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Themenschwerpunkt: Simulation im Konsumgüterbereich
11Infoplaner 01/2010
Finite elements modeling provides an im-portant contribution to
the developmentprocess at Audemars Piguet & Cie SA. Vir-tual
prototyping is used to anticipate di-mensioning problems and
therefore reducethe number of prototypes.
Amongst the wide range of componentsconstituting a bracelet
watch, three keymechanisms are presented below as exam-ples where
numerical simulation is usednowadays.
Optimization of a Date Mechanismwith ANSYS/LS-DYNAFigures 1a and
1b show the mechanismthat allows changing the date display every24
hours. This mechanism is composed ofthree main parts;
The mechanisms used in this study belong exclusively to
Audemars Piguet & Cie SA (www.audemarspiguet.com).
Pictures: Audemars Piguet & Cie SA
1. The display disc2. The trigger bloc (that stores
energy and transfers it to the display disc)
3. The jumper bloc (that brakes the display disc)
A cycle of this mechanism starts withthe loading of the trigger
spring.When the date has to change, thecam blocking spring releases
the pinand the potential energy stored in thetrigger spring rotates
the cam and itsfinger that pushes a tooth of the dis-play disc. The
resulting rotation of thedisplay disc is braked by the jumper
and its spring so that only one tooth passesthe jumper and
therefore the date chan-ges by only one increment.
The complexity of this mechanism residesin the need of setting
and balancing theway the energy is released by the triggerstring
and the way the energy is dissipatedin the jumper bloc so that the
date changeoccurs instantaneously to the eye (typical-ly within
0.015 s) but robustly enough sothat the display never jumps a
date.
The geometrical shapes of the jumperspring, the jumper and the
trigger springwere optimized with a three dimensional
Time for ANSYSDimensioning and Optimization of Flexible Watch
Industry MechanicalComponents with ANSYS/LS-DYNA, ANSYS Workbench
and optiSLang
Watch industry mechanisms involve a large number of high
precision flexible pre-constrained mechanicalcomponents. Using
traditional prototyping, the definition of non-deformed geometries
for production is acostly manual iterative process. The use of
non-linear finite elements modeling improves this process and
thecoupling of the finite elements codes to a stochastic
optimization toolbox like optiSLang makes it automaticand more
robust.
Fig. 1a: Loaded date mechanism at time t=0.01s.
(display disc diameter = 12 mm
Fig. 1b: Zoom on the trigger bloc (cam diameter = 2.2 mm)
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12 Infoplaner 01/2010
Themenschwerpunkt: Simulation im Konsumgüterbereich
dynamic model created with ANSYS/LS-DYNA (fig. 1). The
calculated angularvelocity of the display disc (fig. 2) shows
apositive acceleration of the disc by the trig-ger bloc (a-b), a
sudden reversed accele-
ration when tooth 2 (fig. 1a) bounces onthe steep face of the
jumper (b-c), a posi-tive acceleration again when tooth 1 tou-ches
the jumper again (c-d) and a final sta-bilization between teeth 1
and 2 (d-e). Theloading moment of the trigger spring wasmeasured
experimentally and is in goodagreement with the simulated values
(fig.3). Furthermore, the pre-series mechanismthat was produced
based on the designobtained with ANSYS/LS-DYNA has fulfil-led
acceptance criteria and allowed laun-ching production without any
further pro-totype.
Force Tuning of a Set Time Mechanismwith ANSYS Workbench and
optiSLangFigure 4 shows the set time mechanismconnected to the
pull-out button of awatch. The button actuates a winding shaftthat
can be pulled up to its stop position;its rotation then allows time
setting. Thewinding shaft is connected to the pull-out
ces set within 2% of the required valuewhile the maximum stress
was 10% smal-ler than the value calculated with the in-itial
geometry (fig. 5). The mechanism wasproduced and fulfilled
expectations.
Robust Design Optimization of a Glass Driving Process with ANSYS
Work-bench and optiSLangTightness between glass and watch-caseis
ensured by a flexible joint (fig. 6). Theforce needed to remove the
glass has tobe maximized whereas the force requiredto drive the
glass should be minimized.Plastic deformations in the joint (fig.
7) aswell as stresses in the glass and watch-caseshould also be
minimized.
piece via a pin. The pull-out piece can ro-tate on a fixed axis
but is constrained bythe spring that pushes on a pin at its endand
therefore sets its actuation moment.The maximum traction force on
the win-
ding-shaft has to be 5N in order to ensurea good sensitivity
when pulling with thefingers on the set time button. At the
sametime, stresses in the spring have to remainbelow the yield
strength.
A two dimensional parametric model ofthe spring and its
non-linear frictionalcontact with the pin of the pull-out piecewas
created with ANSYS Workbench andcoupled to optiSLang via the
optiPlug in-terface. This allowed to run an automaticparametric
optimization of the spring‘sshape based on eight geometrical
inputparameters and three objectives:1.Set the traction force2.Set
the pulling force3.Minimize structural stresses
The optimization algorithm chosen was anadaptive response
surface method. After91 automatic design evaluations, the
re-sulting design had traction and pulling for-
Fig.5: Spring initial shape (left) and tuned shape (right).
The
pin of the pull-out piece is at the force inversion position
where stresses reach their maximum. The positioning and
angle of the two flat contact faces of the spring determine
the pull and push forces.
Fig. 2: Display disc angular velocity. A positive velocity
means
a clockwise rotation on figure 1a.
Fig. 3: Comparison between simulated and measured trigger
bloc moments.
Fig. 4:
Set time mechanism
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Themenschwerpunkt: Simulation im Konsumgüterbereich
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A quasistatic two dimensional axisymmetricparametric model was
created with ANSYSWorkbench and coupled to optiSLang inorder to run
three different analyses on themodel:
1.A sensitivity analysis2.A Pareto optimization3.A robustness
analysis
Amongst a list of 16 geometrical inputparameters (dimensions of
glass, joint andwatch body), the sensitivity analysis de-livered a
list of 8 most important geome-trical dimensions. According to the
statis-tical linear coefficient of importance cal-culated by
optiSLang, these parameters de-termine 86% of the maximal
withdrawalforce, 77% of the maximum glace stressand 65% of the
joint maximum plasticstrain. In addition to the selection of
asubset of most relevant parameters,the sensitivity analysis
allowed togain understanding of the physi-cal system. For instance,
the cor-relations between outputs canbe seen at a glimpse in
theoptiSLang post processing.In this case, output va-lues that have
to be mi-nimized (stresses andstrain) and the out-put value that
hasto be maximized(removal force) arepositively correla-ted between
eachother, which meansthat attempting to maxi-mize the force will
also maxi-mize the stresses and strains.
In addition to this intuitive qualitative sta-tement, optiSLang
delivered quantitativecorrelation values that helped defining
ob-jective functions for the optimization.
Due to these output parameters correla-tions, a Pareto
optimization with twoobjective functions was chosen; the
firstobjective is a weighted function of theremoval force and of
the difference bet-ween driving and removal force. The se-cond
objective function is simply the sumof stresses in the watch-case
and in theglass. After 209 design evaluations, theresult of this
optimization, based on anevolutionary algorithm, is a Pareto
frontwith designs that minimize both objectives(fig. 8). In this
case, the choice of a best
design along this front is motivated by theneed to increase the
force (move towardsthe left) while maintaining the stress lowenough
(move down on the graph).
After having selected a candidate designon the Pareto front, a
robustness analysiswas run for this design. Probability densi-ty
functions were defined for each inputparameter, including material
properties.The resulting output parameter probabili-ty density
functions could then be inte-grated in optiSLang in order to get
the pro-bability of being higher than a given stressthreshold. This
failure probability givesquantitative information on whether
thedesign is sufficiently robust or not. In thiscase, the failure
probability of designnumber 203 was 20% for gold (inaccep-table)
and negligible for steel (see 250 MPalimit on the probability
density function offig. 8).