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Nonlinear forced vibration analysis of functionally graded carbon nanotube-reinforced composite Timoshenko beams

机译：功能梯度碳纳米管增强复合Timoshenko梁的非线性强迫振动分析

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摘要

This research deals with the forced vibration behavior of nanocomposite beams reinforced by single-walled carbon nanotubes (SWCNTs) based on the Timoshenko beam theory along with von Karman geometric nonlinearity. For the carbon-nanotube reinforced composite (CNTRC) beams, uniform distribution (UD) and three types of functionally graded (FG) distribution patterns of SWCNT reinforcements are considered. It is assumed that the material properties of FG-CNTRC beams are graded in the thickness direction and estimated through the rule of mixture. The nonlinear governing equations and corresponding boundary conditions are derived based on the Hamilton principle and discretized by means of the generalized differential quadrature (GDQ) method. After that, a Galerkin-based numerical technique is employed to reduce the set of nonlinear governing equations into a time-varying set of ordinary differential equations of Duffing type. Since the nanobeam responds periodically to harmonic excitations, a set of periodic differential matrix operators is introduced to discretize the Duffing equations on the time domain using the derivatives of a periodic base function. The vectorized form of final nonlinear parameterized equations is then solved through the use of pseudo-arc length continuum technique. Numerical results are presented to examine the effects of different parameters such as nanotube volume fraction, slenderness ratio, dimensionless damping parameter, dimensionless transverse force, CNT distributions and boundary conditions on the natural frequencies and frequency responses of FG-CNTRC beams.
机译：这项研究基于Timoshenko束理论以及von Karman几何非线性，研究了单壁碳纳米管（SWCNTs）增强的纳米复合梁的强迫振动行为。对于碳纳米管增强复合材料（CNTRC）梁，考虑了SWCNT增强材料的均匀分布（UD）和功能梯度（FG）的三种类型。假设FG-CNTRC梁的材料特性沿厚度方向分级，并通过混合法则进行估算。非线性控制方程和相应的边界条件是根据汉密尔顿原理导出的，并通过广义差分正交（GDQ）方法进行离散化。之后，采用基于Galerkin的数值技术将非线性控制方程组简化为Duffing型时变常微分方程组。由于纳米束周期性地响应谐波激励，因此引入了一组周期性微分矩阵算子，以使用周期性基函数的导数在时域上离散Duffing方程。然后通过使用伪弧长连续谱技术来求解最终非线性参数化方程的矢量化形式。数值结果表明了纳米管体积分数，细长比，无量纲阻尼参数，无量纲横向力，碳纳米管分布和边界条件等不同参数对FG-CNTRC梁固有频率和频率响应的影响。

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