考虑基频约束的压电柔顺机构拓扑优化方法
TOPOLOGY OPTIMIZATION OF PIEZO-EMBEDDED COMPLIANT MECHANISM CONSIDERING FUNTAMENTAL FREQUENCY CONSTRAINTS
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摘要: 随着压电柔顺机构在航空航天及精密加工等高频振动环境中的广泛应用, 亟需发展大行程、高频响压电柔顺机构的拓扑优化方法. 现有的拓扑优化方法侧重于提高柔顺机构的输出行程, 而忽略了其动态特性. 文章提出一种考虑基频约束的压电柔顺机构拓扑优化方法. 首先, 为了更准确地描述压电驱动器与柔顺机构间的相互作用, 将压电驱动器直接耦合进柔顺机构的优化分析模型中, 建立了压电柔顺机构机-电耦合优化分析模型. 其次, 以输出位移为目标函数, 基于变密度法, 建立了考虑结构基频约束的压电柔顺机构拓扑优化方法. 采用p-norm凝聚函数来近似结构基频, 避免了迭代过程中特征值重复和模态切换所引起的不可导问题, 并利用伴随法和链式求导法推导了目标函数和基频约束对设计变量的灵敏度. 最后, 通过数值算例验证了所提优化方法的有效性, 讨论了基频约束对优化结果的影响. 数值算例结果表明, 所提方法迭代过程快速, 收敛性好, 能够在满足基频约束的前提下, 实现输出行程最大化的设计. 与仅考虑柔顺机构输出位移的优化结果相比, 采用本文提出的方法所得优化构型的材料分布明显不同, 且随着基频约束的增大, 优化构型的基频相应变大, 而输出位移变小.Abstract: With the widespread application of piezoelectric compliant mechanisms in high-frequency vibration environments such as aerospace engineering and precision machining, there is an urgent need to develop a topology optimization method for designing piezoelectric compliant mechanisms with large strokes and high-frequency responses. Current topology optimization methods for compliant mechanisms primarily focus on improving the output stroke while neglecting their dynamic characteristics. In this paper, a topology optimization method for piezo-embedded compliant mechanisms is proposed, which considers both output stroke and dynamic characteristics. First of all, to more accurately describe the interaction between the piezoelectric actuator and the compliant mechanism, the piezoelectric actuator is directly coupled into the analysis model by establishing a mechanical-electrical coupling optimization analysis model for the piezo-embedded compliant mechanism. Secondly, based on the density-based method, a topology optimization method of piezo-embedded compliant mechanisms, considering the fundamental frequency constraint, is established with the aim of maximizing the output stroke of the mechanism. The p-norm approximation function is adopted to alleviate the non-differentiability issue arising from repeated eigenvalues and mode switching during the iterative process. Furthermore, using the adjoint method and chain derivation method, the sensitivities of the objective function and constraints with respect to design variables are derived. Finally, several numerical examples are provided to verify the effectiveness of the proposed optimization method and to demonstrate the influence of fundamental frequency constraints on the optimized results. Numerical results show that the proposed method can achieve a stable iterative process, fast convergence, and effectively provide a high output stroke design while satisfying fundamental frequency constraints. Compared to the optimized results that solely maximize output displacement, the configurations of the optimized designs obtained through the proposed method are notably different. As the fundamental frequency constraint increases, the fundamental frequency of the optimized designs correspondingly rises, while the output displacement decreases.