粘弹性多孔功能梯度纳米梁的非线性振动

Nonlinear Vibration of Viscoelastic Porous Functionally Graded Nanobeams

摘要: 针对经典连续介质力学及标准整数阶模型通常忽略的关键尺寸效应和复杂能量耗散机制，本文提出了一种综合考虑尺寸依赖性的非线性动力学模型。利用该模型研究了热–力耦合作用下功能梯度多孔纳米梁的非线性振动行为，其中融合了非局部弹性理论和分数阶粘弹性地基模型。基于Hamilton原理推导了粘弹性多孔功能梯度纳米梁的分数阶控制方程，采用多尺度法解析得到了系统的稳态幅频响应，并讨论了各参数对纳米梁振动特性的影响。结果表明：分数阶导数阶次会非单调地调节结构刚度，在某一特定阶次处振动频率达到最大。此外，升高的温度和非局部效应会导致结构软化，从而放大振动幅值，而非线性地基刚度则提供了强劲的恢复力。

Abstract: Aiming at the critical size effects and complex energy dissipation mechanisms often ignored by classical continuum mechanics and standard integer-order models, this paper proposes a nonlinear dynamic model that comprehensively considers size-dependent behavior. Using this model, the nonlinear vibration of functionally graded porous nanobeams under thermo-mechanical coupling is investigated, incorporating nonlocal elasticity theory and a fractional-order viscoelastic foundation model. The fractional governing equation of the viscoelastic porous functionally graded nanobeam is derived based on Hamilton’s principle. The steady-state amplitude-frequency response of the system is analytically obtained using the method of multiple scales, and the effects of various parameters on the vibration characteristics of the nanobeam are discussed. The results show that the fractional derivative order non-monotonically modulates the structural stiffness, and the vibration frequency reaches a maximum at a specific order. Furthermore, elevated temperature and nonlocal effects induce structural softening, thereby amplifying the vibration amplitude, while the nonlinear foundation stiffness provides a strong restoring force.