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.