T5型数控钻孔攻牙机纵向进给机构设计7张CAD图
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MACHINE TOOLExpedient modeling of ball screw feed drivesS. FreyA. DadalauA. VerlReceived: 15 March 2011/Accepted: 27 February 2012/Published online: 9 March 2012? German Academic Society for Production Engineering (WGP) 2012AbstractBall screw feed drives are the most commonlyused mechanism to provide linear motion in high speedmachine tools. Position accuracy and the achievable closedloop bandwidth of such drive systems are usually limitedby the structural vibration modes of the mechanical com-ponents. Higher order plant models allow for a betterunderstanding of the system dynamics, improve the designprocess of feed drives and are essential for the developmentof sophisticated control strategies. The ball screw shaft,describing a complex flexible structure, is probably themost significant component concerning structural vibrationmodes of a feed drive. In this paper, the behavior of theshaft and its dominant influence at different operating andcoupling conditions is particularly addressed. Using ahybrid modeling technique, the main characteristics of theshaft are derived and projected onto a clearly arranged andversatile lumped mass model. Simulative and experimentalexaminations are conducted and a parameter analysis isperformed. The presented model proofs to be accurate for agreat range of parameters and in addition allows for aphysical interpretation of the dominant structural vibrationmodes of a feed drive.KeywordsMachine tool ? Simulation ? Ball screw1 IntroductionDynamical feed drives commonly used in machine toolsandproductionunitsrepresentcomplexmechanicalstructures with multiple degrees of freedom (DOF) andvarious eigenmodes. Numerous scientific works have dealtwith the simulation of such drive systems, using Finite-Element-Methods (FEM) and complex modeling tech-niques in order represent the dynamical behavior asdetailed as possible 13. Especially when describing thecomplex behavior of the ball screw shaft, FEM-basedmodeling techniques are commonly used. In contrary, theapproach presented in this paper aims at identifying thedominant effects of the ball screw shaft and including theminto a simple lumped mass model, capable of expressing themost relevant characteristics of feed drives with respect totheir application in machine tools.Generally spoken, a ball screw drive can be charac-terized as a system with three different types of eigen-modes: axial, rotational and flexural modes 4. Typicallythe first axial and rotational mode of the ball screw showa dominant influence on the overall dynamics while therelevance of higher order modes for most technicalapplications is rather small. The excitation of the flexuralmodes and their effect on the feed motion greatly dependson fabrication and mounting tolerances as well as on theoperating conditions of the drive system. Presumingproper mounting and operating conditions, the excitationof the flexural modes are considerably small and thereforewill be neglected in the course of this paper. The twodominant vibrational DOF, namely the axial and therotational mode, are primarily influenced by the geo-metrical dimensions and the pitch of the ball screw shaftand the table position. Depending on the transmissionratio as well as on the operating conditions, the systemcan behave qualitatively and quantitatively very different.In most cases however, the configuration of a feed driveallowsforasimpleapproximationofthesystemscharacteristics.S. Frey (&) ? A. Dadalau ? A. VerlInstitute for Control Engineering of Machine Toolsand Manufacturing Units (ISW), Stuttgart, Germanye-mail: siegfried.freyisw.uni-stuttgart.de123Prod. Eng. Res. Devel. (2012) 6:205211DOI 10.1007/s11740-012-0371-0While the characteristic of the axial mode of a ballscrew feed drive and its dependency to changing operatingconditions are generally known, this work focuses on therotational eigenmode and its integration into a lumpedmass model. In the following, a practical and application-oriented modeling technique for ball screw feed drives willbe derived. The physical context is explained and anextensive parameter analysis is provided. Finally the modelis verified by measurement results conducted on anexperimental test bench.2 Ball screw feed drives for machine toolsFigure 1 shows a typical ball screw assembly with its rel-evant components. In this case servomotor and bearings arefixed to the machine base and the torque of the servomotoris transmitted through an elastic coupling onto the ballscrew shaft. Not depicted in Fig. 1 are the linear guidewaysconstraining the movement of the machine table in axialdirection. The transformation from the rotational into thelinear motion is realized by the ball screw system with itstransmission ratio i, defined as the distance of travel hduring one revolution of the shaft.i h2p1Manufacturers provide a great variety of ball screwsystems with different constructive designs, dimensionsand transmission ratios. The available pitches nowadays liebetween 0.05 and 3 times the ball screw diameter, whilefor most applications in machine tools values between 0.25and 1 are being used.As already mentioned, the dynamical behavior of such adrive system is greatly influenced by the physical dimen-sions of the ball screw shaft. Diameter and length of theshaft have a decisive impact on the overall rigidity andtherefore on the structural vibration modes of the drivesystem. Figure 2 shows a measured frequency response ofa ball screw drive, derived from the comparison of motorand machine table velocity.Clearly noticeable are the two dominant eigenfrequen-cies f1and f2, characteristical for ball screw feed drives.The first eigenfrequency in Fig. 2 represents an axialoscillation of the machine table while the second eigen-frequency can be primarily ascribed to an angular deflec-tion of the rotational system. Although each mode is aresult of axial and angular deformations, it is convenient toclassify an axial and a rotational mode depending on thepredominant deformation. In both cases, the flexibility ofthe ball screw shaft has a major influence on the absolutevalue of the eigenfrequencies. This is due to the fact thatthe shaft represents the component with the least rigiditywithin the drive system 5. At this point it should be noted,that for most technical applications the eigenfrequency ofthe axial mode is articulately smaller than that of therotational mode. On this account the axial mode is oftenreferred to as the first eigenfrequency of a ball screw feeddrive. The interaction between the dominant eigenfre-quencies of a ball screw drive is defined by the transmis-sion ratio, describing the coupling between the axial andthe rotational motion. Furthermore the properties of theball screw shaft are not constant but change over the travelrange of the machine table. This leads to a complex overallsystem with numerous variations and dependencies.The structural vibration modes of the mechanical systemareofparticularimportancewhenconsideringthedynamical behavior of a position controlled feed drive.Especially in high speed applications, high accelerationand jerk values can excite the mechanical structure andtherefore have to be considered when tuning the controller.Fig. 1 Schematic configuration of a ball screw feed driveFig. 2 Measured frequency response of a ball screw feed drive206Prod. Eng. Res. Devel. (2012) 6:205211123When using cascaded P-position PI-velocity controller, theclosed loop bandwidth and tracking accuracy of a feeddrive are directly limited by the first eigenfrequency of themechanical system 6, 7. In similar matter, externalexcitations e.g. cutting forces can excite the mechanicalstructure and cause unwanted oscillations and stabilityproblems. On this account, the second eigenfrequency isequally important to the performance of feed drives. Sus-ceptible to oscillations, the rotational mode has a decisiveimpact on the velocity control loop of a feed drive, andtherefore often has to be compensated for by additionalnotch filters 13. Figure 3 exemplarily shows the open-loop Nyquist plot for a PI velocity controller, measured onan actual feed drive. The characteristics of the two eigen-modes are clearly noticeable and the effect of the rotationalmode on the phase margin of the control system becomesevident.But especially when using more sophisticated controlstrategies, the relevance of accurate plant models becomesevident. A good understanding of the mechanical systemand a detailed plant model are often basic requirementswhen implementing such control systems 8, 9. Thedemand for even higher dynamics and better trackingaccuracy therefore goes along with highly efficient andaccurate feed drive models, equally valid for differentoperating conditions.3 Comparison of two different modeling techniquesIn order to examine the influence of the shaft on the overalldynamical behavior of a ball screw drive, two simplifiedmodels with different degrees of complexity are beingintroduced and compared to one other. The first model isbuilt up in FEM using a combination of springs, dampers,mass-elements and Timoshenko beam elements. We callthis model a hybrid model since we are using discrete andcontinuous formulation elements. The relevant axial androtational flexibilities of the system, namely the axialrigidity of the fixed bearing including the mounting kbearing,the rigidity of the ball screw nut knutand the torsionalrigidity of the coupling kcouplingare being expressed aslinear springs. The individual damping values are treated inequal measure. The shaft including both, relevant axial andtorsional characteristics, is modeled with Timoshenkobeam elements and coupled with the surrounding springsand dampers. This provides the possibility of an exactexamination of the structural behavior of the shaft and itsinfluence on the overall dynamical behavior. In concor-dance to the real mechanical system, rotational and axialmotion are coupled by the transmission ratio i, which ismodeled using constraint equations on the involved DOF.Figure 4 depicts the composition of this hybrid model.Following well established techniques 5, 12, 14, thesecond model entirely uses lumped physical features, and istherefore referred to as discrete model. Due to its simplicityit can be directly modeled with CACE-Tools (Computer-Aided-Control-Engineering-Tools), using available stan-dard blocks like gains, integrators and basic mathematicaloperations. While discrete modeling approaches typicallyneglect the influence of the shaft on the rotational mode ofthe drive system, in this case, the dominant effects areexplicitly included into the lumped mass model. Therefore,the shaft is separated into two different branches, an axialand a rotational branch while the coupling once more isrealized using constrained equations. Since all componentsare expressed by discrete springs and dampers, the rigidityvalues of shaft, coupling and bearing are combined to anoverall axial kaxand rotational value krot:Fig. 3 Measured open loop Nyquist plot for the velocity controller ofa ball screw feed driveFig. 4 Hybrid model of a ball screw driveProd. Eng. Res. Devel. (2012) 6:205211207123krot1kshaftrot1kcoupling?12kax1kshaftax1kbearing?13Figure 5 shows the principle set up of the lumped massmodel. The individual parameters in each model such asthe mass of the machine table mt, the mass of the ball screwshaft ms, the rotary inertia of shaft Jsand motor Jm, as wellas the rigidity and damping factors correspond to theirphysical values.Precondition for this approach is the definition of ade-quate, alternative values for the axial and rotational rigidityof the shaft. For this purpose the shaft is considered as asolid steel rod (Young modulus E = 210e9 N/m2, shearmodulus G = 81e9 N/m2, density q = 7,800 kg/m2). Withthe dimensions of the ball screw, namely equivalentdiameter d, length l and effective length leff, the axial androtational rigidity values of the shaft kshaftaxand kshaftrotcanbe approximated using Eqs. 4 through 6.kshaftaxEAleffp4leffd2E4kshaftrot Js2pfs2p32d4lp2pfs25fs14lffiffiffiffiGqs6Notice that the alternative rotational rigidity of the shafthas been approximated using the eigenfrequency fsof asolid, on one side fixed rod which can according to 10 becalculated using Eq. 6.With this approach, the axial rigidity of the shaftdepends on the distance between fixed bearing and tableposition leff, while the rotational rigidity is consideredconstant throughout the travel range of the machine table.In the following, both models have been examined with anidentical set of parameters according to Table 1.It should be noted, that the assumption of the shaft beinga solid rod is a simplified approximation. In reality allgeometric parameters of the ball screw, including the screwpitch, have an effect on the axial and torsional stiffnessvalues which in turn can be regarded in form of anequivalent diameter 11.The simulation results, namely the first two eigenfre-quencies of the feed drive at different operating and cou-pling conditions, are presented in Table 2.As expected, the value of the first eigenfrequency greatlydepends on the mass and the position of the machine table:The higher the mass and the greater the distance betweenmachine table and fixed bearing the smaller the value of thefirst eigenfrequency. In comparison to that, the secondeigenfrequency seems to change rather insignificantly withdifferent operating conditions. Both models show that therotational mode remains almost constant for a great range ofparameters. The difference between hybrid and discretemodel expressed in Fig. 6 can be ascribed to the simplifiedrepresentation of the shaft in the discrete model.From Fig. 6 it can be seen that the absolute difference aswell as the parameter sensitivity of the eigenfrequenciesgreatly depend on the interaction between axial and rota-tional mode, determined by the transmission ratio of theball screw. With increasing transmission ratio the influenceof the rotational rigidity of the shaft on the overall axialrigidity increases. The position dependent, effective axialrigidity kaxeffcan be expressed as follows.kaxeff?1kshaftax1kbearing1knut1kshaftroti2 !?17In many cases however, the influence of the rotationalrigidity on the axial mode can be neglected due to thetransmission ratio of the ball screw. With the effectiveaxial rigidity and the knowledge of the consistency of therotational mode, the two characteristic eigenfrequencies ofa ball drive can be approximated using Eqs. 8 and 9.f1?12pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikaxeffmt msr8f212pffiffiffiffiffiffiffikrotJsr9Despite the chosen simplifications the discrete modelshows good results for the first two eigenfrequencies of aFig. 5 Discrete model of a ball screw feed driveTable 1 Parameter values used for the simulationsRotary inertia motor6.40E-03kg m2Rotary inertia coupling6.50E-04kg m2Axial rigidity nut500.00N/lmTorsional rigidity coupling141.00Nm/mradBall screw diameter38.00mmLength shaft2,000.00mmAxial rigidity bearing250N/lm208Prod. Eng. Res. Devel. (2012) 6:205211123ball screw system. This allows for an intuitive and clearlyarranged simulation of the dynamical behavior, valid for agreat range of parameters and well suitable for mostpracticalapplications.Inthefollowingchapterthesimulation results of the discrete model depicted in Fig. 4will be compared to measurements on an actual machinetool feed drive. Especially the numerically observed lowdependency of the rotational eigenfrequency is to bevalidated by measurements.4 Experimental verification of the modelAt the Institute for Control Engineering of Machine Toolsand Manufacturing Units (ISW) at the University ofStuttgart exists a great variety of test benches with differentsize ball screws and numerous transmission ratios. Whilethe research on this topic is based on experiments withdifferent ball screw types, the following measurementsfocus on a typical ball screw set up for machine toolsaccording to the specifications in Table 3. The parametersused for the simulation comply with those of the experi-mental set-up. The values for rigidity and mass are basedon manufacturers data while the damping factors corre-spond to empirical values.With the given feed drive various measurements havebeen conducted and compared to the simulation resultsTable 2 First two eigenfrequencies of the ball screw drive models for different mass, screw pitch and table positionsTable mass(kg)Screw pitch(mm)Table position(mm)1st Eigenfrequency (Hz)2nd Eigenfrequency (Hz)HybridmodelDiscretemodelPercentaldifferenceHybridmodelDiscretemodelPercentaldifference30010500101.2100.8-0.4375.4345.0-8.1300101,00090.790.4-0.3375.7344.2-8.4300101,50083.182.7-0.4375.9343.8-8.63002050099.798.6-1.1376.6349.1-7.3300201,00088.888.90.0377.9346.3-8.4300201,50081.081.50.6378.7344.5-9.03004050094.491.1-3.5380.9363.2-4.6300401,00082.483.41.1385.3353.4-8.3300401,50074.277.34.2387.6346.8-10.56001050072.171.9-0.3375.4344.9-8.1600101,00064.764.6-0.1375.7344.2-8.4600101,50059.259.10.0375.9343.8-8.66002050071.170.4-1.0376.5349.0-7.3600201,00063.363.50.2377.9346.3-8.4600201,50057.758.31.0378.6344.4-9.06004050067.365.0-3.4380.8362.9-4.7600401,00058.759.51.4385.2353.3-8.3600401,50052.855.24.6387.6346.8-10.5Fig. 6 Percental difference between hybrid and discrete model of theball screw driveTable 3 Specifications of the test benchNominal diameter ball screw, d040mmBall screw lead, Ph20mmDynamical load value, Cn54,600NNominal ball diameter, Dw6mmNumber of ball track turns5(-)Travel range700mmMass machine table300600kgProd. Eng. Res. Devel. (2012) 6:205211209123from the discrete model. Figures 7 and 8 exemplarily showsimulated and measured frequency responses for the givenfeed drive at different operating conditions.The discrete model used to describe the mechanicalsystem shows good concordance with the behavior of theactual feed drive. Despite of the chosen simplifications andthe impreciseness of the manufacturers data, a good pre-dictability could be achieved. Furthermore, the predictedparameter dependency of the eigenfrequencies is confirmedby the measurement results. The frequency of the axialmode changes depending on machine table mass andposition while the rotational mode of the ball screw driveFig. 7 Simulated and measured frequency response for the feed drive at different machine table positionsFig. 8 Simulated and measured frequency response for different machine table masses210Prod. Eng. Res. Devel. (2012) 6:205211123remains almost constant. The invariance of the rotationalmode towards operating conditions, as already detected inthe foregoing simulations, can be explained by the low passcharacteristic of the mechanical system. After the firsteigenfrequency the system experiences an amplitude drop-off of 20 dB/decade. Hence the energy imposed by theservo drive at higher frequencies is merely transformedonto the machine table. The axial motion of the table at thesecond eigenfrequency is therefore quite small. Due to thekinematic coupling, the angular deflection of the shaft willstill lead to an axial displacement which in this case ispredominantly executed by the deflection of the shaft,bearing and mounting. Considering the minor changes intable position at the rotational mode of the ball screwsystem, it is intelligible that neither the mass nor theeffective position of the table have a relevant influence onthe second eigenfrequency of the oscillating system.The predictability and simulation of the dominanteigenfrequencies of a ball screw drive and their depen-dency to different operating conditions provides a betterunderstanding of the system dynamics and is beneficial forthe design process of the mechanical components as well asfor the implementation of adequate feedback controllers. Inparticular the consistency of the rotational eigenmode is animportant inside, allowing for an adequate design of filterstructures and compensational networks, and hence bene-ficialforboththefieldofresearchandindustrialapplications.5 ConclusionIn this paper two well-structured and clearly arrangedmodels for ball screw feed drives have been presented: ahybrid (discrete-continuum) and a lumped mass (discrete)model. The used modeling techniques are state of the art,however the chosen combination of the lumped physicalfeatures, the integration of additional system properties andthe assignment of the physical parameters to the modeldescribe a new approach. Of central interest to this workare the characteristics of the ball screw shaft and itsdominant influence on the vibrational modes of a feeddrive. By separating the physical characteristics of the shaftinto an axial and a rotational system, the dominant effectsof this complex flexural body could be projected onto thesimple lumped mass model. Through comparison withthe more complex hybrid model it could be shown, that thesimplified discrete model is well suitable for predicting therelevant eigenfrequencies of a feed drive over a great rangeof parameters. Different coupling and operating conditionshave been analyzed and the effect on the dominanteigenmodes has been illustrated. While the axial vibrationmode shows the well know dependencies to the operatingconditions, the rotational vibration mode proofs to be littlesensitive with respect to table mass and table position.Finally the simulation res
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