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1
Content available Travelling wave solutions of the non-linear wave equations
Haider Jamil A., Rahman Jamshaid U., Zaman Fiazud D., Gul Sana
Acta Mechanica et Automatica
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2023
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Vol. 17, no. 2
239--245
EN
This article focuses on the exact periodic solutions of nonlinear wave equations using the well-known Jacobi elliptic function expansion method. This method is more general than the hyperbolic tangent function expansion method. The periodic solutions are found using this method which contains both solitary wave and shock wave solutions. In this paper, the new results are computed using the closed-form solution including solitary or shock wave solutions which are obtained using Jacobi elliptic function method. The corresponding solitary or shock wave solutions are compared with the actual results. The results are visualised and the periodic behaviour of the solution is described in detail. The shock waves are found to break with time, whereas, solitary waves are found to be improved continuously with time.
2
Content available Improved algorithm for periodic steady-state analysis in nonlinear electromagnetic devices
Sobczyk T. J., Radzik M.
Bulletin of the Polish Academy of Sciences. Technical Sciences
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2019
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Vol. 67, nr 5
863--869
EN
This paper presents the improved methodology for the direct calculation of steady-state periodic solutions for electromagnetic devices, as described by nonlinear differential equations, in the time domain. A novel differential operator is developed for periodic functions and the iterative algorithm determining periodic steady-state solutions in a selected set of time instants is identified. Its application to steady-state analysis is verified by an elementary example. The modified algorithm reduces the complexity of steady-state analysis, particularly for electromagnetic devices described by high-dimensional nonlinear differential equations.
3
Content available Application of novel discrete differential operator of periodic function to study electromechanical interactions
Sobczyk T. J., Radzik M.
Bulletin of the Polish Academy of Sciences. Technical Sciences
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2018
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Vol. 66, nr 5
645--653
EN
This paper investigates an algorithm for finding steady-states in electromechanical systems for the cases of their periodic nature. The algorithm enables to specify the steady-state solution identified directly in time domain. The basis for such an algorithm is a discrete differential operator that specifies the values of the first derivative of the periodic function in the selected set of points on the basis of the values of that function in the same set of points. It creates algebraic equations describing the steady-state solution for the nonlinear differential equations describing electromechanical systems. In this paper, the direct time-domain approach is tested for the simple converter considering. The algorithm used in this paper is competitive with respect to the one known in literature an approach based on the harmonic balance method operated in frequency domain.
4
Content available A mixed shooting - harmonic balance method for unilaterally constrained mechanical systems
Schreyer F., Leine R. I.
Archive of Mechanical Engineering
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2016
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Vol. LXIII, nr 2
297--314
EN
In this paper we present a mixed shooting – harmonic balance method for large linear mechanical systems on which local nonlinearities are imposed. The standard harmonic balance method (HBM), which approximates the periodic solution in frequency domain, is very popular as it is well suited for large systems with many degrees of freedom. However, it suffers from the fact that local nonlinearities cannot be evaluated directly in the frequency domain. The standard HBM performs an inverse Fourier transform, then calculates the nonlinear force in time domain and subsequently the Fourier coefficients of the nonlinear force. The disadvantage of the HBM is that strong nonlinearities are poorly represented by a truncated Fourier series. In contrast, the shooting method operates in time-domain and relies on numerical time-simulation. Set-valued force laws such as dry friction or other strong nonlinearities can be dealt with if an appropriate numerical integrator is available. The shooting method, however, becomes infeasible if the system has many states. The proposed mixed shooting-HBM approach combines the best of both worlds.
PL
W artykule przedstawiono metodę będącą połączeniem metody strzałów i metody równowagi harmonicznych zastosowaną do dużych systemów mechanicznych, w których występują lokalne nieliniowości. Standardowa metoda równowagi harmonicznych (HBM), w której aproksymuje się rozwiązanie okresowe w dziedzinie częstotliwości, jest bardzo popularna, gdyż dobrze nadaje się do dużych systemów o wielu stopniach swobody. Niemniej, jej wadą jest to, że lokalne nieliniowości nie mogą być bezpośrednio ocenione w dziedzinie częstotliwości. W standardowej metodzie HBM wykonuje się odwrotną transformację Fouriera, potem oblicza nieliniową siłę w dziedzinie czasu, a następnie wyznacza współczynniki Fouriera siły nieliniowej. Silne nieliniowości są źle reprezentowane przez obcięty szereg Fouriera, co jest wadą tej metody. W przeciwieństwie do niej, metoda strzałów działa w dziedzinie czasu i opiera się na symulacji numerycznej przebiegów czasowych. Metoda działa skutecznie gdy prawa sił są oparte na wartościach zadanych, tak jak dla tarcia suchego i innych silnie nieliniowych, pod warunkiem, że dysponuje się odpowiednim integratorem numerycznym. Metoda strzałów nie daje się jednak stosować gdy system ma wiele stanów. Proponowana metoda mieszana, strzałów i równowagi harmonicznych, łączy zalety obydwu podejść.
5
Content available remote Rigorous Numerics for PDEs with Indefinite Tail : Existence of a Periodic Solution of the Boussinesq Equation with Time-dependent Forcing
Czechowski A., Zgliczyński P.
Schedae Informaticae
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2015
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Vol. 24
143--158
EN
We consider the Boussinesq PDE perturbed by a time-dependent forcing. Even though there is no smoothing effect for arbitrary smooth initial data, we are able to apply the method of self-consistent bounds to deduce the existence of smooth classical periodic solutions in the vicinity of 0. The proof is non-perturbative and relies on construction of periodic isolating segments in the Galerkin projections.
6
Content available Impulsive and periodic class of solutions for hydrodynamic theory of lubrication
Wierzcholski K., Miszczak A.
Journal of KONES
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2012
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Vol. 19, No. 4
651-658
EN
In this paper, two different classes of lubricant flow conditions are basically indicated, i.e. for periodic solutions when pressure values and other flow parameters change periodically and for non-periodic solutions in the case of lubricating in the conditions of impulses and strokes. The Authors formulate the primary problem in the form of a system of 9 nonlinear non-homogeneous partial differential equations with variable and random coefficients in a curvilinear orthogonal system of coordinates which is supplemented with suitable constitutive dependences and conjugated with magnetic field equations with the Ohm equation. These equations include the following: three conservation equations of the lubricating liquid momentum, the stream continuity equation, the energy conservation equation, three equilibrium equations of the thin elastic superficial layer that are reduced to the differential equation in displacements, the heat transfer equation of the superficial layer that is flown around by the lubricating liquid. The following include those equations that describe constitutive dependences the Rivlin-Ericksen equation of physical dependences for viscoelastic ferrofluid, the equation of the physical dependences of the superficial layer of the surfaces lubricated, the equation of the physical dependences of the magnetic field. The unknown values of the material coefficients shall be determined experimentally. The following 9 unknowns are determined from the system of partial differential equations: three velocity components of the lubricating liquid, the hydrodynamic pressure, the temperature in the lubricating liquid, the three components of the superficial layer displacement and the temperature in the superficial layer. The mathematical solution of the problem presented requires a number of boundary conditions to be imposed. The Authors foresee quasi-analytical solutions of the system described of partial differential equations.
7
Content available On the global attractivity and the periodic character of a recursive sequence
Elsayed E. M.
Opuscula Mathematica
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2010
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Vol. 30, no. 4
431-446
EN
In this paper we investigate the global convergence result, boundedness, and periodicity of solutions of the recursive sequence [formula], where the parameters a, b, c, d and e are positive real numbers and the initial conditions x-2, x-1 and x0 are positive real numbers.
8
Content available On periodic and stable solutions of the Lasota equation in different phase spaces
Dawidowicz A. L., Poskrobko A.
Opuscula Mathematica
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2008
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Vol. 28, no. 4
453-461
EN
We study properties of the Lasota partial differential equation in two different spaces: V∞ (Hölder continuous functions) and Lp. The aim of this paper is to generalize the results of [1].
9
Content available Chaotic dynamics in the Volterra predator-prey model via linked twist maps
Pireddu M., Zanolin F.
Opuscula Mathematica
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2008
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Vol. 28, no. 4
567-592
EN
We prove the existence of infinitely many periodic solutions and complicated dynamics, due to the presence of a topological horseshoe, for the classical Volterra predator-prey model with a periodic harvesting. The proof relies on some recent results about chaotic planar maps combined with the study of geometric features which are typical of linked twist maps.
10
Content available remote Periodic solutions for noncoercive superquadratic Hamiltonian systems
Timoumi M.
Demonstratio Mathematica
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2007
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Vol. 40, nr 2
331-346
EN
In this note, we study the existence of periodic solutions for the noncoer- cive Hamiltonian system J ˙ x - u*A(t)u(x) + u*G'(t, u(x)) = 0, where the function G satisfies a new superquadratic condition which includes the case G(t, y) = |y|2(ln(1+|y|p))q, p, q > 1. By using a linking theorem, we obtain some new results.
11
Content available remote Computing periodic solutions of linear differential-algebraic systems with nonsinusoidal excitations
Trzaska Z.W., Marszałek W.
Archives of Electrical Engineering
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2006
|
Vol. 55, nr 3-4
255-271
EN
In the paper the Kronecker-Weierstrass form of linear matrix pencils is applied to analyse periodic solutions of linear differential algebraic equations that are frequently used as models of electric circuits as well as of mechanical and chemical systems. The input variables are periodic, bounded and may have discontinuities. The obtained solutions are exact and all the drawbacks of the Fourier series approach (e.g. the Gibbs phenomenon) are avoided. Two illustrative examples are also presented.
PL
W artykule przedstawiona jest efektywna metoda wyznaczania okresowych rozwiązań układów liniowych równań różniczkowo-algebraicznych, które są często stosowane w modelowaniu zarówno obwodów elektrycznych, jak też układów mechanicznych oraz procesów chemicznych. Rozpatrzono układy z wieloma źródłami niesinusoidalnych sygnałów o tym samym okresie zmian w czasie ich wartości chwilowych i z możliwymi nieciągłościami spełniającymi warunki Dirichleta. Przedstawiono efektywny sposób dokładnej reprezentacji niesinusoidalnych sygnałów okresowych bez stosowania szeregu Fouriera, co pozwoliło ustrzec się od niedogodności powodowanych efektem Gibbsa. Zastosowane zostało przekształcenie Kroneckera-Weierstrassa w odniesieniu do liniowego pęku macierzowego, co w efekcie pozwoliło na rozseparowanie równań różniczkowo-algebraicznych na dwa podukłady: regularny oraz singularny. Dla stanu ustalonego wyznaczono dokładne rozwiązania otrzymanych podukładów równań. Podano kryteria istnienia oraz algorytm określania tych rozwiązań. Dla poszczególnych źródeł zostały zdefiniowane i szczegółowo przedyskutowano pętle określające dokładnie ich energię jednookresową na fazowej płaszczyźnie energii. Przedstawione zostały odpowiednie schematy służące do jednolitej reprezentacji histerezowych pętli energii jednookresowej dla każdego źródła. Zrealizowane zostały stosowne symulacje komputerowe a ich reprezentatywne wyniki odniesione są do wybranych układów, które często występują w praktyce.
12
Content available remote Periodicity of the Population Size of an Age-Dependent Model With a Dominant Age Class by Means of Impulsive Perturbations
Covachev V.
International Journal of Applied Mathematics and Computer Science
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2000
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Vol. 10, no 1
147-155
EN
For an age-dependent model with a dominant age class an w-periodic regime of the population size is sought by means of impulsive perturbations. For both noncritical and critical cases of first order the problem is reduced to operator systems solvable by a convergent simple iteration method.
13
Stability analysis of nonlinear cellular neural networks with hysteresis in the output dynamics
Slavova A.
Applied Mechanics and Engineering
|
1998
|
Vol. 3, no 2
305-322
EN
In this work, hysteresis of cellular neural network is investigated. The general state equation with linear and nonlinear templates is introduced. Moreover, the output equation is own dynamics where the nonlinearity in the feedback is allowed to exibit hysteresis. Stability analysis of this cellular neural network is made. New approach for studying such dynamical systems is introduced using Lyapunov's majorizing equations.
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