A coupled nonlinear nonlocal strain gradient theory (NSGT) is developed for functionally graded Timoshenko nanoscale (FGTN) beams. This highly nonlinear-scale-dependent continuum model is solved numerically for dynamical responses for the first time. Hamilton’s energy minimisation technique is used for rotation-equation of the nanobeam in addition to axial-displacement and transverse-displacement equations; these equations are coupled both nonlinearly and linearly, where decoupling is not possible. The functional nature of the material of the nanobeam is incorporated via the Mori–Tanaka mixture scheme. Small-scale influences are taken into account via the NSGT which is able to incorporate both softening/hardening effects. The coupled/nonlinear model is reduced to a high-dimensional system using Galerkin’s method. The dynamics of the FGTN beam is analysed and the influence of functionally graded nature as well as the softening/hardening effects due to the NSGT is highlighted.

针对功能梯度Timoshenko纳米级（FGTN）光束开发了耦合非线性非局部应变梯度理论（NSGT）。首次以数值方式求解了这种高度依赖于非线性比例的连续体模型，以进行动力学响应。汉密尔顿的能量最小化技术除了用于轴向位移和横向位移方程之外，还用于纳米束的旋转等式。这些方程非线性和线性耦合，不可能去耦。纳米束材料的功能特性是通过Mori–Tanaka混合方案纳入的。通过NSGT可以考虑到小范围的影响，该功能可以同时包含软化/硬化效果。使用Galerkin方法将耦合/非线性模型简化为高维系统。