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Analytical Determination of the PZTs Distribution in Active Beam Vibration Protection Problem

100%
Brański A., Lipiński G.
Acta Physica Polonica A
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2011
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vol. 119
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issue 6A
936-941
EN
The paper concerns an active vibration protection (p-reduction) of the structure via piezoelectric transducers; p-reduction corresponds to an active vibration reduction (a-reduction). The quantity and effectiveness of the (a- or p-) reduction, among other parameters, depend on the piezoelectric transducers distribution on the structure. The best results are obtained bonding piezoelectric transducers to the structure in the sub-domains with the largest curvatures; it is so-called quasi-optimal distribution of the piezoelectric transducers. Up to now, the quasi-optimal distribution was determined based on heuristic reasons only. The aim of the paper is to confirm quasi-optimal distribution in analytical way. The beam clamped at one end, vibrating with first three modes separately, is chosen as the research object. It is assumed that the piezoelectric transducers are exactly the same. Demanding the vibration amplitude to be equal to zero (i.e. p-reduction condition), the general formula for interacting forces piezoelectric transducers-beam is derived. Next, such an appropriate distribution of piezoelectric transducers is searched analytically, that the minimal forces are achieved; it leads to the best reduction effectiveness. It turned out that the analytical method pointed out quasi-optimal distribution of the piezoelectric transducers. The validation of theoretical considerations is confirmed numerically.
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Effectiveness Analysis of the Beam Modes Active Vibration Protection with Different Number of Actuators

86%
Brański A.
Acta Physica Polonica A
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2013
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vol. 123
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issue 6
1123-1127
EN
This paper deals with an active vibration protection (p-reduction) of the beam-actuators mechanical system, hence it concerns separate modes. The paper's aim is an effectiveness analysis of the p-reduction assuming different number of actuators. It is assumed a priori that actuators are bonded to the beam in the sub-domain with the largest curvatures and they are exactly the same. The beam clamped at one end is chosen as the research object. Next, as required by the p-reduction condition, the number and distribution of actuators are changed. It turns out that the best reduction effectiveness, measured via any effectiveness coefficient, is obtained for one actuator bonded in the sub-domain with the largest curvature. The validation of theoretical considerations is confirmed numerically.
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Analytical Determination of Optimal Actuators Position for Single Mode Active Reduction of Fixed-free Beam Vibration Using the Linear Quadratic Problem Idea

86%
Żołopa E., Brański A.
Acta Physica Polonica A
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2014
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vol. 125
|
issue 4A
A-155-A-158
EN
An analytical solution is obtained, based on linear quadratic problem well-known in the control theory. The problem is formulated for fixed-free beam vibration (fourth order partial differential equation) in Hilbert space and the point control and distributed output is considered. Beam deflection at any point is chosen as a criterion of optimization. In this case it means the linear quadratic problem. Up to now, the linear quadratic problem was formulated many times, but only for the time-dependent equation. The aim of the paper is to obtain the value of the cost functional formulated as the function of distribution of actuators. The minimum of this function leads to the optimal actuators location. The results obtained with this method confirm the results obtained in heuristic way and pure analytical one for separate mode; it is pointed out that the actuators ought to be bonded on the beam sub-regions in which the mode curvatures take their local maximums and the highest value.
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