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1
Content available The use of synthesis methods in position optimisation and selection of tuned mass damper (TMD) parameters for systems with many degrees of freedom
Dymarek Andrzej, Dzitowski Tomasz
Archives of Control Sciences
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2021
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Vol. 31, No. 1
185--211
EN
The paper formulates and formalises a method for selecting parameters of the tuned mass damper (TMD) for primary systems with many degrees of freedom. The method presented uses the properties of positive rational functions, in particular their decomposition, into simple fractions and continued fractions, which is used in the mixed method of synthesis of vibrating mechanical systems. In order to formulate a method of tuning a TMD, the paper discusses the basic properties of positive rational functions. The main assumptions of the mixed synthesis method is presented, based on which the general method of determining TMD parameters in the case of systems with many degrees of freedom was formulated. It has been shown that a tuned mass damper suppresses the desired resonance zone regardless of where the excitation force is applied. The advantages of the formulated method include the fact of reducing several forms of the object’s free vibration by attaching an additional system with the number of degrees of freedom corresponding to the number of resonant frequencies reduced. In addition, the tuned mass damper determined in the case of excitation force applied at a single point can be attached to any element of the inertial primary system without affecting the reduction conditions in this way. It results directly from the methodology formalised in the paper. As part of the paper, numerical calculations were performed regarding the tuning of the TMD to the first form of free vibration of a system with 3 degrees of freedom. The parameters determined were subjected to analysis and verification of the correctness of the calculations carried out. For the considered case of a system with 3 degrees of freedom together with a TMD, time responses of displacement, from each floor, were generated to excitation induced by a harmonic force equal to the first form of vibration of the basic system. In addition, in the case of the parameters obtained, the response of the inertial element system to which the TMD was attached to random white noise excitation was determined.
2
Content available Rozkład na ułamki proste
Brociek Rafał, Pawlik Bartłomiej
MINUT Matematyka i Informatyka na Uczelniach Technicznych
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2019
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Nr 1
56-62
PL
Ważnym elementem rozwiązywania całek z funkcji wymiernych jest rozkład wyrażenia wymiernego na ułamki proste. Doświadczenie dydaktyczne Autorów podpowiada, że studenci często niepotrzebnie komplikują sobie to zadanie poprzez tworzenie nadmiernie skomplikowanego układu równań. Autorów w tym przekonaniu utwierdza również przegląd obecnej literatury objaśniającej tą metodą całkowania, jak również zapoznanie się z najpopularniejszymi materiałami e-learningowymi dostępnymi w polskim Internecie. Niniejszy artykuł ma na celu przybliżenie metody, która często w znaczny sposób ułatwia dokonanie rozkładu wyrażenia wymiernego na ułamki proste.
3
Content available On associative rational functions with multiplicative generators
Domańska K.
Scientific Issues of Jan Długosz University in Częstochowa. Mathematics
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2015
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Vol. 20
31--38
EN
We consider the class of rational functions defined by the formula F(x, y) = φˉ¹(φ(x)φ(y)), where φ is a homographic function and we describe associative functions of the above form.
4
Content available On associative rational functions with additive generators
Domańska K.
Scientific Issues of Jan Długosz University in Częstochowa. Mathematics
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2013
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Vol. 18
7--10
EN
We consider the class of rational functions defined by the formula F(x, y) = ϕ −1 (ϕ(x) + ϕ(y)), where ϕ is a homographic function and we describe all associative functions of the above form.
5
Content available On some addition formulas for homographic type functions
Domańska K.
Scientific Issues of Jan Długosz University in Częstochowa. Mathematics
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2012
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Vol. 17
17-24
EN
We deal with the functional equation (so called addition formula) of the form f(x + y) = F(f(x),f(y)), where F is an associative rational function. The class of associative rational functions was described by A. Chéritat [1] and his work was followed by a paper of the author. For function F defined by F(x,y) = ϕ−1(ϕ(x) + ϕ(y)), where ϕ is a homographic function, the addition formula is fulfilled by homographic type functions. We consider the class of the associative rational functions defined by formula F(u,v) =uv αuv + u + v, where α is a fixed real numer.
6
Content available remote Character sums and pair correlations
Vâjâitu M., Zaharescu A.
Demonstratio Mathematica
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2002
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Vol. 35, nr 2
225-232
7
Content available remote Rigidity of tame rational functions
Przytycki F., Urbański M.
Bulletin of the Polish Academy of Sciences. Mathematics
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1999
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Vol. 47, no 2
163-181
EN
We introduce and establish some basic properties of the tame rational functions. The class of these functions contains all the rational functions with no recurrent critical points in their Julia sets. For tame non-exceptional functions we prove that the Lipschitz conjugacy, the same spectra of moduli of derivatives at periodic orbits and conformal conjugavcy are mutually equivalent. We prove also the following rigidity result: If h is a Borel measurable invertible map which conjugates two tame functions f and g a.e. and if h transports conformal measure m[sub f] to a measure equivalent to m[sub g] then h extends from a set of full measure m[sub f] to a conformal homeomorphism of neighbourhoods of respective Julia sets. This extends D. Sullivan's rigidity theorem for holomorphic expanding repellers. We provide also a few lines proof of E. Prado's theorem that two generalized polinomial-like maps at zero Teichmueller's distance are holomorphically conjugate.
8
Content available remote An outline of the linear control system synthesis by a proper, stable rational functions approach
Ładziński R.
Archives of Control Sciences
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1998
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Vol. 7, no. 3/4
241-265
EN
In the process of designing controllers for linear multivariable plants specially effective are algebraic methods which require from the transfer matrices of both, the plant and the controller to be presented in coprime fractional form with factorization carried on with respect to the ring of exponentially-stable, proper real-rational functions. The main objective of the paper is to show that this form of representation with simultaneous parametrization of all linear controllers that provide internal stability of the closed-loop system can be achieved in the simplest and most natural way by analysing the system shown in Fig. 3 - the so-called basic structure. Problems of choosing the parameter to meet some important design specifications, viz. a robust asymptotic tracking of the reference signal with disturbance and noise rejection are also considered and illustrated by two representative examples covering the area of continuous- and discrete-time systems.
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