An Object-Space Based Machining Simulation in Milling: Part 2 — By Toroidal Surface

2009 ◽  
Author(s):  
Eyyup Aras
Keyword(s):  
Linear Equations ◽  
Nonlinear Function ◽  
Intersection Point ◽  
Machining Process ◽  
End Mill ◽  
Root Finding ◽  
Toroidal Surface ◽  
Non Linear ◽  
Cartesian Coordinate ◽  
Tool Motion

This is the second part of a two-part paper presenting an efficient parametric approach for updating the in-process workpiece represented by the Z-map. With the Z-map representation, the machining process can be simulated by intersecting z-axis aligned vectors with cutter swept envelopes. In this paper the vector-envelope intersections are formulized for the toroidal section of a fillet-end mill which may be oriented arbitrarily in space. For a given tool motion a toroidal surface generates more than one envelope. In NC machining because the torus is considered as one of the constituent parts of a fillet-end mill, only some parts of the torus envelopes, called contact envelopes, can intersect with Z-map vectors. In this paper an analysis is developed for separating the contact-envelopes from the non-contact ones. When a fillet-end mill has an orientation along the vertical z-axis of the Cartesian coordinate system, which happens in 2 1/2 and 3-axis machining, the number of intersections between a Z-map vector and the swept envelope of a toroidal section of the fillet-end mill is maximum one. For finding this single intersection point one of the numerical root finding methods, i.e. bisection, can be applied to the nonlinear function obtained from vector-envelope intersections. On the other hand when a fillet-end mill has an arbitrary orientation, the number of intersections can be more than one and therefore the numerical root finding methods cannot be applied directly. Therefore for addressing those multiple intersections, a system of non-linear equations in several variables, obtained by intersecting a Z-map vector with the envelope surface of the toroidal section of a fillet-end mill, is transformed into a single variable non-linear function. Then developing a nonlinear root finding analysis which guarantees the root(s) in the given interval, those intersections are obtained.

An Object-Space Based Machining Simulation in Milling: Part 1 — By Natural Quadric and Flat Surfaces

2009 ◽  
Author(s):  
Eyyup Aras
Keyword(s):  
Linear Equations ◽  
Machining Process ◽  
Machining Simulation ◽  
Flat Surfaces ◽  
Root Finding ◽  
Planar Surfaces ◽  
Several Variables ◽  
Non Linear ◽  
The Given ◽  
Non Linear Equations

This two-part paper presents an efficient parametric approach to updating workpiece surfaces represented by the Z-map vectors. The methodology is developed for up to 3 1/2 1/2-axis machining in which a tool can be arbitrarily oriented. In calculations the Automatically Programmed Tool (APT)-type milling cutters represented by the natural quadrics, planar and the toroidal surfaces are used. The machining process is simulated through calculating the intersections between the Z-map vectors and the tool envelope surface which is modeled by using a tangency function. Part 1 of this two-part paper presents the methodology for the cutters with natural quadrics and planar surfaces. For those surfaces intersection calculations are performed analytically. The geometric complexity of a torus is higher than those of the natural quadric and planar surfaces. Furthermore if the torus has an arbitrary orientation then the intersection calculations for the torus present great difficulties. In NC machining typically a torus is considered as one of the constituent parts of a cutter. In this case only some parts of the torus envelopes, called contact-envelopes, can intersect with Z-map vectors. For this purpose in Part 2 of this two-part paper an analysis is developed for separating the contact-envelopes from the non-contact envelopes. Then a system of non-linear equations in several variables, obtained from intersecting Z-map vectors with contact envelopes, is transformed into a single variable non-linear function. Later using a nonlinear root finding analysis which guarantees the root(s) in the given interval, those intersections are addressed.

scholarly journals An Iterative Hybrid Algorithm for Roots of Non-Linear Equations

Eng ◽  
2021 ◽  
Vol 2 (1) ◽  
pp. 80-98
Author(s):  
Chaman Lal Sabharwal
Keyword(s):  
Hybrid Algorithm ◽  
Analytic Solution ◽  
Linear Equations ◽  
Root Finding ◽  
Engineering Sciences ◽  
Non Linear ◽  
False Position ◽  
Newton Raphson ◽  
And Performance ◽  
Non Linear Equations

Finding the roots of non-linear and transcendental equations is an important problem in engineering sciences. In general, such problems do not have an analytic solution; the researchers resort to numerical techniques for exploring. We design and implement a three-way hybrid algorithm that is a blend of the Newton–Raphson algorithm and a two-way blended algorithm (blend of two methods, Bisection and False Position). The hybrid algorithm is a new single pass iterative approach. The method takes advantage of the best in three algorithms in each iteration to estimate an approximate value closer to the root. We show that the new algorithm outperforms the Bisection, Regula Falsi, Newton–Raphson, quadrature based, undetermined coefficients based, and decomposition-based algorithms. The new hybrid root finding algorithm is guaranteed to converge. The experimental results and empirical evidence show that the complexity of the hybrid algorithm is far less than that of other algorithms. Several functions cited in the literature are used as benchmarks to compare and confirm the simplicity, efficiency, and performance of the proposed method.

scholarly journals Rootfinding : An Iterative Tool to Solve Different Problems

2015 ◽  
Vol 5 ◽  
pp. 121-125
Author(s):  
Iswarmani Adhikari
Keyword(s):  
Numerical Analysis ◽  
Iterative Methods ◽  
Linear Equations ◽  
Secant Method ◽  
Root Finding ◽  
Non Linear ◽  
Iteration Methods ◽  
False Position ◽  
The Common ◽  
Non Linear Equations

The aim of this paper is to apply the iteration methods for the solution of non-linear equations. Among the various root finding techniques, two of the common iterative methods Regula-falsi (false position) and the Secant method are used in two different problems to show the applications of numerical analysis in different fields. The Himalayan Physics Vol. 5, No. 5, Nov. 2014 Page: 121-125

scholarly journals A master exceptional field theory

2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
Guillaume Bossard ◽  
Axel Kleinschmidt ◽  
Ergin Sezgin
Keyword(s):  
Field Theory ◽  
Gauge Invariance ◽  
Sigma Model ◽  
Linear Equations ◽  
Field Theories ◽  
Gauge Invariant ◽  
Non Linear ◽  
Non Linear Equations

Abstract We construct a pseudo-Lagrangian that is invariant under rigid E11 and transforms as a density under E11 generalised diffeomorphisms. The gauge-invariance requires the use of a section condition studied in previous work on E11 exceptional field theory and the inclusion of constrained fields that transform in an indecomposable E11-representation together with the E11 coset fields. We show that, in combination with gauge-invariant and E11-invariant duality equations, this pseudo-Lagrangian reduces to the bosonic sector of non-linear eleven-dimensional supergravity for one choice of solution to the section condi- tion. For another choice, we reobtain the E8 exceptional field theory and conjecture that our pseudo-Lagrangian and duality equations produce all exceptional field theories with maximal supersymmetry in any dimension. We also describe how the theory entails non-linear equations for higher dual fields, including the dual graviton in eleven dimensions. Furthermore, we speculate on the relation to the E10 sigma model.

A shallow-liquid theory in magnetohydrodynamics

1960 ◽  
Vol 7 (1) ◽  
pp. 81-107 ◽  
Author(s):  
L. E. Fraenkel
Keyword(s):  
Magnetic Field ◽  
Mechanical Energy ◽  
Fluid Velocity ◽  
Linear Equations ◽  
Qualitative Difference ◽  
Meridional Plane ◽  
Plane Flow ◽  
Non Linear ◽  
Moderate Magnetic Field ◽  
Approximate Equations

The non-linear and linear ‘shallow-water’ theories, which describe long gravity waves on the free surface of an inviscid liquid, are extended to the case of an electrically conducting liquid on a horizontal bottom, in the presence of a vertical magnetic field. The dish holding the liquid, and the medium outside it, are assumed to be non-conducting. The approximate equations are based on a small ratio of depth to wavelength, on the properties of mercury, and on a moderate magnetic field strength. These equations have a ‘magneto-hydraulic’ character, for in the shallow liquid layer the horizontal fluid velocity and current density are independent of the vertical co-ordinate.Some explicit solutions of the linear equations are obtained for plane flows and for axi-symmetric flows in which the velocity vector lies in a vertical, meridional plane. The amplitudes of waves in a dish, and the amplitudes behind wave fronts progressing into undisturbed liquid, are found to be exponentially damped, the mechanical energy associated with a disturbance being dissipated by Joule heating.The approximate non-linear equations for plane flow are studied by means of characteristic variables, and it appears that, because of the magnetic damping effect, there is less qualitative difference between solutions of the non-linear and linear approximate equations at large times than is the case when the magnetic field is absent. In particular, the characteristic curves depart only a finite distance from their ‘undisturbed positions’.

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