Nonlinear vibration of a fractional viscoelastic micro-beam is investigated in this paper. The Euler–Bernoulli beam theory and the nonlinear von Kármán strain are used to model the beam. The small-scale effects are considered by employing the Modified Couple Stress Theory (MCST). The viscoelastic material of the beam is modeled via the fractional Kelvin–Voigt model. Utilizing the Hamilton’s Principle, a partial fractional differential equation is derived as the governing equation of motion. The Finite Difference Method (FDM) and the Galerkin method are used together for solving the partial fractional differential equation. The FDM is utilized to discretize the time domain, and the Galerkin method is employed to discretize the space domain. In this paper, the FDM and the Shooting method are coupled together to find the periodic solution of the fractional micro-beam and draw the corresponding amplitude–frequency curve. The effects of the order of the fractional derivative, viscoelastic model, and the micro-scale are studied numerically here in this study. Numerical simulations suggest that, the effect of the fractional derivative is very strong and must be considered for modeling the viscoelastic behavior; especially when the amplitude is high. Results also show that the effects of the nonlinear viscoelastic part are considerable when the amplitude is high; this may happen when the excitation frequency is near the natural frequency, at which the maximum amplitude occurs.

本文研究了分数粘弹性微梁的非线性振动。Euler-Bernoulli 梁理论和非线性 von Kármán 应变用于对梁进行建模。小规模效应是通过采用修正偶应力理论 (MCST) 来考虑的。梁的粘弹性材料通过分数 Kelvin-Voigt 模型进行建模。利用哈密顿原理，推导出偏分数阶微分方程作为运动控制方程。有限差分法（FDM）和伽辽金法一起用于求解偏分数阶微分方程。FDM用于时域离散，Galerkin方法用于空间域离散。在本文中，FDM 和 Shooting 方法耦合在一起，找到分数微束的周期解并绘制相应的幅频曲线。本研究以数值方式研究了分数阶导数的阶数、粘弹性模型和微观尺度的影响。数值模拟表明，分数阶导数的影响非常强，在模拟粘弹性行为时必须考虑；尤其是当振幅很高时。结果还表明，当振幅较大时，非线性粘弹性部分的影响相当大；当激励频率接近自然频率时可能会发生这种情况，在该频率上出现最大振幅。本研究对粘弹性模型和微观尺度进行了数值研究。数值模拟表明，分数阶导数的影响非常强，在模拟粘弹性行为时必须考虑；尤其是当振幅很高时。结果还表明，当振幅较大时，非线性粘弹性部分的影响相当大；当激励频率接近自然频率时可能会发生这种情况，在该频率上出现最大振幅。本研究对粘弹性模型和微观尺度进行了数值研究。数值模拟表明，分数阶导数的影响非常强，在模拟粘弹性行为时必须考虑；尤其是当振幅很高时。结果还表明，当振幅较大时，非线性粘弹性部分的影响相当大；当激励频率接近自然频率时可能会发生这种情况，在该频率上出现最大振幅。结果还表明，当振幅较大时，非线性粘弹性部分的影响相当大；当激励频率接近自然频率时可能会发生这种情况，在该频率上出现最大振幅。结果还表明，当振幅较大时，非线性粘弹性部分的影响相当大；当激励频率接近自然频率时可能会发生这种情况，在该频率上出现最大振幅。