$\mathrm{C}$ | Capacitance per unit length $(\mathrm{F}/\mathrm{m})$ |

${C}_{0}$ | Coefficient damping of bearing on the motor side $(\mathrm{Nms}/\mathrm{rad})$ |

${C}_{1}$ | Coefficient damping of bearing on the load side $(\mathrm{Nms}/\mathrm{rad})$ |

${C}_{ms}$ | Equivalent viscous damping of the drive system $(\mathrm{Nms}/\mathrm{rad})$ |

${C}_{dp}$ | Equivalent viscous damping coefficient along the drill pipe $(\mathrm{Nms}/\mathrm{rad})$ |

${C}_{l}$ | Damping coefficient of the drill pipe per unit length $(\mathrm{Nms}/\mathrm{rad})$ |

${C}_{eb}$ | Equivalent viscous damping of the bottom hole assembly $(\mathrm{Nms}/\mathrm{rad})$ |

${d}_{0,dp}$ | Outer diameter of the drill pipe $(\mathrm{mm})$ |

${d}_{i,dp}$ | Inner diameter of the drill pipe $(\mathrm{mm})$ |

${d}_{0,dc}$ | Outer diameter of the drill collar $(\mathrm{mm})$ |

${d}_{i,dc}$ | Inner diameter of the drill collar $(\mathrm{mm})$ |

${d}_{0,hdp}$ | Outer diameter of the HWDP $(\mathrm{mm})$ |

${d}_{i,hdp}$ | Inner diameter of the HWDP $(\mathrm{mm})$ |

$G$ | Conductance per unit length $({\mathrm{Nm}}^{2})$ |

${G}_{sh}$ | Shear modulus of steel $({\mathrm{Nm}}^{2})$ |

${G}_{0}$ | Shear modulus of distributed shaft $({\mathrm{Nm}}^{2})$ |

${i}_{m}$ | Current of armature of the motor $(\mathrm{A})$ |

$I(x,t)$ | Current at distance x and time t $(\mathrm{A})$ |

${J}_{0}$ | Inertia of distributed shaft $({\mathrm{m}}^{4})$ |

${J}_{m}$ | Mass moment of inertia of the motor $({\mathrm{kgm}}^{2})$ |

${J}_{rt}$ | Mass moment of inertia of the rotary table $({\mathrm{kgm}}^{2})$ |

${J}_{ms}$ | Equivalent mass moment of inertia at motor side $({\mathrm{kgm}}^{2})$ |

${J}_{bh}$ | Equivalent mass moment of inertia of the BHA $({\mathrm{kgm}}^{2})$ |

${J}_{ds}$ | Mass moment of inertia of the drill string $({\mathrm{kgm}}^{2})$ |

${J}_{hdp}$ | Mass moment of inertia of the HWDP $({\mathrm{kgm}}^{2})$ |

${J}_{dp}$ | Mass moment of inertia of the drill pipe $({\mathrm{kgm}}^{2})$ |

${J}_{dc}$ | Mass moment of inertia of the drill collar $({\mathrm{kgm}}^{2})$ |

${J}_{0}$ | Mass moment of inertia of distributed shaft $({\mathrm{kgm}}^{2})$ |

${J}_{1}$ | Mass moment of inertia of load $({\mathrm{kgm}}^{2})$ |

$J$ | Shaft polar moment of inertia $\mathsf{\pi}/32{\mathrm{d}}^{4}({\mathrm{m}}^{4})$ |

${k}_{b}$ | The emf constant $(\mathrm{vs})$ |

${k}_{m}$ | Motor constant related to physical properties of the motor $(\mathrm{vs})$ |

${k}_{dp}$ | Equivalent torsional stiffness of the drill pipe $(\mathrm{Nm}\u2044\mathrm{rad})$ |

$L$ | Inductance per unit length of electrical transmission line |

${L}_{m}$ | Armature inductance of the motor $(\mathrm{H})$ |

${l}_{dp}$ | Length of drill pipe $(\mathrm{m})$ |

${l}_{hdp}$ | Length of HWDP $(\mathrm{m})$ |

${l}_{0}$ | Length of torsional distributed shaft $(\mathrm{m})$ |

$n$ | A combined gear ratio of gear box and bevel gear |

${R}_{m}$ | Armature resistance of the motor $(\mathsf{\Omega})$ |

$r$ | Radius of shaft $(\mathrm{mm})$ |

${R}_{b}$ | Radius of the bit $(\mathrm{m})$ |

$R$ | Resistance per unit length $(\mathsf{\Omega}/\mathrm{m})$ |

${T}_{m}$ | Torque of motor $(\mathrm{Nm})$ |

${T}_{rt}$ | Torque of the rotary table $(\mathrm{Nm})$ |

${T}_{tob}$ | Torque on the bit (TOB) $(\mathrm{Nm})$ |

${T}_{vb}$ | Viscosity damping torque over the BHA $(\mathrm{Nm})$ |

${T}_{fb}$ | Friction torque on the bit $(\mathrm{Nm})$ |

${T}_{ab}$ | External torque applied by drill string on the bit $(\mathrm{Nm})$ |

${T}_{sb}$ | Static friction torque on the bit $(\mathrm{Nm})$ |

${T}_{cb}$ | Sliding friction torque (cutting torque) $(\mathrm{Nm})$ |

${T}_{1}$ | Input torque to distributed shaft from a prime mover $(\mathrm{Nm})$ |

${T}_{1,db}$ | Input torque to drill pipe from a rotary table $(\mathrm{Nm})$ |

${T}_{2}$ | Output torque from distributed shaft to load $(\mathrm{Nm})$ |

${T}_{2,db}$ | Output torque from drill pipe to the BHA $(\mathrm{Nm})$ |

${\mathrm{V}}_{\mathrm{b}}$ | Back electric motive force (back-emf) of armature $(\mathrm{v})$ |

${\mathrm{V}}_{\mathrm{s}}$ | Supply armature voltage $(\mathrm{v})$ |

$\mathrm{V}(\mathrm{x},\mathrm{t})$ | Voltage at distance x and time t $(\mathrm{v})$ |

${\mathrm{W}}_{\mathrm{ob}}$ | Weight on a bit (WOB) $(\mathrm{N})$ |

$\mathsf{\delta}\mathsf{\omega}$ | Limit velocity interval $(\mathrm{rpm})$ |

$\mathsf{\gamma}$ | Shear strain |

${\mathsf{\gamma}}_{\mathrm{b}}$ | Positive constant defining the decaying velocity of ${\mathrm{T}}_{\mathrm{fb}}$ |

$\mathsf{\Gamma}$ | Propagation constant of the electrical transmission line $(\mathrm{s}/\mathrm{m})$ |

${\mathsf{\Gamma}}_{0}$ | Propagation constant of the distributed $(\mathrm{s}/\mathrm{m})$ |

$\mathsf{\zeta}$ | Characteristic impedance of electrical transmission line $(\mathrm{v}/\mathrm{A})$ |

${\mathsf{\zeta}}_{0}$ | Characteristic impedance of distributed shaft $(\mathrm{Nms})$ |

${\mathsf{\zeta}}_{1}$ | Characteristic impedance of the drill pipe $(\mathrm{Nms})$ |

${\mathsf{\mu}}_{\mathrm{b}}$ | Friction coefficient at the bit |

${\mathsf{\mu}}_{\mathrm{cb}}$ | Coulomb friction coefficient |

${\mathsf{\mu}}_{\mathrm{sb}}$ | Static friction coefficient |

${\mathsf{\theta}}_{\mathrm{m}}$ | Angular displacement of the motor $(\mathrm{rad})$ |

${\mathsf{\theta}}_{\mathrm{rt}}$ | Angular displacement of the rotary table $(\mathrm{rad})$ |

${\mathsf{\theta}}_{\mathrm{b}}$ | Angular displacement of the bit $(\mathrm{rad})$ |

$\mathsf{\theta}$ | Angle of twist $(\mathrm{rad})$ |

$\mathsf{\rho}$ | Density of steel $(\mathrm{kg}/{\mathrm{m}}^{3})$ |

${\mathsf{\rho}}_{0}$ | Density of distributed shaft $(\mathrm{kg}/{\mathrm{m}}^{3})$ |

${\mathsf{\omega}}_{\mathrm{m}}$ | Angular velocity of the motor shaft $(\mathrm{rpm})$ |

${\mathsf{\omega}}_{\mathrm{r}}$ | Angular velocity of the rotary table $(\mathrm{rpm})$ |

${\mathsf{\omega}}_{\mathrm{b}}$ | Angular velocity of the bit $(\mathrm{rpm})$ |

${\mathsf{\omega}}_{0}$ | Angular velocity at the inlet of a distributed shaft $(\mathrm{rpm})$ |

${\mathsf{\omega}}_{1}$ | Angular velocity at the outlet of a distributed shaft $(\mathrm{rpm})$ |