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[×ÊÔ´] Mechanics_of_Moving_Materials.pdf

1 Introduction ........................................ 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Modelling of Systems with Travelling Continuum. . . . . . . . . . . 3
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Travelling Strings, Beams, Panels, Membranes and Plates....... 9
2.1 Out-of-Plane Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.1 Travelling Strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.1.2 Travelling Panels . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.3 Travelling Membranes and Plates . . . . . . . . . . . . . . . . . 13
2.2 In-Plane Tensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3 Stability Analysis ..................................... 23
3.1 Historical View of Stability Investigations . . . . . . . . . . . . . . . . 23
3.2 Linear Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.3 Dynamic Analysis of Moving Membranes and Plates. . . . . . . . . 30
3.3.1 Dynamic Stability of Membranes . . . . . . . . . . . . . . . . . 31
3.3.2 Dynamic Analysis of Small Transverse Vibrations
and Elastic Stability of Isotropic Plates . . . . . . . . . . . . . 33
3.4 Divergence Instability of Isotropic Plates . . . . . . . . . . . . . . . . . 36
3.4.1 Eigenvalue Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.4.2 Analytical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.4.3 Properties of Analytical Solution . . . . . . . . . . . . . . . . . 40
3.5 Divergence Instability of Orthotropic Plates . . . . . . . . . . . . . . . 45
3.5.1 Eigenvalue Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.5.2 Non-Negativeness of Eigenvalues . . . . . . . . . . . . . . . . . 48
3.5.3 Analytical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.5.4 Properties of Analytical Solution . . . . . . . . . . . . . . . . . 53
3.5.5 Analysis of Solution Properties . . . . . . . . . . . . . . . . . . . 58
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4 Non-Homogeneous Tension Profile ........................ 69
4.1 Dynamic Analysis of Axially Moving Plates. . . . . . . . . . . . . . . 69
4.2 Transverse Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.3 Solution of Eigenvalue Problem . . . . . . . . . . . . . . . . . . . . . . . 76
4.3.1 Transformation to Ordinary Differential Equation . . . . . . 76
4.3.2 Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5 Travelling Panels Made of Viscoelastic Material .............. 89
5.1 Modelling of Moving Viscoelastic Panels. . . . . . . . . . . . . . . . . 89
5.2 Numerical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.3 Numerical Illustrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6 Travelling Panels Interacting with External Flow ............. 113
6.1 Flow Problem of Surrounding Air . . . . . . . . . . . . . . . . . . . . . . 113
6.1.1 Aerodynamic Reaction. . . . . . . . . . . . . . . . . . . . . . . . . 115
6.1.2 Slip Boundary Condition . . . . . . . . . . . . . . . . . . . . . . . 117
6.1.3 Fluid Velocity Potential . . . . . . . . . . . . . . . . . . . . . . . . 121
6.1.4 Complex Analysis Approach . . . . . . . . . . . . . . . . . . . . 124
6.1.5 Properties of Aerodynamic Kernel . . . . . . . . . . . . . . . . 129
6.1.6 Added-Mass Approximation . . . . . . . . . . . . . . . . . . . . . 134
6.1.7 Notes and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 138
6.2 Behaviour of Travelling Panels Submerged in Ideal Fluid . . . . . 139
6.2.1 Numerical Considerations . . . . . . . . . . . . . . . . . . . . . . 144
6.2.2 Static Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . 149
6.2.3 Dynamical Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . 164
6.2.4 Eigenfrequency Analysis . . . . . . . . . . . . . . . . . . . . . . . 174
6.2.5 Flutter Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
7 Fracture and Fatigue of Travelling Plates ................... 197
7.1 Travelling Plates Under Fracture and Instability Constraints . . . . 197
7.1.1 Safe Range of Velocities for the Case
of Homogeneous Tension. . . . . . . . . . . . . . . . . . . . . . . 197
7.1.2 Travelling Plates Under Non-Homogeneous Tension . . . . 201
7.2 Cyclic Tension and Constraints on Longevity and Instability . . . 203
7.2.1 Fatigue Crack Growth and Critical Conditions . . . . . . . . 203
7.2.2 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 206
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
8 Some Optimization Problems ............................ 213
8.1 Optimization of Moving Plates Subjected to Instability
and Fracture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
8.1.1 Optimization Criterion and Constraints . . . . . . . . . . . . . 214
8.1.2 Finding Optimal Solution. . . . . . . . . . . . . . . . . . . . . . . 219
8.1.3 Dependence of Optimal Solution
on Problem Parameters . . . . . . . . . . . . . . . . . . . . . . . . 221
8.2 Pareto Optimal Solutions for Good Runnability. . . . . . . . . . . . . 224
8.2.1 Multicriteria Optimization . . . . . . . . . . . . . . . . . . . . . . 225
8.2.2 Maximizing Critical Velocity and Safety Criterion . . . . . 228
8.2.3 Maximizing Critical Velocity and Process
Effectiveness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
8.2.4 Maximizing Safety and Process Effectiveness. . . . . . . . . 231
8.2.5 Some Illustrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
8.3 Optimization with Uncertainties . . . . . . . . . . . . . . . . . . . . . . . 233
8.3.1 Uncertainty in Initial Crack Length . . . . . . . . . . . . . . . . 237
8.3.2 Uncertainty in Tension . . . . . . . . . . . . . . . . . . . . . . . . 240
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
Index ................................................ 249Mechanics_of_Moving_Materials.pdf
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