Approximating piecewise nonlinearities in dynamic systems with sigmoid functions: advantages and limitations
Approximating piecewise nonlinearities in dynamic systems with sigmoid functions: advantages and limitations
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DOI：
10.1007/s11071-023-08293-1
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发表时间：
2023-02
影响因子：
5.6
通讯作者：
C. Martinelli;Andrea Coraddu;A. Cammarano
中科院分区：
文献类型：
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作者：
C. Martinelli;Andrea Coraddu;A. Cammarano
在工业领域，对轻质结构日益严格的要求正在暴露机械系统最终的非线性本质。对于具有移动部件和松动固定装置（显示分段刚度行为）的系统来说尤其如此。然而，具有理想分段数学特性的系统的数值解与耗时的过程和高计算负担相关。平滑函数可以方便地简化此类系统的数学形式，但很少有研究评估它们对多自由度系统机械响应的影响。为了研究这个问题，通过数值延拓和数值积分程序研究了具有软分段约束的微阻尼机械二自由度系统。采用Sigmoid函数来逼近约束条件，并通过将逼近系统的结果与理想分段对应系统的结果进行比较来探讨这种逼近的效果。数值结果表明，当采用上述两种技术时，Sigmoid 函数可以正确捕捉所提出系统的非常复杂的动态。此外，在近似情况下观察到计算负担的减少以及数值鲁棒性的增加。
In the industry field, the increasingly stringent requirements of lightweight structures are exposing the ultimately nonlinear nature of mechanical systems. This is extremely true for systems with moving parts and loose fixtures which show piecewise stiffness behaviours. Nevertheless, the numerical solution of systems with ideal piecewise mathematical characteristics is associated with time-consuming procedures and a high computational burden. Smoothing functions can conveniently simplify the mathematical form of such systems, but little research has been carried out to evaluate their effect on the mechanical response of multi-degree-of-freedom systems. To investigate this problem, a slightly damped mechanical two-degree-of-freedom system with soft piecewise constraints is studied via numerical continuation and numerical integration procedures. Sigmoid functions are adopted to approximate the constraints, and the effect of such approximation is explored by comparing the results of the approximate system with the ones of the ideal piecewise counter-part. The numerical results show that the sigmoid functions can correctly catch the very complex dynamics of the proposed system when both the above-mentioned techniques are adopted. Moreover, a reduction in the computational burden, as well as an increase in numerical robustness, is observed in the approximate case.