Dynamic Test Study on Self Aligning System of Pinion

Zhai Yuyi, Yang Shiliang, Liu Liang

Shanghai University 149 Yanchang Road, Shanghai.200072, China

E-mail: yyzhai@mail.shu.edu.cn

Abstract

A practical testbed that can help study the dynamic analysis results of self aligning pinion system is presented in this paper. This testbed serves as an auxiliary mechanism for the final pinion self aligning system which could relief vibration, impact and so on other problems during low speed and heavy load transmission. The dynamic performance of this gear system is analysed in this paper. Besides, the relationship between stress and stain under different flexible rotation angle is also presented. Finally, practical analysis results are educed, which can provide theoretical basis to select and optimize design parameters of this device. 1. Introduction

One principal character of low speed and heavy load gear transmission system is large torque and severe bias load on the final stage. Pinion device with self aligning can effectively relief this problem. But vibration and impact sill exist in pinion transmission system. Vibration and impact from heavy load gear transmission has many bad effects on equipments. They deteriorate the dynamic performance and affect the original precision, production efficiency and service life. Furthermore, the noise caused by vibration also pollutes the environment. Thus, it’s important to study the dynamic behaviours and working performance of gear system, which may have great effect to all kinds of machine and mechanism. This paper focuses on dynamic property of crowned gear system equipped with a self aligning pinion device. 2. Working principle of self aligning pinion system

Fig.1 shows the final pinion device with self aligning. An external gear on the shaft meshes with the internal pinion and the pinion meshes with the final gear to transmit the torque. The external gear is a crowned gear and its surface is like a spherical face, so the pinion can

revolve round its symcenter along spherical face and change its position by deflecting a small angle along with position change of large gear. Thus we call it a self aligning pinion which can compensate the bias load of final stage gear and avoid its bad effects such as instability transmission and deadlocking and so on. Though its self aligning range is limited, they still can meet requirements of general drive under low speed and heavy load. Applications of this kind of device in industry are prospective. In this device, the capable angle of self aligning pinion is very important. We develop an experimental device to test the changes of transmission characteristic under different angles.

Fig.1 Schematic diagram of pinion system with

self aligning There are three aspects to comprehensively study

kinetics: dynamic excitation, system design and response characteristics, which are to research the mechanism, characteristics and influencing factors of vibration and noise caused in gear system. One main application field of kinetic research is the guide role of taking measure to reduce the vibration and noise of gear system[3, 4, 5].

3. Design of dynamic testbed

Because the actual volume of this transmission device

is very large, it is difficult to make this device with a 1:1 proportion. Besides, the actual load torque is very large (several hundred KNM) and it is impossible to load under experimental conditions. We scale-down the volume of the device and its load torque, making a small size testbed

2009 International Conference on Artificial Intelligence and Computational Intelligence

978-0-7695-3816-7/09 $26.00 © 2009 IEEE

DOI 10.1109/AICI.2009.86

329

to simulate this transmission device. In this paper, not only the experimental conditions are considered, but also the characteristics of heavy load and dynamic parameters should not to be changed.

The main feature of this pinion device is using a crown gear to transmit the torque and to do the self aligning. We make an emphasis on the design of this crowned gear system. Further experiment was conducted to gain the dynamic characteristics of drum tooth gear system.

Testbed is used to test the dynamic characteristics of self aligning pinion device and the change of transmission characteristics under different angles.

The testbed we designed is shown as Fig.2. The bearing supporter of large gear side can revolve a small angle round large gear’s symcenter along bottom board, which simulate the desaxis caused by bias load of the final stage. Installation precision is the premise of this experiment’s accuracy.

Fig.3 shows motion dynamic transmission model of multi-rigid-body System. ±α are the rotation angles of two directions.

4. Analysis of flexible multi-body dynamics

Flexible Multi-body Dynamics (FMD) is combination

of Multi-Rigid-Body System Dynamics and Flexible Multi-body System Dynamics. In Multi-Rigid-Body System Dynamics, every component is abstracted to a rigid body (But elasticity, damping etc. of each connection point are accounted to the analysis model). FMD further considers the deformation of each component. It studies on the coupling of deformation event and rigid motion, and the dynamic effects caused in this process. The key feature of FMD is that deformation and rigid motion appear and simultaneously and coupling at the same time. So FMD is different from Multi-Rigid-

Body System Dynamics or Structural Dynamics. In fact, dynamic equation of FMD is combination and extension of Multi-Rigid-Body System Dynamics equation and Structural Dynamics equation.

As this pinion device can have flexible rotation angle,

we see it a flexible component and use FMD to analysis it. And FMD is the natural extension and development of Multi-Rigid-Body System Dynamics. In this paper, we

firstly build the dynamic model by the means of Multi- Rigid-Body System Dynamics, and then introduce the FMD model. And through experimental verification, the dynamic model of this system will be perfected.

After transforming, the relationship of base vectors of connector between large gear and drum gear is:

2 01 12 02 22 21 10 00 22 03 3

T

e e

e H H H H e

e e

⎧ ⎫ ⎧ ⎫⎪ ⎪ ⎪ ⎪

= • • • •⎨ ⎬ ⎨ ⎬⎪ ⎪ ⎪ ⎪⎩ ⎭ ⎩ ⎭

(1)

In which:

330

021 2 2 21 1 1

2 2 1 1

1 0 0 1 0 00 cos sin 0 cos sin0 sin cos 0 sin cos

H Hθ θ θ θθ θ θ θ

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥= −⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦

, (2)

θ1 is the axial rotation angle of pinion andθ2 is the axial rotation angle of large gear.

010 1 1 10 0 0

1 1 0 0

1 0 0 1 0 00 cos sin 0 cos sin0 sin cos 0 sin cos

H Hθ θ θ θθ θ θ θ

⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥= −⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥−⎣ ⎦ ⎣ ⎦

. (3)

θ0is the axial rotation angle of drum gear.

And 2 21 12 22 22 22 23 3

T

e h

e H h

e h

⎧ ⎫ ⎧ ⎫⎪ ⎪ ⎪ ⎪

=⎨ ⎬ ⎨ ⎬⎪ ⎪ ⎪ ⎪⎩ ⎭ ⎩ ⎭

, (4)

0 01 10 02 00 2

0 03 3

h e

h H eh e

⎧ ⎫ ⎧ ⎫⎪ ⎪ ⎪ ⎪

=⎨ ⎬ ⎨ ⎬⎪ ⎪ ⎪ ⎪⎩ ⎭ ⎩ ⎭

, (5)

H22, H00 are orientation matrixes，212223

e

e

e

⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭

，

010203

e

ee

⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭

are

vectors of connector. When we analysis this system with FMD, hinges are

supposed to be rigid which means axis of rotation and slip doesn’t have shape change in the motion process. Because of component’s deformation (in pinion system, it’s the self aligning), transformation matrix of hinge coordinate and connector base is not constant matrix. So the hinge coordinates obtained by Multi-Rigid-Body System Dynamics is:

2 11 12 12 21 2

1233

h h

h H h

hh

⎧ ⎫ ⎧ ⎫⎪ ⎪ ⎪ ⎪⎪ ⎪ =⎨ ⎬ ⎨ ⎬⎪ ⎪ ⎪ ⎪

⎩ ⎭⎪ ⎪⎩ ⎭

. (6)

But actually deformations exist, so 22

112 22 22 23 3

hh

h h

h h

⎧ ⎫⎧ ⎫ ⎪ ⎪⎪ ⎪ ⎪ ⎪≠⎨ ⎬ ⎨ ⎬⎪ ⎪ ⎪ ⎪⎩ ⎭ ⎪ ⎪⎩ ⎭

. (7)

We need to introduce deformation matrix of rotation B11, then:

22 1

11 12 2 12 11 2 11 21 22 123 33

hh hh B h B H h

h hh

⎧ ⎫⎧ ⎫ ⎧ ⎫⎪ ⎪⎪ ⎪ ⎪ ⎪⎪ ⎪= =⎨ ⎬ ⎨ ⎬ ⎨ ⎬⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩ ⎭ ⎩ ⎭⎪ ⎪⎩ ⎭

. (8)

5. Different torque fluctuation at different rotation angle

Elastic deformation not only occurred in rotate

direction but also in transverse direction which causes the torque fluctuation.

Fig.4 shows different torque fluctuation at different rotation angle (α ＝1°、2°、3°). The average value of torque is about 10N．M. We can see that the larger rotation angle (α), the bigger torque fluctuation is. Results are tenable in microscopic view when we make the sampling frequency higher (showed in Fig.5).

6. Conclusion

By building an experimental platform and applying

FMD theories, this paper made a study on the dynamic properties of the self aligning pinion system, which provides theoretical basis to the selection and optimization of the design parameters of crowned gear. Changes of dynamic characteristics at different rotation angle are measured and analyzed. This testbed offers the dynamic properties of the self aligning pinion system and also gives the theoretical basis for the pinion system’s application in low speed and heavy load transmissions and its related fields.

7. References

[1] Lu Lesheng, Base of Multi-Rigid-Body System Dynamics,

Harbin Institute of Technology Press, 1995. [2] Lu Youfang, Flexible Multi-body Dynamics, Higher

Education Press, 1993. [3] He Yuancheng, Luo Huailin, Luo Xudong, “Pinion device

with self aligning used in calciner transmission,” Sichuan college of science and engineering journal, vol. 19, no. 1, pp. 1-3, February 2001.

[4] He Yuancheng, Zheng Ziqiu, et al, “Mechanism of gear transmission of heavy load and low speed drive” Modern Manufacturing Engineering, vol. 10, 2002.

[5] Zheng Ziqiu, et al, “DMGH transmission,” Symposium of 2004 annual meeting of Chinese Mechanical Engineering Society: China Machine Press, 2004.

[6] He Yuancheng, Zeng Huanglin, Feng Yanbin, Zheng Ziqiu, Luo xudong, “Study of last stage pinion device with self aligning of heavy load and low speed drive,” Proceedings of the International Conference on Mechanical Transmissions, Published by Science Press,2006, PP. 1065~1068

331

332

333