Te Open Mecanical Engineering Journal, 8,, 3-39 3 Open Access Vibration in a Cracked Macine Tool Spindle wit Magnetic Bearings Huang-Kuang Kung and Bo-Wun Huang Department of Mecanical Engineering, Ceng Siu University, Taiwan Abstract: Due to manufacturing flaws or cyclic loading, cracks frequently appear in a rotating spindle system. Tese cracks markedly affect te dynamic caracteristics in iger modes of te rotating macinery. For faster rotational speeds, especially for super-ig-speed cutting, a spindle wit magnetic bearings is necessary. However, most investigations into spindle system dynamic caracteristics ave been confined to ball-bearing-type spindles. Te dynamic response of rotating cracked spindle systems wit magnetic bearings is examined in tis article. A Euler-Bernoulli beam of circular cross section is used to approximate te spindle and te Hamilton principle is employed to derive te equation of motion for te spindle system. Te effects of crack dept, rotation speed and bearing lengt on te dynamic response of a rotating magnetic bearing spindle system are studied. INTRODUCTION Cracks frequently appear in rotating macinery due to manufacturing flaws or cyclic fatigue during operation. Numerous cracks can be observed after severe operating conditions, especially in ig speed spindles [, ]. ocal structural irregularities caused by cracks in te spindle may significantly cange te dynamic beavior of a rotating macinery system. Te effects of cracks on te dynamic and static beaviors of structures ave been studied by a number of researcers [3-5]. Te effects of cracks on spindle dynamics, saft and rotor systems, were also studied by researcers [6-9]. Wen a spindle rotates, te vibrational response is altered by te crack opening and closing in eac cycle. Most investigations were motivated by te ypotesis tat only opening cracks markedly cange te spindle dynamics. Tis paper focuses on te dynamics of a spindle wit a transverse crack. Hig speed macining is one of te most modern manufacturing engineering tecnologies. In a macining system, te spindle is te most critical element tat affects te dynamic performance and capabilities of te system in te macining process. However focusing exclusively on te spindle system is insufficient because te bearings can cange te dynamics of a macining spindle system. Hence, te bearing effects on te spindle system must also be considered. Bearings are used in many rotating macines to brace te rotating spindles and rotors. In te past, te required rotor speed was low, allowing ball and roller bearings to be used in rotating macinery. Hig temperatures are generated wit ball-bearing spindle systems operating at ig speeds. Te ig temperatures often bring about macine failure. To attain greater complexity and accuracy, modern engineering tecnologies demand macinery tat can be run at ig speeds. To avoid te ig temperatures generated by te contact between te spindles and bearings, non-contact magnetic bearings are used for te spindle and rotor in ig speed rotating macinery. Traditionally, ball bearings ave been used to support te spindle systems wen te rotational speed was not ig. Previous investigations on bearing spindle systems were confined to spindles wit ball bearings. In some studies, te focus was on te dynamic response of a spindle supported by bearings [, ]. At iger speeds, tis bearing canges te stiffness of te entire spindle system and significantly alters te system properties [-5]. Precise macining requires iger spindle speeds, making te magnetic-bearing spindle necessary. Investigations as [6-] studied te performance and dynamic properties of magnetic bearings. Most studies deal wit a magnetic ring for a radiaagnetic bearing used as an unlimited one long magnetic bar for a permanent magnetic bearing. Investigation as [] studied te bearing capacity and stiffness of radiaagnetic bearings. Tus far, most investigations as [-4] on te dynamic caracteristics of a cracked spindle system were limited to ball-bearing-type spindles. Tis study examines te crack effects on te dynamic response of a rotating spindle system wit magnetic bearings. A Euler-Bernoulli beam of circular cross section was used to approximate te spindle model. Te equations of motion for te bearing-spindle system were derived using te Galerkin metod and Hamilton principle. A model te size of an actual spindle system was used. To simplify te calculations, massless springs were employed to model te stiffness of te magnetic bearings. Te effects of crack dept, rotational speed and bearing lengt on te dynamic response of a spindle system were investigated. Address correspondence to tis autor at te Department of Mecanical Engineering, Ceng Siu University, Taiwan; E-mail: uangbw@csu.edu.tw a spindle supported by bearings 874-55X/8 8 Bentam Open
Vibration in a Cracked Macine Tool Spindle wit Magnetic Bearings Te Open Mecanical Engineering Journal, 8, Volume 33 k x : te bearing stiffness in u deflection at a position z, k y : te bearing stiffness in v deflection at a position z, k x : te bearing stiffness in u deflection at a position z, k y : te bearing stiffness in v deflection at a position z, : density, A : cross section area, : rotating speed, : Dirac delta fuction, z : te first located position of bearings, z : te second located position of bearings. For convenience, te dimensionless equations of motion for tis spindle are: a simple model of bearing spindle system Fig. (). A rotating spindle wit bearings sceme. η u A v + u + u 4 A 4 + k x u z z +k x u z z } = (3) R Fig. (). Geometry of a cracked spindle. Teory Tis paper considers a spindle supported by magnetic bearings, as sown in Fig. (a), to elucidate te dynamic response of a spindle system. Fig. (b) presents a simple model for tis bearing-spindle system. In tis model, massless springs are employed to simulate te stiffness of te magnetic bearing and support te spindle. Te rotational speed of te spindle cannot be ignored in te rotating macinery bearing application. In tis study, te deflection components (z,t), and u(z,t) denote te two transverse flexible deflections of te spindle system. E and I represent te Young s Modulus and area inertia of te spindle, respectively. Only te transverse flexible deflections are studied in tis article. According to [5], te governing equations of te spindle system are displayed as: A u A v A u + u + k x u ( z z )+ k x u ( z z )= A v + A u A v + v + k y v ( z z )+ k y v ( z z )= were a o dξ ξ () () v + A u + v + v 4 A 4 + k y v z z + k y v z z } = were te dimensionless parameters are given using: z = z, z = z, z = z, = u ( z )= u z k x = k x 3, v ( z )= v z, k y = k y 3, k x = k x 3, k y = k y 3 and te boundary conditions are: u = u = u = v = u = v = (4) A 4, (5), (6) (7) v =, at z = (8) v =, at z = (9) Te Galerkin metod is employed to derive te spindle equations of motion in matrix form. Terefore, te solutions for Eqs. (3) and (4) can be assumed to be: m u ( z,t)= i ( z )p i () t () i= m v ( z,t)= i ( z )q i () t () i= were i ( z ), i ( z ) are comparison functions for te spindle system, and p i (), t q i () t are te time coefficients to be determined for te system. Te exact solution for a beam wit free-free boundary conditions is considered, and five comparison function modes are used. i ( z )= i ( z )= ( i z ) ( cos i z cos i z ) ()
34 Te Open Mecanical Engineering Journal, 8, Volume Kung and Huang i = i 3 ( u i ), i =,, 3, (3) were u is te unit step function. Substituting Eqs. () and () into Eqs. (3) and (4) respectively, te equations of motion in matrix form for te spindle system can be derived as: () M pt G + p t M qt () G q() t K + e pt () K + K e qt () pt () K qt () K + s pt () K + K s qt () s pt () = K s qt () () (4) were = A. 4 Te elements of te matrices in te above equation are given as follows, ( m ij ) = i ( m ij ) = i ( g ij ) = i ( g ij ) = i ( k e ) ij ( k e ) ij ( k s ) ij ( k s ) ij ( k s ) ij ( k s ) ij ij j dz = k ij j dz = k (5) (6) j dz (7) j dz (8) = = i i = k x i z = k y i z = k x i z j dz (9) j dz () { } j ( z ) { } j ( z ) { } j ( z ) { } T () { } T () { } T (3) { } j ( z ) { } T (4) = k y i z For te sake of convenience, Eq. (4) can be rewritten as, M { X }+ G { X }+ K { X }= (5) were M = M M (6) G = G G K e K = K e + K K K s + K s + K s K s (7) (8) A space vector is introduced in Eq. (5) to solve te eigenvalue problem for te system. X { V }=. (9) X Substituting Eq. (9) into Eq. (5), te equation can be rearranged as; M { V }+ G K { K K V }= (3) Te non-dimensional frequency n in Eq. (3), i.e., te natural frequency of te spindle system, is defined as: n = n / for n =,,... (3) 4 A In industry, ball bearings are frequently used to support rotating spindles in rotating macinery. Recently, magnetic bearings ave been employed increasingly to support spindles because tey must rotate at iger speeds. Few investigations focused on te dynamic responses of defective spindle systems wit magnetic bearings. Terefore, tis investigation addresses te dynamic response of a cracked spindle supported by magnetic bearings. Crack Effect Considering a crack located at z = z on tis spindle, te strain energy of te defective spindle will include te released energy caused by te crack. Fig. () sows te geometry of a cracked spindle. Te released energy caused by a crack, as noted in [6], wit a dept of a may be expressed as: b μ U c = K E I d (3) b were b = R ( R a) and μ is te Poisson's ratio of te spindle, K I is te stress intensity factor under a mode I load and R is te radius of te spindle. In tis case, te stress intensity factors K I can be approximated as K I = 4M b R F R 4 ( ) (33) were, M b is te bending moment, and = R (34)
Vibration in a Cracked Macine Tool Spindle wit Magnetic Bearings Te Open Mecanical Engineering Journal, 8, Volume 35 = a + R R (35) F = tan.93+.99 sin cos 4 (36) Te notations a and R are te maximum crack dept and radius of te spindle, respectively. Based on te investigations in [6, 7], alterations of te elastic deformation energy caused by lateral bending moments are te only important canges in te case of slender beams wit a crack. Te released energy of te crack wit respect to due to te bending moment is obtained as: a b U c = E ( v μ ) z z b z ( R ) F d d dz (37) Similarly, te released energy of te crack wit respect to is derived as follows, U c = E ( μ ) were F F a b u z z z b (38) d d dz = tan.75 +. +.37 sin cos 3 (39) For simplification, te dimensionless equations are employed as: U c = 4R ( μ u ) z z Q z R, R dz (4) U c = 4R ( μ v ) z z Q z R, R dz (4) were Q Q R, arbr R = R R F (4) d R d R d R d R R, ar br R = R R F (43) b R Te bearing-spindle wit a crack can be obtained as: u A v + u + u 4 A 4 8R μ ( )Q R, R u z z +k x u ( z z )+ k x u ( z z )} = v + A u + v + v 4 A 4 8R μ ( )Q R, R v ( z z ) + k y v ( z z )+ k y v ( z z )} = (44) (45) Similarly, te equations of motion for te defective spindle, i.e. Eq. 3, can be rearranged in matrix form using Galerkin s metod as follows: () M pt G + p t M qt () G q() t K + e pt () K + K e qt () K pt () K qt () c pt () K + K c qt () s K s pt () K + qt () s pt () = K s qt () were ( k c ) ij ( k c ) ij 8R μ = 8R μ = Q Q R, R R, R i i z () j ( z ) z j ( z ) (46) (47) z =z (48) z =z Supported by Magnetic Bearing Few investigations on radiaagnetic bearings were found, so tese bearings were selected for tis article. Te rotor is kept in te desired position by a magnetic bearing stator using a magnetic field induced by permanent magnets. According to [], te bearing force F r is derived as follows, F r = B B r r S (49) 4μ Were B r : remanence of te externaagnetic loop for te magnetic bearing; B r : remanence of te internaagnetic loop for te magnetic bearing; μ : permanence in vacuum and, (5) S = S 3 + S 4 S 3 S 4 + ( e + R 3 cos R cos ) (5) Rx = x + x x + R 3 sin R sin
36 Te Open Mecanical Engineering Journal, 8, Volume Kung and Huang + ( e + R 3 cos R cos ) (5) Rx = x + x x + R 3 sin R sin + ( e + R 4 cos R cos ) (53) Rx 3 = x + x x + R 4 sin R sin + ( e + R 4 cos R cos ) (54) Rx 4 = x + x x + R 4 sin R sin S 3 = R R 3 ( e + R 3 cos R cos )dddxdx Rx 3/ (55) size actually used in engineering applications are addressed and a magnetic bearing is considered in tis work. Te dimensions R=.m and =.m of a rotating spindle are assumed. Te bearings positions are assumed to be z = and z =. A spindle system braced by a magnetic bearing is important in engineering applications, especially for igspeed rotationaacinery. For te above-mentioned spindle dimensions, te important magnetic bearing parameters were selected as follows. Nd-Fe-B material was employed to model te elements of a permanent magnet in te radiaagnetic bearings. For tis material, te remanence B r = B r =.3 wb m can be sown. Corresponding to te spindle imension, te lengt of te magnetic = mm and te clearance mg =.m were assumed. Q S 3 = l l m m ( + ) R R e R cos R cos dddxdx 3 3 3/ Rx (56) Q S 4 = R R 4 ( e + R 4 cos R cos )dddxdx Rx 3 3/ (57).. S 4 = R R 4 ( e + R 4 cos R cos )dddxdx Rx 4 3/ (58) were R : external radius of te externaagnetic loop R+ g + mg R : internal radius of te externaagnetic loop R+ g + mg R 3 : external radius of te internaagnetic loop R+ R 4 : internal radius of te internaagnetic loop R : lengt of te magnetic loop mg : clearance between te internal and externaagnetic loops g : te tickness of te magnetic loop e : is eccentric of te magnetic bearing Consequently, te stiffness of a radiaagnetic bearing can be derived as follows: k m = F r e = B r B r 4μ S e g (59) ANAYSIS AND DISCUSSION In ultra-ig-speed macining, using magnetic bearings to support te spindle is necessary [8]. Te dynamic properties of a multi-mode spindle wit bearings of te....4.6.8 Crcak dept a/r Fig. (3). Te variations in te crack flexibility wit te crack dept ratio. Te variation in te crack flexibility wit te crack dept ratio is plotted in Fig. (3). Te numerical analysis reveals tat te crack flexibility increases as te crack dept is increased. From te results, te crack dept markedly affects te saft stiffness. As a wole, tese results and tose from previous investigation [9] are identical. Fig. (4) presents te natural frequencies of a spindle bearing system wit and witout cracks. Te logaritmic scale was employed to study iger modes in tis figure. At lower modes, te natural frequencies of te spindle bearing system cange sligtly regardless of weter tere is a crack or not in te system. However, at iger modes, te natural frequencies of a spindle bearing system decrease wen tere is a crack in tis spindle system. Wit magnetic bearings, te crack effect on te dynamics of a spindle system as more influence at te iger modes tan at te lower modes. Te effect of te crack dept on te natural frequencies is considered in Fig. (5). In tis figure, te lower and iger mode natural frequencies, te st and 5 t modes, are studied togeter. Bot te st and 5 t natural frequencies decrease wit increasing crack dept. If te crack dept were., te crack size would ave little influence on te natural frequencies of a rotating blade system. However, te natural frequencies are depressed significantly wen te crack size is larger tan..
Vibration in a Cracked Macine Tool Spindle wit Magnetic Bearings Te Open Mecanical Engineering Journal, 8, Volume 37 no crack wit a crack ad=.5. 3 4 5 6 7 Mode Number Fig. (4). Te natural frequencies of a magnetic-bearing spindle wit and witout a crack, ( =.5, a D=.5, z =.5 )..4.475 natural frequencies of te system is te lowest wen te crack is located at te middle of te spindle. Te penomena illustrated in Fig. (6a) and (6b) are similar. Te effect of rotation speed on te dynamics of a cracked spindle wit magnetic bearings is plotted in Fig. (7). Double roots for te natural frequencies of a cracked spindle wit magnetic bearings are observed only if tis system as no rotation speed. Wit rotation speed, te natural frequencies of tis system are divided into two parts, te forward and backward frequencies. Only te st and nd natural frequencies are sown in tis figure. Te st natural frequency of a cracked spindle wit magnetic bearings increases as te rotational speed increases. However, it was found tat if te rotational speed increases, te nd natural frequency decreases..4.495.49.485.48.475.45.45.47...3.4.5.6.7.8.9 Crack location z (a) te st mode.4 5 5 5.5..5..5.3 Crack dept a/d (a) te st mode.5..5..5.3 Crack dept a/d (b) te 5t mode Fig. (5). Te natural frequencies of a magnetic-bearing spindle wit different crack depts, ( =.5, z =.5 ). Crack location dramatically canges te dynamics of a spindle wit magnetic bearings. Fig. (6) sows te variation in te natural frequencies of a spindle wit different crack locations. As mentioned above, only te st and 5 t natural frequencies were examined. Wen te crack is located at bot ends, te natural frequencies are almost te same as a system witout a crack. Te value of te 4 8 6 4...3.4.5.6.7.8.9 Crack location z (b) te 5t mode Fig. (6). Te natural frequencies of a magnetic-bearing spindle wit different crack locations, ( =.5, a D=.5 ). For radiaagnetic bearings, te lengt of te bearings remarkably alters te system stiffness. Fig. (8) illustrates te variations in natural frequencies for a cracked spindle wit different magnetic bearing lengts. Te results indicate tat te natural frequencies of a magnetic-bearing spindle increase as te magnetic bearing lengt increases. Finally, te frequency responses of a spindle system wit and witout cracks are illustrated in Fig. (9). Bot te lower and iger frequency domains were examined. In te lower frequency domain, te figure sows tat te frequency response of a bearing-spindle system witout a crack is almost te same as one wit a crack. Te peak frequency response values for a spindle wit magnetic bearings are depressed in te iger frequency
38 Te Open Mecanical Engineering Journal, 8, Volume Kung and Huang domain if tere is a crack in te spindle. As above, te effect of a crack may significantly influence te dynamics of a spindle system wit magnetic bearings at iger modes..8.6.4 te st mode te nd mode 3 - wit a crack witout cracks..8.6.4....3.4.5.6.7 Rotational speed Ω Fig. (7). Te natural frequencies of a magnetic-bearing spindle wit different rotational speeds, ( z =.5, a D=.5 )..9.8.7.6.5.4.3.....3.4.5 engt of bearing m Fig. (8). Te natural frequencies of a rotating cracked spindle wit different lengts of magnetic bearing, ( z =.5, a D=.5 ). CONCUSIONS Te effect of cracks on te dynamics of a spindle supported by magnetic bearings was studied. Te most significant observations in tis study are summarized as follows:. Wit magnetic bearings, te natural frequencies of a spindle system decrease as te dept of te crack increases.. At iger modes, a crack wilarkedly affect te dynamics of a spindle wit magnetic bearings. However, at lower modes, te natural frequencies of te spindle bearing system cange just sligtly no matter weter tere is a crack or not. 3. Te rotational speed and magnetic bearing lengt will significantly influence te dynamics of a spindle wit magnetic bearings. - -3-4 -5-45 -4-35 -3-5 - 5 6 7 8 9 3 Frequency (b) a iger frequency domain Fig. (9). Te frequency response of a rotating spindle wit and witout a crack, ( =.5, a D=.5, z =.5 ) 4. Te crack will dramatically affect te dynamic caracteristics of a spindle wit magnetic bearings if te crack is located at te middle of te spindle. REFERENCES..4.6.8..4.6.8 Frequency (a) a lower frequency domain [] S. C. Huang, Y. M. Huang and S. M.Sie, Vibration and Stability of a Rotating Saft Containing a Transverse Crack, Journal of Sound and Vibration, vol.6, pp. 387-4, 993. [] S. K. Baumik, R. Rangaraju, M. A. Venkataswamy, T. A. Baskaran and M. A. Parameswara, Fatigue fracture of cranksaft of an aircraft engine, Engineering Failure Analysis, vol. 9(3), pp. 55-63,. [3] P. F. Rizos, N. Aspragatos, A. D. Dimarogonas, Dimarogonas Identification of Crack ocation and Magnitude in a Cantilever Beam from te Vibration Mode, Journal of Sound and Vibration, vol. 38, pp. 38-388, 99. [4] D. Broek, Elementary Engineering Fracture mecanics, Martinus Nijoff Publisers, 986. [5] H. Tada, P. Paris and G. Irwin, Te Stress Analysis of Crack Handbook, Hellertown, Pennsylvania: Del Researc Corporation, 973. [6] B. Grabowski, Te Vibrational Beavior of a Turbine Rotor Containing a Transverse Crack, ASME, Journal of Mecanic Design, vol., pp. 4-46, 98. [7] A. S. Sekar, Effects of cracks on rotor system instability, Mecanism and Macine Teory, vol. 35(), pp. 657-674,. [8] N. Bacscmid, P. Pennacci, E. Tanzi and A. Vania, Identification of transverse crack position and dept in rotor systems, Meccanica, vol. 6, pp. 563-58,.
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