Analytical Stability Prediction and Design of Variable Pitch Cutters

1999 ◽  
Vol 121 (2) ◽  
pp. 173-178 ◽  
Author(s):  
Y. Altıntas¸ ◽  
S. Engin ◽  
E. Budak
Keyword(s):  
Time Delay ◽  
Depth Of Cut ◽  
Analytical Prediction ◽  
Time Varying ◽  
Chatter Stability ◽  
Variable Pitch ◽  
Analytical Stability ◽  
The Stability ◽  
Time Invariant ◽  
Regenerative Phase

An analytical prediction of stability lobes for milling cutters with variable pitch angles is presented. The method requires cutting constants, number of teeth, and transfer function of cutter mounted on the machine tool as inputs to a chatter stability expression. The stability is formulated by transforming time varying directional cutting constants into time invariant constants. Constant regenerative time delay in uniform cutters is transformed into nonuniform multiple regenerative time delay for variable pitch cutters. The chatter free axial depth of cut is solved from the eigenvalues of stability expression, whereas the spindle speed is identified from regenerative phase delays. The proposed technique has been verified with extensive cutting tests and time domain simulations. The practical use of the analytical solution is demonstrated by an optimal tooth spacing design application which increases the chatter free depth of cuts significantly.

A frequency-domain solution for efficient stability prediction of variable helix cutters milling

2014 ◽  
Vol 228 (15) ◽  
pp. 2702-2710 ◽  
Author(s):  
Gang Jin ◽  
Qichang Zhang ◽  
Houjun Qi ◽  
Bing Yan
Keyword(s):  
Frequency Domain ◽  
Order Approximation ◽  
Effective Means ◽  
Taylor Series Expansion ◽  
Depth Of Cut ◽  
Time Varying ◽  
Variable Pitch ◽  
Variable Helix ◽  
The Stability ◽  
Axial Depth

The utilization of variable pitch or helix cutters is an effective means to prevent chatter vibration during milling. In this paper, a frequency-domain solution to efficiently predict the stability for variable helix cutters milling is presented. This method is based on the principles of variable pitch model developed by Altintas and only considers the zero-order approximation of time-varying directional cutting constants. After discretizing the axial depth of cut into finite elements and modeling each element as a variable pitch cutter, time-varying regenerative delays in the case of variable helix cutters are transformed into multiple constant regenerative times. The chatter free axial depth of cut is solved from a stability expression in which the regenerative delay terms are approximated by the Taylor series expansion, whereas the spindle speed is identified from regenerative phase delays. Compared with time-averaged semidiscretization method, the accuracy and efficiency of the proposed technique has been verified. The results show that the proposed method has high computational efficiency. It is suited to calculate optimal geometries of milling tools and beneficial for application.

scholarly journals Robust Time-Varying Output Formation Control for Swarm Systems with Nonlinear Uncertainties

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Xiaofeng Chai ◽  
Qing Wang ◽  
Yao Yu ◽  
Changyin Sun
Keyword(s):  
Stability Problem ◽  
Formation Control ◽  
Directed Network ◽  
Control Problems ◽  
Time Varying ◽  
Robust Controller ◽  
Nonlinear Uncertainties ◽  
The Stability ◽  
High Order Time ◽  
Time Invariant

Time-varying output formation control problems for high-order time-invariant swarm systems are studied with nonlinear uncertainties and directed network topology in this paper. A robust controller which consists of a nominal controller and a robust compensator is applied to achieve formation control. The nominal controller based on the output feedback is designed to achieve desired time-varying formation properties for the nominal system. And the robust compensator based on the robust signal compensator technology is constructed to restrain nonlinear uncertainties. The time-varying formation problem is transformed into the stability problem. And the formation errors can be arbitrarily small with expected convergence rate. Numerical examples are provided to illustrate the effectiveness of the proposed strategy.

scholarly journals Leader–follower formation with reduction of the off-tracking and velocity estimation under visibility constraints

2021 ◽  
Vol 18 (6) ◽  
pp. 172988142110576
Author(s):  
C. Mauricio Arteaga-Escamilla ◽  
Rafael Castro-Linares ◽  
Jaime Álvarez-Gallegos
Keyword(s):  
Formation Control ◽  
Kinematic Model ◽  
Velocity Estimation ◽  
Lyapunov Theory ◽  
Time Varying ◽  
Absolute Position ◽  
Nonholonomic Mobile Robots ◽  
The Stability ◽  
Time Invariant ◽  
Explicit Communication

This article addresses the time-varying leader–follower formation control problem for nonholonomic mobile robots, under communication and visibility constraints. Although the leader–follower formation control under visibility constraints has been studied, the elimination of the off-tracking effect has not been widely addressed yet. In this work, a new method to eliminate the off-tracking effect, considering the time-invariant formation as a tractor–trailer system, for unknown and circular tractor paths, taking into account the visibility constraints, is proposed. For a time-varying formation with not circular tractor’s path, the proposed method significantly reduces the off-tracking. Only the relative position and the relative orientation, provided by the on board monocular camera, are required. Thus, both the leader robot’s absolute position and the leader robot’s velocities are not needed. Furthermore, to avoid explicit communication among the robots, an extended state observer is implemented to estimate both the translational and the rotational leader’s velocity. In this way, the desired tasks are executed and achieved in a decentralized manner. For a time-varying formation, with constant leader robot’s velocities, the proposed control strategy, based on the kinematic model, guarantees that the formation errors asymptotically converge to the origin. Based on the Lyapunov theory, the stability proof of the formation errors dynamics is shown. Simulation results, considering time-varying leader robot’s velocities, show the efficiency of the proposed scheme.

Optimum Conditions for Variable Pitch Milling

2006 ◽  
Author(s):  
Hassan Fazelinia ◽  
Nejat Olgac
Keyword(s):  
Metal Removal ◽  
Linear Time ◽  
Removal Rate ◽  
Optimization Procedure ◽  
Milling Process ◽  
Depth Of Cut ◽  
Variable Pitch ◽  
Linear Time Invariant ◽  
Characteristic Roots ◽  
Time Invariant

From the perspective of regenerative chatter, variable-pitch milling process offers a mathematically very challenging task. It can be reduced to the problem of stability assessment on a linear time-invariant dynamics (LTI) which has more than one independent delays. This mathematically notorious problem is uniquely solved by a recent paradigm. It is called Cluster Treatment of Characteristic Roots (CTCR). This paper presents a process optimization procedure using CTCR over a special milling operation with variable pitch cutters. The optimization is based on maximizing the metal removal rate while avoiding the onset of chatter, which, in turn, enables production of the parts with a desirable surface quality. The end result is a powerful tool to determine some important geometrical and operational features of the process: (i) the pitch angle selection on the tool (i.e., variable pitch cutter vs. uniform pitch cutter), (ii) the optimum cutting conditions (i.e., depth of cut and the spindle speeds).

The determinants of parental childrearing behavior trajectories: The effects of parental and child time-varying and time-invariant predictors

2012 ◽  
Vol 36 (3) ◽  
pp. 186-196 ◽  
Author(s):  
Isabelle Roskam ◽  
Jean Christophe Meunier
Keyword(s):  
Self Efficacy ◽  
Externalizing Behavior ◽  
Efficacy Beliefs ◽  
Time Varying ◽  
Child Externalizing Behavior ◽  
Parental Educational Level ◽  
The Stability ◽  
Evidence Based Programs ◽  
Time Invariant ◽  
The Way

“Why do parents parent the way they do?” remains an important question since it concerns both scientific issues, such as the stability or change of childrearing behavior, and clinical issues, such as the way to promote positive parenting in evidence-based programs. Using an accelerated design, the aim of this study was to examine several parental and child predictors of childrearing behavior trajectories among 373 mothers and 356 fathers of 2- to 9-year-old children. Hypotheses were drawn from Belsky (1984) and subsequent studies of the determinants of parenting. The parental and child predictors were assessed and analyzed as time-varying (parental self-efficacy beliefs and child externalizing behavior) or time-invariant (parental educational level and personality traits) predictors, according to their conceptual properties. The results show a linear decrease in both supportive and controlling childrearing behavior in mothers and an improvement in supportive but a decrease in controlling childrearing behavior in fathers over time. Moreover, the results support the idea that childrearing behavior is determined by multiple factors, in particular the parents’ self-efficacy beliefs and the child’s behavior. Finally, the results confirm the hypothesis of a greater influence of child predictors than of parental ones in the case of mothers, while the reverse hypothesis of a greater predictive power of parental variables than of child ones is confirmed for fathers. The results are discussed both for research and clinical purposes.

scholarly journals Outer Synchronization between Complex Networks with Nonlinear Time-Delay Characteristics and Time-Varying Topological Structures

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Xinli Fang ◽  
Qiang Yang ◽  
Wenjun Yan
Keyword(s):  
Time Delay ◽  
Complex Networks ◽  
Lyapunov Stability Theory ◽  
Time Varying ◽  
Outer Synchronization ◽  
Topological Structures ◽  
Synchronization Problem ◽  
Adaptive Control Strategy ◽  
And Control ◽  
Time Invariant

This paper exploits the network outer synchronization problem in a generic context for complex networks with nonlinear time-delay characteristics and nonidentical time-varying topological structures. Based on the classic Lyapunov stability theory, the synchronization criteria and adaptive control strategy are presented, respectively, by adopting an appropriate Lyapunov-Krasovskii energy function and the convergence of the system error can also be well proved. The existing results of network outer synchronization can be obtained by giving certain conditions, for example, treating the coupling matrices as time-invariant, and by applying the suggested generic synchronization criteria and control scheme. The numerical simulation experiments for networks scenarios with dynamic chaotic characteristics and time-varying topologies are carried out and the result verifies the correctness and effectiveness of the proposed control solution.

Active Control of Linear Time-Varying Structures by Confinement of Vibrations

2005 ◽  
Vol 11 (1) ◽  
pp. 89-102 ◽  
Author(s):  
S. Choura ◽  
A. S. Yigit
Keyword(s):  
Control Strategy ◽  
Linear Time ◽  
Steady States ◽  
Time Varying ◽  
Linear Time Invariant ◽  
Slowing Down ◽  
Rapid Removal ◽  
Linear Time Invariant Systems ◽  
The Stability ◽  
Time Invariant

We propose a control strategy for the simultaneous suppression and confinement of vibrations in linear time-varying structures. The proposed controller has time-varying gains and can also be used for linear time-invariant systems. The key idea is to alter the original modes by appropriate feedback forces to allow parts of the structure reach their steady states at faster rates. It is demonstrated that the convergence of these parts to zero is improved at the expense of slowing down the settling of the remaining parts to their steady states. The proposed control strategy can be applied for the rapid removal of vibration energy in sensitive parts of a flexible structure for safety or performance reasons. The stability of the closed-loop system is proven through a Lyapunov approach. An illustrative example of a five-link manipulator with a periodic follower force is given to demonstrate the effectiveness of the method for time-varying as well as time-invariant systems.

A Generalized Runge-Kutta Method for Stability Prediction of Milling Operations With Variable Pitch Tools

2014 ◽  
Author(s):  
Jinbo Niu ◽  
Ye Ding ◽  
Limin Zhu ◽  
Han Ding
Keyword(s):  
Helix Angle ◽  
Depth Of Cut ◽  
Multiple Delays ◽  
Machining Parameters ◽  
Kutta Method ◽  
Computational Accuracy ◽  
Variable Pitch ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
The Stability

This paper extends the generalized Runge-Kutta method (GRKM) to predict the machining stability of milling systems with variable-pitch tools. Different from the uniform cutters with fixed pitch angles, the variation of tooth distribution angles of variable pitch cutters significantly affects the stability diagrams of the milling systems. From the viewpoint of the regenerative chatter, the milling system with non-uniform tools is governed by a delayed differential equation (DDE) with multiple delays. Afterwards, the GRKM, an approach verified with high computational accuracy and efficiency for DDEs with a single delay, is extended to tackle the milling systems with multiple delays based on Floquet theory. Besides the pitch angles, other geometry parameters of the cutter are also taken into consideration, such as the helix angle, which is proved with limited influence on the stability lobes. With the objective of maximizing productivity, the resultant stability charts provide valuable reference for the geometry design of variable-pitch cutters and for the choice of machining parameters, i.e. the spindle speed and the depth of cut.

scholarly journals Discrete-time systems with time-varying time delay: Stability and stabilizability

2006 ◽  
Vol 2006 ◽  
pp. 1-10 ◽  
Author(s):  
El-Kébir Boukas
Keyword(s):  
Time Delay ◽  
Discrete Time ◽  
Sufficient Conditions ◽  
Time Varying ◽  
Design Algorithm ◽  
Time Varying Delay ◽  
Discrete Time Systems ◽  
Varying Time Delay ◽  
The Stability ◽  
Time Systems

This paper deals with the class of linear discrete-time systems with varying time delay. The problems of stability and stabilizability for this class of systems are considered. Given an upper bound and a lower bound on the time-varying delay, sufficient conditions for checking the stability of this class of systems are developed. A control design algorithm is also provided. All the results developed in this paper are in the LMI formalism which makes their solvability easier using existing tools. A numerical example is provided to show the effectiveness of the established results.

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