資源目錄里展示的全都有，所見即所得。下載后全都有,請(qǐng)放心下載。原稿可自行編輯修改=【QQ：401339828 或11970985 有疑問(wèn)可加】 International Journal of Fatigue 25 (2003) force spectra of CNC machine tools and theirapplications, part two: reliability design of elementsYiqiang Wang, Guixiang Shen, Yazhou JiaCollege of Mechanical Engineering, Jilin University, Changchun, 130025, Peoples Republic of ChinaAbstractThis paper deals with the reliability design of elements of computerized numerical control (CNC) machine tools and the practicalapplications of multidimensional force spectra of CNC machine tools described in part one of this paper. To illustrate the applicationof multidimensional force spectra, part two considers the design example of an S1-273 CNC lathe. First the force distribution oftransmission elements is calculated. Then the equivalent fatigue force with specified reliability for the important transmissionelements of the lathe under various working conditions is calculated. Use of the equivalent fatigue force determined to design theCNC lathe element enables obtaining the expected reliability. 2003 Elsevier Science Ltd. All rights reserved.Keywords: CNC machine tools; Multidimensional force spectra; Reliability design; Equivalent fatigue force1. IntroductionThe traditional engineering approaches are to designsafety margins, or safety factors, into the equipment. Thesafety factor (SF) is defined as the ratio of the capabilityof the system to the force placed on the system. Thesafety margin (SM) is the difference between the systemcapability and the force. Failure will occur if the safetyfactor is less than 1 or the safety margin becomes nega-tive. This is often a deterministic approach that ignoresthe variability present in both the forces placed on a sys-tem and the systems ability to react the force 1.By the late 1930s both forces and strengths were beingcommonly expressed as statistical distributions. Theprobability and statistics theories were employed intoengineering design 2. It is difficult to implement theprobabilistic design in engineering for the sake of thedetermination of forces and strengths distributions. It hasbeen shown that the distribution of material resistance(such as yield strength, tensile strength, etc.) generallyfollows a normal distribution; However the distributionof applied forces acting on the element is difficult to beCorresponding author. Tel.: +86-431-5601482.E-mail address: , http:/ (Y. Wang).0142-1123/03/$ - see front matter 2003 Elsevier Science Ltd. All rights reserved.doi:10.1016/S0142-1123(02)00136-6determined 37. The determination of force distri-bution has become the barrier to the reliability design.In recent years, the force spectra of a number ofmachines, such as airplanes, automobiles and so on,were investigated 811.To a large degree, reliability is an inherent attributeof a system, component, or product. As such, it is animportant consideration in the engineering design pro-cess. Reliability design is an iterative process that beginswith the specification of reliability goals consistent withcost and performance objectives. Once the reliabilitygoals have been established, these goals must be trans-lated into individual component, subcomponent, and partspecifications 1,12. After individual component andpart requirements have been determined, various designmethods can be applied in order to meet the goals. Partone of this paper proposed the multidimensional forcespectra of CNC machine tools. This paper determinesthe distribution of forces acting on the element based onthe multidimensional force spectra and the transmissionchain diagram of a specific CNC machine tool. The equi-valent fatigue force of the element can then be determ-ined with specified reliability.448Y. Wang et al. / International Journal of Fatigue 25 (2003) 4474522. Brief introduction to the S1-273 CNC latheThe S1-273 lathe studied in this paper is an all-func-tioned CNC lathe fabricated by a Chinese machine-building mill. A schematic diagram of the main trans-mission is shown in Fig. 1. The lathe is driven by acontinuous speed regulation alternating current (AC)motor by means of a timing belt and two pairs of shiftingslide gears. The timing belt transmits the power andmotion from the motor to shaft I in the headstock. Thediameters of the driving and the driven pulley are 125and 230 mm respectively. The motion through the gearpairs with teeth 28/70 and 49/49 is transmitted fromshaft I to shaft II, and through the gear pairs with teeth59/47 and 30/76 from shaft II to shaft III, i.e. the spindle.The rated speed of the motor is 1500 rev/min, itsmaximum speed is 3000 rev/min and rated power is 11kW. The rated speed is defined as the speed at whichthe motor may output the maximum torque or power, ingeneral, it is the lowest speed that can deliver the ratedpower 13,14. Conventionally, the speed range from therated speed to the maximum speed is called the constantpower region; the speed range below the rated speed iscalled constant torque region. The spindle speeds of thislathe range from 20 to 2000 rev/min. It is obvious thatthe use of shifting slide gears is not to enlarge the speedrange but to enlarge sufficiently the constant powerregion within the whole speed range.The speed diagram of the lathe is shown in Fig. 2.The vertical coordinate is the speed in logarithmic scale.The transmission shafts are shown as vertical parallellines at equal distances from each other. The Romannumerals at the bottom of the diagram correspond to thenumbers in Fig. 1. The points on each vertical line indi-cate the margins of speed ranges for each speed level.Fig. 1.Main transmission of the CNC lathe.Fig. 2.Speed diagram of the CNC lathe.The lines jointing the speeds between two axes indicatethe transmission ratios. The numbers beside the oblique(or horizontal) lines are the numbers of teeth of themeshing gears, corresponding to Fig. 1. Equal gear ratiosare represented by parallel lines.The speed diagram indicates directly not only theactual speed ranges of the various shafts and gears, butalso the intermediate gear pairs through which thesespeeds are obtained. For example, shaft I has one levelof speed range from 125 to 1630 rev/min obtained fromthe motor through the pulley pair 125/230; the spindlespeed range from 20 to 250 rev/min is obtained fromthe motor through the pulley pair 125/230, the gear pair28/70 between shafts I and II and the gear pair 30/76between shafts II and III.The spindle speed range is divided into four speedlevels, i.e., level M21: 20250 rev/min, level M22: 250630 rev/min, level M23: 315800 rev/min and levelM24: 8002000 rev/min. M21, M22, M23 and M24 aremiscellaneous codes of NC program and instruct eachof the four speed levels respectively. It should be notedthat level M22 and M23 overlap within the speeds of315630 rev/min.The main transmission expression is given asMotor?125?230?I ?49492870?II(1)449Y. Wang et al. / International Journal of Fatigue 25 (2003) 447452?59473076?III (Spindle)For four speed levels mentioned above, their individ-ual transmission ratios and corresponding spindle speedsare given in Table 1. nsis the spindle speed (in rev/min),nmis the motor speed (in rev/min), which changes con-tinuously from 0 to 3000 rev/min, u1, u2, u3, u4are thetransmission ratios of each level respectively, njis thebasic speed of spindle for each level corresponding tothe rated speed of the motor.3. Determination of the force distribution oftransmission elementTake the gear with 28 teeth mounted on shaft I as anexample. From Figs. 1 and 2, it can be seen that thereare two routes from shaft I to shaft III while the 28-toothgear on shaft I is working. One is that the motion isobtained from the motor through the pulley pair 125/230,the gear pair 28/70 between shafts I and II and the gearpair 30/76 between shafts II and III. The correspondingspindle speeds range from 20 to 250 rev/min, relativespeeds are from 0.01 to 0.125. The transmission ratiofrom shaft I to shaft III is u1? = 0.158. The relative torqueof spindle is x = TS/TR, TSis the torque of the spindle,TRis the rated torque of the spindle. The torque of thegear with 28 teeth is TZ= u1?TS, The relative torque ofthe gear with 28 teeth corresponding to the rated torqueof the spindle is z = TZ/TR= u1?TS/TR= u1?x.Another is that the motion is obtained from the motorthrough the pulley pair 125/230, the gear pair 28/70between shafts I and II and the gear pair 59/47 betweenshafts II and III. The corresponding spindle speeds rangefrom 63 to 800 rev/min, speeds from 315 to 800 rev/minare used in practical, the relative speeds are from 0.1575to 0.4. It should be noted that the speeds from 315 to630 overlap with those of another speed level; the rela-Table 1Spindle speed of each levelLevel M21Level M22Level M23Level M24Transmission ratio, u1u1=12523028703076= 75/874 u2=12523049493076= 375/u3=12523028705947= 590/u4=12523049493076= 1475/174821622162Spindle speed, ns(rev/min)ns= u1nm,20 ? 250ns= u2nm,250 ? 630ns= u3nm,315 ? 800ns= u4nm,800 ? 2000Basic speed, nj(rev/min)1293224091023Range ratio, Rn= nmax/nmin12.52.52.52.5tive speeds are from 0.1575 to 0.315. The transmissionratio from shaft I to shaft III is u2? = 0.502. The torqueof the gear with 28 teeth is TZ= u2?TS, The relativetorque of the gear with 28 teeth corresponding to therated torque of the spindle is z = TZ/TR= u2?TS/TR=u2?x. To simplify the calculation, we assume that thechances of occurrences of these two speed levels areequal, that is, the probability of occurrence of each speedlevel is 0.5 within the speeds from 315 to 630 rev/min.Thus, the probability density function of force distri-bution for the gear with 28 teeth mounted on shaft I is,f(z) ? a?0.1250.01f(z/u1,y)dy ?0.3150.157512f(z/u2,y)dy(2)?0.40.315f(z/u2,y)dy?,where, f(x,y) is the multidimensional force spectrum ofthe medium-sized CNC lathes.a is the normalized factor, that satisfies the follow-ing equation,?0a?0.1250.01f(z/u1,y)dy ?0.3150.157512f(z/u2,y)dy(3)?0.40.315f(z/u2,y)dy?dz ? 1The probability density function of force distributionfor the gear with 28 teeth mounted on shaft I is shownin Fig. 3.By the similar way, the force distributions of otherelements can be obtained.4. Design exampleThe references 1317 have described the reliabilitydesign of universal machine tools. An important problemin probabilistic reliability for machine tool element450Y. Wang et al. / International Journal of Fatigue 25 (2003) 447452Fig. 3.Force distribution of Z28gear.fatigue is the determination of the equivalent fatigueforce with a specified reliability. The difference ofreliability design between non-CNC and CNC machinetools is that the speeds and forces of CNC machine toolschange continuously.The fatigue strength is generally less than would beobserved under a static force. Fatigue testing results inexperimental data relating the number of cycles to failure(C) to the magnitude of the cyclical stress (S) or force(T). The fatigue strength is the maximum stress ampli-tude for a specified number of cycles until failure. Math-ematically, an SC curve may take the following form1,TmC ? const.(4)where m ? 0 is constant determined experimentally inlaboratory tests that duplicate the amplitude, frequency,and pattern of specific stresses. T is the force or stressof element, C is the number of cycles to failure.Under the fixed speed and power,TmC ? T?mC0or T? ? Tm?CC0? TKT(5)where, C0is the standard number of cycles to failure, T?is the limit force corresponding to C0, KTis the servicelife factor, determined by,KT? m?CC0? m?60ntmC0(6)where, n is the speed of element in rev/min; tmis theexpected service life in hours.While the speed and force of element are variables,we plot the SC curve and force spectrum in the samecoordinate system, shown as Fig. 4. We divide the forceinterval into N equal segments, for a specific force Ti,the cumulative number of cycles in the area of Tito Ti+ ?T, can be expressed into Ci?T. Based on the Minerhypothesis of cumulative damage, the following equ-ation can be obtained,?Ni ? 1TmiCi?T ? TmC0? (KSTR)mC0(7)where, Ciis the number of cycles corresponding to Ti,TRis the rated force of the element, KSis called theequivalent force factor, thus,KS?Ni ? 1?TiTR?mCiC0?T?1/m(8)?C?C0?1/m?Ni ? 1?TiTR?mCiC?T?1/mwhere, C?is the sum of number of cycles, C?=60Ent; t is the expected service life (in hour); En isthe expectation of the speed ( in rev/min).Therefore,KT?C?C0?1/m?60EntC0?1/m(9)KPKn?Ni ? 1?TiTR?mCiC?T?1/m(10)? ?0zmf(z)dz1/mwhere, z = Ti/TRis the relative force of the element; f(z)is the probability density function of relative force of theelement, KPand Knare called power utilization factorand speed change factor respectively.For the gear with 28 teeth mounted on shaft I, substi-tuting the probability density function of relative forceFig. 4.S-C curve and force distribution.451Y. Wang et al. / International Journal of Fatigue 25 (2003) 447452f(z) into Eq. (10), while m = 6.6, we obtain KPKn=0.432.The expectation of the speed of the gear with 28 teethcan be calculated by,En ?0.1250.01ynmaxu1f(y)dy ?0.3150.157512ynmaxu2f(y)dy(11)?0.40.315ynmaxu2f(y)dywhere, f(y) is the speed spectrum of the CNC lathe; nmaxis the maximum speed of the CNC lathe in rev/min.Substituting u1= 0.158, u2= 0.502 and nmax= 2000r/min into Eq. (11), we obtain En = 506 r/min. Sup-pose t = 20,000 hours, m = 6.6 and C0= 107, thus,KT? m?60EntC0? 6.6?60 506 20000107(12)? 1.86Thus the equivalent force factor of the gear with 28teeth is,KS? KT Kp Kn? 0.805(13)The above-mentioned standard number of cycles, C0is obtained under the assumption of deterministic fatiguelife, i.e., with the reliability R = 0.5. While supposingthe fatigue life as stochastic variable, Eq. (9) becomes,KR?60EntC0R?1/m(14)where, KRis called reliability factor; C0Ris the fatiguelife with the specific reliability R.For example, while the fatigue life of element followsWeibull distribution, the reliability R(C) becomes 16.R(C) ? exp?(C/Ca)b(15)where, Ca is the size parameter of Weibull distribution;b is the shape parameter of Weibull distribution.C0Rcan be calculated by the following equation 16,C0R? C0?lnRln0.5?1/b(16)Thus, the equivalent fatigue force factor with thespecified reliability becomes,KS? KR Kp Kn(17)1Yiqiang Wang, Professor, College of Mechanical Engineering,Jilin University, Changchun, 130025, P. R. China. M.Sc. (Jilin Univer-sity of Technology), PhD (JUT). His research interest is Mechatronicsand Reliability. Tel: +86 431 5705428, Email: meyqwanghot-; http:/ the equivalent fatigue force with thespecified reliability can be calculated by the followingequation,Te? KSTR(18)For the S1-273 CNC lathe, the rated force is,TR? 9550Ph/nB? 745 N.m(19)where, h is the mechanical efficient factor, in theexample, h = 0.88, nB= 125 rev/min is the basic speedof the spindle, P = 11 kW is the rated power of the lathe.After force spectrum of the element is determined, ifthe distribution of fatigue life of the material used isknown, the equivalent fatigue force factor and equivalentfatigue force can be calculated through the above-men-tioned equations.5. ConclusionTo a large degree, reliability is an inherent attributeof a CNC machine tool. As such, it is an important con-sideration in the engineering design process. Reliabilitydesign is an iterative process that begins with the speci-fication of reliability goals consistent with cost and per-formance objectives. Once the reliability goals have beenestablished, these goals must be translated into individ-ual component, subcomponent, and part specifications.After individual component and part requirements havebeen determined, various design methods can be appliedin order to meet the goals.CNCmachinetoolsfeaturehighspeed,highefficiency, high accuracy and high automation. Theymust not only be extremely accurate and highly auto-mated, but also work reliably for long periods of service.Only by means of the modern design theory andreliability method can CNC machine tools work withdesired reliability.To carry out reliability design, the distribution offorces acting on the element must be determined first.The determination of force spectra and establishment offorce spectra database lay a foundation for implementingreliability design for CNC machine tools.1References1 Ebeling CE. An introduction to reliability and maintainabilityengineering. Singapore: The McGraw-Hill Companies, Inc, 1997.2 Tait NRS. 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