Derived Nodes Approach for Improving Accuracy of Machining Stability Prediction
Derived Nodes Approach for Improving Accuracy of Machining Stability Prediction
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DOI：
10.1115/1.4038947
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发表时间：
2018-06
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影响因子：
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通讯作者：
Le Cao;Xiaoming Zhang;Tao Huang;H. Ding
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文献类型：
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作者：
Le Cao;Xiaoming Zhang;Tao Huang;H. Ding
加工过程动力学可以用状态空间延迟微分方程（DDE）来描述。为了对过程稳定性进行数值预测，通常采用不同的分段多项式插值方法将连续的动态微分方程离散为一组线性离散方程。离散逼近的精度通常取决于如何处理分段多项式。然而，由于龙格现象，稳定性预测精度的提高并不总是由高阶多项式保证。在这项研究中，分段多项式的导数连续的联合节点被考虑。为了提高微分方程的离散化精度，我们提出了一种用于状态变量插值逼近的导出节点递推估计方法。两种不同的时间离散化方法，即，以二阶全离散化和状态空间时态有限方法为例，说明了应用所提出的方法提高精度的有效性。数值仿真结果表明，该方法在稳定叶的精度和收敛速度上都有很大的提高，与以往的同阶插值多项式方法相比，具有更高的收敛速度.
Machining process dynamics can be described by state-space delayed differential equations (DDEs). To numerically predict the process stability, diverse piecewise polynomial interpolation is often utilized to discretize the continuous DDEs into a set of linear discrete equations. The accuracy of discrete approximation of the DDEs generally depends on how to deal with the piecewise polynomials. However, the improvement of the stability prediction accuracy cannot be always guaranteed by higher-order polynomials due to the Runge phenomenon. In this study, the piecewise polynomials with derivative-continuous at joint nodes are taken into consideration. We develop a recursive estimation of derived nodes for interpolation approximation of the state variables, so as to improve the discretization accuracy of the DDEs. Two different temporal discretization methods, i.e., second-order full-discretization and state-space temporal finite methods, are taken as demonstrations to illustrate the effectiveness of applying the proposed approach for accuracy improvement. Numerical simulations prove that the proposed approach brings a great improvement on the accuracy of the stability lobes, as well as the rate of convergence, compared to the previous recorded ones with the same order of interpolation polynomials.