谐波平衡法与复平均法计算强非线性系统稳态响应的区别

Recently, many scholars have introduced intentional strong nonlinearities into the studied objects to enhance vibration control or energy harvesting. The responses of strongly-nonlinear systems deserve more accurate evaluation in order to predict nonlinear effects in theory. Besides, owing to its cubic nonlinear stiffness, the nonlinear energy sink (NES) can transfer or absorb the broadband vibration energy of a structure at both high and low frequencies. In this study, both the harmonic balance method and the complexification-averaging technique are adopted and compared for acquiring steady-state responses of the nonlinear dynamic system. The studied strongly-nonlinear system contains a two-degree-of-freedom primary structure, which is subjected to a harmonic excitation, and a NES-piezo device. The NES-piezo device is composed of a NES and a piezoelectric energy harvester. The two kinds of approximate analytical results are compared with the numerical results. It is found that the two methods only have slight difference in the steady-state responses when the nonlinearity of the system is not very strong. However, if the system shows strong nonlinearity, the complex-averaging technique fails to obtain the high-order responses, while the harmonic balance method can. Based on the conclusion, the harmonic balance method is adopted to analyze the effect of NES-piezo device on broadband vibration reduction. Two vibration cases are compared: one with the NES-piezo device applied, and the other only with the NES. The results indicate that the use of NES-piezo device does not deteriorate the efficiency of broadband vibration reduction. Furthermore, the responses near the first-order resonant frequency can be improved slightly. The results also demonstrate the integration of broadband vibration reduction and piezoelectric energy harvesting by an appropriate NES-piezo device. This research work shows similarities and differences of two approximate methods in solving problems of strong nonlinearity. It will contribute to the selection of appropriate approaches when facing different nonlinear situations. The integration of broadband vibration reduction and piezoelectric energy harvesting is of theoretical significance.