Modeling and Simulation of a Hexapod Machine Tool for the Dynamic Stability Analysis of Milling Processes. C. Henninger, P.

Transcription

1 Smpack User Meetng 27 Modelng and Smulaton of a Heapod Machne Tool for the Dynamc Stablty Analyss of Mllng Processes C. Hennnger, P. Eberhard Insttute of Engneerng project funded by the DFG wthn the framework of the SPP 199 Parallel Knematc Machne Tools

2 Outlne motvaton: PKM machne tools and dynamc stablty of cuttng processes modelng of the machne structure modelng of the mllng process analyss of dynamc stablty technques for effcent computaton of stablty dagrams pose-dependent stablty dagrams for the heapod machne tool Paral Insttute of Engneerng

3 PKM Machne Tool PARALIX spatal assembly of lnks hgh statc stffness no bendng moments n lnks, only normal forces lght-weght desgn of struts IfW, Unversty of Stuttgart equal components n dentcal lnks reducton of producton costs 6-DOF heapod machne tool struts wth constant length sledges drven by lnear drect drves 5-DOF hgh-speed mllng of metal Insttute of Engneerng nonlnear knematcs and dynamcs nonlnear poston control necessary pose-dependency of stffness force/velocty transmsson and dynamc propertes knematc sngulartes loss of control of DOF, reducton of workspace

4 characterstcs of PKM: lght-weght structures pose-dependent dynamcs Dynamc Stablty of Cuttng Processes for PKM? eternal ectaton unbalance ectaton nerta forces process forces machne structure process relablty dynamc stablty surface qualty dsplacements mechansms of ectaton eternal and self-ected vbratons Insttute of Engneerng process process parameters spndle speed, feed, depth of cut,...

5 machne structure jont elastctes Elastc Multbody System of PKM Paral rheonomc jonts tool + toolholder elastc struts k m c y y z k y c y sledges, platform as rgd bodes fnte element modelng of struts (beam elements) elastc Cardan jonts and lnear jonts modal reducton of strut model (9 dof) Insttute of Engneerng modal reducton of complete model for each pose n workspace

6 Mode 1 y-z Mode 2 y-z Egenfrequences of Machne Structure n Workspace Mode 3 y-z Mode 1 α-β Mode 2 α-β Mode 3 α-β Insttute of Engneerng

7 measured tool-tp frequency repsonse Role of Machne Structure n Tool-Tp Dynamcs bmodal system representng machne structure and tool-toolholder-spndle Insttute of Engneerng

8 nfluence on egenfrequences and tool-tp frequency response Pose-dependency of Dynamc Behavour Insttute of Engneerng

9 two-dmensonal mllng process Modelng of Mllng Process Insttute of Engneerng lnear cuttng force model F = F k bh F = F k bh t t + ct r r + dynamc depth of cut wth regeneratve effect cr [( ξ(t) ξ(t τ) ) sn( ) ( μ( t) μ(t τ) ) cos( ϕ) ] h = h( ϕ) + g( ϕ) ϕ dynamc process force equaton ( y(t) (t )) F = F ( t) + bkc(t) y τ overall system consderng machne structure and process dynamcs M && y( t) + D y& (t) + ( K + Kc ) y(t) Kc y(t τ) = F (t) + delay-dfferental equaton (DDE) wth perodc coeffcents dynamc stablty?

10 Insperger & Stépán, 23: dscretzaton of system hstory Analyss of Dynamc Stablty by Sem-Dscretzaton Method homogeneous perodc tme-varant DDE & = A( t) (t) + Q(t) (t τ) appromatng ODE on [ ] & t t + 1 ( w b w ) = A ( t) + Q m + a m+ 1 appromatons on (t τ) w b m + (t w + Δt / 2 τ) a [, ] t t + 1 m+ 1 A ( t) A(t = A + Δt / 2) : + Δt / 2) : Q ( t) Q(t = Q + 1 = P + war m+ 1 + wbr m wth P = ep( AΔt ) Insttute of Engneerng R = (ep( AΔt) I) A 1 Q

11 Insttute of Engneerng Stablty Analyss transton matr for t Δ = m 1 m 2 1 T b T a 1 m ~ ~ ~ ~ w w ~ ~ ~ ~ I I I C R C R C P M L M M O M M M L L L L M transton matr for system perod T 1 2 k 1 k Φ Φ Φ Φ Φ = L Floquet theory for stablty of perodc tme-varant lnear systems ( ) < => 1 11 ) eg( ma Φ unstable stablty boundary stable ~, C = vector of delay-actve states, d n C n d <

12 Stablty Boundary Dagram dynamc stablty dependng on process parameters spndle speed N and aal depth of cut a p up-mllng process of alumnum alloy wth sngle-bladed cutter, a/d=.5 f [Hz] 722 D [-].87 m [kg].2 unstable k ct [MPa] 644 k cr [MPa] 238 Insttute of Engneerng stable

13 S mplctly defned stablty boundary ( N,a ) ma eg( (N,a )) { ( ) 1} = Φ p p = Effcent Computaton of Stablty Dagrams full dscretzaton equdstant dscretzaton of complete doman computatonally epensve curve trackng local search method adaptve dscretzaton hgh computatonal effcency etra treatment of cusps and near-branch zones necessary dscretzaton of dsconnected regons not possble Sptzkehre Engstelle Insttute of Engneerng

14 Insttute of Engneerng Curve Trackng

15 Comparson: Curve Trackng vs. Full Dscretzaton full dscretzaton equdstant grd 5555 ponts 325 functon evaluatons curve trackng (2952 functon eval.) full dscretzaton (325 functon eval.) curve trackng 27 curve ponts 2952 functon evaluatons equvalent to full dscretzaton wth 2525 = 625 ponts Insttute of Engneerng

16 dea: ntal dscretzaton wth coarse mesh successve refnement of subspaces beng crossed by curve Subspace Dvson advantages: robust algorthm dscretzaton of dsconnected curves possble eample: base dscretzaton: 57 ponts 7 refnement levels 666 functon evaluatons equvalent grd: ponts (98945 ponts) Insttute of Engneerng

17 nfluence of rotaton on dynamc stablty Pose-dependent Stablty Charts Insttute of Engneerng

18 nfluence of translaton on dynamc stablty Pose-dependent Stablty Charts Insttute of Engneerng

19 Conclusons strong pose-dependency of dynamc behavor at tool-tp for parallel knematc machne tool Paral multbody system of machne structure can be used for analyss of dynamc stablty of cuttng processes mass dstrbuton between machne structure and tool-toolholder-spndle s responsble for rato of resonance ampltudes methods for effcent computaton of stablty charts have been developed Insttute of Engneerng dynamc stablty of process s nfluenced by pose of tool platform n workspace