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1

Podatność dynamiczna obustronnie podpartej belki z tłumieniem w ruchu unoszenia

Żółkiewski S.

Modelowanie Inżynierskie

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2009

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T. 7, nr 38

321-328

PL

Podatność dynamiczna jest jedną z szeroko stosowanych metod analizy dynamicznej [1-3, 6-7], dzięki której możliwe staje się określenie wzajemnych relacji pomiędzy amplitudą drgań a częstotliwością wymuszenia. W pracy zamodelowano układ obustronnie podpartej belki, znajdującej się na obrotowym stole, wraz z uwzględnieniem w modelu sił tłumiących amplitudy przemieszczenia. Belka znajduje się na obrotowym stole obracającym się ze stałą prędkością kątową. Rozważany ruch ograniczono do ruchu płaskiego, natomiast belkę do belki o przekroju symetrycznym, stałym na jej całej długości.

EN

One of the most widely used method of dynamical analysis is the dynamical flexibility method [1-3, 6-7], thanks to this method we can determine mutual relations between an amplitude of vibrations and a frequency of load. In this thesis there was modeled the two-sided supported beam being on the rotational table. In this model there were took into consideration the damping forces. The beam is on the rotational table that rotates with constant angular velocity. Considered motion is limited to plane motion and the beam has a symmetrical cross-section constant on its whole length.

2

Longitudinal vibrations of mechanical systems with the transportation effect

Buchacz A., Żółkiewski S.

Journal of Achievements in Materials and Manufacturing Engineering

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2009

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Vol. 32, nr 1

29-36

EN

Purpose: this thesis purpose is a new way of modelling systems working with high speeds of mechanisms. Systems are analyzed with taking into consideration the rotational movement and with criterions of using materials with high flexibility and high precision of work. The dynamical analysis was done with giving into consideration the interaction between working motion and local vibrations. During the motion a model is loaded by longitudinal forces. Design/methodology/approach: equations of motion were derived by the Lagrange method, with generalized coordinates and generalized velocities assumed as orthogonal projections of individual quantities of the rod and manipulators to axes of the global reference frame. Findings: the model of longitudinally vibrating systems in plane motion was derived, after that the model can be transformed to the dynamical flexibility of these systems. Derived equations are the beginning of analysis of complex systems, especially can be used in deducing of the substitute dynamical flexibility of multilinked systems in motion. Research limitations/implications: mechanical systems vibrating longitudinally in terms of rotation were considered in this thesis. Successive problem of the dynamical analysis is the analysis of systems in spatial transportation and systems loaded by transversal forces. Practical implications: effects of presented calculations can be applied into machines and mechanisms in transportation such as: high speed turbines, wind power plant, water-power plants, manipulators, aerodynamics issues, and in different rotors etc. Originality/value: the contemporary analysis of beams and rods were made in a separate way, first working motion of the main system and next the local vibrations. A new way of modelling took into consideration the interaction between those two displacement. There was defined the transportation effect for models vibrating longitudinally in this paper.

3

Dynamical flexibility of the free-free damped rod in transportation

Żółkiewski S.

Journal of Achievements in Materials and Manufacturing Engineering

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2009

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Vol. 35, nr 1

71-78

EN

Purpose: of this thesis is derivation of dynamical flexibility of the free-free rod system in transportation. The well-known problem of dynamical analysis of systems in rotational transportation was developed in this work to systems with taking into consideration damping forces. Design/methodology/approach: The dynamical flexibility method was used to analysis of the free-free rod’s vibrations. Mathematical models derived in previous articles were used to derivation of the dynamical flexibility. Considerations were done by the Galerkin’s method. Findings: There were considered systems in rotational motion treated in this thesis as main transportation. Dynamical characteristics in form of dynamical flexibility as function of frequency and mathematical models were presented in this work. Research limitations/implications: Analyzed systems were simple linear homogeneous not supported rods. Working motion was limited to plane rotational motion. Future works would consider complex systems and nonlinearity. Practical implications: of derived dynamical characteristics can easily support designing process and can be put to use in stability analysis and assigning stability zones. Thank to derived mathematical models the numerical applications can be implemented and some calculations can be automated. Originality/value: Analyzing models are rotating flexible free-free rods with taking into consideration the damping forces.

4

Modelling of dynamical systems in transportation using the modyfit application

Żółkiewski S.

Journal of Achievements in Materials and Manufacturing Engineering

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2008

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Vol. 28, nr 1

71-74

EN

Purpose: of this paper is to present a numerical application for analysis and modelling dynamical flexible systems in transportation. This application enables controlling and regulation of rotating systems with the interaction between the working motion and local vibrations of elements. Design/methodology/approach: Numerical calculations are based onto mathematical models derived in other publications. The objectives of making this application were connected with emerging wants of analyzing and modelling rotating systems with taking into consideration relation between main and local motions. Theoretical considerations were made by classical methods and by the Galerkin's method. Findings: In way of increasing the value of angular velocity we can observe creating additional poles in the characteristic of dynamical flexibility and after increasing it is evident that created modes are symmetrically propagated from the original mode. It is evident, instead of modes there are created zeros. Research limitations/implications: Analyzed systems were limited to simple linear type beams and rods. Main motion is plane motion. Future research should consider complex systems and nonlinearity. Practical implications: of the application are possibilities of numerical analysis of beam and rod systems both the free-free ones and fixed ones. Engineers thank to this application can derived the stability zones of analyzed systems and can observe eigenfrequencies and zeros in the way of changing the value of angular velocity. In practice we should implement more adequate models such as those presented in this paper. Originality/value: This paper consist the description of the application called the Modyfit. The Modyfit is an implementation of derived models in a numerical environment. Those models are rotating flexible systems with consideration the transportation effect.

5

Dynamical flexibility of rod and beam systems in transportation

Żółkiewski S.

Journal of Achievements in Materials and Manufacturing Engineering

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2008

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Vol. 29, nr 2

171-174

EN

Purpose: Purpose of this paper is to present a mathematical model of rod and beam systems in transportation. The mathematical model is presented in the form of dynamical flexibility of systems. Systems are considered as flexible rotating rods and flexible rotating beams. In solution there was took into account the interaction between the main motion and local vibrations of elements. Design/methodology/approach: The dynamical flexibility was derived by the approximate method, the Galerkin's method. The example dynamical characteristics were presented in form of attenuation-frequency characteristics. The dynamical flexibility was derived on the basis of known equations of motion derived in other publications. Findings: There can be observed so called the transportation effect. This effect consist in that when analyzed system rotates with some angular velocity in the characteristic of dynamical flexibility we can notice additional poles and after increasing angular velocity it is noticeable that created modes are symmetrically propagated from the original mode. It is also palpable fact, instead of the original mode there is created zero. Research limitations/implications: Analyzed systems are simple linear type beams and rods in rotational motion. Motion was restricted to plane motion. Future research ought to consider complex systems, damped models and also nonlinearity. Practical implications: Practical implications of derived mathematical models of beam and rod systems both the free-free ones and fixed ones is a possibility of derivation of the stability zones of analyzed systems and derivation of eigenfrequencies and zeros in the way of changing the value of angular velocity. Originality/value: Presented models apply to rotating flexible rod and beam systems with taking into consideration the transportation effect. It is a new approach of analyzing rod and beam systems and can be put to use in modelling and analyzing machines and mechanisms in rotational transportation.

6

Dynamical flexibilities of mechanical rotational systems

Żółkiewski S.

Journal of Achievements in Materials and Manufacturing Engineering

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2008

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Vol. 31, nr 2

602-609

EN

Purpose: of this work is to present dynamical flexibilities of rotational beams and rods systems. The results of mathematical calculations were presented in the form of dynamical flexibility of analyzed systems. In final solution there were took into consideration the interactions between the major motions and local vibrations of subsystems. Design/methodology/approach: The dynamical flexibilities were derived by the Galerkin's method. The dynamical flexibilities for example numerical cases were presented onto charts of attenuation-frequency characteristics. The mathematical models were derived on the basis of known equations of motion derived in previous thesis's. Findings: After analysis of characteristics we can observe the transportation effect. We can notice additional poles on the characteristic of dynamical flexibility characteristics and after increasing angular velocity created modes symmetrically propagate from the origin mode and instead of the original mode there is created a zero's amplitude. Research limitations/implications: Analyzed systems are beams and rods in rotational motion. Motion was limited to plane motion. Future works will be connected with consideration of complex systems. Practical implications: of derived dynamical flexibilities of free-free and fixed beams and rods systems is a possibility of derivation of the stability zones of analyzed systems and derivation of eigenfrequencies and zeros in the function of angular velocity of work motion. Originality/value: Models analyzed in this thesis apply to rotating rod and beam systems with taking into consideration the transportation effect. This new approach of analyzing rod and beam systems can be put to use in modelling, analyzing and designing machines and mechanisms with rotational elements.

7

Analysis of mechanical systems with transversal vibrations in transportation

Buchacz A., Żółkiewski S.

Journal of Achievements in Materials and Manufacturing Engineering

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2008

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Vol. 31, nr 2

434-441

EN

Purpose: of this article are modelling and dynamic analysis of mechanical systems during the rotational movement. Nowadays technical problems are tied with high speeds of mechanisms, high precision of work, using lower density materials, and many other high demands for elements of work. Objective of this paper was the analysis with giving into consideration the interaction between working motion and local vibrations. The model is loaded by transverse forces and transformed to the global reference frame. Design/methodology/approach: derived equations of motion were made by the Lagrange equations method with generalized coordinates and generalized velocities assumed as orthogonal projections of individual coordinates and velocities of each beam to axes of the global reference frame. Findings: systems of equations of motion of transversally vibrating systems in two-dimensional motion will be put to use to derivation of the dynamical flexibility of these systems and complex systems. Those equations are the beginning of the analysis of complex systems. They can also be used to derivation of the substitute dynamical flexibility of n-linked systems. Research limitations/implications: mechanical systems vibrating transversally in terms of two-dimensional motion were considered in the thesis. The consecutive problem of dynamical analysis is modelling of systems in spatial motion and also the analysis of systems loaded by longitudinal forces. Practical implications: mathematical effects of this article can be put to use into many mechanisms and machines running in rotational transportation. For example applications are: high speed turbines, wind power plants, rotors, manipulators and in aerodynamics issues, etc. Of course results should be adopted and modified to appropriate system. Originality/value: High demands for parameters of work of mechanisms and machines are the postulation for new research and new ways of modelling and analyzing those type systems. The example way of solution such systems is presented in this thesis. The transportation effect for models vibrating transversally was defined.

8

Analysis and modelling of rotational systems with the Modyfit application

Żółkiewski S.

Journal of Achievements in Materials and Manufacturing Engineering

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2008

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Vol. 30, nr 1

59-66

EN

Purpose: of this article is to present an application to analysis and modelling of rotational systems. Independent modules of application enables controlling and regulation of characteristics of systems in transportation. The base problem of analyzing systems in transportation is expressing the interaction between the main motion and local vibrations of subsystems. Design/methodology/approach: Mathematical models derived in previous articles were used to implement the mathematical models in numerical calculations. The objectives of creating the Modyfit application were connected with analyzing and modelling rotating systems with taking into account relations between major and local motions. Considerations were done by the Galerkin's method. Findings: There are many effects of analyzing systems with rotation consideration. Main of them are connected with analyzing systems in function of increasing the value of angular velocity. In such way of analyzing we can observe creating additional poles in the dynamical characteristics and we can also observe that instead of modes there are created zeros. Research limitations/implications: Analyzed systems are simple linear homogeneous beams and rods. Working motion is limited to plane motion. Future research would consider complex systems, damping and nonlinearity. Practical implications: of the application are numerical analysis of beams and rods in rotational transportation and designing such systems. Thank to the Modyfit we can derived the stability zones of analyzed beams and rods and we can observe eigenfrequencies and zeros in function of angular velocity. Originality/value: The Modyfit is a forerunner implementation of derived models in a numerical environment of dynamical flexibility. Analyzing models are rotating flexible systems with consideration the rotational transportation effect.

9

Mechanical systems vibrating longitudinally with the transportation effect

Buchacz A., Żółkiewski S.

Journal of Achievements in Materials and Manufacturing Engineering

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2007

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Vol. 21, nr 1

63-66

EN

Purpose: High work speeds of mechanisms, using materials with high flexibility, high precision of work, etc. are the cause of searching of the new ways of modelling. One of these ways is presented in this thesis. The main purpose of this thesis is the dynamical analysis with taking into consideration the interaction between main motion and local vibrations during the model is loaded by longitudinal forces. Design/methodology/approach: Derived equations of motion were made by classical methods, with generalized coordinates and generalized velocities assumed as orthogonal projections of individual coordinates and velocities of the rod and manipulators to axes of the global inertial frame. Findings: Mathematical model of the longitudinally vibrating systems in terms of plane motion can be put to use to derivation of the dynamical flexibility of these systems, and also those equations are the starting point to the analysis of complex systems, especially we can use those equations to derivation of the substitute dynamical flexibility of n-linked systems in transportation. Research limitations/implications: In the thesis were considered mechanical systems vibrating longitudinally in terms of rotation. Next problem of dynamical analysis is the analysis of systems in non-planar transportation and systems loaded by transversal forces. Practical implications: Results of this thesis can be put to use into machines and mechanisms in transportation such as: wind power plant, high speed turbines, rotors, manipulators and in aerodynamics issues, etc. Originality/value: Up to now there were analyzed beams and rods in a separate way, first main motion of the system and after that the local vibrations. The new approach of modelling were presented by authors of this thesis, a new modelling took into consideration the interaction between those two displacements. There was defined the transportation effect for models vibrating longitudinally in this thesis.

10

Dynamic analysis of the mechanical systems vibrating transversally in transportation

Buchacz A., Żółkiewski S.

Journal of Achievements in Materials and Manufacturing Engineering

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2007

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Vol. 20, nr 1-2

331-334

EN

Purpose: Purpose of this paper is analysis and modelling of mechanical systems in transportation. The contemporary technical problems are lashed with high work demands such as high speeds of mechanisms, using lower density materials, high precision of work, etc. The main objective of this thesis was the dynamical analysis with taking into consideration the interaction between main motion and local vibrations during the model is loaded by transverse forces. Design/methodology/approach: Equations of motion were derived by classical methods, the Lagrange equations with generalized coordinates and generalized velocities assumed as orthogonal projections of individual coordinates and velocities of the beam and manipulators to axes of the global inertial frame. Findings: Presented mathematical model of the transversally vibrating systems in planar transportation can be put to use to derivation of the dynamical flexibility of these systems, moreover those equations are the starting point to the analysis of complex systems. In particular we can use those equations to derivation of the substitute dynamical flexibility of multibody systems. Research limitations/implications: There were considered mechanical systems vibrating transversally in terms of plane motion. Next problem of dynamical analysis is the analysis of systems in non-planar transportation and systems loaded by longitudinal forces. Practical implications: Results of this thesis can be put to use into all machines and mechanisms running in transportation such as wind power plants, high speed turbines, rotors, manipulators and in aerodynamics issues, etc. Some results ought to be modified and adopted to appropriate models. Originality/value: High requirements applying to parameters of work of machines and mechanisms are caused the new research and new ways of modelling and analyzing those systems. One of these ways are presented in this thesis. There was defined the transportation effect for models vibrating transversally.