Construction of 12 DOFs spur gear coupling dynamic model

Huanchao Lv1 , Zhengminqing Li2 , Wenlin Zhu3 , Xin Tang4 , Jie Gao5 , Rupeng Zhu6

1, 2, 6National Key Laboratory of Science and Technology on Helicopter Transmission, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China

3, 4, 5Science and Technology on Helicopter Transmission Laboratory, AECC Hunan Aviation Powerplant Research Institute, Zhuzhou, 412002, China

2Corresponding author

Vibroengineering PROCEDIA, Vol. 28, 2019, p. 241-245. https://doi.org/10.21595/vp.2019.21028
Received 17 September 2019; accepted 1 October 2019; published 19 October 2019

Copyright © 2019 Huanchao Lv, 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.
Creative Commons License
Abstract.

A 12-degree-of-freedom (DOF) spur gear dynamic model is constructed, which is coupled by the mesh gear pair and the gearbox. The construction method of spur gear coupling dynamic model, based on lumped mass method, is better than finite element method, due to higher modeling efficiency. The work would be benefit to spur gear coupling dynamic modeling and analyses.

Highlights
  • A dynamic model coupled by the mesh gear pair and the gearbox is constructed.
  • The construction method of spur gear coupling dynamic model is more efficient.
  • Using modal assurance criterion to verify the accuracy of the proposed method.

Keywords: spur gear, coupling dynamic model, 12 DOFs.

1. Introduction

Gear dynamic models are focused by many scholars. There is an extensive body of literatures on it [1-9]. Jin et al. established gear dynamic models coupled with bending-torsion-axis-swing of mesh pairs based on lumped mass method [10]. Zhu et al. constructed finite element models of the gear transmission, and evaluated dynamic behavior of the system [11, 12]. Ren et al. proposed a construction method of gear dynamic models based on substructure method [13-15]. However, the gear coupling dynamic models associated with mesh pairs and gearbox supports are few studied. Thus, in the paper, a 12 DOFs spur gear coupling dynamic model, based on lumped mass method, is proposed. The work would be helpful to the spur gear coupling dynamic analyses.

2. Construction of 12DOFs dynamic model

The gear transmission system is mainly composed of two spur gears, bearings and gearbox supports. When modeling with the finite element method, it is inefficient because of the complexity of the gearbox supports. Therefore, a 12 DOFs coupling dynamic model based on lumped mass method is established, as shown in Fig. 1.

Fig. 1. The coupling dynamic model

As illustrated in Fig. 1, subscript p and g express driving gear and driven gear, respectively, k is a bending stiffness, c is a bending damping, km is a mesh stiffness, cm is a mesh damping, e is a static transmission errors (STE), Ti is the input torsion, To is the output torsion, kb is the support stiffness, cb is the support damping. Moreover, mpb1, mpb2, mgb1 and mgb2 are the equivalent masses of the gearbox supports.

As given in Fig. 1, the mathematical equations of the meshing pair could be derived by:

(1)
m p l ¨ p + c p l ˙ p + k p l p - F m = 0 , m g l ¨ g + c g l ˙ g + k g l g + F m = 0 , I i θ ¨ i + c 1 θ ˙ i - θ ˙ p + k 1 θ i - θ p = T i , I p θ ¨ ˙ p - c 1 θ ˙ i - θ ˙ p - k 1 θ i - θ p - r p F m = 0 , I g θ ¨ g + c 2 θ ˙ g - θ ˙ o + k 2 θ g - θ o + r g F m = 0 , I o θ ¨ o - c 2 θ ˙ g - θ ˙ o - k 2 θ g - θ o = - T o ,

where subscript i and o express motor and load, respectively, θ is a torsion degree, l is a bending degree, m is a mass, r is a base circle radius, I is a moment of inertia, k1 and k2 are torsional stiffness of the shaft, c1 and c2 are torsional damping of the shaft, and Fm could be deduced as:

(2)
F m = k m r g θ g - r p θ p + e + l g - l p + c m r g θ ˙ g - r p θ ˙ p + e ˙ + l ˙ g - l ˙ p .

The gearbox supports dynamic equivalent model is proposed, as shown in Fig. 2.

Fig. 2. The gearbox supports dynamic equivalent model

a) Equivalent model of the driving gear

b) Equivalent model of the driven gear

As illustrated in Fig. 2, the equivalent mass of the gears at the bearing fulcrum could be deduced as:

(3)
m p 1 = m p b a + b , m p 2 = m p a a + b , m g 1 = m g b a + b , m g 2 = m g a a + b ,

where a and b are the distance from the gear to the bearing fulcrum.

As given in Fig. 2, the mathematical equations of the support structure could be derived by:

(4)
m p 1 l ¨ p 1 + c p l ˙ p 1 - l ˙ p b 1 + k p l p 1 - l p b 1 - F p 1 = 0 , m p b 1 l ¨ p b 1 + c b l ˙ p b 1 + k b l p b 1 - c p l ˙ p 1 - l ˙ p b 1 - k p l p 1 - l p b 1 = 0 , m g 1 l ¨ g 1 + c g l ˙ g 1 - l ˙ g b 1 + k g l g 1 - l g b 1 - F g 1 = 0 , m g b 1 l ¨ g b 1 + c b l ˙ g b 1 + k b l g b 1 - c g l ˙ g 1 - l ˙ g b 1 - k g l g 1 - l g b 1 = 0 , m p 2 l ¨ p 2 + c p l ˙ p 2 - l ˙ p b 2 + k p l p 2 - l p b 2 - F p 2 = 0 , m p b 2 l ¨ p b 2 + c b l ˙ p b 2 + k b l p b 2 - c p l ˙ p 2 - l ˙ p b 2 - k p l p 2 - l p b 2 = 0 , m g 2 l ¨ g 2 + c g l ˙ g 2 - l ˙ g b 2 + k g l g 2 - l g b 2 - F g 2 = 0 , m g b 2 l ¨ g b 2 + c b l ˙ g b 2 + k b l g b 2 - c g l ˙ g 2 - l ˙ g b 2 - k g l g 2 - l g b 2 = 0 .

According to the deformation coordination relationship, as shown in Fig. 3, the deformation coordination equations could be derived by:

(5)
l p = b l p 1 + a l p 2 a + b , l g = b l g 1 + a l g 2 a + b .

Fig. 3. The coordination relationship

According to the deformation coordination Eq. (5), Eq. (1) and Eq. (4), a 12 DOFs coupling dynamic model, based on lumped mass method, is established.

3. Simulations

In order to verify the accuracy of the proposed method, the parameters of an example case are listed in Table 1.

Table 1. Parameters of system

Symbol name
Value
Unit
Modulus / m
4
mm
Pressure angle / α
20
°
Tooth number of driving gear / z1
23
Tooth number of driven gear / z2
69
Addendum coefficient / ha*
1
Clearance coefficient / c*
0.25

According to the 12 DOFs coupling dynamic model and the parameters listed in Table 1, the natural frequencies of the example case are simulated. Part of the results are shown in Fig. 4.

In the case of Fig. 4, the natural vibration mode vector of the first-order non-zero natural frequency (second frequency: 1403 Hz) is:

(6)
ϕ A = 0.3211 , - 0.0149,0.0050 , - 0.9469 , - 0.0004 , - 0.0002 ,           - 0.0004 , - 0.0002 , - 0.0004 , - 0.0002 , - 0.0004 , - 0.0002

Fig. 4. Natural frequency simulations

a) First frequency: 0 Hz

b) Second frequency: 1403 Hz

According to the simulation result based on finite element model, as shown in Fig. 5, the natural vibration mode vector could be expressed as:

(7)
ϕ B = - 0.2524 ,   0 ,   0 ,   0.7573 ,   0 ,   0 ,   0 ,   0 ,   0 ,   0 ,   0 ,   0 .

Fig. 5. Natural vibration mode based on FEM (natural frequency: 1282.9 Hz)

According to the modal assurance criterion (MAC), Eq. (6) and Eq. (7), the natural vibration mode vector correlation can be derived by:

(8)
M A C = ϕ B T ϕ A 2 ϕ B T ϕ B ϕ A T ϕ A .

According to Eq. (8), the MAC value of the example case is 0.9997, namely, the natural vibration mode shown in Fig. 4(b) and the natural vibration mode shown in Fig. 5 are the same-order physical mode. The relative error of the natural frequencies between two methods is calculated, as shown in Table 2.

Table 2. The relative error of the natural frequencies between two methods

Value
Unit
The natural frequency based on lumped mass method
1403
Hz
The natural frequency based on FEM
1282.9
The relative error
9.36
%

In the case of Table 2, the relative error of the natural frequencies between two methods is 9.36 %, namely, the proposed method is accurate and feasible.

4. Conclusions

In the issue, a 12 DOFs spur gear coupling dynamic model, based on lumped mass method, is proposed. The construction method of spur gear coupling dynamic model is better than finite element method, because it enables rapid modeling of complex gearbox and makes dynamic modeling more efficient. This contribution would be helpful to the spur gear coupling dynamic modeling and analyses.

Acknowledgements

The authors are grateful for the financial support provided by the National Natural Science Foundation of China under No. 51775264, and by National Key Laboratory of Science and Technology on Helicopter Transmission (Nanjing University of Aeronautics and Astronautics) (Grant No. HTL-A-19G06). In addition, the authors declare that there is no conflict of interests regarding the publication of this article.

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