Dynamics of Flexible Multibody Systems Rigid Finite Element Method:Dynamics of Flexible Multibody Systems: Rigid Finite Element Method (Foundations of Engineering Mechanics)
by Edmund Wittbrodt, Iwona Adamiec-Wójcik, Stanislaw Wojciech,
Publisher : Springer Number Of Pages: 225 Publication Date: 2006-06-02 Sales Rank: 3945302 ISBN / ASIN: 3540323511 EAN: 9783540323518 Binding: Hardcover Manufacturer: Springer Studio: Springer
Book Description:
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.
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Homogenous Transformations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1 Transformation of Coordinates and Homogenous
Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Velocity and Acceleration of a Rigid Body . . . . . . . . . . . . . . . . . . 15
2.3 Description of Geometry of Rigid Links . . . . . . . . . . . . . . . . . . . . 17
2.4 Kinetic Energy and Lagrange Operators . . . . . . . . . . . . . . . . . . . . 19
2.5 Potential Energy of Gravity Forces . . . . . . . . . . . . . . . . . . . . . . . . 25
2.6 Generalised Forces and Equations of Motion . . . . . . . . . . . . . . . . 26
2.7 Generalisation of the Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3 The Rigid Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.1 Division of the Flexible Link into Rigid Finite Elements and
Spring–Damping Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2 Kinetic Energy of the Flexible Link . . . . . . . . . . . . . . . . . . . . . . . . 45
3.3 Energy of Deformation and Dissipation of Energy of Link p . . . 53
3.4 Synthesis of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.5 L inear Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.6 Parameters of Rigid Finite Elements and
Spring–Damping Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4 Modification of the Rigid Finite Element Method. . . . . . . . . . 83
4.1 Generalised Coordinates and Transformation Matrices . . . . . . . 83
4.2 Kinetic Energy of the Flexible Link and Its Derivatives . . . . . . . 86
4.3 Potential Energy of the Flexible Link . . . . . . . . . . . . . . . . . . . . . . 90
4.4 Synthesis of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.5 L inear Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5 Calculations for a Cantilever Beam and Methods of
Integrating the Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . 103
5.1 Equations of the Free Vibrations of a Beam . . . . . . . . . . . . . . . . . 103
5.1.1 Classical Rigid Finite Element Method
Nonlinear Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
VI Contents
5.1.2 Classical Rigid Finite Element Method
L inear Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.1.3 Modified Rigid Finite Element Method
Nonlinear Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.1.4 Modified Rigid Finite Element Method
L inear Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.2 Integrating the Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . 119
5.2.1 Newmark Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.2.2 Euler and Runge–Kutta Methods . . . . . . . . . . . . . . . . . . . . 123
5.2.3 Step-Size Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
5.2.4 Stiff Systems of Differential Equations . . . . . . . . . . . . . . . 131
5.3 Numerical Effectiveness of Models and Methods of Integrating
the Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
6 Verification of the Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
6.1 Vibrations of Whippy Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
6.1.1 Frequencies of Free Vibrations for a Uniform Beam . . . . 143
6.1.2 Linear and Non-linear Vibrations
of a Viscoelastic Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
6.1.3 Kane’s Manipulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
6.1.4 Analysis of Large Deflections . . . . . . . . . . . . . . . . . . . . . . . 163
6.2 Experimental Verification of the Method . . . . . . . . . . . . . . . . . . . 166
6.2.1 Large Amplitude Vibrations of a Fixed
Whippy Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
6.2.2 Sandia Manipulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
7.1 Offshore Crane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
7.1.1 Discretisation of Flexible Links
and the Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . 183
7.1.2 Numerical Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
7.2 Telescopic Rapier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
7.2.1 Discretisation of the Internal Rapier
and the Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . 194
7.2.2 Numerical Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
7.3 A-Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
7.3.1 Classical Rigid Finite Element Method
L inear Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
7.3.2 Modified Rigid Finite Element Method
L inear Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
7.3.3 Description of Programmes and Results
of Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
7.4 Further Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
Dynamics of Flexible Multibody Systems Rigid Finite Element Method
ص
[ 本帖最后由 drjiachen 于 2008-12-28 10:29 编辑 ]
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