Computational techniques of rotor dynamics with the finite element method

288 Pages·2012·20.1 MB·English

by  Komzsik

Mechanical, Aerospace, and Electrical Engineering KV
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Computational Techniques of Rotor Dynamics zan
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with the Finite Element Method
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For more than a century, we have had a firm grasp on rotor dynamics involving rigid 
bodies with regular shapes, such as cylinders and shafts. However, to achieve an equally  Rotor Dynamics
solid understanding of the rotational behavior of flexible bodies—especially those with 
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irregular shapes, such as propeller and turbine blades—we require more modern tools  O
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Computational Techniques of Rotor Dynamics with the Finite Element Method  U
explores the application of practical finite element method (FEM)-based computational  T

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techniques and state-of-the-art engineering software. These are used to simulate  wT Finite Element Method
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behavior of rotational structures that enable the function of various types of tI
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machinery—from generators and wind turbines to airplane engines and propellers.  thN
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The book’s first section focuses on the theoretical foundation of rotor dynamics, and  F AL
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the second concentrates on the engineering analysis of rotating structures. The authors  N T
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explain techniques used in the modeling and computation of the forces involved in the  T
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rotational phenomenon. They then demonstrate how to interpret and apply the results to  EH

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improve fidelity and performance. N
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Coverage includes: Q
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• Use of FEM to achieve the most accurate computational simulation of all E
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  gyroscopic forces occurring in rotational structures S
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• Details of highly efficient and accurate computational and numerical o
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  techniques for dynamic simulations  R
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• Interpretation of computational results, which is instrumental to developing
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  stable rotating machinery HO
• Practical application examples of rotational structures’ dynamic response OR
  to external and internal excitations D D

• An FEM case study that illustrates the computational complexities Y
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  associated with modeling and computation of forces of rotor dynamics
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• Assessment of propellers and turbines that are critical to the transportation
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  and energy industries
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Useful to practicing engineers and graduate-level students alike, this self-contained S
volume also serves as an invaluable reference for researchers and instructors in this field.
K12120
ISBN: 978-1-4398-4770-1 Arne Vollan
90000
Louis Komzsik
9 781439 847701
Computational

Techniques
of
Rotor Dynamics
with the

Finite Element Method

Computational

Techniques
of
Rotor Dynamics
with the

Finite Element Method
Arne Vollan
Louis Komzsik
Boca Raton London New York
CRC Press is an imprint of the
Taylor & Francis Group, an informa business
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Version Date: 20130913
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Contents
Preface ......................................................................................................................ix
Acknowledgments .................................................................................................xi
About the Authors ..............................................................................................xiii
Part I Theoretical Foundation of Rotor Dynamics
1 Introduction to Rotational Physics .............................................................3
1.1 Fixed Coordinate System .....................................................................3
1.2 Rotating Coordinate System ................................................................5
1.3 Forces in the Rotating System .............................................................6
1.4 Transformation between Coordinate Systems ..................................7
1.5 Kinetic Energy Due to Translational Displacement .......................10
1.6 Kinetic Energy Due to Rotational Displacement............................12
1.7 Equation of Motion in Rotating Coordinate System ......................17
1.8 Equation of Motion in the Fixed Coordinate System .....................23
2 Coupled Solution Formulations ................................................................31
2.1 Matrix Formulation of Lagrange’s Equations .................................31
2.2 Coupling Nodal Translations to the Stationary Part .....................32
2.3 Simultaneous Coupling of Translations and Rotations .................35
2.4 Full Coupling of the Stationary and Rotating Parts ......................38
2.5 Time-Dependent Terms of Equations ..............................................44
3 Finite Element Analysis of Rotating Structures ....................................49
3.1 Potential Energy of Structure ............................................................50
3.2 Dissipative Forces ................................................................................52
3.3 Nondissipative Forces ........................................................................57
3.4 Finite Element Equation Assembly ..................................................58
3.5 Coupled Equilibrium Equation Assembly ......................................59
3.6 Analysis Equilibrium Equations.......................................................62
4 Computational Solution Techniques ........................................................67
4.1 Direct Time Domain Solution of the Equilibrium Equation ........67
4.2 Direct Frequency Domain Solution ..................................................69
4.3 Direct Free Vibration Solution...........................................................70
4.4 Modal Solution Technique .................................................................73
4.5 Static Condensation ............................................................................77
4.6 Dynamic Reduction ............................................................................79
v
vi Contents
5 Numerical Solution Techniques ................................................................83
5.1 The Lanczos Method ..........................................................................83
5.2 Orthogonal Factorization ...................................................................87
5.3 The Block Lanczos Method ...............................................................88
5.4 Solution of Periodic Equations ..........................................................90
Part II Engineering Analysis of Rotating Structures
6 Resonances and Instabilities ......................................................................99
6.1 Analysis Type vs. Modeling Approach............................................99
6.2 Resonances and Instabilities ...........................................................100
6.3 Critical Speed of Rotating Mass ......................................................102
6.4 The Laval Rotor .................................................................................105
6.5 Influence of Damping .......................................................................108
6.6 Unsymmetric Effects of Bearing and Rotor ..................................110
6.7 A Rotating Tube .................................................................................113
6.8 Rotating Model with Flexible Arms ...............................................118
6.9 The Ground Resonance ....................................................................125
7 Dynamic Response Analysis ...................................................................131
7.1 Frequency Response without Rotation ..........................................131
7.2 Frequency Response with Rotation ................................................135
7.3 Transient Response without Rotation ............................................139
7.4 Transient Response with Rotation ..................................................144
8 A Finite Element Case Study ....................................................................149
8.1 Turbine Wheel with Shaft and Blades ............................................149
8.2 Engineering Analysis .......................................................................151
8.3 Computational Statistics ..................................................................153
8.4 The Journal Bearing ..........................................................................157
8.5 Active External Loads ......................................................................166
9 Analysis of Aircraft Propellers ................................................................171
9.1 A Propeller Blade ..............................................................................171
9.2 Quasi-steady Aerodynamics of Blade ............................................179
9.3 Unsteady Aerodynamics of Blade ..................................................186
9.4 Propeller with Four Blades ..............................................................193
10 Analysis of Wind Turbines .......................................................................203
10.1 An Example Wind Turbine ..............................................................203
10.2 Modeling and Analysis of Wind Turbine Blade ...........................204
Contents vii
10.3 Wind Turbine with Three Blades ....................................................226
10.4 Response Analysis of Wind Turbines ............................................243
10.5 Horizontal Axis Wind Turbines with Two Blades .......................256
Appendix ............................................................................................................267
References ...........................................................................................................271

Preface
Rotor dynamics is both a classical and a modern branch of engineering sci-
ence. The rotation of rigid bodies, mainly those with regular shapes such
as cylinders and shafts, has been well understood for more than a century.
However, the rotational behavior of flexible bodies, especially those with
irregular shapes like propeller and turbine blades, requires more modern
tools such as finite elements, hence the title and focus of the book. The book
is composed of two parts, the first focusing on the theoretical foundation
of rotor dynamics and the second on the engineering analysis of rotating
structures.
The rotational phenomenon itself is instrumental in our lives. The effects
of the rotational phenomenon range from the well-known centrifugal force,
through the Coriolis force, and to the lesser known Euler force. Chapter 1
of the book reviews these concepts in detail to establish a foundation for
the discourse. The computation of these forces is a very important aspect of
rotor dynamics. Industrial structures contain both stationary and rotating
components, and the coupling of these is another important aspect of rotor
dynamics and the topic of Chapter 2.
The computational simulation of these effects is most accurately done with
the finite element technique, which is the subject of Chapter 3. Rotor dynamic
simulations need efficient computational and accurate numerical techniques
that will also be explored in detail in Chapters 4 and 5 of the book.
In the second part of the book, Chapter 6 starts with the interpretation
of the computational results that is instrumental in establishing the stable
operational range of rotating machinery, which is discussed in detail. The
dynamic response of rotational structures to external and internal excita-
tions is explained by practical application examples in Chapter 7. The com-
putational complexity of rotor dynamics is illustrated by a finite element case
study in Chapter 8. Finally, the analysis of propellers that are important in
the transportation industry and turbines that are instrumental in the energy
industry is the specific focus of the concluding Chapters 9 and 10.
This book is a self-contained volume supporting practicing engineers in
all engineering fields encountering rotational phenomena. It is also a text for
a graduate-level audience, but undergraduates studying this area of engi-
neering will also find it useful. Researchers and members of academia teach-
ing various aspects of the topic may also find the book a useful reference.
ix