Engineering structures usually operate in some specific frequency bands. An effective way to avoid resonance is to shift the structure’s natural frequencies out of these frequency bands. However, in the optimization procedure, which frequency orders will fall into these bands are not known a priori. This makes it difficult to use the existing frequency constraint formulations, which require prescribed orders. For solving this issue, a novel formulation of the frequency band constraint based on a modified Heaviside function is proposed in this paper. The new formulation is continuous and differentiable; thus, the sensitivity of the constraint function can be derived and used in a gradient-based optimization method. Topology optimization for maximizing the structural fundamental frequency while circumventing the natural frequencies located in the working frequency bands is studied. For eliminating the frequently happened numerical problems in the natural frequency topology optimization process, including mode switching, checkerboard phenomena, and gray elements, the “bound formulation” and “robust formulation” are applied. Three numerical examples, including 2D and 3D problems, are solved by the proposed method. Frequency band gaps of the optimized results are obtained by considering the frequency band constraints, which validates the effectiveness of the developed method.

工程结构通常在某些特定频段内运行。避免共振的有效方法是将结构的固有频率移出这些频带。然而，在优化过程中，先验未知哪些频率阶将落入这些频带。这使得难以使用需要规定顺序的现有频率约束公式。为了解决这个问题，本文提出了一种基于修正的Heaviside函数的频带约束新公式。新的公式是连续的和可区分的；因此，可以导出约束函数的灵敏度，并将其用于基于梯度的优化方法中。研究了拓扑优化，以最大程度地消除结构基本频率，同时避开工作频带中的固有频率。为了消除固有频率拓扑优化过程中经常发生的数值问题，包括模式切换，棋盘现象和灰色元素，应用了“约束公式”和“鲁棒公式”。该方法解决了包括2D和3D问题在内的三个数值示例。通过考虑频带约束获得优化结果的频带间隙，这验证了所开发方法的有效性。棋盘格现象，灰色元素，“约束公式”和“稳健公式”被应用。该方法解决了包括2D和3D问题在内的三个数值示例。通过考虑频带约束获得优化结果的频带间隙，这验证了所开发方法的有效性。运用了棋盘格现象和灰色元素，并使用了“约束公式”和“稳健公式”。该方法解决了包括2D和3D问题在内的三个数值示例。通过考虑频带约束获得优化结果的频带间隙，这验证了所开发方法的有效性。