Efficient mechatronic evaluation of machine tool designs using model reduction

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1 Mechatroncs 2010, June 28 30, Swss Federal Insttute of Technology ETH, Zurch Swtzerland 1 Effcent mechatronc evaluaton of machne tool desgns usng model reducton P. Magle *, R. Carbn, S. Wekert, K. Wegener Insttute for Machne Tools and Manufacturng IWF Swss Federal Insttute of Technology ETH Zurch, Swtzerland ABSTRACT Ths paper addresses the use of a reduced FE-representaton based on a model that was orgnally created and used n a standard FEM-Analyss envronment for statc and dynamc nvestgatons. The appled reducton method takes nto account the specfc requrements for machne tools: the modellng of gudeway systems and drve components. In order to obtan a complete model whch a) ncludes the entre machne structure, b) offers nformaton about the resultng behavour between tool and work-pece and c) ncorporates the propertes of the control loop, the followng steps are carred-out: After exportng the needed model nformaton (system matrces and couplng characterstcs) from a commercal FE-software, a wdely automated reducton procedure s appled. The structural axs components are then transferred nto a state-space representaton where, frstly, the global structural behavour s analysed and secondly, the control system s added. Based on ths complete mechatronc model, a varety of nvestgatons can be performed, coverng tme- and frequency-doman. Based on measurements on an exstng test-bed, the calculaton results are compared wth realty and nterpreted. 1. INTRODUCTION Wth the ncreasng demand for hgh-productvty and hgh-accuracy machne tools, the evaluaton of the mechatronc behavour of these systems s requred n order to verfy the system s ablty to meet the gven requrements [1]. The use of a complete FE-model wth a fne mesh, ncludng all the desgn detals, s not yet possble wth acceptable efforts n tme and calculaton power. For the nvestgaton of the system s behavour n the tme- and frequency-doman, the use of a reduced FE-representaton of the machne tool structure s proposed. In ths paper, the dervaton of the reduced structural model for a 2-axes machne tool s shown and dscussed n detal. The dfferent steps of ths applcaton-orented method are explaned, showng the advantages of a hghly automated process. Only the essental parameters and confguraton nputs are requred from the ANSYS Workbench user n order to obtan the reduced model usng the stand-alone algorthm MOR for ANSYS. The resultng reduced system matrces are then mported nto MATLAB, where the desred analyses are carred out n a small fracton of the tme needed for a full FE-smulaton. 2. MODEL BUILDING IN ANSYS WORKBENCH The CAD-nterfacng and pre-processng capabltes of ANSYS Workbench are used n a frst step n order to mport the geometry of a machne tool structure and prepare t for smulaton. In ts smulaton envronment, the dfferent axes of the structure, as well as ther couplng propertes need to be defned COMPONENT DEFINITION The structure of a machne tool s composed of several parts, whch are ether soldly joned by standard fxngs (screws, bolts, etc ) n case they belong to a same body, or joned by rotatng bearngs resp., by lnear gudes when defnng the nterface between two axes wth a relatve degree of freedom. In the example covered n ths paper, the structure s dvded nto three assembles correspondng to the bass of the machne (Volume_2), a lnear axs X (Volume_1) and a lnear axs Z (Volume_3). In Fgure 1 these model components are hghlghted n red and the component names can be found on the project tree on the left. * Pascal Magle: Tel.: +41 (0) ; Fax: +41 (0) ; E-mal: magle@wf.mavt.ethz.ch

2 12 th Mechatroncs Forum Bennal Internatonal Conference The area components defnng the contact faces between a gude carrage and ts gudeway, respectvely between a drve nut and ts ball screw, are

These area components are used durng the model preparaton phase to automatcally create two mass elements at each couplng locaton and set the correspondng net-shaped constrant equatons whch connect

Model components Volume_3 Volume_2 Volume_1 X X Z Y Areas on ball screw Areas on drve nut Couplng Constrants Z Y Areas on gudeway Areas on carrage Smulaton scrpts Fgure 1: Illustraton of the

One scrpt sets all the parameters of the couplng propertes: for each nterface a sxdmensonal vector has to be defned whch contans the three translatonal and the three rotatonal stffness values between

2 2 12 th Mechatroncs Forum Bennal Internatonal Conference The area components defnng the contact faces between a gude carrage and ts gudeway, respectvely between a drve nut and ts ball screw, are shown on the rght n Fgure 1. These area components are used durng the model preparaton phase to automatcally create two mass elements at each couplng locaton and set the correspondng net-shaped constrant equatons whch connect one mass element to the movng part (carrage or drve nut) and one mass element to the fxed part (gudeway or ball screw). Model components Volume_3 Volume_2 Volume_1 X X Z Y Areas on ball screw Areas on drve nut Couplng Constrants Z Y Areas on gudeway Areas on carrage Smulaton scrpts Fgure 1: Illustraton of the component defnton n an ANSYS Workbench smulaton model 2.2. SIMULATION SCRIPTS Two smulaton scrpts then need to be ncluded n the modal analyss pre-processng ppelne (see the project tree n fgure 1). One scrpt sets all the parameters of the couplng propertes: for each nterface a sxdmensonal vector has to be defned whch contans the three translatonal and the three rotatonal stffness values between the two correspondng mass elements. The second scrpt then executes a seres of selfprogrammed ANSYS macros whch use the nputs from the named components (volumes and areas) created n the smulaton model and the parameter values defned n the parameter scrpt. 3. EXPORTING THE MODEL The standard ANSYS smulaton run s stopped before enterng the soluton phase and the resultng outputs of the analyss are the so-called.full fles contanng the system matrces of the model (stffness, dampng and mass). For each nterface locaton between two axes, one.full fle per load drecton has to be exported n order to be able to transfer every force and moment contrbuton between two adjacent bodes (see Fgure 2). Addtonally to these fles, a set of text fles contanng the couplng parameters (physcal stffness and dampng values) and the correspondng node numbers are automatcally exported.

3 Mechatroncs 2010, June 28 30, Swss Federal Insttute of Technology ETH, Zurch Swtzerland 3 Body_B_C_L.full: Fx Fy Fz B: body number C: couplng number L: load drecton Body_B_C_Fx.full Body_B_C_Fy.full Body_B_C_Fz.full Mx My Mz Body_B_C_Mx.full Body_B_C_My.full Body_B_C_Mz.full Fgure 2: Load defnton for the couplng locatons: one.full fle per load drecton 4. MODEL REDUCTION The goal of the model order reducton s to produce systems representng the relevant dynamcal propertes of the orgnal systems, but wth much smaller dmenson. For ths purpose, MOR for ANSYS [2], a standalone program based on the Krylov-Method and mplementng the Arnold algorthm, has been used: startng from the frst order equaton system n equaton 1: Ax& ( t) = x( t) + Bu( t) sax ( s) = X ( s) + BU ( s) y( t) = C T x( t) Y ( s) = C T X ( s) (1) The transfer functon can be formulated as follows n equaton 2: T 1 G( s) = Y ( s) U ( s) = C ( I sa) B (2) The Taylor seres expanson of G(s) about the expanson pont s 0 = 0 s gven by equaton 3: T 2 2 G ( s) = C ( I + sa + s A +...) B = ms (3) Wth m = -C T A B the so-called moments about s 0. The basc dea behnd the Krylov-subspace based block-arnold algorthm s to fnd a reduced system whose transfer functon G r (s) has the same moments as G(s) up to a chosen degree. Once all the needed fles from ANSYS are grouped nto one drectory, the order reducton of each axs can be performed separately, leadng to a number of reduced systems correspondng to the number of bodes n the machne tool. For the structure consdered above, whch s composed of three bodes, three state-space systems are generated, each havng ts own reduced dmenson. In order to be able to analyse the system globally, the three systems need to be combned back nto one unque global state-space system (see fgure 3) [3]:

4 4 12 th Mechatroncs Forum Bennal Internatonal Conference B1 x(0) u(t) B2 B3. x(t) x(t) C1 C2 C3 C1 C2 C3 C y(t) A1 B A2 A3 Fgure 3: Combnaton of the three state-space systems (A 1,B 1,C 1,D 1 ), (A 2,B 2,C 2,D 2 ) and (A 3,B 3,C 3,D 3 ) nto one global state-space space system by assemblng the ndvdual matrces A 5. ANALYSIS IN MATLAB The combnaton of the state-space systems descrbed above contans the reduced dynamcal characterstcs, as well as the nput/output nformaton of the sngle bodes. In addton to ths, n order to correctly set the connectons between the machne axes, the physcal stffness and dampng values, contaned n the couplng parameters text fle, have to be ncorporated nto the system. Wth help from a Sprng-Damper matrx (fgure 4 left), the dsplacement and velocty outputs are automatcally returned to the correspondng system nputs as forces (fgure 4 rght): Force nputs for harmonc response smulaton Dsplacement outputs for harmonc response Internal forces Sprng-Damper matrx Fgure 4: Sprng-Damper matrx (left) and Smulnk model (rght) for a harmonc response smulaton ncludng the assembled state-space system and the addtonal Sprng-Damper matrx specfyng the nternal couplng characterstcs At frst glance, ths whole method of reducng the sngle axes separately can appear to be laborous. If one s nterested n the frequency response between two ponts of the global system, prevous studes have shown that t s actually more adequate to operate a sngle model order reducton of the whole system (see fgure 5) [4]. However ths brngs a certan number of dsadvantages: frstly for complex machnes havng up to several mllons degrees of freedom, the reducton of the full system matrces can be extremely tme consumng and requres notable computng resources. Another drawback of a global model order reducton s the mpossblty of ntegratng addtonal effects/functons wthn the reduced system. The connectons between the movng axes are especally senstve

Mechatroncs 2010, June 28 30, Swss Federal Insttute of Technology ETH, Zurch Swtzerland 5 n ths regard. Most FE-analyses (modal and harmonc analyses e.g.) are performed under the assumpton that the system s lnear.

consttutes a lnearzed approxmaton affectng the dynamcal behavour of the whole machne tool.

envronment of the reduced system: carrages for example can be complemented by non lnear phenomena lke hysteress, or transmsson gears can ncorporate backlash effects.

5 Mechatroncs 2010, June 28 30, Swss Federal Insttute of Technology ETH, Zurch Swtzerland 5 n ths regard. Most FE-analyses (modal and harmonc analyses e.g.) are performed under the assumpton that the system s lnear. Ths s acceptable for the structural nvestgaton of the sngle bodes, but f couplngs are nvolved (gudeways, drves, transmssons or any other system mplyng a relatve moton between two parts) t only consttutes a lnearzed approxmaton affectng the dynamcal behavour of the whole machne tool. Fgure 5: Standard approach consstng n performng a global model order reducton of a machne tool By reducng sngle bodes separately, more detaled modellng steps can be added subsequently n the envronment of the reduced system: carrages for example can be complemented by non lnear phenomena lke hysteress, or transmsson gears can ncorporate backlash effects. In our case, the nterest of the presented method conssts n havng the possblty of ntegratng a control algorthm to the sngle machne tool axes (fgure 6) and thus beng able to study the structural behavour under moton and the correspondng behavour at the Tool Centre Pont (TCP). Ths allows the nvestgaton of dynamcal effects n functon of velocty, acceleraton and jerk settngs, as well as the optmzaton of the control system layout n an early stage of the development of a machne wth drastcally reduced computng tme. Target path X Target path Z Machne tool structure Control system Fgure 6: The complete Smulnk model contanng the model of the machne tool structure, as well as two basc cascaded control systems for the two moton drectons X and Z.

6 6 12 th Mechatroncs Forum Bennal Internatonal Conference 6. RESULTS The frst step to valdate the presented reducton method, leadng from the FE-model orgnated n ANSYS to the state-space model desgned n MATLAB/Smulnk, s to compare the results of two basc analyses MODAL ANALYSIS AND HARMONIC RESPONSE ANALYSIS On the left n fgure 7 the frst ten egenfrequences of a modal analyss are shown for a full FEM computaton n ANSYS and for a reduced computaton n MATLAB. The depcted table outlnes the qualty of the reduced model, whch seems to be able to qute exactly descrbe the modal behavour of the structure up to ~300Hz. On the rght n fgure 7 the outcome of a full harmonc analyss n ANSYS and of a reduced smulaton n MATLAB are superposed n one dagram. The graphc represents the X-dsplacement of the TCP under a load n X-drecton appled at the same locaton. The results reveal an nsgnfcant error between the two methods over the entre consdered frequency range. Modal analyss Harmonc response analyss Mode number FEM Reduced Hz Hz Hz Hz Hz Hz Hz Hz Hz Hz Hz Hz Hz Hz Hz Hz Hz Hz Hz Hz Ampltude [mm/n] Frequency Response TCPX Full Harmonc Analyss "MOR for ANSYS" Frequency [Hz] Fgure 7: Comparson between full ANSYS smulaton and reduced MATLAB smulaton The above consderatons pont out the effcency and accuracy of the method. The qualty of the results of the reduced model shows that t s possble to acheve an enormous dmnuton of computaton tme wthout deteroratng the dynamcal behavour of the machne tool, even n the case of a complex structure as the one llustrated n the prevous chapters. In order to quantfy the benefts obtaned by a reduced smulaton, table 1 summarzes the tmes requred for a full harmonc analyss n ANSYS and the tmes requred for the dfferent steps of the reduced analyss. The advantages of the method are tangble, and become even more sgnfcatve f one consders the fact that, for each further smulaton run n ANSYS the tme requred s 144 mnutes, whereas t remans lmted to less than one mnute for each further analyss n MATLAB. Computng Tmes Full smulaton Reduced smulaton Matrces wrtng from ANSYS - 5 mn Order reducton wth MOR for ANSYS mn State-Space model buldng n - 0 mn MATLAB Harmonc response analyss 144 mn 0.5mn Total 144 mn 8 mn Table 1: Comparson of smulaton tmes for the full ANSYS model and the reduced model

7 Mechatroncs 2010, June 28 30, Swss Federal Insttute of Technology ETH, Zurch Swtzerland CROSS-TALK The basc mechansms of nertal straghtness devatons (cross-talk) are explaned below: fgure 8 shows the confguraton parameters leadng to the basc equatons for a smplfed 2D case (see equaton 4). Regardng the magntude of translatonal dsplacements at the TCP, the followng factors need to be consdered [5]: - amount of nerta force F a due to mass and acceleraton - offset of the nerta Y Fa to the drvng force - offset Z TCP of the slde s centre of gravty to TCP n drecton of moton - tlt-stffness k rot,a of the gudeway system orthogonal to drecton of moton gven by the translatonal and rotatonal stffness values k Y, and k A, of the gudeway elements - offsets of the gudeway elements Z (quadratc nfluence) a gude-way system mass, M Y Fa k k Y2 Y1 F a Z k1 Z k2 Z TCP S EYZ TCP Y Z X Fgure 8: Schematc of the nerta tlt effects on the TCP EYZ nerta F = a Y k Fa Z rot, A TCP = M a Y k Y, z Fa 2 Z + TCP k A, (4) By means of cross-grd measurements on an exstng machne, whose model has been used to demonstrate the reduced smulaton method of the present paper, the effects descrbed above should be evdenced. The requred measurng devces were mounted on the machne as showed n fgure 9. An offset of X = 300mm has been added to the TCP n the drecton of moton n order to magnfy the cross-talk effects. A path of s = 100mm was fed to the X-drve, and whle keepng the programmed velocty at a constant value of v = 15mmn -1 and the jerk value at j = 300ms -3, the acceleraton value has been successvely set to 1ms -2, 2ms -2 and 3ms -2, n order to evaluate the effects of the nertal loads on the cross-talk devatons. 300mm F drve EYX Z X Y Fgure 9: Cross-grd measurement set-up

8 8 12 th Mechatroncs Forum Bennal Internatonal Conference The cross-talk measurements have then been reproduced usng the reduced smulaton model ntroduced n fgure 6. The settngs of the model controller have been approxmately set to the values of the real controller and the three correspondng acceleraton values were assgned to the path generator n Smulnk. v = 15m/mn, a = 1m/s2, j = 300m/s3 v = 15m/mn, a = 2m/s2, j = 300m/s3 v = 15m/mn, a = 3m/s2, j = 300m/s3 Measured Cross-Talk EYX [mm] 3.4µ m 6.0µm 8.0µm Smulated Cross-Talk EYX [mm] 3.5µ m 5.8µm 7.6µm X-Path [100mm] X-Path [100mm] X-Path [100mm] Fgure 10: Measured and smulated cross-talk values by varable acceleraton values Fgure 10 outlnes two thngs: frstly, the quas-lnear correspondence between acceleraton and cross-talk values, whch has already been emphaszed n several occasons on analogue structures, and secondly, assertng the concluson drawn n chapter 6.1, the valdty of the reducton method. The reduced model s capable of matchng the measured cross-talk effects n the acceleraton phase, despte rather naccurate controller parameters, some modellng smplfcatons of the machne tool at CAD-level and the assumpton that the global dynamc behavour of the structure s constant over the consdered X-poston moton range. 7. SUMMARY Ths paper presents the step-by-step process leadng from an elaborate and complex FE-model of a machne tool to ts smplfed and effcent reduced representaton. After relatng all the phases requred to acheve the fnal model, the ncontestable advantages brought by a greatly automated and powerful method are ponted out through the dfferent analyses possbltes. The effcency of ths approach n terms of computng tme and resources s supported by the valdated comparsons of the results: on the one hand between the full fnteelement and the reduced smulatons (modal and harmonc analyses) and on the other hand between the crosstalk effects measured on the real machne and those obtaned wth the reduced smulaton model wth ntegrated controller. REFERENCES [1] G. Kehl: Integrerte Smulaton von Strukturdynamk und Regelungstechnk an Bearbetungszentren, Symposum Smulaton von Werkzeugmaschnen, ETH Zurch, 2008 [2] E. Rudny: [3] O. Zrn, R. Montavon: Gekoppelte Smulaton von FE- und Mehrkörpermodellen für Werkzeugmaschnen, ASIM, 2008 [4] P. Magle: Ensatz von Reduktonsmethoden für de Smulaton von Werkzeugmaschnen, 13. Schwezer CADFEM Users Meetng, 2008 [5] S. Wekert, S. Bosson, K. Wegener: Evaluaton of machne tool concepts under frcton nfluences, Proceedngs of the ASPE 2008 Annual Meetng, 2007

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