Performance Enhancement of Automotive Silencer Using Finite Element Analysis
Presentation
1. Alessandro Mango Furnari
Model Order Reduction: Comparison of Techniques
and Application to a Slewing Flexible Beam
1
Academic Year 2014/2015
DEPARTMENT OF INDUSTRIAL ENGINEERING
Master’s Thesis in Mechanical Engineering for Design and Production
Supervisors: Candidate:
Prof. Ing. Stanislao Patalano Alessandro Mango Furnari
Prof. Ing. Régis Plateaux
Prof. Ing. Jean-Luc Dion
2. 2
OUTLINE
Alessandro Mango Furnari
Model Order Reduction (MOR)
MOR
Structural
Dynamics
Systems & Control
Numerical
Mathematics
Detailed Presentation
ClassificationQualitative Comparison
Slewing
Flexible
Beam
Reduction
Techniques
Quantitative Comparison
Study
Modeling
3. 3
1. INTRODUCTION
Alessandro Mango Furnari
What is Model Order Reduction about ?
Continuous Model:
Partial Differential
Equations
Reduced-Order
Model:
m Ordinary Differential
Equations
(𝑚 ≪ 𝑛)
Discrete Model:
n Ordinary Differential
Equations
Physical
System
Discretization
MOR
Simulation
Control
Stanford Bunny [Harvard University, Microsoft Research]High-Order
Dynamical Systems
4. 4
2. MOR IN DIFFERENT FIELDS
Alessandro Mango Furnari
Differences:
• First-Order Form versus Second-Order Form
• Input-Output Behaviour versus Global Behaviour
• Interpretation of Reduction Space
• Frequency Domain versus Time Domain Based Reduction
• Automatic versus User-Dependent Model Reduction
Model Order Reduction in Different Fields
Structural Dynamics
(Modal Truncation)
Systems & Control
(Balanced Truncation)
Numerical Mathematics
(Krylov Subspace Projection)
Model Order Reduction
6. 6Alessandro Mango Furnari
First-Order Form vs Second-Order Form
2. MOR IN DIFFERENT FIELDS
First-OrderForm
Full-Order Model Reduced-Order Model
Coordinate
Transformation
Second-OrderForm
Full-Order
Model
Reduced-Order
Model
Coordinate
Transformation
7. 7Alessandro Mango Furnari
2. MOR IN DIFFERENT FIELDS
Legend
Comparison aimed to a motivated choice
Summary Table
Desired Properties Retained in the Reduced Model
8. 8
3. MODELING OF A SLEWING FLEXIBLE BEAM
Alessandro Mango Furnari
The robot weight to payload ratio has decreased
from 15 to 5 during the last 20 years.
This development leads to the presence of
vibration modes inside the control bandwidth
and enhances the need for accurate models [1].
Slewing Flexible Beams: Industrial and Research Stakes
Industrial Robot (IRB4600 from ABB)
Experimental Setup of a
Single-Link Flexible Manipulator
Helicopter Rotor Blades
[1] S. Moberg et al. “Modeling and Parameter Estimation of Robot Manipulators using Extended Flexible Joint Models”. In:
Journal of Dynamic Systems, Measurement, and Control 136.3 (2013).
9. 9Alessandro Mango Furnari
Experimental Setup
3. MODELING OF A SLEWING FLEXIBLE BEAM
• DC Electric Motor
• Speed Reducer
• Strain Gauge
• Dynamo Tachometer
• Voltage Amplifier
• dSPACE Controller Board
System Components
Slewing
Flexible
Beam
Continuous
Model
Discretization
Comparison
Modeling
FE Model
Reduced
FE Model
Experimental
Validation
Full
Model
MOR
* Reduction of Rotational
DOF by means of Guyan
Condensation Method
MOR*
10. 10Alessandro Mango Furnari
Time-Domain Validation: Strain Response to an
Angular Velocity Step Input
3. MODELING OF A SLEWING FLEXIBLE BEAM
zoom
Coordinate
Transformation
Discretization:
Assumed Modes Method
Hypothesis:
• Low Angular Velocity
• High Value of 𝐼
Continuous Model of the Slewing Flexible Beam
Analytical Mechanics
11. 11Alessandro Mango Furnari
Comparison of three Beam FE Models
Correlated Coefficient For Modal Vector
Percent Error of natural frequency
3. MODELING OF A SLEWING FLEXIBLE BEAM
Consistent-Mass Model with 2 DOF for each node
Consistent-Mass Model with 1 transl. DOF for each node
Lumped-Mass Model with 1 transl. DOF for each node
•
•
•
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4. FREQUENCY-DOMAIN BASED COMPARISON
Alessandro Mango Furnari
Static Condensation
masters: 1 through 20 (S. C.)
masters: 15, 35, 50 (S. C.)
Observations:
• A 3-DOF model can be much
more accurate than a 20-DOF
model if the masters are appro-
priately chosen;
• This means that the accuracy of
the reduced model highly de-
pends on the analyst's choice of
masters.
Reduction Procedure:
• Full Model: FE Model with 50 translational DOF
• Frequency Domain Based Comparison
13. 13Alessandro Mango Furnari
Fixed Interface Method vs Static Condensation
4. FREQUENCY-DOMAIN BASED COMPARISON
Observations:
• Fixed Interface Method may be
considered as an improvement of
Static Condensation Method.
• The valid frequency range of Fixed
Interface Method is wider than that
of Static Condensation Method.
Transformation Matrix:
masters: 50 (S. C.)
masters: 50
modes: 1, 2
(F. I. M.)
