Comparison of the effectiveness of minimizing cost function parameters for active control of vibrational energy transmission in a lightly damped structure

Abstract

The success of an active control of vibration system depends upon both the cost function used and the positions of the controlling actuators. The cost function used also affects the best actuator positions since their performance is judged on the attenuation of this parameter. However, the physical success will be dependent on how well the cost function represents the actual physical vibration. Sometimes the most meaningful cost function can be calculated in a theoretical model but is difficult to measure in practice, and a compromise to a more practical one is often made. In this paper four cost functions are considered with the aim of reducing the vibration transmitted from the base to the end of a lightweight cantilever two-dimensional structure, and their performances compared with a view to evaluating the true success in using other cost function parameters in reducing the vibrational energy.
Of the four cost functions studied, two are energy-based: one representing the total vibrational energy and one using only the flexural energy level. The other two cost functions are based on velocity measurements: the sum of the squares of the translational velocity components, and one additionally using rotational velocity measurements. An initial study confirms that the total vibrational energy is the cost function which most comprehensively represents the beam vibration and is used as the reference in a comparison of the other cost functions.
Then, a ranking of the best actuator positions on the structure is determined to achieve the best reductions in each cost function. For each of these sets of actuator positions the consequential attenuation in the total vibrational energy is evaluated whilst minimizing the other cost functions. Thus, the effectiveness of these cost functions in reducing the total vibrational energy is evaluated.

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