Bifurcation analysis of a forced delay equation for machine tool vibrations

Published on Dec 1, 2019in Nonlinear Dynamics5.022
· DOI :10.1007/S11071-019-04984-W
János Lelkes1
Estimated H-index: 1
(BME: Budapest University of Technology and Economics),
Tamás Kalmár-Nagy18
Estimated H-index: 18
(BME: Budapest University of Technology and Economics)
Sources
Abstract
A machining tool can be subject to different kinds of excitations. The forcing may have external sources (such as rotating imbalance, misalignment of the workpiece or ultrasonic excitation), or it can arise from the cutting process itself (e.g., periodic chip formation). We investigate the classical one-degree-of-freedom tool vibration model, a delay-differential equation with quadratic and cubic nonlinearity, and periodic forcing. The method of multiple scales is used to derive the slow flow equations. Stability and bifurcation analysis of equilibria of the slow flow equations is presented. Analytical expressions are obtained for the saddle-node and Hopf bifurcation points. Bifurcation analysis is also carried out numerically. Sub- and supercritical Hopf, cusp, fold, generalized Hopf (Bautin), Bogdanov–Takens bifurcations are found. Limit cycle continuation is performed using MatCont. Local and global bifurcations are studied and illustrated with phase portraits and direct numerical integration of the original equation.
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References58
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#1János Lelkes (BME: Budapest University of Technology and Economics)H-Index: 1
#2Tamás Kalmár-Nagy (BME: Budapest University of Technology and Economics)H-Index: 18
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#1János Lelkes (BME: Budapest University of Technology and Economics)H-Index: 1
#2Tamás Kalmár-Nagy (BME: Budapest University of Technology and Economics)H-Index: 18
Abstract A machining tool can be subject to different kinds of excitations. The forcing may have external sources (such as rotating imbalance or misalignment of the workpiece) or it can arise from the cutting process itself (e.g. chip formation). We investigate the classical tool vibration model which is a delay-differential equation with a quadratic and cubic nonlinearity and periodic forcing. The method of multiple scales gave an excellent approximation of the solution. The resonance curves fo...
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#1Tamas G. Molnar (BME: Budapest University of Technology and Economics)H-Index: 9
#2Zoltan Dombovari (BME: Budapest University of Technology and Economics)H-Index: 18
Last. Gábor Stépán (BME: Budapest University of Technology and Economics)H-Index: 59
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The single-degree-of-freedom model of orthogonal cutting is investigated to study machine tool vibrations in the vicinity of a double Hopf bifurcation point. Centre manifold reduction and normal form calculations are performed to investigate the long-term dynamics of the cutting process. The normal form of the four-dimensional centre subsystem is derived analytically, and the possible topologies in the infinite-dimensional phase space of the system are revealed. It is shown that bistable paramet...
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#1Jinchen Ji (UTS: University of Technology, Sydney)H-Index: 24
#2Jin Zhou (SHU: Shanghai University)H-Index: 20
Abstract Two coexisting families of sub-harmonic resonances can be induced at different forcing frequencies in a time-delayed nonlinear system having quadratic nonlinearities. They occur in the region where two stable bifurcating periodic solutions coexist in the corresponding autonomous system following two-to-one resonant Hopf bifurcations of the trivial equilibrium. The forced response is found to demonstrate small- and large-amplitude quasi-periodic motion under the family of sub-harmonic re...
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#1Jinchen Ji (UTS: University of Technology, Sydney)H-Index: 24
#2Terry Brown (UTS: University of Technology, Sydney)H-Index: 6
A time-delayed quadratic nonlinear mechanical system can exhibit two coexisting stable bifurcating solutions (SBSs) after two-to-one resonant Hopf bifurcations occur in the corresponding autonomous time-delayed system. One SBS is of small-amplitude and has the Hopf bifurcation frequencies (HBFs), while the other is of large-amplitude and contains the shifted Hopf bifurcation frequencies (the shifted HBFs). When the forcing frequency is tuned to be the sum of two HBFs or the sum of two shifted HB...
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#1Karim M. Masri (SUNY: State University of New York System)H-Index: 3
#2Shuai Shao (SUNY: State University of New York System)H-Index: 8
Last. Mohammad I. Younis (SUNY: State University of New York System)H-Index: 37
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In this paper, we study the effect of a delayed feedback controller on stabilizing microelectromechanical systems (MEMS) resonators when undergoing large amplitude motion. A delayed feedback velocity controller is implemented through modifying the parallel plate electrostatic force used to excite the resonator into motion. A nonlinear single-degree-of-freedom model is used to simulate the resonator response. Long-time integration is used. Then, a finite difference technique to capture periodic m...
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#1Jinchen Ji (UTS: University of Technology, Sydney)H-Index: 24
Abstract Two stable bifurcating periodic solutions are numerically found to coexist in a time-delayed nonlinear oscillator by using different initial conditions, after the trivial equilibrium loses its stability via two-to-one resonant Hopf bifurcations. These two coexisting solutions have different amplitudes and frequency components with one having the frequencies of Hopf bifurcations while the other containing different frequencies from those of Hopf bifurcations. The dynamic interaction of t...
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#1Gizem S. OztepeH-Index: 3
#2S. Roy ChoudhuryH-Index: 2
Last. Ashish BhattH-Index: 2
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#1Shuning Deng (Hunan University)H-Index: 3
#2Jinchen Ji (UTS: University of Technology, Sydney)H-Index: 24
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Abstract Inherent time delays are often neglected in the modeling and dynamic analysis of centrifugal governor systems for the sake of simplicity, yet they can have a significant effect on the dynamic behavior of the governor systems. This paper investigates the effect of time-delay on the dynamics of a hexagonal centrifugal governor system through a comparative study on the stability and bifurcation of the equilibrium for the system with and without delay considered. It is found that the presen...
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