Displacement compensation of beam vibrations caused by rigid-body motions

The present contribution is concerned with the active suppression of plane flexural vibrations of a slender, cantilever linear elastic beam. The vibrations of the beam are considered to be due to a prescribed large rigid-body motion of the beam support, as well as imposed forces. The rigid-body motion under consideration defines a floating reference configuration with respect to which the vibrations are studied. We assume these vibrations to take place in the moderately large strain regime. The beam is considered to be additionally equipped with distributed piezoelectric actuators, which are perfectly bonded to the beam. It is the scope of the present paper to derive a spatial shape of the latter actuators, such that the above vibrations can be completely suppressed by the piezoelectric actuation. This problem is also known as vibration compensation or shape control by piezoelectric actuation. In the present paper, an analytic solution for shape control is presented within the Bemoulli-Euler-von Karman theory of a slender cantilever beam, taking into account the so-called stress-stiffening effect. The presented solution for shape control makes the initial boundary value problem under consideration homogeneous, such that the vanishing of vibrations indeed represents a solution. Possible instabilities, such as parametrically excited vibrations, can be studied in the usual manner, this not being the content of the present paper. For the dynamically stable example of a rotating beam, the influence of the stress stiffening effect in the presence of a distributed piezoelectric actuation is studied in some detail. It turns out that the stress stiffening effect must indeed be taken into account for an adequate modelling. The corresponding analytic solution for shape control is validated by means of non-linear 3D finite element computations, showing excellent evidence of the appropriateness of the presented analytic beam-type solution for vibration compensation.