Dynamic behavior analysis of planetary gear transmission system with bolt constraint of the flexible ring gear

Abstract

In order to investigate the influence of the bolt constraint parameters on the dynamic characteristic of the planetary gear transmission(PGT) system with fuselage excitation, a dynamic model of the PGT system with flexible ring gear is established with the consideration of time-varying meshing stiffness with meshing phase, manufacturing error, installation error, fuselage excitation and bolt constraint. The Newmark-beta method is used to solve the dynamic model, then the dynamic load coefficient and load sharing coefficient is obtained. The results indicate that the parameters of the bolt have effects on the dynamic performance of the PGT system. Reducing the bolt number can can help to improve the dynamic load performance and load sharing performance of the system when bolt stiffness is small. Moreover, reducing bolt stiffness can also improve the dynamic load performance and load sharing performance of the system effectively.

Appendix

1.1 Strain energy

$$ U_{m} = \sum\limits_{i = 1}^{{N_{p} }} {(\frac{1}{2}k_{sp} d_{spi}^{2} + \frac{1}{2}k_{rp} d_{rpi}^{2} )} $$

$$ U_{b} = \frac{1}{2}[k_{bs} (x_{s}^{2} + y_{s}^{2} ) + k_{bc} (x_{c}^{2} + y_{c}^{2} )] + \sum\limits_{i = 1}^{Np} {\frac{1}{2}k_{bp} [(x_{pi} - x_{bc} )^{2} + (y_{pi} - y_{bc} )^{2} ]} $$

1.2 Kinetic energy

$$ T_{s} = \frac{1}{2}m_{s} (\dot{x}_{s}^{2} + \dot{y}_{s}^{2} ) + \frac{1}{2}J_{s} \dot{\phi }_{s}^{2} $$

$$ T_{c} = \frac{1}{2}m_{c} (\dot{x}_{c}^{2} + \dot{y}_{c}^{2} ) + \frac{1}{2}J_{c} \dot{\phi }_{c}^{2} $$

$$ T_{p} = \sum\limits_{i = 1}^{Np} {[\frac{1}{2}m_{p} (\dot{x}_{pi}^{2} + \dot{y}_{pi}^{2} ) + \frac{1}{2}J_{p} \dot{\phi }_{pi}^{2} ]} $$

1.3 Dissipation energy

$$ D_{m} = \sum\limits_{i = 1}^{{N_{p} }} {(\frac{1}{2}c_{sp} \dot{d}_{spi}^{2} + \frac{1}{2}c_{rp} \dot{d}_{rpi}^{2} )} $$

$$ U_{b} = \frac{1}{2}[c_{bs} (\dot{x}_{s}^{2} + \dot{y}_{s}^{2} ) + c_{bc} (\dot{x}_{c}^{2} + \dot{y}_{c}^{2} )] + \sum\limits_{i = 1}^{Np} {\frac{1}{2}c_{bp} [(\dot{x}_{pi} - \dot{x}_{bc} )^{2} + (\dot{y}_{pi} - \dot{y}_{bc} )^{2} ]} $$

1.4 Total mass matrix

$$ M = M_{r} + M_{PGT} $$

1.5 Ring gear mass matrix

$$ M_{r} = T_{r}^{T} \cdot m_{r} \int_{0}^{2\pi } {\left[ {\gamma (\theta )^{T} \gamma (\theta ) + \psi (\theta )^{T} \psi (\theta )} \right]d\theta } \cdot T_{r} $$

$$ T_{r} = \left[ {\begin{array}{*{20}c} {I_{{N_{r} \times N_{r} }} } & {Z_{{N_{r} \times 3(N_{s} + N_{c} + N_{p} )}} } \\ \end{array} } \right] $$

1.6 PGT system mass matrix

$$ M_{PGT} = T_{PGT}^{T} \cdot diag\left( {M_{s} ,M_{c} ,M_{p} ,M_{p} ,M_{p} } \right) \cdot T_{PGT} $$

$$ M_{j} = \left[ {\begin{array}{*{20}c} {m_{j} } & 0 & 0 \\ 0 & {m_{j} } & 0 \\ 0 & 0 & {J_{j} } \\ \end{array} } \right],j = s,c,p $$

$$ T_{PGT} = \left[ {\begin{array}{*{20}c} {Z_{{N_{r} \times N_{r} }} } & {Z_{{N_{r} \times 3(N_{s} + N_{c} + N_{p} )}} } \\ {sym} & {I_{{3(N_{s} + N_{c} + N_{p} ) \times 3(N_{s} + N_{c} + N_{p} )}} } \\ \end{array} } \right] $$

1.7 Total stiffness matrix

$$ K(t) = K_{r} + K_{PGT} (t) + K(\omega_{c}^{2} ) $$

1.8 Ring gear stiffness matrix

$$ K_{r} = T_{r}^{T} \cdot k_{bend} \int_{0}^{2\pi } {\left[ {(\gamma^{^{\prime}} (\theta ) - \psi^{^{\prime\prime}} (\theta ))^{T} (\gamma^{^{\prime}} (\theta ) - \psi^{^{\prime\prime}} (\theta ))} \right]} d\theta \cdot T_{r} $$

1.9 PGT system stiffness matrix

$$ K_{PGT} (t) = K_{sp} (t) + K_{rp} (t) + K_{bolt} + K_{bs} + K_{bc} + K_{pc} $$

1.10 External meshing stiffness matrix

$$ K_{sp} (t) = \sum\limits_{i = 1}^{3} {K_{sp}^{i} (t)} $$

$$ K_{sp}^{i} (t) = (T_{sp3} T_{sp2} T_{sp1} )^{T} \cdot k_{sp} (t) \cdot (T_{sp3} T_{sp2} T_{sp1} ) $$

$$ T_{sp3} = \left[ {\begin{array}{*{20}c} {cos\alpha_{s} } & {sin\alpha_{s} } & {r_{bs} } & { - cos\alpha_{s} } & { - sin\alpha_{s} } & { - r_{bp} } \\ \end{array} } \right] $$

$$ T_{sp2} = \left[ {\begin{array}{*{20}c} {cos(\omega_{c} t + \varphi_{i} )} & { - sin(\omega_{c} t + \varphi_{i} )} & 0 \\ {sin(\omega_{c} t + \varphi_{i} )} & {cos(\omega_{c} t + \varphi_{i} )} & \vdots \\ 0 & \cdots & {Z_{4 \times 4} } \\ \end{array} } \right] $$

$$ T_{sp1} = \left[ {\begin{array}{*{20}c} {\overbrace {{\begin{array}{*{20}c} 0 & \cdots & 0 \\ \end{array} }}^{{N_{r} }}} & {I_{3 \times 3} } & {\overbrace {{\begin{array}{*{20}c} 0 & \cdots & 0 \\ \end{array} }}^{{3 \times (N_{c} + N_{p} )}}} \\ {\underbrace {{\begin{array}{*{20}c} 0 & \cdots & 0 \\ \end{array} }}_{{3 \times (N_{r} + N_{s} + N_{p} + i - 1)}}} & {I_{3 \times 3} } & {\underbrace {{\begin{array}{*{20}c} 0 & \cdots & 0 \\ \end{array} }}_{{3 \times (N_{p} - i)}}} \\ \end{array} } \right] $$

$$ K_{rp} (t) = \sum\limits_{i = 1}^{3} {K_{rp}^{i} (t)} $$

1.11 Internal meshing stiffness matrix

$$ K_{rp}^{i} (t) = (T_{rp3} T_{rp2} T_{rp1} )^{T} \cdot k_{rp} (t) \cdot (T_{rp3} T_{rp2} T_{rp1} ) $$

$$ T_{rp3} = \left[ {\begin{array}{*{20}c} { - cos\alpha_{r} } & { - sin\alpha_{r} } & { - cos\alpha_{r} } & {sin\alpha_{r} } & {r_{bp} } \\ \end{array} } \right] $$

$$ T_{rp2} = \left[ {\begin{array}{*{20}c} {\gamma (\theta )} & 0 \\ {\psi (\theta )} & \vdots \\ 0 & {I_{3 \times 3} } \\ \end{array} } \right] $$

$$ T_{rp1} = \left[ {\begin{array}{*{20}c} {I_{{N_{r} \times N_{r} }} } & {\overbrace {{\begin{array}{*{20}c} 0 & \cdots & 0 \\ \end{array} }}^{{3 \times (N_{s} + N_{c} )}}} & {\overbrace {{\begin{array}{*{20}c} 0 & \cdots & 0 \\ \end{array} }}^{{3 \times N_{p} }}} \\ {\underbrace {{\begin{array}{*{20}c} 0 & \cdots & 0 \\ \end{array} }}_{{N_{r} + 3 \times (N_{s} + N_{c} + i - 1)}}} & {I_{3 \times 3} } & {\underbrace {{\begin{array}{*{20}c} 0 & \cdots & 0 \\ \end{array} }}_{{3 \times (N_{p} - i)}}} \\ \end{array} } \right] $$

1.12 Bolt stiffness matrix

$$ K_{bolt} = \sum\limits_{i = 1}^{{N_{sp} }} {K_{bolt}^{i} } $$

$$ K_{bolt}^{i} = T_{bolt}^{^{\prime}} \cdot k_{bolt} [\gamma (\theta_{i} )^{T} \gamma (\theta_{i} ) + \psi (\theta_{i} )^{T} \psi (\theta_{i} )] \cdot T_{bolt} $$

$$ T_{bolt} = \left[ {\begin{array}{*{20}c} {I_{{N_{r} \times N_{r} }} } & {Z_{{N_{r} \times 3(N_{s} + N_{c} + N_{p} )}} } \\ \end{array} } \right] $$

1.13 Sun gear support stiffness matrix

$$ K_{bs} = T_{bs}^{T} \cdot k_{bs} \cdot T_{bs} $$

$$ T_{bs} = \left[ {\begin{array}{*{20}c} {Z_{{2 \times N_{r} }} } & {I_{2 \times 2} } & {Z_{{2 \times (1 + 3(N_{c} + N_{p} ))}} } \\ \end{array} } \right] $$

1.14 Carrier support stiffness matrix

$$ K_{bc} = T_{bc}^{T} \cdot k_{bc} \cdot T_{bc} $$

$$ T_{bc} = \left[ {\begin{array}{*{20}c} {Z_{{2 \times 3(N_{r} + N_{s)} }} } & {I_{2 \times 2} } & {Z_{{2 \times (1 + 3N_{p} )}} } \\ \end{array} } \right] $$

1.15 Planet gear support stiffness matrix

$$ K_{bp} = K_{bpx} + K_{bpy} $$

$$ K_{bpx} = (T_{bpx1} T_{bpx2} T_{bpx3} )^{T} k_{bp} (T_{bpx1} T_{bpx2} T_{bpx3} ) $$

$$ T_{bpx1} = \left[ {\begin{array}{*{20}c} 1 & {r_{c} } & { - 1} \\ \end{array} } \right] $$

$$ T_{bpx2} = \left[ {\begin{array}{*{20}c} {cos(\omega_{c} t + \varphi_{i} )} & { - sin(\omega_{c} t + \varphi_{i} )} & 0 \\ 0 & 0 & {I_{2 \times 2} } \\ \end{array} } \right] $$

$$ T_{bpx3} = \left[ {\begin{array}{*{20}c} {I_{3 \times 3} } & {\overbrace {{\begin{array}{*{20}c} 0 & \cdots & 0 \\ \end{array} }}^{{3 \times (N_{s} + N_{c} )}}} & {\overbrace {{\begin{array}{*{20}c} 0 & \cdots & 0 \\ \end{array} }}^{{3 \times N_{p} }}} \\ {\underbrace {{\begin{array}{*{20}c} 0 & \cdots & 0 \\ \end{array} }}_{{N_{r} + 3 \times (N_{s} + N_{c} + i - 1)}}} & 1 & {\underbrace {{\begin{array}{*{20}c} 0 & \cdots & 0 \\ \end{array} }}_{{2 + 3 \times (N_{p} - i)}}} \\ \end{array} } \right] $$

$$ K_{bpy} = (T_{bpy1} T_{bpy2} T_{bpy3} )^{T} k_{bp} (T_{bpy1} T_{bpy2} T_{bpy3} ) $$

$$ T_{bpy1} = \left[ {\begin{array}{*{20}c} 1 & { - 1} \\ \end{array} } \right] $$

$$ T_{bpy2} = \left[ {\begin{array}{*{20}c} {sin(\omega_{c} t + \varphi_{i} )} & {cos(\omega_{c} t + \varphi_{i} )} & 0 & 0 \\ 0 & 0 & 0 & 1 \\ \end{array} } \right] $$

$$ T_{bpy3} = \left[ {\begin{array}{*{20}c} {I_{3 \times 3} } & {\overbrace {{\begin{array}{*{20}c} 0 & \cdots & 0 \\ \end{array} }}^{{3 \times (N_{s} + N_{c} )}}} & {\overbrace {{\begin{array}{*{20}c} 0 & \cdots & 0 \\ \end{array} }}^{{3 \times N_{p} }}} \\ {\underbrace {{\begin{array}{*{20}c} 0 & \cdots & 0 \\ \end{array} }}_{{N_{r} + 3 \times (N_{s} + N_{c} + i - 1) + 1}}} & 1 & {\underbrace {{\begin{array}{*{20}c} 0 & \cdots & 0 \\ \end{array} }}_{{2 + 3 \times (N_{p} - i) - 1}}} \\ \end{array} } \right] $$

1.16 Gyroscopic stiffness matrix

$$ K(\omega_{c}^{2} ) = K_{s} (\omega_{c}^{2} ) + K_{c} (\omega_{c}^{2} ) $$

$$ K_{s} (\omega_{c}^{2} ) = T_{s}^{T} \cdot m_{s} \omega_{c}^{2} \left[ {\begin{array}{*{20}c} { - 1} & 0 \\ 0 & { - 1} \\ \end{array} } \right] \cdot T_{s} $$

$$ T_{s} = \left[ {\begin{array}{*{20}c} {\overbrace {{\begin{array}{*{20}c} 0 & \cdots & 0 \\ \end{array} }}^{{N_{r} }}} & {I_{2 \times 2} } & {\overbrace {{\begin{array}{*{20}c} 0 & \cdots & 0 \\ \end{array} }}^{{1 + 3 \times (N_{c} + N_{p} )}}} \\ \end{array} } \right] $$

$$ K_{c} (\omega_{c}^{2} ) = T_{c}^{T} \cdot m_{c} \omega_{c}^{2} \left[ {\begin{array}{*{20}c} { - 1} & 0 \\ 0 & { - 1} \\ \end{array} } \right] \cdot T_{c} $$

$$ T_{s} = \left[ {\begin{array}{*{20}c} {\overbrace {{\begin{array}{*{20}c} 0 & \cdots & 0 \\ \end{array} }}^{{N_{r} + 3}}} & {I_{2 \times 2} } & {\overbrace {{\begin{array}{*{20}c} 0 & \cdots & 0 \\ \end{array} }}^{{1 + 3 \times N_{p} }}} \\ \end{array} } \right] $$

1.17 Total damping matrix

$$ C(t) = C_{r} + C_{PGT} (t) + C(\omega_{c}^{2} ) $$

Ring Gear Damping Matrix$C_{r} = \delta_{r} \cdot K_{r}$.

1.18 PGT system damping matrix

$$ C_{PGT} (t) = C_{sp} (t) + C_{rp} (t) + C_{bolt} + C_{bs} + C_{bc} + C_{pc} $$

1.19 External meshing damping matrix

$$ C_{sp} (t) = \sum\limits_{i = 1}^{3} {C_{sp}^{i} (t)} $$

$$ C_{sp}^{i} (t) = (T_{sp3} T_{sp2} T_{sp1} )^{T} \cdot c_{sp} \cdot (T_{sp3} T_{sp2} T_{sp1} ) $$

$$ T_{sp3} = \left[ {\begin{array}{*{20}c} {cos\alpha_{s} } & {sin\alpha_{s} } & {r_{bs} } & { - cos\alpha_{s} } & { - sin\alpha_{s} } & { - r_{bp} } \\ \end{array} } \right] $$

$$ T_{sp2} = \left[ {\begin{array}{*{20}c} {cos(\omega_{c} t + \varphi_{i} )} & { - sin(\omega_{c} t + \varphi_{i} )} & 0 \\ {sin(\omega_{c} t + \varphi_{i} )} & {cos(\omega_{c} t + \varphi_{i} )} & \vdots \\ 0 & \cdots & {Z_{4 \times 4} } \\ \end{array} } \right] $$

$$ T_{sp1} = \left[ {\begin{array}{*{20}c} {\overbrace {{\begin{array}{*{20}c} 0 & \cdots & 0 \\ \end{array} }}^{{N_{r} }}} & {I_{3 \times 3} } & {\overbrace {{\begin{array}{*{20}c} 0 & \cdots & 0 \\ \end{array} }}^{{3 \times (N_{c} + N_{p} )}}} \\ {\underbrace {{\begin{array}{*{20}c} 0 & \cdots & 0 \\ \end{array} }}_{{3 \times (N_{r} + N_{s} + N_{p} + i - 1)}}} & {I_{3 \times 3} } & {\underbrace {{\begin{array}{*{20}c} 0 & \cdots & 0 \\ \end{array} }}_{{3 \times (N_{p} - i)}}} \\ \end{array} } \right] $$

1.20 Internal meshing damping matrix

$$ C_{rp} (t) = \sum\limits_{i = 1}^{3} {C_{rp}^{i} (t)} $$

$$ C_{rp}^{i} (t) = (T_{rp3} T_{rp2} T_{rp1} )^{T} \cdot c_{rp} \cdot (T_{rp3} T_{rp2} T_{rp1} ) $$

$$ T_{rp3} = \left[ {\begin{array}{*{20}c} { - cos\alpha_{r} } & { - sin\alpha_{r} } & { - cos\alpha_{r} } & {sin\alpha_{r} } & {r_{bp} } \\ \end{array} } \right] $$

$$ T_{rp2} = \left[ {\begin{array}{*{20}c} {\gamma (\theta )} & 0 \\ {\psi (\theta )} & \vdots \\ 0 & {I_{3 \times 3} } \\ \end{array} } \right] $$

$$ T_{rp1} = \left[ {\begin{array}{*{20}c} {I_{{N_{r} \times N_{r} }} } & {\overbrace {{\begin{array}{*{20}c} 0 & \cdots & 0 \\ \end{array} }}^{{3 \times (N_{s} + N_{c} )}}} & {\overbrace {{\begin{array}{*{20}c} 0 & \cdots & 0 \\ \end{array} }}^{{3 \times N_{p} }}} \\ {\underbrace {{\begin{array}{*{20}c} 0 & \cdots & 0 \\ \end{array} }}_{{N_{r} + 3 \times (N_{s} + N_{c} + i - 1)}}} & {I_{3 \times 3} } & {\underbrace {{\begin{array}{*{20}c} 0 & \cdots & 0 \\ \end{array} }}_{{3 \times (N_{p} - i)}}} \\ \end{array} } \right] $$

1.21 Bolt damping matrix

$$ C_{bolt} = \sum\limits_{i = 1}^{{N_{sp} }} {C_{bolt}^{i} } $$

$$ C_{bolt}^{i} = T_{bolt}^{^{\prime}} \cdot c_{bolt} [\gamma (\theta_{i} )^{T} \gamma (\theta_{i} ) + \psi (\theta_{i} )^{T} \psi (\theta_{i} )] \cdot T_{bolt} $$

$$ T_{bolt} = \left[ {\begin{array}{*{20}c} {I_{{N_{r} \times N_{r} }} } & {Z_{{N_{r} \times 3(N_{s} + N_{c} + N_{p} )}} } \\ \end{array} } \right] $$

1.22 Sun gear support damping matrix

$$ C_{bs} = T_{bs}^{T} \cdot c_{bs} \cdot T_{bs} $$

$$ T_{bs} = \left[ {\begin{array}{*{20}c} {Z_{{2 \times N_{r} }} } & {I_{2 \times 2} } & {Z_{{2 \times (1 + 3(N_{c} + N_{p} ))}} } \\ \end{array} } \right] $$

1.23 Carrier support damping matrix

$$ C_{bc} = T_{bc}^{T} \cdot c_{bc} \cdot T_{bc} $$

$$ T_{bc} = \left[ {\begin{array}{*{20}c} {Z_{{2 \times 3(N_{r} + N_{s)} }} } & {I_{2 \times 2} } & {Z_{{2 \times (1 + 3N_{p} )}} } \\ \end{array} } \right] $$

1.24 Planet gear support damping matrix

$$ C_{bp} = C_{bpx} + C_{bpy} $$

$$ C_{bpx} = (T_{bpx1} T_{bpx2} T_{bpx3} )^{T} c_{bp} (T_{bpx1} T_{bpx2} T_{bpx3} ) $$

$$ T_{bpx1} = \left[ {\begin{array}{*{20}c} 1 & {r_{c} } & { - 1} \\ \end{array} } \right] $$

$$ T_{bpx2} = \left[ {\begin{array}{*{20}c} {cos(\omega_{c} t + \varphi_{i} )} & { - sin(\omega_{c} t + \varphi_{i} )} & 0 \\ 0 & 0 & {I_{2 \times 2} } \\ \end{array} } \right] $$

$$ T_{bpx3} = \left[ {\begin{array}{*{20}c} {I_{3 \times 3} } & {\overbrace {{\begin{array}{*{20}c} 0 & \cdots & 0 \\ \end{array} }}^{{3 \times (N_{s} + N_{c} )}}} & {\overbrace {{\begin{array}{*{20}c} 0 & \cdots & 0 \\ \end{array} }}^{{3 \times N_{p} }}} \\ {\underbrace {{\begin{array}{*{20}c} 0 & \cdots & 0 \\ \end{array} }}_{{N_{r} + 3 \times (N_{s} + N_{c} + i - 1)}}} & 1 & {\underbrace {{\begin{array}{*{20}c} 0 & \cdots & 0 \\ \end{array} }}_{{2 + 3 \times (N_{p} - i)}}} \\ \end{array} } \right] $$

$$ C_{bpy} = (T_{bpy1} T_{bpy2} T_{bpy3} )^{T} c_{bp} (T_{bpy1} T_{bpy2} T_{bpy3} ) $$

$$ T_{bpy1} = \left[ {\begin{array}{*{20}c} 1 & { - 1} \\ \end{array} } \right] $$

$$ T_{bpy2} = \left[ {\begin{array}{*{20}c} {sin(\omega_{c} t + \varphi_{i} )} & {cos(\omega_{c} t + \varphi_{i} )} & 0 & 0 \\ 0 & 0 & 0 & 1 \\ \end{array} } \right] $$

$$ T_{bpy3} = \left[ {\begin{array}{*{20}c} {I_{3 \times 3} } & {\overbrace {{\begin{array}{*{20}c} 0 & \cdots & 0 \\ \end{array} }}^{{3 \times (N_{s} + N_{c} )}}} & {\overbrace {{\begin{array}{*{20}c} 0 & \cdots & 0 \\ \end{array} }}^{{3 \times N_{p} }}} \\ {\underbrace {{\begin{array}{*{20}c} 0 & \cdots & 0 \\ \end{array} }}_{{N_{r} + 3 \times (N_{s} + N_{c} + i - 1) + 1}}} & 1 & {\underbrace {{\begin{array}{*{20}c} 0 & \cdots & 0 \\ \end{array} }}_{{2 + 3 \times (N_{p} - i) - 1}}} \\ \end{array} } \right] $$

1.25 Gyroscopic damping matrix

$$ C(\omega_{c}^{2} ) = C_{s} (\omega_{c}^{2} ) + C_{c} (\omega_{c}^{2} ) $$

$$ C_{s} (\omega_{c}^{2} ) = T_{s}^{T} \cdot 2m_{s} \omega_{c} \left[ {\begin{array}{*{20}c} 0 & { - 1} \\ 1 & 0 \\ \end{array} } \right] \cdot T_{s} $$

$$ T_{s} = \left[ {\begin{array}{*{20}c} {\overbrace {{\begin{array}{*{20}c} 0 & \cdots & 0 \\ \end{array} }}^{{N_{r} }}} & {I_{2 \times 2} } & {\overbrace {{\begin{array}{*{20}c} 0 & \cdots & 0 \\ \end{array} }}^{{1 + 3 \times (N_{c} + N_{p} )}}} \\ \end{array} } \right] $$

$$ C_{c} (\omega_{c}^{2} ) = T_{c}^{T} \cdot 2m_{c} \omega_{c} \left[ {\begin{array}{*{20}c} 0 & { - 1} \\ 1 & 0 \\ \end{array} } \right] \cdot T_{c} $$

$$ T_{s} = \left[ {\begin{array}{*{20}c} {\overbrace {{\begin{array}{*{20}c} 0 & \cdots & 0 \\ \end{array} }}^{{N_{r} + 3}}} & {I_{2 \times 2} } & {\overbrace {{\begin{array}{*{20}c} 0 & \cdots & 0 \\ \end{array} }}^{{1 + 3 \times N_{p} }}} \\ \end{array} } \right] $$

1.26 General force

$$ F = F_{load} + F_{excitation} $$

$$ F_{load} = \left[ {\begin{array}{*{20}c} {\overbrace {{\begin{array}{*{20}c} 0 & \cdots & 0 \\ \end{array} }}^{{N_{r} + 2}}} & {T_{input} } & 0 & 0 & {T_{output} } & {\overbrace {{\begin{array}{*{20}c} 0 & \cdots & 0 \\ \end{array} }}^{{3 \times N_{p} }}} \\ \end{array} } \right] $$

$$ F_{excitation} = F_{esp} + F_{esp - diff} + F_{erp} + F_{erp - diff} + F_{bolt} $$

$$ F_{esp} = \sum\limits_{i = 1}^{{N_{p} }} {F_{esp}^{i} } ,F_{esp - diff} = \sum\limits_{i = 1}^{{N_{p} }} {F_{esp - diff}^{i} } $$

$$ F_{esp}^{i} = k_{sp} \cdot (T_{sp3} T_{sp2} T_{sp1} )^{T} e_{sp} ,F_{esp - diff}^{i} = c_{sp} \cdot (T_{sp3} T_{sp2} T_{sp1} )^{T} e_{sp - diff} $$

$$ T_{sp3} = \left[ {\begin{array}{*{20}c} {cos\alpha_{s} } & {sin\alpha_{s} } & {r_{bs} } & { - cos\alpha_{s} } & { - sin\alpha_{s} } & { - r_{bp} } \\ \end{array} } \right] $$

$$ T_{sp2} = \left[ {\begin{array}{*{20}c} {cos(\omega_{c} t + \varphi_{i} )} & { - sin(\omega_{c} t + \varphi_{i} )} & 0 \\ {sin(\omega_{c} t + \varphi_{i} )} & {cos(\omega_{c} t + \varphi_{i} )} & \vdots \\ 0 & \cdots & {Z_{4 \times 4} } \\ \end{array} } \right] $$

$$ T_{sp1} = \left[ {\begin{array}{*{20}c} {\overbrace {{\begin{array}{*{20}c} 0 & \cdots & 0 \\ \end{array} }}^{{N_{r} }}} & {I_{3 \times 3} } & {\overbrace {{\begin{array}{*{20}c} 0 & \cdots & 0 \\ \end{array} }}^{{3 \times (N_{c} + N_{p} )}}} \\ {\underbrace {{\begin{array}{*{20}c} 0 & \cdots & 0 \\ \end{array} }}_{{3 \times (N_{r} + N_{s} + N_{p} + i - 1)}}} & {I_{3 \times 3} } & {\underbrace {{\begin{array}{*{20}c} 0 & \cdots & 0 \\ \end{array} }}_{{3 \times (N_{p} - i)}}} \\ \end{array} } \right] $$

$$ F_{erp} = \sum\limits_{i = 1}^{{N_{p} }} {F_{erp}^{i} } ,F_{erp - diff} = \sum\limits_{i = 1}^{{N_{p} }} {F_{erp - diff}^{i} } $$

$$ F_{erp}^{i} (t) = k_{rp} \cdot (T_{rp3} T_{rp2} T_{rp1} )^{T} e_{rp} ,F_{erp - diff}^{i} (t) = c_{rp} \cdot (T_{rp3} T_{rp2} T_{rp1} )^{T} e_{rp - diff} $$

$$ T_{rp3} = \left[ {\begin{array}{*{20}c} { - cos\alpha_{r} } & { - sin\alpha_{r} } & { - cos\alpha_{r} } & {sin\alpha_{r} } & {r_{bp} } \\ \end{array} } \right] $$

$$ T_{rp2} = \left[ {\begin{array}{*{20}c} {\gamma (\theta )} & 0 \\ {\psi (\theta )} & \vdots \\ 0 & {I_{3 \times 3} } \\ \end{array} } \right] $$

$$ T_{rp1} = \left[ {\begin{array}{*{20}c} {I_{{N_{r} \times N_{r} }} } & {\overbrace {{\begin{array}{*{20}c} 0 & \cdots & 0 \\ \end{array} }}^{{3 \times (N_{s} + N_{c} )}}} & {\overbrace {{\begin{array}{*{20}c} 0 & \cdots & 0 \\ \end{array} }}^{{3 \times N_{p} }}} \\ {\underbrace {{\begin{array}{*{20}c} 0 & \cdots & 0 \\ \end{array} }}_{{N_{r} + 3 \times (N_{s} + N_{c} + i - 1)}}} & {I_{3 \times 3} } & {\underbrace {{\begin{array}{*{20}c} 0 & \cdots & 0 \\ \end{array} }}_{{3 \times (N_{p} - i)}}} \\ \end{array} } \right] $$

$$ F_{bolt} = \sum\limits_{i = 1}^{{N_{sp} }} {F_{bolt}^{i} } $$

$$ F_{bolt}^{i} = k_{bolt} T_{bolt}^{T} \cdot [\gamma (\theta_{i} )^{T} e_{bolt - \theta } + \psi (\theta_{i} )^{T} e_{bolt - r} ] $$

$$ T_{bolt} = \left[ {\begin{array}{*{20}c} {I_{{N_{r} \times N_{r} }} } & {Z_{{N_{r} \times 3(N_{s} + N_{c} + N_{p} )}} } \\ \end{array} } \right] $$