This work presents an optimization framework for tailoring the nonlinear dynamic response of lightly damped mechanical systems using Spectral Submanifold (SSM) reduction. We derive the SSM-based backbone curve and its sensitivity with respect to parameters up to arbitrary polynomial orders, enabling efficient and accurate optimization of the nonlinear frequency-amplitude relation. We use the adjoint method to derive sensitivity expressions, which drastically reduces the computational cost compared to direct differentiation as the number of parameters increases. An important feature of this framework is the automatic adjustment of the expansion order of SSM-based ROMs using user-defined error tolerances during the optimization process. We demonstrate the effectiveness of the approach in optimizing the nonlinear response over several numerical examples of mechanical systems. Hence, the proposed framework extends the applicability of SSM-based optimization methods to practical engineering problems, offering a robust tool for the design and optimization of nonlinear mechanical structures.

翻译：本文提出了一种利用谱子流形降阶技术定制轻阻尼机械系统非线性动态响应的优化框架。我们推导了基于谱子流形的骨架曲线及其对参数直至任意多项式阶数的灵敏度，从而实现对非线性频率-幅值关系的高效精确优化。采用伴随方法推导灵敏度表达式，相较于直接微分法，该方法在参数数量增加时能显著降低计算成本。该框架的一个重要特征是在优化过程中可根据用户定义的误差容限自动调整基于谱子流形的降阶模型的展开阶数。我们通过多个机械系统数值算例验证了该方法在优化非线性响应方面的有效性。因此，所提出的框架将基于谱子流形的优化方法扩展到实际工程问题中，为非线性机械结构的设计与优化提供了鲁棒工具。