Dynamic analysis of planetary gear train system with double moduli and pressure angles

The planetary gear transmission with double moduli and pressure angles gearing is proposed for meeting the low weight and high reliability requires. A dynamic differential equation of the NGW planetary gear train system with double moduli and pressure angles is established. The 4-Order Runge-Kutta numerical integration method is used to solve the equations from which the result of the dynamic response is got. The dynamic load coefficients are formulated and are compared with those of the normal gear train.The double modulus planetary gear transmission is designed and manufactured. The experiment of operating and vibration are carried out and provides.


Introduction
Planetary gear train is widely used in the helicopters, automobiles, aircraft engines and ships.It has numerous advantages in size, weight, gear ratio, efficiency and load capacity.Many scholars have made a lot of researches on it.M. Inalpolat and A. Kahraman [1] proposed a simplified mathematical model of the planetary gear sets.A special experimental planetary gear set-up is developed and planetary gear sets from three of these five groups are procured.A nonlinear time-varying dynamic model is proposed to predict modulation sidebands of planetary gear sets [2].Yichao Guo and Robert G. Parker [3] systematically study the mesh phase relations of general compound planetary gears.Experimental modal analysis techniques are applied to characterize the planar dynamic behavior of two spur planetary gears [4].The highly structured modal properties of planetary gears having diametrically opposed planets and an elastic ring gear are illustrated and mathematically proved [5].The dynamic responses of a planetary gear are analyzed when component gears have time-varying pressure angles and contact ratios caused by bearing deformations [6].In [7] a dynamic differential equation of the NGW planetary gear train system with unequal moduli and pressure angles is established.In [8] the nonlinear dynamic model of one planetary gear transmission is set and the experiment is carried out.A nonlinear lateral-torsional coupled vibration model of a planetary gear system was established by taking transmission errors, time varying meshing stiffness and multiple gear backlashes into account [9].In this paper, a dynamic differential equation of the NGW planetary gear train system with double moduli and pressure angles gearing is established.The 4-Order Runge-Kutta numerical integration method is used to solve the equations from https://doi.org/10.1051/matecconf/201821117003VETOMAC XIV which the result of the dynamic response is got.The dynamic load coefficients are formulated and are compared with those of the normal gear train.The planetary gear transmission with double moduli and pressure angles gearing is designed and manufactured.In addition, the experiment is carried out.

System parameters
Fig. 1 illustrates a single stage planetary gear train consisting of a sun gear, a ring gear, several planets and a carrier.Sun gear is selected as the input component and the carrier as the output component.There are two double moduli and pressure angles gearings in the planetary gear train.One is external gear pair zs-zpi, the parameters of which meet cos cos Where, ms and mpi are moduli of sun and planet gears respectively; αs and αpi are pressure angles of sun and planet gears respectively.The other is internal gear pair zpi-zr, the parameters of which meet cos cos Where, mpi and mr are moduli of planet and ring gears respectively; αpi and αr are pressure angles of planet and ring gears respectively.

Dynamics model of planetary gear system
The dynamic model of the system is shown in Fig. 2. Planet carrier is the reference coordinate system of the model.The three degrees of freedom of sun gear are fixed on the planet carrier and the origin is coincident with the planet carrier center.The planet gear center is chosen as the origin of coordinate system and is fixed on the planet carrier, too.
The system has (6+3N) degrees of freedom and the generalized coordinate X can be expressed as follows: where  The equivalent accumulated meshing errors espi of sun gear, planetary gear and internal gear eccentric error and displacement equivalent errors erpi on the meshing line of each gear are as follows: where φspi and φrpi are initial phase of the teeth frequency errors Espi and Erpi respectively; Es, Epi and Er are eccentric error of sun gear, planet gear and internal gear, the φs, φpi and φr are initial phase of those.ω is meshing frequency of epicyclic train; ωsc, ωpc and ωrc are angular velocity relative to planet carrier of sun gear, planetary gear and internal gear, respectively.The relative displacement on the planetary gear transmission system line of action caused by rotation displacement are: The forces Fspi between sun gear and planetary gear and the force Frpi between planetary gear and internal gear are respectively The damping coefficient Csp of sun gear with planetary gear is and the damping coefficient Crp of planetary gear with internal gear is According dynamic model of planetary gear transmission system in Fig. 2, system dynamics equations are in the form: where, α is the pressure angle; rb is the radius of gear base circle; ms, mpi and mc are equivalent mass of the sun gear, planet gears and planet carrier, respectively, with mk=Jk/rk 2 (k=s, pi, c); Md, Ms, Mpi, Mc and Ml are mass of the input component, sun gear, planet gears, planet carrier and load; Ks and Kp are support stiffness of sun gear and planet gears; rc is radius of base circle of planet carrier; Pd and Pl is the input and output force, respectively.

Solution of dynamically differential equation
The differential equations were solved by 4-Order Runge-Kutta method.The relevant parameters are as follow: P=10kW, n=1200r/min, xs =0.5417, xpi =-0.416, xr =0.2, zs =30, zpi =31, zr =91, αr =24º, ms =mpi =3.08mm, mr =3.12mm, The dynamic load coefficients of planetary gears are The time-domain graph of dynamic loads of the double moduli and the normal planetary gears (α=24º, m=3.12mm) were obtained as shown in Figs.3-4.The dynamic load coefficients of the double moduli and the normal planetary gears (α=24º, m=3.12mm) system were obtained as shown in Table 1.  1, it is clear that the dynamic load and its coefficients of the double moduli planetary gears are smaller than those of the normal one.That means the dynamic behavior of the double moduli planetary gears is better than the normal one.

Experimental verification
The gear cutter for experimental double moduli planetary gears is shown in Fig. 5. Gear cutter hob is used to cut the sun gear and the planet gear and the gear shaper cutter for the ring gear.     2 shows the load-sharing coefficients of the test and the theoretical analysis and the errors between them.

Fig. 6 .Fig. 7 .
Fig. 6.The test rig schematic.Test gears with strain gauges are shown in Fig.7.In Fig.7(a) is the sun gear with strain gauges and in Fig.7(b) the ring gear with strain gauges.

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Magnetic powder brake MATEC Web of Conferences 211, 17003 (2018) https://doi.org/10.1051/matecconf/201821117003VETOMAC XIV Table where ζ1 and ζ2 are damping ratios of sun gear with planetary gear and planetary gear with internal gear; ksp and krp are average action stiffnesses of sun gear with planetary gear and planetary gear with internal gear.The damping force Dspi of sun gear with planetary gear and Drpi of planetary gear with internal gear respectively are

Table 1 .
The dynamic load coefficients of the double moduli and the normal one

Table 2 .
The load-sharing coefficients of the test and the theoretical analysis.