1. Using reduced system models for
vibration design and validation
Etienne Balmès
SDTools
Arts et Métiers ParisTech
AREVA Technical day, December 10, 2010
1
2. A system = I/O representation
System
In Out
Environment
Design point
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
3. Model complexity
Spot weld 1 gun
Simulation
• Geometry (nominal, variability, …) Spot weld 2 gun
Clamped end
• Material behavior (viscoelastic,
contact/friction, …) Welded plates
• 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
9. System models of structural dynamics
Large/complex FEM
Simple linear time invariant system
Sensors
When
Where
Extensions
• Coupling (structure, fluid,
Modal analysis control, multi-body, …)
• Optimization, variability,
Superelements damping, non linearity, …
CMS, … 9
10. Component mode synthesis
Reduction (Ritz analysis) based on
T qR restrictions :
{q}N= • Excitation (space & freq)
• Responses
• Coupling …
Nx NR
+
σ(x,t) u(x,t)
f(x,t)
+ σ(x,t)
Coupling : state dependent loads
10
11. Moving complexity in the coupling part
In Reduced model Sensors
• Coupling : test/FEM, fluid/structure
active control, …
• Local non-linearities : machining, bearings,
contact/friction, …
• Optimization / uncertainty
11
12. 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
12
13. Example : structural dynamics modification
System : identified response
In
Feedback :
modification
Motivation:
• System model very costly (no blue-print, internal complexity)
• Need to predict impact before implementing solution
PhD ECP. Corus 2002, Groult 2008
14. Test model limitations
• Very limited if non-linear System : identified
• Typically inconsistent
– Channel dependent noise
– Not exactly reciprocal
– Residual terms, not well
excited modes
• Spatially incomplete
– Few inputs
– Limited outputs
PhD ECP. Corus 2002, Groult 2008 14
15. Hybrid test/FEM using expansion
Problem : know outputs but states
(DOF) needed for coupling
Solution
Structure under test
• Local model
Instrumented area •Covers instrumented area
} Local model •Includes the modification
• Expansion
FEM of modification
•model based estimation
•gives knowledge of states
Extended SDM handles
• Spatial inconsistence
• Mass/stiffness/damping modifications
But requires consistent, linear model of tested
system
PhD Corus 2002, Groult 2008
17. Interfaces for coupling
Classical CMS : continuity coupling
• Reduced independently
• All interface motion (or interface modes)
• Assembly by continuity
Difficulties
• 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
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18. 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
19. Component mode tuning method
• Reduced model is sparse
• Free mode amplitudes are DOFs
Disc
1 ωj2
OuterPad
Inner Pad
Anchor
Caliper
Piston
Knuckle
Hub
[M] [Kel] [KintS] [KintU]
• Reduced model has exact nominal modes
(interest 1980 : large linear solution, 2010 : enhanced
coupling)
• Change component mode frequency ⇔ change the diagonal
terms of Kel
20. CMT & design studies
• One reduced model /
multiple designs
Component redesign
Examples Sensitivity
energy analysis
• impact of modulus change
• damping real system or component mode
+10 +20
Nom % %
.
-
20%
20
21. Revised notion of interface
Classical CMS (Craig-Bampton)
• System is brake without contact area
• Reduction : modes of system and
interface loads
• Many interface DOFs needed
heavily populated matrix
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 21
22. Exact system modes + local NL
Full system transient simulation
• 800e3 DOF FEM
modes can’t be used because of
contact area
200e3 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
PhD ECP. Vermot Jan 2011 22
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 23
25. Parametric families & reanalysis
• Evolutions of frequencies
with uncertain parameters
System • Effective stiffness of a
In Out
damping strut
• Campbell diagram
Design space (p) •…
Reduction basis T can be fixed
for range of parameters
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26. Bases for parametric studies
[T(p1) T(p2) … ]
• Multi-model
Orthogonalization
[T]
[Tk] Rdk=K-1 R(q(Tk))
• Other + residue iteration
Orthog [Tk Rdk]
Example water filled tank
• Example : strong coupling
With heavy fluids : modes of structure & fluid give
poor coupled prediction Without residual With residual
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27. Conclusion 1
Reduced / equivalent models T qR
• Reduction gives access to states : typically
superior if local detail needed
Nx NR
Reduction methods :
• Rely on a approximation of subspaces using System
bases that can be piece-wise in space In
Out
and/or time
• Basic tools to build subspaces
•Krylov iterations, static response Environment
•Conjugate gradient/Lanczos Design point
•Eigenvalue/SVD/POD/PGD
• In vibration validity & model complexity
depends on assumptions on loads and
frequency range : not FEM model size
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28. Conclusion 2
Linear 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
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Products : SDT, OpenFEM, Visco, Rotor, Runtime
for use within MATLAB 28
30. 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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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
M K
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