CURVE FITTING IN MATLAB – TRIGONOMETRIC POLYNOMIAL

Článek se zabývá odvozením, algoritmizací a popisem konstrukce trigonometrického polynomu. Jsou zde popsány a vysvětleny základní výpočetní postupy týkající se této problematiky, nejprve je proveden teoretický rozbor, pak následuje řešený příklad a výpisy funkcí v Matlabu s vysvětlujícím komentářem.The article deals with derived, algorithm design and description of the trigonometric polynomial. There are described and explained the basic computational procedures regarding this issue, first is always a theoretical analysis, followed by solved examples and extracts functions in Matlab with explanatory commentary

Poloha vlastních čísel v Gaussově komplexní rovině v závislosti na změně koeficientů homogenní lineární diferenciální rovnice v dopravní aplikaci s použitím Matlabu

The mathematical solution of vibration of a single-degree-of-freedom dynamical system always leads to the construction and solution of a second-order linear ordinary differential equation with constant coefficients. The coefficients of this equation correspond to the mass of the body, the damping coefficient of the damper, and the stiffness of the spring in a given system. The paper examines how changes of these coefficients influence the position of eigenvalues in the Gaussian complex plane. For the eigenvalues of the second-order homogeneous linear differential equation, it is derived and proved that the product of their distances from the origin of the Gaussian complex plane is constant and equal to the numerical value of the natural circular frequency of the corresponding mass-damper-spring system. It is further shown and proved that these eigenvalues follow the rules of conformal mapping of circular inversion with respect to a reference circle with its center at the origin of the Gaussian complex plane and a radius equal to the square root of the natural circular frequency of the corresponding system. Furthermore, third and higher order homogeneous linear differential equations are also investigated and a similar property is derived and proved, namely that the product of the absolute values of the eigenvalues is linearly dependent on the coefficients of the differential equation. The Matlab system environment is used for modeling.Tento článek se zabývá určením polohy vlastních čísel v Gaussově komplexní rovině v závislosti na změně parametrů dynamického systému s jedním stupněm volnosti. Matematické řešení kmitání tohoto systému vždy vede k sestavení a řešení obyčejné lineární diferenciální rovnice druhého řádu. Konstantní koeficienty této rovnice odpovídají v daném systému hmotnosti tělesa, tlumícímu koeficientu tlumiče a tuhosti pružiny. V článku je zkoumán vliv změny těchto koeficientů na polohu vlastních čísel odpovídající charakteristické rovnice v Gaussově komplexní rovině, která jsou důležitá pro sestavení homogenního řešení. Pro odpovídající vlastní čísla je odvozena a dokázána jejich důležitá vlastnost, že součin jejich vzdáleností od počátku Gaussovy komplexní roviny je konstantní a je roven číselné hodnotě vlastní kruhové frekvence daného systému. Jinými slovy, odpovídající si vlastní čísla jsou svoje vzory a obrazy v kruhové inverzi s řídící kružnicí, která má střed v počátku Gaussovy roviny komplexních čísel. Poloměr této kružnice je roven druhé odmocnině vlastní kruhové frekvence daného systému. Celá problematika je matematicky zobecněna a dokázána. Pro výpočty a modelování polohy vlastních čísel je použito prostředí systému Matlab

Electro acoustic analogies in MatLab environment

Analogy is a cognitive process of transferring information from a specific object (source) to another specific object (goal), in this case means a mathematical similarity between different physical systems and processes. This article deals with the method of calculating the resonant frequency acoustic system on the basis of electro acoustic analogy, based on isomorphism various physical systems - electrical, mechanical, hydraulic and electro acoustic. Acoustic systems can be created by combining their individual components, like electrical circuits or mechanical and hydraulic systems. The acoustic resonance in the system always takes place when it has alternating power and its impact coincides with the natural frequency of vibration. The article created electro acoustic model of flute, which is studied using a computer model in an environment of Matlab and Simulink. The model works by calculating the acoustic impedance, which is determined by the resonant frequencies. Resonance occurs always in zero point graph of a function, the impedance changes its value from negative to positive. This model leads to very decent results when we compared the theoretical values of resonant frequencies with calculated. The possibility of experimental investigation of acoustic systems, the equivalent electrical circuit is in economic terms, time and according to the results and accuracy of better solutions than the investigation of their own systems. The model can be used to design dimensions of the flutes and other acoustic systems, or to improve their debugging

CURVE FITTING IN MATLAB – NEWTON INTERPOLATION POLYNOMIAL

Článek se zabývá odvozením, algoritmizací a popisem konstrukce Newtonova interpolačního polynomu. Jsou zde popsány a vysvětleny základní výpočetní postupy týkající se této problematiky, nejprve je proveden teoretický rozbor, pak následuje řešený příklad a výpisy funkcí v Matlabu s vysvětlujícím komentářem.The article deals with derived, algorithm design and description of the Newton interpolation polynomial. There are described and explained the basic computational procedures regarding this issue, first is always a theoretical analysis, followed by solved examples and extracts functions in Matlab with explanatory commentary

Furierova analýza v dopravní aplikaci s využitím Matlabu

Fourier series arose during the eighteenth century as a formal solution to the classic wave equation. Later on, it was used to describe physical processes in which occurrences recur in a regular pattern. Fourier’s theorem provides the mathematical language which enables us to exactly describe the periodical processes. The aim of this paper is to show the application of the Fourier series in transport problem - the emission of the noise in the tire/pavement contact. Based on the tire imprint, the profile function was created, which was applied to the calculation of the coefficients of the Fourier series. The amplitude and frequency spectrum of the noise were assembled and calculated its performance using this coefficients. This issue has been used in the teaching of the applied mathematics and numerical methods in transport at the Jan Perner Transport Faculty, University of Pardubice.Fourierova řada vznikly během osmnáctého století jako formální řešení klasické vlnové rovnice. Později byl použit pro popis fyzikálních procesů v událostí, které se vracejí do pravidelného vzoru. Fourierova analýza poskytuje matematický jazyk, který nám umožňuje přesně popsat periodické procesy. Cílem tohoto článku je ukázat použití Fourierovy řady v dopravního problému - emise hluku při styku pneumatika / vozovky. Z otisku pneumatiky je získána profilová funkce, která byla použita pro výpočet koeficientů Fourierovy řady. Bylo stanoveno frekvenční spektrum hluku a amplituda a vypočítán jeho výkon s použitím tohoto koeficientů. Tento problém byl použit při výuce aplikované matematiky a numerických metod v dopravě na Dopravní fakultě Jana Pernera Univerzity Pardubice

Matice ve vzdělávání budoucích dopravních inženýrů s podporou softwaru Matlab

This study directly follows on the articles dealing with the similar topics, published by authors at ICERI 2016 and 2018 conferences. The studying of the matrices is an attractive illustration of the application of the ideas and techniques of linear algebra in our everyday live. The basic matrix operations in the matrix theory, such as addition, subtraction, multiplication, inverse, calculation of the determinant of the matrix and the like, are separately quite easy to understand for students. Unfortunately, the application of these techniques are already difficult and the students have a big problems with solving of the practical examples, in which the matrix operations appears. Consequently, the aim of this paper is to show some simple practical applications which are suitable for teaching the theory of the matrices and call attention to the most common student’s mistakes in solving of these problems. All applications will be solved using the matrices and it will be also accompanied by a Matlab script and if possible also presented graphically. All presented applications were used in teaching of mathematics especially linear algebra at Faculty of Transport Engineering, University of Pardubice and they can be used as an inspiration for teaching at the other universities with the same or similar focus.Tato studie přímo navazuje na články zabývající se podobnými tématy, publikované autory na konferencích ICERI 2016 a 2018. Studium matic je atraktivní ilustrací aplikace myšlenek a technik lineární algebry v našem každodenním životě. Základní maticové operace v teorii matic, jako je sčítání, odčítání, násobení, inverze, výpočet determinantu matice a podobně, jsou pro studenty obzvlášť snadno srozumitelné. Aplikace těchto technik je bohužel již obtížná a studenti mají velké problémy s řešením praktických příkladů, ve kterých se maticové operace objevují. V důsledku toho je cílem této práce ukázat některé jednoduché praktické aplikace, které jsou vhodné pro výuku teorie matic, a upozornit na nejčastější chyby studentů při řešení těchto problémů. Všechny prezentované aplikace byly použity ve výuce matematiky, zejména lineární algebry na Dopravní fakultě Jana Pernera Univerzity Pardubice a mohou být použity jako inspirace pro výuku na dalších univerzitách se stejným nebo podobným zaměřením

Fourierova analýza ve výuce aplikované matematiky v dopravě s využitím MATLABU

The Fourier’s theorem, which was published during the eighteenth century as a formal solution to the classic wave equation, can be also used for description the physical and technical processes in which occurrences recur in a regular pattern. It provides the mathematical language which enables us to exactly describe the periodical processes. The transport engineers are always looking for techniques to optimize systems with the smallest cost and the highest efficiency which are useful in practice. The aim of this paper is to demonstrate such application of the Fourier series in the transport problem and to show how this issue has been used in the teaching of the applied mathematics and numerical methods in transport at the Jan Perner Transport Faculty, University of Pardubice. Using Matlab, it was revealed that the higher performance of the engine could be obtained without extra work and fuel by the tuning of the length of the intake pipe of a combustion four-stroke single-cylinder gasoline engine. The analysis described in this paper is very valuable for the future automotive engineers.The Fourier’s theorem, which was published during the eighteenth century as a formal solution to the classic wave equation, can be also used for description the physical and technical processes in which occurrences recur in a regular pattern. It provides the mathematical language which enables us to exactly describe the periodical processes. The transport engineers are always looking for techniques to optimize systems with the smallest cost and the highest efficiency which are useful in practice. The aim of this paper is to demonstrate such application of the Fourier series in the transport problem and to show how this issue has been used in the teaching of the applied mathematics and numerical methods in transport at the Jan Perner Transport Faculty, University of Pardubice. Using Matlab, it was revealed that the higher performance of the engine could be obtained without extra work and fuel by the tuning of the length of the intake pipe of a combustion four-stroke single-cylinder gasoline engine. The analysis described in this paper is very valuable for the future automotive engineers

Výhody a nevýhody zavedení systému počítačem podporovaného hodnocení do výuky matematiky

In contrast to the majority of subjects, where laptops and tablets are gradually replacing paper and pencil, in teaching and learning mathematics, paper and pencil as well as classic blackboard with chalk still prevail. The development of information technology, however, leads to the development of computer-aided learning and assessment systems aimed for the use in mathematics education. An example of such system is the platform Maple T. A. (Testing and Assessment), which was established by integrating computational capabilities of computer algebra system Maple with Computer-Aided Assessment (hereinafter referred to as “CAA”) system. This and similar instruments allow to test and practise students´ knowledge in mathematics using different types of questions, which in addition to the standard test questions include also questions where students work with graphs and formulate answers in the form of algebraic expressions or numerical values. New possibilities of these platforms in the area of symbolic computation have many uses in mathematics education at various types of schools. The paper presents the meta-analysis of the published studies dealing with the use of CAA in mathematics and brings an overview of the software used in teaching calculus at Czech universities. On the basis of analysing the current state of the studied problem, the paper provides a comprehensive view of the existing assessment software, experience with this software in the Czech educational environment as well as abroad, with an emphasis on the advantages and disadvantages of these systems.In contrast to the majority of subjects, where laptops and tablets are gradually replacing paper and pencil, in teaching and learning mathematics, paper and pencil as well as classic blackboard with chalk still prevail. The development of information technology, however, leads to the development of computer-aided learning and assessment systems aimed for the use in mathematics education. An example of such system is the platform Maple T. A. (Testing and Assessment), which was established by integrating computational capabilities of computer algebra system Maple with Computer-Aided Assessment (hereinafter referred to as “CAA”) system. This and similar instruments allow to test and practise students´ knowledge in mathematics using different types of questions, which in addition to the standard test questions include also questions where students work with graphs and formulate answers in the form of algebraic expressions or numerical values. New possibilities of these platforms in the area of symbolic computation have many uses in mathematics education at various types of schools. The paper presents the meta-analysis of the published studies dealing with the use of CAA in mathematics and brings an overview of the software used in teaching calculus at Czech universities. On the basis of analysing the current state of the studied problem, the paper provides a comprehensive view of the existing assessment software, experience with this software in the Czech educational environment as well as abroad, with an emphasis on the advantages and disadvantages of these systems

EXAMINATION OF STUDENTS IN ONLINE TEACHING

Příspěvek je zaměřen na problematiku on-line zkoušení studentů na univerzitě. Výzkum byl realizován v rámci zápočtů a zkoušek z předmětu Matematika 2. Dosažené výsledky z období pandemice Covid 19 byly porovnány s výsledky zápočtů a zkoušek z předcházejících letThe contribution is focused on the issue of online testing of students at the university. The research was carried out as part of credits and exams from the subject Mathematics 2. The results achieved during the Covid 19 pandemic period were compared with the results of credits and exams from previous year