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1
Content available Recenzja. Język, którym mówi Bóg
Heller Michał
Postępy Fizyki
|
2021
|
T. 72, z. 2
41--43
PL
Analiza matematyczna, czyli rachunek różniczkowy i całkowy (po angielsku po prostu calculus) jest naszym narzędziem na co dzień. Trudno byłoby znaleźć roboczy dzień w życiu fizyka teoretyka, w którym nie musiałby czegoś zróżniczkować albo scałkować. Wielu z nas przez lata wkładało innym do głowy zasady obliczania różniczek i całek. Czy można jeszcze z podręczników analizy wyczytać coś odświeżającego? Jeżeli mamy co do tego wątpliwości, to weźmy do ręki książkę Stevena Strogatza „Potęga nieskończoności.”
2
Content available Optimal input signal design for fractional-order system identification
Jakowluk W.
Bulletin of the Polish Academy of Sciences. Technical Sciences
|
2019
|
Vol. 67, nr 1
37--44
EN
The optimal design of excitation signal is a procedure of generating an informative input signal to extract the model parameters with maximum pertinence during the identification process. The fractional calculus provides many new possibilities for system modeling based on the definition of a derivative of noninteger-order. A novel optimal input design methodology for fractional-order systems identification is presented in the paper. The Oustaloup recursive approximation (ORA) method is used to obtain the fractional-order differentiation in an integer order state-space representation. Then, the presented methodology is utilized to solve optimal input design problem for fractional-order system identification. The fundamental objective of this approach is to design an input signal that yields maximum information on the value of the fractional-order model parameters to be estimated. The method described in this paper was verified using a numerical example, and the computational results were discussed.
3
Content available The analysis of non-linear free vibration of FGM nano-beams based on the conformable fractional non-local model
Rahimi Z., Sumelka W., Shafiei S.
Bulletin of the Polish Academy of Sciences. Technical Sciences
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2018
|
Vol. 66, nr 5
737--745
EN
Continuum models generalized by fractional calculus are used in different mechanical problems. In this paper, by using the conformable fractional derivative (CFD) definition, a general form of Eringen non-local theory as a fractional non-local model (FNM) is formulated. It is then used to study the non-linear free vibration of a functional graded material (FGM) nano-beam in the presence of von-Kármán non-linearity. A numerical solution is obtained via Galerkin and multiple scale methods and effects of the integer and non-integer (fractional) order of stress gradient (in the non-local stress-strain relation) on the ratio of the non-local non-linear natural frequency to classical non-linear natural frequency of simply-supported (S-S) and clamped-free (C-F) FGM nano-beams are presented.
4
Content available Stability conditions for fractional-order linear equations with delays
Mozyrska D., Ostalczyk P., Wyrwas M.
Bulletin of the Polish Academy of Sciences. Technical Sciences
|
2018
|
Vol. 66, nr 4
449--454
EN
The problem of stability of the Gr¨unwald-Letnikov-type linear fractional-order discrete-time systems with delays is discussed. For the stability analysis of the considered systems the Z -transform is used. The sufficient conditions for the asymptotic stability of the considered systems are presented. Using conditions related to eigenvalues of the matrices defining the linear difference systems, one can determine the regions of location of eigenvalues of matrices associated to the systems in order to guarantee the asymptotic stability of the considered systems. Some of these regions are illustrated with relevant examples.
5
Content available Fractional variable order anti-windup control strategy
Sierociuk D., Malesza W.
Bulletin of the Polish Academy of Sciences. Technical Sciences
|
2018
|
Vol. 66, nr 4
427--432
EN
In this paper, a novel anti-windup strategy is presented. It is based on using fractional variable order integrator instead of integer order one in PID controller. It is shown that among four different types of variable order derivative definitions, only one gives satisfactory results – comparable, and even slightly better than the classical back-calculation anti-windup algorithm. Results are also presented in the form of simulation plots.
6
Content available Economic growth in the European Union modelled with fractional derivatives: first results
Tejado I., Pérez E., Valério D.
Bulletin of the Polish Academy of Sciences. Technical Sciences
|
2018
|
Vol. 66, nr 4
455--465
EN
This paper presents models of economic growth for all states of the European Union (EU), since either 1970 or the year of accession to the EU. Both integer and fractional order models are obtained, where the gross domestic product (GDP) is a function of the country’s land area, gross capital formation (GCF), exports of goods and services, and average years of school attendance.
7
Content available Different fractional order models for an experimental smart beam system
Markowski K. A., Birs I., Muresan C. I., Prodan O.
Bulletin of the Polish Academy of Sciences. Technical Sciences
|
2018
|
Vol. 66, nr 4
485--493
EN
The applicability of fractional calculus in system engineering outperforms classic identification techniques due to its ability to depict physical phenomena with increased accuracy. The present study explores the increased accuracy and flexibility of a fractional order model applied to an experimental smart beam depicting an airplane wing. The paper details the fractional order system identification of the beam and explores the possibility of realization of the model.
8
Content available A local truncation error estimation for a SubIval solver
Sowa M.
Bulletin of the Polish Academy of Sciences. Technical Sciences
|
2018
|
Vol. 66, nr 4
475--484
EN
The paper concerns an analysis for SubIval (the subinterval-based method for fractional derivative computations in initial value problems). A time step size adaptive solver is discussed, for which the formula of a local truncation error is derived. A general form for a system of linear equations is given for the considered class of problems (for which the analysis is performed in the paper). Two circuit examples are introduced to display the usefulness of the SubIval solver. For the examples that have been chosen it is possible to obtain referential solutions through completely different methods. The results obtained through the numerical solver are compared with evaluations of the referential solutions. The error estimation results obtained for the time steps of the SubIval solver are compared with the actual errors, being the differences between the numerical solutions and the referential solutions. The paper also contains a comparison of the accuracy of results obtained through the SubIval solver with the accuracies of other solvers.
9
Content available Analysis of dynamic characteristics of selected pneumatic systems with fractional calculus used in telematics
Luft M., Łukasik Z., Pietruszczak D.
Archives of Transport System Telematics
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2018
|
Vol. 11, iss. 4
37--44
EN
The section of the paper on simulation studies presents the application of fractional calculus to describe the dynamics of pneumatic systems used in telematics. In the construction of mathematical models of the analyzed dynamic systems, the Riemann-Liouville definition of non-integer order was used. For the analyzed model, transfer function of integer and non-integer order was determined. Functions describing characteristics in time and frequency domains were determined, whereas the characteristics of the analyzed systems were obtained by means of computer simulation. The section of the paper on laboratory research presents the results of the laboratory tests of the injection system of the internal combustion engine with special attention to the verification of simulated tests of selected pneumatic systems described with the use of fractional calculus.
10
Content available Fractional-order models: The case study of the supercapacitor capacitance measurement
Lewandowski M., Orzyłowski M.
Bulletin of the Polish Academy of Sciences. Technical Sciences
|
2017
|
Vol. 65, nr 4
449--457
EN
At the beginning of the paper, the fractional calculus is briefly presented. Then, the models of dielectric relaxation in supercapacitors are described. On the basis of the Cole-Cole model, a fractional-order model of supercapacitor impedance is formulated. The frequency characteristics of selected supercapacitors and their voltage response to a current step are assumed as a basis for the analysis of their dynamics. An example of the fractional dynamic model application was used for the critical assessment of the IEC standard recommendation on the conditions of supercapacitor capacitance measurements. The presented study shows some imperfections of the IEC standard recommendations, which probably result from the use of an inaccurate dynamics model. At the end of the paper, the authors propose a solution to this problem by changing the measurement conditions and introducing a concept of dynamic capacitance. The conclusions of the paper indicate that the models of fractional-order dynamics may be useful not only for the control purposes but also in other domains.
11
Content available Dodatnia pochodna Grünwalda-Letnikova jako pochodna funkcji drogi
Cioć R.
Autobusy : technika, eksploatacja, systemy transportowe
|
2017
|
R. 18, nr 12
786--791, CD
PL
W artykule przeanalizowano pochodną Grünwalda-Letnikova ƒ(ƞ)(t) w odniesieniu do klasycznego zagadnienia prędkości, jako pierwszej pochodnej funkcji drogi w czasie ƒ(1)(t). Autor argumentuje, że dodatnia pochodna Grünwalda-Letnikova nie spełnia twierdzenia Lagrange’a, co wiąże się z problemami jednoznacznej fizycznej jej interpretacji.
EN
The paper analyses Grünwald-Letnikov ƒ(ƞ)(t) derivative in space of first order derivative ƒ(1)(t) and also analyses the classical interpretation of derivative of path function as velocity. The author argues that the Grünwald-Letnikov positive derivative does not fulfil the Lagrange Theorem (Mean-Value Theorem for Derivatives) and this problem causes not clear physical interpretation of the Grünwald-Letnikov positive derivative.
12
Content available Analiza właściwości dynamicznych wybranych układów pneumatycznych za pomocą rachunku różniczkowego niecałkowitych rzędów. Cz. 2, Badania laboratoryjne
Luft M., Nowocień A., Pietruszczak D.
Autobusy : technika, eksploatacja, systemy transportowe
|
2017
|
R. 18, nr 12
1056--1060, CD
PL
W artykule przedstawiono wyniki badań laboratoryjnych układu wtryskowego, silnika spalinowego, pod kątem weryfikacji badań symulacyjnych wybranych układów pneumatycznych opisanych rachunkiem różniczkowo-całkowym niecałkowitych rzędów (ang. fractional calculus), przeprowadzonych w Części 1 artykułów. Badania wykonano Instytucie Eksploatacji Pojazdów i Maszyn na Wydziale Mechanicznym Uniwersytetu Technologiczno-Humanistycznego im. Kazimierza Pułaskiego w Radomiu.
EN
The Part 2 of these papers, deals with laboratory tests of a pneumatic transducer. Measurements have been taken and a real pneumatic system is analyzed and compared with a mathematical model. The analysis uses fractional calculus and is carried out with respect to time and frequency.
13
Content available Analiza właściwości dynamicznych wybranych układów pneumatycznych za pomocą rachunku różniczkowego niecałkowitych rzędów. Cz. 1, Badania symulacyjne
Luft M., Nowocień A., Pietruszczak D.
Autobusy : technika, eksploatacja, systemy transportowe
|
2017
|
R. 18, nr 12
1050--1055, CD
PL
W artykule przedstawiono zastosowanie rachunku różniczkowego niecałkowitych rzędów (ang. fractional calculus) do opisu dynamiki zjawisk układów pneumatycznych. W budowie modeli matematycznych, analizowanych układów dynamicznych, wykorzystano definicję Riemanna-Liouville’a pochodno-całki niecałkowitego rzędu. Dla analizowanego modelu, wyznaczono transmitancję operatorową całkowitego i niecałkowitego rzędu. Wyznaczono zależności opisujące charakterystyki czasowe i częstotliwościowe, na drodze symulacji komputerowej uzyskano charakterystyki analizowanych układów. Do badań symulacyjnych wykorzystano oprogramowanie Microsoft Office Excel oraz MATLAB.
EN
Rapid development of technology which occurred at the turn of the 19th and 20th century has enforced development of new models describing reality, especially the technical objects working in extreme conditions or characterized by extreme parameters (a super-capacitor, a transmission line of infinite length, etc.). To describe such objects the classical calculus is insufficient. To make mathematical models which describe dynamics of the phenomena occurring in objects of extreme parameters, it is indispensable to use the differential and integral calculus of non-integer order (fractional calculus). This statement was an inspiration for carrying out research into the use of fractional calculus for describing technical objects working in extreme conditions or characterized by extreme parameters. This paper presents a method of describing a pneumatic system working in extreme conditions by means of fractional calculus and with the use of the Reimann-Liouville definition of non-integer ordered derivatives. This research leads to putting forward the following research hypothesis: The fractional calculus-based method of describing dynamic properties of pneumatic systems enables analysis of properties of a wide spectrum of pneumatic systems of integer and non-integer order. The paper (Part 1) describes fractional calculus. The Riemann–Liouville fractional derivatives is defined. A transfer function and spectral transmittance of non-integer order are described. The Part 2 of these papers, deals with laboratory tests of a pneumatic transducer. Measurements have been taken and a real pneumatic system is analyzed and compared with a mathematical model. The analysis uses fractional calculus and is carried out with respect to time and frequency.
14
Content available Modeling of logistics and safety processes using differential equations – selected examples
Ryczyński J., Spustek H.
TTS Technika Transportu Szynowego
|
2015
|
R. 22, nr 12
1331--1334, CD
EN
The article describes the use of differential calculus to determine the form of differential equations family of curves. Form of differential equations obtained by eliminating the parameters of the equations describing the different family of curves. Elimination of the parameters has been performed several times by differentiation starting equations. Received appropriate form of differential equations for the case of family circles, family of curves of the second degree and the families of the logistic function.
PL
W artykule przedstawiono zastosowanie rachunku różniczkowego do wyznaczania postaci równań różniczkowych rodziny krzywych. Postać równań różniczkowych otrzymano w wyniku eliminacji parametrów z równań opisujących rodzinę poszczególnych krzywych. Eliminacja parametrów została wykonana w wyniku kilkukrotnego różniczkowania równań wyjściowych. Otrzymano stosowne postaci równań różniczkowych dla przypadku rodziny okręgów, rodziny krzywych drugiego stopnia oraz rodziny funkcji logistycznej.
15
Content available Some remarks on fluid flow in hourglasses
Bryll A., Sochacki R.
Scientific Issues of Jan Długosz University in Częstochowa. Mathematics
|
2014
|
Vol. 19
29--36
EN
In the paper the authors analyse different shapes of an hourglass for the linearity of their graduation. We also assume that any hourglass (more precisely, each of the two congruent parts) has the shape of a solid of revolution and any cross section at height h of this hourglass depends on the base radius r, i.e. h = ƒ(r).
16
Content available Model analityczno-numeryczny parametrów pracy korbowodu i mechanizmu przegubowego
Czajkowski A. A., Czajkowska E. T.
Problemy Nauk Stosowanych
|
2014
|
T. 2
69--82
PL
Wstęp i cele: W pracy przedstawiono dwa zagadnienia mechaniczne, a mianowicie analizę prędkości korbowodu oraz badanie prędkości wznoszenia się mechanizmu przegubowego. Głównym celem pracy jest opracowanie modelu analitycznego oraz numerycznego dla analizy prędkości korbowodu i przegubu. Materiał i metody: Materiał stanowi model mechaniczny korbowodu i przegubu. W opracowaniu stosuje się metodę analityczną oraz numeryczną z zastosowaniem programu Mathematica. Wyniki: Dla obu zagadnień opracowano modele analityczne z zastosowaniem rachunku różniczkowego. Opracowano algorytmy w programie Mathematica kreujące wykresy 2D i 3D ruchu punktu korbowodu po torze eliptycznym. Wniosek: Pokazanie technicznego zastosowania pochodnej funkcji jednej zmiennej, a w szczególności wyznaczania ekstremum funkcji, w wybranych zagadnieniach mechaniki technicznej, sprzyja lepszemu zrozumieniu i stosowaniu podstawowych pojęć rachunku różniczkowego.
EN
Introduction and aims: The paper presents two mechanical problems, namely the analysis of the velocity of the connecting-rod and the test velocity of bar link-age mechanism. The main aim of this work is to develop the analytical and numerical model for the analysis of the velocity of the connecting-rod and bar link-age. Material and methods: Material is a mechanical model of a connecting-rod and bar link-age. The study used an analytical method and numerical by using Mathematica program. Results: For both problems has been developed analytical models using differential calculus. Also have been written some algorithms in Mathematica program that creates 2D and 3D graphs of connecting-rod movement of the point on the elliptical path. Conclusion: Showing technical application of a function of one variable, in particular case for determining extreme of function, in technical mechanics of selected problems, promotes better understanding and applying the fundamental concepts of calculus.
17
Content available remote On the canonical connection for smooth envelopes
Moreno G.
Demonstratio Mathematica
|
2014
|
Vol. 47, nr 2
459--464
EN
A notion known as smooth envelope, or superposition closure, appears naturally in several approaches to generalized smooth manifolds, which were proposed in the last decades. Such an operation is indispensable in order to perform differential calculus. A derivation of the enveloping algebra can be restricted to the original one, but it is a delicate question if the the vice–versa can be done as well. In a physical language, this would correspond to the existence of a canonical connection. In this paper, we show an example of an algebra which always possesses such a connection.
18
Content available Selected issues of fractional calculus in mathematical modelling of measuring transducers used in transportation facilities
Luft M., Łukasik Z., Szychta E., Cioć R., Pietruszczak D.
Zeszyty Naukowe / Akademia Morska w Szczecinie
|
2012
|
nr 29 (101)
109-116
EN
The paper presents the possibility of modelling transducers used in transport facilities using fractional calculus and analyses of the dynamic properties in terms of time and frequency for the measuring transducer with a seismic mass. Pointed out the benefits of fractional calculus in the description of the dynamics of transducers used in transportation facilities. Simulation studies were performed in the development environment of MATLAB&Simulink.
PL
W artykule przedstawiono możliwość modelowania przetworników pomiarowych stosowanych w obiektach transportowych przy zastosowaniu rachunku różniczkowo-całkowego rzędów niecałkowitych (ang. fractional calculus). W pracy dokonano analizy właściwości dynamicznych w ujęciu czasowym i częstotliwościowym dla przetwornika pomiarowego z masą sejsmiczną. Wskazano na zalety zastosowania rachunku różniczkowo-całkowego rzędów niecałkowitych w opisie dynamiki przetworników stosowanych w obiektach transportowych. Badania symulacyjne wykonano w środowisku programistycznym MATLAB&Simulink.
19
Content available Derivative of a function at a point and integral by professor Igor Kluvánek
Gunčaga J.
Scientific Issues of Jan Długosz University in Częstochowa. Mathematics
|
2009
|
Vol. 14
37-50
EN
Professor Igor Kluvánek had developed a unique course of calculus (mathematical analysis) to teach students the differential and integral calculus. In the present paper, this concept is briefly outlined. The notion of derivative is introduced via continuity. The definition of integral given in this article applies an idea of Archimedes.
20
Content available remote On mistakes connected with differentiating
Sochacki R., Bryll G.
Scientific Issues of Jan Długosz University in Częstochowa. Mathematics
|
2007
|
Vol. 12
399-405
EN
Many students have problems with solving tasks concerning the existence of the derivative of a function at a point. In this paper we discuss some of them.
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