Enhancing structural dynamics of a V10 crankshaft using multi-criterial numerical optimisation Thomas Hinderer, BMW M GmbH Gerald Hofmann, Intec GmbH November, 20 2007 Bad Godesberg
Enhancing structural dynamics of a V10 crankshaft using multi-criterial numerical optimisation
Thomas Hinderer, BMW M GmbH
Gerald Hofmann, Intec GmbH
November, 20 2007
Bad Godesberg
Thomas Hinderer20.11.2007Seite 2
Ambition
S85, M5 engine
10 cylinders, V-90°
507 HP (~8000 rpm)
Optimisation objectives
Optimum balancing mass distribution
Radial bearing force peaks reduction
Shaft tilting and displacement reduction
Total mass and inertia minimisation
Thomas Hinderer20.11.2007Seite 3 Crank shaft
Flexible body, combined rigid and flexible body
Main bearingsHydrodynamic model (Impedance, Online-EHD)
Manual optimisationStrategyObjectivesProceedingResults
Numerical Optimisation (PSO)RequirementsJob definitionProcedureResults
Agenda
Thomas Hinderer20.11.2007Seite 4
Crank shaftCombined rigid and flexible body
Thomas Hinderer20.11.2007Seite 5
Crank shaftFlexible shaft without counterweights
Thomas Hinderer20.11.2007Seite 6
Force applicationPoint mass counterweights
Thomas Hinderer20.11.2007Seite 7
0 DOF joint
Mass
Inertia tensor
Center of mass
Mass, inertia, center of gravity = f (geometric properties)
RBE3
Point mass counterweightsAttachment of the point mass to the flexible structure
mi
Thomas Hinderer20.11.2007Seite 8
( )2
22 ρbαRR=m Öia
GGW
⋅⋅⋅−
( )( )22
33
32
sin4
iaÖ
Öia
GGWS, RRα
αRR
=r−⋅⋅
⋅−⋅
( )4
44iaÖ
GGWxx,
RRbαρ=I
−⋅⋅⋅
GGW : Gegengewicht (counterweight)Ö : Öffnungswinkel (apex angle)
Varying counterweight geometry Geometry parameters
Thomas Hinderer20.11.2007Seite 9
1st bending, flexible 1st bending, combined
Eigenmodes and Eigenfrequencies Flexible shaft compared to combined shaft
Thomas Hinderer20.11.2007Seite 10
Crank shaftFlexible body, semi-flexible body
Main bearingsHydrodynamic model (Impedance, Online-EHD)
Manual optimisationStrategyObjectivesProceedingResults
Numerical Optimisation (PSO)RequirementsJob definitionProcedureResults
Agenda
Thomas Hinderer20.11.2007Seite 11
Time integration
Input parameter
Position / Velocity
(Shaft, Sleeve)
Output
Bearing forces
Minimal gap
Maximal pressure
Interpolation Impedance Charts
Hydrodynamic forces
Oil gap and states
So So= ( , )ε ϑ
TO
WER
MK
S h
yd
rod
yn
am
ics
• Kinematics
• Kinetics
• Elasticity
External load
Integration
Str
ukt
ur
SIM
PA
CK
Width
Diameter
Play
Viscosity
Rev.speed
NodesTim
e in
dep
end
ent
par
am
eter
s
Solving Reynolds differential equations for
every integration timestep
( ) ( )
3 3
1 2
12 12
1( )
2
+ =
+ +
h p h p
x x z z
u u h hx t
∂ ρ ∂ ∂ ρ ∂∂ η ∂ ∂ η ∂
∂ ∂ρ ρ∂ ∂
Hydrodynamic bearings
Thomas Hinderer20.11.2007Seite 12
Crank shaftFlexible body, combined rigid and flexible body
Main bearingsHydrodynamic model (Impedance, Online-EHD)
Manual optimisationStrategyObjectivesProceedingResults
Numerical Optimisation (PSO)RequirementsJob definitionProcedureResults
Agenda
Thomas Hinderer20.11.2007Seite 13
Tilting
Max. radial fo
rce
Max. lateral force
DisplacementBearing forces
“Schiefex-Wert“
Combined Displacement-Tilting rating
Optimisation evaluation parametersBearing forces, tilting, displacement
Thomas Hinderer20.11.2007Seite 14
Displacement Tilting Schiefex
Optimisation evaluation parametersSchiefex-rating
Thomas Hinderer20.11.2007Seite 15
Optimisation evaluation parametersCounterweight relative angle
Thomas Hinderer20.11.2007Seite 16
FGGW,3
FGGW,2
FGGW,2
FGGW,3
FRes,2&3
FRes,2&3
Frot,GGW2
Frot,GGW3
Counterweight relative anglesConcept
Thomas Hinderer20.11.2007Seite 17
Hubzapfen 2
Displacement GGW3
Displacement GGW2
Reference angle GGW3
Reference angle GGW2
pivot 1
pivot 2
Counterweight relative angle variationsUnified setup
Thomas Hinderer20.11.2007Seite 18
Hz 1
GGW 2
Hz 2
GGW 3
Hz 2Hz 3
GGW 4
GGW 5
GGW 6
GGW 7
Hz 3
Hz 4 Hz 4Hz 5
GGW 8
GGW 9
Counterweight arrangementReference design
Hz : Hubzapfen (Pivot)GGW : Gegengewicht (counterweight)
Thomas Hinderer20.11.2007Seite 19
αÖ,GGW1, αMitte,GGW1
αÖ,GGW2 - ΔαÖαÖ,GGW3 - ΔαÖαÖ,GGW4 - ΔαÖαÖ,GGW5 - ΔαÖαÖ,GGW6 - ΔαÖαÖ,GGW7 - ΔαÖαÖ,GGW8 - ΔαÖαÖ,GGW9 - ΔαÖ
αÖ,GGW10, αMitte,GGW10
αÖ,GGW1, αMitte,GGW1
αÖ,GGW2 - ΔαÖαÖ,GGW3 - ΔαÖαÖ,GGW4 - ΔαÖαÖ,GGW5 - ΔαÖαÖ,GGW6 - ΔαÖαÖ,GGW7 - ΔαÖαÖ,GGW8 - ΔαÖ
αÖ,GGW9
αÖ,GGW10, αMitte,GGW10
αÖ,GGW1, αMitte,GGW1
αÖ,GGW2 - ΔαÖαÖ,GGW3 - ΔαÖαÖ,GGW4 - ΔαÖαÖ,GGW5 - ΔαÖαÖ,GGW6 - ΔαÖαÖ,GGW7 - ΔαÖ
αÖ,GGW8αÖ,GGW9 - ΔαÖ
αÖ,GGW10, αMitte,GGW10
Variation 1 - 3 (of 28 = 256)
Manual optimisationDesign of experiments
ΔαÖ = „Optimisation stepsize“ (!)
resulting from balancing
ΔαÖ not appliedΔαÖ applied
Thomas Hinderer20.11.2007Seite 20
αÖ,GGW1, αMitte,GGW1
αÖ,GGW2αÖ,GGW3αÖ,GGW4αÖ,GGW5αÖ,GGW6αÖ,GGW7
αÖ,GGW8 - ΔαÖαÖ,GGW9
αÖ,GGW10, αMitte,GGW10
αÖ,GGW1, αMitte,GGW1
αÖ,GGW2αÖ,GGW3αÖ,GGW4αÖ,GGW5αÖ,GGW6αÖ,GGW7αÖ,GGW8
αÖ,GGW9 - ΔαÖ
αÖ,GGW10, αMitte,GGW10
αÖ,GGW1, αMitte,GGW1
αÖ,GGW2αÖ,GGW3αÖ,GGW4αÖ,GGW5αÖ,GGW6αÖ,GGW7αÖ,GGW8αÖ,GGW9
αÖ,GGW10, αMitte,GGW10
Variation 254 - 256 (of 256)
Manual optimisationDesign of experiments
Thomas Hinderer20.11.2007Seite 21
Manual optimisation3D-plot of maximal bearing force of one iteration
Only 60 of 256 displayed
Thomas Hinderer20.11.2007Seite 22
Manual optimisationResult evaluation after one iteration step
„Manual Hill-climbing“
Thomas Hinderer20.11.2007Seite 23
Design of experiments
(Maple)
Evaluation of paramererised model
(SIMPACK)
Results visualisation
(Matlab)
Ref
ere
nce
de
sig
n
Ce
nte
r a
ngle
dis
trib
utio
n
Ape
x a
ng
le d
istr
ibut
ion
Manual evaluation
Pick optimum from 256 solutions
Manual optimisationSemi-automatic optimisation loop
Thomas Hinderer20.11.2007Seite 24
Massenverläufe
20,620,8
2121,221,421,621,8
2222,222,422,622,8
2323,223,423,623,8
2424,224,4
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Optimierungsschleife [-]
Ku
rbel
wel
len
mas
se [
kg]
Vierteldrehung (Kraftorientiert)
Halbdrehung (Kraftorientiert)
Serienmasse
Nulldrehung
ViertelHalbdrehung
Manual optimisationFinal results
Crank shaft total mass
Various manual approaches to set up counterweigh angles
Reference
Thomas Hinderer20.11.2007Seite 25
Lagerkraftverläufe
4200043000440004500046000470004800049000500005100052000530005400055000560005700058000590006000061000
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Optimierungsschleife [-]
Ma
xim
ale
La
ge
rkrä
fte
[N
]
Halbdrehung (Kraftorientiert)
Vierteldrehung (Kraftorientiert)
Serienlagerkraft
Nulldrehung
ViertelHalbdrehung
Bearing radial force
Manual optimisationFinal results
Reference
Selected solution
Thomas Hinderer20.11.2007Seite 26
Crank shaftFlexible body, combined rigid and flexible body
Main bearingsHydrodynamic model (Impedance, Online-EHD)
Manual optimisationStrategyObjectivesProceedingResults
Numerical Optimisation (PSO)RequirementsJob definitionProcedureResults
Agenda
Thomas Hinderer20.11.2007Seite 27
Numeric optimisationConditions and requirements
Conditions
Highly nonlinear system characteristics
Parameter space constraints
Reference design of unknown quality
Parameter space of dimension 16
8 optimisation objectives to minimise
Time for one evaluation ~15min
Requirements
No gradient based algorithm (Jabobian matrix, time complexity)
Quick convergence, (semi-) heuristic method
Not vulnerable to local minima
Parallel function evaluation
No predefined weighting or „magic“ factors
Thomas Hinderer20.11.2007Seite 28
Particle swarm optimisationHistory and features
Idea / Functional principle
Agent-based, stochastic method
Simple rules of decision, basing on common knowledge
Distributed artificial intelligence
Self-organising collective intelligence
Emerging behaviour
History
First implemented 1995 by J. Kennedy and R. C. Eberhart
Optimiser used
www.gpsopt.de
Thomas Hinderer20.11.2007Seite 29
Particle swarm optimisationParameters and settings
Swarm
Given number of particles (dynamically de-/activated to save CPU time)
Particles
Parameter vector („Position in the parameter space“)
Velocity vector („Search direction“)
Swarm knowledgeOver-all best solution for every
single objective
Over-all best solution for the
global optimum
Thomas Hinderer20.11.2007Seite 30
Particle swarm optimisationParticle movement model
v=f(k)
Thomas Hinderer20.11.2007Seite 31
Particle swarm optimisationExample: Three poles in a cubic parameter space
Thomas Hinderer20.11.2007Seite 32
Particle swarm optimisationResult visualisation
Example: Three dimensions (of 16) selected for visualisation
Total 3D parameter space snapshots: 16! / ( 3! * (16-3)! ) = 560
Thomas Hinderer20.11.2007Seite 33
Result evaluationMulti-objective result comparison
Thomas Hinderer20.11.2007Seite 34
Result evaluationMulti-objective result comparison
selected
Thomas Hinderer20.11.2007Seite 35
Update parameter files
Run time integration
Update optimum mapDe-/activate particles
Update direction pointersUpdate positions / velocity
Check break conditions
Check parameter constraints
Evaluate resultsCalculate fitness values
Static balancing
per
part
icle
mul
ti-th
read
ed e
valu
atio
n
Particle swarm optimisationProcedure
Thomas Hinderer20.11.2007Seite 36
9°18°0°36°9351°342°0°324°818°0°36°324°754°72°36°108°6
162°144°180°108°5198°216°180°252°4225°234°216°252°3207°198°216°180°2
Achtel-drehung
Viertel-drehung
Halb-drehung
Null-drehungGGW
24.7°174.46°107.14°93.51°9
343.76°71.52°810.26°38.02°748.81°33.09°6
165.22°36.01°5195.25°28.17°4216.88°55.45°3199.23°116.37°2-165.02°131.89°1
Mitten-winkel
Öffnungs-winkelGGW
Result evaluationConfiguration found by PSO compared to manual approaches
Manual design approaches Solution found by PSO
PSO solution not related to any manual approach
Thomas Hinderer20.11.2007Seite 37
Total mass reduction
Max. bearing force peak reduction
Number of iterations
Works without user interaction
Manual gpsopt
14,85% 14,8%
+/- 0 % -14 %
20000 5000
No Yes
Multi-criterial evaluation No Yes
Result summary
PSO found new designs, not covered by classic (manual) design rules.
Thomas Hinderer20.11.2007Seite 38
Thank you for your attention !Thank you for your attention !