A New Model of the Fractional Order Dynamics of the Planetary Gears

Abstract

A theoretical model of planetary gears dynamics is presented. Planetary gears are parametrically excited by the time-varying mesh stiffness that fluctuates as the number of gear tooth pairs in contact changes during gear rotation. In the paper, it has been indicated that even the small disturbance in design realizations of this gear cause nonlinear properties of dynamics which are the source of vibrations and noise in the gear transmission. Dynamic model of the planetary gears with four degrees of freedom is used. Applying the basic principles of analytical mechanics and taking the initial and boundary conditions into consideration, it is possible to obtain the system of equations representing physical meshing process between the two or more gears. This investigation was focused to a new model of the fractional order dynamics of the planetary gear. For this model analytical expressions for the corresponding fractional order modes like one frequency eigen vibrational modes are obtained. For one planetary gear, eigen fractional modes are obtained, and a visualization is presented. By using MathCAD the solution is obtained.

Oscillators: Phenomenological mappings and analogies: Second part: Structural analogy and chains

“…Eigen time functions in each eigen amplitude form of coupled multi-bodies (plates, beams, belts or membranes) in a hybrid system are multi-frequency with accordance with a number of coupled bodies that correspond to the system dynamics of a chain with corresponding number of degrees of freedom. Analysis gives us structural, mathematical and qualitative analogies based on the elements of mathematical phenomenology [26][27][28] and [29][30]. Then all results of the analysis of linear and nonlinear dynamics with graphical presentation in our paper [14]: Oscillators: Phenomenological mappings and analogies -First part: Mathematical analogy and chains -, are useful to use for explanation nonlinear phenomena [15][16][17][8][9] of corresponding eigen time function in corresponding eigen amplitude form of transversal hybrid system that consists of multi bodies coupled by a discrete continuum layer, as in References [20], [4][5][6], [11][12][13][14] and [18][19].…”

“…The theory of Mihailo Petrović Alas, presented in two books and a paper [26][27][28], contains the elements of mathematical phenomenology and phenomenological mapping. Both publications were published in Serbian and only a small number of his contemporaries were able to read and understand this theory.…”

“…In References [26][27], authored by Rašković D., the series of examples of electromechanical analogous vibration chain systems are presented. Mathematically described homogeneous chain dynamics and corresponding system of linear ordinary differential equations are solved by a trigonometric method for different cases of both end conditions.…”

Analysis of Dynamic Characteristics of a Fractional-Order Spur Gear Pair With Internal and External Excitations

“…In terms of the dynamics of gear, a two-step transmission model with three degrees-of-freedom is established by using Riemann-Liouville (R-L) fractional-order derivative under the assumption that the connection between the teeth of the gears is with the range from ideal classic to viscoelastic by Hedrih [28]. Besides, Nikolic-Stanojevic [29] built R-L fractional-order planetary gear dynamics model and obtained the analytic expression of this model. Liu [30] introduced the fractional-order derivative to the model of a spur gear pair when considering additional viscoelastic materials, obtained the periodic solution based on incremental harmonic balance (IHB) method, and discussed the influence of internal excitation and external dynamic excitation on gear mesh vibration.…”