scholarly journals State-Dependent, Non-Smooth Model of Chatter Vibrations in Turning

2015 ◽  
Author(s):  
Dávid Lehotzky ◽  
Tamás Insperger ◽  
Gábor Stépán
Keyword(s):  
Orthogonal Cutting ◽  
Tangential Direction ◽  
Degree Of Freedom ◽  
Global Dynamics ◽  
State Dependency ◽  
Modeling And Analysis ◽  
State Dependent ◽  
Smooth Model ◽  
The One ◽  
Delay Differential

This paper deals with the modeling and analysis of the cutting tool’s global dynamics in the orthogonal cutting process of turning operations considering the effect of state dependency and fly-over in one model. In particular, the one-degree-of-freedom non-smooth model, presented by Wahi and Chatterjee in 2008, is extended by the consideration of vibrations in the direction perpendicular to the feed velocity. This results in the state-dependency of the model and gives an additional direction in which fly-over can occur. The constructed mathematical model consists of a nonlinear PDE, which describes the evolution of the surface height of the workpiece and a two-degree-of-freedom ODE, which governs the motion of the tool. The PDE is connected to the solution of the ODE by a non-local, non-smooth boundary condition. For the case when the tool is within the cut, this model gives the conventional model of turning governed by delay-differential equations with state-dependent delays. In order to study the effect of vibrations in the tangential direction numerical simulations are carried out and their results are compared to the model presented by Wahi and Chatterjee (2008).

Modeling and analysis of a predator–prey model with state-dependent delay

2018 ◽  
Vol 11 (02) ◽  
pp. 1850026 ◽  
Author(s):  
Yunfei Lv ◽  
Yongzhen Pei ◽  
Rong Yuan
Keyword(s):  
Immature Stage ◽  
Global Dynamics ◽  
Modeling And Analysis ◽  
Predator Prey ◽  
Interior Equilibrium ◽  
Predator Prey Model ◽  
Prey Model ◽  
State Dependent ◽  
State Dependent Delay ◽  
Threshold Amount

We propose and study a predator–prey model with state-dependent delay where the prey population is assumed to have an age structure. The state-dependent delay appears due to the mature condition that the prey must spend an amount of time in the immature stage sufficient to accumulate a threshold amount of food. We perform a qualitative analysis of the solutions, which includes studying positivity and boundedness, existence and local stability of equilibria. For the global dynamics of the system, we discuss an attracting region which is determined by solutions, and the region collapses to the interior equilibrium in the constant delay case.

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.

Global Stability and Hopf Bifurcation for a Stage Structured Model with Competition for Food

2021 ◽  
Vol 31 (10) ◽  
pp. 2150145
Author(s):  
Yunfei Lv ◽  
Yongzhen Pei ◽  
Rong Yuan
Keyword(s):  
Differential Equation ◽  
Hopf Bifurcation ◽  
Global Stability ◽  
Delay Differential Equation ◽  
Complete Analysis ◽  
Global Dynamics ◽  
Structured Model ◽  
State Dependent ◽  
State Dependent Delay ◽  
Delay Differential

Considering the mature condition of any individual to have eaten a specific amount of food during the entire period that it can spend at its immature stage, we propose a size-structured model by a first-order quasi-linear partial differential equation. The model can be firstly reduced to a single state-dependent delay differential equation and then to a constant delay differential equation. The state-dependent delay represents intra-specific competition among individuals for limited food resources. A complete analysis of the global dynamics on the positivity and boundedness of solutions, global stability for each equilibrium and Hopf bifurcation is carried out. Our results imply that the delay leads to instability that is shown by a simple example of a certain structured population model.

Stability analysis of uncertain delay differential equations

2021 ◽  
pp. 1-11
Author(s):  
Jian Wang ◽  
Yuanguo Zhu
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Liu Process ◽  
Uncertain Dynamical System ◽  
Analytic Expression ◽  
Functional Differential ◽  
The One ◽  
Delay Differential

Uncertain delay differential equation is a class of functional differential equations driven by Liu process. It is an important model to describe the evolution process of uncertain dynamical system. In this paper, on the one hand, the analytic expression of a class of linear uncertain delay differential equations are investigated. On the other hand, the new sufficient conditions for uncertain delay differential equations being stable in measure and in mean are presented by using retarded-type Gronwall inequality. Several examples show that our stability conditions are superior to the existing results.

NUMERICAL BIFURCATION ANALYSIS OF DIFFERENTIAL EQUATIONS WITH STATE-DEPENDENT DELAY

2001 ◽  
Vol 11 (03) ◽  
pp. 737-753 ◽  
Author(s):  
TATYANA LUZYANINA ◽  
KOEN ENGELBORGHS ◽  
DIRK ROOSE
Keyword(s):  
Differential Equations ◽  
Bifurcation Analysis ◽  
Delay Differential Equations ◽  
Constant Delay ◽  
State Dependent ◽  
State Dependent Delay ◽  
Numerical Bifurcation ◽  
Delay Differential ◽  
Steady State Solutions ◽  
Numerical Bifurcation Analysis

In this paper we apply existing numerical methods for bifurcation analysis of delay differential equations with constant delay to equations with state-dependent delay. In particular, we study the computation, continuation and stability analysis of steady state solutions and periodic solutions. We collect the relevant theory and describe open theoretical problems in the context of bifurcation analysis. We present computational results for two examples and compare with analytical results whenever possible.

The Effect of Air Injector on the Performance of Airlift

2012 ◽  
Vol 516-517 ◽  
pp. 1022-1027 ◽  
Author(s):  
Dong Hu ◽  
Chuan Lin Tang ◽  
Feng Hua Zhang
Keyword(s):  
Flow Rate ◽  
Marked Effect ◽  
Pipe Wall ◽  
Tangential Direction ◽  
Air Injection ◽  
Air Flow Rate ◽  
Injection Angle ◽  
Injection Methods ◽  
The One ◽  
Air Lift

In order to investigate the air injection method on the performance of an airlift. For this purpose an air lift system with a riser 2000 mm long and 80 mm in diameter, was designed and tested. Seven different air injection methods were used at a constant submergence. The experimental results showed a marked effect on the airlift performance when operated with different air injection methods. The arrangement of five nozzles gives the best performance, and the one nozzle is the worst. Although the injection angle has a little effect on the airlift performance, but view the general conclusions as a whole, the best lifting efficiency can be obtained when the angle of the nozzle placed along the tangential direction of pipe wall is equal to 10º at a given air flow rate QG =37m3/h.

scholarly journals On the optimal control force applied to tuned mass dampers for multi-degree-of-freedom systems

2004 ◽  
Vol 26 (1) ◽  
pp. 1-10
Author(s):  
Nguyen Dong Anh ◽  
Nguyen Chi Sang
Keyword(s):  
Optimal Control ◽  
Numerical Study ◽  
Degree Of Freedom ◽  
Control Force ◽  
Linear Quadratic ◽  
Tuned Mass Dampers ◽  
Coloured Noise ◽  
Optimal Theory ◽  
The One ◽  
Better Than

The design of active TMD for multi-degree-of-freedom systems subjected to second order coloured noise excitation is considered using the linear quadratic optimal theory. A detailed numerical study is carried out for a 2-DOF system. It is shown that the effectiveness of active TMD is better than the one of passive TMD.

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