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Review Article

The Motion Constants of Linear Autonomous Dynamical Systems

[+] Author and Article Information
Andrew J. Sinclair

Associate Professor
Aerospace Engineering Department,
Auburn University,
Auburn, AL 36849
e-mail: sinclair@auburn.edu

John E. Hurtado

Associate Professor
Aerospace Engineering Department,
Texas A&M University,
College Station, TX 77843
e-mail: jehurtado@tamu.edu

Manuscript received January 23, 2013; final manuscript received June 4, 2013; published online August 29, 2013. Assoc. Editor: Harry Dankowicz.

Appl. Mech. Rev 65(4), 040803 (Aug 29, 2013) (9 pages) Paper No: AMR-13-1007; doi: 10.1115/1.4024769 History: Received January 23, 2013; Revised June 04, 2013

The time-independent integrals, here referred to as motion constants, for general nth-order linear autonomous systems are developed. Although it is commonly believed that this topic has been fully addressed, close inspection of the literature reveals that a comprehensive development is missing. This paper provides a complete tutorial treatment of the calculation of these motion constants. The process involves a state transformation to a canonical form of uncoupled real subsystems. Following this, motion constants that are internal to each subsystem are found, after which motion constants that connect the subsystems to each other are computed. Complete sets of n − 1 real single-valued motion constants can be formed for all linear autonomous systems with a single exception. The exception is systems composed of undamped oscillators whose frequency ratio is irrational. Such systems are known to exhibit ergodic behavior and lack a number of analytic motion constants.

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