Areva10 Technical Day

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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

• …

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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

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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

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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

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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• …

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• 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

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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