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.
Gear transmissions have a long history dating back since the time of the first engineering systems. Their practical usage in the present day modern engineering systems is enormous. In accordance with contemporary development of mechanical engineering technics ever growing requirements have been imposed concerning characteristics and working specifications. The machines which utilize high-power duty gear transmissions (excavating machines, crushing mashines, rolling machines, ships, etc.) operate under nonstationary conditions so that the loads of the elements of these gear transmissions are variable. For example, abrupt accelerations and abrupt decelerations of machine parts, that is, masses of the gear transmissions cause inertial forces which, in addition to the conditions of operation, influence the magnitude of actual leads of the elements of gear transmissions. All this, together with the changes of the torque of drive and operating machine, the forces induced by dynamic behaviours of the complete system, and so forth, lead to the simulation where the stresses in the gears are higher than critical stresses; after certain time this may result in breakage of the teeth.
Dynamics of coupled rotors (see Figure
Two models of the gear power transmission with visco-elastic fractional order tooth coupling.
Chaotic clock models, as well as original ideas on a paradigm for noise in machines were presented by Moon (see [
Following up the idea of Mossera that the distance between trajectories be measured maintaining different time scales or “clock” with which time is measured along each motion, Leela (see [
By using examples of the rotor system which rotates about two axes with section or without section, we applied the vector method of the kinetic parameters analysis of the rotors with many axes which is done in [
By using vector equations (see [
Expressions of the kinetic pressures of shaft bearing are determined.
The analogy between motions of heavy material point: on the circle in vertical plane which rotates around vertical axis in the plane (see [
Dynamics of disc on the one, or more, shaft is a classical engineering problem. This problem attracts attention of many researchers and permanently takes place in world scientific and engineering professional literature (see [
By using knowledge of nonlinear mechanics (see [
Using new knowledge in the nonlinear mechanics, theory of chaos and dynamical systems published in [
In the paper [
A trigger of coupled singularities, on an example of coupled rotors with deviational material particles are presented in [
From time to time it is useful to pay attention again to classical models of dynamics of mechanical systems and evaluate possibilities for new approaches to these classical results by using other than the methods usually used in the classical literature.
The interest in the study of vector and tensor methods with applications in the
Also, we can conclude that the impact of different possibilities to establish the phenomenological analogy of different model dynamics expressed by vectors connected to the pole and the axis and the influence of such possibilities to applications allows professors, researchers and scientists to obtain larger views within their specialization fields.
This is the reason to introduce mass moment vectors to presentation of the kinetic parameters of the rotor dynamics and multistep gear transmission. On the basis of this approach we built the first model presented in this paper.
In industry there is an increased need for detailed investigation of the toothed coupling through models that involute the coupling of more than two teeth and for more than two, the systems which give high revolution numbers and others. Relatively new models (see [
In use, gear transmissions are very often exposed to action of forces that change with time (dynamic load). There are also internal dynamic forces present. The internal dynamic forces in gear teeth meshing, are the consequence of elastic deformation of the teeth and defects in manufacture such as pitch differences of meshed gears and deviation of shape of tooth profile. Deformation of teeth results in the so-called collision of teeth which is intensified at greater difference in the pitch of meshed gears. Occurrence of internal dynamic forces results in vibration of gears so that the meshed gears behave as an oscillatory system. This model consists of reduced masses of the gear with elastic and damping connections (see [
Primary dependences between geometrical and physical quantities in the mechanics of continuum (and with gear transmissions as well) include mainly establishing the constitutive relation between the stress state and deformation state of the tooth’s material in the two teeth in contact for each particular case.
Thus, solving this task, it is necessary to reduce numerous kinetic parameters to minimal numbers and obtain a simple abstract model describing main properties for investigation of corresponding dynamical influences. Analytic methods include determination of mathematical functions which detemine the solution in closed form. They are based on the constitutive laws and relations of the stress-strain states in gear’s materials, and they can give solutions for a very small number of boundary tasks. But, always each aproach needs certain assumotions-approximations concerning description of real contours, properties of teeth is contacts and initial conditions. For this reason numerous researchers resort to application of numeric method in solving differential equation of the gear transmission motion. The basic characteristic of the numeric methods is that the fundamental equations of the Elasticity theory, including the boundary conditions, are solved by approximative numeric methods. The solutions obtained are approximate.
Based on previous analysis at starting this part, we take into account that contact between two teeth is possible to be constructed by standard light element with constitutive stress—strain state relations which can be expressed by fractional order derivatives.
For that Reason, Let us make a short survey of the present results published in the literatute.
The monographs [
In series of the papers (see [
In [
In the series of [
Let us consider a model who is based on the three-step coupled rigid rotors but couplings between gear teeth are realized by standard light elements fractional order constitutive stress-strain relations, Figure
Basic elements of multistep gear transmission system are gears in the form of disks with mass axial inertia moments
For each single standard coupling light element of negligible mass, we shall define a particular stress-strain constitutive relation-law of material properties. This means that we will define stress-strain constitutive relation as description relation between forces and deformations of two gears teeth in contact determined and constrained by rotation angles of the gear model in the form of disk and with changes of distances in time, with accuracy up to constants which depend on the accuracy of their determination through experiment.
The accuracy of those constants laws and with them the relation between forces and elongations will depend not only on knowing the nature of object, but also on our having the knowledge necessary for dealing with very complex stress-strain relations in the coupling gears teeth (for details see [
For defined model of the two-step gear transmission fractional order system vibrations, we use three generalized coordinates—angle of gear disks rotation
Kinetic energy of the of the two-step gear transmission fractional order system vibrations is in the form
The first standard light fractional order coupling element is between first gear disk and second and is strained for
Governing system of the double gear transmission fractional order differential equations is in the following form:
Now, for beginning let us consider corresponding basic systems of the differential equations in linear form:
From the previous frequency equation, we can obtain the following three roots
By using the expression for generalized coordinates
Obtained system of the three fractional order differential equations (
Then, for the solutions of the each fractional order differential equations (
Now, we can separate three sets of the two fractional order time components
These three series of the two fractional order time components
Then the solution of the basic system of the fractional order differential equations (
We can see that for fractional order model of the double gear transmission vibrations was transformed by eigen normal coordinates
Relations between eigen amplitudes of eigen main normal modes of corresponding system of the basic linear differential equations (
Relations between eigen amplitudes of eigen main normal modes of corresponding system of the basic linear differential equations (
By using different numerical values of the kinetic and geometrical parameters of the two-step gear transmission model, the series of the graphical presentation of the three sets of the two-time components
First eigen fractional order time components
First fractional order mod for different values
Second mod for different values
The first coordinate for diferent values
The first coordinate for diferent values
First eigen fractional mode
Second eigen fractional mode
Third eigen fractional mode
First and second eigen fractional modes,
In Figure
In Figure
In Figure
In Figure
In Figure
Two approaches to the models of the gear transmission system dynamics with possibility of investigate different properties of the very complex dynamics of the corresponding real gear transmission system are possible.
First approach give a model based on the rigid rotors coupled with rigid gear teeth, with mass distributions not balanced and in the form of the mass particles as the series of the mass debalances of the gears in multistep gear transmission. By very simple model is possible and useful investigation of the nonlinear dynamics of the multistep gear transmission and nonlinear phenomena in free and forced dynamics. This model is suitable to explain source of vibrations and big noise, as well as no stability in gear transmission dynamics. Layering of the homoclinic orbits in phase plane is source of a sensitive dependence nonlinear type of regime of gear transmission system dynamics.
Second approach give a model based on the two-step gear transmission taking into account deformation and creeping and also visco-elastic teeth gears coupling. Our investigation was focused to a new model of the fractional order dynamics of the gear transmissiont. For this model we obtain analytical expressions for the corresponding fractional order modes like one frequency eigen vibrational modes. Generalization of this model to the similar model of the multistep gear transmission is very easy.
The fractional order differential equations from all three
This fractional order differential equation (
For boundary cases, when material parameters
The solutions to equations (C.6) and (C.7) are
For kritical case
Fractional-differential equation (
For the case when
In writing (
See Table
The datas of gear box.
Pinion | Middle 1 | Middle 2 | Output gear | |
---|---|---|---|---|
Number of the teeth | 51 | 72 | 19 | 73 |
Modulus, mm | 1,405 | 1,405 | 2,2175 | 2,2175 |
Face whith, mm | 22,5 | 29 | 20 | 20 |
Inertias | 0 01837 | 0 03837 | 0 00071 | 0 1740 |
Contact ratio | 1,60 | 1,7 | ||
Mean stiffness | ||||
Mesh Phasing | 0 257 | |||
Torque | 100 | 0 | 0 | 258,4 |
Parts of this research were supported by the Ministry of Sciences of Republic Serbia through Mathematical Institute SANU Belgrade Grants no. ON144002 Theoretical and Applied Mechanics of the Rigid and Solid Body. Mechanics of Materials, and also through the Faculty of Mechanical Engineering University of Niš and State University of Novi Pazar.