Computational Stability Analysis of Chatter in Turning

1997 ◽  
Vol 119 (4A) ◽  
pp. 457-460 ◽  
Author(s):  
S.-G. Chen ◽  
A. G. Ulsoy ◽  
Y. Koren
Keyword(s):  
Stability Analysis ◽  
Machine Tool ◽  
Characteristic Equation ◽  
Linear Time ◽  
Stability Index ◽  
Algebraic Equations ◽  
Machining Processes ◽  
Computational Stability ◽  
Linear Time Invariant ◽  
Excited Vibration

Machine tool chatter is one of the major constraints that limits productivity of the turning process. It is a self-excited vibration that is mainly caused by the interaction between the machine-tool/workpiece structure and the cutting process dynamics. This work introduces a general method which avoids lengthy algebraic (symbolic) manipulations in deriving, a characteristic equation. The solution scheme is simple and robust since the characteristic equation is numerically formulated as a single variable equation whose variable is well bounded rather than two nonlinear algebraic equations with unbounded variables. An asymptotic stability index is also introduced for a relative stability analysis. The method can be applied to other machining processes, as long as the system equations can be expressed as a set of linear time invariant difference-differential equations.

scholarly journals On the Stability Analysis of Delay Differential-Algebraic Equations

2018 ◽  
Vol 34 (2) ◽  
Author(s):  
Phi Ha
Keyword(s):  
Stability Analysis ◽  
Exponential Stability ◽  
Linear Time ◽  
Differential Algebraic Equations ◽  
Spectral Condition ◽  
Algebraic Equations ◽  
Linear Time Invariant ◽  
Stability Of Dynamical Systems ◽  
The Stability ◽  
Delay Differential

The stability analysis of linear time invariant delay differentialalgebraic equations (DDAEs) is analyzed. Examples are delivered to demonstrate that the eigenvalue-based approach to analyze the exponential stability of dynamical systems is not valid for an arbitrarily high index system, and hence, a new concept of weakly exponential stability (w.e.s) is proposed. Then, we characterize the w.e.s in term of a spectral condition for some special classes of DDAEs.

scholarly journals Differential-algebraic systems are generically controllable and stabilizable

2021 ◽  
Author(s):  
Achim Ilchmann ◽  
Jonas Kirchhoff
Keyword(s):  
Algebraic Geometry ◽  
Linear Time ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Algebraic Systems ◽  
Linear Time Invariant ◽  
Survey Article ◽  
Link Type ◽  
Invariant Differential ◽  
Time Invariant

AbstractWe investigate genericity of various controllability and stabilizability concepts of linear, time-invariant differential-algebraic systems. Based on well-known algebraic characterizations of these concepts (see the survey article by Berger and Reis (in: Ilchmann A, Reis T (eds) Surveys in differential-algebraic equations I, Differential-Algebraic Equations Forum, Springer, Berlin, pp 1–61. 10.1007/978-3-642-34928-7_1)), we use tools from algebraic geometry to characterize genericity of controllability and stabilizability in terms of matrix formats.

Stability Analysis of Multiple Time Delayed Systems Using the Direct Method

2003 ◽  
Author(s):  
Rifat Sipahi ◽  
Nejat Olgac
Keyword(s):  
Stability Analysis ◽  
Time Delays ◽  
Linear Time ◽  
Direct Method ◽  
Multiple Time ◽  
Delayed Systems ◽  
Practical Applications ◽  
Linear Time Invariant ◽  
Multiple Time Delays ◽  
The Stability

A novel treatment for the stability of a class of linear time invariant (LTI) systems with rationally independent multiple time delays using the Direct Method (DM) is studied. Since they appear in many practical applications in the systems and control community, this class of dynamics has attracted considerable interest. The stability analysis is very complex because of the infinite dimensional nature (even for single delay) of the dynamics and furthermore the multiplicity of these delays. The stability problem is much more challenging compared to the TDS with commensurate time delays (where time delays have rational relations). It is shown in an earlier publication of the authors that the DM brings a unique, exact and structured methodology for the stability analysis of commensurate time delayed cases. The transition from the commensurate time delays to multiple delay case motivates our study. It is shown that the DM reveals all possible stability regions in the space of multiple time delays. The systems that are considered do not have to possess stable behavior for zero delays. We present a numerical example on a system, which is considered “prohibitively difficult” in the literature, just to exhibit the strengths of the new procedure.

General formula for stability testing of linear systems with fractional-delay characteristic equation

Open Physics ◽  
2013 ◽  
Vol 11 (6) ◽  
Author(s):  
Farshad Merrikh-Bayat
Keyword(s):  
Characteristic Equation ◽  
Linear Time ◽  
Stability Testing ◽  
Fractional Powers ◽  
Positive Real ◽  
Linear Time Invariant ◽  
Fractional Delay ◽  
The Right ◽  
Linear Time Invariant System ◽  
Time Invariant

AbstractIn some applications (especially in the filed of control theory) the characteristic equation of system contains fractional powers of the Laplace variable s possibly in combination with exponentials of fractional powers of s. The aim of this paper is to propose an easy-to-use and effective formula for bounded-input boundedoutput (BIBO) stability testing of a linear time-invariant system with fractional-delay characteristic equation in the general form of $$\Delta \left( s \right) = P_0 \left( s \right) + \sum\nolimits_{i = 1}^N {P_i \left( s \right)\exp ( - \zeta _i s^{\beta _i } ) = 0}$$, where P i(s) (i = 0,...,N) are the so-called fractional-order polynomials and ξ i and β i are positive real constants. The proposed formula determines the number of the roots of such a characteristic equation in the right half-plane of the first Riemann sheet by applying Rouche’s theorem. Numerical simulations are also presented to confirm the efficiency of the proposed formula.

scholarly journals Complex Scaling Approach for Stability Analysis and Stabilization of Multiple Time-Delayed Systems

2018 ◽  
Vol 2018 ◽  
pp. 1-14 ◽  
Author(s):  
Jun Zhou ◽  
Ketian Gao ◽  
Xinbiao Lu
Keyword(s):  
Stability Analysis ◽  
Linear Time ◽  
Open Loop ◽  
State Feedback Control ◽  
Multiple Time ◽  
Control Input ◽  
Complex Scaling ◽  
Delayed Systems ◽  
Static State ◽  
Linear Time Invariant

Based on complex scaling, the paper copes with stability analysis and stabilization of linear time-invariant continuous-time systems with multiple time delays in state, control input, and measured output, under state and/or output feedback. More specifically, the paper establishes stability criteria for exponential/asymptotical stability with Hurwitz complex scaling being applied to the related characteristic polynomials. The stability conditions are necessary and sufficient, delay-dependent, and independent of feedback structures and open-loop poles distribution. The criteria can be implemented graphically with locus plotting or numerically without it; moreover, no prior frequency sweeping is involved, and the contour and locus encirclement orientations can be self-defined. Exploiting the complex scaling approach and embracing its technical merits, it is considered to design static state feedback control for robustly stabilizing time-delayed systems. A small-gain interpretation for the suggested stabilization is also elaborated. Examples are included to illustrate the main results.

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