Abstract
The elastohydrodynamic lubrication (EHL) contact model is established, the oil film stiffness is calculated based on DowsonHigginson empirical minimum thickness equation. The vibration model of singlerotor threeinput helicopter main reducer is proposed by lumped mass method, and nonlinear factors like oil film stiffness, dynamic meshing force are taken into account. The influence of oil film stiffness on system natural frequency is analyzed as well. The results show that oil film stiffness of the internal and external meshing pairs in the planetary gear system have greater impact on the natural frequency, which tends to destabilize the system. Therefore, the planetary gear train is the most crucial branch regarding the system splash lubrication. When all the oil film stiffness in the system are greater than 4×10^{9} N/m, the natural frequencies tend to be stable. This study can provide the theoretical reference for the lubrication characteristics in the singlerotor multiinput helicopter.
1. Introduction
To protect the tooth surface, the gears normally work in the lubrication condition. With load and entrainment velocity, the lubricating oil forms a certain film thickness on the contact surface of the gear pair. When working condition changes, the film follows with deformation, the oil film stiffness changes. Study of the influence law of the oil film stiffness on the natural characteristics has important engineering significance.
Within the scope of the multistage helicopter main reducer, especially regarding natural frequency analysis. Kahraman [1] applied eigenvalue solution and the modal summation technique to predict the free and forced vibrations of the multistage gear transmission system. Huang [2] established a finite element model of a parallel threeshaft gearrotor coupling system and analyzed the influences of the mesh stiffness, the installation angle, the helix angle and the bearing stiffness on the natural characteristics. Choy [3] analyzed the modal characteristics by using the matrixtransfer technique. Chen [4] studied the influence of torsional stiffness on natural characteristics of fourstage main transmission system in threeengine helicopter. Li [5] investigated the damping mechanism and EHL characteristics at the interface of the two DOF spur gear pair.
In summary, at present, the influence of oil film stiffness on the multistage system natural characteristics, is not to be investigated, according to the limited published issues. Therefore, it is of great theoretical significance and engineering value to study the mechanism of oil film stiffness on vibration and noise in gear transmission.
2. System dynamic model
2.1. Dynamic modeling
The dynamic model of singlerotor threeinput helicopter main reducer is shown in Fig. 1 [6]. The system includes three input branches, a tail output branch and a planet train branch. ${\theta}_{1}^{\left(j\right)}$, ${\theta}_{2}^{\left(j\right)}$, ${\theta}_{3}^{\left(j\right)}$, ${\theta}_{4}^{\left(j\right)}$, ${\theta}_{5}^{\left(j\right)}$, ${\theta}_{6}$, ${\theta}_{c}$, ${\theta}_{7}$, ${\theta}_{8}$, ${\theta}_{9}$, ${\theta}_{s}$ and ${\theta}_{pi}$ are rotational DOF of gear system in each stage. ${k}_{in}$, ${k}_{23}$, ${k}_{45}$, ${k}_{78}$,_{}${k}_{out}$ and ${k}_{6s}$ are torsional stiffness of shafts connecting each gear pair; ${c}_{12}$, ${c}_{34}$, ${c}_{56}$, ${c}_{spi}$ and ${c}_{rpi}$ are meshing damping of each gear pair; ${k}_{12}$, ${k}_{34}$, ${k}_{56}$, ${k}_{spi}$ and ${k}_{rpi}$ are meshing stiffness of each gear pair; ${k}_{m}$ represents oil film stiffness.
Fig. 1Dynamic model of singlerotor threeinput helicopter main reducer
2.2. Differential equation
For the EHL model, Dowson achieved a large amount of data through numerical simulation and experimental measurements and proposed the empirical equation for calculating the minimum oil film thickness. The DowsonHigginson minimum oil film thickness empirical equation is [7]:
where $U$, $G$, $W$ are dimensionless speed, material, load respectively:
where, $F$ is the load in meshing line, $E\text{'}$ is the equivalent elastic modulus, $R$ is synthetical curvature radius, ${u}_{r}$ is entrainment velocity, ${\alpha}_{l}$ is pressureviscosity coefficient, ${\eta}_{0}$ is dynamic viscosity. ${E}_{1}$, ${E}_{2}$ and ${v}_{1}$, ${v}_{2}$ are the contact elastic modulus and Poisson’s ratio.
The minimum oil film stiffness based on DowsonHigginson oil film thickness can be derived by taking the derivative of Eq. (1):
here, $B$ is contact length.
Timevarying meshing stiffness can be expanded in the Fourier series with the fundamental meshing frequency:
where ${k}_{12}^{\left(j\right)}\left(t\right)$, ${k}_{34}^{\left(j\right)}\left(t\right)$, ${k}_{56}^{\left(j\right)}\left(t\right)$, ${k}_{67}\left(t\right)$, ${k}_{89}\left(t\right)$, ${k}_{spi}\left(t\right)$ and ${k}_{rpi}\left(t\right)$ are timevarying mesh stiffness in each gear pair. ${k}_{m}$ is average meshing stiffness considering oil film stiffness, ${k}_{a}$ is maximum variable meshing stiffness. $\beta $ is initial meshing phase.
The differential equation of the singlerotor multiinput helicopter main reducer can be deduced through Newton’s law, as shown in Eq. (4):
where ${T}_{Ej}$ is the torque of engine $j$ ($j=$1, 2, 3); ${T}_{RR}$ and ${T}_{MR}$ are the torque of tail branch and planet carrier. ${F}^{p}$ and ${F}^{d}$ are dynamic meshing and damping forces. ${J}_{c}$ is carrier moment of inertia.
In addition, after the decomposition and recombination of Eq. (4), it can be expressed with mass matrix, damping matrix and stiffness matrix form. By setting the value of damping item and external excitation item to be zero, the vibration differential equation of the system under free condition can be obtained.
3. System natural characteristics analysis
Gear parameters are listed in Table 1, and EHL parameters are shown in Table 2.
Table 1Gear parameters
Gear  Tooth number  Module  Face width (mm) 
Gear 1  30  4.5  40 
Gear 2  85  4.5  40 
Gear 3  40  5  45 
Gear 4  90  5  45 
Gear 5  25  4.75  40 
Gear 6  142  4.75  40 
Gear 7  40  5  40 
Gear 8  60  5  35 
Gear 9  70  5  35 
Gear 10  68  5  28 
Gear 1116  37  5  28 
Table 2EHL parameters
Parameter  Value 
Dynamic viscosity ${\eta}_{0}$_{}(Pa·s)  0.04 
Pressureviscosity coefficient ${\alpha}_{l}$ (Pa^{1})  2.2×10^{8} 
Elastic modulus ${E}_{1}$ (Pa)  2×10^{11} 
Elastic modulus ${E}_{2}$ (Pa)  2×10^{11} 
Poisson’s ratio ${v}_{1}$  0.3 
Poisson’s ratio ${v}_{2}$  0.3 
In order to explore the influence law of oil film stiffness on natural characteristics of the system, the natural frequencies of the first five orders are calculated and depicted in Fig. 2. Fig. 2(a) and Fig. 2(b) are the influence of oil film stiffness on natural frequency in first stage and the second stage, as shown in the figure, when the oil film stiffness is greater than 2×10^{9}^{ }N/m, the natural frequencies no longer change. When the oil film stiffness is less than 2×10^{9}^{ }N/m, the decrease of natural frequency of the fourth and fifth order is the most obvious. Fig. 2(c) shows the effect of oil film stiffness on the natural frequency in the third stage. The figure shows that when ${k}_{56}$ is less than 2×10^{9}^{ }N/m, the firstorder natural frequency decreases rapidly. Fig. 2(d) and Fig. 2(e) show the impact of oil film stiffness on natural frequency in the tail branch. When ${k}_{67}$ is less than 4×10^{9}^{ }N/m, the firstorder natural frequency decreases sharply; when ${k}_{89}$ is less than 0.5×10^{9}^{ }N/m, the natural frequency of the fifth order decreases while the natural frequencies of other orders do not change. Fig. 2(f) and Fig. 2(g) are the influence law of planetary gear train on the system natural frequency, as seen from the figure, when the internal and external meshing oil film stiffness are equal to 0.7×10^{9}^{ }N/m and 0.9×10^{9}^{ }N/m, the natural frequency of each order has a sudden droppoint, which could easily destabilize the system. Therefore, the planetary gear train is the most important branch in splash lubrication analysis, the oil film stiffness cannot be too small in mechanical design.
4. Conclusions
In this paper, a dynamic model of singlerotor threeinput helicopter main reducer is proposed and influence laws of the oil film stiffness are calculated, the analysis results enable us to draw the following conclusions:
1) The influence laws of the oil film stiffness of the two gear pairs in the input branch are almost the same, the influence laws of the internal and external meshing pairs in the planetary gear system are also very similar, therefore these two branches should consider all components within the branch.
Fig. 2Impact of oil film stiffness on first five orders natural frequency
a)${k}_{12}$
b)${k}_{34}$
c)${k}_{56}$
d)${k}_{67}$
e)${k}_{89}$
f)${k}_{msp}$
g)${k}_{mrp}$
2) When the oil film stiffness is small, the natural frequencies of each order can be variable. When the oil films stiffness of all meshing pairs in the system is more than 4×10^{9} N/m, the natural frequencies tend to be stable.
3) Planetary gear train is the most critical branch of the singlerotor threeinput system regarding splash lubrication, so their oil film stiffness cannot be too small during working condition.
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About this article
This work is supported by the National Natural Science Foundation of PRC (Grant No. 51775265 and 51475226); China Scholarship Council; China Association for Science and Technology.