Breakaway Points Research Articles

Схлопывание присоединенной каверны при малых числах Фруда после отрывного удара плавающего эллиптического цилиндра

The plane problem of the collapse of an attached cavity formed as a result of a separation impact of an elliptical cylinder under the free surface of a heavy fluid is considered. It is assumed that after the impact the cylinder moves horizontally with constant velocity. At small Froude numbers, which correspond to small velocities of the cylinder, the free boundaries of the liquid are slightly perturbed and the process of cavity collapse is mainly reduced to the study of dynamics of breakaway points. The solution to this problem is constructed using asymptotic decompositions on a small parameter, which is the Froude number. In the main asymptotic approximation, the mixed boundary value problem of potential theory with one-sided restrictions on the body surface is formulated. On its basis, the dynamics of detachment points, the shape of a thin cavities, and the reaction of the environment to the body are determined. The results obtained can be used for solving practical problems of marine and ship hydrodynamics.

Formularization Method for Calculating the Breakaway and Break-in Points and the Corresponding Gain of Root Locus Graphs

Break points, break-away and break-in points, are an essential part in root locus technique for single input single output linear invariant control systems. The importance of Break points comes from the fact that at the Break points at least two roots of the characteristic equation of the closed loop control system change their type from real to a complex at the break away point, and from complex to real at break-in point. This change affects the response of the system which can be crucial for some of systems’ applications. The conditions for being a Break point are analysed and a new formulated systematic method for finding the Break points and their corresponding gains is presented. An efficient algorithm was developed and can be solved analytically. There is no mathematical differentiation during calculation, and the algorithm can be programmed easily. The developed algorithm is applicable for any order of transfer function of a linear invariant control system. This method is compared with other common methods to show its merits and effectiveness.

The impact of static and dynamic roughness elements on flow separation

The use of static or dynamic roughness elements has been shown in the past to delay the separation of a laminar boundary layer from a solid surface. Here, we examine analytically the effect of such elements on the local and breakaway separation points, corresponding respectively to the position of zero skin friction and presence of a singularity in the roughness region, for flow over a hump embedded within the boundary layer. Two types of roughness elements are studied: the first is small and placed near the point of vanishing skin friction; the second is larger and extends downstream. The forced flow solution is found as a sum of Fourier modes, reflecting the fixed frequency forcing of the dynamic roughness. Solutions for both the static and dynamic roughness show that the presence of the roughness element is able to move the separation points downstream, given an appropriate choice of roughness frequency, height, position and width. This choice is found to be qualitatively similar to that observed for leading-edge separation. Furthermore, for a negative static roughness a small region of separated flow forms at high roughness depth, although there is a critical depth above which boundary-layer breakaway moves suddenly upstream.

Dynamic characterization of a flexible internally damped spinning shaft with constant eccentricity

In the present study, the dynamic characterization of a flexible spinning shaft with constant eccentricity driven by a non-ideal energy source (DC motor) with external and internal damping has been focused. It is well established that the structural response of a vibratory system to which a non-ideal drive is connected may act as an energy sink under certain conditions such that a part of the energy supplied by the source is spent to vibrate the structure rather than to increase the drive speed. This phenomenon is formally known as the Sommerfeld effect. The Sommerfeld effect characterized by jump phenomena is studied through the steady-state amplitude obtained by instantaneous power balance method and further verified through numerical simulation. Finally, power balance equation is transformed into a characteristic equation through which jump phenomena and Sommerfeld effect are predicted numerically for the first and third modes using root loci method. The break-in and breakaway points of the root loci represent the values of the supply voltage at which jumps in shaft speed and flexural vibration amplitudes take place during coast-up and coast-down operation.

Damage evolution of bi-body model composed of weakly cemented soft rock and coal considering different interface effect.

Considering the structure effect of tunnel stability in western mining of China, three typical kinds of numerical model were respectively built as follows based on the strain softening constitutive model and linear elastic–perfectly plastic model for soft rock and interface: R–M, R–Cs–M and R–Cw–M. Calculation results revealed that the stress–strain relation and failure characteristics of the three models vary between each other. The combination model without interface or with a strong interface presented continuous failure, while weak interface exhibited ‘cut off’ effect. Thus, conceptual models of bi-material model and bi-body model were established. Then numerical experiments of tri-axial compression were carried out for the two models. The relationships between stress evolution, failure zone and deformation rate fluctuations as well as the displacement of interface were detailed analyzed. Results show that two breakaway points of deformation rate actually demonstrate the starting and penetration of the main rupture, respectively. It is distinguishable due to the large fluctuation. The bi-material model shows general continuous failure while bi-body model shows ‘V’ type shear zone in weak body and failure in strong body near the interface due to the interface effect. With the increasing of confining pressure, the ‘cut off’ effect of weak interface is not obvious. These conclusions lay the theoretical foundation for further development of constitutive model for soft rock–coal combination body.

On the stability of adaptation process in active noise control systems

The stability analysis of the adaptation process, performed by the filtered-x least mean square algorithm on weights of active noise controllers, has not been fully investigated. The main contribution of this paper is conducting a theoretical stability analysis for this process without utilizing commonly used simplifying assumptions regarding the secondary electro-acoustic channel. The core of this analysis is based on the root locus theory. The general rules for constructing the root locus plot of the adaptation process are derived by obtaining root locus parameters, including start points, end points, asymptote lines, and breakaway points. The conducted analysis leads to the derivation of a general upper-bound for the adaptation step-size beyond which the mean weight vector of the active noise controller becomes unstable. Also, this analysis yields the optimum step-size for which the adaptive active noise controller has its fastest dynamic performance. The proposed upper-bound and optimum values apply to general secondary electro-acoustic channels, unlike the commonly used ones which apply to only pure delay channels. The results are found to agree very well with those obtained from numerical analyses and computer simulation experiments.

Extension of the root-locus method to a certain class of fractional-order systems

In this paper, the well-known root-locus method is developed for the special subset of linear time-invariant systems commonly known as fractional-order systems. Transfer functions of these systems are rational functions with polynomials of rational powers of the Laplace variable s . Such systems are defined on a Riemann surface because of their multi-valued nature. A set of rules for plotting the root loci on the first Riemann sheet is presented. The important features of the classical root-locus method such as asymptotes, roots condition on the real axis and breakaway points are extended to the fractional case. It is also shown that the proposed method can assess the closed-loop stability of fractional-order systems in the presence of a varying gain in the loop. Moreover, the effect of perturbation on the root loci is discussed. Three illustrative examples are presented to confirm the effectiveness of the proposed algorithm.

Extending the Root‐Locus Method to Fractional‐Order Systems

The well‐known root‐locus method is developed for special subset of linear time‐invariant systems known as fractional‐order systems. Transfer functions of these systems are rational functions with polynomials of rational powers of the Laplace variable s. Such systems are defined on a Riemann surface because of their multivalued nature. A set of rules for plotting the root loci on the first Riemann sheet is presented. The important features of the classical root‐locus method such as asymptotes, roots condition on the real axis, and breakaway points are extended to fractional case. It is also shown that the proposed method can assess the closed‐loop stability of fractional‐order systems in the presence of a varying gain in the loop. Three illustrative examples are presented to confirm the effectiveness of the proposed algorithm.

Root-locus invariance exploiting alternative arrival and departure points

The same complete root locus is generated by all loop transfer functions whose numerator and denominator are linear combinations of the polynomials n(s) and d(s). Therefore, a complete root locus can be constructed starting from different sets of arrival and departure points. This property can be used to find the asymptotes of the negative locus for an exactly proper loop transfer function and to determine the configuration of the locus branches around the breakaway points. The root-locus invariance property can also be exploited to characterize a stabilization procedure leading to an exactly proper loop transfer function whose Nyquist diagram travels along a circle centered at -1 + j0

MATLAB Implementation of the Extended Ash Algorithm for Finding Root Loci

Abstract This paper describes extensions to the Ash algorithm for finding root loci, and the implementation of this Extended Ash algorithm within Matlab. The Ash algorithm is a method for finding the root locus of a linear system that finds each point on the locus within a specified computational area by following each branch in that area from beginning to end. Given a point on the locus and the slope angle of that point, it uses a Newton-Raphson iteration scheme to find the next point. It includes special features for finding breakaway points (points where two or more branches intersect) to high accuracy. The Extended Ash algorithm includes enhancements to the original algorithm which improve its efficiency and accuracy. This report describes the important features of the extended algorithm and describes its Matlab implementation. Also included are the results of several test problems that serve to compare the performance of the Matlab implementation of the Extended Ash algorithm with the built-in Matlab function

A review of blood lactate and ventilatory methods of detecting transition thresholds.

As far back as the 1930s exercise physiologists recognised the existence of critical levels of work intensity above which lactate accumulation increased drastically and energy production was affected. Investigation of these transition points (thresholds) both invasively and non-invasively has led to much recent controversy. Respiratory exchange variables such as Ve, Ve/VO2, VCO2, excess CO2 and blood lactate have been monitored for simple, double and exponential breakaway points to elucidate these critical work intensities. A number of studies have produced high correlations between endurance performance and anaerobic threshold calculations, further demonstrating the potential existence of critical work intensities. Much of the controversy surrounding these phenomena has centered on mechanisms and nomenclature. The term 'anaerobic threshold' has been severely criticised because in addition to the tissues being oxygen insufficient, an imbalance in the energy systems may have resulted. The anaerobic condition or lactate accumulation may be due to changes in lactate production and removal. Muscle fibre type and the fibre type recruitment patterns may also be important factors in threshold transitions. Further examination is made in this review of non-invasive measures for determining transition thresholds and protocols for elucidating the critical points.

Remec test with vanishing leading array coefficients

Remec's algorithm for finding breakaway points on the root locus suffers from a serious limitation when the first column entry of the first row of any s' group of the Remec array vanishes. In this paper, a remedy for such situations is proposed by rearranging the coefficients of the row in which a zero entry has occurred in the first column of the array. The coefficients of the rearranged Remec table are calculated by the usual Remec algorithm.

A remark on the use of Remec's method of finding breakaway points

The use of Remec's method of finding breakaway points is described. Examples of the applications are then given.

Computer-program description. Root-locus plotting by digital computer using operational-array technique

The objective of the paper is to present a general program for computation of the root locus of time-invariant systems. The method adopted for root-locus location is to express the characteristic equation in polynomial form for every chosen value of the static-loop sensitivity; the factorisation then gives the location of the poles. In addition, the breakaway points are also individually computed and graphically superimposed on the already computed and plotted root-locus diagrams. The break points are where the locus departs from the real axis or returns to it; hence the terms breakaway and break-in points, respectively. In more general interpretation, the breakpoints are where two or more root-locus branches come into contact anywhere in the complex plane. The interpretation above is wholly implemented in the program.

Finding root-locus breakaway points with the spirule

Finding root-locus breakaway points with the spirule

A new rule for finding breakaway points of root loci involving complex roots

A new rule for finding breakaway points of root loci involving complex roots

Analytical Representation of the Root-Locus

A technique for analytical representation of the root-locus is developed in this paper for both negative and positive feedback. With the help of the derived general equation for the root-locus, such items as intersection of the asymptotes, break-away points, and intersection of loci with imaginary axis are investigated and defined. Simple examples of possible applications are given. Finally, a number of selected root-loci and their equations are shown.