A Model of Gear Transmission: Fractional Order System Dynamics

A theoretical model of multistep gear transmission dynamics is presented. This model is based on the assumption that the connection between the teeth of the gears is with properties within the range from ideal clasic to viscoelastic so that a new model of connection between the teeth was expressed by means of derivative of fractional order. For this model a two-step gear transmision with three degrees of freedom of motion has been used. The obtained solutions are in the analytic form of the expansion according to time. As boundary cases this model gives results for the case of ideally elastic connection of the gear teeth and for the case of viscoelastic connection of the gear teeth, as well. Eigen fractional modes are obtained and a vizualization is done

Oscilatori - fenomenološko preslikavanje i analogije - prvi deo - matematička analogija i lanci

New analytical and numerical results of dynamics for both linear and nonlinear system with two degrees of freedom are presented. For a mechanical chain system with two degrees of freedom, oscillations are investigated analytically and numerically with corresponding comparing between free and forced oscillatory dynamics of linear and nonlinear system. Also, energy analysis and transient energy between the mass particles in the system are discussed. Using Mihailo Petrović's theory of the mathematical phenomenology elements, phenomenological mappings in vibrations, signals, resonances and dynamical absorptions in models with two degrees of freedom - the abstractions of a different real system dynamics are identified and presented. Mathematical description of a chain mechanical system with two mass particles coupled by linear and nonlinear elastic springs and with two degrees of freedom is given. By analysis of corresponding solutions for free and forced vibrations, series of related two-frequency regimes and resonant states, as well as dynamical absorption states, are identified. Besides, by mathematical analogy and phenomenological mappings, the analysis of series of dynamics of other two degrees of freedom models dynamics (torsional system, double pendulum system, double electrical circuit) is performed.Predstavljeni su novi analitički i numerički rezultati o dinamici linearnih i nelinearnih sistema sa dva stepena slobode kretanja. Za mehaničke lance izučavane su, analitički i numerički sa odgovarajućim poredjenjima svojstava izmedju sopstvenih i prinudnih režima, oscilatorne linearne i nelinearne dinamike u njima. Predstavljena je i energijska analiza dinamika i transfer energije medju podsistemima. Koristeći teoriju Mihaila Petrovića ''Elementi matematičke fonomenologije'' i ''Fenomenološko preslikavanje'', u oscilacijama, signalima, fenomenima rezonancija i dinamičkojapsorbciji, u tim modelima, dinamička apstrakcija različitih modela realnih sistema identifikovane su matematička i kvailitativne analogije. Matematički opis jednog mehaničkog lanca sa dvema materijalnim tačkama spregnutim linearnoelastičnom ili nelinearno elastičnim oprugama i sa dva stepena slobode kretanja je prikazan zajedno sa odgovarajućom alaizom kinetičkih parametara. Analizom odgovarajućih rešenja za sopstvene i prinudne dvofrekventne režime oscilacija i rezonantnih stanja, kao i stanja dinamičke apsorpcije došlo se do novih sazanja o interakciji modova u nelinearnoj dinamici. Korišćenjem matematičke analogije i fenomenološkog preslikavanja svojstvenih fenomena izučavanog mehaničkog sistema, pokazano je da se ta saznanja mogu koristiti za saznanja o fenomenima i svojstvima dinamika drugih apstrakcija realnih sistema modelima sistema sa dva stepena slobode oscilovanja (napr. dvojnog klatna, ili modela torzijskih oscilacija vratila sa dva diska, ili dvojnog električnog kola). I u najkraćem, analitički i numerički rezultati linearnih i nelinearnih dinamika sistema sa dva stepena slobode su prikazani kao univerzalni, koji se mogu preneti i na različite druge sisteme analogijama i preslikavanjima fenomena

Oscilatori - fenomenološko preslikavanje i analogije - drugi deo - strukturna analogija i lanci

New analytical and numerical results of dynamics for both linear and nonlinear system with two degrees of freedom are presented. For the mechanical chain system with two degrees of freedom, oscillations are investigated analytically and numerically with corresponding comparison between free and forced oscillatory dynamics of linear and nonlinear system. Using the Mihailo Petrović's theory of elements of mathematical phenomenology, the phenomenological mappings in vibrations, signals, resonances and dynamical absorptions in models with two degrees of freedom-abstractions of different real system dynamics are identified, as well as in eigen time functions of multi-deformable coupled body system dynamics, and presented. Mathematical description of a chain mechanical system with two mass particles coupled by linear and nonlinear elastic springs and with two degree of freedom is given. By the analysis of corresponding solutions for free and forced vibrations, series of related two-frequency regimes and resonant states, as well as dynamical absorption states, are identified. Phenomenological mappings are used to explain dynamics in two deformable body (beams, plates or membranes) systems. In short, new analytical and numerical results of linear and non-linear dynamics of a system with two-degrees of freedom in analogy with eigen time functions oscillations of transversal vibrations of two body system dynamics are presented. Structural analogies are identified between eigen time functions of different multi-coupled deformable body transversal vibrations (plates, beams, belts or membranes). Mathematical phenomenology elements and phenomenological mappings are applied in the scientific results integration. Mathematical analogy and phenomenological mappings of linear and nonlinear singular phenomena in discrete and multi-body system vibrations (torsional system, multi-pendulum system, coupled electrical circuit and multi-deformable-body oscillations-beams, plates, membranes) are performed. Structural analogy is used to explain phenomenological mappings between displacements of the mass particles in discrete system and eigen time functions in one eigen amplitude mode of dynamics in two coupled deformable body (beams, plates or membranes) systems.Predstavljeni su novi analitički i numerički rezultati o dinamici linearnih i nelinearnih sistema sa sva stepena slobode kretanja. Za mehaničke sisteme sa dva stepena slobode kretanja izučavane su, analitički i numerički sa odgovarajućim poređenjima svojstava između sopstvenih i prinudnih režima, oscilatorne linearne i nelinearne dinamike u njima. Koristeći teoriju Mihaila Petrovića - Elementi matematičke fenomenologije i Fenomenološko preslikavanje, u oscilacijama, signalima, fenomenima rezonancija i dinamičkoj apsorpciji u modelima sa dva stepena slobode dinamika apstrakcija različitih modela realnih sistema identifikovane su matematička i kvailitativne analogije sa sopstvenim vremenskim funkcijama transverzalnih oscilacija sistema više spregnutih deformabilnih tela (ploča, greda, traka ili membrana) i predstavljene. Matematički opis jednog mehaničkog lanca sa dvema matrijalnim tačkama spregnutim linearno elastičnom ili nelinearno elastičnim oprugama i sa dva stepena slobode kretanja je prikazan zajedno sa odgovarajućom analizom kinetičkih parametara u analogiji sa sistemom dvaju spregnutih tela diskretno-kontinualnim spojem. Analizom odgovarajućih rešenja za sopstvene i prinudne dvofrekventne režime oscilacija i rezonantnih stanja, kao i stanja dinamičke apsorpcije došlo se do novih saznanja o interakciji modova u nelinearnoj dinamici. Korišćenjem matematičke analogije i fenomenološkog preslikavanja svojstvenih fenomena izučavanog mehaničkog sistema, pokazano je da se ta saznanja mogu koristiti za saznanja o fenomenima i svojstvima dinamika drugih apstrakcija realnih sistema modelima sistema sa dva stepena slobode oscilovanja (npr. dvojnog klatna, ili modela torzijskih oscilacija vratila sa dva diska, ili dvojnog električnog kola, kao i transverzalnih oscilacija spregnutih dveju greda, ili ploča ili membrana). Fenomenološko preslikavanje je korišćeno za objašnjavanje dinamike sopstvenih vremenskih funkcija u sistemu spregnutih jednakih deformabilnih tela (greda, ploča, ili membrana). I u najkraćem, analitički i numerički rezultati linearnih i nelinearnih dinamika sistema sa dva stepena slobode su prikazani kao univerzalni, koji se mogu preneti i na razlićite druge sisteme analogijama i preslikavanjima linearnih i nelinearnih fenomena

Oscillations and stability of dynamics of hybrid biostructure

Biostructures usually have complex geometrical organisation and oscillatory properties. Using a model of hybrid biostructure free and forced oscillatory modes and their stability regarding the geometrical and kinematic parameters of the model were analysed. The basic model of biostructure is in the form of complex cantilever of circular cross-section with rigidly connected structures and discrete masses and has three degree of freedom. For describing oscillatory behaviour of this hybrid biostructure, system of coupled differential equations and displacement influence coefficients of the cantilever were used. For specific geometrical parameters of the hybrid biostructure one unstable mode of free oscillations was identified. Amplitude-frequency graphs for forced oscillations for specific geometrical parameters for stable and unstable modes are presented. Conditions for resonance and dynamical absorption are discussed. Specific geometrical parameters of the hybrid biostructures can affect stability of their oscillations. Choosing specific geometrical forms in engineering design using biomimetic it is possible to reduced or eliminate undesirable oscillations

Linear and non-linear transformation of coordinates and angular velocity and intensity change of basic vectors of tangent space of a position vector of a material system kinetic point

Starting from the notion of linear and nonlinear transformations, affine and functional-nonlinear mappings of coordinates and coordinate systems, geometrical and kinematical invariants along linear or nonlinear transformations their coordinates from one to other coordinate system are pointed out. In a curvilinear coordinate system, coordinates of a geometrical or kinematical point are not equal as coordinates of its corresponding position vector. Expressions of basic vectors of tangent space of kinematical point vector positions in generalized curvilinear coordinate systems for the cases of orthogonal and nonorthogonal curvilinear coordinate systems are derived. Examples of expressions of basic vectors of tangent space of kinematical point vector position in polar-cylindrical, spherical, parabolic-cylindrical and three-dimensional-three-parabolic system of curvilinear orthogonal coordinates are presented. Next, expressions of change of basic vectors of tangent space of kinematical point vector position with time, also, are done. Geometrical (physical), covariant and contra-variant coordinates of position vector of a kinetic mass point, in a coordinate system determined by basic vectors of tangent space of this kinetic point vector position, in generalized curvilinear coordinate systems, are pointed out and determined. Original expressions of angular velocity and velocity of dilatations of basic vectors of tangent space of kinetic point vector position, in generalized curvilinear coordinate systems as well as in series of special orthogonal curvilinear coordinate system are derived by author of this paper and presented