research article a new model of the fractional order...

$: Research Article A New Model of the Fractional Order ...downloads.hindawi.com/journals/mpe/2013/932150.pdf · a new model of the fractional order dynamics of the ... A nite element$
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013, Article ID 932150, 14 pageshttp://dx.doi.org/10.1155/2013/932150

Research ArticleA New Model of the Fractional Order Dynamics ofthe Planetary Gears

Vera Nikolic-Stanojevic,1 Ljiljana Veljovic,2 and Cemal Dolicanin1

1 State University of Novi Pazar, Novi Pazar, Serbia2 Faculty of Engineering Science, University of Kragujevac, Serbia

Correspondence should be addressed to Vera Nikolic-Stanojevic; [email protected]

Received 9 November 2012; Revised 25 March 2013; Accepted 31 March 2013

Academic Editor: Jocelyn Sabatier

Copyright © 2013 Vera Nikolic-Stanojevic et al.This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

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

1. Introduction

Planetary gears are a great application in modern engineer-ing systems as a replacement for the conventional manualtransmission complex because it has a compact structure andhigh transmission ratios. Due to the structure of planetarygears and the fact that the so-called planetary gear-satellitessimultaneously perform two current trends in the work ofplanetary gears, there are even extreme vibration, that is,dynamic loads, which cause damage to the gears, bearings,and other elements of the transmission. Precise study of thedynamic behavior of planetary gear is often a difficult math-ematical problem, because there are no adequate models. Inthe idealization of the attached planetary transmission andselection of appropriate dynamic models usually first allocateprimary properties, which are maintained in solving the task,and then in futurework neglect less important characteristics.

In the first papers on the dynamic behavior of gears in use,one notes a great simplification, for example, that all changeshave linear character. However, subsequent experimentalstudies have shown that this approach is not realistic and that

the dynamic behavior of gears in the paper is influenced bymany factors that cannot be described by linear relationships[1]. These studies have shown that it is especially importantto separate the effects that occur between the gear teeth inmesh, the dynamic effects that result in the load bearing ofthe engine, dynamic errors in transmission, and so forth.Therefore, a number of important research results of thedynamic behavior of gear transmission will be given, withspecial reference to the planetary gear.

Although gear dynamics has been studied for decades,few studies present a formulation intended for the dynamicresponse of full gear systems that contain multiple gearmeshes, flexible shafts, bearings, and so forth. There arefew reliable computational tools for the dynamic analysis ofgeneral gear configurations. Some models exist, but they arelimited by simplified modeling of gear tooth mesh interfaces,two-dimensional models that neglect out of plane behavior,and models specific to a single gear configuration.

In a series of papers that follow, the fundamental taskof analytical gear research is to build a dynamic model.For different analysis purposes, there are several modelling

2 Mathematical Problems in Engineering

choices such as a simple dynamic factor model, compliancetooth model, torsional model, and geared rotor dynamicmodel, for example, [2, 3].

The simplest models are found in a number of textbooksused in education in this field. So, the teeth in meshingaction can be modelled as an oscillatory system [4–6] andso forth. This model consists of concentrated masses (eachof which represents one gear) connected with elastic anddump element. Applying the basic principles of analyticalmechanics and taking the initial and boundary conditionsinto consideration, it is possible to obtain the system ofequations representing physicalmeshing process between thetwo or more gears. In order to obtain better results, it ispossible to model the elastic element as a nonlinear spring.

Dynamic transmission error is taken as the parameter formodelling of noise in geared transmission. In the last twodecades, there is plenty of work concetrated on modellingof the dynamic transmission error for spur and helical gearsand representing the influence of the dynamic transmissionerrors on the level of noise in the geared transmission. Lately,there has been experiments conducted in order to isolateparticular noise effects like noise coming from bearing,housing noise, meshing action noise, and backlash noisesimply by measuring the dynamic transmission error. Someof the earliest models are represented in [7–10].

Using the free vibration analysis, one calculates criticalparameters such as natural frequencies and vibration modesthat are essential for almost all dynamic investigations. Thefree vibration properties are very useful for further analyses ofplanetary gear dynamics, including eigensensitivity to designparameters, natural frequency veering, planet mesh phasing,and parametric instabilities from mesh stiffness variations[11, 12].

Based on the results of the experiments conducted duringthe gear vibration research, it is to conclude that the excitationis restored every time when a new pair of teeth entersthe mesh. Vibrations with natural frequencies dominate thevibration spectrums. The internal dynamic forces in teethmesh, vibration, and noise are consequences of the change inteeth deformation, teeth impact, gear inertia due to measure,and teeth shape deviation [13].

Paper [14, 15] aims to provide insight into the three-dimensional vibration of gears by investigating the mecha-nisms of excitation and nonlinearity coming from the geartooth mesh.

For different analysis purposes, there are several mod-elling choices such as a simple dynamic factormodel, compli-ance toothmodel, torsional model, and geared rotor dynamicmodel [6]. Using the free vibration analysis one calculatescritical parameters such as natural frequencies and vibrationmodes that are essential for almost all dynamic investigations.The free vibration properties are very useful for further anal-yses of planetary gear dynamics, including eigensensitivity todesign parameters, natural frequency veering, planet meshphasing, and parametric instabilities from mesh stiffnessvariations [16–22]. It is also necessary to systematically studynatural frequency and vibration mode sensitivities and theirveering characters to identify the parameters critical to gearvibration. In addition, practical gears may be mistuned

by mesh stiffness variation, manufacturing imperfections,and assembling errors. For some symmetric structures, suchas turbine blades, space antennae, and multispan beams,small disorders may dramatically change the vibration [18,19].The following articles [10, 23] are related to the nonlinearanalysis of dynamic behavior of gears, using experimentalmethods and the application of finite elementmethod (FEM).

Paper [24, 25] examines the nonlinear dynamics ofplanetary gears by numerical and analytical methods overthemeaningfulmesh frequency ranges. Concise, closed-formapproximations for the dynamic response are obtained byperturbation analysis.

By using three-dimensional finite element analysis, it ispossible to model the whole planetary gear and get adequatesolutions. Such a solution to the classic gear transmissionsis given in the paper [26]. General three-dimensional finiteelement models for dynamic response are rare because theyrequire significant computational effort.This is accomplishedby many time steps required for the transient response todiminish so that steady-state data can be obtained. Thisstudy attempts to fill this gap with a general finite elementformulation that can be used for full gearbox dynamicanalyses.

A finite element formulation for the dynamic responseof gear pairs is proposed in [24, 26, 27] and so forth.Following an established approach in lumped parametergear dynamic models, the static solution is used as theexcitation in a frequency domain solution of the finite ele-ment vibration model. The nonlinear finite element/contactmechanics formulation provides an accurate calculation ofthe static solution and average mesh stiffness that are usedin the dynamic simulation. The frequency domain finiteelement calculation of dynamic response compares well withnumerically integrated (time domain) finite element dynamicresults and previously published experimental results. Sim-ulation time with the proposed formulation is two ordersof magnitude lower than numerically integrated dynamicresults. This formulation admits system level dynamic gear-box response, which may include multiple gear meshes,flexible shafts, rolling element bearings, housing structures,and other deformable components.

In the latest research, light fractional order couplingelement is used to describe the dynamic behavior of gears andset of constitutive relationships, so the fractional calculus canbe successfully applied to obtain results.

The monograph [28–31] contains a basic mathematicaldescription of fractional calculus and some solutions of thefractional order differential equations necessary for appli-cations of the corresponding mathematical description of amodel of gear transmission based on the teeth coupling bystandard light fractional order element.

In the series of references [32–40], the mixed discrete-continuum or continuummechanical systems with fractionalorder creep properties are mathematically described andanalytically solved.

Paper [40] presents two models of the geared transmis-sion with two or more shafts. First approach gives a modelbased on the rigid rotors coupled with rigid gear teeth, withmass distributions not balanced and in the form of the mass

Mathematical Problems in Engineering 3

particles as the series of the mass disturbance of the gears inmultistep gear transmission. Using very simple model it ispossible and useful to investigate the nonlinear dynamics ofthemultistep gear transmission and nonlinear phenomena infree and forced dynamics. This model is suitable to explainsource of vibrations and big noise, as well as no stabilityin gear transmission dynamics. Layering of the homoclinicorbits in phase plane is source of a sensitive dependence non-linear type of regime of gear transmission system dynamics.Second approach gives a model based on the two-step geartransmission taking into account deformation and creepingand also viscoelastic teeth gears coupling. This investigationwas focused to a new model of the fractional order dynamicsof the gear transmission. For this model we obtain analyticalexpressions for the corresponding fractional ordermodes likeone frequency eigen vibrationalmodes. Generalization of thismodel to the similarmodel of themultistep gear transmissionis very easy.

The model in this paper represents dynamic model ofthe planetary gears with four degrees of freedom. Ourinvestigation was focused to a new model of the fractionalorder dynamics of the planetary gears. For this model weobtain analytical expressions for the corresponding fractionalorder modes like one frequency eigen vibrational modes.

2. Mathematical Model of the Planetary Gear

In the practice, planetary gears are very often exposed toaction of forces that change with time (dynamic load).There are also internal dynamic forces present. The internaldynamic forces in gear teeth meshing are the consequence ofelastic deformation of the teeth and defects in manufacturesuch as pitch differences of meshed gears and deviation ofshape of tooth profile. Deformation of teeth results in theso-called collision of teeth which is intensified at greaterdifference in the pitch of meshed gears. Occurrence ofinternal dynamic forces results in vibration of gears so thatthe meshed gears behave as an oscillatory system.This modelconsists of reduced masses of the gear with elastic anddamping connections (see [6, 14, 15, 26, 27]). By applying thebasic principles of mechanics and taking into considerationinitial and boundary conditions, the system of equation isestablished which describes physicality of the gear meshingprocess. On the other hand, extremely cyclic loads (dynamicforces) can result in breakage of teeth, thus causing failure ofthe mechanism or system.

Primary dependences between geometrical and physicalquantities in themechanics of continuum (andwith planetarygear as well) include mainly establishing the constitutiverelation between the stress state and deformation state ofthe tooth’s material in the two teeth in contact for eachparticular case. Thus, solving this task, it is necessary toreduce numerous kinetic parameters to minimal numbersand obtain a simple abstract model describing main proper-ties for investigation of corresponding dynamical influences.

Based on previous, at starting this part, we take intoaccount that contact between two teeth is possible to beconstructed by standard light element with constitutive

stress-strain state relations which can be expressed by frac-tional order derivatives.

The papers [29, 39] analyzed in details the standard lightcoupling elements of negligible mass in the form of axiallystressed rod without bending, which has the ability to resistdeformation under static and dynamic conditions.

Figure 1 shows the model planetary gear when the cou-pling between the teeth (sun-planet and ring-planet meshes)was obtained from a standard light fractional element. Theplanetary gear model consists of three members (the sun, 3planets, and ring).

The motion of the sun gear and the ring gear is given bytranslations that is expressed as 𝑦

𝑖, 𝑖 = 1, 2 ( 𝑟

𝑖, 𝑖 = 1, 2), and

rotations that is expressed as 𝜑𝑖, 𝑖 = 1, 2. The kinetic energy

𝐸𝐾of the planetary stage can be written as

𝐸𝐾=

2

∑

𝑖=1

𝐸𝐾𝑖. (1)

The kinetic energy for the each element is represented by

𝐸𝐾𝑖=1

2𝑚𝑖V𝑖⋅ V𝑖+1

2𝐽𝑖𝜔2

𝑖, 𝑖 = 1, 2, (2)

where𝑚𝑖are masses of the sun gear and ring gear, 𝐽

𝑖are mass

moments of the inertia, V𝑖are velocities of mass centers, and

𝜔𝑖are angular velocities of the sun gear and ring gear.So, the total kinetic energy of the planetary stage is given

by

𝐸𝐾=1

2𝑚1𝑦2

1+1

2𝐽1��2

1+1

2𝑚2𝑦2

2+1

2𝐽2��2

2. (3)

Sun gear is supported with bearing which is modeled aslinear spring 𝑐

10, and planet gear is supported with bearing

which is modeled as linear spring 𝑐20, but the meshes of sun

gear-planet gear and ring gear-planet gear are described bystandard light fractional element with restitution forces 𝑃

1(𝑡)

and 𝑃2(𝑡). Thus, the potential energies of the bearings are

𝐸𝑃=1

2𝑐10𝑦2

1+1

2𝑐20𝑦2

2. (4)

The restitution forces are in the function of elementelongation 𝑥

𝑖(𝑡), and they are in the form

𝑃𝑖 (𝑡) = − {𝑐𝑖𝑥𝑖 (𝑡) + 𝑐𝛼𝐷

/

𝛼[𝑥𝑖 (𝑡)]} , 𝑖 = 1, 2. (5)

The fractional order differential operator𝐷/𝛼[∗] of the𝛼th

derivative with respect to time 𝑡 is given in following form:[32, 33, 39]

𝐷/

𝛼[𝑥𝑖 (𝑡)] =

𝑑𝛼𝑥𝑖 (𝑡)

𝑑𝑡𝛼=

1

Γ (1 − 𝛼)

𝑑

𝑑𝑡∫

𝑡

0

𝑥𝑖 (𝜏)

(𝑡 − 𝜏)𝛼𝑑𝜏, (6)

where 𝑐𝑖, 𝑐𝛼are rigidity momentary and prolonged coeffi-

cients and 𝛼 is rational number (0 < 𝛼 < 1).The equations ofmotion for the planetary gear are derived

from Lagrange’s equation given by well-known form

𝑑

𝑑𝑡(𝜕𝐸𝐾

𝜕 𝑞𝑗

) −𝜕𝐸𝐾

𝜕𝑞𝑗

+𝜕𝐸𝑃

𝜕𝑞𝑗

= 𝑄∗

𝑗−𝜕Φ

𝜕 𝑞𝑗

, 𝑗 = 1, 2, . . . , 4,

(7)


𝑟𝑏1 𝑟𝑏2

𝑐10 𝑐20

𝑦1 𝑦2

𝑃1(𝑡)𝑃2(𝑡)

𝑥

𝜑1

𝜑2

(a)

𝑛

𝑟𝑏1 𝑃1(𝑡) 𝑃2(𝑡)

𝑛

𝑟𝑏2𝜑1

𝜑2

𝜑1

𝜑2

(b)

Figure 1: The model of the planetary gear with viscoelastic fractional order tooth coupling: (a) the frontal plane, (b) the direction of contact.

(a) The initial configuration of planetary gear (b) First mode of the planetary gear

Sun gear Planet gear

(c)


(d)

Figure 2: First rotational-axial modes of the planetary gear system defined in Table 1. Translational (a) and angular (b) displacements-eigenamplitudes for 𝜔

1= 215,546Hz.



(a)


(b)

Figure 3: A pair of degenerate translational (a) and angular (b) displacements-eigen amplitudes for second modes of the planetary gearsystem defined in Table 1. Translational (a) and angular (b) displacements-eigen amplitudes for 𝜔

2= 2901Hz.


(a)


(b)

Figure 4: A pair of degenerate translational (a) and angular (b) displacements-eigen amplitudes for second modes of the planetary gearsystem defined in Table 1. Translational (a) and angular (b) displacements-eigen amplitudes for 𝜔

3= 40890Hz.

where 𝑞𝑗are generalized coordinates, 𝑄∗

𝑗are generalized

forces, and Φ is Rayleigh dissipation function (in our caseRayleigh dissipation function is zero because damping effectsare taken into consideration). Generalized coordinates for thegiven system are 𝑦

1, 𝑦2, 𝜑1, and 𝜑

2.

Therefore, the dynamic behavior will be governed by fourindependent equations of motion. In matrix form they are

M { 𝑞} + C {𝑞} = 𝑄∗𝑗−𝜕Φ

𝜕 𝑞𝑗

, 𝑗 = 1, 2, . . . , 4, (8)

where the matrixM is diagonal inertia matrix and the matrixC is stiffness matrix.

Light standard creep constraint element between sun gearand planet gear is strained for 𝑥

1= 𝑦2−𝑦1+𝑟𝑏2𝜑2−𝑟𝑏1𝜑1, and

light standard creep constraint element between planet gearand ring gear is strained for 𝑥

2= 𝑦2− 𝑟𝑏2𝜑2.

So, due to the constitutive relation of the standard lightfractional order coupling elements, the generalized forces asa function of elongation of elements are

𝑄∗

1= − 𝑐

1𝑥1− 𝑐𝛼𝐷/

𝛼[𝑥1]

= − 𝑐1[(𝑦2+ 𝑟𝑏2𝜑2) − (𝑦

1+ 𝑟𝑏1𝜑1)]

− 𝑐𝛼𝐷/

𝛼[(𝑦2+ 𝑟𝑏2𝜑2) − (𝑦

1+ 𝑟𝑏1𝜑1)] ,

𝑄∗

2= − 𝑐

2𝑥2− 𝑐𝛼𝐷/

𝛼[𝑥2]

= − 𝑐2[𝑦2− 𝑟𝑏2𝜑2] − 𝑐𝛼𝐷/

𝛼[𝑦2− 𝑟𝑏2𝜑2] . (9)

Lagrange equations of motion are obtained followingsubstitution (9) into (7), and they can be expressed as

𝑚1𝑦1+ 𝑐01𝑦1+ 𝑐1[(𝑦1+ 𝑟𝑏1𝜑1) − (𝑦

2+ 𝑟𝑏2𝜑2)]

= 𝑐𝛼𝐷/

𝛼[(𝑦2+ 𝑟𝑏2𝜑2) − (𝑦

1+ 𝑟𝑏1𝜑1)] ,

𝐽1��1+ 𝑐1[(𝑦1+ 𝑟𝑏1𝜑1) − (𝑦

2+ 𝑟𝑏2𝜑2)] 𝑟𝑏1

= 𝑐𝛼𝐷/

𝛼𝑟𝑏1[(𝑦2+ 𝑟𝑏2𝜑2) − (𝑦

1+ 𝑟𝑏1𝜑1)] ,

𝑚2𝑦2+ 𝑐02𝑦2

+ 𝑐1[(𝑦2+ 𝑟𝑏2𝜑2) − (𝑦

1+ 𝑟𝑏1𝜑1)]

+ 𝑐2[(𝑦2− 𝑟𝑏2𝜑2)]

= 𝑐𝛼𝐷/

𝛼[(𝑦1+ 𝑟𝑏1𝜑1) − (𝑦

2+ 𝑟𝑏2𝜑2)]

− 𝑐𝛼𝐷/

𝛼[(𝑦2− 𝑟𝑏2𝜑2)] ,

𝐽2��2+ 𝑐1[(𝑦2+ 𝑟𝑏2𝜑2) − (𝑦

1+ 𝑟𝑏1𝜑1)] 𝑟𝑏2

+ 𝑐2[𝑟𝑏2𝜑2− 𝑦2] 𝑟𝑏2

= 𝑐𝛼𝐷/

𝛼[(𝑦1+ 𝑟𝑏1𝜑1) − (𝑦

2+ 𝑟𝑏2𝜑2)]

+ 𝑐𝛼𝐷/

𝛼[(𝑦2− 𝑟𝑏2𝜑2)] .

(10)



(a)


(b)

Figure 5: Fourth rotational-axial modes of the planetary gear system defined in Table 1. Translational (a) and angular (b) displacements-eigenamplitudes for 𝜔

4= 50000Hz.

The diagonal inertia matrixM is

M = diag (𝑚1, 𝐽1, 𝑚2, 𝐽2) . (11)

The stiffness matrix C is

C =

[[[[[[[[

[

𝑐10+ 𝑐1

𝑐1𝑟𝑏1

−𝑐1

−𝑐1𝑟𝑏2

𝑐1𝑟2

𝑏1−𝑐1𝑟𝑏1

−𝑐1𝑟𝑏1𝑟𝑏2

Symmetric 𝑐20+ 𝑐1+ 𝑐2(𝑐1− 𝑐2) 𝑟𝑏2

(𝑐1+ 𝑐2) 𝑟2

𝑏2

]]]]]]]]

]

. (12)

3. Modal Analysis of the Planetary Gear

The system is tuned, that is, all sun-planet and ring-planetmesh stiffnesses, and their centers of stiffnesses, are identicalamong all planets; the planet bearing stiffnesses, the axiallocations of the planet bearings, and the planet inertias arethe same for all planets.

3.1. Eigenvalue Problem. The proposed solutions are in theform of

{𝑞} = {𝐴} cos (𝜔𝑡 + 𝜀) . (13)

The eigenvalue problem is

(C − 𝜆M) {𝑞} = 0 (14)

with natural frequencies√𝜆.It is known that to have nontrivial solutions thematrix on

the left side must be singular. It follows that the determinantof the matrix must be equal to 0, so

det (C − 𝜆M) = 0 (15)

or, in the developed form,

[[[[[[[[

[

(𝑐10+ 𝑐1) − 𝜆𝑚

1𝑐1𝑟𝑏1

−𝑐1

−𝑐1𝑟𝑏2

(𝑐1𝑟2

𝑏1) − 𝜆𝐽

1−𝑐1𝑟𝑏1

−𝑐1𝑟𝑏1𝑟𝑏2

Symmetric (𝑐20+ 𝑐1+ 𝑐2) − 𝜆𝑚

2(𝑐1− 𝑐2) 𝑟𝑏2

((𝑐1+ 𝑐2) 𝑟2

𝑏2) − 𝜆𝐽

2

]]]]]]]]

]

= 0. (16)

Corresponding frequency equation in the polynomialform is

𝑎4𝜆4+ 𝑎3𝜆3+ 𝑎2𝜆2+ 𝑎1𝜆 + 𝑎0= 0, (17)

where, for instance, 𝑎4= traceM, 𝑎

0= detC, and so forth.

Solving this polynomial four roots 𝜆𝑗, 𝑗 = 1, 2, 3, 4 and

corresponding eigen circular frequencies 𝜔𝑗= √𝜆𝑗, 𝑗 =

1, 2, 3, 4, can be obtained.The solution of basic linear differential equation is

𝑞𝑗 (𝑡) =

4

∑

𝑠=1

𝑞(𝑠)

𝑗(𝑡) =

4

∑

𝑠=1

𝐴(𝑠)

𝑗cos (𝜔

𝑠𝑡 + 𝜀𝑠) , (18)


1.53

0.82

0.11

−0.59

−1.3

0 7.5 15 22.5 30 𝑡

𝜂1

𝛼 = 0

𝛼 = 0.1

𝛼 = 0.2

𝛼 = 0.4𝛼 = 0.5𝛼 = 0.6

𝜉1

2.5

1.38

0.25

−0.88

−2

7

3.75

0.5

−2.75

−6

𝜁1

𝑡0 7.5 15 22.5 30

0 7.5 15 22.5 30 𝑡

𝛼 = 0

𝛼 = 0.1

𝛼 = 0.2

𝛼 = 0.4𝛼 = 0.5𝛼 = 0.6

𝛼 = 0

𝛼 = 0.1

𝛼 = 0.2

𝛼 = 0.4𝛼 = 0.5𝛼 = 0.6

Figure 6: First eigen fractional mode 𝜉1(𝑡) with corresponding first fractional order time components 𝜂

1(𝑡) and 𝜁

1(𝑡) for different system

kinetic and geometric parameter values.

and in matrix presentation

{𝑞 (𝑡)} = R {𝐶𝑠cos (𝜔

𝑠𝑡 + 𝜀𝑠)} , (19)

where R is modal matrix defined by the correspondingcofactors, 𝜉

𝑠= 𝐶𝑠cos(𝜔

𝑠𝑡 + 𝜀𝑠), and 𝑠 = 1, 2, 3, 4 are main

coordinates of the linear system.

With this expression, the system of the fractional differ-ential equation (10) can be transformed in the form of [39]

𝜉𝑠+ 𝜔2

𝑠𝜉𝑠= −𝜔2

𝛼𝑠𝐷/

𝛼[𝜉𝑠] , 𝑠 = 1, 2, 3, 4. (20)

This resulted the system of the fractional differentialequation. Analytical solution of these fractional order differ-ential equations is obtained using the approach presented in


0 2.25 4.5 6.75 9

𝑡

𝜉2

1

0.63

0.25

−0.13

−0.5

3

1.75

0.5

−0.75

−2

𝜁2

1.53

0.82

0.11

−0.59

−1.30 2.25 4.5 6.75 9

0 2.25 4.5 6.75 9

0

𝛼 = 0

𝛼 = 0.1

𝛼 = 0.2

𝛼 = 0.4𝛼 = 0.5𝛼 = 0.6

𝛼 = 0

𝛼 = 0.1

𝛼 = 0.2

𝛼 = 0.4𝛼 = 0.5𝛼 = 0.6

𝛼 = 0

𝛼 = 0.1

𝛼 = 0.2

𝛼 = 0.4𝛼 = 0.5𝛼 = 0.6

𝑡

𝑡

𝜂2

Figure 7: Second eigen fractionalmode 𝜉2(𝑡)with corresponding second fractional order time components 𝜂

2(𝑡) and 𝜁



[37, 39]. Therefore, each fractional differential equation canbe written in the form of

𝜉𝑠 (𝑡) = 𝜉0𝑠

∞

∑

𝑘=0

(−1)𝑘𝜔2𝑘

𝛼𝑠𝑡2𝑘

×

𝑘

∑

𝑗=0

(𝑘

𝑗)

(∓1)𝑗𝜔2𝑗

𝛼𝑠𝑡−𝛼𝑗

𝜔2𝑗

𝑠 Γ (2𝑘 + 1 − 𝛼𝑗)

+ 𝜉0𝑠

∞

∑

𝑘=0

(−1)𝑘𝜔2𝑘

𝛼𝑠𝑡2𝑘+1

×

𝑘

∑

𝑗=0

(𝑘

𝑗)

(∓1)𝑗𝜔−2𝑗


𝜔2𝑗

𝑠 Γ (2𝑘 + 2 − 𝛼𝑗)

, 𝑠 = 1, 2, 3, 4,

(21)


1.53

0.82

0.11

−0.59

−1.33.22.41.60.80

𝑡

𝛼 = 0

𝛼 = 0.1

𝛼 = 0.2

𝛼 = 0.4𝛼 = 0.5𝛼 = 0.6

0.075

0

−0.075

−0.150 0.75 1.5 2.25 3

𝑡

𝛼 = 0

𝛼 = 0.1

𝛼 = 0.2

𝛼 = 0.4𝛼 = 0.5𝛼 = 0.6

𝛼 = 0

𝛼 = 0.1

𝛼 = 0.2

𝛼 = 0.4𝛼 = 0.5𝛼 = 0.6

𝜁3𝜂3

1.5

0.88

0.25

−0.38

−1

𝑡

21.510.50

𝜉3

Figure 8: Third eigen fractional mode 𝜉3(𝑡) with corresponding third fractional order time components 𝜂

3(𝑡) and 𝜁



where 𝜉𝑠(0) = 𝜉

0𝑠and 𝜉𝑠(0) = 𝜉

0𝑠are initial values of main

coordinates defined by initial conditions and 𝛼 is rationalnumber (0 < 𝛼 < 1).

The solution of the basis system [39] can be expressed inthe following form:

𝑞𝑗 (𝑡) =

4

∑

𝑠=1

𝐾(𝑠)

𝑝𝑘𝜉𝑠 (𝑡)

=

4

∑

𝑠=1

𝐾(𝑠)

𝑝𝑘𝜉0𝑠

∞

∑

𝑘=0

(−1)𝑘𝜔2𝑘

𝛼𝑠𝑡2𝑘

×

𝑘

∑

𝑗=0

(𝑘

𝑗)

(∓1)𝑗𝜔2𝑗


𝜔2𝑗

𝑠 Γ (2𝑘 + 1 − 𝛼𝑗)

+

4

∑

𝑠=1

𝐾(𝑠)

𝑝𝑘

𝜉0𝑠

∞

∑

𝑘=0

(−1)𝑘𝜔2𝑘

𝛼𝑠𝑡2𝑘+1

×

𝑘

∑

𝑗=0

(𝑘

𝑗)

(∓1)𝑗𝜔−2𝑗


𝜔2𝑗

𝑠 Γ (2𝑘 + 2 − 𝛼𝑗)

, 𝑠 = 1, 2, 3, 4.

(22)


1.53

0.82

0.11

−0.59

−1.30 0.75 1.5 2.25 3

𝑡

𝛼 = 0

𝛼 = 0.1

𝛼 = 0.2

𝛼 = 0.4𝛼 = 0.5𝛼 = 0.6

0 0.75 1.5 2.25 3

𝛼 = 0

𝛼 = 0.1

𝛼 = 0.2

𝛼 = 0.4𝛼 = 0.5𝛼 = 0.6

𝑡

0.15

0.087

0.025

−0.038

−0.1

𝛼 = 0

𝛼 = 0.1

𝛼 = 0.2

𝛼 = 0.4𝛼 = 0.5𝛼 = 0.6

21.510.50

−0.75

0.75

0

1.5

−1.5

𝜂4𝜁4

𝜉4

𝑡

Figure 9: Fourth eigen fractional mode 𝜉4(𝑡) with corresponding third fractional order time components 𝜂

4(𝑡) and 𝜁



3.2. Numerical Visualisation. Eigensolutions of a sample sys-tem [20, Table 1], with four degrees of freedom are evaluatednumerically to expose the modal properties.

Eigensolutions of a sample system (Table 1) with threeequally spaced planets are evaluated numerically to exposethe modal properties. Four natural frequencies and theircorresponding mode types are given in Figures 2, 3, 4, and5. In Figure 2(a), the initial configuration of planetary gear

is shown, and Figure 2(b) shows the planetary gear firstmode. In order to better consideration ofmodes of individualelements of the gear in the following Figures 2(c), 2(d), 3, 4and 5, separate elements of planetary gear are shown.

The vibration modes exhibit distinctive characteristics.The central member rotates and translates axially and planetsdo same. Regardless of the system parameters the modaldeflection of planet gears are zero for 𝜔

4= 50000Hz.


𝜉1

5

0

0.20.4

0.6

𝛼

𝑡

10203040

0.6

0.4

0.2Pa

ram

eter

om

ega a

lpha

1

10 20 30 40Time 𝑡

0.6

0.4

0.8

5

25

(a) First mode of the planetary gear system 𝜉1(𝛼, 𝑡)

𝛼

𝑡 10203040

0.40.3

0.20.1−4

−2

0

2

4

0.4

0.2

Para

met

er o

meg

a alp

ha 1

10 20 30 40Time 𝑡

0.3

0.1

𝜉2

10

0.4

0.2

0.6

(b) Second mode of the planetary gear system 𝜉2(𝛼, 𝑡)

2222222222222222222222222222222222222222222222222

−4

−2

0

4

0.050.10.15

0.2

𝜉3

𝛼𝑡

1020

3040

0.2

0.15

0.1

0.05

Para

met

er o

meg

a alp

ha 1

10 20 30 40Time 𝑡

1

0

0.20.3

0.4

0.5

Parameter alpha

30

(c) Third mode of the planetary gear system 𝜉3(𝛼, 𝑡)

Figure 10: Continued.


−4

−2

0

4

𝑡1040 30

0.30.2

0.1

2

𝛼

0.3

0.2

0.1Para

met

er o

meg

a alp

ha 1

10 20 30 40Time 𝑡

0.60.4

0.2

2𝜉4

20

10.010

(d) Fourth mode of the planetary gear system 𝜉4(𝛼, 𝑡)

Figure 10: Four eigen fractional modes, 𝜉1(𝛼, 𝑡), 𝜉

2(𝛼, 𝑡), 𝜉

3(𝛼, 𝑡), and 𝜉

4(𝛼, 𝑡) presented by surfaces.

Table 1: Parameters of the planetary gear.

Parameter Sun PlanetBase radius 𝑟

𝑏, mm 24 16

Radial bearing stiffnesses 𝑐10and 𝑐20,

N/m 0.5 × 109 0.5 × 109

Stiffness of teeth 𝑐1and 𝑐2, N/m 2.91 × 108 1.81 × 108

Mass𝑚, kg 0.3 0.20Rotational inertia, 𝐽

1, kgm2 10 × 10−3 100 × 10−6

Based on (18), the first normal mode corresponds toboth masses moving in the opposite direction while angulardisplacements are in the same direction. The second normalmode corresponds to the masses moving in the oppositedirections and angular displacements are in the oppositedirections also. The masses, for 𝜔

3and 𝜔

4, move in the

same direction, but angular displacements are in the oppositedirections or equal zero (fourth mode). The general solutionis a superposition of the normal modes where the initialconditions of the problem must be used.

By using different numerical values of the kinetic andgeometrical parameters of the planetary gear model, theseries of the graphical presentation of the four sets of the twotime components 𝜂

𝑠(𝑡) and 𝜁

𝑠(𝑡), 𝑠 = 1, 2, 3, 4 of the solutions,

by using expressions (21) are obtained. In the series Figures6–10 are presented characteristicmodes for different values ofthe 𝛼 coefficient of the fractional order of the used standardlight fractional order element for describing teeth couplingbetween sun-planet and planet-ring. Time 𝑡 is in sec, and allvalues on the vertical axis are in 𝜇m.

First eigen fractional order mode 𝜉1(𝑡) with corre-

sponding first eigen fractional order time components 𝜂1(𝑡)

and 𝜁1(𝑡) for different system kinetic and geometric parame-

ter values is presented in Figure 6.In Figure 7, we can see second eigen fractionalmode 𝜉

2(𝑡)

with corresponding second fractional order time components𝜂2(𝑡) and 𝜁

2(𝑡) for different system kinetic and geometric

parameter values.In Figure 8, third eigen fractional mode 𝜉

3(𝑡) with corre-

sponding third fractional order time components 𝜂3(𝑡) and

𝜁3(𝑡) for different system kinetic and geometric parameter

values is presented.Fourth eigen fractional mode 𝜉

4(𝑡) with corresponding

third fractional order time components 𝜂4(𝑡) and 𝜁

4(𝑡) for

different system kinetic and geometric parameter values, inFigure 9, is presented.

In Figure 10, first, second, third, and fourth eigenfractional modes 𝜉

1(𝛼, 𝑡), 𝜉

2(𝛼, 𝑡), 𝜉

3(𝛼, 𝑡), and 𝜉

4(𝛼, 𝑡) are

presented by surfaces. Also, the family trajectory in the plane(𝛼, 𝑡) is shown.

Based on the obtained results in this paper, we can con-clude that eigen fractional ordermodes are like one frequencyvibrationmodes similar to single frequency eigenmode of thecorresponding linear system [29, 38, 39, 41].

The fractional order dynamic system is like dumpingsystem. With the increase of the parameter 𝛼, the periodof oscillation increases but the amplitude becomes smaller.So we can say that parameter 𝛼 has the same influence asdumping coefficient in the corresponding system.

4. Conclusions

This paper presents a new dynamic model of a planetarygear. The planetary gear system is represented by a modelthat allows for four degrees of freedom per gear-shaft bodysupported by bearings at arbitrary axial positions and with


standard creep constraint element. The standard light frac-tional order coupling element is between sun-planet andplanet-ring. A novel approach for the planetary gear dynamicanalysis is developed. So, in this paper it is shownhow the newmodel of the fractional order dynamic planetary gear can beapplied to study dynamic behavior. This model simulates thereal behavior of the planetary gear.

With this simple model, it is possible to research the non-linear dynamics of the planetary gear and nonlinear phe-nomena in free and forced dynamics. The model is suitableto explain source of vibrations and big noise, as well as nostability in planetary gear.

A new method, using MATCAD software, is used in thispaper for the obtaining of the eigen values and for analysisresults.

In the literature, similar procedures are presented inintroduction, and they were used as referencematerial for thecomposition and verification of models and results.

On the basis of the numerical results, shown in thispaper, it has been concluded that themethodology developedto study the dynamic behaviour of planetary gear is veryefficient. It gives a lot of possibilities and can be easilyupgraded for analysis of other effects.

The dynamic behavior and analysis of results suggest thatthe gear transmission is very complex and that it is almostimpossible to include all the effects by such and similarresearch. This paper considers planetary gear with 3 planetgears, which makes the problem more complex.

Further research should be directed at studying the effectsof mutual dynamic impact of teeth in mesh, as well as atincluding more effects [42]. So, it is possible to study eigenfrequency of planetary gear with moving excentric masses onthe body of one of the gears or with holes on the body, byusing finite element method.

In accordance with the present trend of application ofnew materials, authors will, in future studies, simulate thedynamic behavior of a gear made of composite materials andstudy the life of the gears at the load. Also, future researchshould focus on the study of planetary gears life using lowcycle fatigue properties and so forth.

Results in this paper can be taken as relevant for furtherresearch, because this model simulates the real behavior ofthe planetary gear, more than earlier models.

Acknowledgments

Parts of this research were supported by the Ministry ofSciences of Republic Serbia through Mathematical InstituteSANU Belgrade Grant no. ON 174001 “Dynamics of hybridsystems with complex structures. Mechanics of materials”and also through the Faculty of Mechanical EngineeringUniversity of Nis and the State University of Novi Pazar.

References

[1] F. Cunliffe, D. J. Smith, and D. B. Welbourn, “Dynamic toothloads in epicyclic gears,” Journal of Engineering For Industry, vol.95, no. 2, pp. 578–584, 1974.

[2] G. W. Blankenship and R. Singh, “Dynamic force transmissibil-ity in helical gear pairs,” Mechanism and Machine Theory, vol.30, no. 3, pp. 323–339, 1995.

[3] G. W. Blankenship and R. Singh, “A new gear mesh interfacedynamic model to predict multi-dimensional force couplingand excitation,” Mechanism and Machine Theory, vol. 30, no. 1,pp. 43–57, 1995.

[4] G. Liu and R. G. Parker, “Impact of tooth friction and its bend-ing effect on gear dynamics,” Journal of Sound and Vibration,vol. 320, no. 4-5, pp. 1039–1063, 2009.

[5] G. Niemann and H. Winter, Maschinen-Elemente II and III,Springer, Berlin, Germany, 1989.

[6] V. Nikolic, Machine Elements, Theory, Calculation, Examples,Faculty ofMechanical Engineering, Prizma, Kragujevac, Serbia,2004.

[7] V. Nikolic,Mechanical Analysis of Gears, Faculty of MechanicalEngineering, Cimpes, Kragujevac, Serbia, 1999.

[8] G. Liu and R. G. Parker, “Nonlinear dynamics of idler gearsystems,” Nonlinear Dynamics, vol. 53, no. 4, pp. 345–367, 2008.

[9] A. Kahraman and G.W. Blankenship, “Experiments on nonlin-ear dynamic behavior of an oscillator with clearance and peri-odically time-varying parameters,” Journal of Applied Mechan-ics, vol. 64, no. 1, pp. 217–226, 1997.

[10] A. Kahraman andR. Singh, “Non-linear dynamics of a spur gearpair,” Journal of Sound and Vibration, vol. 142, no. 1, pp. 49–75,1990.

[11] R.G. Parker, S.M.Vijayakar, andT. Imajo, “Non-linear dynamicresponse of a spur gear pair: modelling and experimentalcomparisons,” Journal of Sound and Vibration, vol. 237, no. 3,pp. 435–455, 2000.

[12] J. Lin and R. G. Parker, “Analytical characterization of theunique properties of planetary gear free vibration,” Journal ofVibration and Acoustics, vol. 121, no. 3, pp. 316–321, 1999.

[13] L. Vedmar andA. Andersson, “Amethod to determine dynamicloads on spur gear teeth and on bearings,” Journal of Sound andVibration, vol. 267, no. 5, pp. 1065–1084, 2003.

[14] M. F.Agemi andM.Ognjanovic, “Gear vibration in supercriticalmesh-frequency range,” FMETransactions, vol. 32, no. 2, pp. 87–94, 2004.

[15] D. Dimitrijevic and V. Nikolic, “Eigenfrequencies analysis forthe deep drilling machine gear set,” The Scientific JournalFACTA UNIVERSATES, vol. 1, no. 5, pp. 629–636, 1998.

[16] D. Dimitrijevic and V. Nikolic, “Eigenfrequence analysis of thespur gear pair with moving excentric masses on the body of oneof the gears,” FME Transactions, vol. 35, no. 3, pp. 157–163, 2007.

[17] H. Vinayak and R. Singh, “Multi-body dynamics and modalanalysis of compliant gear bodies,” Journal of Sound andVibration, vol. 210, no. 2, pp. 171–212, 1998.

[18] J. Lin and R. G. Parker, “Analytical characterization of theunique properties of planetary gear free vibration,” Journal ofVibration and Acoustics, vol. 121, no. 3, pp. 316–321, 1999.

[19] J. Lin and R. G. Parker, “Natural frequency veering in planetarygears,”Mechanics of Structures and Machines, vol. 29, no. 4, pp.411–429, 2001.

[20] J. Lin and R. G. Parker, “Planetary gear parametric instabilitycaused by mesh stiffness variation,” Journal of Sound andVibration, vol. 249, no. 1, pp. 129–145, 2002.

[21] T. Eritenel and R. G. Parker, “Modal properties of three-dimensional helical planetary gears,” Journal of Sound andVibration, vol. 325, no. 1-2, pp. 397–420, 2009.


[22] Y. Guo and R. G. Parker, “Sensitivity of general compoundplanetary gear natural frequencies and vibration modes tomodel parameters,” Journal of Vibration and Acoustics, vol. 132,no. 1, 13 pages, 2010.

[23] T. Hidaka, Y. Terauchi, and K. Nagamura, “Dynamic behaviorof planetary gear,” Bulletin of the JSME, vol. 22, no. 169, pp. 1017–1025, 1979.

[24] R.G. Parker, V. Agashe, and S.M.Vijayakar, “Dynamic responseof a planetary gear system using a finite element/contactmechanics model,” Journal of Mechanical Design, vol. 122, no.3, pp. 304–310, 2000.

[25] C. J. Bahk and R. G. Parker, “Analytical solution for the nonlin-ear dynamics of planetary gears,” Journal of Computational andNonlinear Dynamics, vol. 6, no. 2, Article ID 021007, 15 pages,2011.

[26] T. Sun and H. Hu, “Nonlinear dynamics of a planetary gearsystem with multiple clearances,” Mechanism and MachineTheory, vol. 38, no. 12, pp. 1371–1390, 2003.

[27] V. Nikolic, C. Dolicanin, and D. Dimitrijevic, “Numericalmodelling of gear set dynamic behaviour,” Scientific TechnicalReview, no. 3-4, pp. 48–54, 2010.

[28] V. Nikolic, C. Dolicanin, and D. Dimitrijevic, “Dynamic Modelfor the Stress and Strain State Analysis of a Spur Gear Transmis-sion, StrojniRki vestnik,” Journal ofMechanical Engineering, vol.58, no. 1, pp. 56–67, 2012.

[29] O. A. I. Gorosko and K. Hedrih, Analiticka Dinamika(Mehanika) DiskretnihNaslednih Sistema, (Analytical Dynamics(Mechanics) of Discrete Hereditary Systems), Monograph, Uni-versity of Nis, 2001.

[30] O. A. Gorosko and K. Hedrih, “Advances in development of theanalytical dynamics of the hereditary discrete systems,” Journalof Physics, vol. 96, Article ID 012143, 2008.

[31] B. S. Baclic and T. M. Atanackovic, “Stability and creep of afractional derivative order viscoelastic rod,” Bulletin, no. 25, pp.115–131, 2000.

[32] M. T. Atanackovic, C. D. Dolicanin, and S. Pilipovic, “Forcedoscillations of a single degree of freedom system with fractionaldissipation,” Scientific Publications of the State University of NoviPazar, vol. 3, no. 1, pp. 1–11, 2011.

[33] K. Hedrih and R. Knezevic, “Structural stability of the planetaryreductor nonlinear dynamics phase portrait,” Facta Universi-tatis, vol. 1, no. 7, pp. 911–923, 2000.

[34] K. Hedrih, K. Knezevic, and R. Cvetkovic, “Dynamics of plane-tary reductoe with turbulent damping,” International Journal ofNonlinear Sciences and Numerical Simulations, vol. 2, no. 3, pp.265–262, 2001.

[35] R. G. Parker, “On the eigenvalues and critical speed stability ofgyroscopic continua,” Journal of Applied Mechanics, vol. 65, no.1, pp. 134–140, 1998.

[36] K. Hedrih and L. J. Veljovic, “Nonlinear dynamic of heavy gyro-rotor with two skew rotating axes,” Journal of Physics, vol. 96,Article ID 012221, 2008.

[37] K. Hedrih and L. J. Veljovic, “Nonlinear dynamic of heavygyro-rotor with two rotating axes, facta universitates seriesmechanics,” Automatic Control and Robotics, vol. 14, no. 16, pp.55–68, 2004.

[38] K. R. Hedrih and L. Veljovic, “Vector rotators of rigid bodydynamics with coupled rotations around axes without intersec-tion,” Mathematical Problems in Engineering, vol. 2011, ArticleID 351269, 26 pages, 2011.

[39] K. R. Hedrih, “Dynamics of multi-pendulum systems withfractional order creep elements,” Journal of Theoretical andApplied Mechanics, vol. 46, no. 3, pp. 483–509, 2008.

[40] K. S. Hedrih and V. Nikolic-Stanojevic, “A model of geartransmission fractional order system dynamics,” MathematicalProblems in Engineering, vol. 2010, Article ID 972873, 23 pages,2010.

[41] S. Vijayakar, “Combined surface integral and finite elementsolution for a three-dimensional contact problem,” Interna-tional Journal for Numerical Methods in Engineering, vol. 31, no.3, pp. 525–545, 1991.

[42] S. Y. T. Lang, “Graph-theoretic modelling of epicycle gearsystems,”Mechanism andMachineTheory, vol. 40, no. 5, pp. 511–529, 2005.

Submit your manuscripts athttp://www.hindawi.com

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttp://www.hindawi.com

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttp://www.hindawi.com

Volume 2014 Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Stochastic AnalysisInternational Journal of

research article a new model of the fractional order...

Documents