The Dynamics of Systems of Interacting Rigid Bodies

This chapter discusses virtual work, returning to the Newtonian framework to derive the central Lagrange equation, using d’Alembert’s principle. It starts off with a discussion of generalised force, applied force and constraint force. Holonomic constraints and non-holonomic constraint equations are then investigated. The corresponding principles of Gauss (Gauss’s least constraint) and Jourdain are also documented and compared to d’Alembert’s approach before being generalised into the Mangeron–Deleanu principle. Kane’s equations are derived from Jourdain’s principle. The chapter closes with a detailed covering of the Gibbs–Appell equations as the most general equations in classical mechanics. Their reduction to Hamilton’s principle is examined and they are used to derive the Euler equations for rigid bodies. The chapter also discusses Hertz’s least curvature, the Gibbs function and Euler equations.

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Introductory Rotational Dynamics

10.1093/oso/9780198822370.003.0003 ◽

2018 ◽

Author(s):

Peter Mann

Keyword(s):

Angular Momentum ◽

Angular Displacement ◽

Rigid Bodies ◽

Rotational Dynamics ◽

Circular Motion ◽

Uniform Rotation ◽

Chemical Systems ◽

Inertial Frames ◽

Rotating Frames ◽

Special Case

This chapter discusses the importance of circular motion and rotations, whose applications to chemical systems are plentiful. Circular motion is the book’s first example of a special case of motion using the laws developed in previous chapters. The chapter begins with the basic definitions of circular motion; as uniform rotation around a principle axis is much easier to consider, it is the focus of this chapter and is used to develop some key ideas. The chapter discusses angular displacement, angular velocity, angular momentum, torque, rigid bodies, orbital and spin momenta, inertia tensors and non-inertial frames and explores fictitious forces as well as transformations in rotating frames.

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Modelling of speleothems failure in the Hotton cave (Belgium). Is the failure earthquake induced?

Netherlands Journal of Geosciences – Geologie en Mijnbouw ◽

10.1017/s001677460002391x ◽

2001 ◽

Vol 80 (3-4) ◽

pp. 315-321 ◽

Cited By ~ 19

Author(s):

J.F. Cadorin ◽

D. Jongmans ◽

A. Plumier ◽

T. Camelbeeck ◽

S. Delaby ◽

...

Keyword(s):

Natural Frequencies ◽

Quantitative Information ◽

Rigid Bodies ◽

Tensile Failure ◽

Future Research ◽

Geometrical Parameters ◽

Laboratory Measurements ◽

Ground Acceleration ◽

Strong Ground

AbstractTo provide quantitative information on the ground acceleration necessary to break speleothems, laboratory measurements on samples of stalagmite have been performed to study their failure in bending. Due to their high natural frequencies, speleothems can be considered as rigid bodies to seismic strong ground motion. Using this simple hypothesis and the determined mechanical properties (a minimum value of 0.4 MPa for the tensile failure stress has been considered), modelling indicates that horizontal acceleration ranging from 0.3 m/s2 to 100 m/s2 (0.03 to 10g) are necessary to break 35 broken speleothems of the Hotton cave for which the geometrical parameters have been determined. Thus, at the present time, a strong discrepancy exists between the peak accelerations observed during earthquakes and most of the calculated values necessary to break speleothems. One of the future research efforts will be to understand the reasons of the defined behaviour. It appears fundamental to perform measurements on in situ speleothems.

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Example-based expressive animation of 2D rigid bodies

ACM Transactions on Graphics ◽

10.1145/3072959.3073611 ◽

2017 ◽

Vol 36 (4) ◽

pp. 1-10 ◽

Cited By ~ 2

Author(s):

Marek Dvorožňák ◽

Pierre Bénard ◽

Pascal Barla ◽

Oliver Wang ◽

Daniel Sýkora

Keyword(s):

Rigid Bodies

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ClusterSLAM: A SLAM backend for simultaneous rigid body clustering and motion estimation

Computational Visual Media ◽

10.1007/s41095-020-0195-3 ◽

2021 ◽

Author(s):

Jiahui Huang ◽

Sheng Yang ◽

Zishuo Zhao ◽

Yu-Kun Lai ◽

Shi-Min Hu

Keyword(s):

Motion Estimation ◽

Rigid Body ◽

Iterative Scheme ◽

Dynamic Environments ◽

Rigid Bodies ◽

Factor Graph ◽

Agglomerative Clustering ◽

Multiple Objects ◽

Graph Optimization ◽

Dynamic Objects

AbstractWe present a practical backend for stereo visual SLAM which can simultaneously discover individual rigid bodies and compute their motions in dynamic environments. While recent factor graph based state optimization algorithms have shown their ability to robustly solve SLAM problems by treating dynamic objects as outliers, their dynamic motions are rarely considered. In this paper, we exploit the consensus of 3D motions for landmarks extracted from the same rigid body for clustering, and to identify static and dynamic objects in a unified manner. Specifically, our algorithm builds a noise-aware motion affinity matrix from landmarks, and uses agglomerative clustering to distinguish rigid bodies. Using decoupled factor graph optimization to revise their shapes and trajectories, we obtain an iterative scheme to update both cluster assignments and motion estimation reciprocally. Evaluations on both synthetic scenes and KITTI demonstrate the capability of our approach, and further experiments considering online efficiency also show the effectiveness of our method for simultaneously tracking ego-motion and multiple objects.

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