120kw電機軟啟動隔爆箱優(yōu)化設(shè)計【全套含CAD圖紙、說明書】

資源目錄里展示的全都有，所見即所得。下載后全都有,請放心下載。原稿可自行編輯修改=【QQ：401339828 或11970985 有疑問可加】 ImEnnnlI0na1 Jmlml OF mmuwh PttocE5smG Int. J. Miner. Process. 44-45 (1996) 461-469 New comminution applications using hydrocone crushers with setting regulation in real time Arvid Svensson, Per Hedvall, Max Fjaestad Allis Mineral Systems, Crushing and Screening, Svedala, Sweden Abstract The setting (distance between mantle and concave ring) of the Hydrocone cone crusher is altered, even with the crusher operating at full load, by lifting or lowering the mantle by means of a hydraulic piston (Hydroset). Modem developments in the field of electronics and microcomput- ers have made it possible to design a small, reliable and very sophisticated system for automatic regulation of the setting of a Hydrocone crusher. The system monitors energy consumption, crushing force and setting and continuously adapts the crusher to even small variations in the feed and/or operating conditions. This technology has expanded the range of applications that can be successfully solved with cone crushers. In the paper we will give a general description of modem cone crusher technology and also give some examples of new possible applications. Fine crushing in closed circuit with a screen can, thanks to automation and carefully designed crushing chambers, compete with rod mills. Products smaller than 3 mm are common. Another example is the crushing of a gold ore to - 10 mm, in open circuit, with a Hydrocone crusher. The automation system ensures that the crusher runs at the smallest possible setting and thus ensures that the correct discharge is produced. In some applications it is interesting to operate cone crushers in a wet process. About 20 years ago, Allis Mineral Systems installed the first Hydrocones with water injected together with the feed. These crushers are still in operation and we will present our experiences. Small flexible cone crushers with computer control can be used to create more cost efficient ore crushing plants than the conventional installations with few large machines. The “crushing cassette” idea is presented. 1. Introduction The Hydrocone crusher is characterised by the hydraulic support of the mainshaft. The setting of the crusher (often called CSS, closed side setting) can be adjusted by moving the mainshaft up or down. The eccentric assembly forces the mainshaft in a gyrator!, (not rotating!) movement creating the crushing action between the concave ring and the mantle. See Fig. 1. 0301-7516/96/$15.00 0 1996 Elsevier Science B.V. All rights reserved SSDI 03 31-75 16(95)00052-6 462 A. Svensson et al./Int. J. Miner. Process. 44-45 (1996) 461-469 Fig. 1. Principle for Hydrocone crusher with ASR-plus. The concave ring and mantle are consumable wear parts made from abrasion resistant manganese alloyed steel. The shape of the profile of these liners are essential for high constant production in terms of capacity and reduction. The ideal profile is depending on the size and distribution of the feed particles, among other things (Svensson and Steer, 1990). In order to live up to a wide variety of applications, the Hydrocone crusher has seven different crushing chambers from extra coarse to extra fine. 2. Automatic setting regulation- ASR Allis Mineral Systems (AMS) have manufactured automatic setting regulation (ASR) systems for Hydrocone crushers since 1968. The first types were based on relays and had slow regulation and simple logic. They monitored motor power, hydraulic pressure and oil level in the Hydroset tank (indirectly mainshaft position). The systems were refined and in 1986 after producing 1550 units the computer based ASR-C was introduced. The unit had faster regulation and more accurate control allowing it to adjust the setting of the machine at very tine intervals (0.1 mm). This gave a new and unique opportunity to run the crusher at a selected maximum hydraulic pressure and motor power allowing the automation to find the corresponding “ideal” setting. With feed variables such as work index, granulometry, moisture content etc., constantly changing the “ideal” setting for the crusher will also change. In older designs, a fixed setting was chosen and then the hydraulic (or mechanical) pressure and A. Svensson et al./lnt. J. Miner. Process. 44-4.5 (1996) 461-469 463 consumeId power were allowed to vary. Depending on setting, this leads to frequent overloadmg of the crusher or low power utilisation i.e. inefficient crushing. With difficult feed materials, such as moist ores with clay contamination, both may occur! The cNoncave ring and mantle get worn with use. This means that the setting between them increase due to this. A manually operated crusher must be calibrated to compensate for this. A crusher running with power and hydraulic pressure as guide will not encounter this problem, the setting is automatically adjusted. With abrasive materials, like gold ore or quartzite, the crusher setting can increase 1 mm due to wear in a single shift. If this happens in a fine crusher running at CSS 6-7 mm, the relative impact is drastic. In 19 22, the ASR-C was replaced (after 500 units) with the ASR-plus. This system is quicker and has a better, adaptive regulation algorithm. It is easier to programme and has a colmputer communication interface (RS-485) included as standard. Via a telephone modem and a PC we have the possibility to reprogram an ASR-plus system from long distances. The ASR-plus has a memory which can store five different crushing “recipes” and also data on the performance of the crusher. Some of the 249 variables that are available are: CSS, CSS setpoint, average CSS during given period, same for min and max CSS, power, max power, pressure, max pressure, wear, % remaining of total mainshaft travel, regulation modes for 5 recipes, regulation damping for 5 recipes, total and loaded operation time since new, same since latest liner change, same since latest calibration, energy consumption and total pump time. 3. Fine crushing Traditionally crushers have been used for production of fractions down to - 12 or - 16 mm, suitable for primary grinding in rod mills. Today it is feasible to make a - 3 mm product in closed circuit or a - 10 mm product in open circuit. By u,sing controlled feed analysis, optimised crushing chamber and ASR it is possible to create a pressure zone in the crushing chamber leading to interparticle crushing. Some factors to consider are: 3.1. Feed granulometry It is important that the crushing chamber has sufficient intake opening to swallow the largest particles in the feed with great ease. The feed fraction must also have sufficient amount of voids to avoid packing. 3.2. Work index The hardness of the rock is measured by the Impact Work Index test method developed by Mr. Fred C. Bond of Allis Chalmers (today Allis Mineral Systems). This method is based on particles from 50 to 75 mm and has no correlation to the Grinding Work Index method developed by the same man. In general, we assume that soft rock like lim.estone has WI = 12, medium hard rock like granite has WI = 16 and hard rock such as basalt has WI = 20 (kWh/tonne). 464 A. Svensson et al./Int. J. Miner. Process. 44-45 (1996) 461-469 3.3. Density A conecrushers act similarly to a piston pump in the sense that it crush on a certain volume of rock for each revolution. This means that a heavy rock will have a higher capacity than a comparable one with lower density. 3.4. Moisture The moisture in the feed will be absorbed evenly on the particle surfaces. In practice this means that most of the moisture will be absorbed by the finest particles and increase their adhesion to each other and the crushing surfaces. This means that capacity will drop with increasing content of moisture and with decreasing setting. See Fig. 2. 3.5. Choke feeding For a cone crusher to work in a relatively steady state, it must be choke fed. This means that the volume above the crushing chamber always is full of material which flows into the crushing chamber at a pace decided by the crusher. See Table 1 and Fig. 3. 3.6. Distribution If the feed material is segregated on entry to the crusher, the crusher will have to adjust to the area of the chamber where the most difficult crushing takes place. The rest of the chamber will get the same setting and thus not work to its full potential. The chamber will be partly choke fed and partly starve fed. 3.7. Example of performance from a fine crushing circuit A fine crushing circuit as shown in Fig. 4 can accept a feed of 4-12 mm of dry gold ore and make around 50-55 mtph of - 4 mm product using an installed power of 132 kW. Fig. 2. Capacity reduction due to moisture content. A. Suensson et al./Int. J. Miner. Process. 44-45 (1996) 461-469 465 Table 1 Importance of choke feeding a cone crusher. Crusher: Hydrocone H-36-M, throw 32 mm. Feed material: Gneiss-Diabase. Feed size: 3-25 mm, 50% 3-9 mm CSS (mm) % css % Capacity (l/h) kWh/t -6 mm t/h -6 mm Choke fed Starved 9.5 9.5 72 56 50 34 90 45 3.9 1.8 107 77 1.67 1.73 54 27 3.8. Fine crushing in open circuit A new concept in fine crushing is to use the ASR system as a guarantee for correct final size as opposed to use a screen for this duty. The benefits are obvious, lower installation cost since no final screens or return conveyors for coarse material are needed. A practical example of this is shown in the flowsheet in Fig. 5. This is taken from a Teberebie gold fields in Ghana where the final stage is composed by two H-4000-EF cone crushers with ASR-C control systems. Each crusher is fed with 100 mtph of lo-25 and crushes this down to - 10 mm (P, = 9.3 mm) in open circuit. The crushed ore goes directly to gold benefication by agglomoration, cyanide leaching and active carbon recovery. STOP SIGNAL TO FOFlEGOlNG FEEDER 4-e -_ _ I I / -I # , nMlN. IN SURGE SIN CONTROL SIGNAL TO FEEDER MAX. IN CRUSHER FEE0 HOPPER w Ei 0” 0 aocpo 0” 0000 oq OOOO OOOO SWITCHGEAR UNIT Fig. 3. How to obtain choke fed crusher. 466 A. Suensson et al./lnt. .I. Miner. Process. 44-45 (1996) 461-469 4. Wet crushing In special applications it may be interesting to crush rock or ores in wet conditions. Normally this only takes place when the feed material already is moist or wet from a preceding process, a typical case is crushing of critical particles from a SAG or AG mill. In some cases it is interesting to actually add water to the feed prior to crushing. A typical case could be crushing of materials which cause hazardous dust. The amount of water must be sufficient to give a large quantity of free water and a absolutely soaked feed. This means that the water content is depending on the type and granulometry of the feed. In general we use between 0.75 and 1.5 cubic meters of water per tonne of feed material. In general the following points apply for a wet crushing process with Hydrocone crushers: * Capacity through the crusher is increased due to transport of fine material by the water flow. . The power consumption for a given setting is comparable to that of a dry process. . The discharge product contains less fines in comparison to a dry process. CONE CRUSHER ;m?Mx t-3UOO-EF EXTiEM F/G CRUWI. ASRpl us O-f mm Fig. 4. Flowsheet of fine crushing circuit with H-3000-EF. Fig. 5. Flowsheet from Teberebie Goldfields, Ghana. A. Svensson et al./Int. J. Miner. Process. 44-45 (1996) 461-469 461 2xH-84 3 x H-4000 Motor power (kw) 2 x 300 = 600 3 x 200 = 600 I Crusher weight, incl. - 85 500 each - 17 250 each motor & sub-frame (kg) I I Heaviest Iii (kg) - 28 100 - 17 250 I Relative investment incl. motors, starters, feeders, - 230 - 100 support frames, etc. -J Fig. 6. Comparison of large and small cone crushers. * The crusher can normally be operated at smaller CSS than in a dry process. . The life of wear parts are reduced to approximately 70% due to corrosion. ?The crusher is even more sensitive to feed variations and the ASR are consequently set up for very fast regulation. 5. The crushing cassette system 5.1. Small versus big crushers in large crushing plants Many crushing plants for large tonnages (500-2000 mtph) are built after the philosophy that “big is beautiful”. The main design concern has been to use as few machines as possible, leading to very large machines and simple process solutions. A new approach is a crushing plant based on smaller machines in a more optimised process, Some of the arguments for this development are: 468 A. Svensson er al./lnr. J. Miner. Process. 44-45 (1996) 461-469 Fig. 7. Schematic design of Hydrocone 4000 crushing cassette. The unit can be rolled on tracks into a service facility and be replaced by a unit with new liners. Very large cone crushers (with cone diameter of 2.1 m and bigger) represent less than 15% of cone crusher sales world-wide. This means that smaller crushers are manufac- tured in larger series and therefore can be produced more costeffectively. Small crushers generally give more crushing for the dollar. More or less the same applies for vibrating screens. See Fig. 6 (from Svensson and Steer, 1990). In a plant with a larger number of crushers the output of the plant is less dependent on each individual machine. The long-term average capacity and availability are improved. A crushing process with more machines can allow for the individual crushers to be more optimised for their duties. The main drawback with smaller machines is that they have smaller wear parts and thus need to change liners more often (the actual wear rate in grams per tonne of feed is equal or better for smaller crushers). This is compensated for by the far superior serviceability of smaller crushers. If we compare the practicality of a 84” Hydrocone A. Svensson et al./Int. J. Miner. Process. 44-45 (1996) 461-469 449 (installed power 300 kW) weighing 78 200 kg with two 4000 Hydrocones (2 X 200 kW) each weighing 14000 kg the differences are clear. The topshell (19 600 kg) and mainshaft (23 500 kg) of the big machines are heavier then the entire H-4000. 5.2. Crushing cassette Flexibility was one of the most important design parameters of the modem Hydro- cone crushers. All seven crushing chambers (EF, F, MF, M, MC, C and EC) can be fitted to the same topshell. The eccentric bushings have 3 or 4 keyways cut to provide simple change of eccentric throw. As an example we can transform a H-4000-EC (capable of crushing a - 210 mm feed) into a H-4000-EF (capable of making a product with P, = 6.5 mm from 6-12 mm feed) just by changing wear parts (mantle and concave ring). The beauty of this in a large plant is that virtually identical crushers can be installed in secondary, tertiary and quartenary stages and then after a change of liners go into work in any of the other positions. See Fig. 7. Reference Svensson, A. and Steer, J.F., 1990. New cone crusher technology and developments in comminution circuits. Miner. Eng., (l/2): 83-103. Journal of Materials Processing Technology 211 (2011) 141149Contents lists available at ScienceDirectJournal of Materials Processing Technologyjournal homepage: new in-feed centerless grinding technique using a surface grinderW. Xua, Y. WubaGraduate School, Akita Prefectural University, 84-4 Tsuchiya-ebinokuchi, Yurihonjo, Akita 015-0055, JapanbDepartment of Machine Intelligence and Systems Engineering, Akita Prefectural University, 84-4 Tsuchiya-ebinokuchi, Yurihonjo, Akita 015-0055, Japana r t i c l ei n f oArticle history:Received 15 April 2010Received in revised form15 September 2010Accepted 16 September 2010Keywords:Centerless grindingSurface grinderUltrasonic vibrationIn-feedRoundnessShoea b s t r a c tThis paper deals with the development of an alternative centerless grinding technique, i.e., in-feed cen-terless grinding based on a surface grinder. In this new method, a compact centerless grinding unit,composed of an ultrasonic elliptic-vibration shoe, a blade and their respective holders, is installed ontothe worktable of a surface grinder, and the in-feed centerless grinding operation is performed as arotating grinding wheel is fed in downward to the cylindrical workpiece held on the shoe and theblade. During grinding, the rotational speed of the workpiece is controlled by the ultrasonic elliptic-vibration of the shoe that is produced by bonding a piezoelectric ceramic device (PZT) on a metalelastic body (stainless steel, SUS304). A simulation method is proposed for clarifying the workpiecerounding process and predicting the workpiece roundness in this new centerless grinding, and theeffects of process parameters such as the eccentric angle, the wheel feed rate, the stock removal andthe workpiece rotational speed on the workpiece roundness were investigated by simulation followedby experimental confirmation. The obtained results indicate that: (1) the optimum eccentric angle isaround 6; (2) higher machining accuracy can be obtained under a lower grinding wheel feed rate,larger stock removal and faster workpiece rotational speed; (3) the workpiece roundness was improvedfrom an initial value of 19.90?m to a final one of 0.90?m after grinding under the optimal grindingconditions. 2010 Elsevier B.V. All rights reserved.1. IntroductionIn the manufacturing industry, for high accuracy and highproductivity machining of cylindrical components such as bear-ing raceways, silicon-ingots, pin-gauges and catheters, centerlessgrinding operations have been extensively carried out on spe-cialized centerless grinders. Two types of centerless grinders areavailable commercially; one is with a regulating wheel and theotherwithashoe,andtheyaredifferentfromeachotherinhowtheworkpiece is supported and how the workpiece rotational speedis controlled during grinding. Since the invention of the regu-lating wheel type centerless grinder by Heim in 1915 (Yonetsu,1966), much research has been devoted to enhance machiningaccuracy and efficiency. Rowe and Barash (1964) proposed a com-puter method for investigating the inherent accuracy of centerlessgrinding by taking into account the geometrical considerationsand the elastic deflexion of the machine. Further, Rowe et al.(1965) experimentally obtained the machining elasticity parame-ter.Hashimotoetal.(1982)analyzedtheproblemofsafemachiningoperation by discussing the friction-drive function of regulatingCorresponding author. Tel.: +81 184 272157; fax: +81 184 272165.E-mail addresses: d10s007akita-pu.ac.jp, (W. Xu).wheel. Miyashita et al. (1982) studied the deformation of con-tact area and built a dynamic model for selecting chatter freeconditions. Rowe and Bell (1986) experimentally investigated thehigh removal rate grinding process and optimized the grindingconditions. Wu et al. (1996) clarified the influence of grindingparameters on roundness error through a computer simulationmethod to optimize grinding conditions. Epureanu et al. (1997)analyzed the stability of grinding system through a linearizedmodel that described the formation and evolution of the patternon the ground surface. Guo et al. (1997) studied the geometri-cal rounding of above-center and below-center centerless grindingto assist in the selection of acceptable set-up conditions. Albizuriet al. (2007) proposed a novel method to reduce chatter vibra-tions by using actively controlled piezoelectric actuators. Krajniket al. (2008) developed an analytical mode that assists in efficientcenterless grinding system set-up for higher process flexibilityand productivity. Shoe type centerless grinding has also attractedattention from both industrial and academic researchers. Yang andZhang (1998) designed a flat vacuum-hydrostatic shoe to increasethe load capacity and stiffness for high precision applications ofshoe centerless grinding. Then Yang et al. (1999) and Zhang etal. (1999) analyzed the process stability in vacuum-hydrostaticshoe centerless grinding. In addition, Zhang et al. (2003) devel-oped a geometry model to predict the lobing generation in shoe0924-0136/$ see front matter 2010 Elsevier B.V. All rights reserved.doi:10.1016/j.jmatprotec.2010.09.009142W. Xu, Y. Wu / Journal of Materials Processing Technology 211 (2011) 141149Fig. 1. Three types of centerless grinding using a surface grinder: tangential-feed type (a), in-feed type (b) and through-feed type (c).centerless grinding and used the model for the analysis of grindingprocesses.From the viewpoint of production cost, the two types ofcenterless grinders are highly suitable for small-variety and large-volume production because the loading/unloading of workpiecesis extremely easy and fast. However, the centerless grinder isa special-purpose machine and relatively costly, putting it at adisadvantage for large-variety and small-volume production, thedemand for which has increased rapidly in recent years. As a solu-tion to this problem, one of the authors Wu et al. (2005) proposeda new centerless grinding technique that can be performed on asurface grinder (rather than a centerless grinder) previously. Thismethodisbasedontheconceptofultrasonicshoecenterlessgrind-ing developed by Wu et al. (2003, 2004). In the method, a compactunit consisting mainly of an ultrasonic elliptic-vibration shoe, ablade,andtheirrespectiveholdersisinstalledontheworktableofamultipurposesurfacegrinder.Thefunctionoftheultrasonicshoeistoholdthecylindricalworkpieceinconjunctionwiththeblade,andto control the workpiece rotational speed with the elliptic motionon its upper end-face.According to the relative motion of the workpiece to thegrinding wheel, three types of centerless grinding operations canbe performed in the proposed method as shown in Fig. 1: (a)tangential-feed type in which initially the grinding unit is locatedin the down-left side of the grinding wheel with a distance thatis large enough for loading the workpiece on the upper end faceof ultrasonic shoe, and then the workpiece is fed rightward alongthe tangential direction of the grinding wheel at a feed rate ofvf(Fig. 1(a) to perform the grinding action until the unit reaches thedown-right side of the grinding wheel with a distance that is largeenoughforunloadingthegroundworkpieceofftheultrasonicshoe;(b) in-feed type in which initially the grinding wheel is locatedabove the grinding unit with a distance that is large enough forloading the workpiece on the upper end face of ultrasonic shoe,and then the grinding wheel is fed downward in radial directioninto the workpiece at a feed rate ofvfr(Fig. 1(b) to perform thegrinding action until the required stock removal has been attained,and after a short period for “spark-out” the grinding wheel is liftedup from the ground workpiece with a distance that is large enoughto unloading the workpiece off the ultrasonic shoe; (c) through-feed type in which initially the grinding wheel is set at a givendistance from the upper end face of ultrasonic shoe (as shown inFig. 1(c), and then the workpiece is loaded on the loading guideand fed into the space between grinding wheel and ultrasonic shoealong its axial direction at a feed rate ofvfato perform the grindingaction until it loses the contact with the grinding wheel but beingsupported on the unloading guide for the subsequent unloading.Inourpreviousworks,simulationandexperimentalworkshavebeen conducted for the tangential-feed type (Xu et al., 2010). Theobtained results showed that the workpiece roundness can beFig. 2. Schematic illustration of in-feed centerless grinding using a surface grinder.W. Xu, Y. Wu / Journal of Materials Processing Technology 211 (2011) 141149143improved greatly from the initial value of 23.9?m to the final oneof 0.8?m, thus validating this new method. The objective of thepresent paper is to confirm the in-feed type of centerless grind-ing carried out on a surface grinder. For this purpose, a simulationmethod is proposed to clarify the workpiece rounding processand to investigate the effects of process parameters, such as theworkpiece eccentric angle, the grinding wheel feed rate, the stockremoval, and the workpiece rotational speed on the workpieceroundness. Then a series of grinding experiments are carried outto confirm the simulation results.2. Operation principle of in-feed centerless grinding usinga surface grinderFig. 2 shows the operation principle of in-feed centerless grind-ing using a surface grinder. A grinding unit, composed of anultrasonic elliptic-vibration shoe and its holder, a blade and itsholder,astopper,andabaseplate,isinstalledontotheworktableofa surface grinder at an angle of (hereafter called eccentric angle)(see Fig. 2(a). The workpiece is constrained between the blade, theshoeandthestopper.Asthegrindingwheelisfedinradialdirectionintotheworkpieceatfeedrateofvfr,anin-feedtypedown-grindingoperationisperformed,wheretheworkpieceisrotatedintheoppo-site direction to the wheel. As shown in Fig. 2(b), once the requiredstock removal has been attained, the wheel in-feed is stopped fol-lowed by a dwell for several seconds to allow “spark-out”. Duringgrinding, the workpiece rotational speed nwis controlled by theelliptic motion on the upper end-face of the shoe and the stop-per is used to prevent workpiece from jumping away the grindingarea. In addition, the blade is wedge-shaped with a tilt angle of ?(usually called blade angle) and the value of ? is in general set ataround 60in terms of the optimum workpiece rounding conditiondemonstrated by Harrison and Pearce (2004).In the grinding unit, the shoe is constructed by bonding a piezo-electric ceramic device (PZT) with two separated electrodes ontoa metal elastic body (stainless steel, SUS304). When two amplifiedalternating current (AC) signals (over 20kHz) with a phase differ-ence of , generated by a wave function generator, are applied tothe PZT, bending and longitudinal ultrasonic vibrations are excitedsimultaneously.Thesynthesisofthevibrationdisplacementsinthetwo directions creates an elliptic motion on the end-faces of themetal elastic body. Consequently, the workpiece rotation is con-trolled by the frictional force between the workpiece and the shoe,so that the peripheral speed of the workpiece is the same as thebending vibration speed on the shoe end-face. The workpiece rota-tional speed can be adjusted by changing the value of parameterssuch as the amplitude Vppand frequency f of the voltage appliedto the PZT, because the shoe bending vibration speed varies withthevariationoftheappliedvoltage(seeXuetal.,2009).Inaddition,a pre-load is applied to the shoe at its lower end-face in its longitu-dinal direction using a coil spring to prevent the PZT from breakingdue to resonance.3. Geometrical rounding analysisFig.3showsthegeometricalarrangementoftheshoe,theblade,the workpiece and the grinding wheel in an in-feed centerlessgrinding operation using a surface grinder after grinding for timet. At this moment, the eccentric angle and the workpiece radiusbecome (t) and ?(t), respectively, from their respective initial val-uesof0and?0.Inthemeantime,theworkpieceisheldbytheblade(with a tilt angle of ?) and the shoe at points B and C, respectively,and ground at point A by the grinding wheel rotating at the rota-tional speed of ngas the wheel is fed downward into the workpieceat a feed rate ofvfr.Fig. 3. Geometrical arrangements in in-feed centerless grinding using a surfacegrinder.3.1. Geometrical rounding modelingIn the simulation model (see Fig. 3), several assumptions aremade: (1) the workpiece is in constant contact with the blade andthe shoe at points B and C, respectively, during grinding; (2) thevibration of the entire machine is too small to be regarded, andno chatter occurs on the machine due to the ultrasonic elliptic-vibrationoftheshoe;(3)theworkpiecerotationalmotionisalwaysstable, and no variation of rotational speed occurs during grinding;(4) the wear of the grinding wheel is too small to be recognized,and the grinding wheel radius Rgis kept constant during grinding.Let a XY-coordinate system be located on the worktable. AnoptionalpointOontheworktableisdeterminedastheoriginofthecoordinate system. The X-axis is taken in the horizontal directionand the Y-axis in the vertical direction. Before grinding, the initialXY-coordinatesofthegrindingwheelcenterOg0andtheworkpiececenter Ow0are (XOg0, YOg0) and (XOw0, YOw0), respectively. Thus, theXY-coordinates of the initial blade contact point B (XB0, YB0) andthe shoe contact point C (XC0, YC0) can be obtained from the initialgeometrical arrangement, as follows:?XB0= XOw0 ?0sin?YB0= YOw0+ ?0cos?XC0= XOw0YC0= YOw0 ?0Then, the linear equations representing the blade end-face and theshoe upper end-face in this coordinate system can be written as:For the blade end-face:Y YB0= tan?(X XB0)(1)For the shoe upper end-face:Y YC0= 0(2)Substituting coordinates of point B and C into Eqs. (1) and (2),respectively, gives:PX + QY + R = 0(3)Y YOw0+ ?0= 0(4)whereP = tan?,Q = 1,R = YOw0+ ?0cos? tan?(XOw0?0sin?).During grinding, the coordinates of the workpiece centerOwtand the grinding wheel center Ogtwill vary as the material isremoved. Let the instantaneous workpiece radius in the directionparallel to the X-axis after grinding for time t be ?(t) (see Fig. 3).144W. Xu, Y. Wu / Journal of Materials Processing Technology 211 (2011) 141149At this moment, the workpiece radius at points A, B and C canbe expressed with ?(tTA), ?(tTB) and ?(tTC), respectively,where TA=?+2(t)/4?nw, TB=(?+2?)/4?nwand TC=3/4nware the time delays for points A, B and C. Since the ?(tTB) and?(tTC) are equal to the distances from the workpiece center Owtto the blade end-face and to the shoe upper end-face, respectively,they can be obtained from the geometrical arrangement in Fig. 3by using Eqs. (3) and (4) as follows:?(t TB) =|PXOw(t) + QYOw(t) + R|?P2+ Q2(5)?(t TC) = YOw(t) YOw0+ ?0(6)Solving Eqs. (5) and (6) simultaneously yields the XY-coordinatesof the workpiece center Owtat time t, as follows:XOw(t) =?P2+ Q2?(t TB) QYOw(t) RPYOw(t) = ?(t TC) + YOw0 ?0(7)In this moment, the XY-coordinates of the grinding wheel centerOgtare also obtained from the geometrical arrangement in Fig. 3as:?XOgt= XOg0= XOw0 (Rg+ ?0)sinYOgt= YOg0vfrt = YOw0+ (Rg+ ?0)cos vfrt(8)In addition, the following relationships are established from thegeometrical arrangement in Fig. 3.?XA(t) XOg(t)2+ YA(t) YOg(t)2= R2gYA(t) YOw(t) = cot(t)XA(t) XOw(t)(9)where:cot(t) =YOg(t) YOw(t)XOg(t) XOw(t)(10)Subsequently,theXY-coordinatesofpointAareobtainedasfollow-ing by re-arranging Eqs. (9) and (10).XA(t) =(V ?V2 4UW)2UYA(t) = cot (t)XA(t) XOw(t) + YOw(t)(11)whereU = 1 + cot2(t),V = 2cot(t)YOw(t) cot2(t)XOw(t) cot (t)YOg(t) XOg(t),W = X2Og(t) +cot(t)XOw(t) YOw(t) + YOg(t)2 R2gEventually,thework-piece radius ?(tTA) at the point A after grinding for time t iscalculated from the XY-coordinates of the workpiece center Owtand the grinding point A as follows:?(t TA) =?XA(t) XOw(t)2+ YA(t) YOw(t)2(12)Consequently,theapparentwheeldepthofcutwouldbe?=?(tTAT)?(tTA), where T is the time required for onerevolutionoftheworkpiece.Ifthegrindingsystemhasanidealstiff-ness, the true wheel depth of cut would be equal to the apparentone.However,thegrindingsystemwithstandstheelasticdeforma-tion caused by the grinding force during actual grinding. Rowe etal. introduced a dimensionless parameter called machining elastic-ity parameter k as a measure to indicate the elastic deformation ofcenterless grinding system, which is defined as a quotient betweenthe true depth of cut and the apparent depth of cut with Eq. (13)(Rowe and Barash, 1964; Marinescu et al., 2006).k =truewheeldepthofcutapparentwheeldepthofcut=?(13)Following Rowe et al.s consideration, the true wheel depth of cut? can be calculated as ? =k?in the current work, resulting in thatthe true workpiece radius at point A is:?(t TA) = ?(t TA T) k?(t TA T)?XA(t) XOw(t)2+ YA(t) YOw(t)2(14)However, the wheel depth of cut calculated using these equationsis less than zero occasionally. Obviously, this phenomenon wouldnot happen. Therefore, Eq. (14) should be modified as:?(t TA) = ?(t TA T) k(?(t TA T) ?XA(t) XOw(t)2+ YA(t) YOw(t)2)?(t TA) ?(t TA T)?(t TA) = ?(t TA T)?(t TA) ?(t TA T)(15)3.2. Determination of the machining elasticity parameterAs described above, the machining elasticity parameter kdepends on the stiffness of the grinding system. If the simulationresult is to be trusted, the value of k should be determined forthe given grinding system. For the tangential-feed type centerlessgrindingusingasurfacegrinder,themeasuringmethodofparame-ter k was proposed in our previous work (Xu et al., 2010). However,the proposed method is unsuitable for the in-feed type, since thereisasignificantdifferencebetweenthegeometricalarrangementsinthe two types. Therefore, an alternative method should be devel-oped for obtaining the machining elasticity parameter in in-feedtype. Rowe et al. (1965) proposed a method for the determinationof machining elasticity parameter k in conventional in-feed center-less grinding, in which a parameter proportional to the true wheeldepth of cut, i.e., the grinding power or the grinding force, during“feed-in” or “spark-out” stage is measured to obtain the parameterk. Following this methodology, in the current work an alternativeprocedure is proposed to determine the value of k for the in-feedtype centerless grinding based on a surface grinder as below.In spark-out, the apparent wheel depth of cut is removed justafterafewrevolutionsofworkpiece.Thedecreaserateofthewheeldepth of cut depends on the value of parameter k. If the apparentwheeldepthofcuthasavalueofa0atthecommencementofspark-out, the true depth of cut in the first half-revolution, ae1, and that inthesecondhalf-revolution,ae2,canbecalculatedwithEqs.(16)and(17), respectively, according to Rowe et al. (1965) and Marinescuet al. (2006).ae1= ka0(16)ae2= k(a0 ae1) = (1 k)ae1(17)Hence, the true depth of cut in the ith half-revolution, aei, can beobtained as:aei= k(a0 ae1 ae2 . aei1) = (1 k)i1ae1= .= (1 k)imaem(m = 1, 2, .,i)(18)Since the true depth of cut is proportional to the normal grindingforce Fnin general, the following relationship is obtained from Eq.(18):FniFnm=aeiaem= (1 k)im(19)Solving Eq. (19) yields:k = 1 e?(20)W. Xu, Y. Wu / Journal of Materials Processing Technology 211 (2011) 141149145Fig. 4. Schematic illustration of grinding force measurement method.whereeisthebaseofnaturallogarithmsand?=(lnFnilnFnm)/(im). Consequently, the value of k can bedetermined with Eq. (20) as long as the normal grinding forcesFniand Fnmafter spark-out for i and m revolutions have beenmeasured.Fig.4showsagrindingforcemeasurementmethodproposedforthecurrentwork.A3Ddynamometerisinstalledunderthegrindingunit to record the horizontal component Fxin the X-direction andthe vertical component Fyin the Y-direction of the grinding force.Thus, based on the geometrical arrangement shown in Fig. 4, thefollowing relationship is obtained:?Fncos + Ftsin = FyFnsin Ftcos = Fx(21)where Fnand Ftare the normal and tangential grinding forces,respecti