Dynamics of the Axially Moving Orthotropic Web (eBook)

Contents 6
Fundamental Notations 8
Introduction 10
1.1 Identification of Rheological Parameters of the Paper Web 11
1.1.1 Identification Method 12
1.1.2 Results of the Experimental Identification 14
1.2 Identification of the Corrugated Board as a Composite 15
1.2.1 Homogenization Method 16
1.2.2 Identification Results 18
References 19
State of Knowledge on Dynamics of Axially Moving Systems 20
2.1 String Systems 20
2.1.1 Linear Models 20
2.1.2 Nonlinear String Systems 22
2.2 Beam Systems 26
2.2.1 Linear Models 26
2.2.2 Nonlinear Beam Systems 28
2.3 Plate Systems 34
2.3.1 Numerical Investigations 34
2.3.2 Experimental Investigations of Axially Moving Plate Systems 40
2.4 Final Remarks 47
References 48
Dynamical Analysis of the Undamped Axially Moving Web System 51
3.1 One-Layered Orthotropic Web 51
3.1.1 Formulation of Nonlinear Equations of the Web Motion 51
3.1.2 Solution to the Mathematical Model 55
3.1.3 Results of Comparative Studies 62
3.1.4 Results of Dynamic Investigations of the Moving Paper Web 64
3.2 Multi-Layered Composite Web 73
3.2.1 Mathematical Model of the Axially Moving Multi-Layered Web 73
3.2.2 Solution of the Mathematical Model 78
3.2.3 Results of the Comparative Studies 81
3.2.4 Results of the Dynamic Investigations of the Axially Moving Corrugated Board Web 83
3.3 Final Remarks 90
References 91
Displacements of the Web in Equilibrium States of the Linearized System 92
4.1 Mathematical Models of the Axially Moving Orthotropic Web 92
4.2 Solution to the Mathematical Model of the Web Loaded with Constant Longitudinal Force 94
4.3 Displacements of the Web Loaded with Constant Longitudinal Force 97
4.4 Web Loaded with a Non-Uniform Longitudinal Force 101
4.5 Wrinkling of the Web Loaded with a Non-Uniformly Distributed Longitudinal Force 105
4.6 Final Remarks 108
References 109
Dynamics of the Axially Moving Viscoelastic Web 110
5.1 Two-Dimensional Rheological Model for Viscoelastic Materials 111
5.2 Mathematical Model of the Moving Viscoelastic Web 113
5.3 Solution to the Problem 116
5.4 Results of the Numerical Investigations 117
5.5 Final Remarks 121
References 122
Beam Model of the Moving Viscoelastic Web 123
6.1 Nonlinear Beam Model of the Viscoelastic Web 124
6.1.1 Kelvin-Voigt Model of Material 125
6.1.2 Poynting-Thompson Model of Material 126
6.1.3 Solution to the Problems 128
6.2 Investigations Results of the Model with the K-V Element 129
6.2.1 Linearized System 129
6.2.2 Non-Linear System 130
6.3 Investigations Results of the Model with a P-T Element 135
6.3.1 Linearized System 135
6.3.2 Nonlinear System 136
6.4 Final Remarks 141
References 142
Concluding Remarks 143
Appendix A 146
Appendix B 148
Appendix C 151
Index 154