Journal of Advanced Mechanical Design, Systems, and Manufacturing

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1 Vol., No. 4, 8 Mathematical Model of Linear Motor Stage with Non-Linear Friction Characteritic * Satohi KANEKO **, Ryuta SATO *** and Maaomi TSUTSUMI **** **Pot graduate, Tokyo Univerity of Agriculture and Technology Nakacho -4-6, Koganei, Tokyo, , JAPAN *** Aitant Profeor, Tokyo Univerity of Agriculture and Technology ****Profeor, Tokyo Univerity of Agriculture and Technology tutumi@cc.tuat.ac.jp Abtract Thi paper propoe a mathematical model of a feed drive ytem coniting of a cylindrical linear motor and linear ball guide. The friction model conit of two component; a model for the relationhip between diplacement and friction force under microcopic motion (non-linear pring characteritic), and a model for the relationhip between velocity and friction force (Stribeck curve). The non-linear pring i modeled from the reult of very low frequency imple harmonic motion experiment. The Stribeck curve i modeled from the reult of friction force experiment conducted for variou contant velocitie. The parameter in the model were derived from machine pecification of the feed drive ytem and experimental reult. In addition, in order to account for the quantization error of the linear cale, the controller and amplifier were modeled a a dicrete time ytem. To evaluate the propoed model, tep repone and circular motion under variou condition were meaured and imulated. The influence of the friction characteritic on dynamic behavior wa then invetigated. In the experiment, the friction characteritic were changed and compared uing three greae with differing vicoitie. A a reult, it wa confirmed that difference in greae vicoity trongly influence the damping of vibration for the tep repone. Furthermore, it wa clarified that quadrant glitche do not appear in the microcopic diplacement region. For many imulation reult, it wa verified that the propoed model accurately reflect the real behavior. Key word: Linear Motor Stage, Linear Ball Guide, Mathematical Model, Non-Linear Friction, Friction Model. Introduction *Received Mar., 8 (No. 8-6) [DOI:.99/jamdm..675] Linear ball guide are applied to a wide variety of conventional and preciion machine uch a NC machine tool, emiconductor-fabrication equipment, etc. In feed drive ytem employing linear motor, the friction characteritic of the linear ball guide trongly influence dynamic behavior. However, the friction characteritic of the linear ball guide are not well defined. It i well known that the friction force of the linear ball guide behave a a pring for diplacement up to a few hundred micron, and thu the relationhip between diplacement and friction force give a hyterei loop (). Thi characteritic, referred to a a non-linear pring (), influence the dynamic behavior of the feed drive ytem, epecially in the region of minute diplacement. Specifically, in the cae of high-peed poitioning, the non-linear pring characteritic generate vibration (3). Furthermore, in repone to circular motion tet, it generate the quadrant glitch which i 675

2 Vol., No. 4, 8 aociated with hyterei (4) ~ (6). The friction force in linear ball guide i conitent with the Stribeck effect of liding friction and influence the dynamic behavior of the feed drive ytem at very low velocitie (7) (8). The friction characteritic depend on the lubricant vicoity, ball retainer, pre-load, and guide ize (9). Conequently, it i neceary to clarify the non-linear friction characteritic for higher contouring movement. There ha been a great deal of reearch conducted in relation to the modeling of non-linear friction characteritic. Fukada () etablihed that a non-linear pring characteritic exit between ball crew and nut and went on to propoe an elatic-platic model and imulation logic. Karnopp () propoed a imulation model for the relationhip between velocity and friction force. Thi model can expre the actual characteritic for very low velocity. However, it doe not include the influence of vicou friction. In order to properly cover liding motion, Canuda (8) propoed a dynamic model called the britle model. Thi encompae both the Stribeck effect and non-linear pring characteritic. In addition, Tanaka () (3) propoed a method to determine the parameter for the britle model. It wa then confirmed that the real behavior under particular condition can be expreed uing a imulation model which employ the britle model. Iwaaki (4) modeled the non-linear friction characteritic from dynamic and tatic characteritic for a feed drive ytem coniting of a ball crew and linear ball guide. However, a mathematical model which can expre the real characteritic under large diplacement without changing parameter ha not yet been propoed. Thi paper propoe a mathematical model for a feed drive ytem coniting of a rod type linear motor and linear ball guide. The mathematical model wa developed to accurately expre the dynamic behavior. In order to evaluate the propoed model, tep repone and circular motion under variou experimental condition were meaured and imulated. The propoed model can imulate the tep repone and circular motion for variou tep height, amplitude and feed rate by appropriately adjuting the parameter. Additionally, the parameter only need one adjutment for each greae vicoity. In addition the effect of the bae oil vicoity of greae on the dynamic behavior wa alo invetigated. A a reult, it wa verified that the propoed model can accurately expre the influence of the friction characteritic on the dynamic behavior of the feed drive ytem.. al Set-up Figure how the experimental et-up. It i a one-axi feed drive ytem coniting of a rod type linear motor, a table, a et of linear ball guide and a linear cale. Thi et-up i fixed to a bae plate. The feed drive ytem i controlled by a peronal computer (PC) equipped with a DSP board. The intruction value are input to the controller from the PC, and the poition of the table i detected by a. µm reolution linear cale. Linear ball guide employing retainer are alo ued. The friction characteritic were changed by uing two vicoitie of greae Rod for linear motor Linear ball guide Bae Bae-plate PC with DSP board Amplifier Table Linear motor Linear cale Linear ball guide Fig. al et-up 676

3 Vol., No. 4, 8 (3 cst and cst). Prior to taking the firt meaurement, the linear ball guide were worked a little after being packed with greae. Becaue the inertia force i negligible, x f t the friction force i equivalent to the thrut intruction to the amplifier. M K t B x 3. Modeling of Feed Drive Sytem C t F K Figure how a dynamic model of the experimental et-up hown in Fig.. The dynamic model conit of a table, a rod for linear motor and a bae plate. The parameter definition and aociated value for the model are lited in Table. Figure lead to the following equation of motion: C B C M M B Fig. Dynamic model of experimental et-up x B M x + C ( ) + f = F t t t t B M x + C ( ) + K ( x x ) = F B B M x + C + K x = C ( ) + C ( ) + K ( x xb ) + f B B B B B B t t B B () r + K pp + Kvp T c z z Controller K vi + + DA Amplifier T rg T fil + T i + T i F z T c ( x x ) B t (Linear cale) Fig. 3 Block diagram for controller and amplifier In the experiment, the relative diplacement (x t -x B ) between the table and bae i detected uing a. µm reolution linear cale and the poition of the table i digitally controlled by the PC. Therefore, when modeling the control ytem a a dicrete-time ytem, it i neceary to conider the quantization error of the linear cale (3). The model of the controller and amplifier for the imulation are hown in Fig. 3. In thi, r i the reference poition [m], K pp i the proportional gain of the poition loop [/], K vp i the proportional gain of the velocity loop [/m], K vi i the integral gain of the velocity loop [/m], T c i the control frequency [] (identified from experimental reult of velocity tep repone), DA i the DA converter contant [V], T rg i the torque command gain [Nm/V], T fil i the time contant of the torque intruction filter [], and T i i the time contant of current control loop []. T c i identified from experimental reult of velocity tep repone. 4. Modeling of Non-linear Friction Characteritic 4. Relationhip between velocity and friction force In order to invetigate the effect of the velocity on the friction force, the force wa meaured for variou contant velocitie. Seventeen velocitie from 6.3 mm/min to, mm/min were choen. The friction force at each velocity i an average calculated by 677

4 Vol., No. 4, 8 removing the tranient repone and taking the average between the two direction of the motion. Figure 4 how the relationhip between velocity and friction force. It can be een from thi figure that the friction force take a maximum value at very low velocitie. For velocitie above. m/, the friction force linearly increae with the increaing velocity. Thi behavior i inherent to vicou friction. Thi occur irrepective of the greae vicoity employed. Table Parameter for dynamic model Parameter Unit Value Ma of table M t kg 6 Ma of rod M kg.8 Ma of bae M B kg 39 Viou damping of linear ball guide C t N/m 55.5 Viou damping of rod and upport ytem C N/m Viou damping of bae C B N/m Axial tiffne at rod and opport ytem K N/m Axial tiffne at bae K B N/m.34 6 Friction force at linear ball guide f N - Thrut of motor F N - Referring to Ref. (5), the relationhip between velocity and friction force may be approximated to: f γ = () v C v () β α v v ( v) f ( x) Ctv exp + gn + t v a where, f (x) i friction force for 4 the non-linear pring cst characteritic which i a function only of diplacement. 3 3cSt Becaue movement i in the order of millimeter, f (x) i equal to the contant friction force (Coulomb friction force). A hown in Fig. 4, the Velocity m/ meaured friction characteritic, repreented with red mark, can Fig. 4 Relationhip between velocity and friction force be fitted by adjuting the ix parameter (C t, C t, V a, α, β and γ) in Eq. (). The correponding characteritic generated from Eq. () i plotted by a dahed blue line in the ame figure. It can be een from the figure that the imulated curve are good agreement with the meaured friction characteritic. Differentiating the relationhip between velocity and friction force with repect to velocity yield the vicou friction force a a function of velocity df ( v) / dv. For the imulation model, the coefficient of vicou damping C t in Eq. () can be replaced by df ( v) / dv. 4. Non-linear pring characteritic In order to invetigate the non-linear pring characteritic of the linear ball guide, the friction force wa meaured under imple harmonic motion of minute amplitude. Figure 5 how the hyterei loop produced by the diplacement and friction force. The two loop were meaured, one for each greae vicoity at 5 µm amplitude,.5 Hz frequency. It can be clearly een that, up to µm diplacement, the friction force Friction force N 678

5 non-linearly increae with increaing diplacement. Above thi diplacement, the friction force become contant. Thi characteritic i referred to a the non-linear pring characteritic. It i important to model thi characteritic in order to imulate the dynamic behavior under minute motion condition. In thi paper, the characteritic i modeled a follow: Friction force N Fig Amp.=5µm 3cSt Vol., No. 4, 8 cst Diplacement µm Hyterei loop of friction force due to imple harmonic vibration at very low frequency (.5 Hz) f ( x) = K( x) x (3) where, K(x) i a function of pring contant which expree the non linear pring behavior, and x i diplacement. Therefore, K(x) can be expreed a follow: K K K 3 K ( x) = + + (4) a b c x x x x x x 3 Data calculated from Eq. (3) and (4) are alo plotted in Fig. 5. Thi how that the hyterei loop can be accurately repreented by adjuting the parameter (x, x, x 3, K, K, K 3, a, b, c) in Eq. (4). Thi demontrate the validity of the non-linear friction characteritic expreed in the imulation. It i alo poible to take into account difference in the vicoity of the oil on which the greae i baed. A mentioned above, two non-linear friction characteritic which depend on velocity and diplacement were modeled. Therefore, Eq. () can be rewritten a follow: M t M M B dfv ( v) x + ( ) + f ( x) = F t dv t B x + C ( ) + K ( x x ) = F B B df v ( v) x + C + K x = ( ) B B B B B dv t B + C ( ) + K B ( x x ) + B f ( x) (5) Equation (5) include the non-linear pring characteritic and velocity dependence. 5. al and Reult 5. Step repone A mentioned above, mathematical model incorporating non-linear friction characteritic were developed. The parameter of the friction model were determined by trial and error in order to match the model to the experimental reult (Refer to Fig. 4 and 5), a lited in Table. The controller ervo gain are lited in Table 3. Herein, to evaluate the propoed model, tep repone were examined through experiment and imulation. All of the experiment were repeated three time to confirm the repeatability of meaurement. Figure 6 how the imulation reult in the abence of non-linear friction for a tep input height of 5 µm. The tep repone curve how a damped ocillatory form. Therefore, it may be aid that the non-linear pring behavior under minute motion relate to the vibration damping of the ytem. 679

6 Vol., No. 4, 8 Figure 7 how the imulation and experimental reult for tep repone with input height of 5 µm and µm. Figure 7(a) and (b) how the reult for vicoitie of 3 cst and cst, repectively. Figure 8 how the reult for tep height of µm and 5 Table Parameter for friction model Parameter Unit Value 3 cst cst x m x m x 3 m K N/m K N/m K 3 N/m a b - c C t N/m 5. 9 C t N/m v a m/ α β - γ -.. Table 3 Parameter of ervo-gain for controller Parameter Unit Value Step repone Circular motion tet K pp / 35 K vp /m 5 3 K vi /m 5 3 Diplacement µm cst cst.. Time Fig. 6 reult without coniting non-linear friction characteritic.3 µm, and for the vicoitie of 3 cst and cst, repectively. A hown in Fig. 7, for both vicoitie, the propoed model i a good a repreentation of the experimental reult a the vicou friction model. In addition, it can be clearly een that higher greae vicoity yield higher damping. It can be aid that both model can imulate the experimental reult in the range µm to µm. Referring to Fig. 8, for minute input height, the propoed model approximately agree with experimental reult wherea the vicou friction model differ markedly. The vicou friction model yield an ocillatory repone which depend on the greae vicoity. In addition, there i a very large lag in the repone. Thi i in contrat to the cloe agreement between the propoed model and the experimental reult. Moreover, vibration (tranient overhoot) with low frequency uch a hown in Fig. 7 doe not occur although the ervo gain i unchanged. The velocity for the tep height hown in Fig. 8 i very low (about.4 mm/). A hown in Fig. 4, at thi very low velocity, 68

7 Vol., No. 4, 8 the friction force i trongly non-linear. In ummary, for the minute tep repone, the velocity dependence of the friction force and the non-linear pring characteritic govern the vibration damping of the ytem. Diplacement µm Propoed model Vicou friction model Step height, 5 µm Step height, µm...3 Time (a) 3 cst Diplacement µm Fig. 7 Comparion of tep repone 5 4 Propoed model Vicou friction model 3 Step height, 5 µm Step height, µm...3 Time (b) cst Diplacement µm Fig. 8 Step height, µm Step height, 5 µm Propoed model Vicou friction model Time (a) 3 cst Diplacement µm Step height, µm 8 6 Step height, 5 µm 4 Propoed model Vicou friction model Time (a) cst Comparion of tep repone under microcopic diplacement region r : 5 mm f : m/min r : 5 mm f : m/min (a) 3 cst div.: µm (b) cst r : mm f : 3 m/min r : mm f : 3 m/min (c) 3 cst (d) cst Fig. 9 al and imulation reult of circular trajectory under variou condition 5. Circular motion tet A mentioned above, the propoed model for the friction force accurately reflect the actual tep repone obtained experimentally for variou input height. Thi type of 68

8 Vol., No. 4, 8 repone often occur when XY table are driven with circular motion. In order to further verify the effectivene of the propoed model, it wa applied to the cae of uch circular motion. Radiu : 5 µm Feed rate : 9.4 mm/min Direction : CW Greae type : 3 cst div.: µm Fig. al and imulation reult of circular trajectory in minute diplacement region For the imulation, two linear motion axe with the ame characteritic were employed and a circular trajectory wa generated. For the experiment, only one linear axi wa ued. The experimental circular trajectory wa generated by combining two meaured trajectorie; that i, one i not hifted and the other i hifted to create a 9 degree delay. From the circular trajectorie generated by experiment and imulation, the height of the quadrant glitch wa invetigated in detail. Figure 9 how the experimental and imulated reult for circular motion under variou radii, feed rate and greae vicoitie. A hown in Fig. 9(a) and (b), the height of the quadrant glitch change little, even if the greae vicoity change. A hown in Fig. 9(a) and (c), an important factor affecting the height of quadrant glitch i the centripetal acceleration; the higher the acceleration, the larger the quadrant glitch. Thee characteritic are accurately reflected by the propoed model for variou condition. The experiment went on to evaluate the propoed model in the minute diplacement region. In thi region, the non-linear pring characteritic i dominant. Figure how the experimental and imulated reult for a circular trajectory in the minute region. Quadrant glitche are conpicuouly abent. It follow that the non-linear friction characteritic doe not affect the generation of quadrant glitche in the minute diplacement region. Figure how the relationhip between the centripetal acceleration and the quadrant glitch height. Twelve centripetal acceleration were choen within the variou radii and feed rate. The quadrant glitch height were determined at 9 and 8 degree. The value at 9 degree i defined by maximum radiu in the range from to 9 degree. The value at 8 degree i defined by maximum radiu in the range from to 8 degree. It can be een that the quadrant glitch grow with increaing acceleration up to.3 m/. Subequently, it ettle to a contant value. Thi characteritic i reflected well in the imulated reult. However, the actual meaured height of the quadrant glitch i higher than that imulated. Figure how the relationhip between the radiu of the circular motion and the quadrant glitch height. The acceleration wa maintained at. m/. All experiment and imulation were conducted uing identical ervo gain. It can be een that the quadrant glitch height reduce with decreaing radiu. Again, thi behavior i reflected well in the imulated reult. From the experimental and imulated reult, it can be concluded that the propoed 68

9 Vol., No. 4, 8 mathematical model accurately reflect variou motion of feed drive ytem that are driven by linear motor and upported by ball guide with retainer. Fig. Height of quadrant glitch µm Acceleration m/ Influence of centrifugal acceleration on quadrant glitch Height of quadrant glitch µm Radiu mm Fig. Influence of radiu of circular motion on quadrant glitch 6. Concluion In thi paper, mathematical model for feed drive ytem in the minute diplacement region were dicued. In particular, non-linear pring characteritic were experimentally invetigated in detail. Baed on the experimental reult, a new non-linear friction model wa propoed. The following concluion were obtained: () The propoed non-linear friction model can be ued to accurately imulate the friction characteritic of the experimental et-up. () The propoed mathematical model yield the correct tep repone behavior under variou tep height in the minute diplacement region. (3) The quadrant glitch height for circular motion can be imulated for variou radii and feed rate. (4) Circular motion in minute diplacement region doe not generate quadrant glitche. (5) The propoed model reflect the tendency of quadrant glitch height to increae with increaing radiu of motion. The propoed mathematical model can expre the real behavior of a feed ytem that i driven by a linear motor. Thu, it can be an effective tool for deigning the mechanim and controller. The relation between the friction characteritic and the vibration damping of the ytem will be analyzed uing thi model. Acknowledgement The author would like to expre their thank to THK Co. Ltd. who kindly provided the experimental et-up and linear ball guide. Reference () S. Futami, A. Furutani, Nanometer Poitioning Uing AC Linear Motor and Rolling Guide (nd Report) Tribology of the Rolling Guide, Journal of the Japan Society for Preciion 683

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