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Research Papers

Synchronization of Multiple Chaotic Gyroscopes Using the Fundamental Equation of Mechanics

[+] Author and Article Information
Firdaus E. Udwadia

Departments of Mechanical and Aerospace Engineering, Civil Engineering, Mathematics, Systems Architecture Engineering, and Information and Operations Management,  University of Southern California, 430K Olin Hall, Los Angeles, CA 90089-1453fudwadia@usc.eduDepartment of Mechanical and Aerospace Engineering,  University of Southern California, Los Angeles, CA 90089-1453fudwadia@usc.edu

Byungrin Han

Departments of Mechanical and Aerospace Engineering, Civil Engineering, Mathematics, Systems Architecture Engineering, and Information and Operations Management,  University of Southern California, 430K Olin Hall, Los Angeles, CA 90089-1453Department of Mechanical and Aerospace Engineering,  University of Southern California, Los Angeles, CA 90089-1453

We provide the Lagrangian in Appendix . This is specifically because the Lagrangian given in Ref. 2 is incorrect and, consequently, the equation of motion obtained from it is also invalid. Unfortunately, this error has found its way into the current literature dealing with this topic, as in Refs. 1-5 and Ref. 10.

J. Appl. Mech 75(2), 021011 (Feb 25, 2008) (10 pages) doi:10.1115/1.2793132 History: Received July 13, 2006; Revised July 04, 2007; Published February 25, 2008

This paper provides a simple, novel approach for synchronizing the motions of multiple “slave” nonlinear mechanical systems by actively controlling them so that they follow the motion of an independent “master” mechanical system. The multiple slave systems need not be identical to one another. The method is inspired by recent results in analytical dynamics, and it leads to the determination of the set of control forces to create such synchronization between highly nonlinear dynamical systems. No linearizations or approximations are involved, and the exact control forces needed to synchronize the nonlinear systems are obtained in closed form. The method is applied to the synchronization of multiple, yet different, chaotic gyroscopes that are required to replicate the motion of a master gyro, which may have a chaotic or a regular motion. The efficacy of the method and its simplicity in synchronizing these mechanical systems are illustrated by two numerical examples, the first dealing with a system of three different gyros, the second with five different ones.

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Copyright © 2008 by American Society of Mechanical Engineers
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Figures

Grahic Jump Location
Figure 5

Superimposed plots of (θi,θ̇i), i=1,2,3, of the three synchronized gyros for 50⩽t⩽100. The master gyro is a chaotic system and its Lyapunov exponents (17) are l1≈{0.211,−0.896,0}. Each of the gyros execute the entire motion shown in the plot.

Grahic Jump Location
Figure 12

(A) The errors hij(t) as functions of time for 50⩽t⩽100, showing that they exponentially reduce. (B) The errors hij(t) as functions of time for 150⩽t⩽200. Note the vertical scales. Errors in synchronization of the motion are less than the integration error tolerance used.

Grahic Jump Location
Figure 11

Control forces required to be applied to the four slave gyros. The solid line shows the generalized control force on the second gyro, the dashed line that on the third gyro, the dashed-dotted line that on the fourth gyro, and the dotted line that on the fifth gyro.

Grahic Jump Location
Figure 10

The upper figure shows the motion of the five uncoupled gyros over the first 20s. of response. The lower figure shows the manner in which the synchronization occurs over time, the five gyros following the motions of the master gyro, which in turn is asymptotically attracted to a stable periodic orbit, as shown in Fig. 9.

Grahic Jump Location
Figure 9

(A) (θi,θ̇i), i=1,2,3,4,5, plot for 50⩽t⩽100 of the five gyro systems superimposed on each other showing that the four slaves follow the master gyro. As is seen, the motion of the master is a complex transient motion, which has not yet reached its stable periodic orbit, which is characterized by the Lyapunov exponents l1≈{−0.180,−0.50,0}. (B) (θi,θ̇i), i=1,2,3,4,5, plot for 150⩽t⩽200 of the five gyro systems superimposed on each other showing that the four slaves follow the master gyro. The master gyro has reached a periodic orbit, and the four slaves synchronize with the master’s motion. The motion of the five gyros is shown superposed on each other.

Grahic Jump Location
Figure 8

(θi,θ̇i), i=2,3,4,5 plot for 50⩽t⩽100 of the four uncoupled slave gyro systems showing different dynamical behaviors for each gyro. The lower right figure shows the transient motions of this (i=5) dynamical system, which has not yet attained its regular periodic behavior. The other three dynamical systems (i=2,3,4) exhibit chaotic motions, as indicated by the computed Lyapunov exponents.

Grahic Jump Location
Figure 7

The solid line shows the generalized force f2syn required to be applied to second gyro (i=2) to achieve synchronization with the motion of the master gyro (i=1). The dashed line shows the generalized force f3syn required to be applied to the third gyro (i=3).

Grahic Jump Location
Figure 6

h12(t)=θ1(t)−θ2(t) (solid line), h13(t)=θ1(t)−θ3(t) (dashed line), and h23(t)=θ2(t)−θ3(t) (dashed-dotted line) for 50⩽t⩽100. Note the exponential convergence of the hij’s, as demanded by Eq. 16, and also the vertical scale, which indicates that the error in synchronization is of the order of the numerical integration error tolerance, 10−12.

Grahic Jump Location
Figure 4

(A) First 20s of the response of the uncoupled gyros with the master gyro shown with a solid line, the second gyro shown with a dashed line, and the third gyro shown with a dashed-dotted line. (B) Synchronization of the gyros showing the slave gyros following the master (solid line), as required by the constraint set 16 with δ=1 and k=2.

Grahic Jump Location
Figure 3

The differences in the responses between the three uncoupled, unsynchronized gyros shown for a duration of 60s. h12(t)=θ1(t)−θ2(t) is shown by the solid line, h13(t)=θ1(t)−θ3(t) is shown by the dashed line, and h23(t)=θ2(t)−θ3(t) is shown by the dashed-dotted line.

Grahic Jump Location
Figure 2

(θi,θ̇i) plots showing the dynamics of the three uncoupled gyros for 50⩽t⩽100. The lower right corner shows these plots superposed on one another; the first gyro is shown with a solid line, the second with a dashed line, and the third with a dashed-dotted line.

Grahic Jump Location
Figure 1

Symmetric gyroscope with vertical support excitation d(t)=d0sin(ωt)

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