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		2.6.3— Example of Chaotic Control: The Motion of a Magnetoelastic Ribbon

		These new methods of the control of chaos have also been used to control mechanical systems.

		Objects can be fabricated that have a stiffness that depends on the surrounding magnetic field. A magnetoelastic ribbon looks like a piece of audio recording tape. It is quite flexible. If it is supported only on the bottom, then it bends over. However, when placed in a magnetic field, its stiffness increases, and it stands upright.

		Ditto et al. put a magnetoelastic ribbon in a magnetic field generated by an electromagnet. They varied the magnetic field in a periodic way, by varying the voltage into the electromagnet. They measured the distance Xn of the ribbon from a sensor at the same time during the n-th cycle of the magnetic field.

		They used the control of chaos methods to predict how small additional voltages should be applied to the electromagnet to control the motion of the ribbon. When there was no control, the motion of the ribbon was chaotic. That is, the distance Xn of the ribbon from the sensor changed with subsequent values of n. This appears as a smear on the plot of Xn versus n. When the control was applied, then the motion of the ribbon became periodic. That is, the distance Xn of the ribbon from the sensor had the same value at each cycle of the magnetic field. These repeated values form a horizontal line on the plot of Xn versus n. They could also change the control to double the period of the ribbon. That is, the distance Xn of the ribbon from the sensor had one value on one cycle of the magnetic field and a different value on the next cycle, These two values repeated on alternating cycles. These two values form two horizontal lines on the plot of Xn versus n.

		The control took effect very fast when it was turned on, and it was highly accurate.

 

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