Finite Element Analysis of Rotating Beams (eBook)

Prof. Ranjan Ganguli obtained his M.S. and PhD in Aerospace Engineering from the University of Maryland, College Park, in 1991 and 1994, respectively, and his B.Tech degree in Aerospace Engineering from the Indian Institute of Technology, Kharagpur, in 1989. Following his PhD, he worked at the Alfred Gessow Rotorcraft Center of the University of Maryland as Assistant Research Scientist until 1997 on projects on rotorcraft health monitoring and vibratory load validation for the Naval Surface Warfare Center and United Technology Research Center, respectively. He also worked at the GE Research Lab in Schenectady, New York, and at Pratt and Whitney, East Hartford, Connecticut, from 1997 to 2000. He joined the Aerospace Engineering department of the Indian Institute of Science, Bangalore, as Assistant Professor in July 2000. He was promoted to Associate Professor in 2005 and to Full Professor in 2009. He is currently the Satish Dhawan Chair Professor at the Indian Institute of Science, Bangalore. He has held visiting positions at TU Braunschweig, University of Ulm and Max Planck Institute of Metal Research, Stuttgart, in Germany; University Paul Sabatier and Institute of Mathematics, in Toulouse, France; Konkuk University in South Korea, the University of Michigan, Ann Arbor, in USA, and the Nanyang Technological University, Singapore.Prof. Ganguli's research interests are in helicopter aeromechanics, aeroelasticity, structural dynamics, composite and smart structures, design optimization, finite element methods and health monitoring. He has published 178 articles in refereed journals and over 100 conference papers.  He was awarded the American Society of Mechanical Engineers (ASME) best paper award in 2001, the Golden Jubilee award of the Aeronautical Society of India in 2002, the Alexander von Humboldt fellowship in 2007 and the Fulbright Senior Research fellowship in 2010. Prof. Ganguli is a Fellow of the ASME, a Fellow of the Royal Aeronautical Society, UK, a Fellow of the Indian National Academy of Engineering, a Fellow of the Aeronautical Society of India, an Associate Fellow of the American Institute of Aeronautics and Astronautics and a Senior Member of the Institute of Electrical and Electronic Engineers (IEEE). He has taught courses on flight and space mechanics, engineering optimization, helicopter dynamics, aircraft structures, structural mechanics, aeroelasticity and navigation. He has supervised the thesis of 15 PhD and 35 Master’s degree students. He has written books on “Engineering Optimization” and “Gas Turbine Diagnostics”, both published by CRC Press, New York, and books titled “Structural Damage Detection using Genetic Fuzzy Systems” and “Smart Helicopter Rotors”,  published by Springer.

Preface 6
Contents 8
About the Author 12
1 Introduction 14
1.1 Introduction 14
1.1.1 Elastic Blade 14
1.1.2 Horizontal Force Equilibrium 17
1.1.3 Boundary Conditions 18
1.1.4 Initial Conditions 19
1.1.5 Cantilever Beam Vibrations (Non-rotating) 19
1.1.6 Beam Functions 23
1.1.7 Rotating Beam Vibration 24
1.2 Galerkin Method 25
1.3 Rayleigh--Ritz Method 31
1.4 Finite Element Method 52
1.4.1 Element Properties 53
1.4.2 Energy Expressions 55
1.4.3 Assembly of Elements 58
1.4.4 Cantilever 61
2 Stiff String Basis Functions 75
2.1 Stiff String Equation 75
2.2 Stiff String Basis Functions 77
2.3 Uniform Rotating Beam 80
2.4 Tapered Rotating Beam 81
2.5 Hybrid Basis Functions 84
2.6 Finite Element 89
2.6.1 Uniform Rotating Beam 91
2.7 Tapered Rotating Beam 94
2.8 Summary 99
References 100
3 Rational Interpolation Functions 101
3.1 Governing Differential Equation 101
3.2 Hermite Shape Functions 103
3.3 New Shape Functions 105
3.4 Static Finite Element Analysis 107
3.5 Dynamic Finite Element Analysis 110
3.5.1 Uniform Beam 110
3.5.2 Tapered Rotating Beam 112
3.6 Summary 116
References 117
4 Fourier-p Superelement 118
4.1 Governing Equation of Rotating Beams 119
4.2 Shape Functions 120
4.3 Superelement Matrices 122
4.4 Numerical Results 122
4.4.1 Uniform Rotating Beam 123
4.4.2 Tapered Rotating Beam 123
4.5 Summary 129
References 129
5 Physics Based Basis Functions 131
5.1 Basis Function 132
5.2 Finite Element Analysis 139
5.3 Numerical Results 140
5.3.1 Uniform Beam 140
5.3.2 Tapered Beam 144
5.3.3 Beams with Hub Offset 147
5.4 Summary 148
References 151
6 Collocation Approach 153
6.1 Governing Differential Equation 153
6.2 Point Collocation Approach 156
6.2.1 Collocation Point at a Variable Location Within Beam Element 156
6.2.2 Collocation Point Near the Left Node of Beam Element 159
6.2.3 Collocation Point at the Midpoint of Beam Element 160
6.2.4 Collocation Point Near the Right Node of Beam Element 162
6.2.5 Two Point Collocation 163
6.2.6 Analysis of Shape Functions 168
6.3 Finite Element Formulation 170
6.4 Numerical Results 172
6.4.1 Uniform Rotating Beam 172
6.4.2 Tapered Rotating Beam 176
6.5 Summary 179
References 179
7 Rotor Blade Finite Element 180
7.1 Energy Expressions 182
7.2 Governing Differential Equations 184
7.3 Derivation of the Shape Functions 186
7.3.1 Shape Functions for Flapwise Bending 188
7.3.2 Shape Functions for Lead-Lag Bending 190
7.3.3 Shape Functions for Axial Deflection 192
7.3.4 Shape Functions for Torsion 193
7.4 Finite Element Method 195
7.5 Numerical Results 195
7.5.1 Analysis of Shape Functions 195
7.5.2 Validation Study 203
7.6 Convergence Study of New FEM Element and Polynomials 203
7.7 Summary 207
References 211
8 Spectral Finite Element Method 214
8.1 Governing Differential Equation 215
8.2 Spectral Finite Element Formulation 216
8.2.1 Interpolating Function for SFER 217
8.2.2 Interpolating Function for SFEN 217
8.2.3 Dynamic Stiffness Matrix in Frequency Domain 218
8.3 Free Vibration Results 220
8.3.1 Uniform Beam 220
8.3.2 Tapered Beam 1-Linear Mass and Cubic Flexural Stiffness Variation 223
8.3.3 Tapered Beam 2-Linear Mass and Flexural Stiffness Variation 225
8.4 Wave Propagation Study 228
8.4.1 Convergence Study 228
8.4.2 Numerical Results 230
8.5 Summary 234
References 235
9 Violin String Shape Functions 237
9.1 Timoshenko Rotating Beam and Violin String 238
9.2 Violin String Shape Functions 242
9.3 Results and Discussion 248
9.3.1 Uniform Beam 248
9.3.2 Tapered Beam 250
9.4 Summary 256
References 257
Appendix AStiffness Matrix 259
Appendix BMATLAB Code 280
Appendix CGoverning Equation for RotatingTimoshenko Beam 285