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An active vibration isolation system comprising of a simply supported beam and a rigid mass mounted on an active isolator is analyzed using Finite Element Analysis. The cost function which is minimized is the vibrational power transmitted from the vibrating mass into the beam. The analysis shows that moments can generate negative power transmission values along a translational axis. It is shown that a control strategy which minimizes the power transmission along a translational axis and neglects the power transmission due to moments can produce higher beam vibration levels than without control.
It is shown by example that the minimization of squared acceleration or squared force in the vertical direction at the base of the isolator, performs nearly as well as the minimization of total power transmission (along translational and rotational axes).
It is shown that the cost function of translational power transmission along the vertical axis can have negative values, when rotational moments are present. In these situations, the cost function of squared power transmission along the vertical axis will have a locus of filter weights where the squared power transmission is zero along the vertical axis. The optimum set of filter weights corresponding to the minimization of squared acceleration or squared force along a vertical axis, is a point which lies on this locus. It is shown that a point exists on this locus, where the control effort is also minimized. At this point, the control effort is less than that required when the squared acceleration or squared force along the vertical axis is minimized.
However, at the point where squared power transmission along the vertical axis and the control effort is minimized, the total power transmission is not necessarily minimized and generally not as small as achieved by just minimizing squared force or squared acceleration in the vertical Z direction at the base of the isolator.
Two adaptive control algorithms are suggested for finding the optimum filter weights which minimize the squared power transmission and the control effort. The first algorithm uses Newton's method to minimize the control effort by moving the filter weights along a constant power level on the error surface without causing an increase in the residual error. A second method is suggested which alternates the filter weight updates between a partial leaky filtered-x LMS algorithm and the standard filtered-x LMS algorithm. This results in a zigzag path of the filter weights and a slightly greater residual error than the first algorithm.
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