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<td rowspan="2" class="MPReader_Profiles_SpringerLink_Content_PrimitiveHeadingControlCoverImage"><img alt="On the use of linear graph theory in multibody system dynamics" src="/content/102972/cover-medium.jpg" width="95" height="128" /></td><td><h2 class="MPReader_Profiles_SpringerLink_Content_PrimitiveHeadingControlName">
On the use of linear graph theory in multibody system dynamics
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<td class="labelName">Journal</td><td class="labelValue"><a href="/content/102972/?p=5fa8e5a97fd742fda943635f0096b34e&pi=0">Nonlinear Dynamics</a></td>
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<td class="labelName">Publisher</td><td class="labelValue">Springer Netherlands</td>
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<td class="labelName">ISSN</td><td class="labelValue">0924-090X (Print) 1573-269X (Online)</td>
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<td class="labelName">Issue</td><td class="labelValue"><a href="/content/l70t22k19r71/?p=5fa8e5a97fd742fda943635f0096b34e&pi=0">Volume 9, Numbers 1-2 / February, 1996</a></td>
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<td class="labelName">DOI</td><td class="labelValue">10.1007/BF01833294</td>
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<td class="labelName">Pages</td><td class="labelValue">73-90</td>
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<td class="labelName">Subject Collection</td><td class="labelValue"><a href="/engineering/">Engineering</a></td>
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<td class="labelName">SpringerLink Date</td><td class="labelValue">Thursday, July 07, 2005</td>
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<p></p><div class="Heading1"><a name="title"></a>On the use of linear graph theory in multibody system dynamics</div><p class="AuthorGroup">J. J. McPhee<sup>1</sup></p><table><tbody><tr valign="top"><td><span class="Affiliation"><a name="Aff1"></a>(1) </span></td><td><span class="Affiliation">Systems Design Engineering, University of Waterloo, N2L 3G1 Ontario, Canada</span></td></tr></tbody></table><p class="Affiliation"><strong>Received: </strong>26 August 1994 <strong>Accepted: </strong>19 September 1994 </p><div class="Abstract"><a name="Abs1"></a><span class="AbstractHeading">Abstract </span>Multibody dynamics involves the generation and solution of the equations of motion for a system of connected material bodies. The subject of this paper is the use of graph-theoretical methods to represent multibody system topologies and to formulate the desired set of motion equations; a discussion of the methods available for solving these differential-algebraic equations is beyond the scope of this work. After a brief introduction to the topic, a review of linear graphs and their associated topological arrays is presented, followed in turn by the use of these matrices in generating various graph-theoretic equations. The appearance of linear graph theory in a number of existing multibody formulations is then discussed, distinguishing between approaches that use absolute (Cartesian) coordinates and those that employ relative (joint) coordinates. These formulations are then contrasted with formal graph-theoretic approaches, in which both the kinematic and dynamic equations are automatically generated from a single linear graph representation of the system. The paper concludes with a summary of results and suggestions for further research on the graph-theoretical modelling of mechanical systems.</div><p class="Keyword"><span class="KeywordHeading">Key words </span>Multibody dynamics - linear graph theory - absolute and joint coordinates</p><hr>
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<p></p><h2><a name="Bib1"></a>References</h2>
<table>
<tbody class="Citation">
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<td>1.</td>
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<td><a name="CR5"></a>Trent, H. M., <img src="/content/p64768w654438463/xlarge8216.gif" alt="lsquo" align="BASELINE" BORDER="0">Isomorphisms between oriented linear graphs and lumped physical systems<img src="/content/p64768w654438463/xlarge8217.gif" alt="rsquo" align="BASELINE" BORDER="0">,<i>Journal of the Acoustic Society of America</i>
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</tr>
<tr valign="top">
<td>15.</td>
<td><a name="CR15"></a>Sheth, P. N. and Uicker, Jr., J. J., <img src="/content/p64768w654438463/xlarge8216.gif" alt="lsquo" align="BASELINE" BORDER="0">IMP (Integrated Mechanisms Program), A computer-aided design analysis system for mechanisms and linkage<img src="/content/p64768w654438463/xlarge8217.gif" alt="rsquo" align="BASELINE" BORDER="0">,<i>ASME Journal of Engineering for Industry</i>
<b>94</b>, 1972, 454–464.</td>
</tr>
<tr>
<td> </td>
</tr>
<tr valign="top">
<td>16.</td>
<td><a name="CR16"></a>Li, T. W., <img src="/content/p64768w654438463/xlarge8216.gif" alt="lsquo" align="BASELINE" BORDER="0">Dynamics of rigid body systems: A vector-network approach<img src="/content/p64768w654438463/xlarge8217.gif" alt="rsquo" align="BASELINE" BORDER="0">, M.A.Sc. Thesis, University of Waterloo, Canada, 1985.</td>
</tr>
<tr>
<td> </td>
</tr>
<tr valign="top">
<td>17.</td>
<td><a name="CR17"></a>Branin, Jr., F. H., <img src="/content/p64768w654438463/xlarge8216.gif" alt="lsquo" align="BASELINE" BORDER="0">The relation between Kron's method and the classical methods of network analysis<img src="/content/p64768w654438463/xlarge8217.gif" alt="rsquo" align="BASELINE" BORDER="0">,<i>Matrix and Tensor Quarterly</i>
<b>12</b>, 1962, 69–105.</td>
</tr>
<tr>
<td> </td>
</tr>
<tr valign="top">
<td>18.</td>
<td><a name="CR18"></a>Roberson, R. E. and Wittenburg, J., <img src="/content/p64768w654438463/xlarge8216.gif" alt="lsquo" align="BASELINE" BORDER="0">A dynamical formalism for an arbitrary number of interconnected rigid bodies, with reference to the problem of satellite attitude control<img src="/content/p64768w654438463/xlarge8217.gif" alt="rsquo" align="BASELINE" BORDER="0">, in<i>Proceedings of 3rd IFAC Congress</i>, Vol. 1, Book 3, Paper 46D, Butterworth, London, England, 1966.</td>
</tr>
<tr>
<td> </td>
</tr>
<tr valign="top">
<td>19.</td>
<td><a name="CR19"></a>Wittenburg, J., <img src="/content/p64768w654438463/xlarge8216.gif" alt="lsquo" align="BASELINE" BORDER="0">Graph-theoretical methods in multibody dynamics<img src="/content/p64768w654438463/xlarge8217.gif" alt="rsquo" align="BASELINE" BORDER="0">,<i>Contemporary Mathematics</i>
<b>97</b>, 1989, 459–468.</td>
</tr>
<tr>
<td> </td>
</tr>
<tr valign="top">
<td>20.</td>
<td><a name="CR20"></a>McPhee, J. J., Ishac, M., and Andrews, G. C., <img src="/content/p64768w654438463/xlarge8216.gif" alt="lsquo" align="BASELINE" BORDER="0">Wittenburg's formulation of multibody dynamics equations from a graph-theoretic perspective<img src="/content/p64768w654438463/xlarge8217.gif" alt="rsquo" align="BASELINE" BORDER="0">,<i>Mechanism and Machine Theory</i>, accepted for publication, May 1995.</td>
</tr>
<tr>
<td> </td>
</tr>
<tr valign="top">
<td>21.</td>
<td><a name="CR21"></a>Huston, R. L. and Passerello, C., <img src="/content/p64768w654438463/xlarge8216.gif" alt="lsquo" align="BASELINE" BORDER="0">On multi-rigid-body system dynamics<img src="/content/p64768w654438463/xlarge8217.gif" alt="rsquo" align="BASELINE" BORDER="0">,<i>Computers & Structures</i>
<b>10</b>, 1979, 439–446.</td>
</tr>
<tr>
<td> </td>
</tr>
<tr valign="top">
<td>22.</td>
<td><a name="CR22"></a>Amirouche, F. M. I.,<i>Computational Methods in Multibody Dynamics</i>, Prentice-Hall, New Jersey, 1992.</td>
</tr>
<tr>
<td> </td>
</tr>
<tr valign="top">
<td>23.</td>
<td><a name="CR23"></a>Nikravesh, P. E. and Gim, G., <img src="/content/p64768w654438463/xlarge8216.gif" alt="lsquo" align="BASELINE" BORDER="0">Systematic construction of the equations of motion for multibody systems containing closed kinematic loops<img src="/content/p64768w654438463/xlarge8217.gif" alt="rsquo" align="BASELINE" BORDER="0">, in<i>Proceedings of ASME Design Automation Conference</i>, Montreal, Canada, 1989, pp. 27–33.</td>
</tr>
<tr>
<td> </td>
</tr>
<tr valign="top">
<td>24.</td>
<td><a name="CR24"></a>Kim, S. S. and Vanderploeg, M. J., <img src="/content/p64768w654438463/xlarge8216.gif" alt="lsquo" align="BASELINE" BORDER="0">A general and efficient method for dynamic analysis of mechanical systems using velocity transformations<img src="/content/p64768w654438463/xlarge8217.gif" alt="rsquo" align="BASELINE" BORDER="0">,<i>ASME Journal of Mechanisms, Transmissions, and Automation in Design</i>
<b>108</b>, 1986, 176–182.</td>
</tr>
<tr>
<td> </td>
</tr>
<tr valign="top">
<td>25.</td>
<td><a name="CR25"></a>Pereira, M. S. and Proença, P. L., <img src="/content/p64768w654438463/xlarge8216.gif" alt="lsquo" align="BASELINE" BORDER="0">Dynamic analysis of spatial flexible multibody systems using joint co-ordinates<img src="/content/p64768w654438463/xlarge8217.gif" alt="rsquo" align="BASELINE" BORDER="0">,<i>International Journal for Numerical Methods in Engineering</i>
<b>32</b>, 1991, 1799–1812.</td>
</tr>
<tr>
<td> </td>
</tr>
<tr valign="top">
<td>26.</td>
<td><a name="CR26"></a>Hiller, M., Kecskemethy, A., and Woernle, C., <img src="/content/p64768w654438463/xlarge8216.gif" alt="lsquo" align="BASELINE" BORDER="0">A loop-based kinematical analysis of complex mechanisms<img src="/content/p64768w654438463/xlarge8217.gif" alt="rsquo" align="BASELINE" BORDER="0">,<i>ASME Paper 86-DET-184</i>, 1986.</td>
</tr>
<tr>
<td> </td>
</tr>
<tr valign="top">
<td>27.</td>
<td><a name="CR27"></a>Hwang, R. S. and Haug, E. J., <img src="/content/p64768w654438463/xlarge8216.gif" alt="lsquo" align="BASELINE" BORDER="0">Topological analysis of multibody systems for recursive dynamics formulations<img src="/content/p64768w654438463/xlarge8217.gif" alt="rsquo" align="BASELINE" BORDER="0">,<i>Mechanisms, Structures, and Machines</i>
<b>17</b>, 1989, 239–258.</td>
</tr>
<tr>
<td> </td>
</tr>
<tr valign="top">
<td>28.</td>
<td><a name="CR28"></a>Lai, H. J., Haug, E. J., Kim, S. S., and Bae, D. S., <img src="/content/p64768w654438463/xlarge8216.gif" alt="lsquo" align="BASELINE" BORDER="0">A decoupled flexible-relative co-ordinate recursive approach for flexible multibody dynamics<img src="/content/p64768w654438463/xlarge8217.gif" alt="rsquo" align="BASELINE" BORDER="0">,<i>International Journal for Numerical Methods in Engineering</i>
<b>32</b>, 1991, 1669–1689.</td>
</tr>
<tr>
<td> </td>
</tr>
<tr valign="top">
<td>29.</td>
<td><a name="CR29"></a>Andrews, G. C. and Kesavan, H. K., <img src="/content/p64768w654438463/xlarge8216.gif" alt="lsquo" align="BASELINE" BORDER="0">The vector-network model: A new approach to vector dynamics<img src="/content/p64768w654438463/xlarge8217.gif" alt="rsquo" align="BASELINE" BORDER="0">,<i>Mechanisms and Machine Theory</i>
<b>10</b>, 1975, 57–75.<br><a href="http://dx.doi.org.ezproxy.lib.utexas.edu/10.1016/0094-114X(75)90057-9" target="_blank"><img src="/images/crossref_link.gif" border="0" alt="CrossRef" width="65" height="20"></a></td>
</tr>
<tr>
<td> </td>
</tr>
<tr valign="top">
<td>30.</td>
<td><a name="CR30"></a>Andrews, G. C., Richard, M. J., and Anderson, R. J., <img src="/content/p64768w654438463/xlarge8216.gif" alt="lsquo" align="BASELINE" BORDER="0">A general vector-network formulation for dynamic systems with kinematic constraints<img src="/content/p64768w654438463/xlarge8217.gif" alt="rsquo" align="BASELINE" BORDER="0">,<i>Mechanism and Machine Theory</i>
<b>23</b>, 1988, 243–256.<br><a href="http://dx.doi.org.ezproxy.lib.utexas.edu/10.1016/0094-114X(88)90109-7" target="_blank"><img src="/images/crossref_link.gif" border="0" alt="CrossRef" width="65" height="20"></a></td>
</tr>
<tr>
<td> </td>
</tr>
<tr valign="top">
<td>31.</td>
<td><a name="CR31"></a>McPhee, J. J., <img src="/content/p64768w654438463/xlarge8216.gif" alt="lsquo" align="BASELINE" BORDER="0">Formulation of multibody dynamics equations in absolute or relative coordinates using the vector-network method<img src="/content/p64768w654438463/xlarge8217.gif" alt="rsquo" align="BASELINE" BORDER="0">,<i>Machine Elements and Machine Dynamics</i>, ASME DE-Vol. 71, September 1994, pp. 361–368.</td>
</tr>
<tr>
<td> </td>
</tr>
</tbody>
</table>
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