考虑可控性的压电作动器拓扑优化设计

TOPOLOGY OPTIMIZATION OF PIEZOELECTRIC ACTUATOR CONSIDERING CONTROLLABILITY

压电作动器可以把电能转换成机械能,在结构主动振动控制中具有应用背景.由于压电作动器的布局对振动控制效果影响很大,因此作动器布局优化一直是结构控制研究的关键之一.为了提高压电结构控制能量的利用效率,本文提出了以提高结构可控性为目标的压电作动器的拓扑优化方法.基于经典层合板理论对压电结构进行了有限元建模,并采用模态叠加法将动力控制方程映射到模态空间,推导了基于控制矩阵奇异值的可控性指标.优化模型中,选取可控性指标指数形式为目标函数,将设计变量定义为作动器单元的相对密度,并基于人工密度惩罚模型构造了压电系数惩罚模型,给出了基于控制矩阵奇异值的可控性指标关于设计变量的灵敏度分析方法.优化问题采用基于梯度的数学规划法求解.数值算例验证了灵敏度分析方法和优化模型的有效性,并讨论了主要因素对优化结果的影响.

Piezoelectric actuators can convert electrical energy into mechanical energy, and has application potential in active vibration control of structures. Since the layout of the piezoelectric actuators has a great influence on the vibration control effect, the optimization of the actuators has always been one of the key factors to structural control. In order to improve the efficiency of control energy in the piezoelectric structure, this paper proposes a topology optimization method for the layout design of piezoelectric actuators with the goal of improving structural controllability. The finite elemen modeling of the piezoelectric structure is carried out based on the classical laminate theory. The modal superposition method is used to map the dynamic governing equation to the modal space. The controllability index based on the singular value of the control matrix is derived. In the optimization model, the exponential form of the controllability index is chosen as the objective function, and the design variables are the relative densities of the actuator elements. Based on the Solid Isotropic Material Penalization method, an artificial piezoelectric coefficient penalty model is constructed Sensitivity analysis for the controllability index is proposed based on the singular value of the control matrix. The optimization problem is solved by a gradient-based mathematical programming method. Numerical examples verify the effectiveness of the sensitivity analysis method and the optimization model and show the significance of the layout design of piezoelectric actuators. The influence of some key factors on the optimization results are discussed. It shows that the more piezoelectric materials, the better the controllability;the modes of interest in the objective function has a great influence on the layout of the piezoelectric actuators.