The Coupling Dynamic Analysis and Field Test of TBM Main System under Multipoint Impact Excitation

Damage by excessive vibration is serious engineering problem in TBM boring process. Dynamic characteristic analysis is essential for TBM antivibration design. According to TBM horizontal support structure, a dynamic coupling nonlinear model is established, with consideration of time-varying impact load and multicomponent complex relationship from cutter to gripper shoe. A set of field vibration tests is set up to accurately collect data under extreme work conditions; then, field data is collected from Liaoning northwest engineering. Field data is applied to validate simulation model to make sure time-varying damping stiffness, support cylinder stiffness, and the TBM machine stiffness distribution are reasonable. Simulation indicates the weakest part of TBM in axial and torsional DOF is the cylinder hinge and the connection shaft between motor and pinion, and the horizontal and vertical weak parts are bull gear. It also shows that, in normal excavation conditions, the acceleration amplitude of the cutterhead in three directions ranges from 1.5 g to 2 g. These results provide theoretical basis for the antivibration design and structural optimization of TBM.


Introduction
Full face rock boring machine (TBM) is a symbol product of the equipment manufacturing industry.The design, manufacturing capacity is one of the important symbols to measure a country's development and advanced degree in manufacturing industry [1].In hard rock, the boring process often accompanies with big torque, thrust, and large impact load.The excessive vibration of TBM will cause nonnormal damage in critical components and shorten the life of TBM.Therefore, studying the vibration characteristics of TBM from the angle of dynamics is quite important.Domestic and foreign scholars have done a lot of research works in TBM design.Since 1960s, scholars started studying rock crushing mechanism of cutters from different angles such as shear failure and extrusion failure.The rock crushing process is divided into the process of disc cutters intruding into rocks and the formation of rock fragments between two dis cutters [2,3].In order to accurately predict the force on dis cutters in breaking rocks, scholars established a variety of small scale models and set up some force models and calculation formulas, among which Colorado School of Mines (CSM) model is a representative one [4][5][6].Some scholars have detected and analyzed the force on disc cutters in field test.Samuel and Seow [7] studied the change law of force on disc cutters based on a 4.1 m Robbins cutterhead load test and compared with equipment performance parameters.Zhang et al. [8] performed a load test on a 1.75 m TBM cutterhead, considering the coupling three-axis force, and proposed a mathematical method of load measurement on disc cutters.The optimal placement design on TBM cutters is one of the most important means to improve the performance of TBM.Large numbers of scholars carried out cutterhead optimization design such as cutter spacing design and cutter layout design.Zhang et al. [9,10] determined the principle of cutter spacing.Xia and Tan [11][12][13] carried out simulation on cutters breaking rock and obtained the cutting parameters such as optimal cutter spacing and optimal cutting angle in order.Sun and Huo [14][15][16] proposed cutter layout design method based on genetic algorithm and an idea of coevolutionary, and the optimal design on the main parameters of disc cutters and cutterhead structure was carried out.The researches show that the excessive vibration will not only affect the excavation efficiency but shorten the fatigue life of structural component and lead to the accident [17].Therefore, many scholars started studying the vibration dynamical characteristics of TBM.Zhang et al. [18][19][20] established a coupling dynamical model of shield machine considering redundant drive system, hydraulic propulsion system, geological conditions, and so forth, and the dynamical characteristics of the rotary system were studied based on the dynamical model.Sun et al. [21] established a nonlinear dynamical model of cutterhead system and analyzed the dynamical characteristics of cutterhead.Huo et al. [22] presented a multidegree-of-freedom coupling dynamic model for the blocked TBM cutterhead system based on mass centralized method.
In conclusion, the domestic and foreign researches mainly focus on the cutterhead system.These study subjects mainly include rock crushing mechanism, the structural strength and stiffness of cutterhead, the characteristics of random load on cutterhead, and so forth.The study on vibration characteristics mainly focused on the drive system and cutterhead system, and the study on dynamic vibration characteristics of the TBM complete machine was not sufficient.This paper establishes a dynamic coupling nonlinear model of TBM machine, by developing an equivalent method that can accurately simulate dynamical stiffness such as the time-varying meshing stiffness, time-varying support stiffness of thrust cylinder, and bearing stiffness.The research route is shown in Figure 1.This paper provides theoretical basis for the antivibration design and structural optimization of TBM machine.

The Dynamic Coupling Nonlinear Model of TBM Machine
TBM machine mainly includes cutterhead system and propel system.The cutterhead system includes cutterhead, bull gear, pinion, coupling, variable frequency motor, planetary reducer, and so forth.The propel system includes main frame, support cylinder, and gripper shoe.The component diagram of TBM is shown in Figure 2. , , and  represent horizontal, vertical, and axial direction.The dynamic model of TBM is shown in Figure 3. Multidegree-of-freedom coupled nonlinear dynamic equations are as follows.
The horizontal degree of freedom is as follows: The vertical degree of freedom is as follows: ( The axial degree of freedom is as follows: The torsional degree of freedom is as follows: where   ,   ,   ,

Shock and Vibration
The above nonlinear simultaneous equations can be expressed in matrix form: where ,  represent the total damping matrix and stiffness matrix,  represents the force vector,  represents the displacement vector, and  represents the mass matrix.These simultaneous equations are solved by Newmark method.

The Dynamic Excitation Simulation of TBM System
3.1.The Error Excitation.The errors that affect the load distribution between the bull gear and pinion mainly include the manufacturing error of bull gear   , installation error of bull gear   , the tooth thickness deviation of bull gear   , the tooth error of bull gear  pi , the eccentric error of pinion  pi , the installation error of pinion  pi , and the tooth thickness deviation of pinion  pi .The accumulation error on meshing line can be calculated as where  represents meshing contact angle,   represents phase angle of each pinion,  and  represent the phase angle of each error, and   and   represent the angular velocity of pinion and bull gear.

Elastic Deformation Coordination
Conditions.The dynamic model is established based on the structure of Robbins TBM.
The bull gear is driven by ten parallel pinions which are connected with motor.The geometry elastic relationship between pinion and bull gear is shown in Figure 4.
The deformation of pinion and bull gear is projected onto the meshing line, and the total relative elastic deformation can be calculated as where   ,  pi are the displacement of bull gear and pinion on meshing line,   ,   are the horizontal and vertical displacement of bull gear, and  pi ,  pi are the horizontal and vertical displacement of pinion.
Considering the tooth backlash of gear pair, the meshing clearance nonlinear function can be defined as where  is half of the value of the tooth backlash and  is the total relative elastic deformation.

The Calculation of Time-Varying Meshing Stiffness, Damping, and Meshing
Force.The time-varying meshing stiffness can be defined as where   is amplitude coefficient,   is meshing frequency,  is the phase angle of pinion, and  is average meshing stiffness.
The time-varying meshing damping can be defined as where  is meshing damping ratio,  1 is the mass of bull gear, and  2 is the mass of pinion.The time-varying meshing force can be defined as

The Calculation of the Support Cylinder Time-Varying
Stiffness.The static stiffness of support cylinder is calculated according to the following formula [23]: where  1 ,  2 are the area of two side of the cylinder piston,  1 ,  2 are the hydraulic oil volume in two hydraulic cylinders, and  1 ,  2 are the total volume of hydraulic oil in two hydraulic pipelines.The values of  1 ,  2 are almost constant during a boring stoke, and the angle between support cylinder and main frame decreases continuously.Therefore, the equivalent stiffness of support cylinder also decreases.The value of this time-varying stiffness can be calculated as 0 is the angle between support cylinder and main frame at the beginning of a boring stoke. 0 is the length of support cylinder at the beginning of a boring stoke.  is the elongation velocity of the support cylinder.

The Calculation of Equivalent Hinge Stiffness.
The hinge connection stiffness  1 is calculated by the formula below [24]: 1 ,  2 represent the radius of the pin and the hinge ear. 1 ,  2 are determined by the materials of pin and hinge ear.The hinge stiffness is determined not only by the connection stiffness but also by the structural stiffness of hinge ear.The structural stiffness of hinge ear is estimated by static analysis.The equivalent hinge stiffness can be calculated by the following formula:

The Calculation of Bearing Radial Equivalent Stiffness.
Pinion bearing mainly bear support radial load, and the radial equivalent stiffness is calculated by empirical formula [25]: = 0.34 × 10 4   0.1  0.9  0.8 (cos ) In the formula,   represents the radial load,  represents the number of rolling elements,  represents the effective contact length, and  represents the rolling element contact angle.

The Calculation of Connection Shaft Torsional Stiffness.
The torsional stiffness of connection shaft between pinion and motor is calculated by the following formula: In the formula,   represents inertia moment of the connection shaft,  represents the shear modulus of material, and  represents length of the connection shaft.

The Calculation of the Other Structural Stiffness and
Damping.The equivalent structural stiffness of cutterhead, cutterhead support, main frame, and so forth is determined by finite element method.
The equivalent structural damping is determined by empirical formula: represents the damping ratio;   ,   represent the equivalent mass and equivalent stiffness of corresponding components.

Engineering Calculation
This paper takes the Robbins TBM in Liaoning northwest project as example.The tunneling parameters and cutterhead parameters are shown in Table 1.
According to the parameters and the Ls-dyna simulation results, the load time history curve on cutters is obtained.The load on cutterhead is calculated ignoring the load loss in the transmission process, and the load is revised according to the field test data.

Dynamic Response Analysis.
The calculated loads above are taken as input excitation for the dynamic model, and the vibration responses of cutterhead, cutterhead bearing, and so forth in 10 s are obtained.The response curves are shown in Figure 5.
It shows that (1) the means of axial and horizontal cutterhead vibration reach more than 1 mm, and the mean of vertical vibration is about 50% of the axial and horizontal vibration.This could be due to the vertical support near the shield of Robbins TBM, and the vertical vibration on the front components of TBM is effectively reduced.(2) The results of horizontal and vertical vibration on cutterhead show a periodicity around 10 s which is matched with the bull gear rotational period.Obviously, the meshing vibration has a significant influence on the horizontal and vertical vibration on cutterhead.(3) The means of axial vibration on cutterhead, cutterhead bearing, and front frame are around 1.2 mm,  The spectrogram of input load excitation and response vibration are shown in Figure 6.
It shows that (1) under the time-varying internal and external load excitation, the response frequency of axial and horizontal vibration on cutterhead mainly focuses on 10-20 Hz and 30-40 Hz.The response frequency of vertical vibration mainly focuses on 0-10 Hz and 10-20 Hz.It is obvious that the vibration response frequency distribution contains corresponding frequency components of external load excitation.The contrastive results show that the vibration on cutterhead system is mainly forced vibration.(2) The main responses vibration on cutterhead in 3 directions is less than 40 Hz, and the low order natural frequency of TBM cutterhead system focuses on 50-60 Hz according to the numerical simulation results.Therefore, severe resonance phenomenon will not appear during boring process.

The Dynamic Model Vibration
Based on the Field Test wireless receiver.The composition diagram of the wireless test system is shown in Figure 8.The test acceleration sensors are placed on the cutterhead, cutterhead bearing flange, mid frame, and the end of support cylinder, and the test points are numbered mp1, mp2, mp3, and mp4 as shown in Figure 7.
The field test photos are shown in Figure 8. Considering the bad working conditions inside the cutterhead, the corresponding acceleration sensor is fixed in a special protective cap.Due to the rotation of cutterhead and the complex structure, the signals from the cutterhead will decay sharply.
In test, the positions of the signal transmitting antenna and the signal receiver are appropriately adjusted.It makes the vibration signal from cutterhead be stably received.Thus, the primary vibration data of cutterhead is obtained.

The Analysis of Measured Data.
The test points on cutterhead and mid frame are defined by three-directions acceleration sensors, and the test points on cutterhead bearing flange and the end of the support cylinder are defined by uniaxial acceleration sensor.Take some of the test data as example, and Figure 9 shows the measured values of each component in comparison with the corresponding calculated values.
As can be seen from Figure 9, (1) in normal boring conditions, the three-directions acceleration vibration amplitudes are all around 1.5-2.0g. ( 2) The axial and horizontal acceleration vibration amplitudes are close to 0.6 g, and the vertical acceleration vibration amplitude is significantly smaller, which is about 1/2 of the former.(3) The vibration   condition of support cylinder is close to that of the main frame whose vibration amplitude is around 0.6 g.Some random events often occur during the field test.For example, the dropped rocks shock the cutterhead.These external influences present a series of high peak in the test data.In order to avoid the random events producing great influence on the dynamic analysis, the rain flow count method is used to assess how well the calculated values fit the measured values.Firstly, the measured values and the calculated values are divided into several intervals, and then the peak and average values in each interval are computed.Finally, the mean and amplitude of these statistics are calculated to measure the calculation error.The rain flow count statistics for each test point are listed in Table 2.
It can be seen from the figure that, among the several test point, the maximal calculation error of the mean of vibration amplitude is 36.90%while the maximal calculation error of the MSE of vibration amplitude is 40.41%.These results   show that the dynamic model is accurate and reasonable.Particularly, the small calculation error of test point 4 (the end of support cylinder) proves that the calculation of the cylinder stiffness is reasonable.

The Stiffness Distribution of TBM Machine.
The stiffness distribution of the TBM machine is determined by the dynamic model.The stiffness of each component is shown in Table 3 except the time-varying support cylinder stiffness and the time-varying damping stiffness.
As can be seen from Table 3, (1) the axial stiffness of each component is of the order of 1×10 10 N/m, and the hinge is the weakest part in axial direction with the equivalent stiffness being 1 × 10 8 N/m; (2) the horizontal and vertical stiffness of cutterhead bearing are of the order of 1 × 10 12 N/m; it is the major part to bear the horizontal and vertical load.The bull gear is the weakest part in horizontal and vertical direction; (3) the weakest part in torsional direction is the connection shaft between pinion and motor, and stiffness of this part should be enhanced in the structure design.

Conclusions
A dynamic coupling nonlinear model includes cutterhead system and propulsion system is established by concentrated mass method.The Newmark method is used to solve

Figure 2 :
Figure 2: The component diagram of TBM.

Figure 3 :
Figure 3: The dynamic model of TBM.

Figure 4 :
Figure 4: The geometry elastic relationship between pinion and bull gear.
The axial displacement of front frame

Figure 5 :
Figure 5: The response curve of each component.

Figure 7 :
Figure 7: The composition diagram of the wireless test system.
(a) Cutterhead (b) Cutterhead bearing flange (c) Mid frame (d) End of the support cylinder

Figure 8 :
Figure 8: The field test photos.

Figure 9 :
Figure 9: The measured values and calculated values of each component.

Table 1 :
The tunneling parameters and cutterhead parameters.

Table 2 :
The rain flow count statistics for each test point.