In this paper, based on the nonlocal strain gradient theory (NSGT), the coupled vibrations of nanobeams with axial and spinning motions under complex environmental changes are modeled. A detailed parametric investigation is also performed to determine the effect of size-dependent parameters, boundary conditions, hygro–thermo–magnetic loads, axial and spin velocities on the dynamical behavior and stability regions of the system. Adopting the Galerkin discretization technique, the eigenvalue problem is solved, and natural frequencies, divergence and flutter instability thresholds of the system are extracted accordingly. The acquired outcomes are compared with the reported results in the literature. Besides, the accuracy of the numerical method is compared with analytical approaches and a good agreement is observed. The obtained results demonstrate that considering the nonlocal and hygro–thermal effects in modeling has destabilizing impacts on the dynamical response of the system. While imposing the strain gradient term and magnetic field leads to the enhancement of vibrational frequencies and enlargement of stability areas. In addition, it is concluded that in hygro–thermal environments, by ascending the spin velocity, instead of the occurrence of divergence instability, the system experiences the flutter conditions. Meanwhile, the attained outcomes indicated that the variation of spin velocity does not affect the flutter instability threshold of the system.

中文翻译：

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基于非局部应变梯度理论的同时轴向和自旋运动的纳米梁的湿热磁致振动

在本文中，基于非局部应变梯度理论（NSGT），模拟了复杂环境变化下具有轴向和旋转运动的纳米梁的耦合振动。还进行了详细的参数研究，以确定与尺寸相关的参数、边界条件、湿-热-磁载荷、轴向和自旋速度对系统动力学行为和稳定性区域的影响。采用Galerkin离散化技术，求解特征值问题，提取系统的固有频率、发散度和颤振不稳定阈值。将获得的结果与文献中报告的结果进行比较。此外，数值方法的准确性与解析方法进行了比较，并观察到良好的一致性。获得的结果表明，在建模中考虑非局部和湿热效应会对系统的动态响应产生不稳定影响。同时施加应变梯度项和磁场导致振动频率的增强和稳定区域的扩大。此外，还得出结论，在湿热环境中，通过提高自旋速度，而不是发散不稳定性的发生，系统会经历颤振条件。同时，所得结果表明自旋速度的变化不会影响系统的颤振不稳定阈值。同时施加应变梯度项和磁场导致振动频率的增强和稳定区域的扩大。此外，还得出结论，在湿热环境中，通过提高自旋速度，而不是发散不稳定性的发生，系统会经历颤振条件。同时，所得结果表明自旋速度的变化不会影响系统的颤振不稳定阈值。同时施加应变梯度项和磁场导致振动频率的增强和稳定区域的扩大。此外，还得出结论，在湿热环境中，通过提高自旋速度，而不是发散不稳定性的发生，系统会经历颤振条件。同时，所得结果表明自旋速度的变化不会影响系统的颤振不稳定阈值。