Abstract
The face gear pair has been applied in the main reducer of helicopter. Considering the nonlinear backlash, timevarying meshing stiffness and static transmission error, a six degreeoffreedom lumped parameter dynamic model coupling with flexional, torsional and axial motion of the face gear pair for helicopter is developed. This model takes the floating of the pinion as the boundary condition. The nonlinear dynamic characteristics of the face gear pair are analyzed by the help of bifurcation diagram. The effects of rotational speed and supporting stiffness of the pinion on the dynamic responses of the face gear pair are studied in detail by using numerical method. Numerical examples reveal several dynamic evolution mechanisms and motion states including periodic1, periodic2, chaotic and quasiperiodic motions. Some research results are helpful for vibration control and dynamic design of the face gear pair for helicopter.
1. Introduction
The face gear drive plays more and more important role in the power transmission system of helicopter for its ability of torquesplit and insensitivity to misalignment error [13]. The transmission system of helicopter has the characteristics of high speed, heavy load and complex operating ambient, which make gears prone to damage due to fierce vibration. Therefore, it is imperative to develop accurate and reliable dynamic models of the face gear pair to investigate the failure mechanisms. Actually, research on the dynamic behaviors of gear transmission system has been one of the most important aspect of power transmission design. The study of dynamic response is the basis of fatigue life prediction and health monitoring of gear transmission system.
The face gear drive has been applied on the helicopter transmission since the 1990s. Litvin et al. [4, 5] described the torquesplit face gear transmission design in the first place. And a series of experimental and theoretical studies [610] shown that the application of face gear in the helicopter transmission was feasible. In Ref. [11], Chen et al. investigated the nonlinear dynamic responses of face gear drive system by a 6DOF dynamic model considering backlash, timevarying meshing stiffness, timevarying arm of meshing force and effect of modification. Hu et al. [12] presented a 14DOF dynamic model of face geared rotor system. Effects of backlash and load on the dynamic responses were investigated through numerical simulations.
Limited works have been addressed to the dynamic behaviors of the face gear pair considering its application on the transmission system of helicopter. The operating environment of helicopter is complicated and the transmission system often bears alternating loads due to the factors such as backlash, timevarying meshing stiffness and external excitation which will cause gear faults. Accordingly, it is imperative to reveal the intrinsic relations between these factors and the dynamic characteristics of the face gear pair for helicopter.
2. Nonlinear dynamic model of the face gear pair
The face gear pair investigated in the present study is mainly composed of supporting shafts, spur gear, face gear, bearings and other auxiliary components. A translationaltorsionalaxial coupled 6DOF dynamic model of this type of mechanical transmission system is established. The gear bodies are regarded as rigid disks. As is shown in Fig. 1, the tooth meshes are modeled as a nonlinear springdamper element, and the bearings are also simplified as linear springdamper elements, $b$ denotes half of the total backlash, ${k}_{m}\left(t\right)$ and ${c}_{m}$ are the time varing meshing stiffness and viscus mesh damping coefficient respectively, $e\left(t\right)$ represents the timevarying static transmission error.
Fig. 1The dynamic model of the face gear pair
For analytical convenience, by translating the angular displacement ${\theta}_{i}$$(i=p,g)$ to the LOA direction, the generalized displacement vector can be expressed as:
where ${w}_{i}={R}_{i}{\theta}_{i}$, ${R}_{p}$ is the base circle radius of the pinion, ${R}_{g}$ is the pitch radius of the face gear.
Usually, the nonlinear gear backlash adopts the following expression:
The timevarying meshing force can be derived as:
where $f\left(\cdot \right)$ is the backlash function, the superscript “$\dot{}$” is the differentiation of time $t$.
The equations of motion for the translationaltorsionalaxial coupled 6DOF dynamic model shown in Fig. 1 can be derived by Newton’s second law of motion as:
where ${m}_{i}$ is the mass of gear $i$, ${I}_{i}$ is the mass moment of inertia of gear $i$, ${k}_{i}^{j}$ and ${c}_{i}^{j}$ are the average supporting stiffness and bearing damping of gear $i$ respectively along $j$ direction ($j=y$, $z$).
The nondemensional time $\epsilon $ is defined by $\epsilon =t/{\omega}_{n}$, where ${\omega}_{n}$ is the meshing period of gear teeth. By defining a displacement nominal scale of ${b}_{l}=$ 1 μm, the nondemensional displacements, velocities and accelerations can be expressed as:
${\dot{Y}}_{p}=\frac{{\dot{y}}_{p}{\omega}_{n}}{{b}_{l}},\mathrm{}\mathrm{}\mathrm{}\mathrm{}{\dot{Z}}_{p}=\frac{{\dot{z}}_{p}{\omega}_{n}}{{b}_{l}},\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}{\dot{W}}_{p}=\frac{{\dot{w}}_{p}{\omega}_{n}}{{b}_{l}},\mathrm{}\mathrm{}\mathrm{}\mathrm{}{\dot{Y}}_{g}=\frac{{\dot{y}}_{g}{\omega}_{n}}{{b}_{l}},\mathrm{}\mathrm{}\mathrm{}\mathrm{}{\dot{Z}}_{g}=\frac{{\dot{z}}_{g}{\omega}_{n}}{{b}_{l}},\mathrm{}\mathrm{}\mathrm{}\mathrm{}{\dot{W}}_{g}=\frac{{\dot{w}}_{g}{\omega}_{n}}{{b}_{l}},$
${\ddot{Y}}_{p}=\frac{{\ddot{y}}_{p}{\omega}_{n}^{2}}{{b}_{l}},\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}{\ddot{Z}}_{p}=\frac{{\ddot{z}}_{p}{\omega}_{n}^{2}}{{b}_{l}},\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}{\ddot{W}}_{p}=\frac{{\ddot{w}}_{p}{\omega}_{n}^{2}}{{b}_{l}},\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}{\ddot{Y}}_{g}=\frac{{\ddot{y}}_{g}{\omega}_{n}^{2}}{{b}_{l}},\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}{\ddot{Z}}_{g}=\frac{{\ddot{z}}_{g}{\omega}_{n}^{2}}{{b}_{l}},\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}{\ddot{W}}_{g}=\frac{{\ddot{w}}_{g}{\omega}_{n}^{2}}{{b}_{l}}.$
Imposing the expressions above, the nondemensional form of the governing equations of motion of the translationaltorsionalaxial coupled 6DOF dynamic model can be expressed as:
3. Calculation of the timevarying meshing stiffness when the pinion floats
The concentric torquesplit face gear transmission is shown in Fig. 2, it was first proposed by the Enhanced Rotorcraft Drive System Program (ERDS), which was cooperated by the US Army and Boeing company. The pinion drives in between the two face gears and the input torque is distributed to the upper and lower face gears. The pinion is designed to float, so that the torque can be divided evenly to the two face gears [1, 2]. The stiffness of the upper and lower power transmission paths is asymmetric, and the load distributed to the upper face gear is higher [3], so the pinion floats downward. This results in nonstandard meshing between the pinion and the lower face gear, and the meshing stiffness will change. The meshing stiffness is an important excitation of the gear transmission system, and it is necessary to study the nonlinear vibration characteristics of the lower mesh of pinion under the even load sharing condition.
Under the even load sharing condition, the pinion floats downward, and the timevarying meshing stiffness cannot be calculated according to the method proposed in reference [13]. Moreover, the tooth surface of the face gear is complex, and the tooth thickness varies along the tooth width direction. Thus, the analytical solution of the timevarying meshing stiffness of the face gear transmission is hard to obtain. In this section, the finite element method is used to calculate the timevarying meshing stiffness of the face gear transmission when the pinion floats.
The basic design parameters of the face gear pair for stiffness calculations is shown in Table 1. After the meshing stiffness of all the meshing positions are calculated, the meshing stiffness changes along with time points is shown in Fig. 3.
Fig. 2The concentric torque split face gear transmission
Fig. 3Timevarying meshing stiffness when the pinion floats
Table 1Basic design parameters of the face gear pair for stiffness calculations
Parameter  Pinion  Face gear 
Tooth number  21  142 
Modulus / (mm)  1.95  1.95 
Pressure angle / (°)  25  25 
Tooth with / (mm)  27  25 
Inner diameter / (mm)  –  255 
Outer diameter / (mm)  –  305 
Backlash / (μm)  25  25 
4. Nonlinear dynamic responses
4.1. Effect of rotational speed of pinion
The rotational speed of pinion is one of the main parameters that affect the dynamic responses of the face gear pair. The bifurcation characteristic of the face gear pair in dimensionless displacement with respect to the rotational speed of pinion is illustrated in Fig. 4. From this figure, four kinds of motions are presented as the rotational speed of pinion changes: periodic1 motion, chaotic motion, periodic2 motion and quasiperiodic motion.
Fig. 4Bifurcation diagram of the face gear pair using speed as bifurcation parameter
When the rotational speed is less than 4000 rpm, the system dynamic response is periodic1 motion. The phase plane diagram is just a closed circle, the Poincaré map only has one single point, the time domain spectrum shows a sine curve, and the FFT spectrum also just has onepeak amplitude. The results indicate that the system is in periodic1 motion.
When the rotational speed increases to 4000 rpm but less than 4800 rpm, the system dynamic response changes from periodic1 motion to chaotic motion, which can be seen clearly in Fig. 4. In this region, the bifurcation diagram has numerous points. The Phase plane diagram shows that the curve is full of the space, there are many discrete points in the Poincaré map, the time domain spectrum shows a nonperiodic motion, and the FFT spectrum also has multipeak amplitude. The above results show that the system is in chaotic motion.
When the rotational speed changes from 4800 rpm to 7600 rpm, the system dynamic responses back to periodic1 motion again, which can be seen clearly in Fig. 4. The rated rotational speed of the pinion in the transmission system of the helicopter is 6409.86 rpm. From the dynamic responses analysis above we know that the system is in periodic1 motion. When the rotational speed increases from 7600 rpm to 8800 rpm, the system dynamic response is a periodic2 motion. There are two points in the Poincaré map, two closed circles in the phase plane diagram, and numerous excitation frequencies in the FFT spectrum. Also, the time domain spectrum appears periodic.
When the rotational speed is higher than 8800 rpm, the system dynamic response leaves periodic2 and turns to quasiperiodic motion for a long range, which can be revealed exactly from Fig. 4. The phase plane diagram shows a curve belt with certain width, the Poincaré map has a small region that formed by numerous points, and the time domain spectrum shows a periodic motion. All these results means that the system is in quasiperiodic motion.
4.2. Effect of supporting stiffness
The supporting stiffness of pinion is also one of the main parameters that affects the dynamic responses of the face gear pair. The bifurcation characteristic of the face gear pair in dimensionless displacement with respect to the supporting stiffness of pinion is illustrated in Fig. 5 when the rotational speed is 3000 rpm. There are three kinds of motions, namely periodic1 motion, periodic2 motion and chaotic motion, as can be seen from the bifurcation diagram.
Fig. 5Bifurcation diagram of the face gear pair using pinion’s supporting stiffness as bifurcation parameter
The Bifurcation diagram depicted in Fig. 5 shows that when the supporting stiffness of pinion is less than 9.3×10^{8} N/m, the system dynamic response is a periodic1 motion. With the increasing of the supporting stiffness of the pinion, the system turns to periodic2 motion which remains from 9.3×10^{8} N/m to 9.6×10^{8} N/m. From 9.6×10^{8} N/m to 1.27×10^{9} N/m, the system leaves periodic2 and returns to periodic1 again. When the supporting stiffness of pinion is increased to 1.27×10^{9} N/m, the system dynamic response is changed from periodic1 motion to chaotic motion, and it is continued until 1.5×10^{9} N/m. When the supporting stiffness of pinion is greater than 1.5×10^{9} N/m, the system leaves chaotic and return to periodic1 again for a long range of the pinion stiffness until 1.8×10^{9} N/m.
From the dynamic responses of the face gear pair mentioned above, it can be concluded that the support stiffness of the pinion has an important influence on the motion of the system. When the support stiffness of pinon is less than 8.3×10^{8} N/m, the vibration displacement raises to a relatively large value, and the large amplitude part of the system concentrated in high frequency. While as the support stiffness of the pinion increases, the vibration displacement inclines to decrease, the large amplitude part moves to low frequency.
5. Conclusions
The face gear pair is mainly used in the transmission system of helicopter which requires low vibration and high reliability. The reliable dynamic model of the face gear drive is the basis of reducing vibration and improving reliability for the face gear pair of the helicopter. Considering the backlash and the timevarying meshing stiffness when the pinion floats, this paper presents a dynamic model of the face gear pair. The dynamic responses are obtained through RungeKutta method. The effects of rotational speed and supporting stiffness of pinion on the nonlinear dynamic responses of the face gear pair are analyzed with numerical method.
As mentioned in the numerical examples above, with the changing of the rotational speed and supporting stiffness of pinion, the dynamic responses of system contained four motions: periodic1 motion, periodic2 motion, chaotic motion and quasiperiodic motion. It shows that the changes of rotational speed or supporting stiffness of pinion lead to complex dynamic responses of the system. Especially when the system is in the state of chaotic motion, the interaction between gears becomes complex and the gear vibration becomes fierce, which lead to failure and instability of the face gear pair. To reduce the vibration and increase stability, it is necessary to match the rotational speed of pinion with the reduction ratio, and to select an appropriate supporting stiffness of the pinion based on the optimal design method for the face gear pair.
On the whole, the works in this paper not only present us a method to study the nonlinear dynamic characteristic of the face gear pair, but also provide ideas for vibration control of the main reducer of the helicopter in the future.
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