1. Jungho Park, Ph. D. candidate
Laboratory for System Health and Risk Management(SHRM)
Department of Mechanical and Aerospace Engineering, Seoul National University
hihijung@snu.ac.kr
Analytical Modeling of Vibration
Signals from a Planetary Gear in
Normal and Faulty Conditions

2. CONTENTS
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Introduction01
Analytical Modeling of Vibration
Signals from a Planetary Gear
02
Concluding Remarks05
Planetary Gear in a Normal
Condition
Planetary Gear in a Faulty
Condition

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INTRODUCTION

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INTRODUCTION
Dynamic modeling of a Planetary Gear
Inalpolat, Murat, and Ahmet Kahraman. "A dynamic model to predict modulation sidebands of a planetary gear set having manufacturing
errors." Journal of Sound and Vibration 329.4 (2010): 371-393. Google Citation : 98
Al-Shyyab, A., and A. Kahraman. "A non-linear dynamic model for planetary gear sets." Proceedings of the Institution of Mechanical Engineers,
Part K: Journal of Multi-body Dynamics 221.4 (2007): 567-576.Google Citation : 31

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INTRODUCTION
Dynamic modeling of a Planetary Gear
Motivation : The predicted signals of dynamic models could be different
from vibration signals using accelerometers.
Inalpolat, Murat, and Ahmet Kahraman. "A dynamic model to predict modulation sidebands of a planetary gear set having manufacturing
errors." Journal of Sound and Vibration 329.4 (2010): 371-393. Google Citation : 98
Al-Shyyab, A., and A. Kahraman. "A non-linear dynamic model for planetary gear sets." Proceedings of the Institution of Mechanical Engineers,
Part K: Journal of Multi-body Dynamics 221.4 (2007): 567-576.Google Citation : 31

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Inalpolat, Murat, and A. Kahraman. "A theoretical and experimental investigation of modulation sidebands of planetary gear sets." Journal of
Sound and Vibration 323.3 (2009): 677-696. Google Citation : 161
INTRODUCTION
Motivation & Objective
① ② ③ACC.
• Motivation : The predicted signals of dynamic models could be
different from vibration signals using accelerometers.
• Objective : Literature review will be given with the codes that
simulated the modulated vibration signals of a planetary gear.

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Analytical Modeling of
Vibration Signals from
a Planetary Gear

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Analytical Modeling of Vibration Signals from a Planetary Gear
Configuration of a Planetary Gear
Ring Gear
Planet Gear
Carrier
Sun Gear
Case (i) : Equally spaced planets
and in-phase gear meshes
 Same loads at each planet
Case (ii) : Equally spaced planets
and sequentially phased gear
meshes
 Benefits in reducing vibration
and noises
Case (iii) : Unequally spaced
planets and in-phase gear
meshes
Case (iv) : Unequally spaced
planets and sequentially phased
gear meshes

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Analytical Modeling of Vibration Signals from a Planetary Gear
Difference between Case (i) and (ii)
Case (i) : Equally spaced planets and in-
phase gear meshes
Case (ii) : Equally spaced planets and
sequentially phased gear meshes
. °
• Example) Vr = 93, Vp=31 • Example
Vr = 95, Vp=31, Vs=31,
°
. °
120°
. °
3
120°

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Analytical Modeling of Vibration Signals from a Planetary Gear
Generalized Vibration Signals - 1
∑ cos
• First, the dynamic mesh force for ring-planet i mesh can be expressed in
Fourier series as
: The Fourier coefficient of j-th harmonic of the dynamic force
: The phase angle of the j-th harmonic component
: The phase angle between the ring gear meshes of planet i and planet 1
2 1 /
• However, the modulation from measured
vibration signals should be considered. 1 carrier rotation

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Analytical Modeling of Vibration Signals from a Planetary Gear
cos
∑
Weighting function :
(phase angle considered)
Generalized Vibration Signals - 2
(i)th planet
(i+1)th planet
(i+2)th planet
X
X
X
• Combination of Hanning and step function to simulate weighted vibration
signals measured from the accelerometer (phase angle not considered)

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Analytical Modeling of Vibration Signals from a Planetary Gear
Generalized Vibration Signals - 3
(i)th planet
(i+1)th planet
(i+2)th planet
X
X
X
• Combination of Hanning and step function to simulate weighted vibration
signals measured from the accelerometer (phase angle not considered)

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Analytical Modeling of Vibration Signals from a Planetary Gear
Generalized Vibration Signals – 3 : code implementation - ①
cos

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Analytical Modeling of Vibration Signals from a Planetary Gear
∑
Generalized Vibration Signals – 3 : code implementation - ②

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Analytical Modeling of Vibration Signals from a Planetary Gear
Generalized Vibration Signals – 3 : code implementation - ③

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Analytical Modeling of Vibration Signals from a Planetary Gear
Generalized Vibration Signals – 4 - ①: Case (i)
(a) N=3, Zr=123 and Zs=72, (b) N=4, Zr=124 and Zs=72,
(c) N=5, Zr=125 and Zs=70, (d) N=6, Zr=126 and Zs=72
• For this case, the equation can be simplified like a below equation as there are no phase
differences between planets.
1
2
cos
1
4
cos
1
4
cos
(a) (b)
(c) (d)

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Analytical Modeling of Vibration Signals from a Planetary Gear
Generalized Vibration Signals – 4 - ②: Case (ii)
(a) N=3, Zr=125 and Zs=73, (b) N=4, Zr=126 and Zs=74,
(c) N=5, Zr=126 and Zs=74, (d) N=6, Zr=122 and Zs=70
• The symmetry could be different due to the order of phase differences.
(a) (b)
(c) (d)

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Analytical Modeling of Vibration Signals from a Planetary Gear
Modelling of Faulty Signals - 1
Feng, Zhipeng, and Ming J. Zuo. "Vibration signal models for fault diagnosis of planetary gearboxes." Journal of Sound and Vibration 331.22
(2012): 4919-4939. Google Citation : 96
• In the literature, AM and FM effects due to gear damage as well as transfer
path were considered.
1 cos 2 . cos 2 sin 2 .
AM by faulty gear rotation FM by faulty gear rotation
, ,
 In this case, the faults were simplified as abrupt local peaks.

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Analytical Modeling of Vibration Signals from a Planetary Gear
Modelling of Faulty Signals - 2
• The ring gear local fault was simulated.
• Faulty behaviors in the frequency domain are not obvious.
 Need to define health indices in the time domain, after using filtering
techniques to remove noises.

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Concluding
Remarks

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Concluding Remarks
Concluding Remarks
• Simulated the modulated vibration signals of a planetary gear in a normal
and faulty conditions measured from an accelerometer.
• Observed the difference between normal and faulty conditions in the time
and frequency domain.
• Need to simulate faulty behaviors developed in the previous literature, and
validate using the test-bed data.
① ② ③