A Stability Algorithm for the General Milling Process: Contribution to Machine Tool Chatter Research—7

1968 ◽  
Vol 90 (2) ◽  
pp. 330-334 ◽  
Author(s):  
R. Sridhar ◽  
R. E. Hohn ◽  
G. W. Long
Keyword(s):  
Stability Analysis ◽  
Milling Process ◽  
Stability Boundaries ◽  
Machine Tool Chatter ◽  
Milling Operation ◽  
Cutting Operation ◽  
Differential Difference Equation ◽  
Linear Differential ◽  
The Stability

In this paper, a method of stability analysis for the general milling process is given. The milling operation is described by a linear differential-difference equation with periodic coefficients. An algorithm which can be used in conjunction with the digital computer is developed as a means of analytically determining the stability of this equation. This algorithm will permit the determination of the stability boundaries in the space of controllable parameters associated with a cutting operation and allows more realistic models for milling to be studied than have been attempted up to the present time. The technique is used to predict the stability in an example of a milling operation.

A Stability Algorithm for a Special Case of the Milling Process: Contribution to Machine Tool Chatter Research—6

1968 ◽  
Vol 90 (2) ◽  
pp. 325-329 ◽  
Author(s):  
R. E. Hohn ◽  
R. Sridhar ◽  
G. W. Long
Keyword(s):  
Differential Equation ◽  
Experimental Data ◽  
Machine Tool ◽  
Linear Differential Equation ◽  
Mathieu Equation ◽  
Milling Process ◽  
Machine Tool Chatter ◽  
Linear Differential ◽  
The Stability ◽  
Special Case

In an effort to determine the stability of the milling process, and due to the complexity of its describing equation, a special case of this equation is considered. In this way, it is possible to isolate and study its salient characteristics. Moreover, the simplified equation is representative of a machining operation on which experimental data can be obtained. This special case is described by a linear differential equation with periodic coefficients. A computer algorithm is developed for determining the stability of this equation. To demonstrate the use of the algorithm on an example whose solution is known, the classical Mathieu equation is studied. Also, experimental results on an actual machining operation described by this type of equation are compared to the results found using the stability algorithm. As a result of this work, some knowledge about the stability solution of the general milling process is obtained.

scholarly journals The stability of differential-difference equations with proportional time delay. I. Linear controlled system

2020 ◽  
Vol 16 (3) ◽  
pp. 316-325
Author(s):  
Alexander V. Ekimov ◽  
◽  
Aleksei P. Zhabko ◽  
Pavel V. Yakovlev ◽  
◽  
...  
Keyword(s):  
Stability Analysis ◽  
Difference Equations ◽  
Sufficient Conditions ◽  
Controlled System ◽  
Proportional Delays ◽  
Controlled Systems ◽  
Stabilizing Control ◽  
Differential Difference Equation ◽  
Linear Differential ◽  
The Stability

The article considers a controlled system of linear differential-difference equations with a linearly increasing delay. Sufficient conditions for the asymptotic stability of such systems are known; however, general conditions for the stabilizability of controlled systems and constructive algorithms for constructing stabilizing controls have not yet been obtained. For a linear differential-difference equation of delayed type with linearly increasing delay, the canonical Zubov transformation is applied and conditions for the stabilization of such systems by static control are derived. An algorithm for checking the conditions for the existence of a stabilizing control and for its constructing is formulated. New theorems on stability analysis of systems of linear differential-difference equations with linearly increasing delay are proven. The results obtained can be applied to the case of systems with several proportional delays.

scholarly journals DETERMINATION OF THE STABILITY BOUNDARIES FOR THE HAMILTONIAN SYSTEMS WITH PERIODIC COEFFICIENTS

2005 ◽  
Vol 10 (2) ◽  
pp. 191-204 ◽  
Author(s):  
A. N. Prokopenya
Keyword(s):  
Hamiltonian Systems ◽  
Computer Algebra System ◽  
Many Body ◽  
Stability Boundaries ◽  
Periodic Coefficients ◽  
Stability And Instability ◽  
Existence Of Periodic Solutions ◽  
Linear Differential ◽  
The Stability

We consider the hamiltonian system of linear differential equations with periodic coefficients. Using the infinite determinant method based on the existence of periodic solutions on the boundaries between the domains of stability and instability in the parameter space we have developed the algorithm for analytical computation of the stability boundaries. The algorithm has been realized for the second and the fourth order hamiltonian systems arising in the restricted many-body problems. The stability boundaries have been found in the form of powers series, accurate to the sixth order in a small parameter. All the computations are done with the computer algebra system Mathematica. Nagrinejama Hamiltono tiesiniu diferencialiniu lygčiu su periodiniais koeficientais sistema. Remiantis tuo, kad parametru erdveje stabilumo ir nestabilumo sritis skiriančioje sienoje egzistuoja periodinis sprendinys, sukurtas analitinis minetos sienos apskaičiavimo algoritmas. Algoritmas realizuotas antros ir ketvirtos eiles Hamiltono sistemoms, kylančioms nagrinejant apribotu keleto kūnu uždavinius. Stabilumo srities siena randama laipsnines eilutes pavidalu mažojo parametro šešto laipsnio tikslumu. Skaičiavimai atlikti skaičiavimo algebros paketo Mathematica pagalba.

scholarly journals Surface Integrity Investigation to Determine Rough Milling Effects for Assessment of Machining Allowance for Subsequent Finish Milling of Alloy 718

2021 ◽  
Vol 5 (2) ◽  
pp. 48
Author(s):  
Jonas Holmberg ◽  
Anders Wretland ◽  
Johan Berglund ◽  
Tomas Beno ◽  
Anton Milesic Karlsson
Keyword(s):  
Surface Integrity ◽  
Milling Process ◽  
Depth Of Cut ◽  
Machined Surface ◽  
Milling Operations ◽  
Milling Operation ◽  
Machining Allowance ◽  
Rough Milling

The planned material volume to be removed from a blank to create the final shape of a part is commonly referred to as allowance. Determination of machining allowance is essential and has a great impact on productivity. The objective of the present work is to use a case study to investigate how a prior rough milling operation affects the finish machined surface and, after that, to use this knowledge to design a methodology for how to assess the machining allowance for subsequent milling operations based on residual stresses. Subsequent milling operations were performed to study the final surface integrity across a milled slot. This was done by rough ceramic milling followed by finish milling in seven subsequent steps. The results show that the up-, centre and down-milling induce different stresses and impact depths. Employing the developed methodology, the depth where the directional influence of the milling process diminishes has been shown to be a suitable minimum limit for the allowance. At this depth, the plastic flow causing severe deformation is not present anymore. It was shown that the centre of the milled slot has the deepest impact depth of 500 µm, up-milling caused an intermediate impact depth of 400 µm followed by down milling with an impact depth of 300 µm. With merged envelope profiles, it was shown that the effects from rough ceramic milling are gone after 3 finish milling passes, with a total depth of cut of 150 µm.

A Dynamical Model for Studying the Stability of Thermo-Solutal Convection Under the Effect of Vertical Vibration

2005 ◽  
Author(s):  
Y. P. Razi ◽  
M. Mojtabi ◽  
K. Maliwan ◽  
M. C. Charrier-Mojtabi ◽  
A. Mojtabi
Keyword(s):  
Stability Analysis ◽  
Mechanical Vibration ◽  
Stability Limit ◽  
Lewis Number ◽  
Vertical Vibration ◽  
Wave Number ◽  
Dimensionless Wave Number ◽  
Non Linear ◽  
Linear Differential ◽  
The Stability

This paper concerns the thermal stability analysis of porous layer saturated by a binary fluid under the influence of mechanical vibration. The linear stability analysis of this thermal system leads us to study the following damped coupled Mathieu equations: BH¨+B(π2+k2)+1H˙+(π2+k2)−k2k2+π2RaT(1+Rsinω*t*)H=k2k2+π2(NRaT)(1+Rsinω*t*)Fε*BF¨+Bπ2+k2Le+ε*F˙+π2+k2Le−k2k2+π2NRaT(1+Rsinω*t*)F=k2k2+π2RaT(1+Rsinω*t*)H where RaT is thermal Rayleigh number, R is acceleration ratio (bω2/g), Le is the Lewis number, k is the dimensionless wave-number, ε* is normalized porosity and N is the buoyancy ratio (H and F are perturbations of temperature and concentration fields). In the follow up, the non-linear behavior of the problem is studied via a generalization of the Lorenz model (five coupled non-linear differential equations with periodic coefficients). In the presence or absence of gravity, the stability limit for the onset of stationary as well as Hopf bifurcations is determined.

The Analysis of the Systems Stability of Linear Differential Equations Based on the Transformation of Difference Schemes

2019 ◽  
Vol 20 (9) ◽  
pp. 542-549 ◽  
Author(s):  
S. G. Bulanov
Keyword(s):  
Stability Analysis ◽  
Asymptotic Stability ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Computer Analysis ◽  
Linear Differential Equations ◽  
System Stability ◽  
The Difference ◽  
Linear Differential ◽  
The Stability

The approach to the analysis of Lyapunov systems stability of linear ordinary differential equations based on multiplicative transformations of difference schemes of numerical integration is presented. As a result of transformations, the stability criteria in the form of necessary and sufficient conditions are formed. The criteria are invariant with respect to the right side of the system and do not require its transformation with respect to the difference scheme, the length of the gap and the step of the solution. A distinctive feature of the criteria is that they do not use the methods of the qualitative theory of differential equations. In particular, for the case of systems with a constant matrix of the coefficients it is not necessary to construct a characteristic polynomial and estimate the values of the characteristic numbers. When analyzing the system stability with variable matrix coefficients, it is not necessary to calculate the characteristic indicators. The varieties of criteria in an additive form are obtained, the stability analysis based on them being equivalent to the stability assessment based on the criteria in a multiplicative form. Under the conditions of a linear system stability (asymptotic stability) of differential equations, the criteria of the systems stability (asymptotic stability) of linear differential equations with a nonlinear additive are obtained. For the systems of nonlinear ordinary differential equations the scheme of stability analysis based on linearization is presented, which is directly related to the solution under study. The scheme is constructed under the assumption that the solution stability of the system of a general form is equivalent to the stability of the linearized system in a sufficiently small neighborhood of the perturbation of the initial data. The matrix form of the criteria allows implementing them in the form of a cyclic program. The computer analysis is performed in real time and allows coming to an unambiguous conclusion about the nature of the system stability under study. On the basis of a numerical experiment, the acceptable range of the step variation of the difference method and the interval length of the difference solution within the boundaries of the reliability of the stability analysis is established. The approach based on the computer analysis of the systems stability of linear differential equations is rendered. Computer testing has shown the feasibility of using this approach in practice.

Automated Stability Analysis of a Vehicle in Combined Pitch and Roll

2002 ◽  
Author(s):  
James K. Sprague ◽  
Shyi-Ping Liu
Keyword(s):  
Stability Analysis ◽  
Rigid Body ◽  
Contact Forces ◽  
Design Parameters ◽  
Spherical Coordinates ◽  
Stability Boundaries ◽  
Overturning Stability ◽  
Heading Angle ◽  
The Stability ◽  
Six Degree Of Freedom

This paper presents a rigid body modeling approach using ADAMS™ for an overturning stability analysis of a vehicle stopped at an arbitrary heading angle on a steep grade. The vehicle is modeled as a six-degree-of-freedom rigid body with multiple contact forces acting on the ground. A gravity vector bounded by sets of spherical coordinates is applied to the vehicle to represent the physics of a vehicle stopped on a grade with any arbitrary combination of pitch and roll angles. A design of experiments study is performed to locate the overturning stability boundaries within given levels of design parameters. Results are output using two effective graphical means of depicting the stability regions and magnitude of contact forces.

Linear stability analysis in the numerical solution of initial value problems

Acta Numerica ◽  
1993 ◽  
Vol 2 ◽  
pp. 199-237 ◽  
Author(s):  
J.L.M. van Dorsselaer ◽  
J.F.B.M. Kraaijevanger ◽  
M.N. Spijker
Keyword(s):  
Stability Analysis ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Upper Bounds ◽  
New Developments ◽  
Partial Linear ◽  
Linear Differential ◽  
The Stability ◽  
Initial Boundary Value

This article addresses the general problem of establishing upper bounds for the norms of the nth powers of square matrices. The focus is on upper bounds that grow only moderately (or stay constant) where n, or the order of the matrices, increases. The so-called resolvant condition, occuring in the famous Kreiss matrix theorem, is a classical tool for deriving such bounds.Recently the classical upper bounds known to be valid under Kreiss's resolvant condition have been improved. Moreover, generalizations of this resolvant condition have been considered so as to widen the range of applications. The main purpose of this article is to review and extend some of these new developments.The upper bounds for the powers of matrices discussed in this article are intimately connected with the stability analysis of numerical processes for solving initial(-boundary) value problems in ordinary and partial linear differential equations. The article highlights this connection.The article concludes with numerical illustrations in the solution of a simple initial-boundary value problem for a partial differential equation.

On the Investigation of Machine Tool Chatter in the Milling Process

2007 ◽  
Author(s):  
Mahsa Moghaddas ◽  
Mohammad H. Ghaffari Saadat
Keyword(s):  
Milling Process ◽  
Depth Of Cut ◽  
Non Linear Equation ◽  
Stability And Instability ◽  
Machine Tool Chatter ◽  
Stability Chart ◽  
Periodic Delay ◽  
The Stability ◽  
Two Parameters ◽  
Delay Differential

In this paper, the chatter phenomenon is investigated through a single degree of freedom model of the milling process. In this regard, the non-linear equation of motion obtained from modeling of the milling process, which is a time-periodic delay differential equation, is simulated, and by changing the parameters: spindle speed and depth of cut, and assuming constant quantities for other parameters of the system the stable and instable points for the system are gained according to these two parameters by numerical method. In the end, the stability chart for this system is plotted and the approximate boundaries between the stability and instability regions are obtained numerically.

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