The static responses and displacement control of circular curved beams with piezoelectric actuators

We analyse the static responses of a circular curved beam bonded with a single piezoelectric actuator or symmetric actuators. We also study the characteristics of the control parameters for the displacement control of a cantilever circular curved beam via piezoelectric actuators in this paper. The uniform circular beam is thin and actuated by thin piezoelectric layers. The piezoelectric unimorph or the bimorph beam is equivalent to a single-layer structure. The governing equations of these structures with small curvature are derived based on one-dimensional beam theory. The static analysis of the segmented beam undergoing radial concentrated load and concentrated bending moment is given in detail. For a cantilever beam with single actuator or symmetric actuators, the theoretical results of the displacement responses agree well with experimental results reported early and/or the FEM results. The control problem of a cantilever bimorph is studied as the first example. To control the displacement at the free end by piezoelectric actuation, an explicit expression of the optimal voltage is obtained for the beam undergoing the radial concentrated load at an arbitrary location, which indicates the optimal voltage is independent of the material properties of the middle elastic layer. Numerical results show that the peak control voltage and the negative control voltage will occur with the changes of the load location and the beam length. The cantilever beam with symmetric distributed actuators is studied as the second example. As the beam is loaded by too high a radial concentrated load at the free end, the radial displacement of the free end is controlled to the minimum via actuators with given input voltage. The optimization of the actuators' length and location is implemented. For a given actuator's length, the optimal actuator's location is at the clamped end as the central angle of the beam is smaller than a critical value. After the central angle of the beam is larger than the critical value, the optimal value of the actuator's location is dependent on the central angle of the beam and the central angle of the actuators. As the actuator's location is given, the optimal length of the actuators is the allowable maximum length of the actuators.