Bending moment tracking and the reduction of the axial stress in vibrating beams by piezoelectric actuation

Abstract

This contribution focuses on bending moment tracking in slender beam-type structures that are equipped with piezoelectric actuators. Bending moments are associated with the axial stress, which is the dominant stress component of laterally excited beam structures. If the maximum value exceeds a certain tensile stress limit, the structures will crack or be irreparably damaged. In the present contribution, a piezoelectric bimorph beam is considered and it is investigated in which manner the piezoelectric actuation devices have to be distributed along the beam length, such that a certain bending moment distribution is obtained. This is called bending moment tracking. First, the basic equations of a piezoelectric bimorph beam are recalled and the differential equations for the bending moment are derived. Then a positive semi-definite integral depending on the error of the bending moment is defined, which is the difference between the actual and the desired bending moment. The results of the derivations are conditions for the eigencurvature due to the piezoelectric actuation, such that a certain bending moment distribution is achieved. Approximate solutions for the eigencurvature are also presented for the lower-and for the high-frequency domain. The theory is verified by a support-excited piezoelectric bimorph. First, the frequency response curves for the deflection, the bending moment and the axial stresses are calculated. Then the responses due to a sinusoidal excitation are computed, showing that the suggested control algorithm enables the reduction of the bending moment and also of the axial stress in a satisfactory manner.

Control of stress and damage in structures by piezoelectric actuation: 1D theory and monofrequent experimental validation

“…The first outcomes of these laboratory experiments are presented subsequently, where the corresponding structural system is described in detail below. In contrast to our previous theoretical considerations in Irschik, Irschik et al, Schoeftner and Irschik, and Schoeftner, only a part of the considered system is subjected to the piezoelectric control actuation, and only the stress in that part is controlled. Hence, our subsequent formulations are novel also from a theoretical point of view.…”

“…More recently, Schoeftner and Irschik showed how to achieve a desired stress distribution for a base‐excited one‐dimensional bar with continuously distributed mass and stiffness by a proper temporal and spatial distribution of the piezoelectric actuation. Control of stress resultants, such as of bending moments, in piezoelectrically actuated continuous beam‐type structures has been treated in Schoeftner . In Irschik, Irschik et al, Schoeftner and Irschik, and Schoeftner, first 3D and 1D theoretical and numerical results for the feed‐forward stress control by piezoelectric actuation have been presented, where it has been assumed that the necessary spatial and temporal distribution of the piezoelectric actuation can be realized everywhere in the structure under consideration.…”

“…Control of stress resultants, such as of bending moments, in piezoelectrically actuated continuous beam‐type structures has been treated in Schoeftner . In Irschik, Irschik et al, Schoeftner and Irschik, and Schoeftner, first 3D and 1D theoretical and numerical results for the feed‐forward stress control by piezoelectric actuation have been presented, where it has been assumed that the necessary spatial and temporal distribution of the piezoelectric actuation can be realized everywhere in the structure under consideration. No experimental results have been presented so far to validate these theoretical findings in reality.…”

Shunted optimal vibration energy harvesting control of discontinuous smart beams

“…Moreover, the configuration of the system itself obviously underlies on the varieties of the physical and technical studies as proposed by many researchers. The investigation of the system with either smart beam or plate structure covers wide ranging control strategies such as the structural control [1][2][3][4][5][6][7], shape control [8][9][10][11], feedback gain control [12][13][14], and shunt control [15][16][17]. In particular attentions to the vibration suppression using the shunt control, the resonance of the smart structure with the selected mode can be tuned into the desired frequency resulting in the reduction of the amplitude [18][19].…”