The 3K planetary gear system is a basic planetary transmission structure with many advantages over the 2K-H planetary gear system. However, the vibration characteristics will be more complicated due to the increase of central gears meshing with each planet gear simultaneously. In this paper, a lumped-parameter model for a 3K-II planetary gear set was developed to simulate the dynamic response. The time-varying stiffness of each meshing pair for different gear tooth root crack faults is calculated via the finite element method. By considering the effect of time-varying transmission paths, the transverse synthetic vibrations are obtained. Subsequently, the feasibilities of transverse synthetic vibration signals and output torsional vibration signals as reference for fault diagnosis are analyzed by studying the time-domain and frequency-domain characteristics of these two vibration signals. The results indicate that both the transverse synthetic vibration signals and output torsional vibration signals can be used for fault identification and localization of the 3K-II planetary gear train, and yet they both have their limitations. Some results of this paper are available as references for the fault diagnosis of 3K planetary gear trains.

Planetary gear transmission has the advantages of low weight, small size,
high load capacity and large transmission ratio, etc. Thus, planetary gear
reducers are widely used in mining machinery, wind power generation,
automobiles, ships, and other fields. The more common types of basic
planetary gearing include the 2K-H and 3K (Rao, 2014). The 2K-H type
contains two central gears and a carrier arm, and the 3K type includes three
center gears (sun gear

To monitor the operation of a planetary gearbox, the traditional method is to install a fixed sensor on the housing to measure the vibration signals (Mark and Hines, 2009). However, in this way, the transfer paths of the vibration signals to the sensor change due to the rotation of the carrier; the measured vibration data exists as a modulation phenomenon. McFadden (1991, 1994) and Howard (1991) presented a technique for calculating the time-domain averages of the tooth meshing vibration of the individual planet gears and the sun gear in an epicyclic gearbox. McFadden (1994) also proposed various window functions to acquire the vibration signals of each planet gear. Subsequently, numerous scholars have concentrated on the interpretation of the vibration spectrum modulation phenomenon due to the change of transmission path caused by the rotation of the carrier arm. McNames (2002) made use of Fourier series analysis to account for the asymmetries observed in the spectrum and predicted the possible locations of the major spectral constituents. Inalpolat and Kahraman (2009, 2010) applied a Hanning window function to represent the periodic varying of the transfer path from vibration source to transducer due to the rotation of the carrier and analyzed the modulation sidebands of the planetary gear system in five possible situations. Furthermore, Liang et al. (2015) and Liu et al. (2016) used the Hamming and the Hanning window function with different parameters to indicate the transmission path effects. Recently, Li et al. (2019) tested the time-varying transfer path function of a 2K-H planetary gear system by an experimental method based on the modal reciprocity principle.

The 3K-II planetary gear set:

As a signal gauged from the gear axle, torsional vibration conveys the operational information of the gearbox as well (Addabbo et al., 2019). Assuming that all the gears of a planetary gear set are isotropic on the circumference, the propagation distance from the meshing locations of each gear pair to the torsional vibration sensor is invariant. Therefore, torsional vibration analysis is an optional method of planetary gearbox condition monitoring. Feng and Zuo (2013) offered clear formulas to describe the torsional vibration signals for a planetary gearbox. It was shown that the torsional vibration signal is modulated only by the fault frequency. Zeng et al. (2017) developed instantaneous angular speed (IAS) measurement systems for planetary gear train fault diagnosis and examined its validity by an experimental method. Zhao et al. (2018) introduced a Kurtosis-guided partial multinomial differentiator to evaluate the IAS, and based on that a method for fault detection of planetary gearboxes was presented. Moreover, Xue and Howard (2018) also analyzed the feasibility of torsional vibration signals as a fault diagnosis tool for planetary gearboxes using some general signal processing technologies.

Kinetic analysis is a better approach to study the vibration characteristics of a planetary gear set. Compared to mathematical models (Inalpolat and Kahraman, 2009; Li et al., 2019; Feng and Zuo, 2013), a dynamical model can more accurately characterize the physical parameters of a gear system, such as time-varying meshing stiffness and damping, and it helps to understand the effects of various types of gear faults on planetary gearboxes. Many dynamical models have been proposed to obtain the dynamic performance of epicyclic gearboxes. Kahraman (1994) developed a kinetic model for a 2K-H planetary gear set to study the load distribution properties. Based on the model in Kahraman (1994), Lin and Parker (1999) developed a lumped-parameter dynamics model considering the effects of meshing phase difference, time-varying meshing stiffness, and carrier rotation. Then the free vibration characteristics of a 2K-H planetary gearbox were investigated. Subsequently, the nonlinear dynamical behavior was studied, and the comparison with a 2D finite element model was carried out to prove the effectiveness of the lumped-parameter model (Ambarisha and Parker, 2007). Li et al. (2014) developed a dynamic model for a multistage planetary gear train and analyzed the influences of damping, backlash, and excitation frequency on system dynamic characteristics. Additionally, some scholars have also investigated the effects of manufacturing errors, cracks, wear, backlash, breakage, and other factors on the dynamic characteristics of planetary gearboxes by means of establishing a dynamical model (Chaari et al., 2006; Liu et al., 2018; Sun et al., 2019; Xiang et al., 2018; Wu et al., 2017; Liu et al., 2019).

Literature studies mentioned previously indicate that a large range of studies have been conducted for the dynamic characteristics and vibration signal analysis of planetary gear systems. However, most of them focus on 2K-H. As another basic planetary gearing, the 3K planetary gear system is more compact than the 2K-H planetary gear system and offers a larger transmission ratio range. The vibration properties of a 3K planetary gear set would be more complex as each planet gear meshes with more central gears simultaneously. Few pieces of literature could be found on vibration investigations of the 3K planetary gear set. Song et al. (2009) presented a dynamics model for a 3K-II planetary gear set by decomposing it into two 2K-H planetary gear trains and analyzed the inherent characteristics. Nevertheless, the force direction of the gears was not considered in Song et al. (2009).

In this paper, a dynamic model of the 3K-II planetary gear train is proposed
by considering the force on each gear, as shown in Fig. 1b. Taking into
account the phase difference between each meshing pair, the time-varying
meshing stiffness with different types of tooth root crack faults was
analyzed by the finite element method. Supposing the sun gear

In this section, a mathematical planetary gear transmission model is
developed to help understand the complex dynamic response of a 3K-II
planetary gear system, as shown in Fig. 2. This model contains an input
torque, a sun gear, a carrier arm, three planet gears, two ring gears, and
an output load. Similar to the model used by Liang et al. (2015), each
component of this model has three DOFs (degrees of freedom): one rotation and
two transverse motions in the

Dynamic model of a 3K-II planetary gear set.

It is worth noting that the tangential force of each planet gear applied by
ring gear

Motion equations of the sun gear:

The 3K-II planetary gear set parameters.

All the notations of nomenclature used in this paper are explained in Appendix A, the selected parameters of the 3K-II planetary gear set are listed in Table 1, and the number of the planet gears is three.

Crack is a typical form of failure in gear transmission trains. The existence of a crack will weaken the bearing capacity of the gear tooth and even break the tooth. Stiffness excitation is one of the most important internal excitation forms in gear transmission. When a crack develops on the gear tooth, the mesh stiffness will be reduced, and the vibration properties of the gear transmission train will be changed.

Compared with the stiffness calculated by the potential energy method (PEM),
the finite element method (FEM) results describe the crack characteristics
more accurately. And the finite element method includes coupling flexibility
of adjacent teeth and avoids repeated superposition of tooth base stiffness.
The FEM model of an external gear pair used in this paper is shown in Fig. 3a, and the internal gear pair is displayed in Fig. 3b. The elements of
the teeth are refined to ensure the accuracy of the calculation results. The
hubs of the driven wheel and the driving wheel are rigidly coupled with
their respective centers. The load is applied to the rotating shaft of
the driving gear, and all the degrees of freedom of the driving and driven
gears are limited except the axial rotation of the driving gear. All the
calculations are completed in ANSYS (engineering simulation and 3D design software); then the comprehensive meshing
stiffness is expressed as the following:

Finite element models for analysis:

To present the influence of cracks of different gears on the vibration
characteristics of the system, a linear crack with constant crack angle
(65

Gears with a linear crack:

There are five crack cases in this study: case I, i.e., the sun gear
with a root cracked tooth; case II, i.e., a planet gear with cracked tooth root
(sun gear

Summary of meshing stiffness of each gear pair.

Figure 5a shows the effect of tooth crack on

Meshing stiffness curves affected by crack faults:

Meshing frequency is an important property in the vibration analysis of a
gear transmission train. For the 3K-II planetary gear set, since each planet
gear meshes with all central gears simultaneously, the meshing frequency of
each mesh pair is the same. Considering that the ring gear

Vibration analysis is a common method used in mechanical fault detection.
For planetary reducers, a common approach is using a sensor mounted on the
fixed housing to monitor the gear system (as shown in Fig. 1). In this
section, the vibration properties of each component of the 3K-II planetary
gear system are obtained by solving the differential Eqs. (1)–(5). The
output torque is limited to a constant value equal to 200 N m, and the input
speed is maintained at 4000 rpm. For this operating condition, each
characteristic frequency in Sect. 2.3 is listed in Table 3. For a complete
rotation of the carrier, a transducer fixed on the housing experiences
disturbances from all gear meshes in sequence. Considering the suppression
effect of the great bearing damping and long transmission lengths on
vibration signal, three force transmission paths, as shown in Fig. 1, are
mainly considered in this paper. They can be simplified as follows.

Path 1: gear pair

Path 2: gear pair

Path 3: gear pair

Considering that all the above transfer paths pass through the planet gear
and that each planet gear contacts with the sun gear

Each characteristic frequency for a given operating condition.

With the rotation of the carrier, the distance from each mesh point to the
stationary sensor varies periodically. The period is the carrier rotating
period. When the

One reference (Inalpolat and Kahraman, 2009) used a Hanning window function to
describe this change and assumed that each planet gear's individual effect
on the sensor only persists for a time interval of

Figure 6 shows the effect of coefficient

The effect of coefficient

Using the MATLAB ODE (ordinary differential equation) solver to solve the differential Eqs. (1)–(5), the acceleration signal of each planet gear in a rotational coordinate system fixed to the carrier can be obtained. To obtain the absolute acceleration in a fixed coordinate system fixed to the housing, the coordinate transformation theory is adopted (Hibbeler, 2004). Since the sensor just measures the vibration signal in a single direction, only the vertical acceleration signal is considered in this paper.

Figure 7 shows the vertical synthetic acceleration signals of a 3K-II
planetary set at six different cases (the healthy gear case; failure case I,
sun gear cracking; failure case II, planet gear cracking on the

Synthetic vibration signals in the vertical direction:

Figure 8 illustrates the frequency spectrum of the synthetic vibration signals
in different gear health cases. To highlight the various fault frequency
components, we select a frequency range from 1130 to 1270 carrier rotation
frequency orders to demonstrate the spectrum characteristics. The first
significant observation from these figures is that the amplitude at the mesh
frequency

For failure case I with a cracked sun gear, the amplitudes on the positions
of the sideband frequency (

For failure cases II and III with a cracked planet gear, the frequency
components on the positions of the planet gear characteristic frequency
(

For failure case IV with a cracked ring gear

For failure case V with a cracked ring gear

Frequency spectrum of synthetic vibration signals:

A big challenge of comparing the frequency spectra (Fig. 8a–f) is that there
are so many period components, like the meshing frequency, the carrier
rotational frequency, and four fault characteristic frequencies. It is hard
to determine which periodic features make up these sizable sideband
components. The cepstrum is used to show the frequency components of a
signal from the time dimension to highlight the periodic features which are
not apparent in the original spectrum. As displayed in Fig. 9b, the
amplitude at 0.00625 s (

According to the analysis above, we can see that the transverse vibration signal detected by the sensor fixed to the housing can be used as a basis for distinguishing faults of the 3K-II planetary gear set. However, due to the modulation phenomenon caused by the carrier arm's rotation, the sideband component of the signal frequency domain becomes complicated and the signal fault characteristics are not distinct enough. This makes it troublesome to diagnose faults via housing vibration signals.

Cepstrum of synthetic vibration signals:

Instantaneous angular speed (IAS), as a detectable signal, is commonly
used in operation monitoring and fault detection of rotating machinery such
as motors, gears, bearings, etc. (Kazienko and Chybowski, 2020; Wang et al., 2020;
Liu et al., 2020). Compared to vibration signals detected by sensors fixed
on the case, the IAS signal is not affected by the amplitude modulation
caused by the rotation of the carrier arm. Besides, the IAS signal shows
better sensitivity to different types of mechanical faults involving a large
number of sideband orders. In this section, the same operating conditions in
Sect. 3 are selected to obtain the IAS of the output ring gear

Ring gear

Frequency spectrum of ring gear

Cepstrum of ring gear

As seen in Fig. 10b, the rotational velocity of the ring gear

As illustrated in Fig. 10c–d, for the case with a cracked planet gear,
the angular velocity of the ring gear

When a crack develops on a tooth of ring gear

As for the case with a cracked ring gear

It is seen from the analysis of Figs. 10–12 that the torsional vibration signal can be used to diagnose faults of the 3K-II planetary gear system. In contrast to the transverse synthetic vibration signals in Sect. 3, torsional vibration signals are unaffected by the amplitude modulation due to the carrier's rotation. This makes the fault characteristics in the time domain of a torsional vibration signal more pronounced and the sidebands of fault in the frequency domain easier to identify. However, the torsional vibration signal also has limitations for the fault detection of the 3K-II planetary gear system, like the sun gear local tooth root cracking.

The 3K planetary gear system, as a basic planetary drive structure, is more
compact than the 2K-H and has a wider range of transmission ratios. A
torsional vibration dynamics model of a 3K-II planetary gear system is
developed in this paper. The effect of five crack conditions on the
time-varying meshing stiffness is analyzed. A theoretical model that
considers the modulation effect due to the rotation of carrier is adopted to
obtain the synthetic vibration signals to reveal the housing vibration in
the health and tooth root cracking cases. Subsequently, both the feasibility
of using the synthetic vibration signals and the torsional vibration signals
of the output ring gear as a basis for fault diagnosis of the 3K-II
planetary gear system is analyzed by reference to time-domain spectra,
frequency spectrum, and cepstrum in the healthy and five cracked-tooth cases. The main conclusions are as follows:

The synthetic vibration signal of the planetary housing shows obvious amplitude modulation waves. Furthermore, besides the fault-frequency component, the spectral sidebands also have the carrier rotation frequency component in the frequency domain.

The torsion signal is unaffected by the amplitude modulation of the carrier's rotation, making it possible to demonstrate fault diagnostics more clearly in the time and frequency domains.

Both the synthetic vibration signal and torsional vibration signal can be used for fault diagnosis of the 3K-II planetary gear system. However, both of them have their limitations, i.e., the synthetic vibration signal is influenced by the modulation of the carrier rotation, which makes the fault characteristics not clear enough, and the spectral sideband composition more is complex, thus making it more complicated to determine the specific fault type; for the torsional vibration signal, in the case of a large transmission ratio, it is difficult to identify the sun gear failures by the angular velocity signal of the output ring gear. In practice, the fault detection of the 3K planetary gearbox should be performed by installing both a sensor on the housing and an angle encoder on the output.

All the data used to support the findings of this study are included within in the relevant figures and tables in the article.

MS was responsible for the methodology and writing of the draft of the paper, KH and YX contributed to the review and supervision, GH calculated the time-varying meshing stiffness, and ZC checked the article and made suggestions. All authors read and approved the final article.

The contact author has declared that neither they nor their co-authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The authors thank the reviewers for their critical and constructive review of the article.

This research has been supported by the National Natural Science Foundation of China (grant no. 51775156), the National Key R&D Program of China (grant no. 2017YFB0103201), the Natural Science Foundation of Anhui Province of China (grant no. 1908085QE228), the Key Research and Development Project of Anhui Province (grant no. 202004h07020013), and the Fundamental Research Funds for the Central Universities of China (grant no. PA2020GDSK0091).

This paper was edited by Daniel Condurache and reviewed by two anonymous referees.