This document discusses using reduced 3D models in vibrational design processes. It presents tools for model reduction including variable separation, parametric models, and domain decomposition. These tools combine finite element modeling, experimental modal analysis, and reduced order models to efficiently simulate complex systems for design studies while controlling accuracy.

1. 1
Intégration de modèles 3D réduits dans
le processus de conception vibratoire.
Exemples de différentes industries.
Etienne Balmes
SDTools
Arts et Métiers ParisTech
NAFEMS Simulation des systèmes
3 Juin 2015

2. • FEM simulations
• System models (model reduction, state-space, active control, SHM)
• Experimental modal analysis
• Test/analysis correlation, model updating
Activities
2
CAD/Meshing
FEM
Simulation
Testing
CATIA, Workbench, …
NASTRAN, ABAQUS, ANSYS,...
Adams, Simulink,...
LMS TestLab, ME-Scope, …
Simulation
Validation
SDT : MATLAB based toolbox
Commercial since 1995 > 700 licenses sold
Pantograph/catenary Modal test correlation Track dynamics

3. Outline
• Systems, models, dynamic models
• Tools for model reduction
– Variable separation
– Parametric models
– Domain decomposition
• Conclusion
3

4. 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, manipulate
In Out
Environment/Design point
System

5. Meta/reduced models
5
Full numerical
model
expensive
Meta-model
acceptable cost
Learning
points
Responses
Computation
points
LearningX LearningY
X
Estimations
Yˆ
Validity ?
• Regular relation
• Band-limited
• Spatial position
of inputs
• …
Predictive monitoring
of fuel circuit
Ph.D. of B. Lamoureux
~500 parameters
~100 indicators
~20 Inputs
Data from in
operation
measurements

6. 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
Historical keywords :
Modal analysis
Superelements, CMS, …

7. Ingredient 1 : variable separation
• General transient but
– limited bandwidth
– time invariant system
• Modal Analysis
response well approximated
in spatial sub-space
𝑞(𝑥, 𝑡) = 𝜙𝑗 𝑥 𝑎(𝑡)
𝑁𝑀
𝑗=1
• Space shapes =modes
• Time shapes =
generalized coordinates
7

8. SVD on the time response
• coincides with modes if
isolated resonances
• similar info for NL systems
8
Space / Time decomposition
Squeal limit cycle
PhD Vermot (Bosch)
NL system with impacts, PhD Thénint (EDF)

9. Data/model reduction
9
• SVD = data reduction through variable separation
– Extension to higher dimension variable separation see Chinesta (afternoon)
• Ritz analysis : build reduced dynamic models
– Reduced model = differential/analytic equation for qR(𝑡)/qR(𝑠)
– States qR allow restitution
– Assumptions on loading : band limited 𝑢 𝑡
restricted loads in space 𝑏𝑖 𝑥
F x,t = bi x u t
NA
i=1
– Learning = full FEM static & modes (McNeal, Craig-Bampton, …)
{q}N=
qR
Nx NR
T
𝑀𝑠2 + 𝐶𝑠 + 𝐾 𝑞(𝑠) = 𝑏 𝑢(𝑠)
𝑇 𝑇 𝑍(𝑠) 𝑇 𝑁𝑅×𝑁𝑅 𝑞 𝑅(𝑠) = 𝑇 𝑇 𝑏 𝑢(𝑠)

10. Validity of reduced system models
Test & FEM system models assume
• Input restrictions
– Frequency band (modes)
– Localization (residual terms)
• System
– Time invariant
– Linear
Implemented in all major FEM & Modal Testing software
10
In Out
Environment/Design point
System
qR
Nx NR
T
System=IO relation
System=modal series
Challenge :
account for environment/design change

11. Sample design changes
• Material changes (visco damping)
• Junctions (contact)
• Component/system
Mesh/geometry
11

12. 12
Ingredient 2 : parametric matrices
•Viscoelastic damping
𝐾𝑣 = 𝐾 𝐸(𝜔, 𝑇)
•Rotation induced stiffening
𝐾 𝐺 = 𝐾 Ω
•Contact stiffness evolution with
operating pressure
𝐾 𝑁 = 𝐾 p(x, 𝐹𝐺𝑙𝑜𝑏𝑎𝑙)
Reduction basis T can be fixed
for range of parameters
Speedup : 10-1e5

13. 13
• Multi-model
• Other + residue iteration
• Example : strong coupling
With 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] Rd
k=K-1 R(q(Tk))
Orthog [Tk Rd
k]

14. 1th vertical mode: Main frame and
bow moving in phase
2
Co-simulation
SDToos/OSCAR
MSC/Motion
(VSD 2014)
Ingredient 3 : domain decomposition
• 1D  models coupled by few in/out :
hydraulic circuits, shaft torsion
• 3D FEM : classical uses
– Component Mode Synthesis/ Craig-Bampton
– Multibody with flexible superelements
• For each component base assumptions
remain
– LTI, few band-limited I/O
Two challenges
• Performance problems for large interfaces
• Component/system relation
14
Concept
Requirements &
architecture
Component design
System
operation
AVL Hydsim

15. Basic component coupling
Start : disjoint component models
Coupling relation between disjoint states
• Continuity 𝑞𝐼1 − 𝑞𝐼2 = 0
• Energy
+

16. 16
Coupling + reduction
Classical CMS
• Reduced independently
• All interface motion (or interface modes)
• Assembly by continuity
Difficulties
• Mesh incompatibility
• Large interfaces
• Strong coupling (reduction requires knowledge of coupling)
Physical interface coupling
• Assembly by computation of interface energy
(example Arlequin)
Difficulties
• Use better bases than independent reduction

17. 17
Squeal example : trace of system modes
CMS with trace of system modes
• No reduction of DOFs internal to contact area
• Reduction : trace of full brake modes on
reduced area & dependent DOFs (no need for
static response at interface)
Reduced model with exact system modes
Very sparse matrix for faster for time
integration

18. Component mode tuning method
• Reduced model is sparse
• Component mode amplitudes are DOFs
• Reduced model has exact nominal modes
(interest 1980 : large linear solution, 2015 : enhanced
coupling)
• Change component mode frequency  change the diagonal
terms of Kel
Disc
OuterPad
Inner Pad
Anchor
Caliper
Piston
Knuckle
Hub
wj
21
[M] [Kel] [KintS] [KintU]

19. CMT & design studies
• One reduced model /
multiple designs
Examples
• impact of modulus change
• damping real system or component mode
19
Component redesign
Sensitivity
energy analysis
Nom
.
+10
%
+20
%
-
20%

20. 20
Conclusion
Reduced 3D models combine
• Variable separation
• Solve using generalized/reduced DOFs
–u(t) just assumed band-limited
–Restitution is possible
• Parametric matrices
• Domain decomposition
– Craig-Bampton is very costly
– Generalized coordinates can make sense
Challenges
• Engineering time to manage experiments
• Control data volume (>1e3 of NL runs)
• Control accuracy : develop software / train
engineers
In
Out
Environment
Design point
System
qR
Nx NR
T
www.sdtools.com/publications
Ritz