Transformation Matrices for Finite and Small Rotations
Publication: Journal of Engineering Mechanics
Volume 124, Issue 3
Abstract
The finite displacement transformation formulae are presented for the most general case of rigid body translation and rotation, which is consistent with Chasle's theorem. Then, a conversion is made from the scaler presentation on the x, y, and z axes to the vector presentation. The results are then presented considering the following: (1) the case where the rotation axis passes through the origin and the translation part of the total displacements does not exist, which leads to the finite rotation transformation matrix; and (2) the case of small rotations, which simplifies the results.
Get full access to this article
View all available purchase options and get full access to this article.
References
1.
Altmann, S. L. (1986). Rotations, quaternions, double groups. Clarendon Press, Oxford, England.
2.
Argyris, J. H.(1982). “An excursion into large rotations.”Comput. Methods Appl. Mech. and Engrg., 32, 85–155.
3.
Argyris, J. H., and Symeonidis, S.(1981). “Nonlinear finite element analysis of elastic systems under nonconservative loading-natural formulation: Part I. quasistatic problems.”Comput. Methods Appl. Mech. and Engrg., 26, 75–123.
4.
Bottema, O., and Roth, B. (1979). Theoretical kinematics. North-Holland Publishing Co., Amsterdam, The Netherlands.
5.
Cheng, H., and Gupta, K. G.(1989). “An historical note on finite rotations.”J. Appl. Mech., 56, 139–145.
6.
Das, B. M., Kassimali, A., and Sami, S. (1994). Engineering mechanics dynamics. Irwin, Burr Ridge, Ill.
7.
Elias, Z. M. (1986). Theory and methods of structural analysis. John Wiley & Sons, Inc., New York.
8.
Goldstein, H. (1980). Classical mechanics, 2nd Ed., Addison-Wesley Publishing Co., Reading, Mass.
9.
Pi, Y. L., and Trahair, N. S.(1992). “Prebucking deflections and lateral buckling: I theory.”J. Struct. Engrg., ASCE, 118(11), 2949–2966.
10.
Pi, Y. L., Trahair, N. S., and Rajasekaran, S. (1991). “Energy equation for beam lateral buckling.”Tech. Rep. R635, School of Civ. and Mining Engrg., Univ. of Sydney, Australia.
11.
Pi, Y. L., Trahair, N. S., and Rajasekaran, S. (1992). “Energy equation for beam lateral buckling.”J. Struct. Engrg., ASCE 118(6), 1462–1479.
12.
Pietraszkiewicz, W., and Badur, J.(1983). “Finite rotations in the descriptions of continuum deformation.”Int. J. Engrg. Sci., 21(9), 1097–1115.
13.
Reissner, E. (1973). “On one-dimensional large-displacement finite-strain beam theory.”Studies Appl. Math., LII(2), 87–95.
14.
Rosenberg, R. M. (1977). Analytical dynamics of discrete systems. Plenum Publishing Corp., New York.
15.
Torkamani, M. A. M. (1996). “Transformation matrices for finite and small displacements and rotations.”Tech. Rep. ST-15, Dept. of Civ. and Environ. Engrg., Univ. of Pittsburgh, Pittsburgh, Pa.
16.
Whittaker, E. T. (1988). A treatise on the analytical dynamics of particles and rigid bodies with an introduction to the problem of three bodies, 4th Ed., Dover Publications, Inc., New York.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 124 • Issue 3 • March 1998
Pages: 359 - 362
Copyright
Copyright © 1998 American Society of Civil Engineers.
History
Published online: Mar 1, 1998
Published in print: Mar 1998
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.