scholarly journals Dynamics of a delay differential equation with multiple state-dependent delays

2012 ◽  
Vol 32 (8) ◽  
pp. 2701-2727 ◽  
Author(s):  
A. R. Humphries ◽  
◽  
O. A. DeMasi ◽  
F. M. G. Magpantay ◽  
F. Upham
Keyword(s):  
Differential Equation ◽  
Delay Differential Equation ◽  
State Dependent ◽  
Multiple State ◽  
Delay Differential

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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