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In this paper, an analytical solution on the dynamic mesh forces of planetary gear trains (PGTs) is proposed by investigating a lumped-parameter model. By using the method of multiple-scales (MMS), closed-form expressions of mesh force under the effects of manufacturing and assembly errors are obtained. From these expressions, the effects of several key factors such as the tooth thickness error, pin position error, applied torque, support stiffness of sun gear, and tooth profile modifications (TPM) on dynamic load sharing behaviours are explored. Numerical integration is carried out to verify the validation of the proposed method, and the developed expressions are also validated by comparing the results with previously published predictions. The results for several examined PGT systems show that the key factors abovementioned affect the dynamic load sharing behaviours as both static and dynamic factors. An important new conclusion obtained by this work is that proper tooth profile modifications keep the dynamic load sharing factors almost equal to the results obtained under static conditions. This conclusion provides the possibility to simplify the dynamic analysis to the static analysis on the dynamic load sharing problems.

In the applications of high-power mechanical transmission areas, to guarantee the load on each component within the safe ranges, split-torque transmission systems are widely used. Planetary gear train (PGT) is one of the most popular split-torque transmission systems, because of its advantages such as high-power density, high transmission ratio, low bearing load, and compactness. Theoretically, the input torque should be shared evenly by all the planets. In practical applications, however, each path of a PGT will carry uneven load due to the presence of errors, such as tooth thickness error and pin position error. The load sharing behaviours are affected by a variety of factors such as the gravity, support stiffness of central components, bearing clearance, backlashes, applied torque, and component flexibility [

In the past 20 years, nonlinearities in geared systems caused by factors including backlash [

In order to further predict the load sharing behaviours under practical dynamic working conditions, some researchers investigated the dynamic load sharing by using lumped-parameter models. Kahraman [

In this effort, an analytical solution on the dynamic load sharing behaviours of PGTs is proposed by investigating a lumped-parameter model. Based on the previous works [

A lumped-parameter model of PGTs with a sun gear, a fixed carrier, a ring gear, and _{bs} is the translational stiffness of the sun gear. _{sp} and _{rp} are the mesh stiffness of the _{su}, _{ru}, and _{cu} are the torsional stiffness of sun, ring, and carrier, respectively.

Lumped-parameter model of PGTs.

Figures _{r} is the pressure angle of r-p mesh. _{rp} and _{rp} are respectively the tooth thickness errors and TPM functions on r-p mesh.

Diagram of r-p mesh.

Diagram of s-p mesh.

The tooth separation is modelled by

Backlash and tooth separations.

The sun gear links with _{s} and _{s} are the translational motions of sun gear in the _{s} is the pressure angle of s-p mesh, _{p} and _{s} is the transmitted torque applying on sun gear, and _{sp} and _{sp} are respectively the tooth thickness errors and TPM functions on s-p mesh. From equations (

As shown in Figure _{c} is the length of oo_{1}, as shown in Figure

Diagram of carrier with planet gear.

As shown in Figure

Tooth thickness error and pin position error.

Figure

Contributions of different errors to the equivalent tooth thickness errors.

In equation (_{pr}, _{ps}, and _{pp} represent the tooth thickness errors of ring, sun gear, and the _{cc}, _{cr}, _{cs}, and _{cp} represent the pin position errors of the carrier, ring, sun gear, and the _{s} and _{r} are the tooth number of sun gear and ring gear, respectively.

Then, the additional mesh force introduced by equivalent tooth thickness errors on the

There are several methods to modify gear tooth surfaces, including crowning, tip relief, and root relief with linear or parabolic variations with roll angle. The TPM curves, the magnitude of the relief, and the modification length are the three key factors that determine the effects of the TPM on vibration reduction. Without loss of generality, linear relief is applied to double tooth pair contact areas about the tooth tip and root in this study. As shown in Figure

Tooth profile modification.

From equations (

_{b} is the support stiffness matrix between the PGT and the fixture. _{mv} and _{m0} are the varying part and mean part of stiffness matrix, respectively.

In this study, the rectangle waves [_{sp} is the nondimensional mesh stiffness matrix and can be written as

Time-varying mesh stiffness.

_{rp} has a similar form as _{sp}.

_{t} is the external load vector. _{d} and _{m} are respectively the inner force vectors introduced by the equivalent tooth thickness errors and TPM._{sp} and _{rp} are the component vectors of

The eigenvalue problem associated with the linear free vibration of the PGTs is^{st} mode is rigid mode, the 2^{nd} and (^{th} are rotational modes with distinct natural frequency, and the 3^{rd} to (^{th} modes have equal natural frequency. It has been proved that, the vibration of PGTs system is mainly the superposition of these two rotational modes [

In Figure ^{st} gear mesh ^{st} s-p mesh force

Time history of s-p mesh force for a 5-planet system. (a) Without errors; (b) _{r1} = _{s1} = 10

System parameters of the example PGTs.

Parameters | Values |
---|---|

No. of planets, | 3 ∼ 6 |

Support stiffness of sun gear (N/m) | 1e10 (fixed), 1e6 (float) |

Mean s-p mesh stiffness (N/m) | 0.62e9 |

Mean r-p mesh stiffness (N/m) | 0.85e9 |

First harmonic of s-p mesh stiffness (N/m) | 0.14e9 |

First harmonic of r-p mesh stiffness (N/m) | 0.11e9 |

s-p, r-p mesh phasing angle | 0 |

Pressure angle (deg) | 22.5 |

Input torque to sun, _{s} (Nm) | 1130 |

Sun inertia | 6.21 |

Planet inertia | 4.89 |

Damping ratio | 0.02 |

Tooth number of sun gear in 5-planet system | 34 |

Tooth number of planet gear in 5-planet system | 31 |

Tooth number of ring in 5-planet system | 96 |

Module (mm) | 4 |

Face width (mm) | 40 |

The ideal load condition for a PGT system is that each path will carry an equal load and the dynamic mesh force values are low. To illustrate the dynamic load sharing, the load sharing coefficients of the s-p mesh

The s-p load sharing factor

In order to indicate the maximum mesh force, the dynamic load factor

In the case of the PGTs manufacture perfectly without error, as shown in Figure

To investigate the dynamic load sharing and load factor of PGTs, an approximate solution for the mesh force is sought by using the MMS [

The approximated expression of

Substituting equation (

Let

Substituting equation (

From equation (_{sp}. _{sp} includes the effects of both tooth thickness errors and the pin position errors.

Substituting equation (

The corresponding expressions for r-p meshes are similar to the expressions for s-p meshes.

In this investigation, the effects of the errors, the support stiffness of sun gear, the applied torque, and tooth profile modification on the load sharing factors and the dynamic load factors will be discussed following. The main parameters of the example systems with different number of planet gear are listed in Table

Natural frequencies associated with the first two rotational modes.

No. of planets, | |
---|---|

3 | 2114, 4674 |

4 | 2233, 5249 |

5 | 2329, 5775 |

6 | 2412, 6260 |

To verify the proposed method, assume the 1^{st} planet gear has the time-invariant tooth profile errors, _{r1} = _{s1} = 50 ^{st} planet may come from either the tooth thickness error or the pin position errors of the 1^{st} planet gear. The comparisons of the load coefficients obtained by MMS and NI with varying number of planet gear are shown in Figures

Load sharing coefficients with fixed sun gear. ^{nd} and the (^{th} planets; (□□□), the 3^{rd} and the (^{th} planets; ^{th} planet; (_{.}

Load sharing coefficients with floating sun gear. ^{nd} and the (^{th} planets; (□□□), the 3^{rd} and the ((N) − 1)^{th} planets; ^{th} planet; (

With sun gear fixed, as shown in Figure ^{st} planet gear takes the heaviest load due to the errors. Because of the symmetry positions to the 1^{st} planet gear, the 2^{nd} and ^{th} planet gears carry equal loads; the 3^{rd} and (^{th} planet gears carry equal loads. With sun gear floated, as shown in Figure

One can find the dynamic load sharing coefficients vary with the changing of mesh frequency. This is because the amplitudes of mesh forces are the functions of the modal vibration amplitudes ^{st} planet gear is decreased. However, with this improvement of the load sharing, the dynamic load conditions get worse, as show in Figure

Dynamic load factors for PGT systems with _{r1} = _{s1} = 50

It should be noted that, the softening nonlinearity and vibration jump phenomenon appear in the primary resonance ranges. The frequency-load sharing coefficient curves have three branches near the primary resonance, and the middle branch is unstable.

It is believed that both of the planet pin position errors and tooth thickness errors are parts of the vibration excitations of PGTs [

Figure _{r1} and _{s1} for the 5-planet system versus mesh frequency. With the increasing of _{r1} and _{s1}, the load sharing factors increase and the frequency ranges with contact loss expanded. It is worth noting that the increase of the load sharing factors is almost proportional to the increase of _{r1} and _{s1}, as shown in Figure

Load sharing coefficients versus mesh frequency for 5-planet system with varying _{r1}, _{s1}. (a)

Load sharing coefficients for 5-planet system with varying _{r1}, _{s1}. (a) Quasistatic condition with low rotational speed; (b) sun gear rotational speed 4200 r/min.

It is believed that the error induces vibration and then leads the increase of the dynamic load factor, as shown in Figure

Dynamic load factors versus mesh frequency for 5-planet system with varying _{r1}, _{s1}. (a)

Pin position error of central components is a critical factor affecting the dynamic load sharing. In Figure ^{st} planet gear is maximum. This is because the 1^{st} planet is the nearest one to the sun gear due to the pin position error. This pin position error has unequal effects on each planet gear, because of the different angles between the mesh lines and the direction of the pin position errors. The same pin position error of ring gear has opposite effects on the load sharing, as shown in Figures ^{st} planet is minimum. This is because the 1^{st} planet is furthest to the ring gear with this pin position error. These results also agree well with the conclusions obtained from static analysis in reference [

Effects of the pin position errors on the dynamic load coefficients. (a) _{rp} with _{sp} with _{rp} with _{sp} with

The effects of the pin position errors on dynamic load sharing factors versus mesh frequency are shown in Figure

Effects of the pin position errors on the load sharing factors. (a)

Effects of the pin position errors on the dynamic load factors. (a)

Applied torque is another key factor that affects the dynamic characteristic of PGT system. Figure _{r1} = _{s1} = 50

Load sharing factors for 5-planet system with varying applied torque. (a)

The load sharing coefficients of 5-planet system with varying applied torque in both quasistatic and dynamic conditions are shown in Figure

Load sharing coefficients for 5-planet system with varying applied torque. (a) Quasistatic condition with low rotational speed; (b) sun gear rotational speed 4200r/min.

In order to explore the mechanism of the float sun gear on the improvement of the dynamic load sharing, the comparisons of the translational displacements of sun gear of a five-planet system with/without float sun gear are shown in Figure

Comparisons of the translational displacement of sun gear. (a)

In practical PGTs, the sun gear is not absolutely float or fixed. To further explore the effects of the sun gear support stiffness, the influences of the support stiffness on the load sharing factors and the dynamic load factor of sun gear for the 5-planet system are shown in Figures

Load sharing factors for 5-planet system with varying support stiffness of sun gear. (a)

Dynamic load factors for 5-planet system with varying support stiffness of sun gear. (a)

In the case of low rotational speed, the load sharing condition is close to those in quasistatic condition. As shown in Figure ^{st} planet gear increases. These curves in Figure ^{st} planet gear load sharing coefficient is decreased and the load sharing is improved. As discussed above, floating sun gear decreases the dynamic mesh load and optimizes the load sharing, which is still one of the effective methods with high priority in dynamic conditions despite of larger sun vibrations.

Effects of the support stiffness of the sun gear. (a) Quasistatic condition with low rotational speed; (b) sun gear rotational speed 4200r/min.

TPM has been proved as an effective method to decrease the vibrations of PGT system [

Comparisons of the amplitudes of mesh forces with and without TPMs. (—), the 1^{st} planet; (---), the 2^{nd} and the 5^{th} planets; (…), the 3rd and the 4^{th} planets, (a-b) without TPMs; (c-d) with TPMs.

In Figures

Comparisons of the load sharing factors and dynamic load factors of PGTs with and without TPMs. (—), without TPM; (---), with proper TPM. (a-b) Load sharing factors; (c-d) dynamic load factors.

Since the TPM is effective to decrease the system vibration amplitudes, TPM must be an effective method to decrease the dynamic load factors. As shown in Figures

The static and quasistatic results of the load sharing factors agree well with dynamic analysis, and the static and quasistatic analysis has advantages in analysis efficiency. However, dynamic analysis can offer a better understanding of the dynamic load factors. So, selection of static and dynamic analysis depends on the main focus on the PGTs.

In this study, a simplified discrete model is presented to investigate the load sharing among the planet meshes of PGTs with several type errors. Both of the cases of fixed and float sun gear are investigated to study the effects of the support stiffness of sun gear on the load sharing. Time-varying mesh stiffness and tooth separations are also considered. The method of multiple-scales (MMS) is used to obtain the response and closed-form expressions of mesh force are derived over the important mesh frequency ranges. From these expressions, the effects of several key factors such as the tooth thickness and pin position errors, applied torque, support stiffness of sun gear, and tooth profile modifications on dynamic load sharing behaviours are explored. The validation of MMS is obtained by the results of numerical integration and previously published predictions. Several conclusions are obtained:

The amplitudes of dynamic mesh force are the function of vibration amplitude which is associated with the mesh frequency. That means the load sharing factors and the dynamic load factors are the functions of mesh frequency.

For PGTs with different number of planet gears, the load sharing coefficients have similar trend versus mesh frequency. With equivalent tooth thickness error on the 1^{st} planet gear, the dynamic load sharing factor is proportional to the absolute magnitude of equivalent tooth thickness errors. While with the increasing of the planet gear number, the load sharing factors and dynamic load factors became more sensitive to the equivalent tooth thickness errors.

With equivalent tooth thickness error on the 1^{st} planet gear, for 3-planet gear PGTs, the 2^{nd} and 3^{rd} planet gears carry equal load because of the geometric symmetry of position. For the same reason, the 2^{nd} and 4^{th} planet gears for 4-planet PGTs carry equal load. For 5- and 6-planet systems, the 2^{nd} and ^{th} and 3^{rd} and (^{th} planets carry equal load, respectively.

Floating sun gear decreases the dynamic mesh load while optimizing the load sharing although the vibration amplitude of sun gear increases. That means floating sun gear is one of the effective methods to improve the dynamic load sharing with high priority.

Large applied torque and low support stiffness of sun gear help to compensate the effects of manufacturing errors and to improve the load distribution. But large applied torque increases the nominal transmit mesh forces. Considering the static strength of gear tooth, increasing the applied torque to suppress the effects of equivalent tooth thickness error is not the first option.

Tooth profile modification is effective to eliminate the tooth separation and decrease the vibration amplitudes. The amplitudes of mesh forces are also decreased by proper TPM. These effects further help to suppress the fluctuations of the dynamic load factor versus mesh frequency. With proper TPM, one can approximate the dynamical load sharing factors by the result obtained in quasistatic conditions.

Damping matrix

Dimensionless force vector caused by profile errors and TPM

_{cn}:

Pin position error

_{pn}:

Tooth thickness error

Force vector caused by profile error

Force vector caused by TPM

Force vector of applied torque

Moments of inertia

Stiffness matrix of support

Mean mesh stiffness matrix

Variable mesh stiffness matrix

Nondimensional mesh stiffness matrix

Load sharing coefficients

Load sharing factors

Mass matrix

Number of planet gears

_{s}:

Transmitted torque on sun gear

Modal matrix

Tooth number of gears

_{i}:

Modal vibration amplitude

Additional mesh forces induced by errors

Additional mesh forces induced by TPM

Peak values of mesh forces during a mesh period

Equivalent tooth thickness errors

TPM functions

DOFs index

_{bs}:

Support stiffness of sun gear

Mesh stiffness

Torsional stiffness

mass

Planet gear index

Base radii

Time

Deflection of the gear bodies along the line of action

_{s},

_{s}:

Translational motions of sun gear

Rotate speed

Dynamic load factors

Tooth separation function

Position angle of planet gear

Pressure angles

Angle between pin position errors and the mesh line

Initial phase of errors

Mesh frequency

Natural frequency

Mesh deflection

Damping ratio.

The MATLAB data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

The authors gratefully acknowledge the support by the Scientific Research Fund of High-Level Talents in Nanjing Institute of Technology (YKJ 201951).