Matrix hybrid graphs as models of complex vibrating mechanical systems

The paper deals with the formulation of the matrix hybrid graph (mhg) method to test the dynamics of complex mechanical systems. It contains the definition of the network model and of its essential elements. All graphs used in the paper are matrix graphs with combined features of both flow, polar and category k graphs. Concerning the application of a mhg to models of heterogeneous dynamical systems, an example of the modelling of a mechanical subsystem with discrete distribution of parameters was discussed. A method of generating matrixes of the coordinates transformation was presented, concerning the matrix hybrid sub-graph of a mechanical system. These matrices are automatically generated by a program, on the basis of structural and geometrical information about the system. The algebraic expression of the graph and its transformation into the matrix flow graph (mfg) constitutes the basis for applying the matrix graph to test the dynamics of a vibrating system. The principles necessary for this were formulated; i.e: the cyclomatic principle and the vertex mhg principle, also ways of generating matrices were presented: dissolution of polar and flow variables, dynamical stiffness of chords, dynamical flexibility of the mhg tree branches. Next, the formulated methods of the analysis of dynamical systems were programmed and program packets were implemented to IBM PC computers. Examples of testing the dynamics of a complex physical system by this method were given. A set of the dynamical characteristics of IRb-6 industrial robot and the differential equations of the motion were formed by the use of the model with a discrete distribution of parameters. The systems tested are regarded to be multi-dimensional, multi-input and multi-output dynamical systems which are heterogeneous, due to the configuration of mechanical variables and to the characteristics distribution of the parameters of the elements making up the model. The purpose of this research is to obtain satisfactory information about the dynamics of the object by making its dynamical characteristics. It has been assumed that it is possible to test spatial characteristics. It has been assumed that it is possible to test spatial models of vibrating systems, i.e. models characterized by the linear dynamical couplings, which are both structurally and geometrically heterogeneous and which are subjected to the polar and flow sets of functions at the same time, with an arbitrary configuration of the elements of the system, determined only by the coordinates of the convergence points of the elements and by the angles between the axes.