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

近些年，很多学者致力于利用非线性增强振动响应减少的效果或者能量采集器的效率。因而非线性系统的响应值需要从理论计算方面更准确地预测。另外，根据学者已取得的研究成就，非线性能量汇(NES)中存在的立方刚度非线性可以将结构中宽频域的振动能量传递至非线性振子部分。文章将一种由NES和压电能量采集器组成的NES-piezo装置与两自由度主结构耦合连接，系统受谐和激励作用。文章采用谐波平衡法和复平均法分别推导了系统稳态响应，参照数值结果，对比两种近似解析方法在求解强非线性系统稳态响应时的异同。计算结果表明，系统体现较弱非线性时，二者计算结果差异很小；当系统体现强非线性时，复平均法不能准确地呈现系统高阶响应，提高阶数的谐波平衡法能更准确地表示系统响应值。基于谐波平衡法和数值算法，讨论NES-piezo装置对于系统宽频域减振的影响。与仅加入非线性能量汇情况对比，结果表明NES-piezo装置不会恶化宽频域减振效果，并且在第一阶共振频率附近，可以稍微提高结构减振效率。另外，计算结果也表明，采用恰当的NES-piezo装置可实现宽频域范围的结构减振和压电能量采集一体化。此项研究工作为研究不同情形强非线性系统的响应提供了理论方法的指导。另外，研究结果也为宽频域范围的结构减振和压电能量采集一体化提供了理论依据。

The Difference on Steady State Responses of a Strongly Nonlinear System via Harmonic Balance Method and Complexification-averaging Technique

Recently, many scholars introduce intentional strong nonlinearities in the studied objects to enhance vibration control or energy harvesting. The responses of strongly nonlinear systems deserve to be evaluated more accurately in order to predict nonlinear effects in theory. Besides, as is known, the nonlinear energy sink (NES) can transfer or absorb the broadband vibration energy of a structure at both high and low frequencies its cubic nonlinear stiffness. First, 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 the nonlinear energy sink (NES) and a piezoelectric energy harvester. The two kinds of approximate analytical results are compared with numerical results. It is found that the two methods have a little difference in steady state responses when the nonlinearity of the system is not very strong. But if the system shows strong nonlinearity, the complex-averaging technique fails to obtain the high order’s responses, while the harmonic balance method can do this. Based on the conclusion, harmonic balance method is adopted to analyze the effect of an adding NES-piezo device on broadband vibration reduction. Two vibration cases are compared. One case applies the NES-piezo device and the other case only uses the NES. Results indicate that using the NES-piezo device will not deteriorate the efficiency of broadband vibration reduction. Furthermore, the responses near the first order resonant frequency can be improved slightly. Besides, results also demonstrates 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 choosing an appropriate approach when facing different nonlinear situation. And the integration of broadband vibration reduction and piezoelectric energy harvesting provide a theory significance.