. The state equations of (11) can be given as { F= { { x y x y x y m 1 ( x x )+k 1 (x x )+N 1 F bx1 ] m 1 ( y )+k 1 (y y )+m g+n 1 F by1 ] m 1 ( x x )+c 2 ( x x )+k 1 (x x )+k 2 (x x )+K 11 x +K 12 y F cx + K 1 ] m 1 ( y )+c 2 ( y )+k 1 (y y )+k 2 (y y )+K 21 x +K 22 y F cy +m g+k 2 ] m 2 ( x x )+k 2 (x x )+N 2 F bx2 ] m 2 ( y )+k 2 (y y )+m g+n 2 F by2 ] }. (12) } } The acceleations and velocities ae zeo in (12) when the spindle system is in equilibium. Accoding to the fist appoximate theoy of Lyapunov, the motion stability of the nonlinea system is detemined by the latent oot of the linea appoximation system. Expand the state equation (12) at the equilibium point appoximately, and the Jacobi matix can be obtained. If all Jacobi Matix Eigen values have negative eal pats, the system is asymptotically stable. When pat of the Jacobi Matix Eigen values own negative eal pats while the othes own positive eal pats, the system is unstable. On the othe hand, once the value of a eal pat of the Jacobi Matix Eigen is zeo while the othes ae not, the system is in a citical state and is stating buckling, and then the citical speed of the spindle could be obtained. Due to the stongly nonlinea chaacteistic, the equation must be solved with numeical analysis. The Runge-Kutta method is used to calculate (11) to obtain the dynamic esponses of the system. 3. Expeimental Setup and Measuements 3.1. Paametes. To validate the poposed dynamic model, a dynamic opeating test was caied out on a DX1A- 24/2-type motoized spindle. The maximum ated speed of the spindle is 24 pm. Both the two sets of the beaings ae a pai in beaing combination with constant peload, and the font and ea beaings ae the angula contact ball beaings of 76CYTAHQ1 and 75CYTASP4, espectively, fom HRB-BC. Table 1 lists the othe paametes of the spindle system. It deseves noting that the spindle system can opeate with the speed highe than the maximum ated speed in a shot time, and the oil dipping ate emains constant in the test to obtain the expeimental data with same paametes and conditions Expeimental Setup. A dynamic opeating test is caied out to obtain the values of vibation esponse at diffeent

5 Shock and Vibation 5 Table 1: Paametes of DX1A-24/2-type motoized spindle. Paamete Units Given value m kg.212 m kg.21 m kg.483 k 1 N/mm 121 d i mm 6.35/6.7 d o mm 6.65/7.5 N b 18/18 k 2 N/mm 984 s.1 e μm.879 R g mm 48 L g mm 45 Λ H/mm σ mm.856 F sm A F m A φ φ p 1 μ.3 Γ mm.12 δ mm.97 PC UA3 DX1A LDS y-diection Figue 4: Expeimental setup. LDS x-diection speeds. Due to the shap incease of the vibatoy output of the spindle system at the esonant fequency, the fist citical speed of the system can be obtained fom the vaiation of the vibationesponsesoftheoto.toavoidtheinfluenceofthe themal displacement, each vibation signal with a speed was collected as quickly as possible once the opeating speed is stable,andtheintevalofeachsignalcollectionismoethan half an hou. As shown in Figue 4, the vibation signals in x- andydiections at the font end of oto unde wok conditions wee acquied by using Lase Displacement Senso (LDS), then tansmitted and tansfomed though the signal collecto and analyse UA3, and finally tansmitted to the PC to be handled by the signal acquisition pocedue compiled by Visual C Table 2: Fist citical speeds of DX1A-24/2-type motoized spindle. Numbe of pole-pais Calculation esult (pm) With MS Without MS Measued esult (pm) Discussion of the Calculation and Expeimental Results 4.1. Validation of the Dynamic Model. Figue 5 shows the expeimental vibation signals of the font end of spindle in y-diection with diffeent speeds. When the otational speed is 324 pm, the vibation value of the spindle is lage than othe values with diffeent speeds. Figue 6 shows the expeimental vibation displacement cuve of the font end of the spindle oto in y-diection. It can be found fom the cuve that the fist citical speed of the system is about 324 pm. To discuss the influence of the unbalanced magnetic foce on the dynamic behavios of the system, (12) was studied with and without the magnetic stiffness (MS). The fist citical speeds of the spindle system obtained fom calculation and expeiment ae listed in Table 2. As can be seen, consideing the effect of MS, the calculated value of the fist citical speeddeceasesandhasaneoof 6.37% compaed with the expeimental value. The diffeence in the fist citical speeds between the two calculations is about 14.42%. The diffeence between the calculation and the measued esult in this pape is within 7%, so the data in Table 2 can testify to the accuacy of the poposed dynamic model fo motoized spindles. In addition, the calculated value of the fist citical speed inceases with the numbe of pole-pais, which means themshasalittleinfluenceonthesystemstabilityinmultipole-pais motos. To study the influence of the MS moe diectly, the adial displacement of the oto cente is discussed and its value is (x 2 + y 2 ).5. Figue 7 shows the theoetical adial displacements of the oto cente. When taking the MS into account, the vibation amplitudes in nonesonance and esonance egions incease; meanwhile the width of the speed egion of high vibation amplitudes becomes lage. Combined with the analysis esult fom Table 2, it can be known that the MS educes the system stability. The influence of the adial displacement of the oto cente on the system stability is shown as in Figue 8. Fo thesakeofdiscussion,theelativeadialdisplacementofthe oto cente e c is taken and its value is (x 2 +y2 ).5 /e. The lage the value of e c is, the lowe the citical speed of the system will be, and the downwad tend is moe gentle, which means when the citical speed of the spindle system is low, a little decease of the citical speed would lead to moe dastic whiling motion. Meanwhile, the atio between the linea citical angula velocity ω n and the nonlinea citical angula

6 6 Shock and Vibation Displacement ( m) Displacement ( m) Time (s) Time (s) (a) 12 pm (b) 18 pm 6 12 Displacement ( m) Displacement ( m) Time (s) Time (s) (c) 324 pm (d) 36 pm Figue 5: Expeimental vibation signals of the font end of spindle in y-diection. Amplitude ( m) n (kpm) Figue 6: Expeimental vibation displacement cuve of the font end of spindle in y-diection. (x 2 +y2 ).5 ( m) n (kpm) Figue 7: Theoetical adial vibation displacement cuves of the oto cente. velocity ω c gently inceases with the inceasing of the value of e c and then shaply deceases. Respectively, ω n pesents [{k 1 +k 2 +(K 11 +K 12 +K 21 +K 22 )/4}/m ].5 [26] and ω c is the angula velocity fom of the fist citical speed discussed in Table 2. In a significant extent of the value of e c,thelinea citical speed is about.65 times that of the nonlinea citical speed. Figue 9 shows the Poincaé mapping of the oto cente in y-diection, which is a typical peiod-doubling bifucation. When the angula velocity ω f is at 3 ad/s, thee is a fequency division phenomenon in the system (Figue 9(a)); with the ω f incease to 318 ad/s, the chaos phenomenon occus (Figue 9(b)), while the chaos disappeas with the invese bifucation at 33 ad/s. 4.2.DynamicDesignfoMotoizedSpindles. To instuct the dynamic design of the spindle system, the influence of system

7 Shock and Vibation Citical speed (kpm) n / c e c e c (a) Citical speed (b) Angula velocity atio Figue 8: Theoetical influence of the adial displacement of the oto cente on the system stability Velocity (mm s ) Velocity (mm s ) Displacement (mm) Displacement (mm) (a) 3 ad/s (b) 318 ad/s Velocity (mm s ) Displacement (mm) (c) 33 ad/s Figue 9: Poincaé mapping of the oto cente in y-diection.

8 8 Shock and Vibation Citical speed (kpm) Citical speed (kpm) Magnetic paametes (times) Shaft stiffness (1 4 N/mm) With MS Without MS (a) Magnetic paametes (b) Shaft stiffness Citical speed (kpm) 35 3 Citical speed (kpm) Beaing cleaance ( m) Ball numbe With MS Without MS (c) Beaing cleaance With MS Without MS (d) Ball numbe Figue 1: Theoetical influence of system paametes on citical speed. paametes on the fist citical speed is discussed and the theoetical esults ae shown as in Figue 1. (a) The magnetic paametes pesent R g L g Λ / in Figue 1(a), and the unit of the abscissa is the multiple of R g L g Λ /. With the incease of the inne adius of stato R g and the effective length of oto L g and the decease of unifom ai gap δ, the citical speed deceases nonlinealy. When the magnetic paametes ae 1.5 times that of the design value, the citical speed is close to the minimum ated speed. So the magnetic paametes ae vital to the dynamic design of the spindle system, and thei effects on the dynamic chaacteistic of motoized spindles cannot be ignoed. Besides, the incease of the unifom ai gap is of advantagetotheotostability,butthegapcannotbetoolage and should be optimized. (b) As shown in Figue 1(b), with the incease of the shaft stiffness, the fist citical speed of the system inceases, and the diffeence between the two calculations deceases, which means that high shaft stiffness can weaken the influence of the MS. (c) Figue 1(c) shows the impact of the beaing cleaance γ on the citical speed. Peload makes the beaing in a negative cleaance state and impoves the stability of the spindle system, which also can educe the effect of MS. (d) Figue 1(d) shows that the citical speed ises with theinceaseofballnumbe.thisisbecausetheinceaseofball numbe leads to the impovement of the suppot stiffness and the fequency of the paametic vibation fo ball beaings and meanwhile undemines the fequency multiplication and hamonic components of the paametic vibation, which benefits the stability of system. Without the consideation of the waviness on the ace, the beaings which have moe balls can effectively impove the opeational stability of the spindle and decease the influence of MS. Based on the esults of Figues 1(b), 1(c), and 1(d), it can be known that with the natual popeties of the system incease,suchasshaftstiffnessandsuppotstiffness,the influence of MS to the stability of the system deceases. That means the highe the stability of the system is, the lowe the influence of MS to the stability of the system is.

9 Shock and Vibation 9 5. Conclusions This pape deals with the poblem of analysing the dynamic behavios of unbalanced high speed motoized spindles suppotedonballbeaings.animpoveddynamicmodel which takes into account the unbalanced magnetic foce is developedfothepesentedsystem.themodelisstudied by using the Lyapunov nonlinea vibation theoy. Then, the validation test on a DX1A-24/2-type motoized spindle is pefomed. Fom the calculated and expeimental esults the following conclusions ae dawn. (a) The poposed model accuately pedicted the citical speed obseved in the expeiment and allows exploation of the magnetic foce contibution within the context of its assumptions, which can be used fo analysing the dynamic popeties of high speed machine tool motoized spindle systems. (b) The unbalanced magnetic foce in the machine tool motoized spindle system inceases the vibation esponse of theotoandeducestheciticalspeedofthesystem.the influence of the MS on the stability of the system deceases with inceasing the numbe of pole-pais. With the incease of e c, the citical speed of system deceases, while the atio ω n /ω c inceases and then shaply deceases. The peioddoubling bifucation occus with the angula velocity ω f being about 32 ad/s. (c) The system paametes geatly influence the citical speed of the system. The incease of the shaft stiffness and the ball numbe of beaings is of advantage to the stability, but the incease of the magnetic paametes and beaing cleaance wouldleadtothedeceaseoftheciticalspeedofthesystem. If the spindle system has highe natual popeties, the influence of MS on the stability of the system will decease. Competing Inteests The authos declae that they have no competing inteests. 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