Modeling and Dynamic Analysis of Cutterhead Driving System in Tunnel Boring Machine

Failure of cutterhead driving system (CDS) of tunnel boring machine (TBM) often occurs under shock and vibration conditions. To investigate the dynamic characteristics and reduce system vibration further, an electromechanical coupling model of CDS is established which includes the model of direct torque control (DTC) system for three-phase asynchronous motor and purely torsional dynamic model of multistage gear transmission system. The proposed DTC model can provide driving torque just as the practical inverter motor operates so that the influence of motor operating behavior will not be erroneously estimated. Moreover, nonlinear gear meshing factors, such as time-variant mesh stiffness and transmission error, are involved in the dynamic model. Based on the established nonlinear model of CDS, vibration modes can be classified into three types, that is, rigid motion mode, rotational vibration mode, and planet vibration mode. Moreover, dynamic responses under actual driving torque and idealized equivalent torque are compared,which reveals that the ripple of actual driving torquewould aggravate vibration of gear transmission system. Influence index of torque ripple is proposed to show that vibration of system increases with torque ripple. This study provides useful guideline for antivibration design and motor control of CDS in TBM.


Introduction
Tunnel boring machine (TBM) is a large and high-tech construction equipment which is widely used in transport, municipal, and water diversion projects due to its advantages such as highly integrated functions, high tunneling speed, and environment-friendly construction [1].Cutterhead is a key component of TBM, which can crush and cut rock with disc cutters mounted on its panel.As shown in Figure 1, cutterhead driving system (CDS) is a complex electromechanical coupling system, which is mainly composed of inverter motor, planetary reducer, pinion, and ring gear.
During tunneling process, driving torque provided by multiple inverter motor is transported from planetary reducer and pinions to ring gear, which is fixedly connected with cutterhead.Under the enlarged driving torque, cutterhead rotates and breaks rock.In CDS, variable frequency speed control system such as vector control (VC) and direct torque control (DTC) system are often applied to make inverter motor respond quickly to the variable load.
Due to complex and changing geological conditions, cutterhead and its driving system often suffer large impact load with drastic fluctuation during tunnel construction [2].In addition, the multistage gear transmission system is a time varying and strong coupling system which causes periodic internal excitation owning to the nonlinear factors such as time-variant mesh stiffness, transmission error, and backlash [3].Under such internal excitation formed in gear transmission system and external excitation caused by geological condition and inverter motor, server vibration in CDS often occurs and results in failures such as excessive wear, breakage of gear tooth and shaft, imbalance of driving torque, and so on [4].To resolve these problems and improve the design of CDS, the dynamic characteristics of gear transmission system and the operating characters of inverter motor ought to be investigated primarily and imperatively.In recent years, a large amount of work has been done on load sharing and vibration reduction of CDS in TBM.Numerous researches have focused on dynamic analysis of CDS.Wei et al. established a dynamic model of multigear driving system and studied the effects of inertia on loadsharing characteristic [5,6].Sun et al. established a dynamic model of cutterhead driving system based on hierarchical modeling method and obtained dynamic response [7].Zhang et al. analyzed dynamic characteristic of TBM in mixedface conditions [8].Qin and Zhao built multiobjective optimization model based on dynamic analysis and optimized parameters of gear transmission system to reduce vibration [9].Besides, multimotor synchronization control method of CDS also attracts more and more attentions.Liu et al. studied load-sharing characteristic of multiple motors and proposed an adaptable control approach to improve the compliance ability of CDS [10][11][12].All these researches have made fruitful efforts on design of CDS through dynamic analysis and optimization of multimotor control strategy.However, these studies did not consider the influence of external excitation provided by inverter motor, which just replaced the actual driving torque with an idealized constant value.A number of researches have shown that torque ripple caused by variable frequency speed control system is an inevitable factor which may influence the dynamic performance of transmission mechanism [13].Without considering the ripple of actual driving torque, dynamic analysis of gear transmission may be erroneously investigated.Therefore, it needs to take operating characters of inverter motor into account for building the electromechanical coupling model and studying the dynamic characteristic of multistage gear transmission system in CDS.
In this paper, dynamic analysis of CDS in TBM is studied to explore the failure reasons of key components.An electromechanical coupling model of CDS is established which includes a dynamic model of DTC driving system and a purely torsional dynamic model of multistage gear transmission system.By taking the nonlinear factors of gear meshing and the operating characters of inverter motor into account, dynamic characteristics of multistage gear transmission system under the actual driving torque are analyzed.It provides data support for gear antivibration design and motor control of CDS in TBM.

Mathematical Modeling of TBM Cutterhead Driving System
2.1.Dynamic Model of DTC Driving System.DTC system is particularly applied to CDS with large inertia which needs rapid torque response.Based on Bang-Bang control method, DTC system regulates stator flux and provides heavy starting torque for CDS.In DTC driving system, - phase static coordinate system is chosen as the reference frame of mathematical model of three-phase asynchronous motor, and hence the voltage equation can be expressed as follows: Flux Equation Torque Equation where  2. The u-i model is chosen as the stator flux observer which can be expressed as follows:    According to (3)-( 4), torque and stator flux observer model is established as shown in Figure 3.In DTC system, the amplitude of stator flux   is kept constant and the angle of stator flux   is regulated to control the electromagnet torque as shown in Figure 4.The asynchronous motor is controlled by switch status of voltage space vector in inverter.Driving signals are selected from the optimal switching table after directly calculating stator flux and torque.The location of stator flux in - phase static coordinate system can be calculated by comparing the observed values of   and   with the given value of  *  and  *  .Based on the model of DTC driving system, frequency control process of inverter motor can be simulated and electromagnet torque   can be obtained to drive the multistage gear transmission system.
Based on the lumped mass method, a purely torsional dynamic model of multistage gear transmission system is established.Each component is regarded as a rigid body.The direction of displacement along the meshing line is supposed to be positive when the tooth surface is under pressure.Based on Newton's Second Law, the equivalent mathematic model of the multistage gear transmission system can be expressed as follows:
,   , and   are mesh dampings which can be expressed as follows: where  is gear mesh damping ratio ( = 0.03-0.17)and   and   are masses of two meshing gears.As shown in Figure 7, transmission error   is approximated as superposition of harmonic function of mesh frequency and rotation frequency of shaft [15].where   is total cumulative pitch error;    is tangential tolerance of single tooth;   and   are rotation frequency and mesh frequency; and   and   are initial phase of shaft and mesh phase.

Dynamic Analysis of Electromechanical
Coupling Model of CDS  1.
According to these parameters, the model of three-phase asynchronous motor is chosen as Table 2 shows and the control parameters of DTC system are set.In this paper, the multiple inverter motors are supposed to be synchronous and TBM cutterhead is chosen to work under the rotational speed   = 2.1 rpm.Thus, load torque of motor can be calculated based on the mean value of load torque on cutterhead which can be expressed as (10) shows.
where   is rated power;  I is gear ratio of reducer;  II is gear ratio of ring-pinion gears;   is rated speed of cutterhead; and  is number of pinions.Field test data of external load torque is shown in Figure 9.In actual tunneling process, load torque   is unstable and changes abruptly as geological condition varies.On the basis of (10), rated   is 1120 N⋅m under rated rotational speed   = 2.1 rpm, which corresponds to the actual   near 310 s in Figure 8.Thus, taking a 1 s-length (of) actual   between 314.2 s and 315.2 s as an example,   in the first 0.2 s keeps stable near rated torque and then rises sharply to 1700 N⋅m at 314.4 s.After 314.5 s,   remains roughly stable near 1700 N⋅m with little fluctuations.To study the operating characters of inverter motor under shocking load, the 1 s-length of   between 314.2 s and 315.2 s is chosen to be simulated as a piecewise function.In DTC driving system, load torque   is simulated for 2 s.   is set to be 1100 N⋅m before 1.35 s and   is equal to 1700 N⋅m during 1.35 s and 2 s.
The actual driving torque of DTC system is obtained and shown in Figure 9.In the start-up phase, inverter motor operates with the maximum torque to accelerate to the rated speed quickly.After operating for 1 s, electromagnetic torque   fits the actual load torque under rated speed.The fitting result shows that DTC driving system responds quickly according to the changing load torque   .However, electromagnetic torque   has high torque ripple which is about 120 N⋅m, which can be expressed in discrete form as follows [16]:   where  (+1)  and  ()  are electromagnetic torques at  + 1 and  moment; Δ ()  1 is torque attenuation caused by stator and rotor resistance; Δ ()  2 is torque variation caused by voltage space vector;   is sampling time;  is constant which is related to   ,   , and   ; and   is speed of rotor.
Based on (11), torque ripple is inevitable and influenced by sampling time, motor speed, flux, and voltage vector which are closely related to computing power of digital controller and switching frequency [17].Therefore, as the external excitation of gear transmission system, torque ripple of electromagnetic torque   may be higher in actual motor driving process and influence the dynamic characteristics of gear transmission system.

Modal Property of Multistage Gear Transmission System.
In multistage gear transmission system, one-stage pinionring gears consist of several pinions  1 and one ring gear  2 .The size of ring gear  2 is much bigger than other gears and the inherent properties of planetary reducer cannot be clearly presented under the influence of ring gear  2 .Therefore, the modal properties of planetary reducer are chosen to be analyzed in this paper.
Based on ( 5), equivalent mathematic model of planetary reducer can be expressed in the form of matrix:  q () +  q () +  () =  () , (12) where () is vibration displacement vector;  is mass matrix;  is damping matrix;  is stiffness matrix; and () is excitation vector.Since the variation range of mesh stiffness is not big, mesh stiffness is simplified as average stiffness.In the same stage, all external mesh stiffness and all internal mesh stiffness are the same separately.The influence of damping is also ignored to obtain the natural frequencies.Thus, the eigenvalue problem of ( 12) can be expressed as follows: where   is i-order natural frequency;  is average stiffness matrix; and   is i-order vibration mode vector as (1)   ,  (1)  1 ,  (1)  2 ,  (1)  3 ,  (1)   ,  (2)   ,  (2)  1 ,  (2)  2 ,  (2)  3 ,  (2)  4 ,  (2)   ,  (3)   ,  (3)  1 ,  (3)  2 ,  (3)  3 ,  (3)  4 ,  (3)   ] . ( According to the main parameters of planetary reducer listed in Table 3, natural frequencies and vibration modes can be obtained by solving (13).Natural frequencies are listed in Table 4 and vibration modes are shown in Figure 10.Based on the inherent properties, planetary reducer operates in three types of vibration modes: rigid motion mode, rotational vibration mode, and planet vibration mode.In rigid motion mode, natural frequency  1 = 0Hz and all components just operate on the basis of transmission ratio without vibration.In rotational vibration mode, natural frequencies f are distinct and f ̸ = 0 Hz.All components have rotational vibration and planet gears in each stage operate with the same vibration.In planet vibration mode, natural frequencies  5 =  6 =  7 = 3805 Hz,  10 =  11 =  12 = 5266 Hz, and  14 =  15 = 7056 Hz.All central components such as sun gears and planet carriers have no vibration except planet gears.

Vibration Displacement.
Vibration displacement is one of the most important elements in dynamic response, which denotes the vibration degree of gear transmission system.Based on the parameters listed in Tables 1, 2, and 3, vibration displacement can be obtained by solving the electromechanical coupling model.As shown and discussed above, torque ripple of inverter motor is unavoidable and may influence the dynamic response of gear transmission system.Therefore, vibration displacements under electromagnetic torque   with ripple and idealized piecewise torque without ripple are calculated separately.
To ensure the accuracy of results and spare calculation time, Runge-Kutta integration method is chosen to solve the equivalent mathematic model in 1 s.Dynamic responses of sun gears are taken as an example.Vibration displacements of sun gear in each stage are shown in Figure 11.Sun gears vibrate near the equilibrium position and vibration amplitudes decrease as driving torque rises.Vibration amplitude of 2ndstage sun gear is the smallest and significantly smaller than the amplitudes of other sun gears which are approximately equal.Therefore, in the antivibration design process of 3stage gear transmission system in CDS, 1st-stage and 3rdstage gears should be the primary design targets.
For a comparison of dynamic responses under two kinds of driving torque,   herein is defined as the vibration displacement of sun gear under electromagnetic torque   and   herein is defined as the vibration displacement of sun gear under idealized piecewise torque.In the case of 1ststage sun gear for 0.35 s and 1 s, mean values of   and   are the same and equal to 0.0286, which means that actual driving torque of inverter motor has no effect on equilibrium position.However, standard deviation of   is 0.0092 and standard deviation of   is 0.0045, which indicates that the vibration amplitude under electromagnetic torque   is bigger than the one under idealized piecewise torque.Thus, it is tempting to conclude that the actual driving torque of inverter motor may aggravate vibration of gear transmission system owing to the torque ripple.

Dynamic Meshing
Force.Dynamic meshing force directly influences the failure of gear transmission system such as wear or pitting of gear teeth.Meshing force can be expressed based on (1) as follows: where   and   are external/internal meshing forces;   and   are time-variant mesh stiffnesses;   is displacement along the meshing line between the sun gear and each planet gear; and   is displacement along the meshing line between the ring gear and each planet gear.
Under the external excitation of electromagnetic torque   , dynamic meshing forces in each stage are calculated and a part of them are shown in Figures 13 and 14.In time domain, external meshing forces increase abruptly as electromagnetic torque   changes at 0.35 s and meshing forces increase by stage according to gear ratio.Meshing forces of 1st-stage planet gears fluctuate more apparently than the other two stages at changing point which can be probably attributed to the fact that 1st-stage sun gear is directly under the influence of external excitation.In the same stage, meshing forces of planet gears are also different from each other.As shown in Figure 12, load-sharing level of 3rd stage is the highest and load-sharing level of 1st stage is the lowest, which may be caused by phase difference of mesh stiffness and transmission error.
Spectral analysis of external meshing force in each stage is shown in Figure 13.Herein,   ( = 2, 3) donates the i-order natural frequency, and   ( = 1, 2, 3) donates the j-stage mesh frequency.As shown in Figure 13, meshing forces in each stage vibrate in the low frequency domain which is near   and its multiple frequencies.Furthermore, low-order natural frequency ( 2 = 308,  3 = 529) also exist in the internal excitations and  2 possesses the largest amplitude.

Further Discussion
As shown in Figure 11, vibration of gear transmission system is increased under electromagnetic torque   compared with idealized driving torque.The increases of vibration on each component may be related to electromagnetic torque   and its torque ripple.To assess the impact of electromagnetic torque   on each component's vibration, an influence index  of torque ripple is proposed based on the vibration displacements as (16) expresses: where   and   denote the deviation value from equilibrium position under electromagnetic torque   and idealized torque, respectively;   max is the maximum of   which represents vibration degree; and   and   can be expressed as follows: where   is the vibration displacement of one component under electromagnetic torque   and idealized torque and   is mean value of   which represents equilibrium position.Influence index  of torque ripple on all the components is calculated under actual driving torque shown in Figure 9.  on different components in the same stage are shown and compared in Figure 14. on the same component in different stages are shown and compared in Figure 15.In time domain,  on all the components are greater than zero, which means vibrations of all the components are aggravated by torque ripple of electromagnetic torque   .To each component,  increases as load torque   changes from 1100 N⋅m to 1700 N⋅m.In the same stage,  on sun gear is the largest and the impact of electromagnetic torque   on planet carrier is the smallest.In different stages,  on sun gear in 2nd stage is the smallest and the impacts on sun gears in 1st stage and 3rd stage are similar.Thus, as an important performance measure, the influence index  on sun gear in 1st stage or 3rd stage can be taken as the optimization objective to minimize the impact of torque ripple.
To study the impact of torque ripple on vibration further, a series of electromagnetic torque   with different torque ripples are simulated as load torque   is 1700 N⋅m and dynamic responses under such torques are obtained.Maximum of influence index  max is chosen to represent the overall impact of electromagnetic torque   with different torque ripples and  max on all components are shown in Figure 16.It can be seen that vibration degrees of all the components are aggravated more severely as torque ripple increases and tendencies of the impact on each component are similar.Therefore, the ripple of electromagnetic torque   should be controlled to be as small as possible.As shown and discussed above, torque ripple is influenced by several parameters.Since the asynchronous motor is chosen according to the tunneling conditions, parameters of motor are fixed and cannot be adjusted.Thus, in the process of optimizing control method of inverter motor, torque ripple should be reduced by rectifying parameters of speed controller in DTC system.Furthermore, on the premise of meeting tunneling requirements, motor speed can be reasonably controlled to minimize the torque ripple.

Conclusion
In this paper, an electromechanical coupling model of TBM cutterhead driving system (CDS) is established which includes a simulation model of direct torque control (DTC) driving system and a purely torsional dynamic model of multistage gear transmission system.Based on this coupling model, electromagnetic torque   is obtained and torque ripple is analyzed.Modal property and dynamic response of gear transmission system are calculated and the impact of torque ripple on vibration is analyzed.The specific conclusions of this study are as follows.
(1) DTC driving system responds quickly as load torque changes and electromagnetic torque   has high torque ripple which is about 120 N⋅m.Torque ripple is influenced by sampling time, motor speed, flux, and voltage vector which are closely related to computing power of digital controller and switching frequency.
(2) Based on the dynamic analysis of gear transmission system, vibration modes of transmission system can be classified into three types: rigid motion mode, rotational vibration mode, and planet vibration mode.For a comparison of vibration displacements, the vibration amplitude of 2ndstage component is the smallest among all the three stages.Meshing forces mainly vibrate in the low frequency domain which approaches to mesh frequency and low-order natural frequency ( 2 = 308 Hz,  3 = 529 Hz).Moreover, meshing forces increase by stage according to gear ratio and meshing forces of 1st-stage planet gears fluctuate more apparently than the other two stages at changing point of load torque.
(3) Compared with the dynamic responses under idealized piecewise torque, vibration displacements of gear transmission system under electromagnetic torque   are aggravated owing to the torque ripple.Dynamic index  is proposed and discussed to show the impact of electromagnetic torque   .In the same stage,  of sun gear is  ripple increases.Thus, torque ripple should be minimized by optimizing the control method of inverter motor.

Figure 2 :
Figure 2: Direct torque control system of three-phase asynchronous motor.

Figure 4 :
Figure 4: Control principle of DTC system.

Figure 5 :
Figure 5: Purely torsional dynamic model of multistage gear transmission system.

Figure 8 :Figure 9 :
Figure 8: Field test data of external load torque.

Figure 11 :
Figure 11: Dynamic response of sun gears.

Figure 12 :
Figure 12: External meshing force in time domain.

Table 1 :
Technical parameters of TBM cutterhead driving system.

Table 2 :
Parameters of three-phase asynchronous motor.