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In this chapter, we shall look at integration algorithms designed to deal with system descriptions containing second-order derivatives in time. Such system descriptions occur naturally in the mathematical modeling of mechanical systems, as well as in the mathematical modeling of distributed parameter systems leading to hyperbolic partial differential equations.
In this chapter, we shall concentrate on mechanical systems. The discussion of partial differential equations is postponed to the next chapter.
Whereas it is always possible to convert second derivative systems to state-space form, integration algorithms that deal with the second derivatives directly may, in some cases, offer a numerical advantage.
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5.10 References
Klaus-Jürgen Bathe. Finite Element Procedures in Engineering Analysis. Prentice-Hall, 1982.
Christopher Paul Beamis. Solution of Second Order Differential Equations Using the Godunov Integration Method. Master’s thesis, Dept. of Electrical & Computer Engineering, University of Arizona, Tucson, Ariz., 1990.
François E. Cellier. Continuous System Modeling. Springer Verlag, New York, 1991. 755p.
Leonhard Euler. De integratione aquationum differentialium per approximationem. In Opera Omnia, volume 11 of first series, pages 424–434. Institutiones Calculi Integralis, Teubner Verlag, Leipzig, Germany, 1913.
Javier Garcia de Jalón and Eduardo Bayo. Kinematic and Dynamic Simulation of Multibody Systems-The Real-Time Challenge-. Wiley, 1994.
John C. Houbolt. A Recurrence Matrix Solution for the Dynamic Response of Elastic Aircraft. Journal of Aeronautical Science, 17:540–550, 1950.
Nathan M. Newmark. A Method of Computation for Structural Dynamics. ASCE Journal of the Engineering Mechanics Division, pages 67–94, 1959.
Raymond T. Stefani, Clement J. Savant Jr., Bahram Shahian, and Gene H. Hostetter. Design of Feedback Control Systems. Saunders College Publishing, Orlando, Florida, 1994. 819p.
Edward L. Wilson. A Computer Program for the Dynamic Stress Analysis of Underground Structures. Technical Report SESM Report, 68-1, University of California, Berkeley, Division of Structural Engineering and Structural Mechanics, 1968.
5.11 Bibliography
Edda Eich-Söllner and Claus Führer. Numerical Methods in Multibody Dynamics. Teubner-Verlag, Stuttgart, Germany, 1998.
Michel Géradin and Alberto Gardona. Flexible Multibody Dynamics: A Finite Element Approach. John Wiley & Sons, Chichester, New York, 2001.
Parviz E. Nikravesh. Computer-aided Analysis of Mechanical Systems. Prentice Hall, Englewood Cliffs, New Jersey, 1988. 370p.
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(2006). Second Derivative Systems. In: Continuous System Simulation. Springer, Boston, MA. https://doi.org/10.1007/0-387-30260-3_5
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DOI: https://doi.org/10.1007/0-387-30260-3_5
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