simpack user meeting 2007 modeling and simulation of a ... fileinstitute of engineering and...

Institute of Engineering and Computational Mechanics

University of Stuttgart, GermanyProf. Dr.-Ing. Peter Eberhard

Simpack User Meeting 2007

Modeling and Simulation of a Hexapod Machine Tool for the Dynamic Stability Analysis

of Milling Processes

C. Henninger, P. Eberhard

project funded by the DFG within the framework of the SPP 1099‚Parallel Kinematic Machine Tools‘



Outline

motivation: PKM machine tools and dynamic stability of cutting processes

modeling of the machine structure

analysis of dynamic stability

pose-dependent stability diagrams for the hexapod machine tool Paralix

modeling of the milling process

techniques for efficient computation of stability diagrams



PKM Machine Tool PARALIX

• 6-DOF hexapod machine tool

• struts with constant length

• sledges driven by linear direct drives

• 5-DOF high-speed milling of metal

IfW, University of Stuttgart

no bending moments in links, only normal forces→ light-weight design of struts

spatial assembly of links→ high static stiffness

equal components in identical links→ reduction of production costs

nonlinear kinematics and dynamicsnonlinear position control necessary

pose-dependency of stiffnessforce/velocity transmission and dynamic properties

kinematic singularitiesloss of control of DOF,reduction of workspace



Dynamic Stability of Cutting Processes for PKM

machine structure

process

characteristics of PKM:light-weight structurespose-dependent dynamics

external excitationunbalance excitationinertia forces

process forces displacements

process reliabilitydynamic stabilitysurface quality

process parametersspindle speed,feed,depth of cut, ...

mechanisms of excitationexternal and self-excited vibrations

?



Elastic Multibody System of PKM Paralix

machine structure

elastic struts

joint elasticities

tool + toolholder

mkx

x

y

cx

ky cy

rheonomic joints

sledges, platform as rigid bodies

finite element modeling of struts (beam elements)

modal reduction of strut model (9 dof)elastic Cardan joints and linear joints

xyz

modal reduction of complete model for eachpose in workspace



Eigenfrequencies of Machine Structure

in Workspace

Mode 1 y-z

Mode 2 y-z

Mode 3 y-z

Mode 1 α-β

Mode 2 α-β

Mode 3 α-β



Role of Machine Structure in Tool-Tip Dynamicsmeasured tool-tip frequency repsonse

bimodal system representingmachine structure and tool-toolholder-spindle



Pose-dependency of Dynamic Behaviour

influence on eigenfrequenciesand tool-tip frequency response



Modeling of Milling Process

linear cutting force modelbhkFF ct0tt += bhkFF cr0rr +=

two-dimensional milling process

dynamic depth of cut with regenerative effect

( )[ )sin()t()t()(g)(hh 0 ϕτ−ξ−ξϕ+ϕ=

( ))t()t()t(b)t( c0 τ−−⋅+= yyKFFdynamic process force equation

overall system considering machine structure and process dynamics)t()t()t()()t()t( 0cc FyKyKKyDyM =τ−⋅−⋅++⋅+⋅ &&&

delay-differential equation (DDE) with periodic coefficients

dynamic stability?

( ) ])cos()t()t( ϕτ−μ−μ+



Analysis of Dynamic Stability by Semi-Discretization

Methoddiscretization of system history

)t()t()t()t( τ−⋅+⋅= xQxAx&homogeneous periodic time-variant DDE

( )1miamib ww)t( +−− +⋅+⋅= xxQxAx&

approximating ODE on [ ]1ii tt +

1miamib

i

ww)2/tt()t(

+−− +≈τ−Δ+≈τ−

xxxx

approximations on

ii :)2/tt()t( AAA =Δ+≈

[ ]1ii t,t +

ii :)2/tt()t( QQQ =Δ+≈

miib1miiaii1i wwP −+−+ ⋅+⋅+⋅=→∫ xRxRxx with )texp( ii Δ= AP

i1

iii ))t(exp( QAIAR ⋅⋅−Δ= −

Insperger & Stépán, 2003:



Stability Analysistransition matrix for tΔ

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

⋅

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−

+−

−

−

+−

−

−

+

mi

1mi

2i

1i

iT

ibT

iai

1mi

2i

1i

i

1i

~~

~~

ww

~

~~~

xx

xxx

0I000

00I00000I00000C

CRCR00P

x

xxx

x

M

LMMOMMM

L

L

LL

M

transition matrix for system period T 012k1k ΦΦΦΦΦ ⋅⋅= −− L

Floquet theory for stability of periodic time-variant linear systems ( )

⎪⎩

⎪⎨⎧

<=>

111

)(eigmax Φunstablestability boundarystable

,~ xCx ⋅=vector of delay-active states ,nd×∈C nd <



Stability Boundary Diagram

up-milling process of aluminum alloywith single-bladed cutter, a/D=0.05

dynamic stability depending on process parameters spindle speed Nand axial depth of cut ap

0.020.0087722

m [kg]D [-]f0 [Hz]

238644

kcr [MPa]kct [MPa]

unstable

stable



Efficient Computation of Stability Diagrams

implicitly defined stability boundary

( ) ( )( ){ }1)a,N(eigmaxa,NS pp == Φ

full discretization

• equidistant discretization of complete domain• computationally expensive

curve tracking• local search method• adaptive discretization• high computational efficiency• extra treatment of cusps and

near-branch zones necessary• discretization of disconnected

regions not possible

Spitzkehre Engstelle



Curve Tracking



Comparison: Curve Tracking vs. Full Discretization

full discretization

• equidistant grid 55x55 points• 3025 function evaluations

curve tracking

• 270 curve points• 2952 function evaluations• equivalent to full discretization with

250x250 = 62500 pointscurve tracking(2952 function eval.)full discretization(3025 function eval.)



Subspace Division• initial discretization with coarse mesh• successive refinement of subspaces

being crossed by curve

idea:

advantages:

• robust algorithm• discretization of disconnected

curves possible

example:

• base discretization: 5x7 points• 7 refinement levels• 6066 function evaluations• equivalent grid: 257x385 points

(98945 points)



Pose-dependentStability Charts

influence of rotationon dynamic stability



Pose-dependentStability Charts

influence of translationon dynamic stability



Conclusions

strong pose-dependency of dynamic behavior at tool-tipfor parallel kinematic machine tool Paralix

mass distribution between machine structure and tool-toolholder-spindle is responsible for ratio of resonance amplitudes

dynamic stability of process is influenced by pose of tool platform in workspace

methods for efficient computation of stability charts have been developed

multibody system of machine structure can be used foranalysis of dynamic stability of cutting processes

simpack user meeting 2007 modeling and simulation of a ... fileinstitute of engineering and...

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