State-dependent distributed-delay model of orthogonal cutting

Tamás G. Molnár, T. Insperger, G. Stépán

Research output: Contribution to journalArticle

4 Citations (Scopus)

Abstract

In this paper, we present a model of turning operations with state-dependent distributed time delay. We apply the theory of regenerative machine tool chatter and describe the dynamics of the tool-workpiece system during cutting by delay differential equations. We model the cutting force as the resultant of a force system distributed along the rake face of the tool, which results in a short distributed delay in the governing equation superimposed on the large regenerative delay. According to the literature on stress distribution along the rake face, the length of the chip–tool interface, where the distributed cutting force system is acting, is function of the chip thickness, which depends on the vibrations of the tool-workpiece system due to the regenerative effect. Therefore, the additional short delay is state dependent. It is shown that involving state-dependent delay in the model does not affect linear stability properties, but does affect the nonlinear dynamics of the cutting process. Namely, the sense of the Hopf bifurcation along the stability boundaries may turn from sub- to supercritical at certain spindle speed regions.

Original languageEnglish
Pages (from-to)1-10
Number of pages10
JournalNonlinear Dynamics
DOIs
Publication statusAccepted/In press - Dec 24 2015

Fingerprint

State-dependent Delay
Distributed Delay
Cutting Force
Face
Distributed Time Delay
Chatter
Dependent
Machine Tool
Linear Stability
Stress Distribution
Delay Differential Equations
Hopf Bifurcation
Nonlinear Dynamics
Hopf bifurcation
Distributed Systems
Governing equation
Chip
Vibration
Machine tools
Model

Keywords

  • Delay differential equation
  • Distributed delay
  • Hopf bifurcation
  • Metal cutting
  • State-dependent delay
  • Turning

ASJC Scopus subject areas

  • Applied Mathematics
  • Mechanical Engineering
  • Aerospace Engineering
  • Ocean Engineering
  • Electrical and Electronic Engineering
  • Control and Systems Engineering

Cite this

State-dependent distributed-delay model of orthogonal cutting. / Molnár, Tamás G.; Insperger, T.; Stépán, G.

In: Nonlinear Dynamics, 24.12.2015, p. 1-10.

Research output: Contribution to journalArticle

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N2 - In this paper, we present a model of turning operations with state-dependent distributed time delay. We apply the theory of regenerative machine tool chatter and describe the dynamics of the tool-workpiece system during cutting by delay differential equations. We model the cutting force as the resultant of a force system distributed along the rake face of the tool, which results in a short distributed delay in the governing equation superimposed on the large regenerative delay. According to the literature on stress distribution along the rake face, the length of the chip–tool interface, where the distributed cutting force system is acting, is function of the chip thickness, which depends on the vibrations of the tool-workpiece system due to the regenerative effect. Therefore, the additional short delay is state dependent. It is shown that involving state-dependent delay in the model does not affect linear stability properties, but does affect the nonlinear dynamics of the cutting process. Namely, the sense of the Hopf bifurcation along the stability boundaries may turn from sub- to supercritical at certain spindle speed regions.

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