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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‘
Institute of Engineering and Computational Mechanics
University of Stuttgart, GermanyProf. Dr.-Ing. Peter Eberhard
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
Institute of Engineering and Computational Mechanics
University of Stuttgart, GermanyProf. Dr.-Ing. Peter Eberhard
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
Institute of Engineering and Computational Mechanics
University of Stuttgart, GermanyProf. Dr.-Ing. Peter Eberhard
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
?
Institute of Engineering and Computational Mechanics
University of Stuttgart, GermanyProf. Dr.-Ing. Peter Eberhard
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
Institute of Engineering and Computational Mechanics
University of Stuttgart, GermanyProf. Dr.-Ing. Peter Eberhard
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
Institute of Engineering and Computational Mechanics
University of Stuttgart, GermanyProf. Dr.-Ing. Peter Eberhard
Role of Machine Structure in Tool-Tip Dynamicsmeasured tool-tip frequency repsonse
bimodal system representingmachine structure and tool-toolholder-spindle
Institute of Engineering and Computational Mechanics
University of Stuttgart, GermanyProf. Dr.-Ing. Peter Eberhard
Pose-dependency of Dynamic Behaviour
influence on eigenfrequenciesand tool-tip frequency response
Institute of Engineering and Computational Mechanics
University of Stuttgart, GermanyProf. Dr.-Ing. Peter Eberhard
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+
Institute of Engineering and Computational Mechanics
University of Stuttgart, GermanyProf. Dr.-Ing. Peter Eberhard
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:
Institute of Engineering and Computational Mechanics
University of Stuttgart, GermanyProf. Dr.-Ing. Peter Eberhard
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 <
Institute of Engineering and Computational Mechanics
University of Stuttgart, GermanyProf. Dr.-Ing. Peter Eberhard
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
Institute of Engineering and Computational Mechanics
University of Stuttgart, GermanyProf. Dr.-Ing. Peter Eberhard
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
Institute of Engineering and Computational Mechanics
University of Stuttgart, GermanyProf. Dr.-Ing. Peter Eberhard
Curve Tracking
Institute of Engineering and Computational Mechanics
University of Stuttgart, GermanyProf. Dr.-Ing. Peter Eberhard
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.)
Institute of Engineering and Computational Mechanics
University of Stuttgart, GermanyProf. Dr.-Ing. Peter Eberhard
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)
Institute of Engineering and Computational Mechanics
University of Stuttgart, GermanyProf. Dr.-Ing. Peter Eberhard
Pose-dependentStability Charts
influence of rotationon dynamic stability
Institute of Engineering and Computational Mechanics
University of Stuttgart, GermanyProf. Dr.-Ing. Peter Eberhard
Pose-dependentStability Charts
influence of translationon dynamic stability
Institute of Engineering and Computational Mechanics
University of Stuttgart, GermanyProf. Dr.-Ing. Peter Eberhard
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