We model and investigate the response of a nonlinear cantilever beam under principal parametric excitation. The design is initially assessed, optimized, and tuned using three-dimensional finite element analysis (FEA) to ensure the presence of fundamental parametric resonance and the absence of other internal and higher-order parametric resonances. The derived governing differential equation represents a modified generalized parametrically excited dynamic system under principal parametric excitation. The nonlinear dynamic system is developed and presented in the context of resonators with extensive applications in developing sensors, filters, and switches. The quadratic and cubic nonlinearities include second- and third-order deflection, velocity, acceleration terms describing stiffness, damping, and inertial nonlinearities. To explore and investigate the corresponding generalized nonlinear Mathieu equation, the method of multiple-scales along with the reconstitution method are used and modulation equations are obtained and solved to obtain closed-form amplitude and phase equations. The quadratic damping is modeled and approximated using a Fourier series and analytical models are generated in both Cartesian and Polar frames. To further explore the dynamic system and its applications, a resonator is designed to measure external acceleration and investigated for two cases. It is discussed and shown how the external acceleration modifies the dynamic system, the corresponding reduced-order model, and the modulation equations. The external acceleration affects the amplitude, phase, and frequency of oscillation providing means to estimate the input. These results indicate that the proposed resonator design (dynamic system) is able to significantly improve the dynamic range of shock/acceleration sensors.

我们建模和研究非线性悬臂梁在主要参数激励下的响应。最初使用三维有限元分析（FEA）对设计进行评估，优化和调整，以确保基本参数共振的存在以及其他内部和更高阶参数共振的存在。导出的控制微分方程表示主参数激励下的修正广义参数激励动力学系统。非线性动态系统是在谐振器的环境中开发和提出的，在开发传感器，滤波器和开关方面具有广泛的应用。二次和三次非线性包括二阶和三阶挠度，速度，描述刚度，阻尼和惯性非线性的加速度项。为了探索和研究相应的广义非线性Mathieu方程，使用了多尺度方法和重构方法，获得并求解了调制方程，以获得闭合形式的振幅和相位方程。使用傅立叶级数对二次阻尼进行建模和近似，并在笛卡尔坐标系和极坐标系中生成分析模型。为了进一步探索动态系统及其应用，设计了一种谐振器来测量外部加速度并研究了两种情况。讨论并显示了外部加速度如何修改动态系统，相应的降阶模型和调制方程。外部加速度会影响振荡的幅度，相位和频率，从而提供估算输入的手段。