Fold Bifurcation in the State-Dependent Delay Model of Milling: Analytical and Numerical Solutions

2011 ◽  
Author(s):  
Daniel Bachrathy ◽  
Gabor Stepan
Keyword(s):  
Differential Equation ◽  
Time Delay ◽  
Delay Differential Equation ◽  
Milling Process ◽  
The State ◽  
Delay Model ◽  
State Dependent ◽  
Fold Bifurcation ◽  
State Dependent Delay ◽  
Delay Differential

The standard models of the milling process describe the surface regeneration effect by a delay-differential equation with constant time delay. In this study, an improved two degree of freedom model is presented for milling process where the regenerative effect is described by an improved state dependent time delay model. The model contains exact nonlinear screen functions describing the entrance and exit positions of the cutting edges of the milling tool. This model is valid in case of large amplitude forced vibrations close to the near-resonant spindle speeds. The periodic motions of this nonlinear system are calculated by a shooting method. The stability calculation is based on the linearization of the state-dependent delay differential equation around these periodic solutions by means of the semi-discretization method. The results are validated by an advanced numerical time domain simulation where the chip thickness is calculated by means of Boolean algebra.

State Dependent Regenerative Delay in Milling Processes

2005 ◽  
Author(s):  
Ta´mas Insperger ◽  
Ga´bor Ste´pa´n ◽  
Ferenc Hartung ◽  
Janos Turi
Keyword(s):  
Differential Equation ◽  
Time Delay ◽  
Delay Differential Equation ◽  
Linear Equations ◽  
Additional Term ◽  
Milling Process ◽  
Constant Time ◽  
State Dependency ◽  
State Dependent ◽  
Delay Differential

Traditional models of regenerative machine tool chatter use constant time delays assuming that the period between two subsequent cuts is a constant determined definitely by the spindle speed. These models result in delay-differential equations with constant time delay. If the vibrations of the tool relative to the workpiece are also included in the surface regeneration model, then the resulted time delay is not constant, but it depends on the actual and a delayed position of the tool. In this case, the governing equation is a delay-differential equation with state dependent time delay. Equations with state dependent delays can not be linearized in the traditional sense, but there exists linear equations that can be associated to them. This way, the local behavior of the system with state dependent delays can be investigated. In this study, a two degree of freedom model is presented for milling process. A thorough modeling of the regeneration effect results in the governing delay-differential equation with state dependent time delay. It is shown through the linearization of the nonlinear equation that an additional term arises in the linearized equation of motion due to the state-dependency of the time delay.

scholarly journals An analysis on the stability of a state dependent delay differential equation

2016 ◽  
Vol 14 (1) ◽  
pp. 425-435 ◽  
Author(s):  
Sertaç Erman ◽  
Ali Demir
Keyword(s):  
Differential Equation ◽  
Delay Differential Equation ◽  
Existence And Uniqueness ◽  
Uniqueness Of Solutions ◽  
Delay Term ◽  
State Dependent ◽  
State Dependent Delay ◽  
The Stability ◽  
Delay Differential ◽  
Equation With Delay

AbstractIn this paper, we present an analysis for the stability of a differential equation with state-dependent delay. We establish existence and uniqueness of solutions of differential equation with delay term $\tau (u(t)) = \frac{{a + bu(t)}}{{c + bu(t)}}.$ Moreover, we put the some restrictions for the positivity of delay term τ(u(t)) Based on the boundedness of delay term, we obtain stability criterion in terms of the parameters of the equation.

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