Abstract:
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

Abstract:
Purpose: of this paper is the application of the synthesis method according to realization of mobility or immobility function into partial fraction when the level of the denominator of characteristic is higher than the level of its numerator.Design/methodology/approach: was formulated and formalized by the problem of obtaining the “new” discrete vibrating mechanical systems with branched structures. The models of mechanical systems were represented by polar graphs. The reverse problem of dynamics of defined class of vibrating mechanical systems was also formalized and solved.Findings: this study is that the same class of polar graph is a model of mechanical system with “old” and “new” branched structure. Obtaining structures of graphs as models of mechanical systems are a physical realization of dynamical characteristics, which may be considered as the immobility or mobility function.Research limitations/implications: is that the linear vibrating discrete mechanical systems with branched structures were considered.Practical implications: of this researches was that another approach was presented, which that means unclassical method of modeling of different structures of mechanical systems in form of polar graphs was used. The used method of synthesis and the obtained results can be of some value for designers of designated class of vibrating mechanical systems.Originality/value: of this paper is that the only one polar graph is obtained as a model of not only one mechanical system. The results are obtained after distribution of two dynamical characteristics into partial fraction of finding structures of mechanical systems. This approach is other than those considered elsewhere.

Abstract:
Purpose: Application of approximate method was main purpose of work to solution task of assignment of frequency-modal analysis and characteristics of mechatronical system.Design/methodology/approach: The problem in the form of set of differential equation of motion and state equation of considered mechatronical model of object has been formulated and solved. Galerkin’s method to solving has been used. The considered torsionally vibrating mechanical system is a continuous bar of circular cross-section, clamped on one end. A ring transducer, which is the integral part of mechatronical system, extorted by harmonic voltage excitation is assumed to be perfectly bonded to the bar surface.Findings: Parameters of the transducer have important influence of values of natural frequencies and on form of characteristics of considered mechatronical system. The poles of dynamical characteristic calculated by mathematical exact method and the Galerkin’s method have approximately the same values. The results of the calculations were not only presented in mathematical form but also as a transients of examined dynamical characteristic which are function of frequency of assumed excitation.Research limitations/implications: In the paper the linear mechatronical system has been considered, but for this kind of systems the approach is sufficient.Practical implications: In article other approach is presented, that means in domain frequency spectrum the analysis has been considered.Originality/value: The mechatronical system created from mechanical and electrical subsystems with electromechanical bondage has been considered. This approach is other from considered so far. Using methods and obtained results can be value for designers of mechatronical systems.

Abstract:
Purpose: of this paper is the application of the approximate method called Galerkin’s method to solve the task of assigning the frequency-modal analysis and characteristics of a mechatronic system.Design/methodology/approach: was the formulated and solved as a problem in the form of a set of differential equations of the considered mechatronic model of an object. To obtain the solution, Galerkin’s method was used. The discussed torsionally vibrating mechatronic system consists of mechanical system, which is a continuous bar of circular cross-section, clamped on its ends. The electrical subsystem of the considered mechatronic system is a ring transducer to be perfectly bonded to the bar surface.Findings: this study is that the parameters of the transducer have an important influence on the values of natural frequencies and on the form of the characteristics of the said mechatronic system. The results of the calculations were not only presented in a mathematical form but also as transients of the examined dynamical characteristic which are a function of frequency of the assumed excitation.Research limitations/implications: is that the linear mechatronic system was considered, for this type of systems, such approach is sufficient.Practical implications: of this researches was that another approach is presented, that means in the domain of frequency spectrum analysis. The method used and the obtained results can be of some value for designers of mechatronic systems.Originality/value: of this paper is that the mechatronic system, created from mechanical and electrical subsystems with electromechanical bondage was examined. This approach is other than those considered elsewhere.

Abstract:
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.

Abstract:
Purpose: The application of approximate method for solving the task of assignment the frequency-modal analysis and characteristics of flexibly vibrating mechatronic system, because for considered case of boundary conditions exact and approximate methods for the coordinates are equivalent.Design/methodology/approach: Formulate and solve the problem in the form of a set of differential equations of motion and state equations of the considered mechatronic model of an object Galerkin’s method was used. The considered flexibly vibrating mechanical system is a continuous beam, clamped at one of its ends. An integral part of the mechatronic system is a transducer perfectly bonded to the beam surface.Findings: The parameters of the transducer exert an important influence on the values of natural frequencies and on the form of the characteristics of the discussed mechatronic system.Research limitations/implications: The linear mechanical subsystem and linear electrical subsystem of the mechatronic system were analyzed and the theory Euler-Bernoulli is used for the beam; however, this approach is sufficient for such systems.Practical implications: Global approach is presented in the domain of frequency spectrum analysis. The methods of analysis and the obtained results my give grounds for designing and investigating this type of mechatronic systems.Originality/value: The mechatronic system created from mechanical and electric subsystems with electromechanical bondage has been considered. This approach is different from those considered so far.

Abstract:
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.Findings: 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/value: 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.

Abstract:
Purpose: The purpose of this paper is application of approximate method of solving the task of assignment the frequency-modal analysis and characteristics of flexibly vibrating mechatronic system.Design/methodology/approach: The main approach of the subject was to formulate and solve the problem in the form of set of differential equation of motion and state equation of considered mechatronic model of object. Galerkin’s solving method has been used. The considered flexibly vibrating mechanical system is a continuous beam, clamped at one of its end. Integral part of mechatronic system is a transducer, extorted by harmonic voltage excitation, to be perfectly bonded to the beam surface.Findings: The parameters of the transducer have important influence on values of natural frequencies and on form of characteristics of the discussed mechatronic system.Research limitations/implications: In the paper the linear mechanical subsystem and linear electric subsystem of mechatronic system has been considered, however for this kind of systems the approach is sufficient.Practical implications: The methods of analysis and obtained results can be base on design and investigation for this type of mechatronic systems.Originality/value: The mechatronic system formed from mechanical and electric subsystems with electromechanical bondage has been considered. This approach is different from those considered so far.

Abstract:
Purpose: The purpose of this paper is application of approximate method of solving the task of assignment the frequency-modal analysis and characteristics of mechatronic system.Design/methodology/approach: The main approach of the subject was to formulate and solve the problem in the form of set of differential equation of motion and state equation of considered mechatronic model of object. Galerkin’s method to solving has been used. The considered torsionally vibrating mechanical system is a continuous bar of circular cross-section, clamped at one of its end. Integral part of mechatronic system is a ring transducer, extorted by harmonic voltage excitation, to be perfectly bonded to the bar surface.Findings: The parameters of the transducer have important influence of values of natural frequencies and on form of characteristics of the discused mechatronic system. The results of the calculations were not only presented in mathematical form but also as a transients of examined dynamical characteristic which were function of frequency of the excitation.Research limitations/implications: In the paper the linear mechanical subsystem and linear electric subsystem of mechatronic system has been considered, however for this kind of systems the approach is sufficient.Practical implications: The methods of analysis and obtained results can be base of design and investigation for this type of mechatronic systems.Originality/value: The mechatronic system created from mechanical and electric subsystems with electromechanical bondage has been considered. This approach is different from those considered so far.

Abstract:
Purpose: of this paper is the application of the approximate method to solve the task of assigning the frequencymodalanalysis and characteristics of a mechatronic system.Design/methodology/approach: was the formulated and solved as a problem in the form of a set of differentialequations of motion and state equations of the considered mechatronic model of an object. To obtain thesolution, Galerkin’s method was used. The discussed torsionally vibrating mechanical system is a continuousbar of circular cross-section, clamped on its ends. A ring transducer, which is an integral part of the mechatronicsystem is assumed to be perfectly bonded to the bar surface.Findings: this study is that the parameters of the transducer have an important influence on the values ofnatural frequencies and on the form of the characteristics of the said mechatronic system. The poles of thedynamical characteristic calculated with the use of mathematical exact method and Galerkin’s method haveapproximately the same values. The results of the calculations were not only presented in a mathematicalform but also as transients of the examined dynamical characteristic which are a function of frequency of theassumed excitation.Research limitations/implications: is that the linear mechatronic system was considered, but for this type ofsystems, such approach is sufficient.Practical implications: of this researches was that another approach is presented, that means in the domain offrequency spectrum analysis. The method used and the obtained results can be of some value for designers ofmechatronic systems.Originality/value: of this paper is that the mechatronic system, created from mechanical and electrical subsystemswith electromechanical bondage was examined. This approach is other than those considered elsewhere.