Vibration attenuation and control is a typical topic in mechanical, civil and aeronautical engineering. In recent years, there has been extensive research on smart materials and among all of them, the piezoelectrics seem to be the most attractive for passive and active vibration damping applications. Furthermore if multiple modes are concurrently excited, as in case of turbomachinery blades, active damping systems may remarkably increase their life-cycle and outweigh the shortcomings of implementing such systems. However the damping efficiency of the piezoelectric actuators is strictly bound to their driving voltage, size and location on the structure.

In this work, a cantilever piezoelectric bimorph beam under base motion is considered and the analytical expression of the electric potential that nullifies the elastic tip displacement of the beam is derived in case of single and bi-modal excitations.

The model allows to identify for every bi-modal excitations a set of solutions, each of them represented by three parameters: voltage amplitude, left and right corner positions of the piezoelectric actuators pair. As a result, designers can choose the best solution for their specific application demands. For example, if the supply voltage must to be kept as low as possible, then wider actuators should be used and vice versa. It was also found out that the control parameters do not depend on the spectral distribution between the two excited modes. Hence, even if the spectral distribution between the two coupled modes changes over time, it is not necessary to adjust either the voltage or the position of the actuator pair.

The analytical predictions were compared with the results of FEM multi-physics simulations for several base motion excitations and a fair agreement was observed.

减振和控制是机械、土木和航空工程中的典型课题。近年来，对智能材料进行了广泛的研究，其中，压电材料似乎对被动和主动减振应用最具吸引力。此外，如果同时激发多种模式，例如在涡轮机械叶片的情况下，主动阻尼系统可能会显着增加其生命周期，并克服实施此类系统的缺点。然而，压电致动器的阻尼效率严格受限于其驱动电压、尺寸和结构上的位置。

在这项工作中，考虑了基础运动下的悬臂压电双压电晶片梁，并在单模态和双模态激发的情况下推导出使梁的弹性尖端位移无效的电势的解析表达式。

该模型允许为每个双模态激励识别一组解决方案，每个解决方案由三个参数表示：电压幅度、压电致动器对的左右角位置。因此，设计人员可以选择适合其特定应用需求的最佳解决方案。例如，如果电源电压必须保持尽可能低，则应使用更宽的执行器，反之亦然。还发现控制参数不依赖于两种激发模式之间的光谱分布。因此，即使两个耦合模式之间的光谱分布随时间变化，也没有必要调整致动器对的电压或位置。

将分析预测与几个基本运动激励的 FEM 多物理场模拟结果进行比较，观察到相当一致。