Research on Dynamic Models and Performances of Shield Tunnel Boring Machine Cutterhead Driving System

Abstract

A general nonlinear time-varying (NLTV) dynamic model and linear time-varying (LTV) dynamic model are presented for shield tunnel boring machine (TBM) cutterhead driving system, respectively. Different gear backlashes and mesh damped and transmission errors are considered in the NLTV dynamic model. The corresponding multiple-input and multiple-output (MIMO) state space models are also presented. Through analyzing the linear dynamic model, the optimal reducer ratio (ORR) and optimal transmission ratio (OTR) are obtained for the shield TBM cutterhead driving system, respectively. The NLTV and LTV dynamic models are numerically simulated, and the effects of physical parameters under various conditions of NLTV dynamic model are analyzed. Physical parameters such as the load torque, gear backlash and transmission error, gear mesh stiffness and damped, pinions inertia and damped, large gear inertia and damped, and motor rotor inertia and damped are investigated in detail to analyze their effects on dynamic response and performances of the shield TBM cutterhead driving system. Some preliminary approaches are proposed to improve dynamic performances of the cutterhead driving system, and dynamic models will provide a foundation for shield TBM cutterhead driving system's cutterhead fault diagnosis, motion control, and torque synchronous control.

1. Introduction

The shield TBM is the large-scale underground equipment which is used to excavate tunnels and subways. The shield TBM has the advantages of high safety and reliability, low manpower, minor environmental damages, and rapid excavation speed compared with other excavation methods. Therefore, the shield TBM is widely used in underground tunnel and subway projects. The shield TBM is mainly constituted by three core systems which include the cutterhead driving system, hydraulic thrust system, and attitude positioning system. The cutterhead driving system, hydraulic thrust system, and attitude positioning system play different important roles in shield TBM, and they are shown in Figures 1, 2, and 3, respectively. The cutterhead driving system will make the shield TBM cutterhead rotate at certain speed. The hydraulic thrust system will thrust the shield TBM cutterhead along the tunnel axis at certain speed. The attitude positioning system will make the shield TBM cutterhead follow the tunnel excavation direction. The cutterhead usually has two driven modes. One is the hydraulic-driven model and the other is motor-driven mode. This paper focuses on studying motor-driven mode shield TBM cutterhead driving system.

Figure 1:

Shield TBM cutterhead driving system.

Figure 2:

Hydraulic thrust system of shield TBM cutterhead.

Figure 3:

Attitude positioning system of shield TBM cutterhead.

The shield TBM cutterhead cuts rocks and soil. Different shield TBM cutterhead forms can adapt to different geology conditions. Figure 4 shows different shield TBM cutterhead forms which mainly include the spokes form (Figure 4(a), [1, 2]), panel form (Figures 4(b)–4(c)), and hybrid form (Figure 4(d), [1, 2]). Figure 5 shows shield TBM cutterhead profile and inner mechanical transmission structure. Figures 5(a)–5(c) [1, 2] and Figure 5(d) [1, 2] are the cutterhead profile and inner mechanical transmission structure, respectively. The transmission structure in Figure 5(d) shows that multiple pinions mesh a central large gear, each pinion is driven by an induction motor, and this structure is called the multiple gears transmission structure. The multiple gears transmission structure is employed to synthesize the driving motors' electrical magnetic torque (EMT). The motor-driven shield TBM cutterhead driving system's physical components include driving motors, couplings, reducers, bearings, main bearing and gearbox, and frequency inverters. Figure 6 [1] shows the induction motor, reducer, and coupling of the cutterhead driving system. The reducer is the torque converter which slows down the motor speed, thus the reducer's output torque is amplified. The overall schematic of the cutterhead driving system is shown in Figure 7. The cutterhead and central large gear have identical central shaft, thus the large gear and cutterhead have the same rotation speed. Once the central large gear rotates, then the cutterhead is driven immediately.

Figure 4:

Shield TBM cutterhead forms ((a) spokes form, (b) and (c) panel form, and (d) hybrid form).

Figure 5:

Shield TBM cutterhead profile and its inner mechanical transmission structure.

Figure 6:

Induction motor, reduction gear box, and coupling.

Figure 7:

Shield TBM cutterhead driving system (motor-driven mode).

At present, many publications [3–16] usually introduce or study the cutterhead load torque, hydraulic thrust system, cutterhead driving structure and composition, and shield TBM overall composition. In [12], the hydraulic-driven cutterhead driving system is studied for hydraulic-driven mode shield TBM. In [13–16], shield TBM cutterhead's load torque under various geology conditions is investigated in detail. In [17, 18], the cutterhead hydraulic thrust system is studied for hydraulic-driven mode shield TBM. In [1], a nonlinear time-varying dynamic model of the shield TBM cutterhead driving system is established and simulated. The effects of parts of physical parameters on some system performance items of the shield TBM cutterhead driving system are analyzed. In [1], the gear mesh damped is not considered. In addition, the nonlinear dynamic model in [1] considers that the shield TBM cutterhead driving system has identical gear backlash and transmission error. In [2], a linear time-varying model of shield TBM cutterhead driving system is established and analyzed, and the multiple gears transmission structure is studied by automatic dynamic analysis of mechanical systems (ADAMS) software. However, the linear dynamic model in [2] is not directly simulated.

This paper mainly focuses on studying dynamic models and system performances of the shield TBM cutterhead driving system. In this paper, different gear mesh damped, gear backlashes, and transmission errors are considered in the shield TBM cutterhead driving system, thus the nonlinear and linear dynamic models are presented again. The general NLTV dynamic model of this paper considers different mesh damped, gear backlashes, and transmission errors compared with previous dynamic models in [1, 2]. Furthermore, all the effects of physical parameters of the NLTV dynamic model on dynamic response and system performances of shield TBM cutterhead driving system are analyzed in detail. In addition, the nonlinear dynamic model and linear dynamic model are simulated under different conditions, and some preliminary approaches are proposed to improve dynamic performances of the shield TBM cutterhead driving system. Cutterhead driving system's potential problems may be recognized by studying dynamic model. The dynamic models provide theoretical foundation for design, analysis, and control of shield TBM cutterhead driving system in the future. This paper is arranged as follows. In Section 2, the preliminary and preparation is presented. In Section 3, a general nonlinear dynamic model and linear time-varying dynamic model of shield TBM cutterhead driving system are presented, and the linear dynamic models are analyzed. In Section 4, dynamic models are simulated and analyzed under conditions of different physical parameters, and the simulation results and system performances are investigated in detail. In Section 5, study contents and results are briefly reviewed and concluded, and some conclusions are made.

2. General NLTV Dynamic Model of Cutterhead Driving System

2.1. Preliminary and Preparation

Gear backlash and transmission error are inevitable due to manufacturing and installing process variations, mesh requirement differences, and other uncertainties. Therefore, gear mesh process is a nonlinear transmission process. Moreover, transmission error naturally appears in the driving process and usually varies with time. As mesh point diversifies, the actual backlash varies with time. Such gear backlash variation is not considered in this paper. To establish the nonlinear dynamic model, gear mesh process's backlash of cutterhead driving system is simplified and shown in Figures 8(a)–8(b) [1]. The gear nonlinear mesh model and linear mesh model are shown in Figures 8(c) and 8(d), respectively. Figure 8(c) shows that gear backlash and transmission error are considered in the cutterhead driving system. When gear backlash and transmission error are not considered, the gear mesh process can be seen as ideal transmission. The spring and damper are usually used to depict the gear mesh process. In fact, the mesh stiffness k and mesh damped c are the time-varying system parameters.

Figure 8:

Gear backlash diagram and its nonlinear and linear model.

Based on gear mesh dynamic in [19–24], the relative position function, elastic mesh force, and mesh torque are obtained, respectively. Where e _i (t) and $2{\mathrm{\Delta }}_b^i$ are the transmission error and total backlash of the ith gear pair, respectively,

$\begin{array}{c}{p}_i=\{\begin{array}{cc}{r}_i*{\theta }_{p,i}-r_m*{\theta }_m-e_i\left(t\right)-{\mathrm{\Delta }}_b^i,\hfill & \hfill \\ \hspace{1em}\hspace{1em}\text{for}\mathrm{\hspace{0.17em}\hspace{0.17em}}{r}_i*{\theta }_{p,i}-r_m*{\theta }_m-e_i\left(t\right)\ge {\mathrm{\Delta }}_b^i,\hfill & \hfill \\ \mathrm{0},\hspace{1em}\hspace{1em}\text{for}\mathrm{\hspace{0.17em}\hspace{0.17em}}-{\mathrm{\Delta }}_b^i<r_i*{\theta }_{p,i}-r_m*{\theta }_m-e_i\left(t\right)<{\mathrm{\Delta }}_b^i,\hfill & \hfill \\ r_i*{\theta }_{p,i}-r_m*{\theta }_m-e_i\left(t\right)+{\mathrm{\Delta }}_b^i,\hfill & \hfill \\ \hspace{1em}\hspace{1em}\text{for}\mathrm{\hspace{0.17em}\hspace{0.17em}}{r}_i*{\theta }_{p,i}-r_m*{\theta }_m-e_i\left(t\right)\le -{\mathrm{\Delta }}_b^i,\hfill & \hfill \end{array}\\ F_i=f\left(p_i\right)=k_i{p}_i+c_i{\dot{p}}_i,\\ M_{c,i}=F_i*r_i=k_i{p}_i{r}_i+c_i{\dot{p}}_i{r}_i.\end{array}$ (1)

Then, the elastic mesh force F _i and mesh torque M_{c, i} could be expressed as

$\begin{array}{c}{F}_i=\{\begin{array}{cc}{k}_i\left(r_i*{\theta }_{p,i}-r_m*{\theta }_m-e_i\left(t\right)-{\mathrm{\Delta }}_b^i\right)\hfill & \hfill \\ \hspace{1em}+\hspace{0.17em}{c}_i\left(r_i*{\dot{\theta }}_{p,i}-r_m*{\dot{\theta }}_m-{\dot{e}}_i\left(t\right)\right),\hfill & \hfill \\ \hspace{1em}\hspace{1em}\text{for}\mathrm{\hspace{0.17em}\hspace{0.17em}}{r}_i*{\theta }_{p,i}-r_m*{\theta }_m-e_i\left(t\right)\ge {\mathrm{\Delta }}_b^i,\hfill & \hfill \\ \mathrm{0},\hspace{1em}\hspace{1em}\text{for}\mathrm{\hspace{0.17em}\hspace{0.17em}}-{\mathrm{\Delta }}_b^i<r_i*{\theta }_{p,i}-r_m*{\theta }_m-e_i\left(t\right)<{\mathrm{\Delta }}_b^i,\hfill & \hfill \\ k_i\left(r_i*{\theta }_{p,i}-r_m*{\theta }_m-e_i\left(t\right)+{\mathrm{\Delta }}_b^i\right)\hfill & \hfill \\ \hspace{1em}+\hspace{0.17em}{c}_i\left(r_i*{\dot{\theta }}_{p,i}-r_m*{\dot{\theta }}_m-{\dot{e}}_i\left(t\right)\right),\hfill & \hfill \\ \hspace{1em}\hspace{1em}\text{for}\mathrm{\hspace{0.17em}\hspace{0.17em}}{r}_i*{\theta }_{p,i}-r_m*{\theta }_m-e_i\left(t\right)\le -{\mathrm{\Delta }}_b^i,\hfill & \hfill \end{array}\\ M_{c,i}=\{\begin{array}{cc}{k}_i{r}_i\left(r_i*{\theta }_{p,i}-r_m*{\theta }_m-e_i\left(t\right)-{\mathrm{\Delta }}_b^i\right)\hfill & \\ \hspace{1em}+c_i{r}_i\left(r_i*{\dot{\theta }}_{p,i}-r_m*{\dot{\theta }}_m-{\dot{e}}_i\left(t\right)\right),\hfill & \\ \hspace{1em}\hspace{1em}\text{for}\mathrm{\hspace{0.17em}\hspace{0.17em}}{r}_i*{\theta }_{p,i}-r_m*{\theta }_m-e_i\left(t\right)\ge {\mathrm{\Delta }}_b^i,\hfill & \\ \mathrm{0},\hspace{1em}\hspace{1em}\text{for}\mathrm{\hspace{0.17em}\hspace{0.17em}}-{\mathrm{\Delta }}_b^i<r_i*{\theta }_{p,i}-r_m*{\theta }_m-e_i\left(t\right)<{\mathrm{\Delta }}_b^i,\hfill & \\ k_i{r}_i\left(r_i*{\theta }_{p,i}-r_m*{\theta }_m-e_i\left(t\right)+{\mathrm{\Delta }}_b^i\right)\hfill & \\ \hspace{1em}+\hspace{0.17em}{c}_i{r}_i\left(r_i*{\dot{\theta }}_{p,i}-r_m*{\dot{\theta }}_m-{\dot{e}}_i\left(t\right)\right),\hfill & \\ \hspace{1em}\hspace{1em}\text{for}\mathrm{\hspace{0.17em}\hspace{0.17em}}{r}_i*{\theta }_{p,i}-r_m*{\theta }_m-e_i\left(t\right)\le -{\mathrm{\Delta }}_b^i.\hfill & \end{array}\end{array}$ (2)

Therefore, the elastic mesh force and torque of ith pinion can be equivalently expressed as

$\begin{array}{c}{F}_i=\{\begin{array}{cc}{k}_{f,i}\left({\theta }_{p,i}-i_{m,i}*{\theta }_m-\frac{e_i\left(t\right)}{r_i}-{\mathrm{\Delta }}_i\right)\hfill & \hfill \\ \hspace{1em}+c_{f,i}\left({\dot{\theta }}_{p,i}-i_{m,i}*{\dot{\theta }}_m-\frac{{\dot{e}}_i\left(t\right)}{r_i}\right),\hfill & \hfill \\ \hspace{1em}\hspace{1em}\text{for}\mathrm{\hspace{0.17em}\hspace{0.17em}}{\theta }_{p,i}-i_{m,i}*{\theta }_m-\frac{e_i\left(t\right)}{r_i}\ge {\mathrm{\Delta }}_i,\hfill & \hfill \\ \mathrm{0},\hspace{1em}\hspace{1em}\text{for}\mathrm{\hspace{0.17em}\hspace{0.17em}}-{\mathrm{\Delta }}_i<{\theta }_{p,i}-i_{m,i}*{\theta }_m-\frac{e_i\left(t\right)}{r_i}<{\mathrm{\Delta }}_i,\hfill & \hfill \\ k_{f,i}\left({\theta }_{p,i}-i_{m,i}*{\theta }_m-\frac{e_i\left(t\right)}{r_i}+{\mathrm{\Delta }}_i\right)\hfill & \hfill \\ \hspace{1em}+c_{f,i}\left({\dot{\theta }}_{p,i}-i_{m,i}*{\dot{\theta }}_m-\frac{{\dot{e}}_i\left(t\right)}{r_i}\right),\hfill & \hfill \\ \hspace{1em}\hspace{1em}\text{for}\mathrm{\hspace{0.17em}\hspace{0.17em}}{\theta }_{p,i}-i_{m,i}*{\theta }_m-\frac{e_i\left(t\right)}{r_i}\le -{\mathrm{\Delta }}_i,\hfill & \hfill \end{array}\\ M_{c,i}=\{\begin{array}{cc}{k}_{t,i}\left({\theta }_{p,i}-i_{m,i}*{\theta }_m-\frac{e_i\left(t\right)}{r_i}-{\mathrm{\Delta }}_i\right)\hfill & \hfill \\ \hspace{1em}+c_{t,i}\left({\dot{\theta }}_{p,i}-i_{m,i}*{\dot{\theta }}_m-\frac{{\dot{e}}_i\left(t\right)}{r_i}\right),\hfill & \hfill \\ \hspace{1em}\hspace{1em}\text{for}\mathrm{\hspace{0.17em}\hspace{0.17em}}{\theta }_{p,i}-i_{m,i}*{\theta }_m-\frac{e_i\left(t\right)}{r_i}\ge {\mathrm{\Delta }}_i,\hfill & \hfill \\ \mathrm{0},\hspace{1em}\hspace{1em}\text{for}\mathrm{\hspace{0.17em}\hspace{0.17em}}-{\mathrm{\Delta }}_i<{\theta }_{p,i}-i_{m,i}*{\theta }_m-\frac{e_i\left(t\right)}{r_i}<{\mathrm{\Delta }}_i,\hfill & \hfill \\ k_{t,i}\left({\theta }_{p,i}-i_{m,i}*{\theta }_m-\frac{e_i\left(t\right)}{r_i}+{\mathrm{\Delta }}_i\right)\hfill & \hfill \\ \hspace{1em}+c_{t,i}\left({\dot{\theta }}_{p,i}-i_{m,i}*{\dot{\theta }}_m-\frac{{\dot{e}}_i\left(t\right)}{r_i}\right),\hfill & \hfill \\ \hspace{1em}\hspace{1em}\text{for}\mathrm{\hspace{0.17em}\hspace{0.17em}}{\theta }_{p,i}-i_{m,i}*{\theta }_m-\frac{e_i\left(t\right)}{r_i}\le -{\mathrm{\Delta }}_i.\hfill & \hfill \end{array}\end{array}$ (3)

The system physical parameters are defined as

$\begin{array}{c}{\mathrm{\Delta }}_i=\frac{{\mathrm{\Delta }}_b^i}{r_i},\hspace{1em}{k}_{f,i}=k_i{r}_i,\hspace{1em}{k}_{t,i}=k_{f,i}{r}_i=k_i{r}_i^{\mathrm{2}},\\ c_{f,i}=c_i{r}_i,\hspace{1em}{c}_{t,i}=c_{f,i}{r}_i=c_i{r}_i^{\mathrm{2}},\\ i_{m,i}=\frac{r_m}{r_i}\hspace{1em}\left(i=\mathrm{1,2},\dots ,n\right),\end{array}$ (4)

where Δ _i is the relative backlash and is a dimensionless parameter, k_{f, i} and c_{f, i} are elastic mesh force coefficients, k_{t, i} and c_{t, i} are the mesh torque coefficients, and i_{m, i} is the gear transmission ratio. The mesh stiffness k _i , mesh damped c _i , and transmission error e _i (t) are time-varying parameters which can be expressed as the Fourier series form

$\begin{array}{c}{k}_i\left(t\right)=k_{i,\mathrm{0}}+\sum _{j=\mathrm{1}}^{\infty }{k}_{i,j}\text{cos}\left(j\mathrm{2}\pi f_{z,i}t+{\varphi }_{i,j}^k\right),\\ c_i\left(t\right)=c_{i,\mathrm{0}}+\sum _{j=\mathrm{1}}^{\infty }{c}_{i,j}\text{cos}\left(j\mathrm{2}\pi f_{z,i}t+{\varphi }_{i,j}^c\right),\\ e_i\left(t\right)=e_{i,\mathrm{0}}+\sum _{j=\mathrm{1}}^{\infty }{e}_{i,j}\text{cos}\left(j\mathrm{2}\pi f_{z,i}t+{\varphi }_{i,j}^e\right),\end{array}$ (5)

where f_{z, i} is the mesh frequency, k_{i, 0}, c_{i, 0}, and e_{i, 0} are the mean mesh stiffness, mean mesh damped, and mean transmission error, respectively, and ϕ_{i, j} ^k , ϕ_{i, j} ^c , and ϕ_{i, j} ^e are the phase angle of mesh stiffness, mesh damped, and transmission error, respectively. Nonlinear parameters are introduced to simplify elastic mesh force and torque, thus the nonlinear parameters φ _i and β _i are defined as

$\begin{array}{c}{\phi }_i=\{\begin{array}{cc}\mathrm{1},\hfill & \left|{\theta }_{p,i}-i_{m,i}*{\theta }_m-\frac{e_i\left(t\right)}{r_i}\right|\ge {\mathrm{\Delta }}_i,\hfill \\ \mathrm{0},\hfill & \left|{\theta }_{p,i}-i_{m,i}*{\theta }_m-\frac{e_i\left(t\right)}{r_i}\right|<{\mathrm{\Delta }}_i,\hfill \end{array}\\ {\beta }_i=\{\begin{array}{cc}-\frac{e_i\left(t\right)}{r_i}-{\mathrm{\Delta }}_i,\hfill & {\theta }_{p,i}-i_{m,i}*{\theta }_m-\frac{e_i\left(t\right)}{r_i}\ge {\mathrm{\Delta }}_i,\hfill \\ \mathrm{0},\hfill & -{\mathrm{\Delta }}_i<{\theta }_{p,i}-i_{m,i}*{\theta }_m-\frac{e_i\left(t\right)}{r_i}<{\mathrm{\Delta }}_i,\hfill \\ -\frac{e_i\left(t\right)}{r_i}+{\mathrm{\Delta }}_i,\hfill & {\theta }_{p,i}-i_{m,i}*{\theta }_m-\frac{e_i\left(t\right)}{r_i}\le -{\mathrm{\Delta }}_i.\hfill \end{array}\end{array}$ (6)

Finally, the elastic mesh force and torque can be written as the following general form:

$\begin{array}{c}{F}_i=k_{f,i}{\phi }_i\left({\theta }_{p,i}-i_{m,i}*{\theta }_m+{\beta }_i\right)+c_{f,i}{\phi }_i\left({\dot{\theta }}_{p,i}-i_{m,i}*{\dot{\theta }}_m-\frac{{\dot{e}}_i\left(t\right)}{r_i}\right),\\ M_{c,i}=k_{t,i}{\phi }_i\left({\theta }_{p,i}-i_{m,i}*{\theta }_m+{\beta }_i\right)+c_{t,i}{\phi }_i\left({\dot{\theta }}_{p,i}-i_{m,i}*{\dot{\theta }}_m-\frac{{\dot{e}}_i\left(t\right)}{r_i}\right).\end{array}$ (7)

2.2. Nonlinear Dynamic Model

To establish a general nonlinear dynamic model and not to lose generality, it assumes that the cutterhead is driven by n motors. If the coupling mass is far less than the motor rotor mass, the coupling inertia can be ignored. In fact, the motor rotor mass cannot be larger than the coupling mass significantly, so the coupling inertia is not ignored. According to torque balance principle, the torque balance equations of the first induction motor yield

$T_{e,\mathrm{1}}=J_{r,\mathrm{1}}{\stackrel{‥}{\theta }}_{\mathrm{1}}+b_{r,\mathrm{1}}{\dot{\theta }}_{\mathrm{1}}+T_{o,\mathrm{1}},$ (8) $T_{o,\mathrm{1}}=J_{z,\mathrm{1}}{\stackrel{‥}{\theta }}_{\mathrm{1}}+b_{z,\mathrm{1}}{\dot{\theta }}_{\mathrm{1}}+M_{\mathrm{1,1}}.$ (9)

For the second induction motor, the corresponding torque balance equations are obtained:

$T_{e,\mathrm{2}}=J_{r,\mathrm{2}}{\stackrel{‥}{\theta }}_{\mathrm{2}}+b_{r,\mathrm{2}}{\dot{\theta }}_{\mathrm{2}}+T_{o,\mathrm{2}},$ (10) $T_{o,\mathrm{2}}=J_{z,\mathrm{2}}{\stackrel{‥}{\theta }}_{\mathrm{2}}+b_{z,\mathrm{2}}{\dot{\theta }}_{\mathrm{2}}+M_{\mathrm{1,2}}.$ (11)

Likewise, the torque balance equations of the nth induction motor will yield

$T_{e,n}=J_{r,n}{\stackrel{‥}{\theta }}_n+b_{r,n}{\dot{\theta }}_n+T_{o,n},$ (12) $T_{o,n}=J_{z,n}{\stackrel{‥}{\theta }}_n+b_{z,n}{\dot{\theta }}_n+M_{\mathrm{1},n},$ (13)

where T _{o, i} is the output torque of the ith motor and M_1,i is the input torque of the ith reducer (i = 1, 2, …, n). The equations (9), (11), and (13) are substituted into (8), (10), and (12), respectively, and then the following equations will be obtained:

$\begin{array}{c}{T}_{e,\mathrm{1}}=\left(J_{r,\mathrm{1}}+J_{z,\mathrm{1}}\right){\stackrel{‥}{\theta }}_{\mathrm{1}}+\left(b_{r,\mathrm{1}}+b_{z,\mathrm{1}}\right){\dot{\theta }}_{\mathrm{1}}+M_{\mathrm{1,1}},\\ T_{e,\mathrm{2}}=\left(J_{r,\mathrm{2}}+J_{z,\mathrm{2}}\right){\stackrel{‥}{\theta }}_{\mathrm{2}}+\left(b_{r,\mathrm{2}}+b_{z,\mathrm{2}}\right){\dot{\theta }}_{\mathrm{2}}+M_{\mathrm{1,2}},\\ T_{e,n}=\left(J_{r,n}+J_{z,n}\right){\stackrel{‥}{\theta }}_n+\left(b_{r,n}+b_{z,n}\right){\dot{\theta }}_n+M_{\mathrm{1},n}.\end{array}$ (14)

The system parameters are defined as J_{d, i} = J_{r, i} + J_{z, i}, b_{d, i} = b_{r, i} + b_{z, i} (i = 1, 2, …, n). Then (14) can be rewritten as the general form

$T_{e,i}=J_{d,i}{\stackrel{‥}{\theta }}_i+b_{d,i}{\dot{\theta }}_i+M_{\mathrm{1},i}\hspace{1em}\left(i=\mathrm{1,2},\dots ,n\right).$ (15)

The input and output relationships of the ith reducer can be described by the equation

${\theta }_i=q{\theta }_{p,i},\hspace{1em}{M}_{\mathrm{2},i}=q{M}_{\mathrm{1},i}\hspace{1em}\left(i=\mathrm{1,2},\dots ,n\right),$ (16)

where M_2,i is the output torque of the ith reducer. Torque balance equation of the reduce-i or pinion-i will yield

$M_{\mathrm{2},i}=J_{w,i}{\stackrel{‥}{\theta }}_{p,i}+b_{w,i}{\dot{\theta }}_{p,i}+T_{p,i}\hspace{1em}\left(i=\mathrm{1,2},\dots ,n\right),$ (17) $T_{p,i}=J_{p,i}{\stackrel{‥}{\theta }}_{p,i}+b_{p,i}{\dot{\theta }}_{p,i}+M_{c,i}\hspace{1em}\left(i=\mathrm{1,2},\dots ,n\right),$ (18)

where T _{p, i} is the input torque of pinion-i and M _{c, i} is the elastic mesh torque of pinion-i or gear pair-i. Substituting (18) into (17), then the torque balance equation can be written as

$\begin{array}{c}{M}_{\mathrm{2},i}=\left(J_{p,i}+J_{w,i}\right){\stackrel{‥}{\theta }}_{p,i}+\left(b_{p,i}+b_{w,i}\right){\dot{\theta }}_{p,i}+M_{c,i}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\left(i=\mathrm{1,2},\dots ,n\right).\end{array}$ (19)

The system parameters are defined as J_{c, i} = J_{p, i} + J_{w, i}, b_{c, i} = b_{p, i} + b_{w, i} (i = 1, 2, …, n). Then, (19) can be rewritten as the following general form:

$M_{\mathrm{2},i}=J_{c,i}{\stackrel{‥}{\theta }}_{p,i}+b_{c,i}{\dot{\theta }}_{p,i}+M_{c,i}\hspace{1em}\left(i=\mathrm{1,2},\dots ,n\right).$ (20)

The elastic mesh torque of the pinion-i or gear pair-i has been presented in Section 2.1, and the mesh torque of pinion-i will be obtained as follows:

$M_{c,i}=k_{t,i}{\phi }_i\left({\theta }_{p,i}-i_{m,i}*{\theta }_m+{\beta }_i\right)+c_{t,i}{\phi }_i\left({\dot{\theta }}_{p,i}-i_{m,i}*{\dot{\theta }}_m-\frac{{\dot{e}}_i\left(t\right)}{r_i}\right)\hspace{1em}\left(i=\mathrm{1,2},\dots ,n\right).$ (21)

The torque balance equations of the large gear can be obtained as

$\begin{array}{c}{M}_m=J_m{\stackrel{‥}{\theta }}_m+b_m{\dot{\theta }}_m+T_L,\\ M_m=i_{m,\mathrm{1}}{M}_{c,\mathrm{1}}+i_{m,\mathrm{2}}{M}_{c,\mathrm{2}}+·+i_{m,n}{M}_{c,n}=\sum _{k=\mathrm{1}}^n{i}_{m,k}{M}_{c,k},\end{array}$ (22)

where T _L is total load torque of the shield TBM cutterhead. Many factors such as the cutterhead's design parameters, physical structure, and geology conditions directly affect the load torque. The cutterhead design parameters and physical structure include the cutting depth, cutterhead opening ratio, excavation diameter, cutter tools distribution, and cutterhead forms. Geology conditions include soil properties, rocks properties, and boulders. The actual load torque will change with the geology conditions. The shield TBM cutterhead's load torque is estimated and calculated in [13–16]. The load torque consists of the following six parts which are presented in [15]:

$T_L=T_{\mathrm{1}}+T_{\mathrm{2}}+T_{\mathrm{3}}+T_{\mathrm{4}}+T_{\mathrm{5}}+T_{\mathrm{6}}=\sum _{j=\mathrm{1}}^{\mathrm{6}}{T}_j,$ (23)

where T₁ is the resistant torque of soil and rocks, T₂ is the friction torque of soil and rocks which chafes with the front of the cutterhead, T₃ is the friction torque of soil and rocks which chafes with the back of the cutterhead, T₄ is the friction torque of soil and rocks which chafes with bulkhead, T₅ is the stirring torque of soil and rocks by cutterhead stirring rod, and T₆ is the friction torque of cutterhead bearings and sealed chamber. Detailed derivations and descriptions of the cutterhead load torque and its parameters can be found in [15]. Therefore, calculation formulas of the load torque are directly presented here as

$\begin{array}{c}{T}_{\mathrm{1}}=\frac{D^{\mathrm{2}}{\nu }_s{q}_u}{\left(\mathrm{8}{N}_c\right)},\hspace{1em}\hspace{1em}{T}_{\mathrm{2}}=\frac{\pi \gamma H_{\mathrm{0}}\mu D^{\mathrm{3}}}{\mathrm{12}},\\ T_{\mathrm{3}}=\frac{\pi \gamma H_{\mathrm{0}}{\mu }_{\mathrm{1}}\left(D^{\mathrm{3}}-D_{\mathrm{1}}^{\mathrm{3}}\right)}{\mathrm{12}},\\ T_{\mathrm{4}}=\frac{\pi \gamma H_{\mathrm{0}}{\mu }_{\mathrm{1}}B{\left(D_{\mathrm{1}}+D_{\mathrm{2}}\right)}^{\mathrm{2}}}{\left(\mathrm{8}\text{cos}\alpha \right)},\hspace{1em}\hspace{1em}\text{tan}\alpha =\frac{\left(D_{\mathrm{1}}-D_{\mathrm{2}}\right)}{\left(\mathrm{2}B\right)},\\ T_{\mathrm{5}}=\gamma H_{\mathrm{0}}{\mu }_{\mathrm{1}}{D}_b{L}_b{R}_b,\hspace{1em}\hspace{1em}{T}_{\mathrm{6}}=\frac{\left(\mathrm{1.414}{G}_{\mathrm{0}}+K_{H}H\right){\mu }_{\mathrm{2}}{D}_{g\mathrm{0}}}{\mathrm{2}}.\end{array}$ (24)

Equations (15)–(23) constitute the NLTV dynamic model of shield TBM cutterhead driving system. Therefore, the general NLTV dynamic model is obtained as

$\begin{array}{c}{T}_{e,i}=J_{d,i}{\stackrel{‥}{\theta }}_i+b_{d,i}{\dot{\theta }}_i+M_{\mathrm{1},i},\hspace{1em}\hspace{1em}{\theta }_i=q{\theta }_{p,i},\\ M_{\mathrm{2},i}=q{M}_{\mathrm{1},i},\hspace{1em}\hspace{1em}{M}_{\mathrm{2},i}=J_{c,i}{\stackrel{‥}{\theta }}_{p,i}+b_{c,i}{\dot{\theta }}_{p,i}+M_{c,i},\\ M_{c,i}=k_{t,i}{\phi }_i\left({\theta }_{p,i}-i_{m,i}*{\theta }_m+{\beta }_i\right)+c_{t,i}{\phi }_i\left({\dot{\theta }}_{p,i}-i_{m,i}*{\dot{\theta }}_m-\frac{{\dot{e}}_i\left(t\right)}{r_i}\right)\hspace{1em}\left(i=\mathrm{1,2},\dots ,n\right),\\ M_m=J_m{\stackrel{‥}{\theta }}_m+b_m{\dot{\theta }}_m+T_L,\\ M_m=\sum _{k=\mathrm{1}}^n{i}_{m,k}{M}_{c,k},\hspace{1em}\hspace{1em}{T}_L=\sum _{j=\mathrm{1}}^{\mathrm{6}}{T}_j.\end{array}$ (25)

The NLTV dynamic model (25) can be combined and simplified, and then the nonlinear dynamic model is expressed as

$\begin{array}{c}q{T}_{e,i}=\left(q^{\mathrm{2}}{J}_{d,i}+J_{c,i}\right){\stackrel{‥}{\theta }}_{p,i}+\left(q^{\mathrm{2}}{b}_{d,i}+b_{c,i}\right){\dot{\theta }}_{p,i}+k_{t,i}{\phi }_i\left({\theta }_{p,i}-i_{m,i}*{\theta }_m+{\beta }_i\right)+c_{t,i}{\phi }_i\left({\dot{\theta }}_{p,i}-i_{m,i}*{\dot{\theta }}_m-\frac{{\dot{e}}_i\left(t\right)}{r_i}\right),\\ J_m{\stackrel{‥}{\theta }}_m+b_m{\dot{\theta }}_m+T_L\hspace{0.17em}=\sum _{k=\mathrm{1}}^n{i}_{m,k}\{\phantom{\left({\dot{\theta }}_{p,k}-i_{m,k}*{\dot{\theta }}_m-\frac{{\dot{e}}_i\left(t\right)}{r_i}\right)}{k}_{t,k}{\phi }_k\left({\theta }_{p,k}-i_{m,k}*{\theta }_m+{\beta }_k\right)+c_{t,k}{\phi }_k\left({\dot{\theta }}_{p,k}-i_{m,k}*{\dot{\theta }}_m-\frac{{\dot{e}}_k\left(t\right)}{r_k}\right)\}\\ \hspace{0.17em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.5em}\left(i=\mathrm{1,2},\dots ,n\right).\end{array}$ (26)

The nonlinear dynamic model can be transformed into the state space dynamic model via selecting suitable state variables, output variables, and control variables. Where x_ρ is a state vector, y_ρ is an output vector, and u_ρ is a control vector,

$\begin{array}{c}{\dot{\theta }}_{p,\mathrm{1}}={\omega }_{p,\mathrm{1}},\dots ,{\dot{\theta }}_{p,n}={\omega }_{p,n},\hspace{1em}{\dot{\theta }}_m={\omega }_m,\\ x_{\rho }={\left(\begin{array}{cccccccccc}{\theta }_{p,\mathrm{1}}& {\theta }_{p,\mathrm{2}}& ·& {\theta }_{p,n}& {\theta }_m& {\omega }_{p,\mathrm{1}}& {\omega }_{p,\mathrm{2}}& ·& {\omega }_{p,n}& {\omega }_m\end{array}\right)}^{\text{T}}∊{ℝ}^{\mathrm{2}n+\mathrm{2}},\\ y_{\rho }={\left(\begin{array}{ccccc}{\omega }_{p,\mathrm{1}}& {\omega }_{p,\mathrm{2}}& ·& {\omega }_{p,n}& {\omega }_m\end{array}\right)}^{\text{T}}∊{ℝ}^{n+\mathrm{1}},\\ u_{\rho }=(\begin{array}{ccccccccc}{T}_{e,\mathrm{1}}& T_{e,\mathrm{2}}& ·& T_{e,n}& T_L& {\beta }_{\mathrm{1}}& {\beta }_{\mathrm{2}}& ·& {\beta }_n\end{array}{\begin{array}{cccc}{\dot{e}}_{\mathrm{1}}\left(t\right)& {\dot{e}}_{\mathrm{2}}\left(t\right)& ·& {\dot{e}}_n\left(t\right)\end{array})}^{\text{T}}∊{ℝ}^{3n+\mathrm{1}}.\end{array}$ (27)

The MIMO state space model of the nonlinear time-varying dynamic model is obtained as

$\begin{array}{c}{\dot{x}}_{\rho }=A_{\rho }{x}_{\rho }+B_{\rho }{u}_{\rho },A_{\rho }=\left(\begin{array}{cc}{A}_{\mathrm{11}}^{\rho }& A_{\mathrm{12}}^{\rho }\\ A_{\mathrm{21}}^{\rho }& A_{\mathrm{22}}^{\rho }\end{array}\right),\hspace{1em}{B}_{\rho }=\left(\begin{array}{ccc}{B}_{\mathrm{11}}^{\rho }& B_{\mathrm{12}}^{\rho }& B_{\mathrm{13}}^{\rho }\\ B_{\mathrm{21}}^{\rho }& B_{\mathrm{22}}^{\rho }& B_{\mathrm{23}}^{\rho }\end{array}\right),\\ y_{\rho }=C_{\rho }{x}_{\rho },\hspace{1em}{C}_{\rho }=\left(\begin{array}{cc}{C}_{\mathrm{11}}^{\rho }& C_{\mathrm{12}}^{\rho }\end{array}\right),\end{array}$ (28)

where A_ρ is the state matrix, B_ρ is a control matrix, and C_ρ is an output matrix. The matrices A_ρ, B_ρ, and C_ρ are presented in Appendix A.

3. General LTV Dynamic Model of Cutterhead Driving System

To establish general linear dynamic model, nonlinear factors in cutterhead driving system are ignored. Based on gear mesh dynamic in [19–24], the relative position, elastic mesh force, and mesh torque of linear dynamic model will be obtained, respectively, as follows:

$\begin{array}{c}{p}_i=r_i*{\theta }_{p,i}-r_m*{\theta }_m=r_i\left({\theta }_{p,i}-i_{m,i}*{\theta }_m\right),\\ F_i=k_i{p}_i+c_i{\dot{p}}_i=k_{f,i}\left({\theta }_{p,i}-i_{m,i}*{\theta }_m\right)+c_{f,i}\left({\dot{\theta }}_{p,i}-i_{m,i}*{\dot{\theta }}_m\right),\\ M_{c,i}=k_i{p}_i{r}_i+c_i{\dot{p}}_i{r}_i=k_{t,i}\left({\theta }_{p,i}-i_{m,i}*{\theta }_m\right)+c_{t,i}\left({\dot{\theta }}_{p,i}-i_{m,i}*{\dot{\theta }}_m\right).\end{array}$ (29)

Therefore, from the previous sections, the torque balance equations of ith induction motor yield

$T_{e,i}=J_{d,i}{\stackrel{‥}{\theta }}_i+b_{d,i}{\dot{\theta }}_i+M_{\mathrm{1},i}\hspace{1em}\left(i=\mathrm{1,2},\dots ,n\right),$ (30)

where system parameters J_{d, i} and b_{d, i} are J_{r, i} + J_{z, i} and b_{r, i} + b_{z, i} (i = 1, 2, …, n), respectively. The input and output relationships of the ith reducer can be described by the equation

${\theta }_i=q{\theta }_{p,i},\hspace{1em}{M}_{\mathrm{2},i}=q{M}_{\mathrm{1},i}\hspace{1em}\left(i=\mathrm{1,2},\dots ,n\right).$ (31)

Torque balance equation of the reduce-i or pinion-i will yield

$M_{\mathrm{2},i}=J_{c,i}{\stackrel{‥}{\theta }}_{p,i}+b_{c,i}{\dot{\theta }}_{p,i}+M_{c,i}\hspace{1em}\left(i=\mathrm{1,2},\dots ,n\right),$ (32)

where system parameters J_{c, i} and b_{c, i} are J_{p, i} + J_{w, i} and b_{p, i} + b_{w, i} (i = 1, 2, …, n), respectively. The mesh torque of pinion-i will be obtained as follows:

$M_{c,i}=k_{t,i}\left({\theta }_{p,i}-i_{m,i}*{\theta }_m\right)+c_{t,i}\left({\dot{\theta }}_{p,i}-i_{m,i}*{\dot{\theta }}_m\right)\left(i=\mathrm{1,2},\dots ,n\right).$ (33)

The torque balance equations of the large gear can be obtained as

$\begin{array}{c}{M}_m=J_m{\stackrel{‥}{\theta }}_m+b_m{\dot{\theta }}_m+T_L,\\ M_m=i_{m,\mathrm{1}}{M}_{c,\mathrm{1}}+i_{m,\mathrm{2}}{M}_{c,\mathrm{2}}+·+i_{m,n}{M}_{c,n}=\sum _{k=\mathrm{1}}^n{i}_{m,k}{M}_{c,k}.\end{array}$ (34)

Equations (30)–(34) constitute the LTV dynamic model of the shield TBM cutterhead driving system. Therefore, the general LTV dynamic model is obtained as

$\begin{array}{c}{T}_{e,i}=J_{d,i}{\stackrel{‥}{\theta }}_i+b_{d,i}{\dot{\theta }}_i+M_{\mathrm{1},i},\hspace{1em}{\theta }_i=q{\theta }_{p,i},\\ M_{\mathrm{2},i}=q{M}_{\mathrm{1},i},\hspace{1em}{M}_{\mathrm{2},i}=J_{c,i}{\stackrel{‥}{\theta }}_{p,i}+b_{c,i}{\dot{\theta }}_{p,i}+M_{c,i}\hspace{1em}\left(i=\mathrm{1,2},\dots ,n\right),\\ M_{c,i}=k_{t,i}\left({\theta }_{p,i}-i_{m,i}\hspace{0.17em}*\hspace{0.17em}{\theta }_{m,i}\right)+c_{t,i}\left({\dot{\theta }}_{p,i}-i_{m,i}\hspace{0.17em}*{\dot{\theta }}_{m,i}\hspace{0.17em}\right),\\ M_m=J_m{\stackrel{‥}{\theta }}_m+b_m{\dot{\theta }}_m+T_L,\hspace{1em}{M}_m=\sum _{k=\mathrm{1}}^n{i}_{m,k}{M}_{c,k}.\end{array}$ (35)

The linear dynamic model (35) can be combined and simplified. Therefore, the linear dynamic model is expressed as

$\begin{array}{c}q{T}_{e,i}=\left(q^{\mathrm{2}}{J}_{d,i}+J_{c,i}\right){\stackrel{‥}{\theta }}_{p,i}+\left(q^{\mathrm{2}}{b}_{d,i}+b_{c,i}\right){\dot{\theta }}_{p,i}+k_{t,i}\left({\theta }_{p,i}-i_{m,i}*{\theta }_m\right)+c_{t,i}\left({\dot{\theta }}_{p,i}-i_{m,i}*{\dot{\theta }}_m\right),\\ J_m{\stackrel{‥}{\theta }}_m+b_m{\dot{\theta }}_m+T_L=\sum _{k=\mathrm{1}}^n{i}_{m,k}\{k_{t,k}\left({\theta }_{p,k}-i_{m,k}*{\theta }_m\right)+c_{t,k}\left({\dot{\theta }}_{p,k}-i_{m,k}*{\dot{\theta }}_m\right)\}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.5em}\left(i=\mathrm{1,2},\dots ,n\right).\end{array}$ (36)

The linear dynamic model can be transformed into the state space model via selecting suitable state variables, output variables, and control variables. Where x_σ is a state vector, y_σ is an output vector, and u_σ is a control vector,

$\begin{array}{c}{\dot{\theta }}_{p,\mathrm{1}}={\omega }_{p,\mathrm{1}},\dots ,{\dot{\theta }}_{p,n}={\omega }_{p,n},\hspace{1em}{\dot{\theta }}_m={\omega }_m,\\ u_{\sigma }={\left(\begin{array}{ccccc}{T}_{e,\mathrm{1}}& T_{e,\mathrm{2}}& ·& T_{e,n}& T_L\end{array}\right)}^{\text{T}}∊{ℝ}^{n+\mathrm{1}},\\ x_{\sigma }={\left(\begin{array}{cccccccccc}{\theta }_{p,\mathrm{1}}& {\theta }_{p,\mathrm{2}}& ·& {\theta }_{p,n}& {\theta }_m& {\omega }_{p,\mathrm{1}}& {\omega }_{p,\mathrm{2}}& ·& {\omega }_{p,n}& {\omega }_m\end{array}\right)}^{\text{T}}∊{ℝ}^{\mathrm{2}n+\mathrm{2}},\\ y_{\sigma }={\left(\begin{array}{ccccc}{\omega }_{p,\mathrm{1}}& {\omega }_{p,\mathrm{2}}& ·& {\omega }_{p,n}& {\omega }_m\end{array}\right)}^{\text{T}}∊{ℝ}^{n+\mathrm{1}}.\end{array}$ (37)

The MIMO state space model of the linear time-varying dynamic model is obtained as

$\begin{array}{c}{\dot{x}}_{\sigma }=A_{\sigma }{x}_{\sigma }+B_{\sigma }{u}_{\sigma },\hspace{1em}{A}_{\sigma }=\left(\begin{array}{cc}{A}_{\mathrm{11}}^{\sigma }& A_{\mathrm{12}}^{\sigma }\\ A_{\mathrm{21}}^{\sigma }& A_{\mathrm{22}}^{\sigma }\end{array}\right),\hspace{0.17em}\hspace{0.17em}{B}_{\sigma }=\left(\begin{array}{c}{B}_{\mathrm{11}}^{\sigma }\\ B_{\mathrm{21}}^{\sigma }\end{array}\right),\\ y_{\sigma }=C_{\sigma }{x}_{\sigma },\hspace{1em}{C}_{\sigma }=\left(\begin{array}{cc}{C}_{\mathrm{11}}^{\sigma }& C_{\mathrm{12}}^{\sigma }\end{array}\right),\end{array}$ (38)

where A_σ is the state matrix, B_σ is a control matrix, and C_σ is an output matrix. The matrices A_σ, B_σ, and C_σ are presented in Appendix B.

3.1. Analysis of General LTV Dynamic Model

The general linear dynamic model of cutterhead driving system is established in the previous section. In order to analyze linear dynamic model in theory, some time-varying system parameters such as mesh stiffness, mesh damped, and load torque are replaced by mean mesh stiffness, mean mesh damped, and mean load torque. Therefore, the linear dynamic model seemed to be the linear time-invariant system. Then, the Laplace transform of (36) will be obtained as follows:

$q{T}_{e,i}\left(s\right)=s^{\mathrm{2}}\left(q^{\mathrm{2}}{J}_{d,i}+J_{c,i}\right){\theta }_{p,i}\left(s\right)+s\left(q^{\mathrm{2}}{b}_{d,i}+b_{c,i}\right){\theta }_{p,i}\left(s\right)+\left(k_{t,i}+c_{t,i}s\right)\left({\theta }_{p,i}\left(s\right)-i_{m,i}*{\theta }_m\left(s\right)\right),$ (39) $J_m{s}^{\mathrm{2}}{\theta }_m\left(s\right)+b_{m}s{\theta }_m\left(s\right)+T_L\left(s\right)\hspace{0.17em}=\sum _{k=\mathrm{1}}^n{i}_{m,k}\left\{\left(k_{t,k}+c_{t,k}s\right)\left({\theta }_{p,k}\left(s\right)-i_{m,k}*{\theta }_m\left(s\right)\right)\right\}\hspace{0.17em}\hspace{1em}\hspace{1em}\hspace{1em}\left(i=\mathrm{1,2},\dots ,n\right).$ (40)

For the symbol simplicity, the system parameters are defined as J _i = q²J_{d, i} + J_{c, i}, b _i = q²b_{d, i} + b_{c, i} (i = 1, 2, …, n). Then, (39) can be equivalently expressed as

$\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\theta }_{p,i}\left(s\right)=\frac{q{T}_{e,i}\left(s\right)+i_{m,i}\left(k_{t,i}+c_{t,i}s\right){\theta }_m\left(s\right)}{J_i{s}^{\mathrm{2}}+\left(b_i+c_{t,i}\right)s+k_{t,i}}\left(J_i=q^{\mathrm{2}}{J}_{d,i}+J_{c,i},b_i=q^{\mathrm{2}}{b}_{d,i}+b_{c,i},\hspace{0.17em}\hspace{0.17em}i=\mathrm{1,2},\dots ,n\right).$ (41)

Substituting (41) into (40), then (40) can be written as

$J_m{s}^{\mathrm{2}}{\theta }_m\left(s\right)+b_{m}s{\theta }_m\left(s\right)+T_L\left(s\right)\hspace{0.17em}=\sum _{k=\mathrm{1}}^n\left(\frac{q{i}_{m,k}\left(k_{t,k}+c_{t,k}s\right)T_{e,k}\left(s\right)}{J_k{s}^{\mathrm{2}}+\left(b_k+c_{t,k}\right)s+k_{t,k}}\right)\hspace{0.17em}-\sum _{k=\mathrm{1}}^n\left(\frac{i_{m,k}\left(k_{t,k}+c_{t,k}s\right)\left(i_{m,k}{J}_k{s}^{\mathrm{2}}+i_{m,k}{b}_{k}s\right){\theta }_m\left(s\right)}{J_k{s}^{\mathrm{2}}+\left(b_k+c_{t,k}\right)s+k_{t,k}}\right).$ (42)

It has the relationship ω _m (s) = sθ _m (s), and then (42) can be further equivalently transformed into the following equations:

$\begin{array}{c}\left(J_{m}s+b_m\right){\omega }_m\left(s\right)\hspace{0.17em}+\sum _{k=\mathrm{1}}^n\left(\frac{i_{m,k}\left(k_{t,k}+c_{t,k}s\right)\left(i_{m,k}{J}_{k}s+i_{m,k}{b}_k\right){\omega }_m\left(s\right)}{J_k{s}^{\mathrm{2}}+\left(b_k+c_{t,k}\right)s+k_{t,k}}\right)\hspace{0.17em}+T_L\left(s\right)=\sum _{k=\mathrm{1}}^n\left(\frac{q{i}_{m,k}\left(k_{t,k}+c_{t,k}s\right)T_{e,k}\left(s\right)}{J_k{s}^{\mathrm{2}}+\left(b_k+c_{t,k}\right)s+k_{t,k}}\right)\end{array}$ (43) $⟺\{\phantom{\sum _{k=\mathrm{1}}^n\left(\frac{i_{m,k}\left(k_{t,k}+c_{t,k}s\right)\left(i_{m,k}{J}_{k}s+i_{m,k}{b}_k\right)}{J_k{s}^{\mathrm{2}}+\left(b_k+c_{t,k}\right)s+k_{t,k}}\right)}\left(J_{m}s+b_m\right)\hspace{1em}\hspace{1em}+\sum _{k=\mathrm{1}}^n\left(\frac{i_{m,k}\left(k_{t,k}+c_{t,k}s\right)\left(i_{m,k}{J}_{k}s+i_{m,k}{b}_k\right)}{J_k{s}^{\mathrm{2}}+\left(b_k+c_{t,k}\right)s+k_{t,k}}\right)\}{\omega }_m\left(s\right)\hspace{0.17em}=\sum _{k=\mathrm{1}}^n\left(\frac{q{i}_{m,k}\left(k_{t,k}+c_{t,k}s\right)T_{e,k}\left(s\right)}{J_k{s}^{\mathrm{2}}+\left(b_k+c_{t,k}\right)s+k_{t,k}}\right)-T_L\left(s\right)$ (44) $⟺{\omega }_m\left(s\right)=\{\phantom{\sum _{k=\mathrm{1}}^n\left(\frac{i_{m,k}\left(k_{t,k}+c_{t,k}s\right)\left(i_{m,k}{J}_{k}s+i_{m,k}{b}_k\right)}{J_k{s}^{\mathrm{2}}+\left(b_k+c_{t,k}\right)s+k_{t,k}}\right)}\left(J_{m}s+b_m\right){+\sum _{k=\mathrm{1}}^n\left(\frac{i_{m,k}\left(k_{t,k}+c_{t,k}s\right)\left(i_{m,k}{J}_{k}s+i_{m,k}{b}_k\right)}{J_k{s}^{\mathrm{2}}+\left(b_k+c_{t,k}\right)s+k_{t,k}}\right)\}}^{-\mathrm{1}}\times \left\{\sum _{k=\mathrm{1}}^n\left(\frac{q{i}_{m,k}\left(k_{t,k}+c_{t,k}s\right)T_{e,k}\left(s\right)}{J_k{s}^{\mathrm{2}}+\left(b_k+c_{t,k}\right)s+k_{t,k}}\right)-T_L\left(s\right)\right\}.$ (45)

Therefore, the cutterhead speed transfer function is finally obtained. The cutterhead speed transfer function shows that the dynamic performances are affected by the number of driving motors and physical parameters. To further simplify transfer function (45), it assumes that all the components of shield TBM cutterhead driving system have identical physical parameters and properties. It is stated as follows:

$\begin{array}{c}{J}_i=J,\hspace{1em}{b}_i=b,\hspace{1em}{J}_{d,i}=J_d,\hspace{1em}{J}_{c,i}=J_c,\\ b_{d,i}=b_d,\hspace{1em}{b}_{c,i}=b_c,\hspace{1em}{k}_{t,i}=k_t,\hspace{1em}{c}_{t,i}=c_t,\\ i_{m,i}=i_m\hspace{1em}\left(i=\mathrm{1,2},\dots ,n,\hspace{0.17em}\hspace{0.17em}J=q^{\mathrm{2}}{J}_d+J_c,\hspace{0.17em}\hspace{0.17em}b=q^{\mathrm{2}}{b}_d+b_c\right).\end{array}$ (46)

Therefore, the cutterhead speed equation (45) could be equivalently expressed as the following:

$\left\{\left(J_{m}s+b_m\right)\left(J{s}^{\mathrm{2}}+\left(b+c_t\right)s+k_t\right)+n{i}_m^{\mathrm{2}}\left(k_t+c_{t}s\right)\left(Js+b\right)\right\}\hspace{0.17em}\times {\omega }_m\left(s\right)\hspace{0.17em}=q{i}_m\left(k_t+c_{t}s\right)\sum _{i=\mathrm{1}}^n{T}_{e,i}\left(s\right)\hspace{0.17em}-T_L\left(s\right)\left(J{s}^{\mathrm{2}}+\left(b+c_t\right)s+k_t\right)\hspace{0.17em}\Rightarrow {\omega }_m\left(s\right)=D{\left(s\right)}^{-\mathrm{1}}\times \left\{N\left(s\right)\sum _{i=\mathrm{1}}^n{T}_{e,i}\left(s\right)-N_L\left(s\right)T_L\left(s\right)\right\},$ (47) $\begin{array}{c}N\left(s\right)=q{i}_m\left(k_t+c_{t}s\right),\hspace{1em}\hspace{1em}{N}_L\left(s\right)=\left(J{s}^{\mathrm{2}}+\left(b+c_t\right)s+k_t\right),\\ D\left(s\right)=\left(J_{m}s+b_m\right)\left(J{s}^{\mathrm{2}}+\left(b+c_t\right)s+k_t\right)+n{i}_m^{\mathrm{2}}\left(k_t+c_{t}s\right)\left(Js+b\right)=J_{m}J{s}^{\mathrm{3}}+\left(J_{m}b+J_m{c}_t+J{b}_m+nJ{i}_m^{\mathrm{2}}{c}_t\right)s^{\mathrm{2}}+\left(J_m{k}_t+b_{m}b+b_m{c}_t+nJ{i}_m^{\mathrm{2}}{k}_t+n{i}_m^{\mathrm{2}}b{c}_t\right)s+\left(b_m{k}_t+n{i}_m^{\mathrm{2}}b{k}_t\right),\end{array}$ (48)

where D(s) is the denominator (characteristic polynomial) of the cutterhead speed equation (47) and N(s) and N _L (s) are the torque numerator and load torque numerator, respectively,

$\begin{array}{c}{\mathrm{\Phi }}_{e,k}\left(s\right)=\frac{{\omega }_m\left(s\right)}{T_{e,k}\left(s\right)}=\frac{N\left(s\right)}{D\left(s\right)}\hspace{1em}\left(k=\mathrm{1,2},\dots ,n\right)\left\{T_{e,j}\left(s\right)=\mathrm{0}\hspace{0.17em}\left(j\ne k,j=\mathrm{1,2},\dots ,n\right),\hspace{0.17em}\hspace{0.17em}{T}_L\left(s\right)=\mathrm{0}\right\},\\ {\mathrm{\Phi }}_{e,t}\left(s\right)=\frac{{\omega }_m\left(s\right)}{T_{e,t}\left(s\right)}=\frac{N\left(s\right)}{D\left(s\right)},\hspace{1em}{T}_{e,t}\left(s\right)={\sum }_{i=\mathrm{1}}^n{T}_{e,i}\left(s\right)\hspace{0.17em}\hspace{0.17em}\left\{T_L\left(s\right)=\mathrm{0}\right\},\\ {\mathrm{\Phi }}_L\left(s\right)=\frac{{\omega }_m\left(s\right)}{T_L\left(s\right)}=\frac{-N_L\left(s\right)}{D\left(s\right)}\hspace{1em}\left\{T_{e,t}\left(s\right)={\sum }_{i=\mathrm{1}}^n{T}_{e,i}\left(s\right)=\mathrm{0}\right\}.\end{array}$ (49)

Thus, the speed torque transfer function and speed load torque transfer function for the shield TBM cutterhead driving system are acquired. Where T_{e, t}(s) is the total torque of all the driving motors, Φ_{e, k}(s) is the cutterhead speed torque transfer function of driving motor-k, Φ_{e, t}(s) is the cutterhead speed total torque transfer function, and Φ _L (s) is thef cutterhead speed load torque transfer function. The transfer function equation (49) shows that the D(s) is the poles polynomial and N(s) and N _L (s) are zeros polynomial. The Routh-Hurwitz stability of poles polynomial D(s) is determined by the following equation:

$\begin{array}{c}D\left(s\right)=a_{\mathrm{0}}{s}^{\mathrm{3}}+a_{\mathrm{1}}{s}^{\mathrm{2}}+a_{\mathrm{2}}s+a_{\mathrm{3}}∊H\left(s\right)⟺a_{\mathrm{0}}>\mathrm{0},\hspace{1em}{a}_{\mathrm{1}}>\mathrm{0},\hspace{1em}{a}_{\mathrm{1}}{a}_{\mathrm{2}}-a_{\mathrm{0}}{a}_{\mathrm{3}}>\mathrm{0},a_{\mathrm{3}}>\mathrm{0},\hspace{1em}{a}_{\mathrm{0}}=J_{m}J,\\ a_{\mathrm{1}}=J_{m}b+J_m{c}_t+J{b}_m+nJ{i}_m^{\mathrm{2}}{c}_t,\\ a_{\mathrm{2}}=J_m{k}_t+b_{m}b+b_m{c}_t+nJ{i}_m^{\mathrm{2}}{k}_t+n{i}_m^{\mathrm{2}}b{c}_t,\\ a_{\mathrm{3}}=b_m{k}_t+n{i}_m^{\mathrm{2}}b{k}_t.\end{array}$ (50)

H(s) is the set of stable Hurwitz polynomial which has negative real part. All physical parameters of the cutterhead driving system are the positive real; therefore, it has relationships a₀ > 0, a₁ > 0, a₂ > 0, and a₃ > 0. It needs to check the relationships a₁a₂ – a₀a₃ > 0, and then the following equation will be obtained:

$\begin{array}{c}{a}_{\mathrm{1}}{a}_{\mathrm{2}}-a_{\mathrm{0}}{a}_{\mathrm{3}}=\left(J_{m}b+J_m{c}_t+J{b}_m+nJ{i}_m^{\mathrm{2}}{c}_t\right)\times \left(J_m{k}_t+b_{m}b+b_m{c}_t+nJ{i}_m^{\mathrm{2}}{k}_t+n{i}_m^{\mathrm{2}}b{c}_t\right)-J_{m}J\left(b_m{k}_t+n{i}_m^{\mathrm{2}}b{k}_t\right)\\ ⟺a_{\mathrm{1}}{a}_{\mathrm{2}}-a_{\mathrm{0}}{a}_{\mathrm{3}}\hspace{0.17em}=J_{m}b\left(J_m{k}_t+b_{m}b+b_m{c}_t+n{i}_m^{\mathrm{2}}b{c}_t\right)+J{b}_m\left(b_{m}b+b_m{c}_t+nJ{i}_m^{\mathrm{2}}{k}_t+n{i}_m^{\mathrm{2}}b{c}_t\right)+\left(J_m{c}_t+nJ{i}_m^{\mathrm{2}}{c}_t\right)\times \left(J_m{k}_t+b_{m}b+b_m{c}_t+nJ{i}_m^{\mathrm{2}}{k}_t+n{i}_m^{\mathrm{2}}b{c}_t\right)>\mathrm{0}.\end{array}$ (51)

The item a₁a₂ – a₀a₃ holds positive real at any time. Thus, the pole polynomial D(s) is a Hurwitz polynomial, and its characteristic roots have negative real part. The pole polynomial D(s) is a third-order system (deg{D(s)} = 3), when all the physical components of shield TBM cutterhead driving system have identical system parameters. To analyze the speed gain of the motor torque, speed torque gain, speed total torque gain, and speed load torque gain are obtained, respectively,

$\begin{array}{c}{K}_{e,i}={\frac{{\omega }_m\left(s\right)}{T_{e,i}\left(s\right)}|}_{s=\mathrm{0}}=\frac{q{i}_m}{b_m+n{i}_m^{\mathrm{2}}b}=\frac{q{i}_m}{b_m+n{i}_m^{\mathrm{2}}\left(q^{\mathrm{2}}{b}_d+b_c\right)}\hspace{1em}\left(i=\mathrm{1,2},\dots ,n\right),\\ K_{e,t}={\frac{{\omega }_m\left(s\right)}{T_{e,t}\left(s\right)}|}_{s=\mathrm{0}}=\frac{q{i}_m}{b_m+n{i}_m^{\mathrm{2}}b}=\frac{q{i}_m}{b_m+n{i}_m^{\mathrm{2}}\left(q^{\mathrm{2}}{b}_d+b_c\right)},\\ K_L={\frac{{\omega }_m\left(s\right)}{T_L\left(s\right)}|}_{s=\mathrm{0}}=\frac{-\mathrm{1}}{b_m+n{i}_m^{\mathrm{2}}b}=\frac{-\mathrm{1}}{b_m+n{i}_m^{\mathrm{2}}\left(q^{\mathrm{2}}{b}_d+b_c\right)},\\ \underset{n\to \infty }{\lim }{K}_{e,i}=\underset{n\to \infty }{\lim }{K}_{e,t}=\underset{n\to \infty }{\lim }\frac{q{i}_m}{b_m+n{i}_m^{\mathrm{2}}b}=\mathrm{0},\\ \underset{n\to \infty }{\lim }{K}_L=\underset{n\to \infty }{\lim }\frac{-\mathrm{1}}{b_m+n{i}_m^{\mathrm{2}}b}=\mathrm{0},\end{array}$ (52)

where K_{e, i} is the speed torque gain, K_{e, t} is the speed total torque gain, and K _L is the speed load torque gain. The speed gain equations show that gain K_{e, i}, K_{e, t}, and K _L will be gradually reduced by increasing the number of driving motors (pinions), large gear viscous damped, motor rotor viscous damped, and pinion viscous damped. The limit of the gain K_{e, i}, K_{e, t}, and K _L equals zero.

If all the driving motors of shield TBM cutterhead have identical electrical magnetic torque T_{e, 1}(s) = T_{e, 2}(s) = · = T_{e, n}(s) = T _e (s), then the transfer function of the cutterhead driving system will be obtained. Where Ω_{e, i}(s), Ω_{e, t}(s), and Ω _L (s) are speed torque transfer function, speed total torque transfer function, and speed load torque transfer function, respectively, when all the driving motors have identical electrical magnetic torque,

$\begin{array}{c}{\mathrm{\Omega }}_{e,k}\left(s\right)=\frac{{\omega }_m\left(s\right)}{T_{e,k}\left(s\right)}=\frac{{\omega }_m\left(s\right)}{T_e\left(s\right)}=\frac{n\times N\left(s\right)}{D\left(s\right)}\hspace{1em}\left(\forall k=\mathrm{1,2},\dots ,n\right)\hspace{0.17em}\hspace{0.17em}\left\{T_L\left(s\right)=\mathrm{0}\right\},\\ {\mathrm{\Omega }}_{e,t}\left(s\right)=\frac{{\omega }_m\left(s\right)}{T_{e,t}\left(s\right)}=\frac{{\omega }_m\left(s\right)}{n{T}_e\left(s\right)}=\frac{N\left(s\right)}{D\left(s\right)},\begin{array}{c}{T}_{e,t}\left(s\right)=n{T}_e\left(s\right)\hspace{1em}\left\{T_L\left(s\right)=\mathrm{0}\right\},\end{array}\\ {\mathrm{\Omega }}_L\left(s\right)=\frac{{\omega }_m\left(s\right)}{T_L\left(s\right)}=\frac{-N_L\left(s\right)}{D\left(s\right)}\hspace{1em}\left\{T_{e,t}\left(s\right)=n{T}_e\left(s\right)=\mathrm{0}\right\}.\end{array}$ (53)

Therefore, the speed torque gain, speed total torque gain, and speed load torque gain are obtained, respectively, when all the driving motors of shield TBM cutterhead have identical torque. Then, the speed torque gain, speed total torque gain, and speed load torque gain will have the relationships

$\begin{array}{c}{\mathrm{\Psi }}_{e,i}={\frac{{\omega }_m\left(s\right)}{T_{e,i}\left(s\right)}|}_{s=\mathrm{0}}={\frac{{\omega }_m\left(s\right)}{T_e\left(s\right)}|}_{s=\mathrm{0}}=\frac{nq{i}_m}{b_m+n{i}_m^{\mathrm{2}}b}=\frac{nq{i}_m}{b_m+n{i}_m^{\mathrm{2}}\left(q^{\mathrm{2}}{b}_d+b_c\right)}\hspace{1em}\left(\forall i=\mathrm{1,2},\dots ,n\right),\\ {\mathrm{\Psi }}_{e,t}={\frac{{\omega }_m\left(s\right)}{T_{e,t}\left(s\right)}|}_{s=\mathrm{0}}={\frac{{\omega }_m\left(s\right)}{n{T}_e\left(s\right)}|}_{s=\mathrm{0}}=\frac{q{i}_m}{b_m+n{i}_m^{\mathrm{2}}b}=\frac{q{i}_m}{b_m+n{i}_m^{\mathrm{2}}\left(q^{\mathrm{2}}{b}_d+b_c\right)},\\ {\mathrm{\Psi }}_L={\frac{{\omega }_m\left(s\right)}{T_L\left(s\right)}|}_{s=\mathrm{0}}=\frac{-\mathrm{1}}{b_m+n{i}_m^{\mathrm{2}}b}=\frac{-\mathrm{1}}{b_m+n{i}_m^{\mathrm{2}}\left(q^{\mathrm{2}}{b}_d+b_c\right)},\\ \underset{n\to \infty }{\lim }{\mathrm{\Psi }}_{e,i}=\underset{n\to \infty }{\lim }\frac{nq{i}_m}{b_m+n{i}_m^{\mathrm{2}}b}=\frac{q}{i_{m}b},\\ \underset{n\to \infty }{\lim }{\mathrm{\Psi }}_{e,t}=\underset{n\to \infty }{\lim }\frac{q{i}_m}{b_m+n{i}_m^{\mathrm{2}}b}=\mathrm{0},\\ \underset{n\to \infty }{\lim }{\mathrm{\Psi }}_L=\underset{n\to \infty }{\lim }\frac{-\mathrm{1}}{b_m+n{i}_m^{\mathrm{2}}b}=\mathrm{0},\end{array}$ (54)

where Ψ_{e, i},Ψ_{e, t}, and Ψ _L are the speed torque gain, speed total torque gain, and speed load torque gain, respectively, when all the driving motors have identical electrical magnetic torque. Therefore, the transfer function and gain equations (53) and (54) under T_{e, 1}(s) = T_{e, 2}(s) = · = T_{e, n}(s) = T _e (s) case show that speed torque gain Ψ_{e, i} and cutterhead speed torque transfer function Ω_{e, i}(s) are the n times of the previous ones. The speed total torque transfer function Ω_{e, t}(s) and cutterhead speed load torque transfer function Ω _L (s) keep the same as the previous ones. The speed gain equation (54) show that speed gain Ψ_{e, i}, Ψ_{e, t}, and Ψ _L will be gradually reduced by increasing large gear viscous damped, motor rotor viscous damped, and pinion viscous damped. The gain Ψ_{e, t} and Ψ _L will be gradually reduced by increasing the number of driving motors (pinions); however, speed torque gain Ψ_{e, i} will be gradually increased by increasing the number of driving motors (pinions). The limit of the gain Ψ_{e, t} and Ψ _L is equal to zero; however, the limit of speed torque gain Ψ_{e, i} is equal to q/(bi _m ). When the number of the driving motors (pinions) is fixed, it can select suitable reducer ratio and transmission ratio to obtain the maximum speed gain. The maximum gain of speed torque gain K_{e, i} and speed total torque gain K_{e, t} will be obtained by solving the following optimization problem; therefore, the corresponding optimal transmission ratio i _m ^* and optimal reducer ratio q^* will be obtained as follows:

$\begin{array}{c}\max \left(K_{e,i}\right)\mathrm{\hspace{0.17em}\hspace{0.17em}}\text{or}\mathrm{\hspace{0.17em}\hspace{0.17em}}\max \left(K_{e,t}\right)⟺\max \left(\frac{q{i}_m}{b_m+n{i}_m^{\mathrm{2}}b}\right)=\max \left(\frac{q{i}_m}{b_m+n{i}_m^{\mathrm{2}}\left(q^{\mathrm{2}}{b}_d+b_c\right)}\right)\\ \Rightarrow i_m^{*}=\sqrt{\frac{b_m}{nb}}=\sqrt{\frac{b_m}{n\left(q^{\mathrm{2}}{b}_d+b_c\right)}},\\ K_{e,i}^{*}=K_{e,t}^{*}=\frac{q{i}_m^{*}}{b_m+nb{\left(i_m^{*}\right)}^{\mathrm{2}}}\\ \Rightarrow q^{*}=\sqrt{\frac{b_m+n{b}_c{i}_m^{\mathrm{2}}}{n{b}_d{i}_m^{\mathrm{2}}}},\\ K_{e,i}^{*}=K_{e,t}^{*}=\frac{q^{*}{i}_m}{b_m+n{i}_m^{\mathrm{2}}\left(b_d{\left(q^{*}\right)}^{\mathrm{2}}+b_c\right)}.\end{array}$ (55)

Finally, the optimal transmission ratio i _m ^* and optimal reducer ratio q^* are obtained. Through analyzing the linear dynamic model, some conclusions could be made for shield TBM cutterhead driving system's linear dynamic model: (i) dynamic performances of the shield TBM cutterhead driving system are affected by physical parameters such as the number of driving motors (pinions), transmission ratio, reducer ratio, large gear inertia and viscous damped, motor rotor inertia and viscous damped, pinion inertia and viscous damped, mesh stiffness, and mesh damped. (ii) System order of the cutterhead speed ω _m (s) or cutterhead driving system is $2n+1(\mathrm{deg}\{{\omega }_m(s)\}=2n+1)$ when system parameters are not equal to each other. The shield TBM cutterhead driving system degenerates into a third-order system $(\mathrm{deg}\{D(s)\}=3)$ when system parameters are equal to each other, and the cutterhead driving system can be seen as a single input system at this time. (iii) The cutterhead driving system is an open-loop stable system with negative zeros, the speed torque gain K_{e, i}, speed total torque gain K_{e, t}, and speed load torque gain K _L will be gradually reduced by increasing the number of driving motors (pinions), large gear viscous damped, motor rotor viscous damped, and pinion viscous damped. In addition, the limit of the gain K_{e, i}, K_{e, t}, and K _L is equal to zero. (iv) When all the driving motors' electrical magnetic torque has relationship with T_{e, 1}(s) = T_{e, 2}(s) = · = T_{e, n}(s) = T _e (s), speed torque gain Ψ_{e, i} and cutterhead speed torque transfer function Ω_{e, i}(s) are the n times of the previous ones. Speed torque gain Ψ_{e, i} will be gradually increased by increasing the number of driving motors (pinions); however, the gain Ψ_{e, t} and Ψ _L will be gradually reduced. The limit of the gain Ψ_{e, t} and Ψ _L is equal to zero; however, the limit of speed torque gain Ψ_{e, i} is equal to q/(bi _m ). (v) It can select suitable reducer ratio q or transmission ratio i _m to heighten the speed torque gain K_{e, i} and speed total torque gain K_{e, t}. The corresponding optimal reducer ratio q^* and optimal transmission ratio i _m ^* are obtained by maximizing the gain K_{e, i} or K_{e, t}.

4. Simulation Results and Analysis of Shield TBM Cutterhead Driving System

There are not good theoretical analysis methods for NLTV dynamic model and LTV dynamic model. To facilitate the simulation and the analysis of dynamic models of the shield TBM cutterhead driving system, some time-varying system parameters such as mesh stiffness, mesh damped, load torque, and transmission error are replaced by the mean mesh stiffness, mean mesh damped, mean load torque, and mean transmission error in the simulation. Time-varying physical parameters mean that physical parameters will change and be affected by many uncertain factors, and these uncertain factors may change with time. Therefore, from the perspective of the timeline, these physical parameters could be seen as time varying. The general NLTV and LTV dynamic models of shield TBM cutterhead driving system are simulated and studied. Effects of the NLTV dynamic model's physical parameters on the dynamic response and performances of shield TBM cutterhead driving system are investigated in detail under different conditions. The differences of dynamic response and performances between NLTV model and LTV model are discussed under different load torque conditions. The simulation parameters are presented in Appendix C. When discussing a specific physical parameter's effect on the dynamic response and performances, other physical parameters are all fixed in the simulation. The corresponding simulation results are shown in Figures 9–21.

Figure 9:

Dynamic response under various load torque conditions.

Figure 10:

Dynamic response under various gear relative backlashes.

Figure 11:

Dynamic response under various transmission errors.

Figure 12:

Dynamic response under various large gear inertia conditions.

Figure 13:

Dynamic response under various conditions of damped of large gear.

Figure 14:

Dynamic response under various motor rotor inertia conditions.

Figure 15:

Dynamic response under various damped of motor rotor conditions.

Figure 16:

Dynamic response under various pinion inertia conditions.

Figure 17:

Dynamic response under various damped of the pinion conditions.

Figure 18:

Dynamic response under various gear mesh stiffness conditions.

Figure 19:

Dynamic response under various gear mesh damped conditions.

Figure 20:

Dynamic response of nonlinear model and linear model in unload case.

Figure 21:

Dynamic response of nonlinear model and linear model under various load cases.

4.1. Physical Parameters' Effects on Dynamic Response and Performances

The effects of all the physical parameters such as load torque, mesh stiffness and mesh damped, motor rotor inertia and viscous damped, pinion inertia and viscous damped, gear backlash, transmission error, and large gear inertia and viscous damped on the dynamic response and performances of the shield TBM cutterhead driving system are analyzed. The simulation results of the NLTV dynamic model are shown and investigated under different conditions.

(1) The load torque of shield TBM cutterhead driving system is determined by outside geology conditions and cutterhead design parameters. The response results of the NLTV dynamic model in various load torque cases are shown in Figures 9(a)–9(f). The results reveal that the cutterhead driving system shows good dynamic response and performances when load torque is increased from 0 KN·m (unload case) to 500 KN·m and 1000 KN·m. The cutterhead steady-state speed is about 17.21 rpm, 14.32 rpm, and 11.33 rpm, respectively, when the load torque is 0 KN·m, 500 KN·m, and 1000 KN·m, respectively. The response results show that dynamic response and performances are deteriorated when the load torque is increased to 2500 KN·m. The cutterhead speed shows severe fluctuations and also shows a lot of nonlinear characteristics which include the irregular jump and oscillation, jump discontinuity, and harmonic phenomenon. The cutterhead speed also shows few nonlinear characteristics in unload case, but nonlinear characteristics do not obviously appear when the load torque is 500 KN·m and 1000 KN·m. The response results in Figures 9(a)–9(f) also show that the oscillation amplitude of cutterhead initial speed will be increased and the steady-state speed amplitude will be reduced when increasing the load torque. Therefore, load torque has large effects on dynamic response and performances of the shield TBM cutterhead driving system.

(2) The gear backlash and transmission error cannot be avoided in the shield TBM cutterhead driving system. The response results of the NLTV dynamic model under various gear backlashs and transmission errors (in unload case) are shown in Figures 10(a)–10(d) and Figures 11(a)–11(f), respectively. The dynamic response results in Figures 10(a)–10(d) show that the cutterhead speed shows peak pulse and jump discontinuity behaviors when gear relative backlash is increased. The response results in Figures 10(a)–10(d) obviously reveal that cutterhead speed shows numerous nonlinear characteristics and behaviors such as irregular jump and jump discontinuity when the gear relative backlash is increased from 0.001 to 0.01, 0.05, and 0.1. However, nonlinear characteristics and behaviors are dramatically reduced when the gear relative backlash is increased from 0.01 to 0.05. Numerous nonlinear behaviors appear again when relative backlash is increased from 0.05 to 0.1. The maximum jump amplitude and minimum jump amplitude of the cutterhead speed are about 26.88 rpm and 5.09 rpm, 16.38 rpm and 7.38 rpm, and 25.56 rpm and 6.13 rpm when the relative gear backlash is 0.01, 0.05, and 0.1, respectively. So, gear relative backlash has complex and important effects on dynamic response and performances of the shield TBM cutterhead driving system.

The response results in Figures 11(a)–11(f) obviously reveal that the transmission error has complex and important effects on the dynamic response and performances of the cutterhead driving system. The results show that cutterhead speed shows numerous nonlinear behaviors such as irregular jump and oscillation, jump peak, and jump discontinuity when the transmission error is increased from 0.001 m to 0.004 m, 0.006 m, and 0.008 m. However, nonlinear behaviors and characteristics of the cutterhead speed are reduced when transmission error is increased from 0.006 m to 0.008 m. The steady-state speed ripples and fluctuations will be increased when the transmission error is increased. The dynamic response results in Figures 11(a)–11(f) also show that oscillation amplitude of cutterhead initial speed will be increased when transmission error is heightened. The oscillation amplitude of cutterhead initial speed is about −3.24 rpm, −12.20 rpm, −18.69 rpm, and −26.95 rpm, respectively, when transmission error is 0.001 m, 0.004 m, 0.006 m, and 0.008 m, respectively. Thus, transmission error has important effects on dynamic response and performances of the shield TBM cutterhead driving system.

The nonlinear behaviors and characteristics can be directly affected by gear relative backlash and transmission error. Hence, nonlinear behaviors would reflect actual working conditions of the cutterhead driving system in some degree. If relative backlash and transmission error exceed a certain level, they will cause noise, speed ripple, and speed discontinuity jump which will cause damage to cutterhead driving system in the long run. The response results indicate that the gear backlash and transmission error will tend to strengthen nonlinearity and worsen dynamic performances. The cutterhead driving system will reap benefits for reducing gear relative backlash and transmission error.

(3) The response results of the NLTV dynamic model under various conditions of inertia and viscous damped of large gear (in unload case) are shown in Figures 12(a)–12(d) and Figures 13(a)–13(d), respectively. The response results in Figures 12(a)–12(d) show that large gear inertia has complex effects on the dynamic response and performances of the cutterhead driving system. The cutterhead speed shows few nonlinear behaviors as the large gear inertia is 56.93 kg m². The cutterhead speed also shows lots of nonlinear behaviors such as irregular jump, irregular oscillation, and jump discontinuity when the large gear inertia is increased form 56.93 kg m² to 86.93 kg m², 126.93 kg m², and 156.93 kg m². The cutterhead speed shows numerous nonlinear behaviors such as irregular jump and jump discontinuity when the large gear inertia is increased from 86.93 kg m² to 126.93 kg m² and 156.93 kg m². Therefore, large gear inertia has important effects on dynamic response and performances of the shield TBM cutterhead driving system.

The response results in Figures 13(a)–13(d) obviously reveal that large gear viscous damped has great and complex effects on the dynamic response and performances of the cutterhead driving system. The cutterhead speed does not obviously show nonlinear behaviors and characteristics such as the irregular jump and jump discontinuity phenomenon when viscous damped is increased from 0.921 kg m²/rad s⁻¹ to 1.921 kg m²/rad s⁻¹. However, the cutterhead speed does show numerous irregular jump and jump discontinuity behaviors when large gear viscous damped is increased from 1.921 kg m²/rad s⁻¹ to 4.921 kg m²/rad s⁻¹, and the jump discontinuity behaviors keep a long time. But the nonlinear behaviors obviously disappear when the large gear viscous damped is increased from the 4.921 kg m²/rad s⁻¹ to 7.921 kg m²/rad s⁻¹. The amplitude of cutterhead steady-state speed is not affected when the large gear viscous damped is increased. Therefore, response results indicate that large gear viscous damped has important and complex effects on dynamic response and system performances of the shield TBM cutterhead driving system.

(4) The response results of the NLTV dynamic model under various motor rotor inertia and viscous damped conditions (in unload case) are shown in Figures 14(a)–14(d) and Figures 15(a)–15(d), respectively. The response results in Figures 14(a)–14(d) reveal that motor rotor inertia has large effects on the dynamic response and performances of the cutterhead driving system. The dynamic response results in Figures 14(a)–14(d) obviously do not show nonlinear behaviors and characteristics such as irregular jump, jump discontinuity, irregular oscillation, and harmonics when the motor rotor inertia is continuously increased from J_{d, i} = 2.10 kg m² (i = 1, 3, 5, 6, 8, 10) and J_{d, k} = 2.11 kg m² (k = 2, 4, 7, 9) to J_{d, i} = 6.10 kg m² (i = 1, 3, 5, 6, 8, 10) and J_{d, k} = 6.11 kg m² (k = 2, 4, 7, 9). The transient performance items of cutterhead driving system such as rise time, delay time, and setting time are gradually increased when the motor rotor inertia is continuously increased. However, the amplitude of cutterhead steady-state speed would not be reduced when the motor rotor inertia is continuously increased. The dynamic response results show that transient performances have been affected significantly. Therefore, motor rotor inertia has large effects on dynamic response and performances of the shield TBM cutterhead driving system.

The response results in Figures 15(a)–15(d) reveal that motor rotor viscous damped has important and complex effects on the dynamic response and performances of the cutterhead driving system. The dynamic response results in Figures 15(a)–15(d) obviously do show numerous nonlinear behaviors and characteristics such as irregular jump, jump discontinuity, and irregular oscillation when the motor rotor viscous damped is increased from b_{d, i} = 0.225 kg m²/rad s⁻¹ (i = 1, 3, 5, 6, 8, 10) and b_{d, k} = 0.250 kg m²/rad s⁻¹ (k = 2, 4, 7, 9) to b_{d, i} = 0.325 kg m²/rad s⁻¹ (i = 1, 3, 5, 6, 8, 10) and b_{d, k} = 0.350 kg m²/rad s⁻¹ (k = 2, 4, 7, 9). However, the dynamic response results also show that numerous nonlinear behaviors and characteristics such as irregular jump and jump discontinuity dramatically vanish when the motor rotor viscous damped is increased from b_{d, i} = 0.325 kg m²/rad s⁻¹ (i = 1, 3, 5, 6, 8, 10) and b_{d, k} = 0.350 kg m²/rad s⁻¹ (k = 2, 4, 7, 9) to b_{d, i} = 0.525 kg m²/rad s⁻¹ (i = 1, 3, 5, 6, 8, 10) and b_{d, k} = 0.550 kg m²/rad s⁻¹ (k = 2, 4, 7, 9) and b_{d, i} = 0.725 kg m²/rad s⁻¹ (i = 1, 3, 5, 6, 8, 10) and b_{d, k} = 0.750 kg m²/rad s⁻¹ (k = 2, 4, 7, 9), respectively. The dynamic response results obviously demonstrate that the amplitude of cutterhead steady-state speed is continuously reduced when the motor rotor viscous damped is continuously increased. Therefore, motor rotor viscous damped has significant and complex effects on dynamic response and performances of the shield TBM cutterhead driving system.

(5) The response results of the NLTV dynamic model under various conditions of pinion inertia and viscous damped (in unload case) are shown in Figures 16(a)–16(d) and Figures 17(a)–17(d), respectively. The response results in Figures 16(a)–16(d) reveal that pinion inertia has complex effects on the dynamic response and performances of the cutterhead driving system. The dynamic response results do not obviously show nonlinear behaviors and characteristics such as irregular jump, jump discontinuity, and irregular oscillation when the pinion inertia is increased from 1.00 kg m² to 10.00 kg m². However, response results show that numerous nonlinear behaviors such as irregular jump and jump discontinuity dramatically appear when the pinion inertia is increased from 10.00 kg m² to 20.00 kg m². But the nonlinear behaviors vanish again when the pinion inertia is increased from 20.00 kg m² to 50.00 kg m². The response results show that the pinion inertia has complex affects on dynamic response and performances of the cutterhead driving system.

The response results in Figures 17(a)–17(d) reveal that pinion viscous damped has complex effects on the dynamic response and performances of the cutterhead driving system. The dynamic response results in Figures 17(a)–17(d) show numerous nonlinear behaviors and characteristics such as irregular jump, jump discontinuity, and irregular oscillation when the pinion viscous damped is increased from b_{c, i} = 0.115 kg m²/rad s⁻¹ (i = 1, 3, 5, 6, 8, 10) and b_{c, k} = 0.125 kg m²/rad s⁻¹ (k = 2, 4, 7, 9) to b_{c, i} = 0.315 kg m²/rad s⁻¹ (i = 1, 3, 5, 6, 8, 10) and b_{c, k} = 0.325 kg m²/rad s⁻¹ (k = 2, 4, 7, 9). The dynamic response results show that numerous nonlinear behaviors and characteristics of the cutterhead speed obviously disappear when pinion viscous damped is increased from b_{c, i} = 0.315 kg m²/rad s⁻¹ (i = 1, 3, 5, 6, 8, 10) and b_{c, k} = 0.325 kg m²/rad s⁻¹ (k = 2, 4, 7, 9) to b_{c, i} = 0.515 kg m²/rad s⁻¹ (i = 1, 3, 5, 6, 8, 10) and b_{c, k} = 0.525 kg m²/rad s⁻¹ (k = 2, 4, 7, 9) and b_{c, i} = 1.515 kg m²/rad s⁻¹ (i = 1, 3, 5, 6, 8, 10) and b_{c, k} = 1.525 kg m²/rad s⁻¹ (k = 2, 4, 7, 9), respectively. These nonlinear behaviors and characteristics dramatically vanish when the pinion viscous damped is further increased. The dynamic response results obviously demonstrate that the amplitude of cutterhead steady-state speed is not reduced when the pinion viscous damped is continuously increased. Therefore, pinion viscous damped has complex and significant effects on dynamic response and performances of the shield TBM cutterhead driving system.

(6) The response results of the NLTV dynamic model under various mesh stiffness and mesh damped conditions (in unload case) are shown in Figures 18(a)–18(f) and Figures 19(a)–19(f), respectively. The response results in Figures 18(a)–18(f) obviously show that mesh stiffness has important and complex effects on the dynamic response and performances of the cutterhead driving system. The cutterhead speed does not obviously show nonlinear behaviors and characteristics such as the jump discontinuity and irregular jump when the gear mesh stiffness is reduced from 10⁵ N·m/rad to 10⁴ N·m/rad. However, the cutterhead speed shows lots of nonlinear characteristics and behaviors such as irregular jump, irregular oscillation, and jump discontinuity when mesh stiffness is increased form 10⁵ N·m/rad to 10⁶ N·m/rad. Furthermore, the cutterhead speed also shows numerous nonlinear behaviors such as irregular jump, jump discontinuity, irregular oscillation, unstable oscillation, and subharmonic behavior when mesh stiffness is increased from 10⁶ N·m/rad to 10⁷ N·m/rad. The response results show that cutterhead speed contains lots of subharmonic and oscillation behaviors when mesh stiffness is increased beyond a certain range. Dynamic performances tend to be worsened by increasing mesh stiffness. Meanwhile, the oscillation amplitude of cutterhead initial speed and cutterhead steady-state speed ripple are increased, and nonlinear characteristics are motivated as well when the mesh stiffness is increased. Therefore, mesh stiffness has important and complex effects on dynamic response and performances of the shield TBM cutterhead driving system.

The response results in Figures 19(a)–19(f) obviously reveal that mesh damped has important and complex effects on the dynamic response and performances of the cutterhead driving system. The cutterhead speed does not obviously show nonlinear behaviors and characteristics such as the irregular jump, irregular oscillation, and jump discontinuity when the mesh damped is reduced from 100 N·m/rad s⁻¹ to 50 N·m/rad s⁻¹. However, the cutterhead speed shows numerous nonlinear characteristics and behaviors such as irregular jump, irregular oscillation, and jump discontinuity when mesh damped is further reduced from 50 N·m/rad s⁻¹ to 10 N·m/rad s⁻¹ and 5 N·m/rad s⁻¹. In addition, cutterhead steady-state speed contains numerous subharmonic and stable oscillation behaviors when mesh damped is reduced to 10 N·m/rad s⁻¹ and 5.0 N·m/rad s⁻¹. Meanwhile, results show that the oscillation amplitude of cutterhead initial speed and cutterhead steady-state speed ripple would be increased, and nonlinear characteristics will be motivated too when mesh damped is gradually reduced. Dynamic performances of the cutterhead driving system tend to be worsened by reducing mesh damped. Therefore, mesh damped has very important and complex effects on dynamic response and performances of the shield TBM cutterhead driving system.

The results indicate that mesh stiffness is increased considerably which tends to aggravate the dynamic performances and increase nonlinearity behaviors. The results also show that the steady-state speed ripple and nonlinear phenomenon such as complex harmonic, irregular jump, and jump discontinuity behaviors will be motivated by increasing mesh stiffness or reducing mesh damped. As a result, the cutterhead steady-state speed ripple, cutterhead harmonic speed, and nonlinear behaviors could be reduced by reducing mesh stiffness or increasing mesh damped. Consequently, the mesh stiffness and mesh damped are critical parameters which should be reasonably designed for the cutterhead driving system.

(7) The differences between the NLTV dynamic model and LTV dynamic model of cutterhead driving system are comparatively studied, and their simulation results under various load torque conditions are shown in Figures 20(a)–20(d) (in unload case) and Figures 21(a)–21(f), respectively. The response results in Figures 20(a)–20(d) show that nonlinear model and linear model nearly have the same dynamic response in unload case. They nearly have the same cutterhead speed and transient response process. The nonlinear model shows a few nonlinear behaviors such as jump discontinuity and irregular jump, whereas linear model does not have these behaviors. The speed response of the nonlinear model shows speed oscillation in initial stage. The response results of nonlinear model and linear model under various load torque conditions are shown in Figures 21(a)–21(f), where response results indicate that nonlinear model and linear model have roughly the same response results when the load torque is 500 KN·m and 1000 KN·m, respectively. Small differences between the nonlinear model and linear model are cutterhead speed response and speed oscillation amplitude in initial stage when the load torque is fixed at 500 KN·m and 1000 KN·m, respectively. However, the response results in Figures 21(a)–21(f) show that nonlinear model and linear model have large differences when load torque is 2500 KN·m. The nonlinear model shows larger cutterhead speed fluctuation and steady-state speed ripple than those of linear model. The cutterhead speed of nonlinear model shows lots of nonlinear behaviors such as irregular jump, jump discontinuity, and subharmonics. Of course, the linear model also shows some speed ripple and jump discontinuity behaviors, but linear model's speed fluctuation and speed ripple are smaller than those of the nonlinear model.

5. Conclusion

In this paper, a general NLTV dynamic model and LTV dynamic model are presented for the shield TBM cutterhead driving system. The general NLTV dynamic model considers different transmission errors, gear backlashes, and gear mesh damped compared with the previous nonlinear dynamic model. MIMO state space forms of the NLTV and LTV dynamic model are presented as well. Through theoretical analysis of linear dynamic model, the analysis results reveal the following.

• Dynamic response and performances of the cutterhead driving system will be affected by the number of driving motors (active pinions), transmission ratio, reducer ratio, large gear inertia and viscous damped, motor rotor inertia and viscous damped, pinion inertia and viscous damped, and mesh stiffness and mesh damped.

• The model order of cutterhead driving system is the 2n + 1 (deg{ω _m (s)} = 2n + 1) when physical parameters are not equal to each other. However, the cutterhead driving system degenerates into a third-order system (deg{D(s)} = 3) when physical parameters are equal to each other, and it can be seen as a single input system at this time.

• The cutterhead driving system is a minimum phase system (open-loop stable system with negative zeros). The speed torque gain, speed total torque gain, and speed load torque gain will be gradually reduced by increasing the number of driving motors (active pinions), large gear viscous damped, motor rotor viscous damped, and pinion viscous damped. The limits of all the speed torque gain, speed total torque gain, and speed torque gain are equal to zero.

• When motor electrical magnetic torque has T_{e, 1}(s) = T_{e, 2}(s) = · = T_{e, n}(s) = T _e (s), speed torque gain and speed torque transfer function are the n times of the previous ones. Speed torque gain will be gradually increased by increasing the number of driving motors (active pinions), but speed total torque gain and speed load torque gain will be gradually reduced. The limits of speed total torque gain and speed load torque gain are equal to zero, but the limit of speed torque gain is equal to q/(bi _m ).

• It can select suitable reducer ratio q or transmission ratio i _m to heighten the speed torque gain and speed total torque gain, and corresponding optimal reducer ratio q^* and optimal transmission ratio i _m ^* are derived, respectively. The optimal TR and RR could help cutterhead driving system to acquire higher rotation speed.

To numerically study and analyze the NLTV dynamic model, the NLTV dynamic model is simulated under various physical parameters conditions. Physical parameters such as the load torque, gear backlash and transmission error, gear mesh stiffness and mesh damped, pinions inertia and damped, large gear inertia and damped, and motor rotor inertia and damped are investigated in detail to analyze their effects on the dynamic response and performances of the shield TBM cutterhead driving system. The response results of NLTV dynamic model show lots of interesting nonlinear characters and behaviors such as irregular jumps, jump discontinuity, irregular oscillations, and harmonics and subharmonics especially for cutterhead driving system under conditions of unload, heavy load, large backlash and transmission error, large mesh stiffness, small mesh damped, great pinion damped, great large gear inertia and damped, and large motor rotor damped. Through numerical studies and analysis of NLTV dynamic model, the numerical simulation results reveal the following.

• The load torque has large effects on dynamic response and performances of cutterhead driving system. By increasing load torque, it will lead to increasing the oscillation amplitude of cutterhead initial speed and reducing the amplitude of cutterhead steady-state speed. When load torque is increased considerably, the cutterhead speed will show numerous nonlinear behaviors and characteristics.

• Gear relative backlash and transmission error have complex and important effects on dynamic response and performances of the cutterhead driving system. The cutterhead speed will show numerous nonlinear characteristic and behaviors such as irregular jump, and jump discontinuity when the relative backlash and transmission error are increased. Increasing transmission error, the oscillation amplitude of cutterhead initial speed will be increased as well.

• The large gear inertia and damped have complex and great effects on dynamic response and performances of the cutterhead driving system. When large gear inertia and damped are increased to a certain extent, the cutterhead speed will show numerous nonlinear behaviors such as irregular jump, irregular oscillation, and jump discontinuity. However, the nonlinear behaviors will disappear when damped is increased beyond a certain level.

• The motor rotor inertia and damped have complex and large effects on dynamic response and performances of the cutterhead driving system. The transient performance items such as rise time, delay time, and setting time will be increased when increasing the motor rotor inertia. The cutterhead speed will show numerous nonlinear behaviors when motor rotor damped is increased to a certain extent. Furthermore, cutterhead steady-state speed will be reduced by increasing the motor rotor damped. However, nonlinear behaviors will disappear when the motor rotor damped is increased beyond a certain level.

• The pinion inertia and damped have complex and large affects on dynamic response and performances of the cutterhead driving system. The cutterhead speed will show lots of nonlinear behaviors and characteristics such as the irregular jump, irregular oscillation, and jump discontinuity when the pinion damped is increased to a certain extent. However, the nonlinear behaviors will disappear when they are increased beyond a certain level.

• The mesh stiffness and damped have complex and great affects on dynamic response and performances of the cutterhead driving system. The cutterhead speed will show lots of nonlinear behaviors and characteristics such as irregular jump, and jump discontinuity when the mesh stiffness is increased to a certain extent. Increasing mesh stiffness, the oscillation amplitude of cutterhead speed in initial stage will be increased as well. When the mesh stiffness is increased beyond a certain level, the cutterhead speed will show severe oscillations and harmonics. The cutterhead speed will show numerous nonlinear behaviors such as irregular jump and jump discontinuity when mesh damped is reduced to a certain extent. Reducing mesh damped, the oscillation amplitude of cutterhead speed in initial stage will be increased as well.

• The nonlinear model and linear model nearly have the same dynamic response in unload case and certain load torque case. The nonlinear model shows a few nonlinear behaviors such as jump discontinuity and irregular jump, whereas linear model does not show these behaviors. The speed response of the nonlinear model shows speed oscillation in initial stage. However, the nonlinear model and linear model show large differences of dynamic response when load torque is beyond a certain level.

Through studying physical parameters' effects on dynamic response and performances, several approaches are proposed to improve system performances. The approaches include the following. It can design reasonable reducer ratio and transmission ratio, pinion inertia and damped, motor rotor inertia and damped, large gear inertia and damped, and mesh stiffness and damped and also reduce gear backlash and transmission error. The NLTV and LTV dynamic models provide the basis for control and fault diagnosis of shield TBM cutterhead driving system.