Dynamics of Thin-Walled Aerospace Structures for Fixture Design in Multi-axis Milling

249 Pages·2009·5.88 MB·English

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Dynamics of Thin-Walled Aerospace
Structures for Fixture Design in Multi-axis
Milling
Mouhab Meshreki
Department of Mechanical Engineering
McGill University
Montreal, Quebec
February 2009
A thesis submitted to McGill University in partial fulfilment of the requirements of the
degree of Doctor of Philosophy
Copyright (cid:13)c Mouhab Meshreki 2009
All Rights Reserved
Abstract
Millingofthin-walledaerospacestructuresisacriticalprocessduetothehighflexibilityof
the workpiece. Available models for the prediction of the effect of the fixture on the dynamic
responseoftheworkpiecearecomputationallydemandingandfailtorepresentpracticalcases
for milling of thin-walled structures. Based on the analysis of typical structural components
encountered in the aerospace industry, a generalized unit-element, with the shape of an
asymmetric pocket, was identified to represent the dynamic response of these components.
Accordingly, two computationally efficient dynamic models were developed to predict the
dynamic response of typical thin-walled aerospace structures. These models were formulated
using Rayleigh’s energy and the Rayleigh-Ritz methods.
In the first model, the dynamics of multi-pocket thin-walled structures is represented
by a plate with torsional and translational springs. A methodology was proposed and
implementedforanoff-linecalibrationofthestiffnessofthespringsusingGeneticAlgorithms.
In the second model, the dynamics of a 3D pocket is represented by an equivalent 2D multi-
span plate. Through a careful examination of the milling of thin-walled structures, a new
formulation was developed to represent the continuous change of thickness of the workpiece
due to the material removal action. Two formulations, based on holonomic constraints and
springs with finite stiffness, were also developed and implemented to take into account the
effect of perfectly rigid and deformable fixture supports.
Allthedevelopedmodelsandformulationswerevalidatednumericallyandexperimentally
for different workpiece geometries and various types of loading. These models resulted in
one to two orders of magnitude reduction in computation time when compared with FE
models, with prediction errors of less than 10%. The experimental validation of the models
was performed through the machining of thin-walled components. The predictions of the
developed models were found to be in excellent agreement with the measured dynamic
responses. The developed models meet the conflicting requirements of prediction accuracy
and computational efficiency. In addition, a novel methodology was proposed and validated
to compensate for the effect of the dynamics of the force measurement system.
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R´esum´e
Le fraisage des structures a´erospatiales a` parois minces est un processus critique duˆ a` la
flexibilit´e ´elev´ee de la pi`ece. Les mod`eles disponibles pour la pr´evision de l’effet du syst`eme
de fixation sur la r´eponse dynamique de la pi`ece sont bas´es sur des m´ethodes num´eriques
tr`es lentes et n’arrivent pas `a repr´esenter les cas pratiques du fraisage des structures a` parois
minces. Bas´e sur une analyse des composants structurels typiques produits dans l’industrie
a´erospatiale, un ´el´ement g´en´eralis´e de base avec la forme d’une poche asym´etrique, a ´et´e
identifi´e pour repr´esenter la r´eponse dynamique de ces composants. En cons´equence, deux
mod`eles dynamiques efficaces ont ´et´e d´evelopp´es pour pr´evoir la r´eponse dynamique des
structures a´erospatiales types a` parois minces. Ces mod`eles ont ´et´e formul´es en utilisant les
m´ethodes de Rayleigh et Rayleigh-Ritz.
Dans le premier mod`ele, les r´eponses dynamiques des structures de poches multiples `a
parois minces sont repr´esent´ees par des plaques avec des ressorts de torsion et de translation.
Unem´ethodologiea´et´epropos´eeetmiseenapplicationpourcalibrerlarigidit´edesressortsen
utilisant les algorithmes g´en´etiques. Dans le deuxi`eme mod`ele, la r´eponse dynamique d’une
`
poche en 3D est repr´esent´ee par une plaque ´equivalente de multi-trav´ees en 2D. A travers
une ´etude approfondie du fraisage des structures a` parois minces, une nouvelle formulation
a ´et´e d´evelopp´ee pour repr´esenter le changement continu de l’´epaisseur de la pi`ece durant
l’usinage. Deux formulations, bas´ees sur des contraintes holonomes et des ressorts avec des
rigidit´es finies, ont ´et´e d´evelopp´ees et mises en application pour simuler l’effet des supports
parfaitement rigides et d´eformables.
Tous les mod`eles et les formulations d´evelopp´es ont ´et´e valid´es num´eriquement et
exp´erimentalement pour des pi`eces de g´eom´etries diff´erentes et divers types d’efforts. Ces
mod`eles ont r´eduit le temps de calcul de un `a deux ordres de grandeur, en comparaison
avec des mod`eles des ´el´ements finis. Les erreurs de pr´evision ´etaient moins de 10%. La
validation exp´erimentale des mod`eles a ´et´e effectu´ee par l’usinage des pi`eces `a parois minces.
Les pr´evisions des mod`eles d´evelopp´es sont en parfait accord avec les r´eponses dynamiques
mesur´ees. Les mod`eles d´evelopp´es r´epondent aux exigences contradictoires de l’exactitude
de pr´evision et de l’efficacit´e du calcul. En outre, une m´ethodologie originale a ´et´e propos´ee
et valid´ee pour compenser l’effet de la dynamique du syst`eme de mesure des efforts.
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Claims of Originality
1. A generalized unit-element with the form of an asymmetric pocket was extracted from
typical thin-walled aerospace structures. This unit-element allows the development
of simplified models for the analysis of the dynamic response of complex thin-walled
structures.
2. A new generalized model was developed and formulated based on an off-line calibration
ofaplatewithtorsionalandtranslationalspringsusingGeneticAlgorithms,torepresent
the dynamics of multi-pocket thin-walled structures. This simplified model allows for
the reduction of the computational time by at least one order of magnitude when
compared to FE models, with prediction errors less than 5%.
3. A model was developed and formulated for the representation of the dynamics of a 3D
pocket using a 2D semi-analytical multi-span plate model. For this model, a new set
of trial functions were developed for the approximation of the multi-span plate mode
shapes. This model allows for the prediction of the dynamic responses of various types
of thin-walled aerospace structures with at least one order of magnitude reduction in
computation time, compared to FE models, and with prediction errors less than 6%.
4. A new formulation was proposed and implemented to simulate possible cases for the
change of the thickness during milling of thin-walled pockets. This formulation was
validated through FE models and experimental machining tests. Excellent agreement
in the results was achieved with errors less than 10%.
5. Anintegratedmodelfortheanalysisoftheeffectofthefixturelayoutonthedynamicsof
thin-walledstructureswasdevelopedwhiletakingintoaccountthecontinuouschangeof
thickness of the workpiece and the effect of rigid and deformable fixture supports. This
integrated model was validated numerically through FE models and experimentally
through machining tests of thin-walled structures.
6. A novel methodology was developed for the compensation of the effect of the dynamics
of the force measurement system, as well as the dynamics of the workpiece.
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Acknowledgment
With all gratitude and respect, I would like to thank my supervisors Dr. Helmi Attia and
Dr. Jo´zsef K¨ovecses for their great support and guidance throughout this research. They
were always available for me for advice regarding technical and non-technical matters.
In addition, I would like to thank deeply Dr. Helmi Attia who always encourages me,
cares for me, and provides me with his experience and his knowledge. In addition, he gave
me a great opportunity to perform my PhD experiments at the National Research Council of
Canada at the Aerospace Manufacturing Technology Centre and to interact with an amazing
team of research officers.
A special thanks goes to all the research officers in the machining group at AMTC, NRC
for their support and cooperation, especially Dr. Nejah Tounsi, for his feedback on my PhD
work and Nick De Palma for his help in performing the experiments.
I would like to express my gratitude for the National Research Council of Canada for
allowing me to use their facilities to perform the experiments. I would like to thank FQRNT
(Le Fonds Quebecois de La Recherche sur la Nature et les Technologies) for financially
supporting my research.
I would like to tell all my family that they were a great support for me. A very sincere
and special thanks goes to my father Adel, my mother Sylvia, and my brother Hakim who
have always been there for me throughout my PhD work and my whole life. I wish that one
day I could offer them a small portion of what they always give me.
Lastbutcertainlynotleast,IwouldliketothankmywifeDianawhohasalwayssupported
me with amazing patience and great love.
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Table of Contents
1 Introduction 1
1.1 Introduction.......................................................................... 1
1.2 Motivation............................................................................ 2
1.3 Scope and Terminal Objective of the Research.................................... 3
1.4 Thesis Outline ....................................................................... 5
2 Literature Review 7
2.1 Introduction.......................................................................... 7
2.2 Fixture Design for Machining Applications........................................ 7
2.2.1 Static Analysis of Fixtures with a Rigid Workpiece ...................... 10
2.2.2 Static Analysis of Fixtures with a Compliant Workpiece................. 11
2.2.3 Dynamic Analysis of Fixtures with a Rigid Workpiece................... 14
2.2.4 Dynamic Analysis of Fixtures with a Compliant Workpiece ............. 15
2.2.5 Computer-Aided Fixture Design and Fixture Optimization.............. 17
2.3 Dynamics of the Milling of Thin-Walled Structures............................... 19
2.3.1 Techniques Used for the Milling of Thin-Walled Structures.............. 19
2.3.2 Mechanistic Force Models Including the Static Deflections of Thin-
Walled Workpieces .......................................................... 20
2.3.3 MechanisticForceModelsIncludingDynamicVibrationsofThin-Walled
Workpieces................................................................... 23
2.3.4 Models for Chatter Analysis of the Workpiece-Tool System ............. 25
2.4 Model-Order Reduction Techniques................................................ 26
2.4.1 Reduction of the Physical Coordinates of the Model ..................... 26
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2.4.2 Modal Superposition Methods.............................................. 27
2.4.3 Load-Dependent Ritz Vector Methods..................................... 29
2.4.4 Component Mode Synthesis ................................................ 29
2.5 Gap Analysis and Limitations of Existing Methods............................... 30
2.6 Detailed Objectives and Approach of the Research ............................... 31
3 Conceptual Development of Dynamic Models for Thin-Walled Aerospace
Structures 33
3.1 Introduction.......................................................................... 33
3.2 Development of a Generalized Unit-Element ...................................... 33
3.2.1 Effect of In-Plane Forces on the Transverse Vibration of Thin-Walls ... 37
3.2.2 Effect of Non-Adjacent Sides on the Response of the Load-Application
Side........................................................................... 38
3.2.3 EffectoftheAdjacentPocketsontheVibrationoftheLoad-Application
Side........................................................................... 40
3.2.4 Effect of the Fixture Layout on the Response of the Load-Application
Side........................................................................... 43
3.3 Conceptual Development for the Generalized Single-Span Plate (GSSP) Model 46
3.3.1 Description of the Proposed Concept for the GSSP Model............... 46
3.3.2 Advantages and Limitations of the GSSP Model ......................... 49
3.4 Conceptual Development for the Multi-Span Plate (MSP) Model ............... 49
3.4.1 Description of the Proposed Concept for the MSP Model................ 50
3.4.2 Advantages and Limitations of the MSP Model........................... 52
3.5 Special Considerations for the Modelling of the Change of Thickness During
Milling of Thin-Walled Pockets..................................................... 53
3.6 ReviewofMethodsandApproximationTechniquesforModellingtheDynamics
of Plates.............................................................................. 56
3.6.1 Models for Plates with Torsional and Translation Springs ............... 57
3.6.2 Models for Multi-Span (Continuous) Plates ............................... 58
3.7 Concluding Remarks ................................................................ 59
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4 Mathematical Formulations of the Proposed Dynamic Models 61
4.1 Introduction.......................................................................... 61
4.2 Mathematical Formulation of a Rectangular Plate Model........................ 61
4.2.1 Rayleigh’s Energy Method.................................................. 62
4.2.2 Rayleigh-Ritz Method....................................................... 65
4.3 Mathematical Formulation of the Generalized Single-Span Plate (GSSP) Model 66
4.3.1 Generation of the Trial Functions for the GSSP Model .................. 67
4.3.2 Dynamic Response for the GSSP Model................................... 72
4.3.3 Genetic Algorithms and Boundary Condition Calibration ............... 73
4.4 Mathematical Formulation of the Multi-Span Plate (MSP) Model .............. 76
4.4.1 Boundary Conditions for the MSP Model ................................. 76
4.4.2 Generation of the Trial Functions for the MSP Model ................... 78
4.4.3 Dynamic Response for the MSP Model.................................... 84
4.5 Change of Thickness (CoT) Formulation .......................................... 84
4.6 Formulation for the Incorporation of the Fixture Constraints.................... 90
4.6.1 Perfectly Rigid Support (PRS) Formulation............................... 90
4.6.2 Finite Stiffness Support (FSS) Formulation ............................... 93
4.7 Concluding Remarks ................................................................ 94
5 Validation of the Dynamic Models Using FEM and Impact Tests 96
5.1 Introduction.......................................................................... 96
5.2 Validation of the Generalized Single-Span Plate Model........................... 96
5.2.1 Set 1 of Tests: Validation of the GSSP Model Using a FE Model of a
Plate.......................................................................... 98
5.2.2 Set 2 of Tests: Prediction of the Response of an Asymmetric Pocket ... 99
5.2.3 Set 3 of Tests: Prediction of the Response of a Highly Flexible Square
Pocket ........................................................................ 105
5.2.4 Set 4 of Tests: Prediction of the Response of a Complex Multiple
Pocket Structure............................................................. 107
5.2.5 Set 5 of Tests: Experimental Validation Through Impact Experiments. 108
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5.3 Validation of the Multi-Span Plate Model......................................... 113
5.3.1 Set 1 of Tests: Validation of the MSP Model Against FE Models of
Multi-Span Plates ........................................................... 113
5.3.2 Set 2 of Tests: Validation of the MSP Model Against FE Models of
Asymmetric Pockets......................................................... 116
5.4 Validation of the Change of Thickness (CoT) Formulation....................... 123
5.5 Validation of the Formulations for the Effect of the Fixture Supports .......... 130
5.5.1 Validation of the Perfectly Rigid Support (PRS) Formulation........... 130
5.5.2 Validation of the Finite Stiffness Support (FSS) Formulation ........... 133
5.5.3 Validation of the PRS and the FSS Formulations while Including the
CoT Formulation ............................................................ 135
5.5.4 Use of the FSS Formulation with the MSP Model to Predict the
Response of a Pocket with Internal Rib-Walls ............................ 137
5.6 Summary of the Results............................................................. 139
6 Experimental Validation of the Dynamic Models Through the Machining
of Thin-Walled Pockets 141
6.1 Introduction.......................................................................... 141
6.2 Description of the Experiment...................................................... 141
6.2.1 Experimental Setup ......................................................... 142
6.2.2 Experimental Procedures ................................................... 146
6.2.3 Parameters Used in the Developed Dynamic Models ..................... 150
6.3 Dynamics of the Force Measurement System...................................... 151
6.3.1 Two Degree of Freedom Model............................................. 153
6.3.2 Proposed Methodology for the Identification of the Transfer Function
of the Force Measurement System.......................................... 155
6.3.3 Details of the Formulation of the Dynamics of the Force Measurement
System........................................................................ 157
6.3.4 Validation of the Proposed Methodology for the Identification of the
Transfer Function of the Force Measurement System..................... 159
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6.4 Error Analysis of the Measurement System ....................................... 161
6.5 Experimental Results................................................................ 164
6.5.1 Validation Test Results under Stable Cutting Conditions................ 164
6.5.2 Validation Test Results under Unstable Cutting Conditions............. 179
6.6 Summary of the Results............................................................. 182
7 Conclusions and Recommendations for Future Research Work 184
7.1 Conclusions........................................................................... 184
7.2 Contributions to Knowledge ........................................................ 186
7.3 Recommendations for Future Research Work...................................... 188
A Boundary Conditions and Trial Functions for a Plate with Torsional and
Translational Springs 210
A.1 Boundary Conditions for a Plate with Torsional and Translational Springs .... 210
A.2 Beam with Torsional and Translational Springs................................... 211
B Characteristic Equations for the Multi-Span Beam Models 214
B.1 Four-Span Beam..................................................................... 214
B.2 Five-Span Beam ..................................................................... 216
B.3 Three-Span Clamped-Free Beam ................................................... 218
C Formulation of the Transfer Function of the Force Measurement System 221
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