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1

Pulse wave propagation along human aorta: a model study

Kizilova Natalya, Mizerski Jeremi, Solovyova Helen

Journal of Theoretical and Applied Mechanics

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2020

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Vol. 58 nr 1

17--34

EN

In the study, wave propagation along aorta is studied for different normal and pathological conditions in distal arteries. The mathematical model is based on the axisymmetric incompressible Navier-Stokes equations for blood and momentum equations for an incompressible viscoelastic arterial wall. The solution has been found as a superposition of forward and backward running waves. The blood pressure and flow curves measured by ultrasound in larger systemic arteries of ten healthy volunteers have been used for identification of the model parameters. It is shown that individual geometry plays an essential role in the location of positive and negative wave reflection sites along the aorta and, thus, in the pressure and flow patterns as well as blood distribution into the side branches. The model is validated by comparative study with the same dependencies computed previously on a 55-tube model as well as on the measurement data. The model can be used for determination of the individual parameters for patient-specific cardiovascular models and further in silico modeling of the outcomes of surgical and therapeutic procedures.

2

Analiza modalna narzędziem mechaniki stosowanej w identyfikacji stanu budowli

Żółtowski M.

Logistyka

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2014

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nr 6

PL

Istnieje potrzeba ciągłego doskonalenia metod badania drganiowego elementów budowlanych i całych konstrukcji narażonych na destrukcję. Wprowadzane do budownictwa nowe materiały i technologie oraz nowe rozwiązania konstrukcyjne, umożliwiają zwiększenie wydajności i jakości wytworów, lecz towarzyszą im często groźne i duże obciążenia dynamiczne. Nowe narzędzia w tym obszarze badań dotyczą możliwości wykorzystania metod analizy modalnej, a także nowoczesnego pozyskiwania i przetwarzania procesu drganiowego. Do oceny jakości inżynierii produkcji konstrukcji i elementów murowych wykorzystuje się ostatnio wiele różnych estymatorów drganiowych, które stanowią podstawę nowych metod badawczych (analiza modalna) co stanowi podstawę rozważań tej pracy.

EN

There is a need for continuous improvement of vibration testing methods of building elements and whole structures exposed to destruction. Introduced for building new materials and technologies and new engineering solutions for performance and quality creations, but often accompanied by severe and large dynamic loads. New tools in this area of research relate to the possibility of use of modal analysis method, as well as a modern acquiring and processing of vibration process. To assess the quality of the design and manufacturing engineering masonry recently used a number of different estimators of vibration, which are the basis a new research methods (modal analysis) as a basis for discussion of this work.

3

Boundary conditions in models of power plant components under thermal loading

Okrajni J., Twardawa M.

Archives of Materials Science and Engineering

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2013

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Vol. 62, nr 1

28--35

EN

Purpose: The main purpose of the work is the description of conditions of fatigue process of power plant components working under mechanical and thermal loading. The work focuses on the chosen component characteristics. The issue of influence of the heat transfer conditions on the component surface on stresses changes in time has been discussed. Design/methodology/approach: The FEM modelling method has been used to describe the behaviour of the chosen component. The models have been validated on the basis of temperature measurements during operation period. Findings: It has been shown that the determination of the effects induced by unsteady conditions of start-up and shut-down of installations requires application of unconventional methods of research and analysis of their results. In such a case, a methodology can be applied consisting in combining the methods of computer modelling of temperature fields with temperature measurements in selected points of the component. Research limitations/implications: The presented analysis is the part of the complex investigation method which main purpose is increasing the accuracy of the thermo-mechanical fatigue process description. In such situation the investigations curried out in the work give the model approach and data for the comparison the real behaviour with the predictions. However the work is focused only on the chosen component and chosen characteristics of loading. Practical implications: The method of the chosen component behaviour analysis used in the paper could be useful in the practical cases when the real components mechanical behaviour would be analysed and their fatigue life would be assessed. Originality/value: The main value of this paper is the own method of the mechanical behaviour analysis of the power plant components. This method includes FEM modelling and assumption that the heat transfer coefficient should be treated as dependent on time. The material stress-strain behaviour has been treated as the local phenomenon that could be modelled.

4

Badania dyfuzyjności cieplnej wody metodą wymuszenia okresoweg

Szczepaniak R., Panas A. J., Nowakowski M., Panas J.

Problemy Mechatroniki : uzbrojenie, lotnictwo, inżynieria bezpieczeństwa

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2012

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Vol. 3, Nr 3 (9)

85-97

PL

W pracy przedstawiono zastosowanie zmodyfikowanej metody wymuszeń okresowych do określenia dyfuzyjności cieplnej wody. Eksperyment objął przedział temperatury od -13,6°C do +12,0°C. Dyfuzyjność cieplną wyznaczono z danych zarówno amplitudowych, jak i fazowych. W odróżnieniu od klasycznego sposobu postępowania, w obliczeniach wykorzystano nieprzybliżone zależności modelu analitycznego generujące równania przestępne. Badania potwierdziły skuteczność zastosowanej metody badań, która może być wykorzystana do zweryfikowania dotychczas publikowanych danych.

EN

The modified periodic temperature oscillation i.e. Angstrom technique at scanning mode operation has been applied to perform continuous measurements of the thermal diffusivity of water. The investigations has been done at cooling from 12,0°C to -13,6°C. The thermal diffusivity has been calculated independently from two transcendental relations based on the measured amplitude attenuation and the measured phase shift of the temperature oscillation respectively. The experiments have proved performance of the applied method for systematic studies of thermophysical properties of ice accretions. Unique results of the thermal diffusivity of water supercooled to about -5°C have been obtained also.

5

Stephen P. Timoshenko : klasyk, uczony i nauczyciel mechaniki teoretycznej i stosowanej

Szcześniak W.

Logistyka

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2012

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nr 3

PL

W referacie plenarnym omówiono wkład S. P. Timoshenki w rozwój mechaniki teoretycznej i przede wszystkim Jego wkład do mechaniki technicznej-stosowanej. Jest to referat okolicznościowy z okazji 40 rocznicy śmierci tego Uczonego. Podano skrócony Ŝyciorys Timoshenki, przedstawiono monografie i podręczniki wychodzące w świat do tej pory. Omówiono osiągnięcia naukowe w mechanice w okresie przedrewolucyjnym w Rosji Carskiej do 1920 roku, następnie na emigracji w Chorwacji i USA. Przedstawiono związki Profesora z Polską i polskimi uczonymi tego okresu. W końcu omówiono kilka prac autora referatu, w których pośrednio lub bezpośrednio wykorzystano idee Timoshenki. Referat przygotowano na podstawie przede wszystkim autobiografii S. P. Timoszenki.

EN

The paper presents the contribution of Timoshenko to the development of theoretical mechanics and his contributions to engineering and applied mechanics. This is a occasional lecture on the 40th anniversary of the death of the scientist. The biography of Timoshenko is presented as well as outgoing monographs and textbooks in the world so far. Discusses the scientific achievements in mechanics during the pre-revolutionary Tsarist Russia until 1920, then in exile in Croatia and the USA. Professor's compounds with Poland and with Polish scholars of the period are presented. Finally, some of the author's papers, in which directly or indirectly Timoshenko's ideas used, are discussed. The paper has been prepared on the basis of Timoshenko's autobiography most of all.

6

Formulating of reverse task of chosen class of mechatronic systems

Buchacz A.

Journal of Achievements in Materials and Manufacturing Engineering

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2012

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Vol. 54, nr 1

75--82

EN

Purpose: of this paper is modelling by means of the first and second category graphs and analysis of vibrating subsystem of mechatronic systems by means of the exact and approximate methods. Design/methodology/approach: Approach was to nominate the relevance or irrelevance between the characteristics obtained by means of the exact method (only for the mechanical subsystem) and the approximate method. Such formulation concerns mostly the relevance of the natural frequencies-poles of the characteristics both mechanical subsystems and mechatronic systems. Findings: are approximate solutions requiring all the conditions for torsionally vibrating mechanical and/or mechatronic systems. It is an introduction to synthesis of these systems modelled by graphs of the considered category. Research limitations/implications: is both torsional vibrating continuous mechanical subsystem and mechatronic systems of the linear continuous type. Practical implications: of this work is to present the introduction to synthesis of considered class of mechatronic bar-systems with a constant changeable cross-section. Originality/value: Originality of such formulation is focused on the use of the different category graphs for modelling and synthesising by means of the continued fraction expansion method represented by graphs of torsionally vibrating bars to the synthesis of discrete-continuous mechatronic systems.

7

Deformation process of the material of mine powered roof supports in low-cycle fatigue conditions

Okrajni J., Plaza M., Jaszczuk M.

Journal of Achievements in Materials and Manufacturing Engineering

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2012

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Vol. 50, nr 2

59--65

EN

Purpose The main purpose of the work is the description of the low-cycle fatigue process of mine powered roof supports working under mechanical high loading. The work focuses on the chosen component strain-stress characteristics. The issue of modelling the stress-strain behaviour of powered roof supports components during low-cycle fatigue has been discussed. Design/methodology/approach: The FEM modelling and Neuber’s method have been used to describe the local stress-strain behaviour of the chosen component. Findings: In the examined devices, variable stress and strain values were calculated for a chosen characteristic load cycle. Diagrams in the form of a hysteresis loop determined using Neuber’s hypothesis and FEM were compared. The values of the range of equivalent strain determined for multiaxial stress states using the finite element method proved to be close to those estimated via Neuber’s method. Research limitations/implications: The presented analysis is the part of the complex investigation method which main purpose is increasing the accuracy of the low-cycle fatigue process description. In such situation the investigations curried out in the work give the model approach and data for the comparison the real behaviour with the predictions. However the work is focused only on the chosen component and chosen characteristics of loading. Practical implications: The method of stress-strain behaviour analysis used in the paper could be useful in the practical cases when the real components mechanical behaviour would be analysed and their fatigue life would be assessed. Originality/value: The main value of this paper is the own method of the mechanical behaviour analysis of the powered roof support component. This method includes FEM modelling and Neuber’s method of the stress-strain characteristics assessment. The material stress-strain behaviour has been treated as the local phenomenon that could be modelled.

8

The supply of formal notions to synthesis of the vibrating discrete-continuous mechatronic systems

Buchacz A.

Journal of Achievements in Materials and Manufacturing Engineering

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2011

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Vol. 44, nr 2

168--178

EN

Purpose of this paper is modeling by different category graphs and analysis of vibrating clamped - free mechatronic system by the approximate method called Galerkin’s method. Such approach considers the frequency - modal analysis and assignment of amplitude - frequence charcteristics of the mechatronic system. Design/methodology/approach: was to nominate the relevance or irrelevance between the characteristics obtained by exact - only for shaft - and considered method. Such formulation especially concerns the relevance the relevance of the natural frequencies-poles of characteristics both of mechanical subsystem and the discrete - continuous clamped - free vibrating mechatronic system. Finding this approach is a fact, that approximate solutions fulfill all conditions for vibrating mechanical and/ or mechatronic systems and can be an introduction to synthesis of these systems modeled by different category graphs. Research limitations/implications: Research limitation is that both torsional vibrating continuous mechanical subsystem and mechatronic discrete - continuous subsystems are linear discrete - continuous are linear systems. Practical implications: of this study is that the main point can be the introduction to synthesis of considered class mechatronic bar-systems with constant changeable cross-section. Originality/valut Originality of such formulation rely on the use of the hypergraph methods of modelling and synthesis of torsionally vibrating bars to the synthesis of discrete-continuous mechatronic systems.

9

Selected aspects of power distribution for 4wd vehicle driven by diagonal power transmission

Baier A., Mydlarz J.

Journal of Achievements in Materials and Manufacturing Engineering

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2011

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Vol. 44, nr 1

73--79

EN

Purpose: Paper describes phenomena related to the vehicle acceleration on the curve, with the high speed, considering inertia forces and driving force. Design/methodology/approach: The phenomena were investigated with three tools: classical analytical mechanics, computer simulation and tests with physical model. Findings: Simulations and physical tests led to similar results. However, new diagonal transmission system showed advantages only on slippery surface of the test track. Nevertheless such a behavior resulted in conclusion that diagonal transmission system would be useful influencing vehicle safety, specifically at slippery roads. Practical implications: There is a possibility of diagonal transmission system application for sport vehicles to improve performance. For utilitarian vehicles increasing safety is the most important aspect, especially in cooperation with ESP system. When it comes to special and military vehicles both mentioned above features are desired. Possible power-pack application is an additional advantage. Originality/value: The goal was the comparison of standard 4x4 transmission system with unconventional patented 4x4 transmission system called diagonal one.

10

Indication of the suitable model of a mechatronic system as an introduction to the synthesis task

Płaczek M.

Journal of Achievements in Materials and Manufacturing Engineering

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2011

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Vol. 49, nr 2

338--349

EN

Purpose: The identification of the optimal mathematical model that meets the assumed criteria is the main purpose of this paper, which is an introduction to the task of synthesis of one-dimensional vibrating mechatronic systems. Assumed criteria are to provide the accurate analysis of the system together with maximum simplification of used mathematical tools and minimize required amount of time. The correct description of a given system by its model during the design phase is a fundamental condition for proper operation of it. Therefore, the processes of modelling, testing and verification of used models were presented. On the basis of carry out analysis the optimal (in case of assumed criteria) model was selected and it will be used to realize the task of synthesis in future works. Design/methodology/approach: A series of mathematical models with different simplifying assumptions was created. Using the created models and corrected approximate Galerkin method the dynamic characteristic of the considered system was designated. The analysis of an influence of parameters of the system’s components on obtained characteristic was conducted. The approximate method was verified to check its accuracy and decide if it can be used to analyse such kind of mechatronic systems. Findings: The main result of the work is an indication of the suitable mathematical model of the considered system. Research limitations/implications: Influence of temperature changes on the transducer’s properties was neglected in developed mathematical models. It will be considered in the future works. Practical implications: Presented method of mechatronic system’s analysis can be use in process of designing of technical devices where both, simply and reverse piezoelectric effects can be used. Originality/value: Development of the mathematical models in which the considered system is modelled as a combined beam.

11

Hypergraphs of simple beams-models of their analysis in synthesis of complex beam-systems

Buchacz A.

Journal of Achievements in Materials and Manufacturing Engineering

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2011

|

Vol. 49, nr 2

233--242

EN

Purpose: of this paper is the analysis of vibrating beam by the exact and approximate methods and creating the hypergraphs of the beam concerning of two methods of analysis. Design/methodology/approach: was to nominate the relevance or irrelevance between the characteristics obtained by considered methods - especially concerning the relevance of the natural frequencies-poles of beams characteristics. The main subject of the research is to solve the continuous free-pinned (F-P) and clamped-sliding (C-S) beams as a subsystems of vibrating beam-system. Findings: this approach is a fact, that approximate solutions fulfill all conditions for vibrating beams and can be introduction to synthesis of these systems modeled by hypergraphs. Research limitations/implications: is that linear continuous transverse vibrating (F-P) and (C-S) beams are considered. Practical implications: of this study is the main point is the introduction to synthesis of transverse vibrating continuous beam-systems. Originality/value: of this approach considers the application Galerkin’s method which concerns the analysis of beams and modeling them of transformed hypergraphs.

12

Fatigue aspects of an evaluation of the operational safety of components working in creep conditions

Okrajni J.

Journal of Achievements in Materials and Manufacturing Engineering

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2011

|

Vol. 45, nr 1

51--58

EN

Purpose: The main purpose is the method of the description of the thermo-mechanical fatigue process of power plant components working under mechanical and thermal loading. The work focuses on the chosen component strain-stress characteristics and their strength. The paper discusses the issue of modelling the heating and cooling processes of components in a power plants in the start-up and shut-down conditions of a boiler. Design/methodology/approach: The FEM modelling has been used to describe the local stress-strain behaviour of the chosen component. Findings: The calculations of stress distribution on the chosen component surface show that the internal pressure induces considerably smaller values of stresses and strains in comparison with the same stresses specified for thermal loads. However, it should be noted that the impact of temperature gradients and thermal stresses is usually short-lived, therefore, its influence on creep processes is less significant in comparison to pressure load. Material fatigue is mainly the effect of thermal stresses. Thus, thermal impacts are responsible for cracks initiation and growth in areas of the greatest intensity of damage accumulation. Research limitations/implications: The presented analysis is the part of the complex investigation method which main purpose is increasing the accuracy of the TMF process description and thermo-mechanical life assessment. The possibility of applying the fatigue durability criteria currently assumed in standards still requires justification and confirmation in laboratory and industrial conditions to be closer to the real components behaviour. In such situation the industrial investigations curried out in the work give the model approach and data for the comparison the real behaviour with the predictions. Practical implications: The method of stress-strain behaviour analysis used in the paper could be useful in the practical cases when the real components mechanical behaviour would be analysed. Originality/value: The main value of this paper is the own method of the mechanical behaviour analysis of the power plant component. This method includes the temperature fields analysis taking into account the boundary conditions based on the operation parameter data and the thermoplastic material model. The material stress-strain behaviour has been treated as the local phenomenon, that could be modelled by FEM.

13

Design of damping systems with required frequency spectrum

Dymarek A., Dzitowski T.

Journal of Achievements in Materials and Manufacturing Engineering

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2011

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Vol. 49, nr 2

269--274

EN

Purpose: The presented work extends the task of synthesis with new methods, with particular focus on the method of active and passive synthesis of vibrating damping mechanical systems. This paper is concerned with the formulation and formalisation of the problem of active and passive synthesis of mechanical damping systems. Design/methodology/approach: The work proposes a combination of the synthesis method of vibrating systems with the method of determining the active force as the method for active synthesis of mechanical damping systems and the values of two-terminal damping in case passive synthesis of mechanical damping systems. Findings: The performed active and passive synthesis of restrained systems has demonstrated an ambiguity of synthesis of damping systems. The resulting structures and set system parameters are not the only ones that can be obtained in the case of the assumed frequencies. Research limitations/implications: The scope of discussion is the synthesis of discrete damping systems. Practical implications: The advantage of the proposed method of determining the active force in the system is the ease of its programming. The proposed method can be included in the method of proportional control with force feedback from the state. Originality/value: The document is also an attempt at identifying new opportunities and a direction of research in the design of machine components with a desired frequency spectrum.

14

Stephen Timoshenko - ojciec mechaniki stosowanej (okres emigracyjny). cz. 2

Dąbrowiecki K.

Nowoczesne Budownictwo Inżynieryjne

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2010

|

Nr 4

70-72

PL

W numerze 1 (22) "Nowoczesnego Budownictwa Inżynieryjnego" z 2009 r. ukazała się pierwsza część materiału poświęconego prof. Stephenowi Timoshence (Stiepanowi Prokofiewiczowi Timoszence) 1878.1972, światowej sławy specjaliście w dziedzinie mechaniki stosowanej Artykuł ten opisywał działalność profesora w okresie przedemigracyjnym. Jednak najwięcej szeroko znanych prac naukowych napisał i opublikował, będąc na emigracji w Stanach Zjednoczonych i właśnie tego okresu jego życia i działalności dotyczy poniższy artykuł.

15

Parameter identification of a full-car model for active suspension design

Guida D., Nilvetti F., Pappalardo C. M.

Journal of Achievements in Materials and Manufacturing Engineering

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2010

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Vol. 40, nr 2

138-148

EN

Purpose: A method for identifying parameters of a full-car model for active suspension design Design/methodology/approach: The method is based on ERA/OKID identification procedure developed by Juang [1]. Findings: Numerical results show a good agreement with real system parameters. Research limitations/implications: Results show that this method works well only with a high signal-to-noise ratio. Practical implications: This method can be used as an aid to active suspension design. Originality/value: This procedure has been tested on the full-car model of suspension system in two cases: a) full-state measurements; b) not full-state measurements.

16

Life and operational safety of power systems and chemical plants

Okrajni J.

Journal of Achievements in Materials and Manufacturing Engineering

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2010

|

Vol. 43, nr 1

38-63

EN

Purpose: The problem addressed in the paper is the description of an effort and durability of components under the conditions of mechanical and thermal interactions. The problems of proper material testing methods have been shown as the important part of the components life assessment methodology. Design/methodology/approach: The FEM modelling has been used to determine the stress and strain fields in the components and to describe their behaviour under mechanical and thermal loading. Findings: An appropriate models description has been developed. So far, experimental verification of the usefulness of the model description to determine the stress and strain patterns in particular object and for chosen operation conditions has been made. Research limitations/implications: The developed description should be useful in problems of behaviour predictions of high temperature components and their durability assessment under different mechanical and thermal loadings in industry practical applications. Originality/value: The method, which more precise description of power industry components behaviour makes possible have been shown in the work. The work is addressed to researchers interested in problems of component behaviour prediction under different loadings that we can meet in the operation practice and to power industry engineering maintenance staff.

17

Graphs different category of subsystem as models to synthesis of transverse vibrating beam - system

Buchacz A.

Journal of Achievements in Materials and Manufacturing Engineering

|

2010

|

Vol. 43, nr 2

644--650

EN

Purpose: of this paper is modeling by different categorygraphs and analysis of vibrating clamped - free beam as subsystem of transverse vibrating beam-system by the exact and approximate methods and creating the hypergraphs of the beams in case of presented methods of analysis. Design/methodology/approach: was to nominate the relevance or irrelevance between the characteristics obtained by considered methods - especially concerning the relevance of the natural frequencies-poles of characteristics of considered beam. The main subject of the research is the continuous clamped - free beam with constant cross sections as a subsystem of transverse vibrating beam - system. Findings: this approach is fact, that approximate solutions fulfill all conditions for vibrating beams and can be introduction to synthesis of these systems modeled by different category graphs. Research limitations/implications: is that linear continuous transverse vibrating clamped-free beam is considered. Practical implications: of this study is the main point is the introduction to synthesis of transverse vibrating continuous beam-systems with constant changeable cross-section. Originality/value: of this approach relies on application approximate methods of analysis of clamped - free beam and modeling the one of transformed hypergraph.

18

Dry friction of bearings on dynamics and control of an inverted pendulum

Guida D., Nilvetti F., Pappalardo C. M.

Journal of Achievements in Materials and Manufacturing Engineering

|

2010

|

Vol. 38, nr 1

80-94

EN

Purpose: Investigation of dry friction parameters on the control system of a cart pendulum. Design/methodology/approach: A geometrical approach has been used to analyze the influence of bearing friction on the cart-pendulum dynamics. This approach was developed by one of authors in the paper “ Influence of the Variation between Static and Kinetic Friction on Stick-Slip Instability” pubblished by WEAR 1993. Findings: Relations that allow us to evaluate how the friction parameters of bearings influence the performances of the control system of a cart-pendulum. Research limitations/implications: The “a priori knowing” of friction parameters is the intrinsic limitation of this method. Originality/value: A novel methodology for designing of more effective control systems.

19

Computer models of steam pipeline components in the evaluation of their local strength

Okrajni J., Essler W.

Journal of Achievements in Materials and Manufacturing Engineering

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2010

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Vol. 39, nr 1

71-78

EN

Purpose: The paper discusses the issue of modelling the heating and cooling processes of T-pipes in a power plant pipeline in the start-up conditions of a boiler. The main purpose of this work is the description of the mechanical behaviour of power plant components working under mechanical and thermal loading and validation of the computer modelling methods. Design/methodology/approach: The FEM modelling has been used to describe the local stress-strain behaviour of the chosen component. Findings: The reasons for the presence of high and variable in time temperature gradients in the components of the main steam pipeline include, among other things, variable values of the coefficient of heat transfer between the pipeline material and the medium flowing inside it, which, at this stage of boiler operation, may change its state. Unsteady operation of a pipeline, especially in case of subsequent boiler start-ups, may induce thermal stresses which exceed the values of allowable stress in components of complex shapes. Research limitations/implications: The possibility of applying the durability criteria currently assumed in standards still requires justification and confirmation in laboratory and industrial conditions to be closer to the real components behaviour. In such situation the presented analysis is the part of the complex investigation method which main purpose is increasing accuracy of the TMF process description and thermo-mechanical life assessment. Practical implications: The calculations carried out may constitute a basis for developing a material test parameters which would bring closer the fatigue conditions appearing locally in the analysed components. The method of stress-strain behaviour analysis used in the paper could be useful in the practical cases when the real components mechanical behaviour would be analysed. Originality/value: The main value of this paper is the own method of the mechanical behaviour analysis of the power plant component. This method includes the temperature fields analysis taking into account the boundary conditions based on the operation parameter data and the thermoplastic material model. The material stress-strain behaviour has been treated as the local phenomenon, that could be modelled by FEM.

20

Analysis of probability as an impulse to synthesis of a beam-system represented by hypergraphs

Buchacz A.

Journal of Achievements in Materials and Manufacturing Engineering

|

2010

|

Vol. 38, nr 1

72-79

EN

Purpose: of this paper is to analyze vibrating beam by an exact and approximate methods and to create hypergraphs of the beam in case of two methods of analysis. Design/methodology/approach: was to nominate the relevance or irrelevance between the characteristics obtained by the considered methods - especially concerning the relevance of the natural frequency-poles of beam characteristics. The main subject of the research is a continuous free beam as a subsystem of vibrating beam-system with constant cross sections. Findings: this approach is that approximate solutions fulfil all conditions for vibrating beams and can be an introduction to synthesis of these systems modelled by hypergraphs. Research limitations/implications: linear continuous flexibly vibrating free beam is considered Practical implications: of this study is mainly the introduction to synthesis of flexibly vibrating continuous beam-systems. Originality/value: of this approach is about application of approximate methods of analysis of a beam and modelling the one of transformed hypergraph.