Mechanical characteristics of a tunnel boring machine cutterhead during rock breaking: Physical model tests and transient dynamic analysis

During tunnel excavation using a tunnel boring machine (TBM), the interaction between the cutter and the rock causes the cutterhead to generate abrupt vibrations, which could lead to engineering problems, such as bearing seal failure and even cutterhead cracking. In this study, large-scale physical model tests with various cutter spacings were performed to study the vibration and load characteristics of the cutterhead during the rock-breaking process. Through a TBM modal comprehensive test bench, the real-time acceleration of the cutterhead was obtained using a monitoring system. The test results showed that the complete time-domain curves were stepped and periodic. When the cutterhead was stuck or rock peeled off during the excavation, the monitoring curves showed abrupt vibrations, which were mainly concentrated at 0–3 Hz and were typical low-frequency vibrations. Taking measured data as the driving parameter, the real-time mechanical characteristics of the cutterhead were determined through transient dynamic analysis. Numerical simulation results showed that the vibration trend during excavation was the superposition of random vibrations generated by rock breaking and tunneling without rock cutting. During the tunneling process, the wear of the edge cutters should be highlighted in actual construction. The related methods and results can provide an essential reference for the selection of cutterheads.


Introduction
With increasing speed in transportation networks, tunnel boring machines (TBMs) have become necessary pieces of equipment for performing tunnel excavation due to their safety and reliability. [1][2][3] Generally, the cutterhead is the most important part of the TBM. After determining the construction environment, several factors should be considered in selecting the cutterhead, such as its structural form, the opening ratio, the drive form, the rotational speed, the torque, the excavation diameter, and the configuration of the cutter. Specifically, the interaction between the cutter and the rock causes the cutterhead to generate continued vibrations borne by the main bearing of the cutterhead. These vibrations could lead to engineering problems, such as bearing seal failure and even cutterhead cracking. [4][5][6] The selection of cutter spacing is one of the critical factors determining the efficiency of rock breaking and the vibration of the cutterhead. According to the vibration law of the cutterhead under various cutter spacings, the optimal cutter spacing can be determined, rock-breaking efficiency can be improved, and wear caused by the system resonance can be further reduced. [7][8][9] Notably, the specific threshold value should be further determined with the actual working conditions and the natural frequency of the TBM.
Existing studies mainly focus on the corresponding mechanism of rocks when a TBM passes through complex geological conditions. [10][11][12] There are few studies on the load characteristics of the cutterhead in combination with the vibration of the cutterhead caused by the rock breaking of the cutter. 13,14 Current cutter spacing research has studied the derivation and verification of the optimal cutter spacing formula or numerical simulation. [15][16][17] However, due to the limited test conditions, there is still a large gap between rock-breaking tests and actual working conditions. [18][19][20] The TBM modal comprehensive test bench used in this study was mainly composed of a hydraulic pump station, tunneling device, rotating device, screw conveyor, and related control system ( Figure 1). This test bench could be used to carry out experiments on various rock samples with different cutter materials, cutter spacing arrangements, and different rock-cutting speeds under vertical and horizontal conditions. The physical model tests of rock breaking under various cutter spacings performed in this study can play a pioneering role in promoting the indoor model test of TBM.
A monitoring system was added to the test bench that can obtain the three-direction acceleration of the cutterhead in real time. 21 The measured data from the tests were close to the actual construction status. Based on these large-scale mechanical model tests and the numerical simulations, this study will further explore the dynamic characteristics of the cutterhead under various cutter spacings.

Setting of various cutter spacings and sensor layout
The calculation equation of TBM disk cutter spacing proposed by Tan et al. 22 is as follows: where s is the cutter spacing (mm) and p is the penetration depth (mm). Combined with the specific cutterhead of the TBM model comprehensive test bench, three typical cutter spacings of 80, 90, and 100 mm were selected. The numerical simulation models of these three cutter spacings are shown in Figure 2.
Data transmission in the test system involves three parts ( Figure 3). Specifically, the wireless acceleration sensor of the cutterhead can be fixed on the cutterhead by a magnet, and its test data were transmitted to the processing software through a wireless gateway. The acceleration sensor was arranged on the frame of the cutterhead. The selection of the sensor position was determined using static analysis, referring to a previous study. 21 Under various cutter spacings, the placement of the sensors was the same, as shown in Figure 2 (yellow rectangles).

Data analysis under various cutter spacings
After 28 days of curing with cement mortar in the rock disk, its uniaxial compressive strength was 32 MPa, as measured by a rebound hammer. The speed of the cutterhead was 1 r/min, and the advancing rate was about 20 mm/min. The sensor arranged on the cutterhead can monitor the acceleration in three directions. In this experiment, the x-direction was the horizontal direction of the cutterhead plane, the y-direction was the vertical direction of the cutterhead plane, and the z-direction was the tunneling direction. A total of four working conditions were performed: tunneling without rock cutting, tunneling with 80 mm cutter spacing, tunneling with 90 mm cutter spacing, and tunneling with 100 mm cutter spacing. The measured time-domain curves and spectrum analysis of different cutter spacings are shown in Figure 4, and the characteristic values of measured data are shown in Table 1. The cutter spacings did not affect the acceleration of the cutterhead without rock cutting. Therefore, a set of typical curves is displayed in Figure 4(a). Under the condition of tunneling without rock cutting, the acceleration in the x-direction and the y-direction cyclically reciprocated between 21 and 1 g. The acceleration curves of the cutterhead in these two directions changed in the form of sine or cosine functions without obvious vibrations and sudden changes. The zdirection was the center axis of tunneling, so it was almost a straight line. Due to structural limitations, the sensor could not be placed in the middle of the cutterhead, and the acceleration in the z-direction could not cause fluctuations. Compared with the monitoring curves of the cutterhead without rock cutting, the acceleration curves underwent abrupt vibrations when performing rock cutting. The experimental phenomenon suggested that vibrations of the time-domain curves were generated when the cutterhead was stuck or the rock was broken.
From the perspective of the changing magnitude of the acceleration, the larger the cutter spacing, the more obvious the vibrations. Specifically, the vibration amplitudes under various cutter spacings in the x-and y-directions were 1.48-2.03 times those under the condition of tunneling without rock cutting. In the tunneling direction, the vibration differences of the cutterhead under various cutter spacings were considerable. The vibration amplitudes of the cutterhead under 80, 90, and 100 mm cutter spacings were 15.12, 44.17, and 58.56 times those of the cutterhead tunneling without rock cutting, respectively. This was because some of the vibrations generated on the surface of the cutterhead were affected by the rotation of the cutterhead. Thus, the vibrations in the tunneling direction can better reveal the dynamic characteristics of the cutterhead without further processing.
The fast Fourier transformation (FFT) method was used to perform spectrum analysis on the time domain curves. The vibration of the cutterhead was mainly concentrated at 0-3 Hz, which is a typical low-frequency vibration. Although the cutter spacing had a small effect on the vibration frequency of the cutterhead, there was a difference in the peak value of the frequency spectrum. Specifically, the peak value of the spectrum gradually increased from 80 to 90 to 100 mm. This trend was particularly obvious in the tunneling direction as with the time domain curves. The FFT peak value of the cutterhead under 80, 90, and 100 mm cutter

Dynamic analysis of cutterhead
To further determine the load characteristics of the cutterhead during the tunneling process, the measured data were used as the driving parameter to carry out the transient dynamic analysis. ANSYS software was adopted to simulate the dynamic response of the cutterhead. 23 The equation of structural dynamics of the cutterhead is where, x 1 (t) and x 2 (t) are the node acceleration vector and node velocity vector, and M, C, K, and q(t) are the mass matrix, damping matrix, stiffness matrix, and node load vector. These parameters are integrated by the respective unit matrix and vector.
When the influence of damping is neglected, the structural dynamics equation can be simplified to The transient dynamic analysis can also be defined as time-history response analysis, a method used to determine the structural dynamic response of a structure under time-varying loads.
According to different systems, transient dynamics analysis can be divided into first-order and secondorder system analyses. A second-order system means that the system has a second-order state in time. For example, in structural analysis, the mass matrix corresponds to the second derivative of displacement, so structural analysis belongs to a second-order system.
Structural transient dynamics analysis belongs to structural dynamics analysis. Therefore, the general equation of the second-order system transient analysis is The measured data were used to perform transient dynamic analysis to obtain the real-time mechanical characteristics of the cutterhead ( Figure 5). The selected measured data were time-domain curves of one cycle under various cutter spacings. Additionally, the measured curves of the cutterhead without rock cutting were also selected for comparison. In the process of numerical simulation, the rear part of the cutterhead was adopted with fixed support, and the dynamic loads were applied to all parts of the cutterhead. Specific numerical calculation settings were applied according to the related literature. 21,24 The real-time deformation and stress-strain of the cutterhead were dynamically determined ( Figure 6). After sorting, the mechanical characteristics of the cutterhead are shown in Table 2.
The dynamic changing trends of the cutterhead without rock cutting were smoother than that with rock breaking. As the relative position of the cutter spacing changed, the mechanical properties of the cutterhead also changed. The increased cutter spacing dispersed the integrity of the cutterhead, causing the vibration load to have a more significant impact on the cutterhead. The deformation and stress-strain of the cutterhead increased with the cutter spacing, and their corresponding values were the largest under the 100 mm cutter spacing. For the cutterhead tunneling without rock cutting, the deformation, stress, and strain of the cutterhead under the 100 mm cutter spacing were 2.16, 1.75 and 1.59 times those of the 80 mm cutter spacing. Correspondingly, for the cutterhead tunneling with rock cutting, the deformation, strain, and stress of the cutterhead under the 100 mm cutter spacing were 2.52, 2.25, and 2.06 times those of the 80 mm cutter spacing. The structural differences caused by cutter spacing were further magnified when the cutterhead was tunneled with rock cutting, and the gap in the mechanical states of the cutterhead became more obvious.
In addition, the randomness of the vibration may cause a sudden change in the mechanical properties of the cutterhead at a specific point, and the magnitude of this sudden change may exceed the existing threshold. For example, under the cutter spacing of 90 mm, the cutterhead had an abnormal amplitude during the rock-breaking process (around 9 s). At this moment, the mechanical state of the cutterhead exceeded the related value of 100 mm cutter spacing. Under actual construction conditions, the geological conditions will be more complex, causing this phenomenon to occur more frequently.

Discussion
The mechanical distributions of the cutterhead in the limit state were extracted for comparison (Figure 7).
The maximum deformation of the cutterhead generally appeared at the position where the edge cutter was located. The farther the distance from the center of the cutterhead, the greater the deformation of the edge cutter. When there was no load on the cutterhead, the maximum deformation of the cutterhead under various cutter spacings was 25.501 s. Meanwhile, the timedomain curves in the x-direction and y-direction intersected, and the superimposed acceleration was the largest. When the cutterhead was tunneling and cutting rocks, the randomness of vibration led to inconsistent moments of maximum deformation under various cutter spacings. However, the maximum deformation occurred when the superimposed acceleration reached the maximum.
There was a difference between the stress-strain and the deformation of the cutterhead. The maximum stress-strain under all conditions was concentrated on the edge cutter at the bottom left. During the boring process, the wear of the edge cutters should be highlighted in actual construction. Furthermore, under the same dynamic load, the moments of maximum stressstrain under 80 mm cutter spacing were later than those of the other two working conditions. This difference was due to the structure of the cutterhead. Under 80 mm cutter spacing, the integrity degree of the cutterhead was higher, and there may be a certain delay in response to the mechanics of vibration.
The time-domain curves of the cutterhead indicated that the vibration trend during excavation was the  superposition of random vibrations generated by the rock breaking by the cutter and unloaded tunneling. The mechanical characteristics of the cutterhead without rock cutting can be used as the basis for predicting the dynamic state of tunneling. Notably, there are more uncontrollable factors in actual working conditions, especially under complex geological conditions, and the random distribution characteristics of the abnormal vibration of the cutterhead are more obvious. The physical model test and numerical simulation in this study were performed under controlled conditions. When analyzing the specific cases, the physical model tests can be carried out according to the geotechnical conditions. Furthermore, on-site monitoring can be used for data validation in numerical simulations.

Conclusions
In this paper, the load characteristics of the cutterhead with various cutter spacings were analyzed through large-scale physical model tests. Taking the measured data as the driving parameters, the real-time load  distribution of the cutterhead was achieved through transient dynamic analysis. As a result, the following conclusions were obtained. The complete time-domain curves were stepped and periodic. When the cutterhead was stuck or the rock peeled off during the rotation, the vibration monitoring curves showed abrupt vibrations. During the tunneling process, the wear of the edge cutters should be highlighted. The FFT method was used to perform spectrum analysis on the time-domain curves. The vibration of the cutterhead was mainly concentrated at 0-3 Hz, a typical low-frequency vibration.
The increase in the distance between the cutters caused the integrity of the cutterhead to be dispersed, which in turn caused the vibration load to have a greater impact on the mechanical properties of the cutterhead. When the cutterhead was tunneling, the structural differences caused by the cutter spacing were further magnified and the gap in the mechanical states of the cutterhead became more obvious.
The time-domain curves of the cutterhead indicated that the vibration trend during excavation was the superposition of random vibrations generated by the rock breaking by the cutter and unloaded tunneling. The mechanical characteristics of the cutterhead without rock cutting can be used as the basis for predicting the dynamic state of tunneling. The related methods and results can provide an essential reference for the selection of cutterheads.

Declaration of conflicting interests
The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.