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# Nonlinear Static and Dynamic Stiffness Characteristics of Support Hydraulic System of TBM

*Chinese Journal of Mechanical Engineering*
**volumeÂ 32**, ArticleÂ number:Â 101 (2019)

## Abstract

Full-face hard rock tunnel boring machines (TBM) are essential equipment in highway and railway tunnel engineering construction. During the tunneling process, TBM have serious vibrations, which can damage some of its key components. The support system,an important part of TBM, is one path through which vibrational energy from the cutter head is transmitted. To reduce the vibration of support systems of TBM during the excavation process, based on the structural features of the support hydraulic system, a nonlinear dynamical model of support hydraulic systems of TBM is established. The influences of the component structure parameters and operating conditions parameters on the stiffness characteristics of the support hydraulic system are analyzed. The analysis results indicate that the static stiffness of the support hydraulic system consists of an increase stage, stable stage and decrease stage. The static stiffness value increases with an increase in the clearances. The pre-compression length of the spring in the relief valve affects the range of the stable stage of the static stiffness, and it does not affect the static stiffness value. The dynamic stiffness of the support hydraulic system consists of a U-shape and reverse U-shape. The bottom value of the U-shape increases with the amplitude and frequency of the external force acting on the cylinder body, however, the top value of the reverse U-shape remains constant. This study instructs how to design the support hydraulic system of TBM.

## 1 Introduction

Full-face hard rock tunnel boring machines (TBM) are widely used in highway and railway tunnel engineering construction because excavating tunnels with TBM has advantages such as little environmental disturbance, high tunnel efficiency, low personnel expenditure, and better working conditions and safety [1, 2]. The basic elements of TBM have serious vibrations because the load changes suddenly due to the rock being broken in a working environment of high hardness, quartz content, and surface stress. The system vibrations generate noise, harm workersâ€™ health, and may damage key components of the system [3,4,5,6]. The vibration energy is transferred to the support system through the main beam, making the support system vibrate [7]. As a key working component of TBM, the support system provides the excavating counter-torque and counter-force during the excavation process [8, 9], which has an important influence on system vibration; therefore, the stiffness characteristics of the support system receive significant concern [10, 11].

Support system stiffness is composed of hydraulic system stiffness, mechanical structure stiffness, and contact stiffness [12, 13]. Wu et al. [14] established a multi-coupling dynamic model for support systems of TBM and carried out a multi-position dynamic strain test based on practical engineering to calibrate the dynamic model. They pointed out that the support directly affected the tunneling efficiency of TBM. Yu et al. [15] studied the relationship between the roughness and fractal parameters of three dimensional tunnel surfaces based on fractal theory and found that increases in the fractal dimension, elastic modulus, normal loads, and number of support boots cause an increase in the interfacial contact stiffness, and an increase in the ratio of soft rocks in the contacting area causes an increase in the contact stiffness. This shows that the contact stiffness is influenced by the support hydraulic system because the normal loads of support shoes can be adjusted by the support hydraulic system. Hence, hydraulic system stiffness characteristics play an important role in the support system [16, 17]. Wu et al. [18] obtained the static stiffness of gripper cylinders and the machine-hydraulicâ€“rock stiffness through the linearization method near the equilibrium point of the simplified model. The dynamic characteristics of the relief valve in the support hydraulic system and the leakage characteristics of the gripper cylinder had not been taken into account in the system model, and linearization may lack some important characteristics. Wang et al. [19] studied the nonlinear dynamic characteristics of moving cylinders by theoretical analysis and experimental verification, and they found that the cylinder took on soft spring and hard spring properties in different working situations; however, this research merely focused on the nonlinear stiffness characteristics of the hydraulic cylinder, and the stiffness characteristics of the hydraulic system and the leakage characteristics of the cylinder were not investigated. Lei et al. [20] analyzed the gripper cylinder internal leakage characteristics of TBM and the influences of the cylinder structure parameters and working conditions parameters on the structure feature of the support hydraulic system. The relief valve model in this research was simplified as a linear relation between input pressure and flowrate, and the dynamic characteristics of the relief valve were neglected. As for the support hydraulic system, the stiffness characteristics depend heavily on the open and close of the relief valve and the dynamic characteristics of the valve. In fact, the static and dynamic characteristics of the poppet relief valve have a significant influence on the efficiency of the hydraulic system [21, 22]. For this purpose, it is necessary to investigate the nonlinear static and dynamic stiffness characteristics of the support hydraulic system while also considering the dynamic characteristics of the relief valve and working conditions.

In this paper, taking into account the dynamic model of the relief valve, the nonlinear dynamic model of the support hydraulic system is established according to the structural feature of the support hydraulic system, and the influences of the component structure parameters and operating conditions parameters on the stiffness characteristics of the support hydraulic system are analyzed.

## 2 Configuration of the Support Hydraulic System

TBM are composed of a main beam, cutterhead, gripper cylinder, shield, rear support leg, and thrust cylinder as shown in FigureÂ 1 [23, 24].

During the excavation process, gripper shoes support the rock, relying on the output force of the gripper cylinder to provide the counter-force and counter-torque needed for excavating, and FigureÂ 2 shows the working state of an gripper cylinder [25].

Based on FigureÂ 2, the support system in TBM includes gripper shoes, hinges, and a support hydraulic system, and the support hydraulic system in the working state is demonstrated in FigureÂ 3.

As shown in FigureÂ 3, the support hydraulic system consists of two gripper cylinders, two check valves, and two relief valves. The two gripper cylinders are fixed together; their rod-less chambers are connected, and the two relief valves are connected to the rod chambers, The rod-less chambers of the gripper cylinders are connected with a pipeline. The support hydraulic system can be simplified and illustrated in FigureÂ 4.

## 3 Dynamic Model of Support Hydraulic System

As shown in FigureÂ 4, the dynamic model of the support hydraulic system consists of the gripper cylinder sub model and the relief valve sub model. The gripper cylinder sub model includes the moment equation and the leakage equation of the cylinder, the relief valve sub model includes the flow model and the force equilibrium equation of the valve core.

### 3.1 Dynamic Model of Gripper Cylinder

As shown in FigureÂ 5 and taking the cylinder body as the research object and neglecting the viscous force, the following equation can be obtained based on Newton second theorem.

where \( m_{c} \) is the mass of the cylinder body, \( x \) is the displacement of the cylinder body, \( A_{c} \) denotes the effective area of the rod chamber, \( F_{lc} \) stands for the external force acting on the cylinder body, \( F_{fc} \) is the friction between the cylinder body and the piston, \( p_{1} \) and \( p_{2} \) are the rod chamber of the two cylinders, respectively, and \( \ddot{x} \), \( \dot{x} \) are the acceleration and velocity of the cylinder body, respectively.

According to the pressure-rise rate equation, the pressure expression of the rod chamber can be obtained [26].

where \( \beta_{e} \) is the bulk modulus of the working fluid media, \( V \) is the initial volume of the rod chamber, \( q_{l1} \) and \( q_{l2} \) are the leakage flowrates from the rod-less chambers to the rod chambers of the two cylinders, respectively, and \( q_{v1} \), \( q_{v2} \) denote the flowrates of the relief valves.

Based on the Navierâ€“Stokes equations and neglecting the eccentricity of the piston in the gripper cylinder, the volume rate of flow between the rod-less chamber and the rod chamber is derived as [27]

where \( d \) denotes the diameter of the cylinder piston, \( \delta_{1} \) and \( \delta_{2} \) are the clearances between the piston and the internal wall of the two cylinders, respectively, \( p_{s} \) is the pressure in the rod-less chamber, \( \mu \) is the absolute viscosity of the working fluid media, and \( l_{c} \) stands for the length of the piston.

### 3.2 Dynamic Model of Relief Valve

The dynamic characteristics of the relief valve play an important role in the stiffness characteristics of the support hydraulic system. FigureÂ 6 shows the schematic of the relief valve in the open position [28].

As shown in FigureÂ 6, the relief valve consists of a pre-compression thread, valve cover, valve core, spring, and valve body.

#### 3.2.1 Flow Equation for Relief Valve

The flow through the orifice can be expressed as [29]

where \( C_{d} \) is the discharge coefficient, \( A(y) \) is the flow-through area, \( y \) is the position of the valve body, \( p \) is the pressure difference between the inlet and the outlet of the relief valve, and \( \rho \) is the density of the fluid media.

FigureÂ 7 shows the geometry of the flow area, and the flow-through area can be approximately given as [26]

where \( D_{s} \) is the diameter of the seat port of the relief valve, and \( \theta \) denotes the half angle of the valve core.

Based on BazsÃ³â€™s [30] experimental results, Eq. (6) can be rewritten as

where \( \upsilon \) denotes the kinematic viscosity of the fluid media.

#### 3.2.2 Mechanical Analysis of Relief Valve Core

The forces acting on the valve core, as shown in FigureÂ 6, include the friction force, spring force, viscous force, and flow force, which are represented symbolically as \( F_{f} \), \( F_{s} \), \( F_{\upsilon } \) and \( F_{fl} \), respectively. The spring force and friction force are given by

where \( k \) is the stiffness of the spring, \( y_{0} \) is the initial pre-compression length of the spring, \( B_{v} \) is the damping coefficient, and \( \dot{y} \) denotes the velocity of the valve body.

The flow forces depend on the flowrate through the orifice and the inlet pressure, and they are an important part of the forces acting on the valve core. FigureÂ 8 shows the control volume for calculating the flow forces acting on the valve core.

According to the Reynolds transport theorem, the *Y*-component of the fluid force acting on the valve core can be expressed as [27, 28, 31]

where \( F_{y} \) stands for the component force in the *y* direction, \( \ddot{y} \) denotes the acceleration of the valve core, and \( A_{s} \) is the seat area.

Based on Newtonâ€™s second theorem, the following equation can be obtained:

where \( m_{v} \) is the mass of the valve core.

Substituting Eqs. (10) and (11) into Eq. (12), under the condition that \( y > 0 \), the kinematic equation of the valve body takes the form

Therefore, the dynamic model of the relief valve can be expressed by Eqs. (8) and (13).

### 3.3 Dynamic Model of Support Hydraulic System

Based on the sub-models of the gripper cylinder and relief valve and neglecting the pipe length between the cylinder and the relief valve, the volume flowrate through the relief valves can be expressed as follows:

where \( y_{1} \) and \( y_{2} \) are the displacements of the relief valve cores.

The dynamic model of the relief valve can be given as follows:

Therefore, the dynamic model of support hydraulic system can be described by Eqs. (1)â€’(5) and (13)â€’(17).

## 4 Nonlinear Stiffness Characteristics Analysis

Based on the dynamic model of the support hydraulic system, the simulation model was established the Simulink package in MATLAB. The parameters employed in the simulation model are listed in TableÂ 1.

Defining the support hydraulic system stiffness according to the simulation model as \( k_{s} = F_{lc} /x \), the nonlinear static and dynamic stiffness characteristics of the support hydraulic system are analyzed as follows. In static stiffness analysis, a single direction of \( F_{lc} \) is investigated for simplicity because the support hydraulic system is symmetrical.

### 4.1 Quasi-static Stiffness Characteristics Analysis

#### 4.1.1 Influence of the Clearances on Static Stiffness

The velocity of the gripper cylinder body is neglected when calculating static stiffness. The influences of the clearances between the piston and the internal wall of the cylinder on static stiffness is analyzed.

For \( \delta_{1} = \delta_{2} \), the static stiffness characteristic of the support hydraulic system at different conditions is plotted in FigureÂ 9.

Based on FigureÂ 9, under the condition that \( \delta_{1} = \delta_{2} \), the static stiffness of the support hydraulic system can be divided into three parts: an increase stage, stable stage, and decrease stage. Static stiffness increases as the external force acting on the cylinder body increases in the increase stage, remains approximately constant in the stable stage, and decreases in the last stage.

The static stiffness increases as the clearances between the piston and the internal wall of the cylinder increase because the fluid working media from the rod-less chamber to the rod one increases the pressure in the rod chamber.

The decrease point of the three stages occurs at about 4.8 MN. At that point, the relief valve connected to the rod chamber of the left cylinder opens, decreasing the static stiffness of the support hydraulic system.

For \( \delta_{1} \ne \delta_{2} \), the static stiffness characteristic of the support hydraulic system under different conditions is illustrated in FigureÂ 10.

As illustrated in FigureÂ 10, the clearance of the left cylinder influences the static stiffness of the support hydraulic system, while the clearance of the right cylinder has little influence on the static stiffness because the working fluid media from the rod-less chamber to the rod one increases the pressure inside the rod chamber of the left cylinder, which provides a counter-force against the external force acting on the cylinder.

#### 4.1.2 Influence of the Pre-Compression Spring Length-in the Relief Valve on Dynamic Stiffness

The static stiffness characteristics with different pre-compressions length of the spring are analyzed, and FigureÂ 11 shows the stiffness characteristics of the support hydraulic system at \( \delta_{1} = \delta_{2} = 0. 0 1 {\text{ mm}} \).

As shown in FigureÂ 11, the pre-compression length of the spring in the relief valve has no influence on the stiffness value of the increase and stable stages. The range of the stable stage increases as the pre-compression length of the spring increases because this also increases the cracking pressure of the relief valve. The pressure inside the rod chamber of the left cylinder decreases when the connected relief valve opens; therefore, the stiffness of the support hydraulic system also decreases.

It must be noted that the output force of the gripper cylinder decreases when the cracking pressure increases when the supply pressure of the support hydraulic system is kept constant.

### 4.2 Dynamic Stiffness Characteristics Analysis

FigureÂ 12 shows the nonlinear dynamic stiffness characteristics of the support hydraulic system when the external force acting on the cylinder body takes the form of \( F_{\text{lc}} = a\sin \omega t \), \( a = 0. 3 {\text{ MN}} \), and \( \omega = 3{\text{ rad/s}} \), and FigureÂ 13 is a partial enlarged view of FigureÂ 12.

As seen in FiguresÂ 12 and 13, the dynamic stiffness switches between the positive and negative extreme values when the direction of the displacement changes, and the dynamic stiffness of the support hydraulic system takes on a U-shape and reversed U-shape, respectively. The U-shape represents the displacement of the cylinder body in the opposite direction of \( x \), and the reversed U-shape denotes the displacement of the cylinder body in the direction of \( x \). The bottom value of the U-shape is about 150Â GN/m, and the top value of the reversed U-shape is about 3.9Â GN/m.

#### 4.2.1 Influence of the External Force Amplitude Acting on the Cylinder Body

The external force acting on the cylinder body takes the form of \( F_{lc} = a\sin \omega t \) in which \( \omega = 3{\text{ rad/s}} \), and \( a \) takes on the values 0.5Â MN, 1Â MN, 3Â MN, and 5Â MN. FigureÂ 14 shows the nonlinear dynamic stiffness characteristics of the support hydraulic system at different amplitudes of the force. FigureÂ 15 is the partial enlarged view of FigureÂ 14.

As shown in FiguresÂ 14 and 15, the top values of the reversed U-shapes remain constant with external force amplitudes. The bottom value of the U-shape increases and the width decreases as the amplitude of the external force increases until \( a = 3 {\text{ MN}} \). When the amplitude is \( 5 {\text{ MN}} \), the bottom value of the U-shape decreases and the width increases because the relief valve opens and the pressure in the rod chamber decreases.

#### 4.2.2 Influence of the External Force Frequency Acting on the Cylinder Body

The external force acting on the cylinder body takes the form \( F_{lc} = a\sin \omega t \) in which \( a = 0. 3 {\text{ MN}} \), and \( \omega \) takes on the values 8Â rad/s, 12Â rad/s, and 18Â rad/s. FigureÂ 16 illustrates the nonlinear dynamic stiffness characteristics when the external force acts on the cylinder body with different frequencies. FigureÂ 17 is the partial enlarged view of FigureÂ 16.

As shown in FiguresÂ 16 and 17, the bottom value of the U-shape increases as the frequency of the external force acting on the cylinder body increases, and the top value of the reverse U-shapes remain constant with varying frequencies. The top value of the U-shape is about 3.12Â GN/m.

## 5 Conclusions

In this study, the nonlinear dynamic model of the support hydraulic system in TBM is established, and the nonlinear static and dynamic stiffness characteristics of the support hydraulic system are analyzed. Based on the analysis results, several conclusions can be obtained.

- (a)
The static stiffness of the support hydraulic system consists of three stages; the increase stage, stable stage, and decrease stage.

- (b)
The static stiffness of the support hydraulic system increases as the clearances between the piston and the internal wall of the gripper cylinder increase.

- (c)
The pre-compression length of the spring in the relief valve has no influence on the static stiffness value of the support hydraulic system; however, the pre-compression length does influence the range of the stable stage of the static stiffness.

- (d)
The dynamic stiffness of the support hydraulic system consists of a U-shape and reverse U-shape.

- (e)
The bottom value of the U-shape increases as the amplitude of the external force acting on the cylinder body increases, while the top value of the reverse remains U-shape constant.

- (f)
The bottom value of the U-shape increases as the frequency of the external force acting on the cylinder body increases, while the top value of the reverse U-shape remains constant.

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## Authorsâ€™ Contributions

JL was in charge of the whole trial and established the numerical model of the relief valve; JT established the numerical model of support system and wrote the manuscript; CL assisted with data analyses and WY established the simulation model of the support hydraulic system of TBM with Simulink. All authors read and approved the final manuscript.

### Authorsâ€™ Information

Jianfeng Tao, born in 1975, is currently an associate professor at *Shanghai Jiaotong University, China*. He received his PhD degree on mechatronics from *Beijing University of Aeronautics and Astronautics, China*, in 2003. His research interests include mechatronics engineering, fluid power transmission and control and robotics.

Junbo Lei, born in 1980, is currently a research assistant at *Shanghai Jiaotong University, China*. He received his PhD degree on mechatronics from *Zhejiang University, China*, in 2012. His research interests include mechatronics engineering, fluid power transmission and control, and hydraulic component design.

Chengliang Liu, born in 1964, is currently a professor and a PhD candidate supervisor at *Shanghai Jiaotong University, China*. He received his PhD degree from *Southeast University, China*, in 1999. His research interests include mechatronics engineering, man-machine system and robotics.

Wei Yuan, born in 1981, is currently an associate professor at *Jiangsu University of Science and Technology, China*. He received his master degree on control engineering from *Jiangsu University of Science and Technology, China*, in 2007. His research interests include nonlinear control systems, robust control and convex optimization.

### Competing Interests

The authors declare that they have no competing interests.

### Funding

Supported by National Key R&D Program of China (Grant No. 2018YFB1702503), National Program on Key Basic Research Project of China (973 Program, Grant No. 2013CB035403) and Startup Fund for Youngman Research at SJTU (SFYR at SJTU)

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Tao, J., Lei, J., Liu, C. *et al.* Nonlinear Static and Dynamic Stiffness Characteristics of Support Hydraulic System of TBM.
*Chin. J. Mech. Eng.* **32**, 101 (2019). https://doi.org/10.1186/s10033-019-0414-y

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DOI: https://doi.org/10.1186/s10033-019-0414-y