Keywords

1 Introduction

Planetary gear can transmit higher power because they use multiple power paths formed by each planet branches. This allows the input torque to be divided between the n planet paths, reducing the force transmitted by each gear mesh.

Each planet path carries an equal load under ideal conditions. Nevertheless, planetary gears have inevitable manufacturing and assembly errors. So, the load is not equally shared amongst the different planet-ring and sun-planet paths, which can be a problem in terms of both durability and dynamic behavior.

Many significant works on the subject of planetary gear load sharing have been done. These works were based on transmission modeling and assessed by experimental tests. Kahraman (1994) used a discrete model to study the influence of carrier pin hole and planet run-out errors on planet load-sharing characteristics of a four-planet system under dynamic conditions. He employed later (Kahraman 1999) a planet load-sharing model to determine the static planet load sharing of four-planet systems and presented experimental data for validation of the model predictions. Iglesias et al. (2013) studied the effect of planet position error on the load sharing and transmission error. Singh (2005) found that the tangential pin position error has a greater effect on the load sharing than the radial pin position error. Ligata et al. (2008) proved experimentally that for the same amount of error, the degree of inequality in the planet load-sharing behavior increases with the number of planets in the system. Guo and Keller (2012) presented a three-dimensional dynamic model which take into account to the addressing gravity, bending moments, fluctuating mesh stiffness, nonlinear tooth contact, and bearing clearance. They validated this model against the experimental data.

All works on the subject of planetary gear load sharing are in the static, quasi-static or stationary operations.

In this paper, the effects of meshing phase and the error position of the pin hole of planets on the load sharing characteristics are studied during the run up regime numerically and validated experimentally through back-to-back planetary gear.

2 Description of the Test Bench

The test bench is composed of two identical planetary gear sets with the same gear ratio allowing the mechanical power circulation (Fig. 1). The first planetary gear is a “test gear set” and the second planetary gear is a “reaction gear set” having a free ring. An arm is fixed on this ring and allowing the introduction of external load (Hammami et al. 2014a, b). The two planetary gears are connected back-to-back: the sun gears of both planetary gear sets are connected through a common shaft and the carriers of both planetary gear sets are connected to each other through a rigid hollow shaft (Hammami et al. 2014c).

Fig. 1
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Back-to-back planetary gear test bench

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Three strains gauges are used in quarter bridge configuration in order to compare the load sharing between the tests planets. They are installed in the pin holes of each planet in the tangential direction of the test carrier (Fig. 1).

The wires from the strain gauges are connected to the Programmable Quad Bridge Amplifier module (PQBA) of the acquisition system “LMS SCADAS 316 system” through a hollow slip ring which is installed with the hollow shaft that connects the carriers (Fig. 2).

Fig. 2
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Instrumentation layout

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Additionally, an optic tachometer (Compact VLS7) which is placed along the hollow carriers’ shaft measure its instantaneous angular velocity.

The data will be processed with the software “LMS Test.Lab” to visualize time history of strains.

3 Numerical Results

In this part, effects of meshing phase and the position error of planets on the load sharing behavior are studied during the run up regime. The variation of speed during this regime is controlled by the frequency converter “Micromaster 440” which commands linearly the variation of the rotational speed of motor as shown in Fig. 3.

Fig. 3
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Evolution of the motor’s speed during run up

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First of all, we define the planet load sharing ratio (LSR) as the ratio of the meshing torque due to sun-planet (i) and ring-planet (i) meshes of planet (i) by the meshing torque of all planets.

$$ L_{Pi} = \frac{{T_{mesh(Pi)} }}{{\sum\nolimits_{i = 1}^{n} {T_{mesh(Pi)} } }}. $$
(1)

3.1 Effect of Meshing Phase

For the case of equally spaced planets and in phase meshes gear (sun planets and ring planets), the planet load sharing factor is equal to 1/N (N: number of planets).

In our case, planets are equally spaced and gear meshes functions are sequentially phased.

$$ \frac{{Z_{j} \psi_{i} }}{2\pi } \ne n\;{\text{and}}\;\sum\limits_{i = 1}^{N} {Z_{j} \psi_{i} } = m\pi \;\;(j = r,s) $$
(2)

\( Z_{j} \) is the number of tooth of the gear (j). \( \psi_{i} \) is the angle position of the planet (i). n and m are integer.

For run up regime, the period of mesh stiffness function decrease as speed increases (Khabou et al. 2011; Viadero et al. 2014). Meshes stiffness between gears are modelled as square functions. Figure 4 shows the evolution of mesh stiffness for ring-planets on the test gear set during run up taking into account mesh phasing.

Fig. 4
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Evolution of the mesh stiffness ring-planets during run-up

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The dynamic response is computed according to the procedure given in (Kahraman 1994) and the load sharing ratio is computed according to Eq. (1). Figure 5 shows the planet load sharing ratio for the nominal position of planets (faultless system) for 100 N.m of input torque.

Fig. 5
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Planet load sharing ratio for the nominal position of planets

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In this case, the LSR for all planets fluctuates slightly around the 1/N value (N = 3: number of planets) because the number of tooth in contact changes. The fluctuation of LSR of each planet is with a phase shift of 2π/3. This phase is induced by the fact that gear mesh sun-planets and ring-planets are sequentially phased. In addition, the period of fluctuation of LSR of each planet decrease with time which is explained by the evolution of the mesh stiffness ring-planets during run-up.

3.2 Effect of Planet Position Error

If a planet has an error “e” on the position of its pin hole, and all other planets are at their ideal position, then the force due to this error is given by (Singh 2010):

$$ F_{e} = K_{eff} .e $$
(4)

Keff is the cumulative stiffness due to meshing stiffness of the contact at the sun–planet Kps and planet–ring Kpr, and the planet bearing stiffness Kb.

Keff is defined as (Ligata et al. 2009):

$$ \frac{1}{{K_{eff} }} = \frac{1}{{K_{b} }} + \frac{1}{{K_{ps} + K_{pr} }} $$
(5)

In our case, the planet 1 has an error “e1 = 60 μm”, the planet 2 has an error “e2 = −5 μm” and the planet 3 has an error “e3 = −60 μm”. The LSR in this case is represented in Fig. 6.

Fig. 6
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Planet load sharing ratio with planets position errors

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The position errors of planets 1 and 2 have an important effect in the LSR. Planet 1 which has a positive error anticipates the contact, being preloaded before planets 2 and 3 begin to transmit load; whereas planet 3 which has a negative error is preloaded after planet 2 and 1. This defects are in tangential direction and they have an effect very important in the LSR (Bodas and Kahraman 2004). In addition, planet load sharing ratio tends towards fair values in the run up regime which is explained by the fact that the transmitted load decrease in each planet as the speed increase.

4 Correlation with Experimental Results

Strain-time histories for three planets system having errors “e1 = 60 μm” and “e2 = −5 μm” and “e2 = −60 μm” are shown in Fig. 7. It is clear that the positioning error has an important effect in the strain of each pin hole of planet. Also, strains tend towards fair values as the speed increase.

Fig. 7
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Variation of the measured planets strains for the speed motor 165 rpm

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The computational results of the load sharing ratio are compared to the measured LSR (Fig. 8). In general, the calculated load sharing agrees with the measured data. In fact, the effect of meshing phase is observed only in the numerical results because the recorded signals presents noise due to the contact between brushes and slip ring. So, a signal processing was necessary.

Fig. 8
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Evolution of measured and calculated LSR

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5 Conclusion

In this paper, the planets load sharing behaviour is studied during the run up regime and correlated with those obtained experimentally using strain gages on planets of a back-to-back planetary gear test bench.

The effect that planets are sequentially phased is included to the model and as results; the load sharing ratio for all planets fluctuates slightly around the 1/N value and the period of fluctuation of each planet decreases with time. The planets position errors have an important effect on the LSR of each planet which tends towards fair values in the run up regime. The final numerical results with the two effects agree with the measured strains on planets.