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Technical Brief

Non-Normal Dynamic Analysis for Predicting Transient Milling Stability

[+] Author and Article Information
Qingzhen Bi

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

Xinzhi Wang

State Key Laboratory of Mechanical System and Vibration,
School of Mechanical Engineering,
Shanghai Jiao Tong University,
Shanghai 200240, China
e-mail: 1130209030@sjtu.edu.cn

Hua Chen

Marine Engineering College,
Dalian Maritime University,
Dalian 116026, China
e-mail: huachen204887@163.com

Limin Zhu

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

Han Ding

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

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT,AND CONTROL. Manuscript received July 19, 2017; final manuscript received December 19, 2017; published online March 7, 2018. Assoc. Editor: Jongeun Choi.

J. Dyn. Sys., Meas., Control 140(8), 084501 (Mar 07, 2018) (7 pages) Paper No: DS-17-1371; doi: 10.1115/1.4039033 History: Received July 19, 2017; Revised December 19, 2017

A transient milling stability analysis method is presented based on linear dynamics. Milling stability is usually analyzed based on asymptotic stability methods, such as the Floquet theory and the Nyquist stability criterion. These theories define stability that can return to equilibrium in an infinite time horizon under any initial condition. However, as a matter of fact, most dynamic processes in milling operations occur on a finite time scale. The transient vibration can be caused by some disturbance in practical milling process. Heavy transient vibrations were observed in existing works, though the machining parameters were selected in the stability zone determined by the asymptotic stability method. The strong transient vibrations will severely decrease the machining surface quality, especially for small workpieces in which the majority of machining process is executed in a short period of time. The analysis method of the transient milling stability is seldom studied, and only some experiments and conjectures can be found. Here the transient milling stability is defined as transient energy growth in a finite time horizon, and the prediction method of transient stability is proposed based on linear dynamics. The eigenvalues and non-normal eigenvectors of the Floquet transition matrix are all used to predict the transient milling stability, while only eigenvalues are employed in the traditional asymptotic stability analysis method. The transient stability is finally analyzed by taking the maximum vibration energy growth and the maximum duration time of transient energy growth in a finite time for optimal selection of processing parameters.

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Figures

Grahic Jump Location
Fig. 1

The recorded displacement of the cutting test [20]

Grahic Jump Location
Fig. 5

The finite time stability lobe diagram for case 1 with 5% radial immersion. Parametric study of the maximum duration time of transient energy growth as a function of the spindle speed and depth of cut. The white zones in the lobe diagrams represent the unstable machining status according to the Floquet theory and the contour levels represent the maximum duration time tmax. Perturbation energy, displacement, and initial displacement for the selected parameter points A (12,500 rpm, 0.9 mm) and B (16,000 rpm, 3.3 mm).

Grahic Jump Location
Fig. 6

The finite time stability lobe diagram for case 1 with 5% radial immersion. Parametric study of the maximum energy growth as a function of the spindle speed and depth of cut. The white zones in the lobe diagrams represent the unstable machining status according to the Floquet theory and the contour levels represent the maximum energy growth Gmax. Perturbation energy, displacement, and initial displacement for the selected parameter points C (12,000 rpm, 0.5 mm) and D (14,300 rpm, 2 mm).

Grahic Jump Location
Fig. 4

Geometric interpretation of transient growth [26]

Grahic Jump Location
Fig. 3

Energy amplification G(k) for 2DOFs milling model with 5% radial immersion

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Fig. 2

2DOF milling system

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