Response Sensitivity Analysis of the Dynamic Milling Process Based on the Numerical Integration Method

As one of the bases of gradient-based optimization algorithms, sensitivity analysis is usually required to calculate the derivatives of the system response with respect to the machining parameters. The most widely used approaches for sensitivity analysis are based on time-consuming numerical methods, such as finite difference methods. This paper presents a semi-analytical method for calculation of the sensitivity of the stability boundary in milling. After transforming the delay-differential equation with time-periodic coefficients governing the dynamic milling process into the integral form, the Floquet transition matrix is constructed by using the numerical integration method. Then, the analytical expressions of derivatives of the Floquet transition matrix with respect to the machining parameters are obtained. Thereafter, the classical analytical expression of the sensitivity of matrix eigenvalues is employed to calculate the sensitivity of the stability lobe diagram. The two-degree-of-freedom milling example illustrates the accuracy and efficiency of the proposed method. Compared with the existing methods, the unique merit of the proposed method is that it can be used for analytically computing the sensitivity of the stability boundary in milling, without employing any finite difference methods. Therefore, the high accuracy and high efficiency are both achieved. The proposed method can serve as an effective tool for machining parameter optimization and uncertainty analysis in high-speed milling.

Variable-step integration method for milling chatter stability prediction with multiple delays

This paper analyzes the stability of milling with variable pitch cutter and tool runout cases characterized by multiple delays,and proposes a new variable-step numerical integration method for efficient and accurate stability prediction. The variable-step technique is emphasized here to expand the numerical integration method,especially for the low radial immersion cases with multiple delays. First,the calculation accuracy of the numerical integration method is discussed and the variable-step algorithm is developed for milling stability prediction for single-delay and multiple-delay cases,respectively. The milling stability with variable pitch cutter is analyzed and the result is compared with those predicted with the frequency domain method and the improved full-discretization method. The influence of the runout effect on the stability boundary is investigated by the presented method. The numerical simulation shows that the cutter runout effect increases the stability boundary,and the increasing stability limit is verified by the milling chatter experimental results in the previous research. The numerical and experiment results verify the validity of the proposed method.

Fast and accurate adaptive collocation iteration method for orbit dynamic problems

For over half a century,numerical integration methods based on finite difference,such as the Runge-Kutta method and the Euler method,have been popular and widely used for solving orbit dynamic problems.In general,a small integration step size is always required to suppress the increase of the accumulated computation error,which leads to a relatively slow computation speed.Recently,a collocation iteration method,approximating the solutions of orbit dynamic problems iteratively,has been developed.This method achieves high computation accuracy with extremely large step size.Although efficient,the collocation iteration method suffers from two limitations:(A)the computational error limit of the approximate solution is not clear;(B)extensive trials and errors are always required in tuning parameters.To overcome these problems,the influence mechanism of how the dynamic problems and parameters affect the error limit of the collocation iteration method is explored.On this basis,a parameter adjustment method known as the“polishing method”is proposed to improve the computation speed.The method proposed is demonstrated in three typical orbit dynamic problems in aerospace engineering:a low Earth orbit propagation problem,a Molniya orbit propagation problem,and a geostationary orbit propagation problem.Numerical simulations show that the proposed polishing method is faster and more accurate than the finite-difference-based method and the most advanced collocation iteration method.

Numerical quadrature for singular and near-singular integrals of boundary element method and its applications in large-scale acoustic problems

The numerical quadrature methods for dealing with the problems of singular and near-singular integrals caused by Burton-Miller method are proposed, by which the conventional and fast multipole BEMs （boundary element methods） for 3D acoustic problems based on constant elements are improved. To solve the problem of singular integrals, a Hadamard finite-part integral method is presented, which is a simplified combination of the methods proposed by Kirkup and Wolf. The problem of near-singular integrals is overcome by the simple method of polar transformation and the more complex method of PART （Projection and Angular ＆ Radial Transformation）. The effectiveness of these methods for solving the singular and near-singular problems is validated through comparing with the results computed by the analytical method and/or the commercial software LMS Virtual.Lab. In addition, the influence of the near-singular integral problem on the computational precisions is analyzed by computing the errors relative to the exact solution. The computational complexities of the conventional and fast multipole BEM are analyzed and compared through numerical computations. A large-scale acoustic scattering problem, whose degree of freedoms is about 340,000, is implemented successfully. The results show that, the near singularity is primarily introduced by the hyper-singular kernel, and has great influences on the precision of the solution. The precision of fast multipole BEM is the same as conventional BEM, but the computational complexities are much lower.

Numerical simulation of a cabin ventilation subsystem in a space station oriented real-time system

An environment control and life support system（ECLSS） is an important system in a space station. The ECLSS is a typical complex system, and the real-time simulation technology can help to accelerate its research process by using distributed hardware in a loop simulation system. An implicit fixed time step numerical integration method is recommended for a real-time simulation system with time-varying parameters. However, its computational efficiency is too low to satisfy the real-time data interaction, especially for the complex ECLSS system running on a PC cluster. The instability problem of an explicit method strongly limits its application in the ECLSS real-time simulation although it has a high computational efficiency. This paper proposes an improved numerical simulation method to overcome the instability problem based on the explicit Euler method. A temperature and humidity control subsystem（THCS） is firstly established, and its numerical stability is analyzed by using the eigenvalue estimation theory. Furthermore, an adaptive operator is proposed to avoid the potential instability problem. The stability and accuracy of the proposed method are investigated carefully. Simulation results show that this proposed method can provide a good way for some complex time-variant systems to run their real-time simulation on a PC cluster.

An accurate RLGC circuit model for dual tapered TSV structure

A fast RLGC circuit model with analytical expression is proposed for the dual tapered through-silicon via （TSV） structure in three-dimensional integrated circuits under different slope angles at the wide frequency region. By describing the electrical characteristics of the dual tapered TSV structure, the RLGC parameters are extracted based on the numerical integration method. The RLGC model includes metal resistance, metal inductance, substrate resistance, outer inductance with skin effect and eddy effect taken into account. The proposed analytical model is verified to be nearly as accurate as the Q3D extractor but more efficient.