Abstract The effect of parametric amplification, where parametric excitation is added to direct excitation, on the performance of a dynamical system is investigated. To model the system, the linear forced Mathieu equation is considered. In order to obtain the response of the system without restrictions such as requiring the system parameters to be small or the parametric excitation frequency to be exactly twice the direct excitation frequency, the method of varying amplitudes (MVA) is used. Employing the MVA, the response of the system and its maximum value for arbitrary values of the system parameters including the excitation frequencies are derived. It is shown that when the ratio between the excitation frequencies is rational, the maximum response of the system is highly dependent on the phase angle between the excitations. However, when the frequency ratio is irrational, the maximum response of the system is not dependent on the phase angle. Consequently, large values of response can be attained by using an irrational frequency ratio regardless of the phase angle. MVA results for peak responses show good agreement with numerical results obtained by direct integration of the equation of motion.

中文翻译：

---

线性参量放大器对任意直接和参量激励的响应

摘要 研究了参数放大对动态系统性能的影响，其中参数激励被添加到直接激励。为了对系统建模，需要考虑线性强制 Mathieu 方程。为了获得系统响应不受限制，例如要求系统参数很小或参数激励频率恰好是直接激励频率的两倍，使用变幅法（MVA）。使用 MVA，可以导出系统的响应及其对包括激励频率在内的系统参数的任意值的最大值。结果表明，当激励频率之间的比率合理时，系统的最大响应高度依赖于激励之间的相位角。然而，当频率比不合理时，系统的最大响应与相角无关。因此，无论相位角如何，都可以通过使用不合理的频率比来获得较大的响应值。峰值响应的 MVA 结果与通过直接积分运动方程获得的数值结果非常吻合。