A new approach is presented for modelling multi-body systems, which constitutes a substantial enhancement of the Rigid Finite Element method. The new approach is based on homogeneous transformations and joint coordinates, and it yields the advantage that equations of motion are automatically generated for systems consisting of alternate rigid and flexible links. Apart from its simple physical interpretation and easy computer implementation, the method is also valuable for educational purposes since it impressively illustrates the impact of mechanical features on the mathematical model. This novel modelling approach is then applied to systems such as offshore-cranes and telescopic rapiers.
Author(s): Edmund Wittbrodt, Iwona Adamiec-Wójcik, Stanislaw Wojciech
Series: Foundations of Engineering Mechanics
Edition: 1
Publisher: Springer
Year: 2006
Language: English
Pages: 231
Contents......Page 4
1 Introduction......Page 6
2.1 Transformation of Coordinates and Homogenous Transformations......Page 10
2.2 Velocity and Acceleration of a Rigid Body......Page 20
2.3 Description of Geometry of Rigid Links......Page 22
2.4 Kinetic Energy and Lagrange Operators......Page 24
2.5 Potential Energy of Gravity Forces......Page 30
2.6 Generalised Forces and Equations of Motion......Page 31
2.7 Generalisation of the Procedure......Page 32
3 The Rigid Finite Element Method......Page 40
3.1 Division of the Flexible Link into Rigid Finite Elements and Spring–Damping Elements......Page 42
3.2 Kinetic Energy of the Flexible Link......Page 50
3.3 Energy of Deformation and Dissipation of Energy of Link p......Page 58
3.4 Synthesis of Equations......Page 63
3.5 Linear Model......Page 66
3.6 Parameters of Rigid Finite Elements and Spring–Damping Elements......Page 74
4.1 Generalised Coordinates and Transformation Matrices......Page 88
4.2 Kinetic Energy of the Flexible Link and Its Derivatives......Page 91
4.3 Potential Energy of the Flexible Link......Page 95
4.4 Synthesis of Equations......Page 97
4.5 Linear Model......Page 99
5.1 Equations of the Free Vibrations of a Beam......Page 108
5.1.1 Classical Rigid Finite Element Method Nonlinear Model......Page 109
5.1.2 Classical Rigid Finite Element Method Linear Model......Page 116
5.1.3 Modified Rigid Finite Element Method Nonlinear Model......Page 118
5.1.4 Modified Rigid Finite Element Method Linear Model......Page 121
5.2 Integrating the Equations of Motion......Page 124
5.2.2 Euler and Runge–Kutta Methods......Page 128
5.2.3 Step-Size Control......Page 130
5.2.4 Stiff Systems of Differential Equations......Page 136
5.3 Numerical Effectiveness of Models and Methods of Integrating the Equations of Motion......Page 140
6.1 Vibrations of Whippy Beams......Page 146
6.1.1 Frequencies of Free Vibrations for a Uniform Beam......Page 148
6.1.2 Linear and Non-linear Vibrations of a Viscoelastic Beam......Page 150
6.1.3 Kane’s Manipulator......Page 158
6.1.4 Analysis of Large Deflections......Page 168
6.2 Experimental Verification of the Method......Page 171
6.2.1 Large Amplitude Vibrations of a Fixed Whippy Beam......Page 172
6.2.2 Sandia Manipulator......Page 178
7.1.1 Discretisation of Flexible Links and the Equations of Motion......Page 188
7.1.2 Numerical Calculations......Page 193
7.2 Telescopic Rapier......Page 197
7.2.2 Numerical Calculations......Page 202
7.3 A-Frame......Page 203
7.3.1 Classical Rigid Finite Element Method Linear Model......Page 210
7.3.2 Modified Rigid Finite Element Method Linear Model......Page 216
7.3.3 Description of Programmes and Results of Calculations......Page 217
7.4 Further Applications......Page 221
References......Page 224
Index......Page 228
