Presented in this work is a novel and easy-to-implement supplemental-frequency harmonic balance (SF-HB) approach to efficiently compute dynamically aperiodic systems, which can be cumbersome to model using the nominal high-dimensional harmonic balance (HDHB) technique. The stability of the time-spectral operator and the SF-HB solver involving multiple excitation frequencies is ensured by introducing a group of supplemental frequencies based on the fact that aperiodic (or almost aperiodic) response would contain all possible frequencies. Together with the excitation (primary) frequencies, the frequency set forms a nearly-harmonic series leading to a small condition number for the Fourier transformation matrices directly impacting the stability of the HDHB solver. In contrast to the similar work reported in the literature where optimal unequally-spaced sub-time levels are determined through a rather complicated procedure, the current approach simply uses equally-spaced sub-time levels spanning a time period that depends on a predicted base frequency. At convergence, the original excitation frequency modes dominate the solution as desired. This new technique is verified for a forced pitching airfoil in the transonic inviscid flow regime subjected to two excitation frequencies. Results of both periodic and aperiodic cases show that with the help of Fourier interpolation, the entire time history of the dynamic response can be obtained through a single run of the SF-HB solver, for which the efficiency and robustness of the traditional HDHB method is preserved.

这项工作提出了一种新颖且易于实现的补充频率谐波平衡（SF-HB）方法，可以有效地计算动态非周期性系统，而使用标称高维谐波平衡（HDHB）技术进行建模可能很麻烦。基于非周期性（或几乎非周期性）响应将包含所有可能的频率这一事实，通过引入一组补充频率，可以确保涉及多个激发频率的时谱算子和SF-HB解算器的稳定性。频率集与激励（主）频率一起形成了近谐波序列，导致傅立叶变换矩阵的条件值小，直接影响了HDHB求解器的稳定性。与文献中报道的类似工作不同，后者通过相当复杂的过程确定了最佳的不等间隔子时间水平，而当前方法只是使用跨一定时间段的等间隔子时间水平，该时间间隔取决于预测的基准频率。收敛时，原始激励频率模式将按需支配解决方案。这项新技术已针对跨音速无粘性流动状态下的强制俯仰翼型受到两个激励频率验证。周期性和非周期性情况的结果均表明，借助傅立叶插值，可以通过单次运行SF-HB解算器来获得动态响应的整个时间历史记录，而传统HDHB方法的效率和鲁棒性是保留。