# Dynamic Characteristics Analysis with Multi-Directional Coupling in a TBM Mainframe

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## Abstract

The cutterhead of a full-face rock tunnel boring machine (TBM) is constantly subjected to varying impact and dynamic loads during tunneling processes, resulting in relatively large vibrations that could easily lead to fatigue cracking of the entire machine and affect the tunneling performance and efficiency. To explore the dynamic characteristics of the TBM mainframe, a TBM from a water-diversion project is investigated in this research. According to the TBM vibration transmission route, an equivalent dynamic model of the TBM mainframe is established using the lumped-mass method in which the relevant dynamic parameters are solved. Additionally, the dynamic response characteristics of the TBM mainframe are analyzed. The results indicate that the vibration levels in three directions are approximately the same, the multi-directional vibration of the cutterhead is more intense than that of other components, and the vibration and external excitation exhibit identical change trends. A set of vibration field tests is performed to analyze the in situ dynamic responses of the mainframe and verify the correctness of the dynamic model. The theoretical and measured acceleration values of the TBM mainframe have the same magnitude, which proves the validity of the dynamic model and its solution. The aforementioned results provide an important theoretical value and practical significance for the design and assessment of the TBM mainframe.

## Keywords

Tunnel boring machine Time-varying excitation Dynamic model Load transfer Dynamic response## 1 Introduction

The tunnel boring machine (TBM) is large-scale underground equipment that is widely used in underground tunnel projects owing to high safety and reliability, low manpower requirement, minor environmental damages, and rapid excavation speed [1, 2]. As a key component of the TBM, the mainframe is composed of a cutterhead system, main drive system, shield, and main girder as well as a supporting-thrust system. Because of complicated geological conditions and variable tunneling parameters, a tunnel often features high strength and high confining pressure as well as a high quartz content, along with the step crushing of rocks. In the tunneling process, the TBM cutterhead is subjected to large thrust and torque caused by the interaction between the cutters and rock [3, 4], which may result in significant damages, such as cracking of the cutterhead and loosening or separation of bolts, and affect the fatigue life of the TBM mainframe.

Studies on the TBM mainframe system have been carried out by several researchers, including those on external loads, thrust, torque, overturning torque, and unbalanced radial force in the tunneling process. Relevant models have also been established to conduct dynamic characteristic analyses of the cutterhead system. Rostami [5] analyzed various factors affecting the rock-breaking load of a disc cutter, established the CSM (Colorado School of Mines) force-estimation model for the cutter, and evaluated the performance of the cutterhead from the perspective of disc cutter layouts. Xia et al. [6] studied the formation mechanism and change pattern of the side force suffered by the center cutter and set up a calculation model for predicting the average side force. Huo et al. [7] subdivided the TBM disc cutter into center, inner, and gauge cutters and presented a multi-stage rock fragmentation load prediction model for disc cutter groups based on the dense-core theory and types of disc cutters. Shi et al. [8] put forward a method for calculating the cutterhead torque by taking into account the cutterhead structure, cutting principle, and geological conditions. Additionally, Zhou et al. [9] and Geng et al. [10] studied the factors that influence the thrust and torque of cutterheads and derived a model for calculating the relevant thrust, torque, overturning torque, and other external loads. Xia et al. [11] modified the cutterhead overturning moment calculation model and analyzed the mechanical performance of a typical TBM cutterhead under different working conditions. Zhao et al. [12] analyzed the composition of earth pressure balance (EPB) TBM cutterhead torque and developed a theoretical TBM torque model for a rock–soil interface mixed ground. Because of the difficulties in obtaining the accurate load of the cutterhead using theoretical analysis, test methods have been applied to acquire the load data of the cutterhead and its change trends. A test of external loads was performed by Entacher et al. [13, 14], who presented a method to detect the three-directional loading of a disc cutter in real-time and conducted a field test in the Austrian Koralm Tunnel. Geng et al. [15] proposed an experimental method to investigate the rock-cutting process of a TBM gauge cutter. According to the theoretical models, Zhang et al. [16, 17] analyzed the total load in an EPB TBM tunneling process and established a predictive model for the total load, which reflected the influence of the geological, operating, and structural parameters. Sun et al. [18, 19] and Li et al. [20] established a coupling nonlinear dynamic model of a cutterhead system to analyze the dynamic characteristics of the cutterhead. Huo et al. [21, 22] established multi-degree-of-freedom coupling dynamic models for the disc cutter and cutter system and revealed the dynamic characteristics of the disc cutter and cutter system. In addition, Huo et al. [23, 24] presented multi-coupling dynamic models for the TBM main drive and supporting systems and carried out a field strain test to verify the models. Festa et al. [25] set up a kinematic model for a TBM based on theoretical and geometrical considerations, and the ground displacements were obtained through the model and verified using the TBM monitoring data. In addition, considering the redundant driven rotary system, propulsion system, and geological conditions, Zhang et al. [26] and Huang et al. [27] established a mutual coupling dynamic model for cutterhead tunneling to analyze the dynamic characteristics of the rotary and supporting-propulsion systems under complex geological conditions.

As mentioned before, efforts have been intensified in the research and design of TBM cutterheads. However, profound studies on the TBM mainframe from the perspective of dynamics have not yet been performed. The existing dynamic model has been simplified to merely analyze the data of the main drive without considering the supporting or propelling forces provided by the main girder and gripper shoes in the mainframe. As a result, the transmission path is not closed. Additionally, differences in the frequency spectrum characteristics exist between the simulation-predicted dynamic and real external loads. Therefore, to investigate the dynamic characteristics of the TBM mainframe, a dynamic model of a TBM mainframe is established, which comprehensively considers the time-varying external excitations, transmission route of the vibration, and three-dimensional forces obtained from rock-breaking experiments. In addition, a set of field tests is carried out to collect field data and validate the correctness of the dynamic model. The present study provides a theoretical basis for evaluating and designing the TBM mainframe structure that can effectively prevent cracking, reduce the vibrations of the TBM mainframe, and prolong the fatigue life of the main bearing as well as improve tunneling efficiency.

## 2 Dynamic Model of TBM Mainframe

*F*

_{z}and

*T*are the thrust force in the tunneling direction and total torque, respectively, and

*M*,

*F*

_{x}, and

*F*

_{y}represent the overturning moment, lateral unbalanced force, and vertical unbalanced force, respectively. Total thrust

*F*

_{t}is provided by the propelling hydraulic cylinders to move the TBM forward. The cutterhead, main drive, and first section of the main girder are mainly backed by the bottom shield, which are tightly attached to the rock, thus bearing the friction resistance force

*F*

_{f}caused by gravity

*G*.

The transmission path of the mainframe external excitations and vibrations is opposite that of the TBM tunneling thrust. The three-directional impact loads of the cutterhead are created by the disc cutters that break the rock during TBM tunneling. Through a connection flange in the cutterhead, the load and vibration are transmitted from the cutterhead to the main bearing and subsequently to the other parts behind the main drive. Then, through the main girder and propelling hydraulic cylinders at both sides of the main girder, the load and vibration are transmitted to the saddle holder of the gripper shoes that grip the rock, providing a forward reaction force and reducing the vibration of the gripper shoes.

Figure 2 shows a dynamic model of the lateral and vertical vibrations of the mainframe. Figure 3 shows a dynamic model that represents the coupling relationship between the tunneling and overturning vibrations in the mainframe. When the overturning vibration is calculated, the connection relationships between the cutterhead and the main bearing are simplified into four spring-damping systems. The whole overturning stiffness of the cutterhead is equal to the tunneling joint stiffness. In the dynamic model, *x*, *y*, and *z* denote the lateral, vertical, and tunneling directions, respectively. *m*_{1}, *m*_{2}, *m*_{3}, *m*_{41}, *m*_{42}, *m*_{43}, and *m*_{5} represent the masses of the cutterhead, main bearing, main drive (including the shield), main girder 1, main girder 2, main girder 3, and gripper shoes, respectively. *x*_{1}, *x*_{2}, *x*_{3}, *x*_{41}, *x*_{42}, *x*_{43}, and *x*_{5} represent the lateral degrees of freedom of the cutterhead, main bearing, main drive, main girder 1, main girder 2, main girder 3, and gripper shoes respectively. *y*, *z*, and *θ* denote the corresponding vertical, tunneling, and overturning degrees of freedom respectively. *k*_{x1}, *k*_{x2}, *k*_{x3}, *k*_{x41}, *k*_{x42}, *k*_{x43}, *k*_{x5}, *k*_{y1}, *k*_{y2}, *k*_{y3}, *k*_{y41}, *k*_{y42}, *k*_{y43}, and *k*_{y5} denote the lateral and vertical equivalent stiffness values of the cutterhead, main bearing, main drive, main girder 1, main girder 2, main girder 3, and gripper shoes, whereas *c* represents the corresponding damping.

*I*

_{1x}and

*I*

_{1y}are the lateral and vertical moments of inertia, respectively.

*k*

_{zx11},

*k*

_{zx12},

*k*

_{zy11}, and

*k*

_{zy12}are the tunneling equivalent stiffness values of the four pieces of cutterhead.

Coupling relationships exist between the tunneling and overturning vibrations. The relationships between the displacement of each cutterhead pieces and that of the cutterhead tunneling vibration are expressed in Eq. (4):

*z*

_{x11},

*z*

_{x12},

*z*

_{y11}, and

*z*

_{y12}are the displacements of each cutterhead piece,

*z*

_{1}is the axial vibration displacement of the cutterhead,

*θ*

_{x1}and

*θ*

_{y1}are the overturning angles of the cutterhead, and

*a*

_{L}is the radius of the cutterhead.

*X*is the displacement vector,

*is the mass matrix,*

**M***is the stiffness matrix,*

**K***is the damping matrix, and*

**C***F*is the external excitation.

## 3 Solution for Dynamic Parameters

### 3.1 External Excitations of the TBM Cutterhead

Based on the radius, position angle, and tilt angle of the cutter as well as the cutter loads in three directions, the total thrust, total torque and change trend of the radial unbalanced force of the cutterhead can be obtained, which can be presented as time-varying external excitations.

### 3.2 Calculation of Equivalent Stiffness and Damping

*L*

_{we}represents the effective length of the rollers and

*Q*denotes the contact load.

*α*is the viscosity pressure coefficient of the lubricant,

*η*

_{0}is the dynamic viscosity of the lubricant,

*u*is the average speed,

*R*is the equivalent curvature radius of the contact area,

*E*is the elastic modulus, and

*E*

_{0}is the equivalent elastic modulus.

*k*

_{H}denotes the Hertz contact stiffness and

*k*

_{L}denotes the contact stiffness of the oil film.

According to the elastic fluid dynamics, the contact damping between the rollers and roller raceway consists of the roller structural and oil-film damping. Considering the effect of the oil-film thickness, the main-bearing damping coefficient is calculated using the following equation [30].

*Z*is the number of rollers,

*n*

_{i}is the rotating speed of the raceway,

*r*is the radius of the roller, and

*R*

_{1}is the radius of the inner raceway.

*c*is the structural damping,

*ζ*is the damping ratio, and

*m*

_{e}and

*k*

_{e}represent the equivalent mass and equivalent stiffness, respectively.

Partial stiffness and damping parameters of mainframe

Cutterhead | Main drive | Main girder 1 | Gripper shoes | |
---|---|---|---|---|

Lateral stiffness (N m) | 1.15 × 10 | 5 × 10 | 6.38 × 10 | 8.68 × 10 |

Overturning stiffness (N m) | 1.56 × 10 | 6.16 × 10 | 5.7 × 10 | 2.11 × 10 |

Tunneling stiffness (N m) | 4.72 × 10 | 2.5 × 10 | 1.8 × 10 | 3.36 × 10 |

Lateral damping (N s/m) | 1.75 × 10 | 1.04 × 10 | 1.47 × 10 | 1.84 × 10 |

Lateral damping (N s/m) | 3.55 × 10 | 7.35 × 10 | 7.9 × 10 | 1.14 × 10 |

Lateral damping (N s/m) | 4.64 × 10 | 3.91 × 10 | 4.36 × 10 | 5.5 × 10 |

## 4 Dynamic Responses of TBM Mainframe

Because the vertical vibration response is similar to the lateral vibration response, this study only analyzes the vibration responses in the lateral, tunneling, and overturning directions of the cutterhead and main girder 1.

### 4.1 Vibration Response in Tunneling Direction

*g*to + 3.5

*g*with maximum acceleration of more than 7

*g*, demonstrating an intense vibration. When the vibration is transmitted to main girder 1, the acceleration changes to the range from − 2.8

*g*to 2.8

*g*with a peak of 5.9

*g*. The frequency responses of the TBM mainframe in the tunneling direction are shown in Figure 8. The vibration energy of the cutterhead is mainly in the frequency range of 0–130 Hz, and the vibration energy of the main girder is mainly concentrated in the frequency range of 0–70 Hz.

### 4.2 Vibration Responses in Overturning Direction

^{2}with the cutterhead rotation. Overall variation periods of angular acceleration and overturning angle are equal to the period of cutterhead rotation. Frequency responses of the mainframe in the overturning direction are shown in Figure 10. The vibration energy occurs mainly in the frequency range of 0–130 Hz.

### 4.3 Vibration Responses in Lateral Direction

*g*to + 1.4

*g*with a maximum of more than 1.45

*g*. The acceleration of the main girder is slightly lower than that of the cutterhead, which can be attributed to the restraint effect of the bottom support shield. The acceleration of the main girder ranges from − 1.0

*g*to + 1.0

*g*with a maximum of 1.25

*g*. The frequency spectrum of the lateral vibration is shown in Figure 12. Vibration energy of the cutterhead is distributed in the frequency range of 0–130 Hz, whereas that of the main girder is distributed in the range of 0–100 Hz.

### 4.4 Transfer Law of Loads

From the abovementioned results, the following conclusions can be drawn. Parallel conditions exist between the vibration responses of the cutterhead and main girder. The cutterhead suffers from more intense vibrations with the frequencies concentrated at approximately 0–130 Hz, whereas the frequencies of the main girder vibration are mainly concentrated at approximately 0–100 Hz. The farther the position is from the cutterhead, the lower are the vibration frequencies. The vibrations in the tunneling direction are more intense than those in the lateral direction. In the load-transfer process, the mean loads remain the same, whereas the change range of the loads expands.

## 5 Validation of Dynamic Model Based on Field Tests

Considering the poor working condition of the cutterhead, acceleration sensors were attached to the manhole of the cutterhead, i.e., point A in Figure 14. The test points in the main drive were located around the B point on the main girder connecting plate, and the test points for the main girder were located on the C, D, and E point positions. The vibration signals in the manhole were transferred in a wireless manner, whereas those of the other test points were directly obtained through the transmission lines. At each test point, three acceleration sensors for the lateral, vertical, and tunneling directions were installed.

### 5.1 Vibration Test of Cutterhead

*g*to + 2

*g*. The mean-square deviation of the lateral-test acceleration is 1.01

*g*, whereas the calculated value is 0.61

*g*. The measured values of the vibration acceleration in the tunneling direction are lower than the calculated values, which mainly range from − 3

*g*to + 3

*g*. The mean-square deviation of the measured tunneling acceleration is 1.92

*g*, and that of the calculated acceleration is 2.1

*g*.

### 5.2 Vibration Test of Main Girder

*g*to + 2

*g*. The mean-square deviation of the lateral measured acceleration is 0.86

*g*, and that of the calculated value is 0.52

*g*. In the tunneling direction, the test values of the vibration acceleration are lower than the calculated values, which mainly range between − 2.3

*g*and 2.3

*g*with the maximum value reaching 5

*g*. Mean-square deviation of the measured tunneling acceleration is 1.51

*g*, and that of the calculated acceleration is 1.72

*g*.

Comparison of the field-test and calculated values of the acceleration in the TBM mainframe reveals that the calculated values of the lateral and vertical vibration acceleration are lower than the test values, whereas the calculated values of the vibration acceleration in the tunneling direction is larger than the measured values. The reasons for the differences in the tunneling and other directions can be attributed to the complex internal structure of each component in the mainframe and the slant stiffened plate of the cutterhead. In the transmission process, partial vibrations in the tunneling direction can be transmuted into other vibrations such as the lateral and vertical vibrations due to the slant stiffened plate and complex internal structure of the mainframe.

The calculated and measured acceleration values of the TBM mainframe have the same magnitude within a relatively acceptable error range. The major error comes from the simplification of the complex internal structure of the TBM mainframe coupled with errors in the calculations of the equivalent stiffness and damping.

## 6 Conclusions

- (1)
In the tunneling process, the lateral, vertical, and tunneling vibrations of the TBM mainframe have the same vibration level in which the changes are similar to those of the external excitations.

- (2)
Parallel conditions exist between the vibration responses of the cutterhead and main girder. The multi-directional vibration of the cutterhead is more intense than that of the main drive, main bearing, and main girder. The vibrations in the tunneling direction are more intense than those in the lateral direction. The frequencies of the cutterhead vibration energy are predominantly concentrated at approximately 0–130 Hz, and those of the main girder vibration energy are mainly concentrated at approximately 0–100 Hz. As the distance from the cutterhead increases, the vibration energy is concentrated in a lower frequency. During the design of the TBM mainframe, measurement of the vibration reduction needs to be considered to prevent fatigue damage and breakdown of the cutterhead and damage to the main bearing.

- (3)
In the process of mainframe load transfer from the tunneling direction, the average load and change trend of the loads remain constant, whereas the range of the loads expands by 1.4 times with the increase in the load amplitude. As a result, when the strength of the components in the back end of the mainframe is designed and checked, the external loads from the cutterhead need to be amplified.

- (4)
The lateral measured acceleration of the TBM mainframe is concentrated in the range from − 2

*g*to + 2*g*, and the measured acceleration in the tunneling direction is mainly concentrated in the range from − 3*g*to + 3*g*. Comparison of the theoretical values with the actual test values from three directions in the mainframe reveals that the measured and calculated vibrations are basically the same within a relatively acceptable range of error. The results have verified the correctness of the modeling and the solution of the dynamic model, which confirm the weakest components of TBM mainframe and provide a theoretical basis for the design and evaluation of the mainframe as well as the external loads for mainframe fatigue-life analysis.

## Notes

### Authors’ Contributions

LL and YX were in charge of the whole research; LL analyzed results and wrote the manuscript; ZL assisted with sampling and field test. CW, YC and QT assisted with modeling, data analyzing and manuscript writing. All authors read and approved the final manuscript.

### Authors’ Information

Laikuang Lin, born in 1988, is currently a postdoc at *College of Mechanical and Electrical Engineering, Central South University, China*. He received his PhD degree from *Central South University, China*, in 2017. His research interests include dynamics, and design theory of TBM cutterhead.

Yimin Xia, born in 1967, is currently a professor at *College of Mechanical and Electrical Engineering, Central South University, China*. He received his PhD degree from *Central South University, China,* in 2006. His research interests include hydraulic transmission and control, mechanical engineering.

Zhengguang Li, born in 1994, received his master degree from *Central South University, China*, in 2018. His current research interests include dynamics, transmission and control technology.

Caizhang Wu, born in 1990, received his master degree from *Central South University, China*, in 2016. His research interests include dynamics and mechanical design.

Yongliang Cheng, born in 1978, is currently a PhD candidate at *Central South University, China*. His research interests include design theory of TBM.

Qing Tan, born in 1955, is currently a professor at *Central South University, China*. His research interests include dynamics, and rock breaking mechanism of disc cutters.

### Competing Interests

The authors declare that they have no competing interests.

### Funding

Supported by National Key R&D Program of China (Grant No. 2017YFB1302603), National Natural Science Foundation of China (Grant No. 51905550), National Basic Research Program of China (Grant No. 2013CB035401), and China Postdoctoral Science Foundation (Grant No. 2019M652795).

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