scholarly journals Analytic solutions of transcendental equations with application to automatics

2016 ◽  
Vol 26 (4) ◽  
pp. 497-513
Author(s):  
Henryk Górecki ◽  
Mieczysław Zaczyk
Keyword(s):  
Characteristic Equation ◽  
Dynamic Error ◽  
Analytic Solutions ◽  
Analytical Formulae ◽  
Transcendental Equations ◽  
The Moment ◽  
Extremal Value

AbstractIn the paper the extremal dynamic error x(τ) and the moment of time τ are considered. The extremal value of dynamic error gives information about accuracy of the system. The time τ gives information about velocity of transient. The analytical formulae enable design of the system with prescribed properties. These formulae are calculated due to the assumption that x(τ) is a function of the roots s1, ..., snof the characteristic equation.

Design of systems with extremal dynamic properties

2013 ◽  
Vol 61 (3) ◽  
pp. 563-567 ◽  
Author(s):  
H. Górecki ◽  
M. Zaczyk
Keyword(s):  
Characteristic Equation ◽  
Chemical Industry ◽  
Dynamic Properties ◽  
Dynamic Error ◽  
Analytical Formulae ◽  
Extremal Values ◽  
Extremal Value ◽  
Zeros And Poles

Abstract In this article the problem of determination of coefficients a1, a2, . . . , an of the characteristic equation which yield required extremal values of the solution x(t) at extremal values τ of time is solved. The extremal values of x(t) and τ are treated as functions of the roots s1, s2, . . . , sn. The analytical formulae enable to design the systems with prescribed dynamic properties. The zeros and poles can be located using the known method. The extremal dynamic error x(t) is the most important property of the behaviour of the system. This extremal value of the dynamic error has fundamental role in the chemical industry where for example overrising temperature or pressure can lead to an explosion. A second very important property is the extremal time τ connected with the extremal value of the error. This property is essential in the electroenergetic system, which can be destroyed by the overvoltages waves.

scholarly journals Design of the oscillatory systems with the extremal dynamic properties

2014 ◽  
Vol 62 (2) ◽  
pp. 241-253 ◽  
Author(s):  
H. Górecki ◽  
M. Zaczyk
Keyword(s):  
Characteristic Equation ◽  
Practical Problem ◽  
Dynamic Properties ◽  
Oscillatory Systems ◽  
Complex Roots ◽  
Analytical Formulae ◽  
Extremal Values ◽  
Extremal Value

Abstract In this article the problem of determination of such coefficients a1, a2, ..., an and eigenvalues s1, s2, ..., sn of the characteristic equation which yield required extremal values of the solution x(t) at the extremal value τ of time is solved. The extremal values of x(τ ) and τ are treated as functions of the roots s1, s2, ..., sn. The analytical formulae enable us to design the systems with prescribed dynamic properties. For solution of the problem the properties of symmetrical equations are used. The method is illustrated by an example of the equation of 4-th degree. The regions of the different kinds of the roots: real, with one pair of complex and two pairs of complex roots are illustrated. A practical problem is shown.

SOLITON INTERACTIONS FOR THE GENERALIZED (3+1)-DIMENSIONAL BOUSSINESQ EQUATION

2012 ◽  
Vol 26 (07) ◽  
pp. 1250062 ◽  
Author(s):  
XIAO-LING GAI ◽  
YI-TIAN GAO ◽  
XIN YU ◽  
ZHI-YUAN SUN
Keyword(s):  
Soliton Solution ◽  
Boussinesq Equation ◽  
Soliton Solutions ◽  
Analytic Solutions ◽  
Propagation Process ◽  
Soliton Propagation ◽  
Soliton Interactions ◽  
Pass Through ◽  
The Moment ◽  
The One

Generalized (3+1)-dimensional Boussinesq equation is investigated in this paper. Through the dependent variable transformation and symbolic computation, the one- and two-soliton solutions are obtained. With the one-soliton solution, the coefficient effects in the soliton propagation process are investigated. Through analyzing the two-soliton solution, two kinds of two-soliton interactions are presented: (i) Two solitons merge into a bigger one whose amplitude increases but does not exceed the sum of the two at the moment of the collision; (ii) Two solitons can pass through each other, and their shapes keep unchanged with a phase shift after the separation. In addition, two kinds of analytic solutions are discussed: (i) "Amplitudes" of the two analytic solutions immediately turn to negative (positive) infinity after the "collision"; (ii) Two analytic solutions are fused into a higher peak (valley) at the moment of "collision", whose "amplitudes" change to negative (positive) infinity after the separation.

scholarly journals Decomposition method and its application to the extremal problems

2016 ◽  
Vol 26 (1) ◽  
pp. 49-67 ◽  
Author(s):  
Henryk Górecki ◽  
Mieczysław Zaczyk
Keyword(s):  
Linear Systems ◽  
Characteristic Equation ◽  
Decomposition Method ◽  
Extremal Problems ◽  
Order System ◽  
Order Linear ◽  
Extremal Value

In the article solution of the problem of extremal value of x(τ) is presented, for the n-th order linear systems. The extremum of x(τ) is considered as a function of the roots s1, s2, ... sn of the characteristic equation. The obtained results give a possibility of decomposition of the whole n-th order system into a set of 2-nd order systems.

Simulation of a hydraulic drive with volumetric regulation of an amphibious vehicle

2021 ◽  
Author(s):  
N.G. Sosnovsky ◽  
V.A. Brusov ◽  
V.H. Nguyen
Keyword(s):  
Mathematical Model ◽  
Dynamic Error ◽  
Hydraulic Drive ◽  
Moment Of Inertia ◽  
Control Signal ◽  
Computational Studies ◽  
Hydraulic Motor ◽  
Drive Power ◽  
Working Volume ◽  
The Moment

The article considers a hydraulic drive designed for the fan transmission, which implements the amphibious vehicle chassis on an air cushion. A mathematical model of the dynamics of the hydraulic rotary drive power section with volumetric regulation has been developed. It is proposed to carry out volumetric regulation by means of a directed change in the working volume of the pump. The dynamics of the output link of the hydraulic drive is calculated when a control signal is applied to change the pump washer angle of inclination. The control signal varied from zero to a signal corresponding to 70% of the maximum, and in the range of 70...100%. The basic and structural diagrams of the hydraulic drive are given; its transient characteristics are obtained when the moment of inertia on the shaft of the hydraulic motor changes when the amphibious vehicle is moving. The simulation study focuses on the change in the moment of inertia on the hydraulic motor shaft under various modes of amphibious vehicle movement. The computational studies of the hydraulic drive determine the time of the transient process and the dynamic error. Computational studies of the hydraulic drive revealed its sufficient performance. The use of the developed mathematical model allows choosing the optimal ratio of the hydraulic drive parameters for an amphibious vehicle.

scholarly journals Analytic results for oscillatory systems with extremal dynamic properties

2014 ◽  
Vol 24 (4) ◽  
pp. 771-784
Author(s):  
Henryk Górecki ◽  
Mieczysław Zaczyk
Keyword(s):  
System Design ◽  
Characteristic Equation ◽  
Extreme Values ◽  
Dynamic Properties ◽  
Extreme Value ◽  
Important Criterion ◽  
Oscillatory Systems ◽  
Analytical Formulae

Abstract The maximal value of the error is the most important criterion in system design. It is also the most difficult one. For that reason there exist many other criteria. The extreme value of the error represents the attainable accuracy which can be obtained and the corresponding extreme time gives information about how fast the transients are. The extreme values of the error and the corresponding time are treated here as functions of the roots of the characteristic equation. The proposed analytical formulae allow designing systems with prescribed dynamic properties.

Implosion of a uniform current sheet in a low-beta plasma

1982 ◽  
Vol 27 (3) ◽  
pp. 491-505 ◽  
Author(s):  
T. G. Forbes
Keyword(s):  
Current Sheet ◽  
Current Density ◽  
Neutral Line ◽  
Finite Thickness ◽  
Fluid Pressure ◽  
Analytic Solutions ◽  
Mhd Equations ◽  
Nonlinear Behaviour ◽  
Electric Discharges ◽  
The Moment

Using ideal, one-dimensional MHD equations, numerical and analytic solutions are presented which describe the nonlinear behaviour of an imploding current sheet in a low-fl plasma. Initially the current density is uniformly distributed in asheet of finite thickness, and the Lorentz force tending to pinch the plasma together is unopposed by any fluid pressure force. As the implosion develops the current density in the sheet is concentrated into a thin layer at the centre of the sheet, and both the current density and the current in this layer become infinite in a finite time if β = 0. At the moment this occurs, fast-mode shocks are produced which propagate outward from the centre of the current sheet, and as the shocks move away an infinitely thin current sheet is left behind. Although the solutions are related to electric discharges, they are also closely related to a problem posed by Dungey concerning the evolution of a uniformly distributed current in the vicinity of an X-type magnetic neutral line. The implications of these solutions for Dungey's problemare discussed.

Electroviscous Transport Problems via Lattice-Boltzmann

1997 ◽  
Vol 08 (04) ◽  
pp. 889-898 ◽  
Author(s):  
Patrick B. Warren
Keyword(s):  
Lattice Boltzmann ◽  
Convective Diffusion ◽  
Analytic Solutions ◽  
Simple Application ◽  
Lattice Boltzmann Methods ◽  
Osmotic Flow ◽  
The Moment ◽  
Transport Problems ◽  
Electro Osmotic Flow ◽  
Diffusion Problems

The application of lattice-Boltzmann methods to electroviscous transport problems is discussed, generalising the moment propagation method for convective-diffusion problems. As a simple application, electro-osmotic flow in a parallel-sided slit is analysed, and the results compared favourably with available analytic solutions for this geometry.

scholarly journals Buckling Resistance of Two-Segment Stepped Steel Columns

Materials ◽  
2021 ◽  
Vol 14 (4) ◽  
pp. 1046
Author(s):  
Bartłomiej Fliegner ◽  
Jakub Marcinowski ◽  
Volodymyr Sakharov
Keyword(s):  
Differential Equations ◽  
Numerical Simulations ◽  
Simple Procedure ◽  
Experimental Tests ◽  
Engineering Practice ◽  
Steel Columns ◽  
Buckling Resistance ◽  
Analytical Formulae ◽  
Resistance Assessment ◽  
Transcendental Equations

Columns of stepwise variable bending stiffness are encountered in the engineering practice quite often. Two different load cases can be distinguished: firstly, the axial force acting only at the end of the column; secondly, besides the force acting at the end, the additional force acting at the place where the section changes suddenly. Expressions for critical forces for these two cases of loading are required to correctly design such columns. Analytical formulae defining critical forces for pin-ended columns are derived and presented in the paper. Derivations were based on the Euler-Bernoulli theory of beams. The energetic criterion of Timoshenko was adopted as the buckling criterion. Both formulae were derived in the form of Rayleigh quotients using the Mathematica® system. The correctness of formulae was verified based on one the of transcendental equations derived from differential equations of stability and presented by Volmir. Comparisons to results obtained by other authors were presented, as well. The derived formulae on the critical forces can be directly used by designers in procedures leading to the column’s buckling resistance assessment. The relatively simple procedure leading to buckling resistance assessment of steel stepped columns and based on general Ayrton-Perry approach was proposed in this work. The series of experimental tests made on steel, stepped columns and numerical simulations have confirmed the correctness of the presented approach.

The High Resolution Scanning Transmission Electron Microscope

1974 ◽  
Vol 32 ◽  
pp. 316-317
Author(s):  
A. V. Crewe
Keyword(s):  
Electron Microscope ◽  
High Resolution ◽  
High Vacuum ◽  
Scanning Transmission Electron Microscope ◽  
Ultra High Vacuum ◽  
Resolving Power ◽  
Imaging Characteristics ◽  
Transmission Electron ◽  
Scanning Transmission ◽  
The Moment

The high resolution STEM is now a fact of life. I think that we have, in the last few years, demonstrated that this instrument is capable of the same resolving power as a CEM but is sufficiently different in its imaging characteristics to offer some real advantages.It seems possible to prove in a quite general way that only a field emission source can give adequate intensity for the highest resolution^ and at the moment this means operating at ultra high vacuum levels. Our experience, however, is that neither the source nor the vacuum are difficult to manage and indeed are simpler than many other systems and substantially trouble-free.

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