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A dynamic model of multiple-stage planetary gear train composed of a two-stage planetary gear train and a one-stage 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 multiple-stage planetary gear train of wind driven generator.

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 lumped-parameter models and deformable or hybrid models of varying complexity [

Although the references available focused on different fields, most of them established mathematical model of one-stage planetary gear train. Dynamic model of multiple-stage planetary gear train is limitedly reported. Few reports about dynamic model of multiple-stage planetary gear train composed of two-stage planetary gear train and one-stage parallel axis and its dynamic load sharing characteristics are concerned.

In this study, a transmission scheme of load-split two-stage 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 load-split multiple-stage gear train composed of a two-stage planetary gear train and a one-stage 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.

The kinematic scheme of load-split two-stage planetary gear is shown in Figure

Kinematic scheme of load-split two-stage planetary gear.

The relationship between rotational speed of sun gear and that of planetary carrier and inner ring of first-stage planetary gear train is shown in

Equation (

We can come to (

The relationship between rotational speed of planetary gear and that of planetary carrier and inner ring is expressed as (

Thus, (

The relationship between the rotational speed of sun gear train and that of planetary carrier and inner ring of second-stage planetary gear is expressed as follows:

Equation (

Using (

Equations (

By connecting (

And the relationship between rotational speeds of planetary gear is expressed as follows:

Similar to (

Substitution of (

Considering the scheme of Figure

Substitution of (

The expressions of input and output rotational speed of load-split two-stage planetary gear train are given by substitution of (

Thus, transmission ratio formula of load-split two-stage planetary gear train is obtained as

General transmission ratio in Figure

Relationship between transmission ratio and characteristic parameters in load-split two-stage planetary gear train is shown in Figure

Relationship between transmission ratio and characteristic parameters of planetary gear train.

A multiple-stage gear train composed of a two-stage planetary gear train and a one-stage parallel axis gear is shown in Figure

Transmission system of load-split multiple-stage planetary gear train.

Dynamic model of Figure

Dynamic model of load-split multiple-stage planetary gear train.

Torsional model of single-stage planetary gear.

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:

The interaction force between sun gear and the

The interaction force between the inner ring and the

The interaction force between sun gear and the

The interaction force between the inner ring and the

The interaction force between the pinion gear and driven gear of the third-stage parallel axis gear along the line of action can be expressed as follows:

Fix the inner ring of the first-stage planetary gear train, and take the number of planetary gears of the planetary gear train as 3; namely,

The equations of the dynamic model are given in the matrix form as

Use numerical integration method for solving the dynamic equation (

When

The paper analyzes the transmission system as shown in Figure

Primary parameters of planetary gear train.

Parameter | Carrier | Ring | Sun gear | Planetary gear |
---|---|---|---|---|

Pitch radius, |
468 | 726 | 210 | 258 |

Base circle, |
— | 682.22 | 197.34 | 242.44 |

Mass, |
2042.77 | 410.34 | 344.91 | 388.39 |

Moment of inertia, ^{2}) |
462.55 | 226.57 | 7.62 | 16.25 |

Pressure angle, |
— | 20 | 20 | 20 |

Pitch radius, |
345 | 550 | 140 | 205 |

Base circle, |
— | 516.83 | 131.56 | 192.64 |

Mass, |
1212.49 | 80.56 | 131.40 | 176.54 |

Moment of inertia, ^{2}) |
151.75 | 25.65 | 1.29 | 5.12 |

Pressure angle, |
— | 20 | 20 | 20 |

Primary parameters of parallel-shaft gears.

Parameter | Pinion gear | Driven gear |
---|---|---|

Pitch radius, |
292 | 100 |

Base circle, |
274.39 | 93.97 |

Mass, |
208.22 | 24.14 |

Moment of inertia, ^{2}) |
8.87 | 0.12 |

Pressure angle, |
20 | 20 |

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

Load-sharing coefficient curves of each planetary gear with different mesh errors.

Each external-meshing first-stage planetary gear

Each internal-meshing first-stage planetary gear

Each external-meshing second-stage planetary gear

Each internal-meshing second-stage planetary gear

Results below can be concluded according to Figure

Each load-sharing coefficient increases with increasing mesh error.

Load sharing coefficient of internal-meshing is different from that of external-meshing under different mesh errors. Maximum external-meshing and internal-meshing load sharing coefficients of first-stage planetary gear are 1.579 and 1.645, respectively, while those of second-stage planetary gear are 1.630 and 1.665, respectively.

Compared to the differences in change rate of each load sharing coefficient of second-stage planetary gear, that of first-stage planetary gear is more evident. The maximum difference in change rate of first-stage planetary gear is 0.101/50

To analyze the influence of revolution speed of the first-stage 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 (

Load sharing coefficient curves of each planetary at different revolution speeds.

Each external-meshing first-stage planetary gear

Each internal-meshing first-stage planetary gear

Each external-meshing second-stage planetary gear

Each internal-meshing second-stage planetary gear

Influence of revolution speed on load-sharing coefficient can be concluded below, according to Figure

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.

At the variation interval of revolution speed, the change rate difference of load-sharing coefficient between internal and external meshing of first-stage planetary gear train is significantly different; those of first-stage planetary gears 1, 2, and 3 are 1.77%, 0.84%, and 1.49%, respectively. Similar result can be concluded in second-stage planetary gear train, and change rate differences of 1.47%, 2.71%, and 2.76% of second-stage planetary gears 1, 2, and 3 are figured out, respectively.

The dynamic model is built to account for the dynamic behavior of multiple-stage planetary gear train used in wind driven generator. The model can provide useful guideline for the dynamic design of the multiple-stage planetary gear train of wind driven generator.

Each load-sharing coefficient of the first-stage planetary gear varies more than that of the second-stage planetary gear. At the same mesh error, second-stage internal-meshing load sharing coefficient is the largest, the first-stage internal-meshing load sharing coefficient is the second largest, and the first-stage external-meshing load sharing coefficient is the minimum.

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 second-stage planetary gear train vary more than those of the first-stage planetary gear train.

Angular displacement of

Gear base radii,

Radius of the circle passing through planet centers

Sun-planet engaging angle

Ring-planet engaging angle

Total number of planet sets for the

Polar mass moment of inertia of

Mass of 1st-stage planetary gear

=

Polar mass moment of inertia of

Mass of 2nd-stage planetary gear

=

Gear base radii of

Sun-planet engaging angle for

Ring-planet engaging angle for

Sun-planet mesh error

Ring-planet mesh error

Sun-planet mesh stiffness

Ring-planet mesh stiffness

Sun-planet mesh damping coefficient

Ring-planet mesh damping coefficient

Angular displacement of

Sun-planet mesh error of

Ring-planet mesh error of

Mesh error of parallel-shaft gears

Sun-planet mesh stiffness of

Ring-planet mesh stiffness of

Mesh stiffness of parallel-shaft gears

Torsional stiffness associated with 1st-stage sun and 2nd-stage carrier

Torsional stiffness associated with 1st-stage carrier and 2nd-stage ring

Torsional stiffness associated with 2nd-stage sun and 3rd-stage gear

Sun-planet mesh damping coefficient of

Ring-planet mesh damping coefficient of

Mesh damping coefficient of parallel-shaft gears

Torsional damping coefficient associated with 1st-stage sun and 2nd-stage carrier

Torsional damping coefficient associated with 1st-stage carrier and 2nd-stage ring

Torsional damping coefficient associated with 2nd-stage sun and 3rd-stage gear

Input torque

Output torque.

The authors declare that they have no conflict of interests regarding the publication of this paper.

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).