Multi-Body Dynamics Modeling and Analysis of Planetary Gearbox Combination Failure Based on Digital Twin

: Existing studies on the dynamic characteristics of planetary gearboxes are generally based on traditional modeling methods. The models were idealized because of various uncertain factors that affect the dynamic characteristics of the planetary gearbox. The inﬂuence of non-mechanical components, such as driving motors, on dynamic analysis, is often ignored. Digital twin technology can not only simulate the data characteristics of physical entities to realize the real-time state evaluation and fault diagnosis of complex equipment but also reﬂect the entity characteristics of the modeled object completely. Therefore, a digital twin multi-body dynamic model of a planetary gearbox is established for detecting the combined fault of the sun gear missing teeth and the planetary gear missing teeth or pitting under different working conditions. The dynamic characteristics of a planetary gearbox subjected to electromechanical coupling were studied. The sun gear fault frequency, planetary gear fault frequency, and both related modulation frequencies appear in the experiment, and the corresponding fault characteristic frequencies also appear in the digital twin simulation. The validity of the proposed model was experimentally veriﬁed. Finally, the root-mean-square index was used to analyze the fault sensitivity of the traditional and digital twin models. The results show that the fault diagnosis effect of the digital twin model has a signiﬁcant advantage, thus verifying the necessity of considering the effect of electromechanical coupling on the dynamic characteristics of the planetary gearbox.


Introduction
Owing to the compact structure and high-power transmission efficiency of the transmission mechanism of the planetary gearbox, the planetary gearbox transmission mechanism has been widely used in wind turbines, automobiles, ships, aerospace, and other machinery.In terms of the planetary gear train design, the literature [1,2] proposes a new method for planetary gear train (PGT) analysis, which simplifies the design process of the planetary gear and provides guidance for practical applications.The literature [3,4] evaluated the effect of the vibration activity of gearbox housings made of composite materials and showed that housings made of composite materials have comparable or lower vibration levels at higher gear speeds and that composite gearbox housings can be a good alternative to steel solutions.The literature [5] discusses the results of a study of the original system condition monitoring of a three-stage transmission, the main objective of which was to establish the frequency characteristics of the gearing, to determine whether it is possible to perform a vibroacoustic diagnosis of the same transmission under conditions of considerable influence and to analyze the influence of the sensor mounting point on the effectiveness of the signal diagnosis.In the context of rigid body entanglement, the literature [6,7] studied the rotational motion of a body fixed at a point, with the former Appl.Sci.2022, 12, 12290 2 of 18 representing this graphically, showing the variation of the parameters of the body during the motion, and the latter using the Krylov-Bogoliubov-Mitropolski (KBM) method to obtain an approximate solution for the kinematic system, a work that is used in physics, engineering, and the aeronautical industry.With regard to the geometric model of planetary gear teeth, due to the more difficult manufacturing of the convex-concave gear profile in comparison to the involute one, an application example was defined that suppressed this disadvantage [8].An improved algorithm for herringbone gear tooth topology modification based on the overlap rate is proposed in the literature [9], and the superiority of the improved topology-modified gear pair is verified in terms of both statics and dynamics.
Planetary gearboxes are prone to failures during long-term operation, and if these failures are not detected and eliminated in time, they will cause uneven forces on the gear mesh, accelerate the wear of the gear teeth, and lead to the failure of other gears meshed with them, i.e., combined failures.Therefore, it is very necessary to establish a planetary gearbox dynamics model to study the effects of different levels of combined failures on the dynamic characteristics of planetary gear systems and to provide the necessary information for fault diagnosis and reliability studies on the planetary gearbox.In the literature [10][11][12][13], gear dynamic models containing different faults were established and the mechanisms of the different faults were analyzed using the established models.The literature [14][15][16] derived the mesh stiffness algorithm of gears under different faults and analyzed the spectral characteristics of gear vibration signals under different fault states and transmission paths using the Fourier transform.The literature [17,18] established a dynamic model of the gear system considering nonlinear factors such as teeth orientation errors, gear manufacturing, and assembly errors, and analyzed the dynamic response of the gear.In the literature [19,20], the torsional vibration model with degrees of freedom and the set total parameter model were established, respectively, and the frequency characteristics of the planetary gearbox under fault-free and different fault conditions were analyzed using the proposed models.A data fusion method based on deep convolutional neural networks (DCNN) was proposed in the literature [21], and the effectiveness of the method was verified by data collected from a planetary gearbox.The quality and reliability of consumables, including gear oil, is also an important factor in ensuring trouble-free gear operation, as oil contamination can affect lubrication efficiency and lead to failures [22].
Although some progress has been made in the dynamic modeling of planetary gear systems, the fact remains that the large number of factors affecting planetary gear systems leads to the need for the idealization of the model during the modeling process.In addition, there are deficiencies in the existing methods for the dynamic modeling of planetary gear systems, i.e., the influence of non-mechanical parts (e.g., drive motors) on the dynamic characteristics is neglected.Therefore, there is a need for a new and well-considered modeling approach.
The digital twin is defined as an integrated multi-physical-field, multi-scale, probabilistic simulation process that builds a complete virtual mapping model of the equipment based on its available high-fidelity physical model, historical data, and real-time sensor update data to portray and reflect the full life-cycle process of the physical system and achieve the prediction of the equipment health status, remaining service life, and mission accessibility.
Therefore, this paper adopts digital twin technology to build a planetary gearbox system model and consider the influence of non-mechanical factors on the model to more closely imitate the real situation and better guide the subsequent work on equipment health state assessment and remaining useful life (RUL) prediction.
In terms of the digital twin, [23,24] applied the digital twin to gears: the former validated the proposed digital twin model based on the provided numerical values and experimental examples; the latter discussed the use of the model in the prediction of the remaining service life of gearboxes.In the literature [25][26][27], a digital twin model was developed for wind turbine drive trains to achieve the remote monitoring of the wind turbine gear unit status and the prediction of downtime due to faults as well as for RUL prediction.In the literature [28,29], a digital twin fault diagnosis method was used to secure the operational safety of nuclear power plants.The works [30,31] developed a digital twin predictive maintenance framework for air handling units (AHUs) and a digital twin model for fault diagnosis in distributed photovoltaic (PV) systems with rooftop building integration.In [32], a digital twin model for predicting impacts on helicopter composite sandwich structures was developed, which can estimate the damage before the inspection.The previous work [33] developed a digital twin model for detecting tool wear that can be used to identify the wear status of blades as well as to predict the development of wear based on future cutting schedules.Moreover, [34] proposes a comprehensive and robust digital twin architecture for power plants that achieve high reliability, availability, and maintainability.Another study [35] proposes a blockchain-based DT framework as a trusted digital twin model for securing cyber-physical systems (CPSs) in response to the complexity of cyber-attacks in CPSs.The works [36,37] established a digital twin model for electric motors to achieve the fault diagnosis and optimization of electric motors.What is more, [38,39] present a digital twin model for improving process and product quality and a digital twin framework that allows for the simultaneous modeling and estimation of multiple dimensions of system health.
In summary, the digital twin technique was applied to bevel gears in [23], and the digital twin modeling in the [24] did not consider the modeling of the non-mechanical parts.The works [23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39] present a detailed description of the current application of digital twin technology, which provides theoretical guidance for the study of this paper.Since planetary gearboxes are widely used, and large industrial equipment contains planetary gearboxes, which are significantly different from the application areas introduced in the above research, and the current lack of models that consider the influence of non-mechanical factors on the dynamic characteristics of planetary gearbox system, it is necessary to apply digital twin technology to planetary gearbox systems.
In this paper, digital twin is used for planetary gearbox fault diagnosis t.The model is by SOLIDWORKS, and simulations were made jointly with ADAMS and MATLAB.This work compares the simulations with the experiment to verify the correctness of the model, which is compared and analyzed using different modeling methods to obtain the root-mean-square in order to verify the superiority of digital twin modeling.

Construction of Digital Twin Model of Planetary Gearbox
Traditional modeling is divided into numerical modeling and physical modeling.Numerical modeling is difficult to analyze theoretically and renders deriving mathematical formulas complex, while physical modeling is relatively simple but often does not contain all entities of the simulation object.For example, in the traditional physical modeling method for planetary gearboxes, the modeling of the drive motor is omitted, resulting in the physical model not accurately reflecting the real dynamics of the simulation object.Based on this, this paper uses the digital twin method to build the physical model of the planetary gearbox system and then models the drive motor based on the planetary gearbox model, so that the planetary gearbox system model is more complete and the simulation results are closer to reality.
The digital twin model of a planetary gearbox is a digital representation of a physical planetary gearbox, a carrier of scenario-aware data for a planetary gearbox, and an important basis for predictive maintenance algorithms.However, the planetary gearbox is located in a complex environment, and the construction of its model requires a comprehensive consideration of the multi-domain characteristics of the planetary gear.Therefore, based on the principles of object-oriented, incremental, and multi-domain unified modeling, the mechanical model of a planetary gearbox and the coupling of multi-domain models were studied to realize the digital representation of the complex characteristics of planetary gearbox, so that the digital twin model could reflect the operation state of planetary gearbox gear.
In the planetary gear system, the key transmission components are the sun gear, the planetary gear, the ring gear, etc.In order to make the simulation results more accurate later, in the process of modeling the components of the planetary gear system, and in order to prevent inaccurate simulation results due to modeling errors, the professional CAD modeling software SOLIDWORKS was used to establish a high-precision geometric model of the components containing teeth in the gear model.The 3D model of other components was also established.Through the data interface between the software and other simulation software, the 3D assembly or part model created in SOLIDWORKS could then be directly imported into the simulation software for the subsequent model simulation and validation work.
The parameters of the constructed gears are given in Table 1 below.After the modeling of the components of the planetary gearbox was completed by SOLIDWORKS, the overall assembly work was performed.The complete mating diagram is shown in Figure 1 below, and the failure modes of each tooth-containing part are given in Figure 2. planetary gearbox, so that the digital twin model could reflect the operation state of planetary gearbox gear.
In the planetary gear system, the key transmission components are the sun gear, the planetary gear, the ring gear, etc.In order to make the simulation results more accurate later, in the process of modeling the components of the planetary gear system, and in order to prevent inaccurate simulation results due to modeling errors, the professional CAD modeling software SOLIDWORKS was used to establish a high-precision geometric model of the components containing teeth in the gear model.The 3D model of other components was also established.Through the data interface between the software and other simulation software, the 3D assembly or part model created in SOLIDWORKS could then be directly imported into the simulation software for the subsequent model simulation and validation work.
The parameters of the constructed gears are given in Table 1 below.After the modeling of the components of the planetary gearbox was completed by SOLIDWORKS, the overall assembly work was performed.The complete mating diagram is shown in Figure 1 below, and the failure modes of each tooth-containing part are given in Figure 2.  In this paper, ADAMS and MATLAB were selected as the joint simulation platform for the digital twin.In the ADAMS multi-body dynamics modeling environment, the part model in X_T format or the part assembly constraints in ADAMS format was read, the properties of the part entities were modified, the new part coordinate systems were added to connect other entities and constraints, and the contact forces and torques in the model were defined.The modification and addition of kinematic pairs according to the actual application purpose could then be enacted to realistically reflect the actual connection relationship between the planetary gearbox components.The physical properties of the kinematic subsets, such as stiffness and damping, and the friction coefficient were also planetary gearbox, so that the digital twin model could reflect the operation stat planetary gearbox gear.
In the planetary gear system, the key transmission components are the sun gear planetary gear, the ring gear, etc.In order to make the simulation results more accu later, in the process of modeling the components of the planetary gear system, and in der to prevent inaccurate simulation results due to modeling errors, the professional C modeling software SOLIDWORKS was used to establish a high-precision geome model of the components containing teeth in the gear model.The 3D model of other c ponents was also established.Through the data interface between the software and o simulation software, the 3D assembly or part model created in SOLIDWORKS could t be directly imported into the simulation software for the subsequent model simula and validation work.
The parameters of the constructed gears are given in Table 1 below.After the mo ing of the components of the planetary gearbox was completed by SOLIDWORKS, overall assembly work was performed.The complete mating diagram is shown in Fig 1 below, and the failure modes of each tooth-containing part are given in Figure 2.  In this paper, ADAMS and MATLAB were selected as the joint simulation platf for the digital twin.In the ADAMS multi-body dynamics modeling environment, the model in X_T format or the part assembly constraints in ADAMS format was read, properties of the part entities were modified, the new part coordinate systems were ad to connect other entities and constraints, and the contact forces and torques in the mo were defined.The modification and addition of kinematic pairs according to the ac application purpose could then be enacted to realistically reflect the actual connection lationship between the planetary gearbox components.The physical properties of the ematic subsets, such as stiffness and damping, and the friction coefficient were In this paper, ADAMS and MATLAB were selected as the joint simulation platform for the digital twin.In the ADAMS multi-body dynamics modeling environment, the part model in X_T format or the part assembly constraints in ADAMS format was read, the properties of the part entities were modified, the new part coordinate systems were added to connect other entities and constraints, and the contact forces and torques in the model were defined.The modification and addition of kinematic pairs according to the actual application purpose could then be enacted to realistically reflect the actual connection relationship between the planetary gearbox components.The physical properties of the kinematic subsets, such as stiffness and damping, and the friction coefficient were also defined.The physical entities were structural steel and the contact forces were simulated using the IMPACT function.Considering that the pure rigid body dynamics simulation can only simulate the dynamics map of the model's center of mass, and cannot extract accurate fault information, it was necessary to introduce a flexible body, and to consider the placement of the sensors so the ring gear is flexible in this paper, in order to better transfer the gear vibration information and allow the sensors to collect data better.In this paper, we use ANSYS APDL software (ANSYS APDL 2020R2) to make the ring gear flexible, import it into ADMAS in MNF file format, replace the original rigid body model, and finally, obtain a rigid-flexible coupled multi-body dynamics model with both rigid and flexible bodies.The rigid-flexible coupled multi-body dynamics model is shown in Figure 3.
defined.The physical entities were structural steel and the contact forces using the IMPACT function.Considering that the pure rigid body dyna can only simulate the dynamics map of the model's center of mass, an accurate fault information, it was necessary to introduce a flexible body the placement of the sensors so the ring gear is flexible in this paper, in transfer the gear vibration information and allow the sensors to collect d paper, we use ANSYS APDL software (ANSYS APDL 2020R2) to make th ible, import it into ADMAS in MNF file format, replace the original rigid b finally, obtain a rigid-flexible coupled multi-body dynamics model wit flexible bodies.The rigid-flexible coupled multi-body dynamics model is 3.We then set up the speed input, the displacement output of the X an the ring gear, and the speed output of the X and Y directions, respective up in ADAMS, we exported the setup model to MATLAB, as shown in and built the motor model in MATLAB, using the PMSM motor module a Bridge module in MATLAB Simulink.We set the specific parameters in according to the actual parameters of the test bench motor, and used the simulate a process change in the motor from standstill to acceleration.T outputs a speed signal into the planetary gearbox module in radians.Th diagram is shown in Figure 5 below.The set motor module was connecte etary gear module, and finally, the digital twin model of the planeta formed, as shown in Figure 6 below.We then set up the speed input, the displacement output of the X and Y directions of the ring gear, and the speed output of the X and Y directions, respectively.After setting up in ADAMS, we exported the setup model to MATLAB, as shown in Figure 4 below, and built the motor model in MATLAB, using the PMSM motor module and the Universal Bridge module in MATLAB Simulink.We set the specific parameters in the motor model according to the actual parameters of the test bench motor, and used the STEP function to simulate a process change in the motor from standstill to acceleration.The motor model outputs a speed signal into the planetary gearbox module in radians.The motor module diagram is shown in Figure 5 below.The set motor module was connected with the planetary gear module, and finally, the digital twin model of the planetary gearbox was formed, as shown in Figure 6   We then set up the speed input, the displacement output of the X the ring gear, and the speed output of the X and Y directions, respec up in ADAMS, we exported the setup model to MATLAB, as shown and built the motor model in MATLAB, using the PMSM motor modu Bridge module in MATLAB Simulink.We set the specific parameters according to the actual parameters of the test bench motor, and used t simulate a process change in the motor from standstill to acceleratio outputs a speed signal into the planetary gearbox module in radians diagram is shown in Figure 5 below.The set motor module was conn etary gear module, and finally, the digital twin model of the plan formed, as shown in Figure 6 below.

Analysis of Planetary Gearbox Vibration Characteristics
The original signal measured on the gearbox is usually a complex composi or several vibration signals superimposed with structural frequencies.If the cr lation component is ignored, the gearbox vibration signal can be expressed in (1), which is taken from [24], as follows: G t is the vibration signal at each rotation frequency in Equation (1).
is the gear meshing frequency modulation signal.
is the modulated signal of each order of the gear's in quency.

• ( ) ( )
is the modulated signal of the intrinsic frequency of the


is the bearing vibration signal.

• ( )
n t represents the other vibration disturbance signals.
• ( ) X t is the carrier signal.

•
is the modulated signal.

Analysis of Planetary Gearbox Vibration Characteristics
The original signal measured on the gearbox is usually a complex compo or several vibration signals superimposed with structural frequencies.If the lation component is ignored, the gearbox vibration signal can be expressed (1), which is taken from [24], as follows: G t is the vibration signal at each rotation frequency in Equation (1).
is the gear meshing frequency modulation signal.
is the modulated signal of each order of the gear's quency.
• ( ) ( ) is the modulated signal of the intrinsic frequency of t


is the bearing vibration signal.

• ( )
n t represents the other vibration disturbance signals.
• ( ) X t is the carrier signal.

•
is the modulated signal.
From Equation ( 1), it can be seen that when a fault occurs in a gear system to the gear mesh frequency and its respective harmonics, the modulation

Analysis of Planetary Gearbox Vibration Characteristics
The original signal measured on the gearbox is usually a complex composition of one or several vibration signals superimposed with structural frequencies.If the cross-modulation component is ignored, the gearbox vibration signal can be expressed in Equation (1), which is taken from [24], as follows: where • G(t) is the vibration signal at each rotation frequency in Equation (1).From Equation (1), it can be seen that when a fault occurs in a gear system, in addition to the gear mesh frequency and its respective harmonics, the modulation of the signal appears in the spectrum: modulation with the gear mesh frequency and its respective harmonics as carriers, resonance modulation with the intrinsic frequency of the gear as a carrier or with the intrinsic frequency of the box as a carrier [40].In the real operating environment of the entity corresponding to the digital twin, amplitude modulation and phase modulation coexist, and the effect is expressed in the time domain curve and the spectrogram, respectively.

Planetary Gearbox Vibration Characteristic Frequency
For planetary gearboxes, the characteristic frequency of gear faults depends on the specific structural parameters and rotational speed of the gearbox.In single-stage planetary gearboxes, the gear meshing frequencies of the two meshing pairs, the sun gear and the planet gear, and the planet gear, and the ring gear are the same.Usually, the ring gear is fixed and the sun gear, planet gear, and planetary carrier rotate, in which case the meshing frequency f m is [41] In Equation ( 2), Z r and Z s are the number of teeth of the ring gear and the sun gear, respectively; f r s is the absolute rotation frequency of the sun gear.Local fault in the gear will cause shock vibration.For planetary gearboxes, the characteristic frequency of local faults in the gear depends not only on the speed of the gearbox, but also on the number of teeth of the sun gear, planetary gear, and ring gear, and the number of planetary gears.
If there is a local fault in the sun gear, the faulty gear will mesh with all planetary gears during one rotation cycle relative to the planetary frame and produce shocks, so the characteristic frequency of local faults in the sun gear is In Equation (3), Z s is the number of sun gear teeth and N is the number of planetary gears.
If there is a local fault in the planetary gear, the faulty gear meshes with both the sun gear and the ring gear during one rotation cycle relative to the ring gear, generating a shock, so the characteristic frequency of the local fault in the planetary gear is In Equation ( 4), Z p is the number of ring gear teeth.

Planetary Gearbox Digital Twin Model Simulation Analysis
The simulation platform used in this section was MATLAB.The basic parameters of the simulated gear are shown in Table 2.The simulation motor drive speed was set to 1800 rpm, the simulation time was 4 s, and the step size was 0.0002 s.According to the structural parameters of the gearbox and the speed, the characteristic frequency of the gear local fault was calculated and is shown in Table 2.

Simulation Analysis of Normal Working Condition of Sun Gear and Planetary Gear
The simulation model was set up according to the above parameters, and the sensor measurement point was set up at the point directly above the ring gear, where the vibration velocity signal in the Y direction of the ring gear was used, and then the simulation data were processed in MATLAB with a sampling frequency of 5000 Hz, and the Hilbert transform was performed on the time domain spectrum, and the final obtained time domain waveform and frequency spectrum are shown in Figure 7 below.The spectrum was taken between 450 Hz and 550 Hz.
Figure 7 shows the time domain waveform and the Fourier spectrum of the planetary gearbox vibration signal under normal conditions.The time domain waveform is uniformly distributed as a whole, without any obvious evidence of shock. 1 and 2 in the figure are f m − 7 f c and f m + 7 f c , respectively.It can be seen from the Fourier spectrum that shocks with frequencies of f m ± n f c (n = 1,2, . . . ) appear in the sideband of the meshing cycle, and the sideband is spaced at the planetary frame rotation frequency.These phenomena indicate that the passing effect of the planetary gear plays a major role in amplitude modulation of the meshing vibration, which is in line with the theoretical expectation of the normal planetary gearbox vibration signal, and also basically in line with the results of the subsequent experimental verification.

Simulation Analysis of Normal Working Condition of Sun Gear and Planetary Gear
The simulation model was set up according to the above parameters, and the sensor measurement point was set up at the point directly above the ring gear, where the vibration velocity signal in the Y direction of the ring gear was used, and then the simulation data were processed in MATLAB with a sampling frequency of 5000 Hz, and the Hilbert transform was performed on the time domain spectrum, and the final obtained time domain waveform and frequency spectrum are shown in Figure 7 below.The spectrum was taken between 450 Hz and 550 Hz.

Simulation Analysis of the Combined Fault Conditions of Sun Gear Missing Teeth and Planetary Gear Pitting
The simulation model was also set up according to the above parameters, the sampling frequency was kept constant, the simulation data were imported into MATLAB for processing, and then the Hilbert transform was performed on the time domain spectrum, and the final time domain waveform and spectrum are shown in Figure 8 below.
Figure 8 shows the time domain waveform and Fourier frequencies of the combined fault vibration signal of the sun gear missing teeth and planetary gear pitting.In the time domain waveform, more obvious evidence of shock appears, and more modulation frequencies appear in the Fourier spectrum.For easy identification, only the meshing frequency and the modulation frequency associated with the fault are marked in the figure .Due to the modulation effect caused by the planetary frame and sun gear self-propagation, the sideband interval f c and its multiplicative correlation modulation sideband appear on both sides of f m .For example, at point 5 is f m − f c .Moreover, due to the presence of three planetary gears, it can be seen in the figure that peaks occur at the positions f m ± n f s /3 ± f c of the meshing frequency, and the sideband is spaced as n f s /3 times the characteristic frequency of the sun gear fault, where n is an integer, 1 is f m − f s /3 − f c .Peaks appear at the positions f m ± n f p ± n f c , related to the characteristic frequency f p associated with the planetary gear, where n is an integer, such that 2, 3, and 4 would be f m − f p , f m + f p , and f m + 2 f p + f c , respectively, so the comprehensive diagnosis is that both the sun gear and the planetary gear are faulty.phenomena indicate that the passing effect of the planetary gear plays a major role in amplitude modulation of the meshing vibration, which is in line with the theoretical expectation of the normal planetary gearbox vibration signal, and also basically in line with the results of the subsequent experimental verification.

Simulation Analysis of the Combined Fault Conditions of Sun Gear Missing Teeth and Planetary Gear Pitting
The simulation model was also set up according to the above parameters, the sampling frequency was kept constant, the simulation data were imported into MATLAB for processing, and then the Hilbert transform was performed on the time domain spectrum, and the final time domain waveform and spectrum are shown in Figure 8 below.Figure 8 shows the time domain waveform and Fourier frequencies of the combined fault vibration signal of the sun gear missing teeth and planetary gear pitting.In the time

Simulation Analysis of Combination Fault of Sun Gear Missing Teeth and Planetary Gear Missing Teeth
Figure 9 shows the time domain waveform and Fourier frequencies of the combined fault vibration signals of the sun gear missing teeth and the planetary gear missing teeth.Compared with the case of the sun gear missing teeth and the planetary gear pitting, there is a more obvious shock in the time domain waveform when the sun gear and the planetary gear are missing teeth at the same time, and the frequency modulation of the whole Fourier spectrum is more obvious.Similarly, due to the modulation effect caused by the planetary frame at both sides of the meshing frequency f m , the sideband appears on both sides of f m with a sideband interval of f c , such that 4 and 5 would be f m − f c and f m + f c , respectively.Due to the presence of three planetary gears, it can also be seen in the figure that peaks at positions such as the meshing frequency f m ± n f s /3 ± f c with the sideband spaced at n f s /3 times the characteristic frequency of the sun gear fault, where n is an integer, such that 1, 3, 7, and 8 would be f m − f s /3 − f c , f m − f s /3, f m + f s /3 + f c , and f m + f s /3 + 2 f c , respectively.Peaks are found at the positions of frequency f m ± n f p ± n f c associated with the planetary gear fault characteristic frequency f p , where n is an integer, such that at 2 and 6 are f m − 2 f p and f m + f p , respectively.Therefore, the comprehensive diagnosis is that both the sun gear and the planetary gear are faulty.

Experimental Verification of Different Combinations of Fault Conditions
The faulty gear is shown in Figure 10.The experimental bench is shown in Figure 11 below.The acceleration sensor was placed directly above the planetary gearbox.The experimental motor speed was set to 1800 rpm, the actual speed was 1782 rpm, the sampling A/(mm/s) f/Hz

Experimental Verification of Different Combinations of Fault Conditions
The faulty gear is shown in Figure 10.The experimental bench is shown in Figure 11 below.The acceleration sensor was placed directly above the planetary gearbox.The experimental motor speed was set to 1800 rpm, the actual speed was 1782 rpm, the sampling frequency was 5000 Hz, the sampling time was 4s, and the actual rotation frequency of the sun gear was 29.7 Hz.The gear-related parameters and the frequency of the gearbox are given in Tables 3 and 4 below.The frequency was taken between 450 Hz and 550 Hz.

Experimental Verification of Normal Working Conditions of Sun Gear and Planetary Gear
Figure 12 shows the time domain waveform and Fourier frequencies of the vibration signals under the normal operating conditions of both the sun gear and the planetary gear.The time domain waveform is uniformly distributed as a whole without obvious shocks, and the Fourier spectrum is at the position where the sideband of the meshing cycle appears at a frequency of f m ± n f c (n = 1,2, . . .), and the sideband interval is the rotation frequency f c of the planetary frame, as shown in the figure at points 1 and 2, which are f m − 7 f c and f m + 7 f c , respectively, and these phenomena are consistent with the theoretical expectations of the normal planetary gearbox vibration signal and are basically consistent with the simulation in the previous section, which verifies the correctness of the simulation model and the credibility of the results.

Experimental Verification of the Combined Fault Conditions of Sun Gear Missing Teeth and Planetary Gear Pitting
Figure 13 shows the time domain waveform and the Fourier spectrum of the experimental vibration signal under the combined fault conditions of the sun gear teeth missing and planetary gear pitting.The time domain waveform shows a more obvious shock, and the peak of the whole spectrum is more obvious.There is a modulation effect of the planetary frame on the meshing frequency, so there are peaks at f m ± n f c on both sides of the meshing frequency f m , where n is an integer, such that 6, 9 and 7, 8 are f m ± f c and f m ± 2 f c , respectively.Due to the presence of three planetary gears, it can also be seen in the figure that peaks appear at the positions f m ± n f s /3 ± n f c of the meshing frequency, with sideband intervals of n f s /3 times the characteristic frequencies of the sun gear fault, where n is an integer, such that 1, 3, 4, 11, 12, and 13 are f m − 2 f s /3, f m − f s /3 − f c , f m − f s /3, f m + f s /3, f m + f s /3 + f c , and f m + 2 f s /3, respectively.Peaks are found at the positions of frequency f m ± n f p /3 ± f c associated with the planetary gear fault characteristic frequency f p , such as at 2, 5, and 10 are f m − 2 f p − f c , f m − f p , and f m + f p , respectively, and it can be seen that the experimental signal is basically consistent with the simulated signal.Therefore, the comprehensive diagnosis is that both the sun gear and the planetary gear are faulty.

Experimental Verification of the Combined Fault Conditionsof Sun Gear Missing Teeth and Planetary Gear Missing Teeth
From Figure 14, it can be seen that the time domain waveform shock is more obvious when the sun gear and planetary gear are missing teeth at the same time.Similarly, peaks f m ± n f c on both sides of the meshing frequency f m , where n is an integer, such as at 3 and 4 are f m ± f c , respectively.Due to the existence of three planetary gears, it can be seen in the figure that peaks appear at the positions f m ± n f s /3 ± n f c of meshing frequency with a sideband interval of n f s /3 times the characteristic frequency of the sun gear fault, where n is an integer.For example, at 2, 6, and 7 are f m − f s /3, f m + f s /3, and f m + f s /3 + f c , respectively.Peaks are found at the positions of frequency f m ± n f p ± n f c associated with the planetary gear fault characteristic frequency f p , such that 1, 5, 8, and 9 are f m − 2 f p − f c , f m + f p , f m + 2 f p , and f m + 2 f p + f c , respectively.The experimental signal is basically consistent with the simulated signal.Therefore, the comprehensive diagnosis is that both the sun gear and planetary gear are faulty.
From Figure 14, it can be seen that the time domain waveform shock is more obvious when the sun gear and planetary gear are missing teeth at the same time.Similarly, peaks From the experiment, it can be seen that when both the sun gear and the planetary gear are faulty, the spectrum peaks are more intensive compared with normal operating conditions.Although the types of fault combinations are different, they are all faulty for the planetary gearbox system, so the spectrum peaks are roughly the same in terms of distribution.In addition, there are manufacturing errors and assembly errors in the test bench, which lead to the appearance of an interference spectrum in the experimental signal, but it can still be found that the frequency distribution of the model signal is basically consistent with the test through the comparison between the simulation and the test, and the frequency spectrum is mostly distributed around the engagement frequency, thus verifying the accuracy of the digital twin model.The accuracy of the digital twin model is From the experiment, it can be seen that when both the sun gear and the planetary gear are faulty, the spectrum peaks are more intensive compared with normal operating conditions.Although the types of fault combinations are different, they are all faulty for the planetary gearbox system, so the spectrum peaks are roughly the same in terms of distribution.In addition, there are manufacturing errors and assembly errors in the test bench, which lead to the appearance of an interference spectrum in the experimental signal, but it can still be found that the frequency distribution of the model signal is basically consistent with the test through the comparison between the simulation and the test, and the frequency spectrum is mostly distributed around the engagement frequency, thus verifying the accuracy of the digital twin model.The accuracy of the digital twin model is verified, and it provides a theoretical basis for the health monitoring and fault diagnosis of the planetary gearbox system.

Comparative Research
When a gear fault occurs, it produces a significant shock, and the response amplitude of the fault location becomes larger as the fault degree increases.In order to further reflect the advantages of digital twin modeling, the traditional modeling method and digital twin modeling method were compared and analyzed, and the root-mean-square was used as an example to analyze the changes in the root-mean-square index throughout the development of the fault.Taking the evolution of the sun gear missing teeth fault as an example, the missing teeth degradation process was modeled conventionally and digitally twinned, respectively, and the gear parameters and simulation settings were the same as in the previous section.The changes in the statistical indicators of the root-mean-square values under different modeling methods are shown in Figure 15.From Figure 15, it can be seen that the root-mean-square indicator shows an increasing trend with the increase in the fault degree, and the root-mean-square indicator is very sensitive to the fault.The comparison of the two models shows that the root-mean-square indicator of the digital twin model is better than the traditional model and better reflects the fault evolution trend.From Figure 15, it can be seen that the root-mean-square indicator shows an increasing trend with the increase in the fault degree, and the root-mean-square indicator is very sensitive to the fault.The comparison of the two models shows that the root-mean-square indicator of the digital twin model is better than the traditional model and better reflects the fault evolution trend.

Conclusions
In this paper, the digital twin technique was used to investigate the vibration spectrum characteristics of planetary gearboxes.The physical model of the system vibration was constructed and the characteristic frequency of the system fault was calculated taking into consideration that the planetary gearbox system is prone to faults.A digital twin model of the planetary gearbox structure was established with reference to the existing primary planetary gearbox test bench in the laboratory.Firstly, a combined fault model of the sun gear and planetary gear was built in SOLIDWORKS, and then a rigid-flexible coupled multi-body dynamic model was built in ADAMS, which was close to the actual operating state of the physical object.Then, the model was imported into MATLAB, and the electrical model was built using the motor module that comes with Simulink.Finally, each module was connected to form the digital twin model of the planetary gearbox.The sun gear fault frequency f s and planetary gear fault frequency f p appear in the experiment, similarly, the sun gear fault frequency f s and planetary gear failure frequency f p appear in the simulation.Through the comparison of the experiment and simulation results, it was verified that the digital twin model can accurately simulate the motion state of the physical object and still has good accuracy even under multi-fault working conditions.Meanwhile, according to the root-mean-square metrics used to characterize the faults, the root-mean-square characteristic metrics of the digital twin model are better than the root-mean-square index of the traditional model.In the future, based on this model, the predictive maintenance of planetary gear can be studied to provide a solution to the problem of obtaining data during the degradation process of actual gear equipment.At the same time, through the ideas of positive derivation and inverse identification, accurate meshing parameters, transmission errors, and other mechanical parameters during the fault evolution can be obtained, based on which a more realistic digital twin gear degradation model can be established as a substitute for partial experiments.This research can be applied in the aerospace and automotive fields, and in large-scale industrial equipment to provide solutions for fault diagnosis and predictive maintenance in the above fields.

Figure 2 .
Figure 2. Fault model diagram for each included tooth construction: (a) Sun gear missing teeth; teeth; (b) Planetary gear missing teeth; (c) Planetary gear pitting.
below.finally, obtain a rigid-flexible coupled multi-body dynamics model flexible bodies.The rigid-flexible coupled multi-body dynamics mode 3.

Figure 6 .
Figure 6.Digital twin model of planetary gearbox.

Figure 6 .
Figure 6.Digital twin model of planetary gearbox.

Figure 6 .
Figure 6.Digital twin model of planetary gearbox.

Figure 7 .
Figure 7. Normal simulation vibration signal of sun gear and planetary gear: (a) Time domain waveform; (b) Fourier spectrum.

Figure 7 7 +
Figure 7 shows the time domain waveform and the Fourier spectrum of the planetary gearbox vibration signal under normal conditions.The time domain waveform is uniformly distributed as a whole, without any obvious evidence of shock. 1 and 2 in the figure are c m f f 7 − and c m f f 7 + , respectively.It can be seen from the Fourier spectrum that shocks with frequencies of c m nf f ± (n = 1,2,...) appear in the sideband of the meshing cycle, and the sideband is spaced at the planetary frame rotation frequency.These

Figure 7 .
Figure 7. Normal simulation vibration signal of sun gear and planetary gear: (a) Time domain waveform; (b) Fourier spectrum.

Figure 8 .
Figure 8. Simulated vibration signal of sun gear missing teeth and planetary gear pitting: (a) Time domain waveform; (b) Fourier spectrum.

Figure 8 .
Figure 8. Simulated vibration signal of sun gear missing teeth and planetary gear pitting: (a) Time domain waveform; (b) Fourier spectrum.

Figure 9 .
Figure 9. Simulated vibration signal of sun gear missing teeth and planetary gear missing teeth: (a) Time domain waveform; (b) Fourier spectrum.

Figure 9 .
Figure 9. Simulated vibration signal of sun gear missing teeth and planetary gear missing teeth: (a) Time domain waveform; (b) Fourier spectrum.

Figure 10 .
Figure 10.Sun gear and planetary gear missing teeth and planetary gear pitting fault: (a) Sun gear missing teeth; (b) Planetary gear pitting; (c) Planetary gear missing teeth.

Figure 12 Figure 10 .
Figure 12 shows the time domain waveform and Fourier frequencies of the vibration signals under the normal operating conditions of both the sun gear and the planetary gear.The time domain waveform is uniformly distributed as a whole without obvious shocks, and the Fourier spectrum is at the position where the sideband of the meshing cycle

Figure 10 .
Figure 10.Sun gear and planetary gear missing teeth and planetary gear pitting fault: (a) Sun gear missing teeth; (b) Planetary gear pitting; (c) Planetary gear missing teeth.

Figure 12 Figure 11 .
Figure12shows the time domain waveform and Fourier frequencies of the vibration signals under the normal operating conditions of both the sun gear and the planetary gear.The time domain waveform is uniformly distributed as a whole without obvious shocks, and the Fourier spectrum is at the position where the sideband of the meshing cycle

7 +Figure 12 .
Figure 12.Experimental vibration signal of sun gear and planetary gear under normal working conditions: (a) Time domain waveform; (b) Fourier spectrum.

Figure 13
Figure13shows the time domain waveform and the Fourier spectrum of the experimental vibration signal under the combined fault conditions of the sun gear teeth missing and planetary gear pitting.The time domain waveform shows a more obvious shock, and the peak of the whole spectrum is more obvious.There is a modulation effect of the planetary frame on the meshing frequency, so there are peaks at c m nf f ± on both sides of

Figure 12 .
Figure 12.Experimental vibration signal of sun gear and planetary gear under normal working conditions: (a) Time domain waveform; (b) Fourier spectrum.
respectively, and it can be seen that the experi- mental signal is basically consistent with the simulated signal.Therefore, the comprehensive diagnosis is that both the sun gear and the planetary gear are faulty.

Figure 13 .
Figure 13.Vibration signal of sun gear missing teeth and planetary gear pitting experiment: (a) Time domain waveform; (b) Fourier spectrum.

Figure 13 .
Figure 13.Vibration signal of sun gear missing teeth and planetary gear pitting experiment: (a) Time domain waveform; (b) Fourier spectrum.

Figure 14 .
Figure 14.Vibration signal of the sun gear missing teeth and planetary gear missing teeth experiment: (a) Time domain waveform; (b) Fourier spectrum.

Figure 14 .
Figure 14.Vibration signal of the sun gear missing teeth and planetary gear missing teeth experiment: (a) Time domain waveform; (b) Fourier spectrum.

Figure 15 .
Figure 15.Comparison of the two models: (a) Root-mean-square value of the digital twin model; (b) Root-mean-square value of the traditional model.

Figure 15 .
Figure 15.Comparison of the two models: (a) Root-mean-square value of the digital twin model; (b) Root-mean-square value of the traditional model.

Table 1 .
Parameters related to teeth-containing components.

Table 1 .
Parameters related to teeth-containing components.

Table 1 .
Parameters related to teeth-containing components.

Table 2 .
Simulation of planetary gearbox characteristic frequency.

Table 3 .
Gear parameters used in the experiment.

Table 4 .
Experiment of planetary gearbox characteristic frequency m f s f

Table 4 .
Experiment of planetary gearbox characteristic frequency m f s f

Table 3 .
Gear parameters used in the experiment.

Table 4 .
Experiment of planetary gearbox characteristic frequency.