Mechanical Engineering Series Frederick F. Ling Editor-in-Chief www.com Mechanical Engineering Series J. Angeles, Fundamentals of Robotic Mechanical Systems: Theory, Methods, and Algorithms, 2nd ed. Jestin, Boilers and Burners: Design and Theory J.
Berthelot, Composite Materials: Mechanical Behavior and Structural Analysis I. Busch-Vishniac, Electromechanical Sensors and Actuators J. Chakrabarty, Applied Plasticity K. Kim, Structural Sensitivity Analysis and Optimization 1: Linear Systems K.
Kim, Structural Sensitivity Analysis and Optimization 2: Nonlinear Systems and Applications G. Chiyssolouris, Laser Machining: Theory and Practice V. Constantinescu, Laminar Viscous Flow G. Costello, Theory of Wire Rope, 2nd Ed.
Czolczynski, Rotordynamics of Gas-Lubricated Journal Bearing Systems M. Darlow, Balancing of High-Speed Machinery J. Doyle, Nonlinear Analysis of Thin-Walled Structures: Statics, Dynamics, and Stability J. Doyle, Wave Propagation in Structures: Spectral Analysis Using Fast Discrete Fourier Transforms, 2nd ed.
Engel, Structural Analysis of Printed Circuit Board Systems AC. Fischer-Cripps, Introduction to Contact Mechanics A. Fischer-Cripps, Nanoindentations, 2nd ed. Garcia de Jalon and E.
Bayo, Kinematic and Dynamic Simulation of Multibody Systems: The Real-Time Challenge W. Gawronski, Advanced Structural Dynamics and Active Control of Structures W. Gawronski, Dynamics and Control of Structures: A Modal Approach G. Genta, Dynamics of Rotating Systems (continued after index) www.
Rowland Intermediate Dynamics: A Linear Algebraic Approach ^ Sprimger www. Howland University of Notre Dame Editor-in-Chief Frederick F. Gloyna Regents Chair Emeritus in Engineering Department of Mechanical Engineering The University of Texas at Austin Austin, TX 78712-1063, USA and Distinguished William Howard Hart Professor Emeritus Department of Mechanical Engineering, Aeronautical Engineering and Mechanics Rensselaer Polytechnic Institute Troy, NY 12180-3590, USA Intermediate Dynamics: A Linear Algebraic Approach ISBN 0-387-28059-6 e-ISBN 0-387-28316-1 Printed on acid-free paper. ISBN 978-0387-28059-2 © 2006 Springer Science+Business Media, Inc.
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Printed in the United States of America.com Dedicated to My Folks www.com Mechanical Engineering Series Frederick F. Ling Editor-in-Chief The Mechanical Engineering Series features graduate texts and research monographs to address the need for information in contemporary mechanical engineering, including areas of concentration of applied mechanics, biomechanics, computational mechanics, dynamical systems and control, energetics, mechanics of materials, processing, pro- duction systems, thermal science, and tribology. Advisory Board/Series Editors Applied Mechanics F. Leckie University of California, Santa Barbara D.
Gross Technical University of Darmstadt Biomechanics V. Mow Columbia University Computational Mechanics H. Yang University of California, Santa Barbara Dynamic Systems and Control/ D. Bryant Mechatronics University of Texas at Austin Energetics J.
Welly University of Oregon, Eugene Mechanics of Materials I. Finnic University of California, Berkeley Processing K. Wang Cornell University Production Systems G. Klutke Texas A&M University Thermal Science A.
Bergles Rensselaer Polytechnic Institute Tribology W. Winer Georgia Institute of Technology www.com Series Preface Mechanical engineering, and engineering discipline bom of the needs of the in- dustrial revolution, is once again asked to do its substantial share in the call for industrial renewal. The general call is urgent as we face profound issues of pro- ductivity and competitiveness that require engineering solutions, among others. The Mechanical Engineering Series is a series featuring graduate texts and re- search monographs intended to address the need for information in contemporary areas of mechanical engineering.
The series is conceived as a comprehensive one that covers a broad range of concentrations important to mechanical engineering graduate education and re- search. We are fortunate to have a distinguished roster of consulting editors, each an expert in one of the areas of concentration. The names of the consulting editors are listed on page vi of this volume. The areas of concentration are applied me- chanics, biomechanics, computational mechanics, dynamic systems and control, energetics, mechanics of materials, processing, thermal science, and tribology.com Preface A number of colleges and universities offer an upper-level undergraduate course usually going under the rubric of "Intermediate" or "Advanced" Dynamics— a successor to the first dynamics offering generally required of all students.
Typically common to such courses is coverage of 3-D rigid body dynamics and Lagrangian mechanics, with other topics locally discretionary. While there are a small number of texts available for such offerings, there is a notable paucity aimed at "mainstream" undergraduates, and instructors often resort to utiliz- ing sections in the first Mechanics text not covered in the introductory course, at least for the 3-D rigid body dynamics. Though closely allied to its planar counterpart, this topic is far more complex than its predecessor: in kinematics, one must account for possible change in direction of the angular velocity; and the kinetic "moment of inertia," a simple scalar in the planar formulation, must be replaced by a tensor quantity. If elementary texts' presentation of planar dynamics is adequate, their treatment of three-dimensional dynamics is rather less satisfactory: It is common to expand vector equations of motion in compo- nents—in a particular choice of axes—and consider only a few special instances of their application {e,g, fixed-axis rotation in the Euler equations of motion).
The presentation of principal coordinates is typically somewhat ad hoc, either merely stating the procedure to find such axes in general, or even more com- monly invoking the "It can be shown t h a t. Machines seem not to exist in 3-D! And equations of motion for the gyroscope are derived indepen- dently of the more general ones—a practice lending a certain air of mystery to this important topic. Such an approach can be frustrating to the student with any degree of curios- ity and is counterproductive pedagogically: the component-wise expression of vector quantities has long since disappeared from even Sophomore-level courses in Mechanics, in good part because the complexity of notation obscures the relative simplicity of the concepts involved. But the Euler equations can be expressed both succinctly and generally through the introduction of matrices.
The typical exposition of principal axes overlooks the fact that this is precisely the same device used to find the same "principal axes" in solid mechanics (ex- plicitly through a rotation); few students recognize this fact, and, unfortunately, few instructors take the opportunity to point this out and unify the concepts. And principal axes themselves are, in fact, merely an application of an even more general technique utilized in linear algebra leading to the diagonalization www.com of matrices (at least the real, symmetric ones encountered in both solid mechan- ics and dynamics). These facts alone suggest a linear algebraic approach to the subject. A knowledge of linear algebra is, however, more beneficial to the scientist and engineer than merely to be able to diagonalize matrices: Eigenvectors and eigenvalues pervade both fields; yet, while students can typically find these quantities and use them to whatever end they have been instructed in, few can answer the simple question "What is an eigenvector?" As the field of robotics becomes ever more mainstream, a facility with [3-D] rotation matrices becomes increasingly important.
Even the mundane issue of solving linear equations is often incomplete or, worse still, inaccurate: "All you need is as many equations as unknowns. If you have fewer than that, there is no solution." (The first of these statements is incomplete, the second downright wrong!) Such fallacies are likely not altogether the students' fault: few curricula allow the time to devote a full, formal course to the field, and knowledge of the material is typically gleaned piecemeal on an "as-need" basis. The result is a fractionated view with the intellectual gaps alluded to. Yet a full course may not be necessary: For the past several years, the Intermediate Dynamics course at Notre Dame has started with an only 2-3 week presentation of linear algebra, both as a prelude to the three-dimensional dynamics to follow, and for its intrinsic pedagogical merit—to organize the bits and pieces of concepts into some organic whole.
However successful the latter goal has been, the former has proven beneficial. With regard to the other topic of Lagrangian mechanics, the situation is perhaps even more critical. At a time when the analysis of large-scale systems has become increasingly important, the presentation of energy-based dynam- ical techniques has been surprisingly absent from most undergraduate texts altogether. These approaches are founded on virtual work (not typically the undergraduate's favorite topic!) and not only eliminate the need to consider the forces at interconnecting pins [assumed frictionless], but also free the designer from the relatively small number of vector coordinate systems available to de- scribe a problem: he can select a set of coordinate ideally suited to the one at hand.
With all this in mind, the following text commits to paper a course which has gradually developed at Notre Dame as its "Intermediate Dynamics" offer- ing. It starts with a relatively short, but rigorous, exposition of linear systems, culminating in the diagonalization (where possible) of matrices—the foundation of principal coordinates. There is even an [optional] section dealing with Jordan normal form, rarely presented to students at this level. In order to understand this process fully, it is necessary that the student be familiar with how the [ma- trix] representation of a linear operator (or of a vector itself) changes with a transformation of basis, as well as how the eigenvectors—in fact the new axes themselves—affect this particular choice of basis.
That, at least in the case of real, symmetric, square inertia matrices, this corresponds to a rotation of axes requires knowledge of axis rotation and the matrices which generate such rotations. This, in turn, demands an appreciation of bases themselves and.com XI particularly, the idea of linear independence (which many students feel deals exclusively with the Wronskian) and partitioned matrix multiplication. By the time this is done, little more effort is required to deal with vector spaces in general. This text in fact grew out of the need to dispatch a [perceived] responsi- bility to rigor {Le.proofs of theorems) without bogging down class presentation with such details.
Yet the overall approach to even the mathematical material of linear algebra is a "minimalist" one: rather than a large number of arcane theorems and ideas, the theoretical underpinning of the subject is provided by, and unified through, the basic theme of linear independence—the echelon form for vectors and [subsequently] matrices, and the rank of the latter. It can be argued that these are the concepts the engineer and scientist can—should— appreciate anyhow. Partitioning establishes the connection between vectors and [the rows/columns of] matrices, and rank provides the criterion for the solution of linear systems (which, in turn, fold back onto eigenvectors). In order to avoid the student's becoming fixated too early on square matrices, this fundamental theory is developed in the context of linear transformations between spaces of arbitrary dimension.
It is only after this has been done that we specialize to square matrices, where the inverse, eigenvectors, and even properties of determi- nants follow naturally. Throughout, the distinction between vectors and tensors, and their representations—one which is generally blurred in the student's mind because it is so rarely stressed in presentation—is heavily emphasized. Theory, such as the conditions under which systems of linear equations have a solution, is actually important in application.