基于线性定常系统等效的分数阶阻尼结构动力响应数值求解

Numerical solving for dynamic response of fractional order damped structurebased on linear time-invariant system equivalence

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。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、2种直接数值解法和该方法,验证该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、工程近似算法和该文方法,进一步验证该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。

Fractional order derivative is a common expression of damper constitutive model, and it is difficult to obtain analytical solutions to dynamic responses of related structures. Efficient time domain numerical solving has important practical significance. Here, aiming at widely used Riemann-Liouville type fractional order damped structure, firstly, Adams-Moulton algorithm based on multi-step predictor-corrector mechanism was introduced to calculate fractional order derivative, and then an equivalent linear time-invariant dynamic system was constructed at each discrete time instant. Thirdly, combined with Newmark- β numerical integration method, explicit formulas for solving dynamic response at each instant were established to realize direct and rapid solving of dynamic response with high accuracy. Taking a single-DOF vibrator subjected to simple harmonic load/unit pulse as an example, the analytical solution, solutions obtained with two direct numerical solving methods, and the numerical solution obtained with the proposed method were compared to verify the high accuracy and stability of the proposed method. Finally, taking a multi-layer damped structure subjected to earthquake action as an example, the iterative numerical algorithm, the engineering approximation algorithm, and the proposed method were compared to further verify comprehensive advantages of the proposed method in terms of computational accuracy and efficiency, and reveal its potential in engineering applications.