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In order to research the vibroimpact characteristics of a planetary gear transmission system under high speed and lightly loaded conditions, a new modeling method is proposed. In the modeling process, linear spring was used to simulate gear mesh elasticity under heavy load cases, and Hertz contact theory was used to calculate the contact force of gear pair under light load cases. Then, effects of the working conditions on the system vibroimpact characteristics are analyzed. The results show that, with input speed growing, the mesh force produced obvious fluctuations on the resonance frequencies of the sun gear and carrier torsion vibration, ring gear’s transverse vibration under the heavy load. Under light load condition, the collision vibration occurs in the gear pair; the changing trend of the contact force shows strongly nonlinear characteristics. The time of mesh-apart in gears pair decreases gradually as the load is increased; until it reaches collision vibration threshold value, the gear pair is no longer mesh-apart. With increasing of the input speed, the time of mesh-apart is decreased gradually; the fluctuation amplitude of contact force shows a linearly increasing trend. The study provides useful theoretical guideline for planetary gear transmission low-noise design.

Planetary gear transmission is characterized by large gear ratio, compact structure, and light weight and has been widely applied to various industries. Planetary gear transmission has become an indispensable key device in main power system in high speed and heavy load transmission device such as helicopter transmission and marine transmission; however, the wear in tooth profile and error in machining are likely to result in gear backlash in gear pair, which will bring repetitive collision phenomenon of contact [

Impact process is a complex process associated with relative speed of objects in contact, geometrical shape of contact surface, duration of contact, and local plastic deformation [

Typical analyses model of planetary gear transmission can be divided into two main groups: finite element model and lumped parameter model. The lumped parameter model is the most common model used to study dynamic behaviors of the planetary gear system. With the development of research, this kind of analytical model has developed from initial purely torsional model to torsion-transverse vibration coupling and torsion-transverse-axial coupling model [

Same scholars use a clearance-type nonlinear restoring function in the model to describe the gear mesh and backlash. Walha et al. found that gear contact is characterized by a periodically changing stiffness and a backlash can lead to loss of the contact [

Through using this kind of model, the gear tooth flank backlash is considered, and the contact loss phenomenon and nonlinear dynamic characteristics are simulated. However, when the gear tooth recontact, the gear pairs impact will happened. The impact force and its influence on the system dynamics cannot be solved by this kind model. Barthod et al. indicate that the rattling noise is produced by gear system vibroimpact, and the rattling noise shall be enhanced evidently with the increase of excitation in frequency and amplitude [

Finite element model has more geometry information of the tooth, so its result is more precise and intuitive. Generally, finite element model is used to analyze the gear loaded tooth contact analysis (LTCA) [

The paper presents vibroimpact analytical model building method of planetary gear transmission system using the lumped parameter method, in which the Hertz contact theory is induced to describe impact of the gear pairs, the influence of operating condition of system on dynamic engaging force is calculated, and the system vibroimpact characteristics are analyzed under the conditions of heavy load and light load.

The planetary gear transmission system given in the paper is as shown in Figure

The planetary gear transmission system sketch and three-dimensional model.

Planetary gear transmission system sketch

Assembly model for planetary gear transmission system

The basic parameters for experimental prototype are as shown in Table

The planetary gear transmission system parameters.

Parameter | Sun gear | Planet wheel | Ring gear |
---|---|---|---|

Number of teeth | 34 | 31 | 96 |

Face width (mm) | 42 | 42 | 42 |

Module (mm) | 3 | 3 | 3 |

Pressure angle (°) | 28 | 28 | 28 |

Modification coefficient | −0.02 | 0.02 | 0.02 |

Mass | 2.7 | 2.5 | 5.3 |

Moment of inertia | 0.008 | 0.007 | 0.178 |

2K-H planetary gear transmission system dynamic model is shown in Figure

The planetary gear transmission system dynamic model.

A number of simplifying assumptions were employed in establishing the gear transmission subsystem dynamic model through lumped parameter method:

The gear body and planet carrier are assumed to be rigid. The flexibilities of the gear teeth at each gear mesh interface are modeled by a spring having periodically time varying stiffness.

The engaging force between gears always exists along the gear line of action. Each support part’s flexibility is included in the form of a linear spring, and the support stiffness is constant.

Planet gears are surrounding the central gear (including sun gear and planet carrier). Every planet gear has the same mass, moment of inertia, and support stiffness. Friction during gear engagement is neglected.

Each planet gear has the same meshing stiffness with the sun gear except for different phase positions. The support stiffness of sun gear, ring gear, and planet carrier in all directions is the same.

The gear-shaft connections were assumed to be rigid, ignoring the stiffness of the connections and any consequent relative torsional motion between the shaft and gear hub.

Then in the system

When load is large enough, the gear pair are always engaged and no contact loss happened, so engaging force generated on two tooth flanks and elastic deformation are in linear proportion. At this time, the gear pair stiffness is replaced by linear spring; the dynamic model for sun gear and planet gear is as shown in Figure

The dynamic model of sun gear and planet gear.

Here, the engaging force is

When transmission system load is light, the load is unable to make two tooth flanks maintain constant engaging, so the gear pair comes into instantaneous impact on two tooth flanks. Therefore, driven gear shall speed up in a second to separate from driving gear to reciprocate in such a way that the gear pair contact loss phenomenon appeared. In this process, the impact force is produced between the contacted gears. The paper describes elastic effect between gear teeth by Hertz contact mechanics. Under this condition, the dynamic model for sun gear and planet gear is as shown in Figure

The vibroimpact model of sun gear and planet gear.

In the gear engagement process, with the influence of errors and structure deformations, the direction of gear engaging force is not strictly along the theoretical action line. Since such errors and deformations are small enough, here we suppose that the gear engaging force is still along theoretical action line, and the two gear teeth in contact shall be taken as two bodies in impact. Taking material damping into consideration, the generalized Hertz formula of the sun gear and planet gear is shown as the following forms:

The speed relation of hysteresis damping coefficient

In addition, impact is energy loss and indicated by

Differential equation for system vibration is built based on stress condition of all parts

When the load act on the gear train is heavy, the tooth flanks of gear pair stay in contact all the time; the dynamic load of system is calculated at the speed of 500 r/min and under the load of 500 N·m.

The dynamic force of sun gear and planet gear is as shown in Figure

The dynamic force of sun gear and planet gear.

Time history of sun gear and planet gear meshing force

Sun gear and planet gear meshing force spectra

The dynamic force of planet gear and ring gear is as shown in Figure

The dynamic force of ring gear and planet gear.

Time history of sun gear and planet gear meshing force

Sun gear and planet gear meshing force spectra

The vibration and excitation of planetary gear transmission system are composed of engaging frequency and multiple frequency; in addition, the relation between excitation component and natural frequency of transmission system also has direct effect on vibration amplitude. The dynamic response of gear box is as shown in Figure

The dynamic force waterfall curve of the sun gear and planet gear.

The dynamic force waterfall curve of the sun gear and planet gear

The dynamic force waterfall curve of the planet gear and ring gear

Although vibration of parametric excitation of the gear system is linear matter, the dynamic load frequency components are complex due to coupled effect of the time varying mesh stiffness and the contact force of tooth profile. Meanwhile, different speed shall have a direct effect on excitation frequency component. In order to reveal the influence of rotating speed on fluctuation in dynamic load of bearing, the 1st harmonics variations of

The dynamic force first-order harmonic component of ring gear and planet gear under different speed.

The first-order harmonic component variation of engaging force is divided into three sections, namely a, b, and c. The first section is from 500 r/min to 1050 r/min,

The research of gear system vibroimpact does not take account of structural permanent deformation, but the process for gear from separated to contact is considered. With effect of relative velocity, the gear system is inevitably to bring impact between gear teeth of the driving and driven gear; then large impulsive load must be generated.

At the condition of the speed being 500 r/min and the load being light and gear contact force being as shown in Figure

The dynamic impact force under different load.

Gear impact force and planet gear angular speed without load

Gear impact force and planet gear angular speed under

In addition, we can see that there are two obvious peak values in every cycle, so the second harmonic component is main frequency component of impact force. With the increase of load, the time for disengagement shall gradually shorten. When load is 21 N·m, the gear pair is in critical state between normal engagement and disengagement, as shown in Figure

The influence of load on vibroimpact of planetary gear transmission system is shown in Figure

The effect of the load on gear vibroimpact characteristics.

At this point, gear shall start vibroimpact of engagement, disengagement, and reengagement [

When load is up to threshold value (in the paper it is 21 N·m), gear engagement shall enter into normal engagement, in which gear shall not disengage again. With the increase of load, amplitude in fluctuation of engaging force shall show linear increase, but its increasing speed is reduced compared with that in vibroimpact stage.

The influence of rotating speed on engagement of planetary gear transmission system is shown in Figure

The effect of the input speed on gear vibroimpact characteristics.

In the process of speed change, since torsional direction of planet gear is unconstrained, with torsional rigidity of sun gear being much smaller than contact rigidity, therefore, the influence of constraint elasticity of two tooth flanks on coefficient of recovery shall be very small; at this time, the impact of gear can be assumed as free body [

The floating trajectory of sun gear and ring gear at different input speed under the condition of the load is 0 as shown in Figure

Floating trajectory of center gear at various speed.

Floating trajectory of the sun gear

Floating trajectory of the ring gear

When the gears tooth are contacted, the impact force which is produced by different speed between two faces of gear teeth is smaller, and the impact energy is also small and will be rapidly consumed; then the driven gear starts to speed down to give rise to second impact, so the system shows periodical vibration. Thus, when the rotating speed is 100 r/min, with effect of the gear impact, every impact between gears shall make floating trajectory generate larger deviation at instant. Therefore, the floating trajectory of center gear is regular and present

When input rotating speed is 2000 r/min, the impact force between sun gear and planet gear shall increase; the system shows subharmonic resonance phenomenon, the time history of the sun gear microdisplacement along

Time history and frequency spectrum for displacement of center gear along

The time history of the sun gear displacement in the

The spectra of the sun gear displacement in the

The time history of the ring gear displacement in the

The spectra of the sun gear displacement in the

This paper presents a novel modeling method of planetary gear system, in which the Hertz contact theory is induced to describe impact of the gear pairs under the condition of the high speed and light load. The vibroimpact characteristics of the system are analyzed; some interesting conclusions are obtained as follows:

As input speed continuously increased under large load condition, the harmonic components of dynamic mesh load show radial distribution. The dynamic load obvious fluctuation appeared when harmonic components are around natural frequency of system.

Under the light load condition, the vibroimpact phenomenon happened in planetary gear system. With increase of load, the main frequency component of impact force between the gears is changed from the second harmonic component into engaging frequency component. Meanwhile, the time of gear mesh-apart is decreased continuously, till load is big enough to make gear system engaged normally.

With increasing of input speed under the light load condition, the gear mesh-apart time shortens gradually; the relative speed of the gear pairs is increased, so vibroimpact of the system increased in intensity, and amplitude of impact force between the gears increased linearly.

With increasing of input speed, the floating trajectory of the center gear is changed from regular leafy curves to the irregularly curves. Nonlinear feature of the center gear displacement can be observed at 2000 r/min.

The authors declare that there is no conflict of interests regarding the publication of this paper.

Thanks must be given to Xinjiang University for giving the authors a good surrounding and Natural Science Foundation of Xinjiang Province (2014211B004, XJEDU2014S009, and BS130120) for giving the authors a good financial backing to finish the most important part of this research.