High-precision semi-analytical solution for the quasi-periodic nanobeam system based on the weight time-domain minimum residual method
High-precision semi-analytical solution for the quasi-periodic nanobeam system based on the weight time-domain minimum residual method
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DOI：
10.1016/j.compstruct.2023.117457
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发表时间：
2023-08
影响因子：
6.3
通讯作者：
Guang Liu;Jike Liu;Zhong-rong Lu
中科院分区：
文献类型：
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作者：
Guang Liu;Jike Liu;Zhong-rong Lu
纳米梁由于其尺寸小、灵敏度高等优点，在各种微机电系统中得到了广泛的应用。在纳米尺度下，由于表面弹性、残余表面张力和几何非线性等表面效应，纳米梁经常发生准周期振动。根据纳米梁系统的二次规划（QP）运动特性，在时域最小残差法（TMRM）的基础上引入权向量，提出了一种权时域最小残差法（WTMRM），并利用WTMRM获得了高精度的半解析QP（SAQP）解.该方法主要包括四个方面。首先，根据纳米梁系统的QP运动特性，提出了一种新的由主频和频率间隔构造的SAQP解。然后，通过推导假设解并将状态变量代入原控制方程，将求解SAQP解的问题转化为非线性最小优化问题，并采用增强响应灵敏度法（ERSA）迭代求解。其次，引入权向量来解决控制方程与初始条件约束之间的残差矛盾。最后，数值结果的龙格-库塔（RK）方法与所提出的方法进行了比较。对比结果表明，WTMRM方法求解的SAQP精度远高于已有的结果。
Nanobeams have been widely used in various micro-electromechanical systems because of their small size and high sensitivity. At the nanoscale, quasi-periodic (QP) vibration often occurs in nanobeams due to the surface effects such as surface elasticity, residual surface tension and geometric nonlinearity. According to the QP motion properties of the nanobeam system, a weight time-domain minimum residual method (WTMRM) is presented in this paper by introducing the weight vector into the original time-domain minimum residual method (TMRM), and the high-precision semi-analytical QP (SAQP) solution is obtained by the WTMRM. This method mainly includes four aspects. Firstly, according to the properties of QP motion of the nanobeam system, a new SAQP solution constructed by the dominant frequency and frequency interval is presented. Then, the problem of solving the SAQP solution is converted into a nonlinear minimum optimization issue by deducing the presumptive solution and substituting the state variables back to the original control equations, and this optimization issue is resolved by the enhanced response sensitivity approach (ERSA) iteratively. Next, the weight vector is introduced to deal with the contradiction of the residual between the control equation and the initial condition constraints. Finally, the numerical results of the Runge–Kutta (RK) method are compared with those of the proposed method. The comparison results show that the precision of the SAQP solution achieved by the WTMRM is much higher than the existing results.