分数阶导数是阻尼器本构模型的常用表达，相关结构动力响应的解析解答难以获得，高效时域数值求解具有重要实践意义。本文针对应用广泛的Riemann-Liouville型分数阶阻尼结构，引入基于多步预估-校正机制的Adams-Moulton算法计算分数阶导数，继而在各离散时刻构造等效线性定常动力系统，最终结合Newmark-β数值积分方案建立各时刻求解显式公式，实现高精度动力响应的直接快速求解。以单自由度振子受简谐荷载和单位脉冲为例，对比考察解析解、两种直接数值解法和本文方法，验证本文方法的高精度和稳定性。最后以多层阻尼减震结构受地震作用为例，对比考察迭代数值算法、工程近似算法和本文方法，进一步验证本文方法在计算精度和计算效率上的综合优势，揭示工程应用潜力。
Abstract
Fractional derivative is a comon expression of the constitutive model of a damper. The analytical solution of the dynamic response of the relevant structure is difficult to obtain, and an efficient time-domain numerical solution is of practical importance. In this paper, for the widely used Riemann-Liouville type fractionally damped structure, the Adams-Moulton algorithm based on multi-step predictor-corrector mechanism is introduced to calculate the fractional derivatives, and then the equivalent linear and time-invariant dynamical system is constructed at each discrete time instant, and finally the Newmark-β numerical integration scheme is combined to establish the explicit computation formula at each time instant to achieve direct and rapid solution of high-accuracy dynamic responses. Taking a SDOF oscillator subjected to harmonic loading and unit impulse as an example, the analytical solution, two direct numerical solutions, and the present method are compared to verify the high accuracy and stability of the present method. Finally, for the example of a multi-layer damped structure subjected to seismic excitation, the iterative numerical algorithm, the engineering approximation algorithm, and the present method are compared and investigated to further verify the combined advantages of the present method in terms of computational accuracy and computational efficiency, and to reveal the potential of engineering applications.
关键词
分数阶阻尼 /
动力分析 /
预估-校正 /
等效线性系统
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Key words
fractional damper /
dynamic analysis /
predictor-corrector /
equivalent linear system
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参考文献
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