Research Article  Open Access
Analysis of Dynamic Behavior of MultipleStage Planetary Gear Train Used in Wind Driven Generator
Abstract
A dynamic model of multiplestage planetary gear train composed of a twostage planetary gear train and a onestage parallel axis gear is proposed to be used in wind driven generator to analyze the influence of revolution speed and mesh error on dynamic load sharing characteristic based on the lumped parameter theory. Dynamic equation of the model is solved using numerical method to analyze the uniform load distribution of the system. It is shown that the load sharing property of the system is significantly affected by mesh error and rotational speed; load sharing coefficient and change rate of internal and external meshing of the system are of obvious difference from each other. The study provides useful theoretical guideline for the design of the multiplestage planetary gear train of wind driven generator.
1. Introduction
The planetary gear train, having advantages of large transmission ratio, simple construction, compactness, and smooth running, has been widely applied in many machines. In spite of these advantages, planetary gears may have undesirable dynamic behavior resulting in much noise, vibration, and other unacceptable performances. A number of papers have been published on planetary gear dynamics which comprise lumpedparameter models and deformable or hybrid models of varying complexity [1–7]. Modal analyses were performed by Lin and Parker [8] and Bodas et al. [9–13], who emphasized the structured modal properties of singlestage drives and showed that only planet, rotational, and translational modes could exist. It is important to understand the fundamental cause of the unequal load sharing behavior in planetary transmissions. The input torque applied should theoretically be shared by each planet in an planet; that is, each sunpinionring path should carry of the total torque. However, in actual transmissions, there is unequal load sharing between the parallel paths. Du et al. [14] found the deformation compatibility equations and the torque balance equations of the 2type planetary transmission system based on the characteristic that the system is composed of a closed loop of power flow. In consideration of the manufacturing error, assembly error, and float of the parts, the load sharing coefficient of each planetary gear was calculated by using the theory of equivalent mesh error and equivalent mesh stiffness. Gu and Velex [15] presented an original lumped parameter model of planetary gears to account for planet position errors and simulate their contribution to the dynamic load sharing amongst the planets. Singh [16] developed the concept of an epicyclic load sharing map to describe the load sharing characteristics of every epicyclic gear set at any positional error and torque level. A comprehensive experimental study [17] was conducted to study the load sharing behavior of a family of epicyclical gear sets with varying number of planets. Experiments were conducted at several error and torque levels. The results clearly showed the influence of positional errors and that the sensitivity of the epicyclical gear set increased as the number of planets increased. A physical explanation [18] has been provided for the load sharing behavior. Load required to produce the needed deformation is the cause of the unequal load sharing. This explains the effectiveness of system float in reducing the load sharing inequality. Lu et al. [19] presented a calculative model for singlestage planetary gear with the dynamic way to study the load sharing behavior of each planetary gear and the relationship between error and load sharing was analyzed. Ye et al. [20] built an analytical model for NGW planetary gear train with unequal modulus and pressure angles and analyzed the load sharing behavior of each planet.
Although the references available focused on different fields, most of them established mathematical model of onestage planetary gear train. Dynamic model of multiplestage planetary gear train is limitedly reported. Few reports about dynamic model of multiplestage planetary gear train composed of twostage planetary gear train and onestage parallel axis and its dynamic load sharing characteristics are concerned.
In this study, a transmission scheme of loadsplit twostage planetary gear used in wind driven generator is proposed. Transmission ratio of the planetary gear train is obtained, as well as the relationship between transmission ratio and characteristic parameter of planetary gear train, according to conversion mechanism method and general relationship among the speed of each unit in planetary gear train. Dynamic model of loadsplit multiplestage gear train composed of a twostage planetary gear train and a onestage parallel axis gear is established on the basis of lumped parameter theory and influence of revolution speed and mesh error on dynamic load sharing characteristic of the system is analyzed.
2. LoadSplit TwoStage Planetary Gear Train
2.1. Kinematic Scheme
The kinematic scheme of loadsplit twostage planetary gear is shown in Figure 1, which is composed of closed planetary gear train and differential planetary gear train. Former basic units 1c (planetary carrier) and 1s (sun gear) are connected to units 2r (inner ring) and 2c (planetary carrier) of differential gear train, respectively. Therefore, load split is realized by firststage and secondstage gear bearing input torque simultaneously. In Figure 1, 1r, 1p1, 1s, and 1c are inner ring, planetary gear, sun gear, and planetary carrier of firststage planetary gear train, respectively, while 2r, 2p1, 2s, and 2c represent corresponding units of secondstage planetary gear train.
2.2. Speed of Each Unit of FirstStage Planetary Gear Train
The relationship between rotational speed of sun gear and that of planetary carrier and inner ring of firststage planetary gear train is shown in
Equation (2.2) is obtained according to general relationship of relative gear ratio among each unit in planetary gear train principle and transmission type and characteristic parameter of firststage planetary gear train:
We can come to (3) by (1) and (2.2):
The relationship between rotational speed of planetary gear and that of planetary carrier and inner ring is expressed as (4), according to relationship of relative rotational speed among each unit in the firststage planetary gear train: Similar to (2.2), (5) is obtained as follows:
Thus, (6) can be obtained using (5) and (4): where is the characteristic parameter of planetary gear train and . , , and are tooth number of inner ring, sun gear, and planetary gear, respectively. and ( = c, s, r, p; = c, s, r, p; = c, s, r, p) represent the rotational speed and relative gear ratio of each unit of firststage planetary gear train, respectively.
2.3. Speed of Each Unit of SecondStage Planetary Gear Train
The relationship between the rotational speed of sun gear train and that of planetary carrier and inner ring of secondstage planetary gear is expressed as follows:
Equation (8) can be obtained according to the transmission characteristic of basic unit of secondstage planetary gear train:
Equations (10) and (11) are obtained by the relative movement relationship of planetary gear train’s units, when inner ring and planetary carrier of secondstage planetary gear train are fixed, respectively:
By connecting (10) and (11) to (9), the relationship of rotational speed between input and output units of secondstage planetary gear train is obtained as follows:
And the relationship between rotational speeds of planetary gear is expressed as follows:
Similar to (2.2), (14) can be obtained as follows:
Substitution of (14) into (13) yields
Considering the scheme of Figure 1, (16) can be given as follows:
Substitution of (16) and (3) into (15) gives where represents the characteristic parameter of planetary line of secondstage planetary gear train and . , , and stand for tooth number of inner ring, sun gear, and planetary gear of second planetary gear train, respectively. is the rotational speed of each unit of secondstage planetary gear train, and (; ; ) is the relative gear ratio of corresponding unit.
2.4. Transmission Ratio of the Planetary Gear Train
The expressions of input and output rotational speed of loadsplit twostage planetary gear train are given by substitution of (16) and (3) into (12), as follows:
Thus, transmission ratio formula of loadsplit twostage planetary gear train is obtained as
General transmission ratio in Figure 1 is related to characteristic parameters of the planetary gear trains and . Too small values of and result in undersizing of the system and decreasing of bearing capacity. Values of characteristic parameters have to be reasonable. Recommended interval of and is .
Relationship between transmission ratio and characteristic parameters in loadsplit twostage planetary gear train is shown in Figure 2. Transmission ratio rises with increasing values of characteristic parameters of the planetary gear train, and maximum transmission ratio in the interval of is 73.
3. Dynamic Model
3.1. Model of the MultipleStage Transmission System
A multiplestage gear train composed of a twostage planetary gear train and a onestage parallel axis gear is shown in Figure 3. 3g1 and 3g2 in Figure 3 stand for pinion gear and driven gear of parallel axis.
Dynamic model of Figure 3 is shown in Figure 4 based on lumped parameter theory. Since the firststage planetary gear train and the secondstage planetary gear train have the same basic structure, they can be represented by the single stage purely torsional model shown in Figure 5.
The linear displacements of all members of the multistage transmission system are shown as follows:
Generalized masses of all members of the multistage transmission system are shown as follows:
3.2. Dynamic Equation of the Multistage Transmission System
The interaction force between sun gear and the th planetary gear of the firststage planetary gear train along the line of action is given as follows:
The interaction force between the inner ring and the th planet gear of the firststage planetary gear train along the line of action can be expressed as follows:
The interaction force between sun gear and the th planetary gear of the secondstage planetary gear train along the line of action is
The interaction force between the inner ring and the th planet gear of the secondstage planetary gear train along the line of action can be expressed as follows:
The interaction force between the pinion gear and driven gear of the thirdstage parallel axis gear along the line of action can be expressed as follows:
Fix the inner ring of the firststage planetary gear train, and take the number of planetary gears of the planetary gear train as 3; namely, . According to the planetary mechanism modeling methods in [13], dynamic equation of the multistage transmission system shown in Figure 4 can be built, as shown in
The equations of the dynamic model are given in the matrix form as where the displacement vector, the mass matrix, the damping matrix, the stiffness matrix, and the load vector are given, respectively, as
4. Load Sharing Characteristic of LoadSplit MultipleStage Planetary Gear Train
4.1. Calculation of Load Sharing Coefficient
Use numerical integration method for solving the dynamic equation (28) of the system, obtain the responses to displacement and velocity of the system, and substitute the responses into (22)–(25), and then obtain the engaging forces, , , and . Make and , respectively, represent the load sharing coefficients of the internal and external meshing of all gear pairs of the firststage planetary gear train and and as those of the internal and external meshing of all gear pairs of the secondstage planetary gear train; then load sharing coefficients are expressed as where , are meshing cycle numbers for internal and external meshing of the planetary gear pair.
When and are used to stand for load sharing coefficient of internal and external meshing of each firststage gear and and for that of each secondstage gear in system period, respectively, the expression can be given as follows:
The paper analyzes the transmission system as shown in Figure 4. The basic parameters of the transmission system are shown in Tables 1 and 2, and other parameters can be determined by [21]. Substitute the relevant parameters of the system into (28) for solution. Use (30) and (31) to obtain the load sharing coefficients of the transmission system.


4.2. Influence of Mesh Error on Load Sharing Coefficient of the System
Load sharing property of planetary gear train is significantly affected by manufacturing error, installation error, and eccentric error, which cannot be neglected in planetary gear train. Considering system’s complexity, it is assumed that equivalent mesh error of each stage planetary gear at the direction of meshing line is equal, and values of 10, 20, 30, 40, and 50 μm are given, respectively. Load sharing properties of multiplestage gear train under these five conditions are studied. Relationships between load sharing coefficient curves of internal and external meshing of firststage and secondstage, which are calculated according to (31), are drawn in Figure 6, with different mesh errors.
(a) Each externalmeshing firststage planetary gear
(b) Each internalmeshing firststage planetary gear
(c) Each externalmeshing secondstage planetary gear
(d) Each internalmeshing secondstage planetary gear
Results below can be concluded according to Figure 6.(1)Each loadsharing coefficient increases with increasing mesh error.(2)Load sharing coefficient of internalmeshing is different from that of externalmeshing under different mesh errors. Maximum externalmeshing and internalmeshing load sharing coefficients of firststage planetary gear are 1.579 and 1.645, respectively, while those of secondstage planetary gear are 1.630 and 1.665, respectively.(3)Compared to the differences in change rate of each load sharing coefficient of secondstage planetary gear, that of firststage planetary gear is more evident. The maximum difference in change rate of firststage planetary gear is 0.101/50 μm, while that of secondstage planetary gear is only 0.003/50 μm.
4.3. Influence of Revolution Speed on Load Sharing Coefficient
To analyze the influence of revolution speed of the firststage planetary gear on load sharing coefficient, the revolution speed is set as 5 r/min, 10 r/min, 15 r/min, 20 r/min, and 25 r/min, respectively. Equation (31) is used to calculate the load sharing coefficient under different conditions, and curves are obtained in Figure 7.
(a) Each externalmeshing firststage planetary gear
(b) Each internalmeshing firststage planetary gear
(c) Each externalmeshing secondstage planetary gear
(d) Each internalmeshing secondstage planetary gear
Influence of revolution speed on loadsharing coefficient can be concluded below, according to Figure 7.(1)Each load sharing coefficient increases with raising the revolution speed, which indicates that load sharing capacity of planetary gear train is weakened and vibration is aggravated with increasing revolution speed.(2)At the variation interval of revolution speed, the change rate difference of loadsharing coefficient between internal and external meshing of firststage planetary gear train is significantly different; those of firststage planetary gears 1, 2, and 3 are 1.77%, 0.84%, and 1.49%, respectively. Similar result can be concluded in secondstage planetary gear train, and change rate differences of 1.47%, 2.71%, and 2.76% of secondstage planetary gears 1, 2, and 3 are figured out, respectively.
5. Conclusion
(1)The dynamic model is built to account for the dynamic behavior of multiplestage planetary gear train used in wind driven generator. The model can provide useful guideline for the dynamic design of the multiplestage planetary gear train of wind driven generator.(2)Each loadsharing coefficient of the firststage planetary gear varies more than that of the secondstage planetary gear. At the same mesh error, secondstage internalmeshing load sharing coefficient is the largest, the firststage internalmeshing load sharing coefficient is the second largest, and the firststage externalmeshing load sharing coefficient is the minimum.(3)Load sharing property is weakened and transmission system’s vibration is aggravated with increasing revolution speed. At each interval of revolution speed, internal and external meshing load sharing coefficients of the secondstage planetary gear train vary more than those of the firststage planetary gear train.
Nomenclature
:  Angular displacement of th member (, , ; , 2, 3) 
:  Gear base radii, ; 
:  Radius of the circle passing through planet centers 
:  thstage radii of the circle passing through planet centers; 
:  Sunplanet engaging angle 
:  Ringplanet engaging angle 
:  Total number of planet sets for the thstage drive train; 
:  Polar mass moment of inertia of th member for 1ststage drive train; 
:  Mass of 1ststage planetary gear 
:  = 
:  Polar mass moment of inertia of th member for 2ndstage drive train; 
:  Mass of 2ndstage planetary gear 
:  = 
:  Gear base radii of th member for thstage drive train; ; 
:  Sunplanet engaging angle for thstage drive train; 
:  Ringplanet engaging angle for thstage drive train; 
:  Sunplanet mesh error 
:  Ringplanet mesh error 
:  Sunplanet mesh stiffness 
:  Ringplanet mesh stiffness 
:  Sunplanet mesh damping coefficient 
:  Ringplanet mesh damping coefficient 
:  Angular displacement of th member for thstage drive train; ; ; 
:  Sunplanet mesh error of th planet gear for thstage drive train; ; 
:  Ringplanet mesh error of th planet gear for thstage drive train; ; 
:  Mesh error of parallelshaft gears 
:  Sunplanet mesh stiffness of th planet gear for thstage drive train; ; 
:  Ringplanet mesh stiffness of th planet gear for thstage drive train; ; 
:  Mesh stiffness of parallelshaft gears 
:  Torsional stiffness associated with 1ststage sun and 2ndstage carrier 
:  Torsional stiffness associated with 1ststage carrier and 2ndstage ring 
:  Torsional stiffness associated with 2ndstage sun and 3rdstage gear 
:  Sunplanet mesh damping coefficient of th planet gear for thstage drive train; ; 
:  Ringplanet mesh damping coefficient of th planet gear for thstage drive train; ; 
Mesh damping coefficient of parallelshaft gears  
:  Torsional damping coefficient associated with 1ststage sun and 2ndstage carrier 
:  Torsional damping coefficient associated with 1ststage carrier and 2ndstage ring 
:  Torsional damping coefficient associated with 2ndstage sun and 3rdstage gear 
:  Input torque 
:  Output torque. 
Conflict of Interests
The authors declare that they have no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors gratefully acknowledge the support of the Chinese National Science Foundation (no. 51175299), the Shandong Provincial Natural Science Foundation, China (no. ZR2010EM012), the Independent Innovation Foundation of Shandong University (IIFSDU2012TS044), and the Graduate Independent Innovation Foundation of Shandong University, GIIFSDU (no. yzc10117).
References
 Z. D. Fang, Y. W. Shen, and Z. D. Huang, “Dynamic characteristics of 2KH planetary gearing,” Journal of Northwestern Polytechnical University, vol. 10, no. 4, pp. 361–371, 1990. View at: Google Scholar
 A. Kahraman, “Planetary gear train dynamics,” Journal of Mechanical Design, Transactions of the ASME, vol. 116, no. 3, pp. 713–720, 1994. View at: Google Scholar
 R. Hbaieb, F. Chaari, T. Fakhfakh, and M. Haddar, “Dynamic stability of a planetary gear train under the influence of variable meshing stiffnesses,” Proceedings of the Institution of Mechanical Engineers D, vol. 220, no. 12, pp. 1711–1725, 2006. View at: Publisher Site  Google Scholar
 Y. Guo and R. G. Parker, “Purely rotational model and vibration modes of compound planetary gears,” Mechanism and Machine Theory, vol. 45, no. 3, pp. 365–377, 2010. View at: Publisher Site  Google Scholar
 S.Y. Wang, Y.M. Song, Z.G. Shen, C. Zhang, T.Q. Yang, and W.D. Xu, “Research on natural characteristics and loci veering of planetary gear transmissions,” Journal of Vibration Engineering, vol. 18, no. 4, pp. 412–417, 2005. View at: Google Scholar
 Z. F. Ma, K. Liu, and Y. H. Cui, “Analysis of the torsional characteristics of planetary gear trains of an increasing gearbox,” Mechanical Science and Technology For Aerospace Engineering, vol. 29, no. 6, pp. 788–791, 2010. View at: Google Scholar
 J. Lin and R. G. Parker, “Analytical characterization of the unique properties of planetary gear free vibration,” Journal of Vibration and Acoustics, Transactions of the ASME, vol. 121, no. 3, pp. 316–321, 1999. View at: Google Scholar
 J. Lin and R. G. Parker, “Structured vibration characteristics of planetary gears with unequally spaced planets,” Journal of Sound and Vibration, vol. 233, no. 5, pp. 921–928, 2000. View at: Publisher Site  Google Scholar
 S. Dhouib, R. Hbaieb, F. Chaari, M. S. Abbes, T. Fakhfakh, and M. Haddar, “Free vibration characteristics of compound planetary gear train sets,” Proceedings of the Institution of Mechanical Engineers C, vol. 222, no. 8, pp. 1389–1401, 2008. View at: Publisher Site  Google Scholar
 S. J. Wu, H. Ren, and E. Y. Zhu, “Research advances for dynamics of planetary gear trains,” Engineering Journal of Wuhan University, vol. 43, no. 3, pp. 398–403, 2010. View at: Google Scholar
 A. Bodas and A. Kahraman, “Influence of carrier and gear manufacturing errors on the static load sharing behavior of planetary gear sets,” JSME International Journal C, vol. 47, no. 3, pp. 908–915, 2004. View at: Publisher Site  Google Scholar
 A. Kahraman, “Natural modes of planetary gear trains,” Journal of Sound and Vibration, vol. 173, no. 1, pp. 125–130, 1994. View at: Publisher Site  Google Scholar
 A. Kahraman, “Free torsional vibration characteristics of compound planetary gear sets,” Mechanism and Machine Theory, vol. 36, no. 8, pp. 953–971, 2001. View at: Publisher Site  Google Scholar
 J. F. Du, Z. D. Fang, B. B. Wang, and H. Dong, “Study on load sharing behavior of planetary gear train based on deformation compatibility,” Journal of Aerospace Power, vol. 27, no. 5, pp. 1166–1171, 2012. View at: Google Scholar
 X. Gu and P. A. Velex, “dynamic model to study the influence of planet position errors in planetary gears,” Journal of Sound and Vibration, vol. 331, pp. 4554–4574, 2012. View at: Google Scholar
 A. Singh, “Epicyclic load sharing map: development and validation,” Mechanism and Machine Theory, vol. 46, no. 5, pp. 632–646, 2011. View at: Publisher Site  Google Scholar
 H. Ligata, A. Kahraman, and A. Singh, “An experimental study of the influence of manufacturing errors on the planetary gear stresses and planet load sharing,” Journal of Mechanical Design, Transactions of the ASME, vol. 130, no. 4, Article ID 041701, 2008. View at: Publisher Site  Google Scholar
 A. Singh, “Load sharing behavior in epicyclic gears: physical explanation and generalized formulation,” Mechanism and Machine Theory, vol. 45, no. 3, pp. 511–530, 2010. View at: Publisher Site  Google Scholar
 J. Lu, R. Zhu, and G. Jin, “Analysis of dynamic load sharing behavior in planetary gearing,” Journal of Mechanical Engineering, vol. 45, no. 5, pp. 85–90, 2009. View at: Publisher Site  Google Scholar
 F.M. Ye, R.P. Zhu, and H.Y. Bao, “Static load sharing behavior in NGW planetary gear train with unequal modulus and pressure angles,” Journal of Central South University, vol. 42, no. 7, pp. 1960–1966, 2011. View at: Google Scholar
 R. F. Li, Wang, and J. J. Vibration, Shock and Noise of Gear Dynamics, Science Press, Beijng, China, 1997.
Copyright
Copyright © 2014 Jungang Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.