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Technical Brief

Differential Quadrature Method for Stability Analysis of Dynamic Systems With Multiple Delays: Application to Simultaneous Machining Operations

[+] 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

Jinbo Niu

State Key Laboratory of Mechanical System
and Vibration,
School of Mechanical Engineering,
Shanghai Jiao Tong University,
Shanghai 200240, China
e-mail: niujinbo@hotmail.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 Digital Manufacturing
Equipment and Technology,
Huazhong University of Science and Technology,
Wuhan 430074, China
e-mail: dinghan@mail.hust.edu.cn

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received March 19, 2014; final manuscript received October 11, 2014; published online November 14, 2014. Assoc. Editor: Philip Bayly.

J. Vib. Acoust 137(2), 024501 (Apr 01, 2015) (8 pages) Paper No: VIB-14-1086; doi: 10.1115/1.4028832 History: Received March 19, 2014; Revised October 11, 2014; Online November 14, 2014

This paper proposes a semi-analytical solution via the differential quadrature method (DQM) for stability analysis of linear multi-degree-of-freedom second-order systems with multiple delays. The vibration systems are reformulated as a delayed differential equation (DDE) in state–space form. With the derivative of the state vector with respect to time at an arbitrary discrete-time point being expressed as a linear weighted sum of the values of the state vector, the original DDE is approximated by a set of algebraic equations, leading to the Floquet transition matrix. Based on Floquet theory, the stability of the systems is then determined by checking the eigenvalues of the transition matrix. The computational accuracy and efficiency are demonstrated through comparison with existing methods via numerical examples. As an application, the proposed method is employed to predict chatter stability in simultaneous machining operations, providing a reference for the choice of machining parameters.

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Figures

Grahic Jump Location
Fig. 1

Stability charts of the Mathieu equation in Eq. (3) with the DQM and first-SDM

Grahic Jump Location
Fig. 2

Algorithm comparison of DQM and first-SDM, in terms of convergence rate and computational efficiency. Boundary lines in gray color serve as reference by DQM with n=80.

Grahic Jump Location
Fig. 3

Schematic diagram of multicutter turning

Grahic Jump Location
Fig. 4

Stability diagram of the multicutter turning system. Symbols (•) and (×) indicate stable points and unstable points, respectively.

Grahic Jump Location
Fig. 5

Displacement response at points F, G, H, and I in Fig. 4. z1 and z2 represent the vibratory displacement of Tool 1 and Tool 2, respectively.

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