In this paper, the free and forced vibrations of a three-dimensional nonplanar nanobeam with initial geometric imperfection are investigated. The Kelvin–Voigt model and nonlocal strain gradient theory (NSGT) are applied to the nanobeam with viscoelastic structural damping and size-dependent effect. A mechanical model of the nanobeam is established by applying Hamilton’s principle, and an initial displacement is used to describe the initial geometric imperfection. The differential quadrature method (DQM) is used to discretize the complex partial differential equation and simulate the mid-span vibration amplitude of the nonplanar nanobeam. Comparisons between the proposed model and published results show good agreement and further demonstrate the validity of the model. Numerical simulations are performed to illustrate how the geometric imperfection, the viscoelasticity, the nonlocal parameter, the strain gradient parameter and lateral motion influence the buckling phenomena, natural frequencies and vibrational responses. It is found that the nonlocal effect can reduce the natural frequencies and enhance the complicated nonlinear phenomena. However, the contrary phenomena are induced by the strain gradient effect. The geometric imperfection causes buckling phenomena which significantly affect the natural frequencies of the nanobeam. Moreover, the viscoelastic effect, imperfect effect and lateral motion can lead to more complicated and rich vibrational responses due to the greater contribution of nonlinearity.

在本文中，研究了具有初始几何缺陷的三维非平面纳米束的自由和受迫振动。Kelvin-Voigt 模型和非局部应变梯度理论 (NSGT) 应用于具有粘弹性结构阻尼和尺寸相关效应的纳米梁。应用Hamilton原理建立纳米束的力学模型，并用初始位移来描述初始几何缺陷。微分求积法（DQM）用于离散复杂的偏微分方程并模拟非平面纳米梁的跨中振动幅度。所提出的模型与已发表的结果之间的比较显示出良好的一致性，并进一步证明了模型的有效性。进行数值模拟以说明几何缺陷如何，粘弹性、非局部参数、应变梯度参数和横向运动影响屈曲现象、固有频率和振动响应。研究发现，非局部效应可以降低固有频率，增强复杂的非线性现象。然而，相反的现象是由应变梯度效应引起的。几何缺陷导致屈曲现象显着影响纳米束的固有频率。此外，由于非线性的贡献更大，粘弹性效应、不完美效应和横向运动会导致更复杂和丰富的振动响应。研究发现，非局部效应可以降低固有频率，增强复杂的非线性现象。然而，相反的现象是由应变梯度效应引起的。几何缺陷导致屈曲现象显着影响纳米束的固有频率。此外，由于非线性的贡献更大，粘弹性效应、不完美效应和横向运动会导致更复杂和丰富的振动响应。研究发现，非局部效应可以降低固有频率，增强复杂的非线性现象。然而，相反的现象是由应变梯度效应引起的。几何缺陷导致屈曲现象显着影响纳米束的固有频率。此外，由于非线性的贡献更大，粘弹性效应、不完美效应和横向运动会导致更复杂和丰富的振动响应。