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numerical integration method for Stability Analysis of Milling With Variable Spindle Speeds

JOURNAL OF VIBRATION AND ACOUSTICS-TRANSACTIONS OF THE ASME 2016年第1期138卷 011010-011010页

作者： Ding, Ye Niu, Jinbo Zhu, Limin Ding, Han Shanghai Jiao Tong Univ Gas Turbine Res Inst Shanghai 200240 Peoples R China Shanghai Jiao Tong Univ Sch Mech Engn State Key Lab Mech Syst & Vibrat Shanghai 200240 Peoples R China Huazhong Univ Sci & Technol State Key Lab Digital Mfg Equipment & Technol Wuhan 430074 Peoples R China

A semi-analytical method is presented in this paper for stability analysis of milling with a variable spindle speed (VSS), periodically modulated around a nominal spindle speed. Taking the regenerative effect into account, the dynamics of the VSS milling is governed by a delay-differential equation (DDE) with time-periodic coefficients and a time-varying delay. By reformulating the original DDE in an integral-equation form, one time period is divided into a series of subintervals. With the aid of numerical integrations, the transition matrix over one time period is then obtained to determine the milling stability by using Floquet theory. On this basis, the stability lobes consisting of critical machining parameters can be calculated. Unlike the constant spindle speed (CSS) milling, the time delay for the VSS is determined by an integral transcendental equation which is accurately calculated with an ordinary differential equation (ODE) based method instead of the formerly adopted approximation expressions. The proposed numerical integration method is verified with high computational efficiency and accuracy by comparing with other methods via a two-degree-of-freedom milling example. With the proposed method, this paper details the influence of modulation parameters on stability diagrams for the VSS milling.

关键词： milling stability variable spindle speed delay-differential equation numerical integration method Floquet theory

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Response Sensitivity Analysis of the Dynamic Milling Process Based on the numerical integration method

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Chinese Journal of Mechanical Engineering 2012年第5期25卷 940-946页

作者： DING Ye ZHU Limin ZHANG Xiaojian DING Han State Key Laboratory of Mechanical System and Vibration Shanghai Jiao Tong University Shanghai 200240 China State Key Laboratory of Digital Manufacturing Equipment and Technology Huazhong University of Science and Technology Wuhan 430074 China

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.

关键词： milling stability sensitivity of the stability boundary numerical integration method

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An improved numerical integration method to predict the milling stability based on the Lagrange interpolation scheme

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INTERNATIONAL JOURNAL OF ADVANCED MANUFACTURING TECHNOLOGY 2021年第7-8期116卷 2111-2123页

作者： Xia, Yan Wan, Yi Luo, Xichun Liu, Zhanqiang Song, Qinghua Shandong Univ Sch Mech Engn Key Lab High Efficiency & Clean Mech Manufacture Minist Educ Jinan 250061 Peoples R China Shandong Univ Natl Demonstrat Ctr Expt Mech Engn Educ Jinan 250061 Peoples R China Univ Strathclyde Ctr Precis Mfg DMEM Glasgow G1 1XJ Lanark Scotland

To predict the milling stability accurately and efficiently, an improved numerical integration method (INIM) is proposed based on the Lagrange interpolation scheme. First, the milling dynamic model considering the regenerative chatter can be described as a delay linear differential equation. The tooth passing period of the milling cutter is divided into the free and forced vibration stages. Then, the forced vibration stage is equally discretized, and the INIMs are built based on the Lagrange interpolation scheme within the discretized intervals to construct the state transition matrix. Finally, the convergence rates and the stability lobes for the benchmark milling systems are calculated and discussed by using the proposed INIMs and the existing methods, respectively. The comparison results reveal that the proposed second-order INIM shows higher computational efficiency and accuracy compared with the related discretization methods, and in the meantime, it is more accurate under just slightly loss of the time cost compared with the existing NIMs.

关键词： Milling stability numerical integration method Lagrange interpolation scheme Convergence rate Stability lobe diagram

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An improved numerical integration method for predicting milling stability with varying time delay

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INTERNATIONAL JOURNAL OF ADVANCED MANUFACTURING TECHNOLOGY 2013年第9-12期68卷 1967-1976页

作者： Liang, Xin-Guang Yao, Zhen-Qiang Luo, Lei Hu, Jun Shanghai Jiao Tong Univ Sch Mech Engn Shanghai 200240 Peoples R China Shanghai Jiao Tong Univ State Key Lab Mech Syst & Vibrat Shanghai 200240 Peoples R China

Varying time delay (VTD) plays an important role in governing the state of chatter vibration in low radial immersion milling. Meanwhile, numerical integration method (NIM) has been proven a highly efficient method in predicting milling stability lobe diagram (SLD) where a constant time delay (CTD) should be assumed as one tooth passing period. In this paper, an improved numerical integration method (INIM) is proposed, focusing on how to extend this efficient method to the condition that VTD has to be considered in low radial immersion milling. Based on an equivalent description of VTD, an offset matrix is constructed to modify the Floquet transition matrix. Thus, the distributions of SLD and system oscillations can be updated simultaneously with the effect of VTD. Then, a chatter excitation test (CET) method for low radial immersion milling is introduced so that SLD can be extracted directly by analyzing the machined surface structure, and the lobe-drifting effect of the SLD can be revealed visually. Comparing the extracted SLDs with the predictions, the accuracy of the proposed INIM can be validated due to the capability of bridging the gap between the lobe-drifting effect and VTD, while the traditional NIM with CTD is powerless. The improvement of the proposed INIM has also been validated by comparing with a peripheral milling test provided in a published literature.

关键词： numerical integration method Milling stability Varying time delay Low radial immersion Lobe-drifting effect

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Updated numerical integration method for stability calculation of Mathieu equation with various time delays (vol 87, pg 2077, 2017)

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NONLINEAR DYNAMICS 2017年第4期87卷 2097-2098页

作者： Zhang, Xiao Jian Xiong, Cai Hua Ding, Ye Ding, Han Huazhong Univ Sci & Technol Sch Mech Sci & Engn State Key Lab Digital Mfg Equipment & Technol Wuhan 430074 Peoples R China Shanghai Jiao Tong Univ Sch Mech Engn State Key Lab Mech Syst & Vibrat Shanghai 200240 Peoples R China

The determination of the stability of systems with time delays is of high importance in many industrial and research applications. In this study, we improve the numerical integration method (NIM) by using the Lagrange form interpolating polynomial to approximate the delayed terms and construct a periodic discrete dynamical map for the damped Mathieu equation with time delays. Hence, we can obtain the Floquet transition matrices without matrix multiplication, which can reduce calculation time when the matrix inversed has a small bandwidth according to the computation based on MATLAB. To compare the calculation efficiency and computational accuracy between the updated and original methods, we compare the stability charts calculated by using three NIM methods for a damped Mathieu equation with multiple delays. To further confirm the efficiency of the presented method, we calculate the stability of the Mathieu equation with distributed delays and time-periodic delays, and then we compare the computational accuracy and calculation time with those of the semi-discretization method.

关键词： Stability calculation Mathieu equation Multiple delays Distributed delays Time-periodic delays numerical integration method

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An improved numerical integration method for prediction of milling stability using the Lagrange-Simpson interpolation scheme

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INTERNATIONAL JOURNAL OF ADVANCED MANUFACTURING TECHNOLOGY 2022年第11-12期120卷 8105-8115页

作者： Xia, Yan Wan, Yi Su, Guosheng Du, Jin Zhang, Peirong Xu, Chonghai Qilu Univ Technol Sch Mech Engn Shandong Acad Sci Jinan 250353 Peoples R China Shandong Univ Sch Mech Engn Key Lab High Efficiency & Clean Mech Manufacture Minist Educ Jinan 250061 Peoples R China

The stability prediction is usually used to avoid the unstable machining in milling process. According to the Lagrange-Simpson hybrid interpolation scheme, this paper improves a numerical integration method (NIM) to perform the milling chatter prediction accurately and efficiently. Firstly, the higher-order numerical integral formulas (NIFs) are constructed based on the Lagrange polynomial. Thus, the third-order and fourth-order NIMs are built and investigated respectively. Then, to improve the calculated performance of the NIMs, the Simpson scheme is introduced to decrease the local discretization error. Finally, the comparisons among the built NIMs and the existing discretization methods are carried out by calculating the convergence rate and the stability boundaries. Compared to the third-order NIM and the third-order full-discretization method, the proposed second-order Lagrange-Simpson NIM shows the better computational performance.

关键词： Milling stability prediction numerical integration method Polynomial interpolation Simpson rule Local discretization error

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A monotone numerical integration method for mean-variance portfolio optimization under jump-diffusion models

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MATHEMATICS AND COMPUTERS IN SIMULATION 2024年 219卷 112-140页

作者： Zhang, Hanwen Dang, Duy-Minh Univ Queensland Sch Math & Phys Brisbane 4072 Australia

We develop a highly efficient, easy-to-implement, and strictly monotone numerical integration method for Mean-Variance (MV) portfolio optimization. This method proves very efficient in realistic contexts, which involve jump-diffusion dynamics of the underlying controlled processes, discrete rebalancing, and the application of investment constraints, namely no-bankruptcy and leverage.A crucial element of the MV portfolio optimization formulation over each rebalancing interval is a convolution integral, which involves a conditional density of the logarithm of the amount invested in the risky asset. Using a known closed-form expression for the Fourier transform of this conditional density, we derive an infinite series representation for the conditional density where each term is strictly positive and explicitly computable. As a result, the convolution integral can be readily approximated through a monotone integration scheme, such as a composite quadrature rule typically available in most programming languages. The computational complexity of our proposed method is an order of magnitude lower than that of existing monotone finite difference methods. To further enhance efficiency, we propose an implementation of this monotone integration scheme via Fast Fourier Transforms, exploiting the Toeplitz matrix *** proposed monotone numerical integration scheme is proven to be both ��infinity-stable and pointwise consistent, and we rigorously establish its pointwise convergence to the unique solution of the MV portfolio optimization problem. We also intuitively demonstrate that, as the rebalancing time interval approaches zero, the proposed scheme converges to a continuously observed impulse control formulation for MV optimization expressed as a Hamilton- Jacobi-Bellman Quasi-Variational Inequality. numerical results show remarkable agreement with benchmark solutions obtained through finite differences and Monte Carlo simulation, underscoring the effectiveness of

关键词： Mean-variance Portfolio optimization Monotonicity numerical integration method Jump-diffusion Discrete rebalancing

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An Efficient Dimension-Adaptive numerical integration method for Stochastic Dynamic Analysis of Structures with Uncertain Parameters

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INTERNATIONAL JOURNAL OF STRUCTURAL STABILITY AND DYNAMICS 2021年第3期21卷

作者： Yu, Helu Wang, Bin Gao, Zongyu Li, Yongle Southwest Jiaotong Univ Dept Bridge Engn Chengdu 610031 Peoples R China Wind Engn Key Lab Sichuan Prov Chengdu 610031 Peoples R China China Railway Major Bridge Reconnaissance & Desig Wuhan 430050 Peoples R China

This paper presents a novel dimension-adaptive numerical integration method for dynamic analysis of structures with stochastic parameters subjected to deterministic excitations. First, an efficient dimension-adaptive algorithm is proposed to detect the importance of each random parameter involved in the structural model, based on which the quadrature nodes used for numerical integration can be collocated more reasonably. Then, the Gaussian quadrature formulas are used to evaluate the structural response statistics. To further improve the robustness and efficiency of the proposed method, the dimension-adaptive integration is only used to calculate the structural displacement response statistics. The velocity and acceleration response statistics are further evaluated using the finite difference formulas based on the concept of stochastic difference. Such a strategy is especially attractive when evaluating the response statistics of the derivative processes requires more quadrature nodes than that of the original process. Finally, two numerical examples encountered in civil engineering, including a shear frame with stochastic parameters subjected to a seismic ground motion and an Euler beam with unidimensional stochastic field of material properties (discretized via the Karhunen-Loeve expansion) subjected to a moving load are studied to illustrate the performance of the proposed method. Via the numerical results, the accuracy and efficiency of the proposed method are verified.

关键词： Stochastic structural dynamics parametric uncertainty numerical integration method dimensional adaptability finite difference

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Research on numerical integration methods Based on LC Binary Equivalent Switch Model in Real-Time Simulation of Power Electronic System 8th

Research on Numerical Integration Methods Based on LC Binary...

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8th International Conference on Life System Modeling and Simulation, LSMS 2024 and 8th International Conference on Intelligent Computing for Sustainable Energy and Environment, ICSEE 2024

作者： Liang, Bowen Chen, Guochu School of Electrical Engineering Shanghai Dianji University Shanghai China

ISBN: (纸本)9789819602247

In real-time simulation of power electronic system, the modeling method and numerical integration method are key factors affecting simulation performance. The LC binary method significantly reduces hardware resource utilization and FPGA workload. However, different integration methods have varying error accuracy and convergence. This paper provides a detailed analysis of the impact of different numerical integration methods on the system and proposes an effective and versatile real-time simulation method for multi-switch circuit, demonstrated through the application case of a single-phase inverter based on LC filtering. Comparative tests with Simulink model under a real-time simulation step size of 1μs show that the proposed method achieves a relative error of less than ± 1% and consumes hardware resources by no more than 30%. This approach demonstrates practical value for efficient real-time simulation of multi-switch power electronic system. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2025.

关键词： LC Binary Equivalent Switch Model numerical integration method Real-Time Simulation

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

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Chinese Journal of Aeronautics 2017年第6期30卷 1809-1817页

作者： Zezheng QIU Liping PANG Guoxiang LI Helin ZHANG School of Aeronautic Science and Engineering Beihang UniversityBeijing 100083China Aviation Key Laboratory of Science and Technology on Aero Electromechanical System Integration Nanjing Engineering Institute of Aircraft SystemsNanjing 211102China

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.

关键词： numerical integration method Real-time simulation Stability THCS Time-variant system

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