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Using reduced system models for vibration design and validation
Etienne BalmèsSDToolsArts et Métiers ParisTechAREVA Technical day, December 10, 2010
A system = I/O representation
Prototype Virtual prototype
☺ all physics (no risk on validity) � limited physics (unknown & long CPU)☺ in operation response � design loads� limited test inputs ☺ user chosen loads� measurements only ☺ all states known� few designs ☺ multiple (but 1 hour, 1 night,
several days, … thresholds)
� Cost : build and operate � Cost : setup, run, manipulate
In Out
EnvironmentDesign point
System
Model complexitySimulation• Geometry (nominal, variability, …)
• Material behavior (viscoelastic, contact/friction, …)
• Input : dynamic environment
• Objectives : static deflection, frequencies, dynamic amplitudes, stresses, cycle counts
• …
Test (modal analysis)• Bandwidth, how many modes
• Number of in/out, reciprocity, residual terms
• Non-linear characterization
• …
3
Welded plates
Clamped end
Spot weld 1 gun
Spot weld 2 gun
Outline• Micro/macro behavior : equivalent behavior
– Homogeneization, updating, …
– Modal damping
• Model reduction : subspace representationCMS, variable separation, POD, PGD, …
– Classical modal synthesis : spatially simple models– Coupling test & FEM models
– Energy coupling & revised CMS
– Design phases / uncertainty
4
Equivalent models : honeycomb example
• Micro : cell walls, glue, face-sheet, viscoelastic material
• Macro : shell/ orthotropicvolume/ shell
• Equivalence: waves/modes
• End result orthotropic law
• Loss of detail
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Detailed 3Dhoneycomb
Shell/volume/shell
Numerical homogeneization
Updating from test
PhD ECP Jan. 2010 : Corine Florens
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Equivalent time domain modal damping• Modal damping = assume viscous damping matrix diagonal in modal basis
• Rayleigh damping:
– Physical domain
– Modal domain
• Modal + piece-wise Rayleigh
Reality
Mass Stiffness
Bianchi ISMA Sep. 2010
Equivalent model building• Homogeneization (equivalent material, equivalent model)
• Updating : identical static,frequency, dissipation (weld spot, screw, beam, …)
• Modal damping
with loss of detail
• Model reduction
with restitution7PhD 2005 Abbadi (PSA)
Outline• Micro/macro behavior : equivalent behavior
– Homogeneization, updating, …
– Modal damping
• Model reduction : subspace representation– Classical modal synthesis : spatially simple models– Coupling test & FEM models
– Energy coupling & revised CMS
– Design phases / uncertainty
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System models of structural dynamics
Simple linear time invariant system
Extensions• Coupling (structure, fluid, control, multi-body, …)
• Optimization, variability, damping, non linearity, …
When
Where
Sensors
Large/complex FEM
Modal analysisSuperelementsCMS, …
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Component mode synthesisReduction (Ritz analysis) based on restrictions :
• Excitation (space & freq)• Responses• Coupling …
σ(x,t)f(x,t)
u(x,t)σ(x,t)
Coupling : state dependent loads
+
+
{q}N=qR
Nx NR
T
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Moving complexity in the coupling part
Reduced model
• Coupling : test/FEM, fluid/structureactive control, …
• Local non-linearities : machining, bearings, contact/friction, …
• Optimization / uncertainty
In Sensors
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CMS current practice• Craig-Bampton (unit displacements + fixed interface modes)
– Very robust, guaranteed independence • McNeal (free modes + static response to loads)
– Tends to have poor conditioning (residual flexibility)
• Well established applications– structural vibrations– multi flexible-bodies– vibroacoustics
• Limits– Very large models– Large interfaces– Parametric design of component– Non local or strong coupling (reduction not independent)
– Hybrid test/analysis– …– Ease of use
Example : structural dynamics modification
In
response
Feedback : modification
System : identified
Motivation: • System model very costly (no blue-print, internal complexity)• Need to predict impact before implementing solution
PhD ECP. Corus 2002, Groult 2008
Test model limitations
• Very limited if non-linear
• Typically inconsistent– Channel dependent noise
– Not exactly reciprocal
– Residual terms, not well excited modes
• Spatially incomplete– Few inputs
– Limited outputs
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System : identified
PhD ECP. Corus 2002, Groult 2008
Hybrid test/FEM using expansion
Instrumented area
Local model}FEM of modification
Structure under test
Problem : know outputs but states (DOF) needed for coupling
Solution• Local model
•Covers instrumented area•Includes the modification
• Expansion•model based estimation•gives knowledge of states
PhD Corus 2002, Groult 2008
Extended SDM handles• Spatial inconsistence• Mass/stiffness/damping modificationsBut requires consistent, linear model of tested system
Outline• Micro/macro behavior : equivalent behavior
– Homogeneization, updating, …
– Modal damping
• Model reduction : subspace representation– Classical modal synthesis : spatially simple models– Coupling test & FEM models
– Energy coupling & revised CMS
– Design phases / uncertainty
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Interfaces for couplingClassical CMS : continuity coupling
• Reduced independently• All interface motion (or interface modes)• Assembly by continuityDifficulties• Mesh incompatibility• Large interfaces• Strong coupling (reduction requires knowledge of coupling)
Disjoint components : energy coupling
• Assembly by computation of interface energy (example Arlequin)
Difficulties• Use better bases than independent reduction
Energy coupling• Disjoint components with interface energy
• Subspace for each component can be arbitrary:valid Rayleigh-Ritz
• Component Mode Tuning method– free/free real modes (explicit DOFs)– trace of the assembled modes on the component
+
Component mode tuning method• Reduced model is sparse• Free mode amplitudes are DOFs
• Reduced model has exact nominal modes(interest 1980 : large linear solution, 2010 : enhanced coupling)
• Change component mode frequency ⇔ change the diagonal terms of Kel
DiscDiscDiscDisc
OuterPadOuterPadOuterPadOuterPad
Inner PadInner PadInner PadInner Pad
AnchorAnchorAnchorAnchor
CaliperCaliperCaliperCaliper
PistonPistonPistonPiston
KnuckleKnuckleKnuckleKnuckle
HubHubHubHub
ωj21
[M] [Kel] [KintS] [KintU]
CMT & design studies
• One reduced model /multiple designs
Examples
• impact of modulus change
• damping real system or component mode
20
Component redesign
Sensitivityenergy analysis
Nom.
+10%
+20%
-20%
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Classical CMS (Craig-Bampton)• System is brake without contact area
• Reduction : modes of system and interface loads
• Many interface DOFs needed heavily populated matrix
Revised notion of interface
Disjoint component with exact modes
• No reduction of DOFs internal to contact area
• Reduction : trace of full brake modes on reduced area (no need for static response at interface)
PhD ECP. Vermot Jan 2011
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Full system transient simulation• 800e3 DOF FEMmodes can’t be used because of contact area200e3 time steps = 1.2 To⇒ Need piece-wise reduction
Local detail accessible• Contact pressure/stiffness• Modal damping for accurate instability study
• Post-processing modal amplitudes, component energy
Exact system modes + local NL
PhD ECP. Vermot Jan 2011
23
Disjoint component bases • Reduction by component : minimize basis
storage
• Use system predictions for correct coupling with minimal number of interface modes
Example full shaft model
• Use cyclic symmetry to build
• CMT for mistuning
PhD ECP. A. Sternchüss 2009
Outline• Micro/macro behavior : equivalent behavior
– Homogeneization, updating, …
– Modal damping
• Model reduction : subspace representation– Classical modal synthesis : spatially simple models– Coupling test & FEM models
– Energy coupling & revised CMS
– Design phases / uncertainty
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Parametric families & reanalysis
Reduction basis T can be fixedfor range of parameters
In Out
Design space (p)
System
• Evolutions of frequencies with uncertain parameters
• Effective stiffness of a damping strut
• Campbell diagram• …
26
• Multi-model
• Other + residue iteration
• Example : strong couplingWith heavy fluids : modes of structure & fluid give poor coupled prediction
Bases for parametric studies
Example water filled tank
With residualWithout residual
[T(p1) T(p2) … ]
Orthogonalization
[T]
[Tk] Rdk=K-1 R(q(Tk))
Orthog [Tk Rdk]
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Conclusion 1Reduced / equivalent models• Reduction gives access to states : typically superior if local detail needed
Reduction methods : • Rely on a approximation of subspaces using bases that can be piece-wise in space and/or time
• Basic tools to build subspaces•Krylov iterations, static response•Conjugate gradient/Lanczos•Eigenvalue/SVD/POD/PGD
• In vibration validity & model complexitydepends on assumptions on loads and frequency range : not FEM model size
In Out
EnvironmentDesign point
System
qR
Nx NR
T
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Conclusion 2Linear time invariant reduced model still allows• Coupling (test/FEM, structure/component, fluid/structure)• Variability/design studies
Top issues• SDTools, as software editor, aware that first cost is model setup ⇒ ease of use
• Equivalent/reduced models rely on assumptions ⇒ how can these be clear and controlled by the user ? (control accuracy)
• Understanding comes from result analysis at system and component level ⇒ handling restitution ?
• Handling design studies ?• Design methods for non-linear vibration
www.sdtools.com/publicationsProducts : SDT, OpenFEM, Visco, Rotor, Runtime for use within MATLAB
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Post-processing with reduced models• Restitution
– Many DOFs a few DEF (energy, strain, …)
– A few DOFs many DEF (animation, test/analysis correlation)
– Time simulation sub-sampling
• Understanding the response– Component energies
– Time/freq SVD
• That’s the real frontier
31
Multi-frontal solvers / AMLS• Graph partionning methods ⇒group DOFs in an elimination tree with separate branches
• Block structure of reduction basis
• Block diagonal stiffness
• Very populated mass coupling
• Multi-frontal eigensolvers introduce some form of interface modes to limit size of mass coupling
KM