四角点支撑纳米板稳态受迫振动解析解

本文基于非局部弹性理论及辛叠加方法，得到放置在黏弹性介质上四角点支撑矩形纳米板稳态受迫振动问题的解析解．将纳米板受迫振动问题导入哈密顿体系，得到哈密顿控制方程，在无需任何预设函数的情况下可直接对哈密顿控制方程进行求解，得到简支纳米板稳态受迫振动问题在辛空间展开形式的解析解．进而通过边界叠加，可求出四角点支撑纳米板稳态受迫振动的解析解．数值算例中验证了本文应用辛叠加方法得到解析解的准确性，并以石墨烯纳米板为例，分析了非局部参数和黏弹性介质参数对四角点支撑石墨烯纳米板稳态受迫振动的影响．结果表明，非局部参数和黏弹性介质参数的变化会影响石墨烯纳米板的共振频率及共振幅值．

Exact Solutions for Steady-State Forced Vibration of Four-Corner Point-Supported Nanoplate

Based on the theory of nonlocal elasticity and symplectic superposition method, this paper presented exact solutions for steady-state forced vibration of four-corner point-supported rectangular nanoplates resting on viscoelastic foundations. By introducing the forced vibration problem of nanoplate into the Hamiltonian system, the governing equations in the Hamiltonian system are obtained. The governing equations are solved directly without any prior assumption. Analytical solutions for steady-state forced vibration of simply-supported nanoplate are obtained in symplectic space. Subsequently, the exact solution for steady-state forced vibration of four-corner point-supported nanoplates is deduced through boundary superposition. The accuracy of the exact solution obtained by using symplectic superposition method is verified by numerical examples. Taking graphene nanoplate as an example, the effects of nonlocal parameter and viscoelastic foundation parameters on the steady-state forced vibration of four-corner point-supported graphene nanoplate are analyzed. The results show that the variation of nonlocal parameter and viscoelastic foundation parameters can affect the resonance frequency and resonance amplitude of the graphene nanoplate.