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Technical Brief

# Harmonic Differential Quadrature Method for Surface Location Error Prediction and Machining Parameter Optimization in Milling

[+] Author and Article Information
Ye Ding

State Key Laboratory of Mechanical System and Vibration,
School of Mechanical Engineering,
Shanghai Jiao Tong University,
Shanghai 200240, China
e-mail: y.ding@sjtu.edu.cn

XiaoJian Zhang, Han Ding

State Key Laboratory of Digital Manufacturing Equipment
and Technology,
Huazhong University of Science and Technology,
Wuhan 430074, China

1Corresponding author.

Contributed by the Manufacturing Engineering Division of ASME for publication in the JOURNAL OF MANUFACTURING SCIENCE AND ENGINEERING. Manuscript received May 15, 2014; final manuscript received August 5, 2014; published online December 12, 2014. Assoc. Editor: Tony Schmitz.

J. Manuf. Sci. Eng 137(2), 024501 (Apr 01, 2015) (6 pages) Paper No: MANU-14-1283; doi: 10.1115/1.4028279 History: Received May 15, 2014; Revised August 05, 2014; Online December 12, 2014

## Abstract

This paper presents a semi-analytical numerical method for surface location error (SLE) prediction in milling processes, governed by a time-periodic delay-differential equation (DDE) in state-space form. The time period is discretized as a set of sampling grid points. By using the harmonic differential quadrature method (DQM), the first-order derivative in the DDE is approximated by the linear sums of the state values at all the sampling grid points. On this basis, the DDE is discretized as a set of algebraic equations. A dynamic map can then be constructed to simultaneously determine the stability and the steady-state SLE of the milling process. To obtain optimal machining parameters, an optimization model based on the milling dynamics is formulated and an interior point penalty function method is employed to solve the problem. Experimentally validated examples are utilized to verify the accuracy and efficiency of the proposed approach.

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## Figures

Fig. 1

Schematic mechanical model of the two degrees of freedom milling process [33]

Fig. 2

Surface location error defined by the desired and the machined surfaces: (a) up-milling and (b) down-milling

Fig. 4

The Pareto optimal set in the criterion space by solving problem P3

Fig. 3

Comparison of the proposed method with the TFEA method for SLE (5% immersion down-milling)

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