Stability and Stabilization of Systems with Time Delay (original) (raw)

State-dependent delay in regenerative turning processes

Nonlinear Dynamics, 2006

Stability of a two degrees of freedom model of the turning process is considered. An accurate modeling of the surface regeneration shows that the regenerative delay, determined by the combination of the workpiece rotation and the tool vibrations, is in fact statedependent. For that reason, the mathematical model considered in this paper is a delay-differential equation with state-dependent time delay. In order to study linearized stability of stationary cutting processes, an associated linear system, corresponding to the statedependent delay equation, is derived. Stability analysis of this linear system is performed analytically.

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

Volume 4: 8th International Conference on Multibody Systems, Nonlinear Dynamics, and Control, Parts A and B, 2011

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.

ORIGINAL ARTICLE Time modeling in high-speed machining of mold pocket

Numerical control milling (NCM) at high speed is the most used machining process in the manufacture of molds because it offers high productivity and workpiece surface quality. The aim of this work is to establish a methodology to evaluate the rough machining time, during high speed milling. In pocket machining, a 2.5D milling has been considered. The proposed approach considers the roughing cutting time as the ratio of the pocket volume by the removed material rate. The pocket is divided into volumes distributed according to the real radial depth. Since the radial depth varies during machining, the removed material rate is not constant. In this paper, an experimental study is carried out to validate models of machining time calculation. The obtained results show that the proposed method offers fast and easy calculation of the machining time of pocket roughing.

Key Variables in the Control of Lead Time in Spinning Mills

Fibres and Textiles in Eastern Europe, 2016

An investigation of various factors which affect the lead time in spinning mills which produce 14.76 tex (40 Ne) carded yarns linear density is reported. For this study, data were collected from 27 mills producing 14.76 tex carded yarns of linear density. The important parameters which affect the lead time were obtained by principal component analysis of the data. Correlation matrix and multiple regression analysis were carried out taking into account the lead time as the dependent variable and HOK (the number of Operative Hours required to produce 100 kg of yarn), FQI (Fibre Quality Index), YQI (Yarn Quality Index) and spindle production as independent variables. The reliability of the data was checked by Cronbach's alpha, which indicated 0.839. Other tests such as the Kruskal-Wallis test, Durbin Watson test, KMO (Kaiser-Meyer Olkin) and Bartlett's test were also done to find out their association. The results show that of all the parameters considered the, lead time exerts maximum influence on spindle production, HOK and the yarn quality index in carded counts.

Effect of Modal Parameters on Both Delay-Independent and Global Stability of Turning Process

Journal of Mechanical Engineering and Automation, 2012

The model fo r regenerative vibration of linear orthogonal turning process is a second order time -invariant delay differential equation. Stability analysis resulted in lobes that combine to give transition curve that separates the paramete r space of spindle speed and depth of cut into stable and unstable subspaces. It is found that there is a subspace of the stable subspace in which the turning process is delay-independent stable. The size of this subspace is found to be a function of modal parameters and increases with damping ratio of the tool. Non -linear analysis of turning by some investigators suggests that subcritical bifurcations always occur thus the need to design a portion of the subspace of delay -independent stability for global stability. The subspace of global stability is also theoretically and quantitatively demonstrated to increase faster than the driving increase in damp ing ratio.

Cycle time prediction in high-speed milling operations for sculptured surface finishing

2006

This work studies the cycle time prediction of high-speed milling for sculptured surfaces with high feed rates. Experiments and predictions were focused on representative surfaces of dies and molds, whose geometric complexity and complexity distribution were modified parametrically. CNC programs for machining these surfaces were executed in a HURON KX-10 machining center with a SIEMENS 840D controller, with different levels of programmed feed rate. Discrepancies between programmed and actual feed rates were evaluated. A mechanistic approach for cycle time evaluation in high-speed milling of sculptured surfaces is proposed. The mechanistic model construction is based on: (a) the frequency distribution (histogram) of linear interpolation path lengths in the CNC program and (b) a characterization of the machine tool for brisk (large changes in tool path direction) and smooth movements (small changes in tool path direction). Two case studies were used to demonstrate the effectiveness of the proposed approach (a set of representative sculptured surfaces with spherical caps and a forging die surface). Comparing the actual cycle time versus ideal cycle time under programmed feed rates up to 16 m/min, discrepancies of 300-800% were found. The proposed model is capable of predicting cycle time with a maximum error of 5-22%.

Analyzing and estimating delays in wood chipping operations

Biomass & Bioenergy, 2009

Forestry a b s t r a c t Productivity studies are still frequently used to describe, understand and improve forest operations. Delays are recognized as being one of the major factors that limit chipper productivity in most operations and are therefore an integral part of most time studies.

Machining time in rough milling

Materials Technology, 2008

Numerical controlled (NC) milling is widely used in the manufacturing industry because of its high productivity and workpiece surface quality. Work to establish a methodology to evaluate the rough machining time, during high speed milling, is reported. In face machining, 2?5-dimensional milling approach has been considered. The objective is to predict optimal values of cutting speed to minimise both time and cost of die production. Optimum and economical values of cutting speed give respectively minimum production time and minimum production cost. An experimental study has been carried out to validate models for production time and cost in high speed milling. The cutting parameters analysed are cutting speed and feed per tooth.

Global Stability Lobes of Turning Processes with State-Dependent Delay

SIAM Journal on Applied Mathematics, 2012

We obtain global stability lobes of two models of turning processes with inherit nonsmoothness due to the presence of state-dependent delays. In the process, we transform the models with state-dependent delays into systems of differential equations with both discrete and distributed delays and develop a procedure to determine analytically the global stability regions with respect to parameters. We find that the spindle speed control strategy that we investigated in [SIAM J. Appl. Math., 72 (2012), pp. 1-24] can provide essential improvement on the stability of turning processes with state-dependent delay, and furthermore we show the existence of a proper subset of the stability region which is independent of system damping. Numerical simulations are presented to illustrate the general results.

Stabilization of turning processes using spindle feedback with state-dependent delay

Discrete & Continuous Dynamical Systems - B

We develop a stabilization strategy of turning processes by means of delayed spindle control. We show that turning processes which contain intrinsic state-dependent delays can be stabilized by a spindle control with state-dependent delay, and develop analytical description of the stability region in the parameter space. Numerical simulations stability region are also given to illustrate the general results.

Time modeling in high-speed machining of mold pocket

The International Journal of Advanced Manufacturing Technology, 2011

Increased Stability of Low-Speed Turning Through a Distributed Force and Continuous Delay Model

Journal of Computational and Nonlinear Dynamics, 2009

This paper investigates the increased stability behavior commonly observed in low-speed machining. In the past, this improved stability has been attributed to the energy dissipated by the interference between the workpiece and the tool relief face. In this study, an alternative physical explanation is described. In contrast to the conventional approach, which uses a point force acting at the tool tip, the cutting forces are distributed over the tool-chip interface. This approximation results in a second-order delayed integrodifferential equation for the system that involves a short and a discrete delay. A method for determining the stability of the system for an exponential shape function is described, and temporal finite element analysis is used to chart the stability regions. Comparisons are then made between the stability charts of the point force and the distributed force models for continuous and interrupted . Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms\_Use.cfm Fig. 8 Stability charts for the distributed force model of continuous turning plotted as a function of the nondimensionalized cutting speed and depth of cut. The damping ratio used is = 0.0038 and the delay ratios used are "a… r = 0.03, "b… r = 0.05, and "c… r = 0.10 "unstable regions shaded….

Delay Impacts onto Turnaround Performance - Optimal Time Buffering for Minimizing Delay Propagation

During 2007, 19% of all European flights were more than 15 min late. One contributor to this delay is the insufficient ground operation performance inducing excessive process durations. Whenever these processes are part of the critical Turnaround (TA) path, such as de-boarding, fuelling, cleaning, catering and boarding, the effects immediately propagate an accumulating delay through the ATM network. Recent studies have investigated into the effects of technical aircraft deficiencies onto TA reliability, and could show that significant potential is given for improvement. Field analyses at German airlines showed that pre-set quality standards for punctuality can actually not be met. This paper extends that analysis by considering the individual inbound delay measured at the gate, revealing the correlation between TA process duration and stability versus a given delay with an analytical model. The concept of dynamically scheduling buffer times to compensate for potential delays into th...

PREDICTION OF MULTI-DIMENSIONAL MILLING BEHAVIOR

Milling process is a multi-dimensional cutting process accompanied by chatter vibrations because of material removal discontinuity. Also, chatter phenomenon has a great impact on final milled products quality. In this paper, an adequate criterion derived from mathematical literature is used to predict stability of multi-dimensional chatter milling. For that, firstly, the chatter milling system is represented by a system of three Degrees Of Freedom (DOF) and its dynamics is modeled by a system of Retarded Differential Equations (RDEs). Afterward, stability in Lyapunov sense is computed on the basis of the quasi-polynomial characteristic function in frequency domain. Finally, this method is used to predict stability of chatter milling process under different machining conditions and the output related to each case is verified by resolving the system of RDEs and visualizing the displacements in time domain using Matlab software.

Analytical models for high performance milling. Part II: Process dynamics and stability

Chatter is one of the most important limitations on the productivity of milling process. In order to avoid the poor surface quality and potential machine damage due to chatter, the material removal rate is usually reduced. The analysis and modeling of chatter is complicated due to the time varying dynamics of milling chatter which can be avoided without sacrificing the productivity by using analytical methods presented in this paper. r

Bifurcations in the axial–torsional state-dependent delay model of rotary drilling

International Journal of Non-Linear Mechanics, 2018

We present a detailed bifurcation analysis of the state-dependent delay model of rotary drilling considering only the axial and torsional modes. This analysis is presented for the general case of independent natural frequencies of these two modes. The regenerative effect accompanying axial vibrations gives rise to a delayed model with the delay determined by the torsional oscillations. It is observed that steady drilling loses stability through a Hopf bifurcation. The nature of bifurcation is ascertained by the method of multiple scales for the general values of system parameters. Analytical results suggest that both supercritical and subcritical bifurcations exist for different choices of operating and system parameters. These analytical findings are further confirmed by numerical simulations. Possible unfoldings of the dynamics near the codimension-2 point, guided by numerical simulations and analytical results for the codimension-1 Hopf branches, are also presented. We find two different scenarios at the primary codimension-2 point viz. both Hopf branches having supercritical bifurcation, and one branch being supercritical while the other being subcritical. Our numerical simulations suggest that the dynamics near the codimension-2 point is dominated by the low-frequency limit cycles in both the scenarios.