Nonlinear vibration response analysis of a rotor-blade system with blade-tip rubbing

Abstract

An improved rotor-blade dynamic model is developed based on our previous works (Ma et al. in J Sound Vib, 337:301–320, 2015; J Sound Vib 357:168–194, 2015). In the proposed model, the shaft is discretized using a finite element method and the effects of the swing of the rigid disk and stagger angles of the blades are considered. Furthermore, the mode shapes of rotor-blade systems can be obtained based on the proposed model. The proposed model is more accurate than our previous model, and it is also verified by comparing the natural frequencies obtained from the proposed model with those from the finite element model and published literature. By simplifying the casing as a two degrees of freedom model, the single- and four-blade rubbings are studied using numerical simulation and experiment. Results show that for both the single- and four-blade rubbings, amplitude amplification phenomena can be observed when the multiple frequencies of the rotational frequency $(f_\mathrm{r})$ coincide with the conical and torsional natural frequencies of the rotor-blade system, natural frequencies of the casing and the bending natural frequencies of the blades. In addition, for the four-blade rubbing, the blade passing frequency (BPF, $4f_\mathrm{r})$ and its multiple frequency components also have larger amplitudes, especially, when they coincide with the natural frequencies of the rotor-blade system or casing; the four-blade rubbing levels are related to the rotor whirl, and the most severe rubbing happens on the blade located at the right end of the whirl orbit.

Appendices

Appendix 1: Vectors and matrices related to the blades

• (1) $\varvec{q}_\mathrm{b}$ is the generalized coordinates vector of the blades, where

  $$\begin{aligned} \varvec{q}_\mathrm{b}= & {} \left[ {{\begin{array}{lll} {\varvec{q}_\mathrm{b}^1}&{}\quad {\cdots \varvec{q}_\mathrm{b}^i \cdots } &{}\quad {\varvec{q}_\mathrm{b}^{N_\mathrm{b} } } \\ \end{array} }} \right] ^{\mathrm{T}},\nonumber \\ \varvec{q}_\mathrm{b}^i= & {} \left[ {{\begin{array}{lll} {{\varvec{U}^{i}}^{\mathrm{T}}}&{}\quad {{\varvec{V}^{i}}^{\mathrm{T}}}&{}\quad {{\varvec{\psi }^{i}}^{\mathrm{T}}} \\ \end{array} }} \right] ^{\mathrm{T}}, \end{aligned}$$

  (24)

  here, $\varvec{U}^{i}=[U_{1 }^i,\ldots ,U_{{N_{\mathrm{mod}}}}^i ]^{\mathrm{T}}, \varvec{V}^{i}=\big [ V_{1}^i,\ldots ,V_{{N_{\mathrm{mod}}}}^i \big ]^{\mathrm{T}}, \varvec{\psi }^{i}=\big [ \psi _{1}^i,\ldots ,\psi _{{N_{\mathrm{mod}} } }^i \big ]^{\mathrm{T}}$.

• (2) $\varvec{M}_\mathrm{b}$ is the mass matrix of the blades

  $$\begin{aligned} \varvec{M}_\mathrm{b} =\text {diag}[{\begin{array}{lllll} {\varvec{M}_\mathrm{b}^1 }&{}\quad \cdots &{}\quad {\varvec{M}_\mathrm{b}^i }&{}\quad \cdots &{}\quad {\varvec{M}_\mathrm{b}^{N_\mathrm{b} } } \\ \end{array} }], \end{aligned}$$

  (25)

  where $\varvec{M}_\mathrm{b}^i$ is the mass matrix of the ith blade, and the elements of $\varvec{M}_\mathrm{b}^i$ are given as follows:

  $$\begin{aligned}&\varvec{M}_\mathrm{b}^i ({m,n})=\sum _{n=1}^{N_{\mathrm{mod}} } {\left[ {\rho _\mathrm{b} \int _0^L {A_\mathrm{b} \phi _{1n} } \phi _{1m} \text {d}x} \right] },\\&\varvec{M}_\mathrm{b}^i \left( {m+N_{\mathrm{mod}},n+N_{\mathrm{mod}} } \right) \\&\quad =\sum _{n=1}^{N_{\mathrm{mod}} } {\left[ {\rho _\mathrm{b} \int _0^L {A_\mathrm{b} \phi _{2n} \phi _{2m} } \text {d}x} \right] },\\&\varvec{M}_\mathrm{b}^i \left( {m+2N_{\mathrm{mod}},n+2N_{\mathrm{mod}} } \right) \\&\quad =\sum _{n=1}^{N_{\mathrm{mod}} } {\left[ {\rho _\mathrm{b} \int _0^L {I_\mathrm{b} \phi _{3m} \phi _{3n} } \text {d}x} \right] }, \end{aligned}$$

  here, $m,n=\xi =1,2,\ldots ,N_{\mathrm{mod}}$, and the surplus elements are all zero.

• (3) $\varvec{G}_\mathrm{b}$ is the Coriolis force matrix of the blade:

  $$\begin{aligned} \varvec{G}_\mathrm{b} =\text {diag}[{\begin{array}{lllll} {\varvec{G}_\mathrm{b}^1 }&{}\quad \cdots &{}\quad {\varvec{G}_\mathrm{b}^i }&{}\quad \cdots &{}\quad {\varvec{G}_\mathrm{b}^{N_\mathrm{b} } } \\ \end{array} }], \end{aligned}$$

  (26)

  where $\varvec{G}_\mathrm{b}^i$ is the Coriolis force matrix of the ith blade, and the elements of $\varvec{G}_\mathrm{b}^i$ are given as follows:

  $$\begin{aligned}&\varvec{G}_\mathrm{b}^i \left( {m,n+N_{\mathrm{mod}} } \right) \nonumber \\&\quad =\sum _{n=1}^{N_{\mathrm{mod}} } {\left[ {-2\dot{\theta }\rho _\mathrm{b} \text {cos}\beta \int _0^L {A_\mathrm{b} \phi _{2n} \phi _{1m} } \text {d}x} \right] },\nonumber \\&\varvec{G}_\mathrm{b}^i \left( {m+N_{\mathrm{mod}},n} \right) \nonumber \\&\quad =\sum _{n=1}^{N_{\mathrm{mod}} } {\left[ {2\dot{\theta }\rho _\mathrm{b} \text {cos}\beta \int _0^L {A_\mathrm{b} \phi _{1n} \phi _{2m} } \text {d}x} \right] }, \end{aligned}$$

  and the surplus elements are all zero.

• (4) $\varvec{K}_\mathrm{b}$ is the stiffness matrix of the blades:

  $$\begin{aligned} \varvec{K}_\mathrm{b} =\text {diag}\left[ {\begin{array}{lllll} {\varvec{K}_\mathrm{b}^1 }&{}\quad \cdots &{}\quad {\varvec{K}_\mathrm{b}^i }&{}\quad \cdots &{}\quad {\varvec{K}_\mathrm{b}^{N_\mathrm{b} } } \\ \end{array} }\right] , \end{aligned}$$

  (27)

where $\varvec{K}_\mathrm{b}^i$ is the stiffness matrix of the ith blade, and the elements of $\varvec{K}_\mathrm{b}^i$ are given as follows:

$$\begin{aligned}&\varvec{K}_\mathrm{b}^i ({m,n})=\sum _{n=1}^{N_{\mathrm{mod}} } \left[ -\dot{\theta }^{2}\rho _\mathrm{b} \int _0^L {A_\mathrm{b} \phi _{1n} \phi _{1m} } \text {d}x\right. \nonumber \\&\quad \left. +\left. {E_\mathrm{b} A_\mathrm{b} {\phi }'_{1n} \phi _{1m} } \right| _{x=L} -\int _0^L E_\mathrm{b} \left( {A}'_\mathrm{b} {\phi }'_{1n} \right. \right. \nonumber \\&\quad \left. \left. +A_\mathrm{b} {\phi }''_{1n} \right) \phi _{1m} \text {d}x \right] ,\\&\varvec{K}_\mathrm{b}^i \left( {m,n+N_{\mathrm{mod}} } \right) \nonumber \\&\quad =\sum _{n=1}^{N_{\mathrm{mod}} } {\left[ {-\ddot{\theta }\text {cos}\beta \rho _\mathrm{b} \int _0^L {A_\mathrm{b} \phi _{2n} \phi _{1m} } \text {d}x} \right] },\\&\varvec{K}_\mathrm{b}^i \left( {m+N_{\mathrm{mod}},n} \right) \nonumber \\&\quad =\sum _{n=1}^{N_{\mathrm{mod}} } {\left[ {\ddot{\theta }\text {cos}\beta \rho _\mathrm{b} \int _0^L {A_\mathrm{b} \phi _{1n} \phi _{2m} } \text {d}x} \right] },\\ \end{aligned}$$

$$\begin{aligned}&\varvec{K}_\mathrm{b}^i \left( {m+N_{\mathrm{mod}},n+N_{\mathrm{mod}} } \right) \nonumber \\&\quad =\sum _{n=1}^{N_{\mathrm{mod}} } {\left[ {\begin{array}{l} \displaystyle \left. {\kappa G_\mathrm{b} A_\mathrm{b} {\phi }'_{2n} \phi _{2m} } \right| _{x=L} -\int _0^L {\kappa G_\mathrm{b} \left( {{A}'_\mathrm{b} {\phi }'_{2n} +A_\mathrm{b} {\phi }''_{2n} } \right) \phi _{2m} } \text {d}x \\ \displaystyle +f_\mathrm{c} (x)\left. {{\phi }'_{2n} \phi _{2m} } \right| _{x=L} -\int _0^L {\left( {{f}'_\mathrm{c} (x){\phi }'_{2n} +f_\mathrm{c} (x){\phi }''_{2n} } \right) \phi _{2m} } \text {d}x \\ \displaystyle +F_\mathrm{n} \left. {{\phi }'_{2n} \phi _{2m} } \right| _{x=L} -\int _0^L {\left( {F_\mathrm{n} {\phi }''_{2n} +{F}'_\mathrm{n} {\phi }'_{2n} } \right) \phi _{2m} } \text {d}x-\dot{\theta }^{2}\text {cos}^{\mathrm{2}}\beta \rho _\mathrm{b} \int _0^L {A_\mathrm{b} \phi _{2n} \phi _{2m} } \text {d}x \\ \end{array}} \right] }, \end{aligned}$$

$$\begin{aligned}&\varvec{K}_\mathrm{b}^i \left( {m+N_{\mathrm{mod}},n+2N_{\mathrm{mod}} } \right) \nonumber \\&\quad =\sum _{n=1}^{N_{\mathrm{mod}} } \left[ -\left. \kappa G_\mathrm{b} A_\mathrm{b} \phi _{3n} \phi _{2m} \right| _{x=L} +\int _0^L \kappa G_\mathrm{b} \left( {A}'_\mathrm{b} \phi _{3n} \right. \right. \nonumber \\&\quad \left. \left. +A_\mathrm{b} {\phi }'_{3n} \right) \phi _{2m} \text {d}x \right] ,\\&\varvec{K}_\mathrm{b}^i \left( {m+2N_{\mathrm{mod}},n+N_{\mathrm{mod}} } \right) \nonumber \\&\quad =\sum _{n=1}^{N_{\mathrm{mod}} } {\left[ {-\int _0^L {\kappa G_\mathrm{b} A_\mathrm{b} {\phi }'_{2n} \phi _{3m} } \text {d}x} \right] }, \end{aligned}$$

$$\begin{aligned}&\varvec{K}_\mathrm{b}^i \left( {m+2N_{\mathrm{mod}},n+2N_{\mathrm{mod}} } \right) \nonumber \\&\quad =\sum _{n=1}^{N_{\mathrm{mod}} } {\left[ {\begin{array}{l} \displaystyle \left. {E_\mathrm{b} I_\mathrm{b} {\phi }'_{3n} \phi _{3m} } \right| _{x=L} -\int _0^L {E_\mathrm{b} \left( {{I}'_\mathrm{b} {\phi }'_{3n} +I_\mathrm{b} {\phi }''_{3n} } \right) \phi _{3m} } \text {d}x \\ \displaystyle +\int _0^L {\kappa G_\mathrm{b} A_\mathrm{b} \phi _{3n} \phi _{3m} } \text {d}x-\dot{\theta }^{2}\rho _\mathrm{b} \int _0^L {I_\mathrm{b} \phi _{3n} \phi _{3m} } \text {d}x \\ \end{array}} \right] }, \end{aligned}$$

and the surplus elements are all zero.

Appendix 2: Coupled vectors and matrices related to rotor-blade systems

1.1 Mass matrix

The mass coupling matrix of rotor-blade systems is

$$\begin{aligned} \varvec{M}_\mathrm{c} =[\varvec{M}_{\mathrm{c1}},\varvec{M}_{\mathrm{c2}},\varvec{M}_{\mathrm{c3}},\varvec{M}_{\mathrm{c4}},\varvec{M}_{\mathrm{c5}},\varvec{M}_{\mathrm{c6}} ].\end{aligned}$$

(28)

• (1) $\varvec{M}_{\mathrm{c1}}$ is the mass coupling term at the disk location in X direction.

  $$\begin{aligned} \varvec{M}_{\mathrm{c}1} =\left[ {{\begin{array}{lllll} {\varvec{M}_{\mathrm{c}1}^1 }&{}\quad \cdots &{}\quad {\varvec{M}_{\mathrm{c}1}^i }&{}\quad \cdots &{}\quad {\varvec{M}_{\mathrm{c}1}^{N_\mathrm{b} } } \\ \end{array} }} \right] ^{\mathrm{T}}, \end{aligned}$$

  (29)

  where the superscript i denotes the ith blade. The elements of $\varvec{M}_{\mathrm{c1}}^i$ are given as follows:

  $$\begin{aligned}&\varvec{M}_{\mathrm{c}1}^i \left( {m,1} \right) =\rho _\mathrm{b} \cos \vartheta _i \int _0^L {A_\mathrm{b} } \phi _{\mathrm{1}m} \text {d}x,\\&\varvec{M}_{\mathrm{c}1}^i \left( {m+N_{\mathrm{mod}},1} \right) \\&\quad =-\rho _\mathrm{b} \sin \vartheta _i \text {cos}\beta \int _0^L {A_\mathrm{b} } \phi _{\mathrm{2}m} \text {d}x,\\&\varvec{M}_{\mathrm{c}1}^i \left( {m+2N_{\mathrm{mod}},1} \right) =0. \end{aligned}$$

• (2) $\varvec{M}_{\mathrm{c2}}$ is the mass coupling term at the disk location in Y direction.

  $$\begin{aligned} \varvec{M}_{\mathrm{c}2} =\left[ {{\begin{array}{lllll} {\varvec{M}_{\mathrm{c}2}^1 }&{}\quad \cdots &{}\quad {\varvec{M}_{\mathrm{c}2}^i }&{}\quad \cdots &{}\quad {\varvec{M}_{\mathrm{c}2}^{N_\mathrm{b} } } \\ \end{array} }} \right] ^{\mathrm{T}}. \end{aligned}$$

  (30)

  The elements of $\varvec{M}_{\mathrm{c2}}^i$ are given as follows:

  $$\begin{aligned}&\varvec{M}_{\mathrm{c}2}^i \left( {m,1} \right) =\rho _\mathrm{b} \sin \vartheta _i \int _0^L {A_\mathrm{b} } \phi _{\mathrm{1}m} \text {d}x,\\&\varvec{M}_{\mathrm{c}2}^i \left( {m+N_{\mathrm{mod}},1} \right) \\&\quad =\rho _\mathrm{b} \cos \vartheta _i \cos \beta \int _0^L {A_\mathrm{b} } \phi _{\mathrm{2}m} \text {d}x,\\&\varvec{M}_{\mathrm{c}2}^i \left( {m+2N_{\mathrm{mod}},1} \right) =0. \end{aligned}$$

• (3) $\varvec{M}_{\mathrm{c3}}$ is the mass coupling term at the disk location in Z direction.

  $$\begin{aligned} \varvec{M}_{\mathrm{c}3} =\left[ {{\begin{array}{lllll} {\varvec{M}_{\mathrm{c}3}^1 }&{}\quad \cdots &{}\quad {\varvec{M}_{\mathrm{c}3}^i }&{}\quad \cdots &{}\quad {\varvec{M}_{\mathrm{c}3}^{N_\mathrm{b} } } \\ \end{array} }} \right] ^{\mathrm{T}}.\end{aligned}$$

  (31)

  The elements of $\varvec{M}_{\mathrm{c3}}^i$ are given as follows:

  $$\begin{aligned}&\varvec{M}_{\mathrm{c}3}^i ({m,1})=0,\\&\varvec{M}_{\mathrm{c}3}^i \left( {m+N_{\mathrm{mod}},1} \right) =\rho _\mathrm{b} \text {sin}\beta \int _0^L {A_\mathrm{b} } \phi _{\mathrm{2}m} \text {d}x,\\&\varvec{M}_{\mathrm{c}3}^i \left( {m+2N_{\mathrm{mod}},1} \right) =0. \end{aligned}$$

• (4) $\varvec{M}_{\mathrm{c4}}$ is the mass coupling term at the disk location in $\theta _{\mathrm{X}}$ direction.

  $$\begin{aligned} \varvec{M}_{\mathrm{c}4} =\left[ {{\begin{array}{lllll} {\varvec{M}_{\mathrm{c}4}^1 }&{}\quad \cdots &{}\quad {\varvec{M}_{\mathrm{c}4}^i }&{}\quad \cdots &{}\quad {\varvec{M}_{\mathrm{c}4}^{N_\mathrm{b} } } \\ \end{array} }} \right] ^{\mathrm{T}}.\end{aligned}$$

  (32)

  The elements of $\varvec{M}_{\mathrm{c4}}^i$ are given as follows:

  $$\begin{aligned}&\varvec{M}_{\mathrm{c}4}^i \left( {m,1} \right) =0,\\&\varvec{M}_{\mathrm{c}4}^i \left( {m+N_{\mathrm{mod}},1} \right) \\&\quad = \rho _\mathrm{b} \sin \vartheta _i \sin \beta \int _0^L {\left( {R_\mathrm{d} +x} \right) A_\mathrm{b} } \phi _{\mathrm{2}m} \text {d}x,\\&\varvec{M}_{\mathrm{c}4}^i \left( {m+2N_{\mathrm{mod}},1} \right) \\&\quad = \rho _\mathrm{b} \sin \vartheta _i \sin \beta \int _0^L {I_\mathrm{b} \phi _{3m} } \text {d}x. \end{aligned}$$

• (5) $\varvec{M}_{\mathrm{c5}}$ is the mass coupling term at the disk location in $\theta _{\mathrm{Y}}$ direction.

  $$\begin{aligned} \varvec{M}_{\mathrm{c}5} =\left[ {{\begin{array}{lllll} {\varvec{M}_{\mathrm{c}5}^1 }&{}\quad \cdots &{}\quad {\varvec{M}_{\mathrm{c}5}^i }&{}\quad \cdots &{}\quad {\varvec{M}_{\mathrm{c}5}^{N_\mathrm{b} } } \\ \end{array} }} \right] ^{\mathrm{T}}.\end{aligned}$$

  (33)

  The elements of $\varvec{M}_{\mathrm{c5}}^i$ are given as follows:

  $$\begin{aligned}&\varvec{M}_{\mathrm{c}5}^i ({m,1})=0,\\&\varvec{M}_{\mathrm{c}5}^i ({m+N_{\mathrm{mod}},1})\\&\quad =- \rho _\mathrm{b} \cos \vartheta _i \sin \beta \int _0^L {\left( {R_\mathrm{d} +x} \right) A_\mathrm{b} } \phi _{\mathrm{2}m} \text {d}x,\\&\varvec{M}_{\mathrm{c}5}^i \left( {m+2N_{\mathrm{mod}},1} \right) \\&\quad =-\rho _\mathrm{b} \cos \vartheta _i \sin \beta \int _0^L {I_\mathrm{b} \phi _{3m} } \text {d}x. \end{aligned}$$

• (6) $\varvec{M}_{\mathrm{c6}}$ is the mass coupling term at the disk location in $\theta _{\mathrm{Z}}$ direction.

  $$\begin{aligned} \varvec{M}_{\mathrm{c}6} =\left[ {{\begin{array}{lllll} {\varvec{M}_{\mathrm{c}6}^1 }&{}\quad \cdots &{}\quad {\varvec{M}_{\mathrm{c}6}^i }&{}\quad \cdots &{}\quad {\varvec{M}_{\mathrm{c}6}^{N_\mathrm{b} } } \\ \end{array} }} \right] ^{\mathrm{T}}.\end{aligned}$$

  (34)

The elements of $\varvec{M}_{\mathrm{c6}}^i$ are given as follows:

$$\begin{aligned}&\varvec{M}_{\mathrm{c}6}^i \left( {m,1} \right) =0,\\&\varvec{M}_{\mathrm{c}6}^i \left( {m+N_{\mathrm{mod}},1} \right) \\&\quad =\rho _\mathrm{b} \text {cos}\beta \int _0^L {\left( {R_\mathrm{d} +x} \right) A_\mathrm{b} } \phi _{\mathrm{2}m} \text {d}x,\\&\varvec{M}_{\mathrm{c}6}^i \left( {m+2N_{\mathrm{mod}},1} \right) \\&\quad =\rho _\mathrm{b} \text {cos}\beta \int _0^L {I_\mathrm{b} \phi _{3m} } \text {d}x. \end{aligned}$$

1.2 Damping matrix

The damping coupling matrix of rotor-blade systems at the disk location is

$$\begin{aligned} \varvec{G}_\mathrm{c} =[\varvec{G}_{\mathrm{c1}},\varvec{G}_{\mathrm{c2}},\varvec{G}_{\mathrm{c3}},\varvec{G}_{\mathrm{c4}},\varvec{G}_{\mathrm{c5}},\varvec{G}_{\mathrm{c6}} ], \end{aligned}$$

(35)

where ${\varvec{G}}_{\mathrm{c}i} =\mathbf 0 (i=1,2{\ldots }5)$, and the expression of $\varvec{G}_{\mathrm{c6}}$ is

$$\begin{aligned} \varvec{G}_{\mathrm{c}6} =\left[ {{\begin{array}{lllll} {\varvec{G}_{\mathrm{c}6}^1 }&{}\quad \cdots &{}\quad {\varvec{G}_{\mathrm{c}6}^i }&{}\quad \cdots &{}\quad {\varvec{G}_{\mathrm{c}6}^{N_\mathrm{b} } } \\ \end{array} }} \right] ^{\mathrm{T}},\end{aligned}$$

(36)

where the elements of $\varvec{G}_{\mathrm{c6}}^i$ are given as follows:

$$\begin{aligned}&\varvec{G}_{\mathrm{c}6}^i ({m,1})=-2\dot{\theta }\rho _\mathrm{b} \int _0^L {A_\mathrm{b} \left( {R_\mathrm{d} +x} \right) \phi _{1m} } \text {d}x,\\&\varvec{G}_{\mathrm{c}6}^i \left( {m+N_{\mathrm{mod}},1} \right) =0,\\&\varvec{G}_{\mathrm{c}6}^i \left( {m+2N_{\mathrm{mod}},1} \right) =0. \end{aligned}$$

1.3 Stiffness matrix

The stiffness coupling matrix of rotor-blade systems related to the acceleration at the disk location is

$$\begin{aligned} \varvec{K}_{\mathrm{ac}} =[\varvec{K}_{\mathrm{ac1}},\varvec{K}_{\mathrm{ac2}} ,\varvec{K}_{\mathrm{ac3}},\varvec{K}_{\mathrm{ac4}},\varvec{K}_{\mathrm{ac5}} ,\varvec{K}_{\mathrm{ac6}} ],\end{aligned}$$

(37)

where ${\varvec{K}}_{\mathrm{ac}i}=\varvec{0 }(i=1,2{\ldots }5)$, and the expression of ${\varvec{K}}_{\mathrm{ac6}}$ is

$$\begin{aligned} \varvec{K}_{\mathrm{ac}6} =\left[ {{\begin{array}{lllll} {\varvec{K}_{\mathrm{ac}6}^1 }&{}\quad \cdots &{}\quad {\varvec{K}_{\mathrm{ac}6}^i }&{}\quad \cdots &{}\quad {\varvec{K}_{\mathrm{ac}6}^{N_\mathrm{b} } } \\ \end{array} }} \right] ^{\mathrm{T}}.\end{aligned}$$

(38)

The elements of $\varvec{K}_{\mathrm{ac6}}^i$ are given as follows:

$$\begin{aligned}&\varvec{K}_{\mathrm{ac}6}^i \left( {m,1} \right) =-\ddot{\theta }\rho _\mathrm{b} \int _0^L {A_\mathrm{b} \left( {R_\mathrm{d} +x} \right) \phi _{1m} } \text {d}x,\\&\varvec{K}_{\mathrm{ac}6}^i \left( {m+N_{\mathrm{mod}},1} \right) =0,\\&\varvec{K}_{\mathrm{ac}6}^i \left( {m+2N_{\mathrm{mod}},1} \right) =0. \end{aligned}$$

The stiffness coupling matrix of rotor-blade systems is

$$\begin{aligned} \varvec{K}_\mathrm{c} =[\varvec{K}_{\mathrm{c1}},\varvec{K}_{\mathrm{c2}},\varvec{K}_{\mathrm{c3}},\varvec{K}_{\mathrm{c4}},\varvec{K}_{\mathrm{c5}},\varvec{K}_{\mathrm{c6}} ], \end{aligned}$$

(39)

where ${\varvec{K}}_{\mathrm{c}i}=\mathbf 0 \,(i=1,2{\ldots }5)$, and the expression of ${\varvec{K}}_{\mathrm{c6}}$ is

$$\begin{aligned} \varvec{K}_{\mathrm{c}6} =\left[ {{\begin{array}{lllll} {\varvec{K}_{\mathrm{c}6}^1 }&{}\quad \cdots &{}\quad {\varvec{K}_{\mathrm{c}6}^i }&{}\quad \cdots &{}\quad {\varvec{K}_{\mathrm{c}6}^{N_\mathrm{b} } } \\ \end{array} }} \right] ^{\mathrm{T}}. \end{aligned}$$

(40)

The elements of $\varvec{K}_{\mathrm{c6}}^i$ are given as follows:

$$\begin{aligned}&\varvec{K}_{\mathrm{c}6}^i ({m,1})=0,\\&\varvec{K}_{\mathrm{c}6}^i ({m+N_{\mathrm{mod}},1})=-\dot{\theta }^{2}\cos \beta \rho _\mathrm{b} \int _0^L {A_\mathrm{b} \left( {R_\mathrm{d} +x} \right) \phi _{2m} } \text {d}x,\\&\varvec{K}_{\mathrm{c}6}^i \left( {m+2N_{\mathrm{mod}},1} \right) =-\dot{\theta }^{2}\cos \beta \rho _\mathrm{b} \int _0^L {I_\mathrm{b} \phi _{3m} } \text {d}x. \end{aligned}$$

Appendix 3: Other vectors and matrices related to the rotor-blade system

• (1) $\tilde{\varvec{M}}_\mathrm{d}$ is the added mass matrix at the disk location.

$$\begin{aligned} \tilde{\varvec{M}}_\mathrm{d} =\left[ {{\begin{array}{llllll} {\tilde{M}_{{XX}} }&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad {\tilde{M}_{X\theta _Z } } \\ 0&{}\quad {\tilde{M}_{{YY}} }&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad {\tilde{M}_{Y\theta _Z } } \\ 0&{}\quad 0&{}\quad {\tilde{M}_{{ZZ}} }&{}\quad {\tilde{M}_{Z\theta _X } }&{}\quad {\tilde{M}_{Z\theta _Y } }&{}\quad 0 \\ 0&{}\quad 0&{}\quad {\tilde{M}_{\theta _X Z} }&{}\quad {\tilde{M}_{\theta _X \theta _X } }&{}\quad {\tilde{M}_{\theta _X \theta _Y } }&{}\quad {\tilde{M}_{\theta _X \theta _Z } } \\ 0&{}\quad 0&{}\quad {\tilde{M}_{\theta _Y Z} }&{}\quad {\tilde{M}_{\theta _Y \theta _X } }&{}\quad {\tilde{M}_{\theta _Y \theta _Y } }&{}\quad {\tilde{M}_{\theta _Y \theta _Z } } \\ {\tilde{M}_{\theta _Z X} }&{}\quad {\tilde{M}_{\theta _Z Y} }&{}\quad 0&{}\quad {\tilde{M}_{\theta _Z \theta _X } }&{}\quad {\tilde{M}_{\theta _Z \theta _Y } }&{}\quad {\tilde{M}_{\theta _Z \theta _Z } } \\ \end{array} }} \right] ,\end{aligned}$$

(41)

where, $\tilde{M}_{{XX}} =\sum _{i=1}^{N_\mathrm{b} } {\int _0^L {\rho _\mathrm{b} } } A_\mathrm{b} \text {d}x, \tilde{M}_{{YY}} =\sum _{i=1}^{N_\mathrm{b} } {\int _0^L {\rho _\mathrm{b} } } A_\mathrm{b} \text {d}x, \tilde{M}_{{ZZ}} =\sum _{i=1}^{N_\mathrm{b} } {\int _0^L {\rho _\mathrm{b} } } A_\mathrm{b} \text {d}x$,

$$\begin{aligned}&\tilde{M}_{\theta _X \theta _X } =\sum _{i=1}^{N_\mathrm{b} } \left( \sin ^{2}\vartheta _i \int _0^L {\rho _\mathrm{b} A_\mathrm{b} \left( {R_\mathrm{d} +x} \right) ^{2}} \text {d}x\right. \nonumber \\&\quad \left. +\sin ^{\mathrm{2}}\beta \int _0^L {\rho _\mathrm{b} I_\mathrm{b} } \text {d}x+\cos ^{2}\vartheta _i \cos ^{\mathrm{2}}\beta \int _0^L {\rho _\mathrm{b} I_\mathrm{b} } \text {d}x \right) ,\\&\tilde{M}_{\theta _Y \theta _Y } =\sum _{i=1}^{N_\mathrm{b} } \left( \cos ^{2}\vartheta _i \int _0^L {\rho _\mathrm{b} A_\mathrm{b} \left( {R_\mathrm{d} +x} \right) ^{2}} \text {d}x\right. \nonumber \\&\quad \left. +\sin ^{\mathrm{2}}\beta \int _0^L {\rho _\mathrm{b} I_\mathrm{b} } \text {d}x+\sin ^{2}\vartheta _i \cos ^{\mathrm{2}}\beta \int _0^L {\rho _\mathrm{b} I_\mathrm{b} } \text {d}x \right) ,\\&\tilde{M}_{\theta _Z \theta _Z } =m_\mathrm{d} e^{2}+\sum _{i=1}^{N_\mathrm{b} } \left( \int _0^L {\rho _\mathrm{b} A_\mathrm{b} \left( {R_\mathrm{d} +x} \right) ^{2}} \text {d}x\right. \nonumber \\&\quad \left. +\int _0^L {\rho _\mathrm{b} I_\mathrm{b} \text {cos}^{\mathrm{2}}\beta } \text {d}x \right) ,\\&\tilde{M}_{X\theta _Z } =\tilde{M}_{\theta _Z X} =\sum _{i=1}^{N_\mathrm{b} } {\int _0^L {-\rho _\mathrm{b} } } A_\mathrm{b} \left( {R_\mathrm{d} +x} \right) \text {sin}\vartheta _i \text {d}x,\nonumber \\&\tilde{M}_{Y\theta _Z } =\tilde{M}_{\theta _Z Y} =\sum _{i=1}^{N_\mathrm{b} } {\int _0^L {\rho _\mathrm{b} } } A_\mathrm{b} \left( {R_\mathrm{d} +x} \right) \text {cos}\vartheta _i \text {d}x,\\&\tilde{M}_{Z\theta _X } =\tilde{M}_{\theta _X Z} =\sum _{i=1}^{N_\mathrm{b} } {\int _0^L {\rho _\mathrm{b} } } A_\mathrm{b} \left( {R_\mathrm{d} +x} \right) \text {sin}\vartheta _i \text {d}x,\nonumber \\&\tilde{M}_{Z\theta _Y } =\tilde{M}_{\theta _Y Z} =\sum _{i=1}^{N_\mathrm{b} } {\int _0^L {-\rho _\mathrm{b} } } A_\mathrm{b} \left( {R_\mathrm{d} +x} \right) \text {cos}\vartheta _i \text {d}x,\\&\tilde{M}_{\theta _X \theta _Y } =\tilde{M}_{\theta _Y \theta _X } \nonumber \\&\quad =\sum _{i=1}^{N_\mathrm{b} } \left( \int _0^L {-\rho _\mathrm{b} } A_\mathrm{b} \left( {R_\mathrm{d} +x} \right) ^{\mathrm{2}}\text {sin}\vartheta _i \cos \vartheta _i \text {d}x\right. \nonumber \\&\quad \left. +\int _0^L {\rho _\mathrm{b} } I_\mathrm{b} \text {sin}\vartheta _i \cos \vartheta _i \text {cos}^{\mathrm{2}}\beta \text {d}x\right) ,\\&\tilde{M}_{\theta _X \theta _Z } =\tilde{M}_{\theta _Z \theta _X } =\sum _{i=1}^{N_\mathrm{b} } {\int _0^L {\rho _\mathrm{b} } } I_\mathrm{b} \text {sin}\vartheta _i \text {sin}\beta \text {cos}\beta \text {d}x,\nonumber \\&\tilde{M}_{\theta _Y \theta _Z } =\tilde{M}_{\theta _Z \theta _Y } =\sum _{i=1}^{N_\mathrm{b} } {\int _0^L {-\rho _\mathrm{b} } } I_\mathrm{b} \cos \vartheta _i \text {sin}\beta \text {cos}\beta \text {d}x. \end{aligned}$$

• (2) $\tilde{\varvec{G}}_\mathrm{d}$ is the added damping matrix at the disk location.

  $$\begin{aligned} \tilde{\varvec{G}}_\mathrm{d} =\left[ {{\begin{array}{llllll} 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0 \\ 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0 \\ 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0 \\ 0&{}\quad 0&{}\quad 0&{}\quad {\tilde{G}_{\theta _X \theta _X } }&{}\quad {\tilde{G}_{\theta _X \theta _Y } }&{}\quad 0 \\ 0&{}\quad 0&{}\quad 0&{}\quad {\tilde{G}_{\theta _Y \theta _X } }&{}\quad {\tilde{G}_{\theta _Y \theta _Y } }&{}\quad 0 \\ 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0 \\ \end{array} }} \right] ,\end{aligned}$$

  (42)

  where $\tilde{G}_{\theta _X \theta _X } =\sum _{i=1}^{N_\mathrm{b} } \big ( 2\dot{\theta }\sin \vartheta _i \cos \vartheta _i \int _0^L \rho _\mathrm{b} A_\mathrm{b} \left( {R_\mathrm{d} +x} \right) ^{2} \text {d}x-2\dot{\theta }\sin \vartheta _i \cos \vartheta _i \cos ^{2}\beta \int _0^L {\rho _\mathrm{b} I_\mathrm{b} } \text {d}x \big )$,

  $$\begin{aligned} \tilde{G}_{\theta _Y \theta _Y }= & {} \sum _{i=1}^{N_\mathrm{b} } \left( -2\dot{\theta }\sin \vartheta _i \cos \vartheta _i \int _0^L {\rho _\mathrm{b} A_\mathrm{b} \left( {R_\mathrm{d} +x} \right) ^{2}} \text {d}x\right. \\&\left. +2\dot{\theta }\sin \vartheta _i \cos \vartheta _i \cos ^{2}\beta \int _0^L {\rho _\mathrm{b} I_\mathrm{b} } \text {d}x \right) ,\\ \tilde{G}_{\theta _X \theta _Y }= & {} \sum _{i=1}^{N_\mathrm{b} } \left( 2\dot{\theta }\sin ^{2}\vartheta _i \int _0^L {\rho _\mathrm{b} A_\mathrm{b} \left( {R_\mathrm{d} +x} \right) ^{2}} \text {d}x\right. \\&\left. +2\dot{\theta }\cos ^{2}\vartheta _i \cos ^{2}\beta \int _0^L {\rho _\mathrm{b} I_\mathrm{b} } \text {d}x \right) ,\\ \tilde{G}_{\theta _Y \theta _X }= & {} \sum _{i=1}^{N_\mathrm{b} } \left( -2\dot{\theta }\cos ^{2}\vartheta _i \int _0^L {\rho _\mathrm{b} A_\mathrm{b} \left( {R_\mathrm{d} +x} \right) ^{2}} \text {d}x\right. \\&\left. -2\dot{\theta }\sin ^{2}\vartheta _i \cos ^{2}\beta \int _0^L {\rho _\mathrm{b} I_\mathrm{b} } \text {d}x \right) . \end{aligned}$$

• (3) $\tilde{K}_\mathrm{d}$ is the added stiffness matrix at the disk location.

  $$\begin{aligned} \tilde{K}_\mathrm{d} =\left[ {{\begin{array}{llllll} 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0 \\ 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0 \\ 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0 \\ 0&{}\quad 0&{}\quad 0&{}\quad {\tilde{K}_{\theta _X \theta _X } }&{}\quad {\tilde{K}_{\theta _X \theta _Y } }&{}\quad 0 \\ 0&{}\quad 0&{}\quad 0&{}\quad {\tilde{K}_{\theta _Y \theta _X } }&{}\quad {\tilde{K}_{\theta _Y \theta _Y } }&{}\quad 0 \\ 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad {\tilde{K}_{\theta _Z \theta _Z } } \\ \end{array} }} \right] ,\nonumber \\\end{aligned}$$

  (43)

where

$$\begin{aligned}&\tilde{K}_{\theta _X \theta _X } =\sum \limits _{i=1}^{N_\mathrm{b} } \left( {\begin{array}{l} \displaystyle \left( {\ddot{\theta }\sin \vartheta _i \cos \vartheta _i -\dot{\theta }^{2}\sin ^{\mathrm{2}}\vartheta _i } \right) \int _0^L {\rho _\mathrm{b} A_\mathrm{b} \left( {R_\mathrm{d} +x} \right) ^{2}} \text {d}x \\ \displaystyle +\left( {-\ddot{\theta }\sin \vartheta _i \cos \vartheta _i \text {cos}^{\mathrm{2}}\beta -\dot{\theta }^{2}\cos ^{\mathrm{2}}\vartheta _i \text {cos}^{\mathrm{2}}\beta } \right) \int _0^L {\rho _\mathrm{b} I_\mathrm{b} } \text {d}x \\ \end{array}} \right) ,\\&\tilde{K}_{\theta _Y \theta _Y } =\sum _{i=1}^{N_\mathrm{b} } \left( {\begin{array}{l} \displaystyle \left( {-\ddot{\theta }\sin \vartheta _i \cos \vartheta _i -\dot{\theta }^{2}\cos ^{2}\vartheta _i } \right) \int _0^L {\rho _\mathrm{b} A_\mathrm{b} \left( {R_\mathrm{d} +x} \right) ^{2}} \text {d}x \\ \displaystyle +\left( {\ddot{\theta }\sin \vartheta _i \cos \vartheta _i \text {cos}^{\mathrm{2}}\beta -\dot{\theta }^{2}\sin ^{\mathrm{2}}\vartheta _i \text {cos}^{\mathrm{2}}\beta } \right) \int _0^L {\rho _\mathrm{b} I_\mathrm{b} } \text {d}x \\ \end{array}} \right) ,\\&\tilde{K}_{\theta _X \theta _Y } =\sum _{i=1}^{N_\mathrm{b} } \left( {\begin{array}{l} \displaystyle \left( {\ddot{\theta }\sin ^{2}\vartheta _i +\dot{\theta }^{2}\sin \vartheta _i \cos \vartheta _i } \right) \int _0^L {\rho _\mathrm{b} A_\mathrm{b} \left( {R_\mathrm{d} +x} \right) ^{2}} \text {d}x \\ \displaystyle +\left( {\ddot{\theta }\cos ^{2}\vartheta _i \text {cos}^{\mathrm{2}}\beta -\dot{\theta }^{2}\sin \vartheta _i \cos \vartheta _i \text {cos}^{\mathrm{2}}\beta } \right) \int _0^L {\rho _\mathrm{b} I_\mathrm{b} } \text {d}x \\ \end{array}} \right) ,\\&\tilde{K}_{\theta _Y \theta _X } =\sum _{i=1}^{N_\mathrm{b} } \left( {\begin{array}{l} \displaystyle \left( {-\ddot{\theta }\cos ^{2}\vartheta _i +\dot{\theta }^{2}\sin \vartheta _i \cos \vartheta _i } \right) \int _0^L {\rho _\mathrm{b} A_\mathrm{b} \left( {R_\mathrm{d} +x} \right) ^{2}} \text {d}x \\ \displaystyle -\left( {\ddot{\theta }\sin ^{2}\vartheta _i \text {cos}^{\mathrm{2}}\beta +\dot{\theta }^{2}\sin \vartheta _i \cos \vartheta _i \text {cos}^{\mathrm{2}}\beta } \right) \int _0^L {\rho _\mathrm{b} I_\mathrm{b} } \text {d}x \\ \end{array}} \right) ,\\&\tilde{K}_{\theta _Z \theta _Z } =\sum _{i=1}^{N_\mathrm{b} } \left( {-\int _0^L {\rho _\mathrm{b} A_\mathrm{b} \left( {R_\mathrm{d} +x} \right) ^{2}\dot{\theta }^{2}} \text {d}x-\int _0^L {\rho _\mathrm{b} I_\mathrm{b} \dot{\theta }^{2}\text {cos}^{\mathrm{2}}\beta } \text {d}x} \right) . \end{aligned}$$

Appendix 4: Element matrices of the rotational shaft

The element consistent mass matrix of Timoshenko beam ${\varvec{M}}^{\mathrm{e}}$ is

$$\begin{aligned} \varvec{M}^{\mathrm{e}}=\rho A_s l\left[ {{\begin{array}{llllllllllll} a&{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad \\ 0&{}\quad a&{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad \\ 0&{}\quad 0&{}\quad {1/3}&{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad \\ 0&{}\quad {-c}&{}\quad 0&{}\quad g&{}\quad &{}\quad &{}\quad &{}\quad &{}\quad {\text {Symmetric}}&{}\quad &{}\quad &{}\quad \\ c&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad g&{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad \\ 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad {J/(3A_s )}&{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad \\ b&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad d&{}\quad 0&{}\quad a&{}\quad &{}\quad &{}\quad &{}\quad &{}\quad \\ 0&{}\quad b&{}\quad 0&{}\quad {-d}&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad a&{}\quad &{}\quad &{}\quad &{}\quad \\ 0&{}\quad 0&{}\quad {1/6}&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad {1/3}&{}\quad &{}\quad &{}\quad \\ 0&{}\quad d&{}\quad 0&{}\quad f&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad c&{}\quad 0&{}\quad g&{}\quad &{}\quad \\ {-d}&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad f&{}\quad 0&{}\quad {-c}&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad g&{}\quad \\ 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad {J/(6A_s )}&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad {J/(3A_s )} \\ \end{array} }} \right] , \end{aligned}$$

(44)

where

$$\begin{aligned} a= & {} \frac{\frac{13}{35}+\frac{7}{10}\phi +\frac{1}{3}\phi ^{2}+\frac{6}{5}(r_g /l)^{2}}{(1+\phi )^{2}},\\ b= & {} \frac{\frac{9}{70}+\frac{3}{10}\phi +\frac{1}{6}\phi ^{2}-\frac{6}{5}(r_g /l)^{2}}{(1+\phi )^{2}},\\ c= & {} \frac{\left[ {\frac{11}{210}+\frac{11}{120}\phi +\frac{1}{24}\phi ^{2}+\left( {\frac{1}{10}-\frac{1}{2}\phi } \right) (r_g /l)^{2}} \right] l}{(1+\phi )^{2}}, \\ d= & {} \frac{\left[ {\frac{13}{420}+\frac{3}{40}\phi +\frac{1}{24}\phi ^{2}-\left( {\frac{1}{10}-\frac{1}{2}\phi } \right) (r_g /l)^{2}} \right] l}{(1+\phi )^{2}},\\ f= & {} \frac{\left[ {\frac{1}{140}+\frac{1}{60}\phi +\frac{1}{120}\phi ^{2}+\left( {\frac{1}{30}+\frac{1}{6}\phi -\frac{1}{6}\phi ^{2}} \right) (r_g /l)^{2}} \right] l^{2}}{(1+\phi )^{2}},\\ g= & {} \frac{\left[ {\frac{1}{105}+\frac{1}{60}\phi +\frac{1}{120}\phi ^{2}+\left( {\frac{2}{15}+\frac{1}{6}\phi +\frac{1}{3}\phi ^{2}} \right) (r_g /l)^{2}} \right] l^{2}}{(1+\phi )^{2}}, \end{aligned}$$

in which the transverse shear parameter $\phi =\frac{12EI}{\kappa _1 A_s Gl^{2}}$; I is the area moment of inertia; $\kappa _1 =\frac{6(1+\upsilon )}{7+6\upsilon }$ is the shape of factor; $A_{{s}}$ is the shaft cross-sectional area; G is shear modulus; l is element length; J is polar area moment of inertia; the radius of gyration $r_g =\sqrt{I/{A_s }}$.

Element stiffness matrix of Timoshenko beam ${\varvec{K}}^{\mathrm{e}}$ is

$$\begin{aligned} \varvec{K}^{\mathrm{e}}=\left[ {{\begin{array}{cccccccccccc} h&{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad \\ 0&{}\quad h&{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad \\ 0&{}\quad 0&{}\quad {A_s E/l}&{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad \\ 0&{}\quad {-i}&{}\quad 0&{}\quad j&{}\quad &{}\quad &{}\quad &{}\quad &{}\quad {\text {Symmetric}}&{}\quad &{}\quad &{}\quad \\ i&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad j&{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad \\ 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad {GJ/l}&{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad \\ {-h}&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad {-i}&{}\quad 0&{}\quad h&{}\quad &{}\quad &{}\quad &{}\quad &{}\quad \\ 0&{}\quad {-h}&{}\quad 0&{}\quad i&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad h&{}\quad &{}\quad &{}\quad &{}\quad \\ 0&{}\quad 0&{}\quad {-A_s E/l}&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad {A_s E/l}&{}\quad &{}\quad &{}\quad \\ 0&{}\quad {-i}&{}\quad 0&{}\quad k&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad i&{}\quad 0&{}\quad j&{}\quad &{}\quad \\ i&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad k&{}\quad 0&{}\quad {-i}&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad j&{}\quad \\ 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad {-GJ/l}&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad {GJ/l} \\ \end{array} }} \right] ,\end{aligned}$$

(45)

where $h=\frac{12EI}{l^{3}(1+\phi )}, i=\frac{6EI}{l^{2}(1+\phi )}, j=\frac{(4+\phi )EI}{l(1+\phi )}$ and $k=\frac{(2-\phi )EI}{l(1+\phi )}$.

Gyroscopic matrix of Timoshenko beam ${\varvec{G}}^{\mathrm{e}}$ is

$$\begin{aligned} \varvec{G}^{\mathrm{e}}=2\dot{\theta }\rho A_s l\left[ {{\begin{array}{cccccccccccc} 0&{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad \\ {-p}&{}\quad 0&{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad \\ 0&{}\quad 0&{}\quad 0&{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad \\ {-q}&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad &{}\quad &{}\quad &{}\quad &{}\quad {\text {Antisymmetric}}&{}\quad &{}\quad &{}\quad \\ 0&{}\quad {-q}&{}\quad 0&{}\quad {-s}&{}\quad 0&{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad \\ 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad \\ 0&{}\quad {-p}&{}\quad 0&{}\quad {-q}&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad &{}\quad &{}\quad &{}\quad &{}\quad \\ p&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad {-q}&{}\quad 0&{}\quad {-p}&{}\quad 0&{}\quad &{}\quad &{}\quad &{}\quad \\ 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad &{}\quad &{}\quad \\ {-q}&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad w&{}\quad 0&{}\quad q&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad &{}\quad \\ 0&{}\quad {-q}&{}\quad 0&{}\quad {-w}&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad q&{}\quad 0&{}\quad {-s}&{}\quad 0&{}\quad \\ 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0 \\ \end{array} }} \right] ,\end{aligned}$$

(46)

where $p=\frac{{6r_g^2 }/5}{l^{2}(1+\phi )^{2}}, q=\frac{-\left( {\frac{1}{10}-\frac{1}{2}\phi } \right) r_g^2 }{l\left( {1+\phi } \right) ^{2}}$, $s=\frac{\left( {\frac{2}{15}+\frac{1}{6}\phi +\frac{1}{3}\phi ^{2}} \right) r_g^2 }{\left( {1+\phi } \right) ^{2}}$ and $w=\frac{-\left( {\frac{1}{30}+\frac{1}{6}\phi -\frac{1}{6}\phi ^{2}} \right) r_g^2 }{\left( {1+\phi } \right) ^{2}}$.

Appendix 5: Nonlinear forces of the rotor-blade system

• (1) $\varvec{F}_{\mathrm{nonlinear,\,b}}^i$ is a nonlinear force vector of ith blade:

  $$\begin{aligned}&\varvec{F}_{\mathrm{nonlinear,\,b}}^i \left( {m,1} \right) =\dot{\theta }^{2}\rho _\mathrm{b} \\&\quad \int _0^L {\left( {R_\mathrm{d} +x} \right) \phi _{1m} A_\mathrm{b} } \text {d}x,\\&\varvec{F}_{\mathrm{nonlinear,\,b}}^i \left( {m+N_{\mathrm{mod}},1} \right) \nonumber \\&\quad =\left( {\begin{array}{l} -\ddot{\theta }\text {cos}\beta \\ -2\sin \vartheta _i \sin \beta \dot{\theta }_{Y\mathrm{d}} \dot{\theta }-\sin \vartheta _i \sin \beta \theta _{Y\mathrm{d}} \ddot{\theta } \\ -\cos \vartheta _i \sin \beta \theta _{Y\mathrm{d}} \dot{\theta }^{2}-2\cos \vartheta _i \sin \beta \dot{\theta }_{X\mathrm{d}} \dot{\theta } \\ -\cos \vartheta _i \sin \beta \theta _{X\mathrm{d}} \ddot{\theta }+\sin \vartheta _i \sin \beta \theta _{X\mathrm{d}} \dot{\theta }^{2} \\ \end{array}} \right) \end{aligned}$$
  $$\begin{aligned}&\quad \rho _\mathrm{b} \int _0^L {A_\mathrm{b} \left( {R_\mathrm{d} +x} \right) \phi _{2m} } \text {d}x, \end{aligned}$$

  (47)
  $$\begin{aligned}&\varvec{F}_{\mathrm{nonlinear},\text {b}}^i \left( {m+2N_{\mathrm{mod}},1} \right) \nonumber \\&\quad =-\ddot{\theta }\text {cos}\beta \rho _\mathrm{b} \int _0^L {I_\mathrm{b} \phi _{3m} } \text {d}x.\end{aligned}$$

  (48)

• (2) $f_{\mathrm{nonlinear},X}$ is a nonlinear force applied on the disk in X direction.

$$\begin{aligned}&f_{\mathrm{nonlinear,}\;X} =em_\mathrm{d} \cos \left( {\theta +\theta _{Z\mathrm{d}} } \right) \left( {\dot{\theta }+\dot{\theta }_{Z\mathrm{d}} } \right) ^{2}\nonumber \\&\quad +em_\mathrm{d} \sin \left( {\theta +\theta _{Z\mathrm{d}} } \right) \left( {\ddot{\theta }+\ddot{\theta }_{Z\mathrm{d}} } \right) \nonumber \\&\quad +\sum _{i=1}^{N_\mathrm{b} } {\sum _{i=1}^{N_{\mathrm{mod}} } {\left( {\begin{array}{l} \displaystyle 2\dot{\theta }\sin \vartheta _i \int _0^L {\rho _\mathrm{b} A_\mathrm{b} \phi _{1m} \dot{U}_m \text {d}x} \\ \displaystyle +\left( {\dot{\theta }^{2}\cos \vartheta _i +\ddot{\theta }\sin \vartheta _i } \right) \int _0^L {\rho _\mathrm{b} A_\mathrm{b} \phi _{1m} U_m \text {d}x} \\ \end{array}} \right) } } \nonumber \\&\displaystyle \quad +\sum _{i=1}^{N_\mathrm{b} } {\sum _{i=1}^{N_{\mathrm{mod}} } {\left( {\begin{array}{l} \displaystyle 2\dot{\theta }\cos \vartheta _i \text {cos}\beta \int _0^L {\rho _\mathrm{b} A_\mathrm{b} \phi _{2m} \dot{V}_m \text {d}x} \\ \displaystyle +\left( {-\dot{\theta }^{2}\sin \vartheta _i +\ddot{\theta }\cos \vartheta _i } \right) \text {cos}\beta \int _0^L {\rho _\mathrm{b} A_\mathrm{b} \phi _{2m} V_m \text {d}x} \\ \end{array}} \right) } } \nonumber \\&\quad +\sum _{i=1}^{N_\mathrm{b} } \left( \left( {\begin{array}{l} 2\dot{\theta }_{Z\mathrm{d}} \dot{\theta }\cos \vartheta _i -\theta _{Z\mathrm{d}} \dot{\theta }^{2}\sin \vartheta _i +\theta _{Z\mathrm{d}} \ddot{\theta }\cos \vartheta _i \\ +\dot{\theta }^{2}\cos \vartheta _i +\ddot{\theta }\sin \vartheta _i \\ \end{array}} \right) \right. \nonumber \\&\quad \left. \int _0^L {\rho _\mathrm{b} A_\mathrm{b} \left( {R_\mathrm{d} +x} \right) \text {d}x} \right) .\end{aligned}$$

(49)

• (3) $f_{{\mathrm{nonlinear}},Y}$ is a nonlinear force applied on the disk in Y direction.

$$\begin{aligned}&f_{\mathrm{nonlinear, }Y} =em_\mathrm{d} \text {sin}\left( {\theta +\theta _{Z\mathrm{d}} } \right) \left( {\dot{\theta }+\dot{\theta }_{Z\mathrm{d}} } \right) ^{2}\nonumber \\&\quad -em_\mathrm{d} \cos \left( {\theta +\theta _{Z\mathrm{d}} } \right) \left( {\ddot{\theta }+\ddot{\theta }_{Z\mathrm{d}} } \right) \nonumber \\&\quad +\sum _{i=1}^{N_\mathrm{b} } {\sum _{i=1}^{N_{\mathrm{mod}} } {\left( {\begin{array}{l} -2\dot{\theta }\cos \vartheta _i \int _0^L {\rho _\mathrm{b} A_\mathrm{b} \phi _{1m} \dot{U}_m \text {d}x} \\ +\left( {\dot{\theta }^{2}\sin \vartheta _i -\ddot{\theta }\cos \vartheta _i } \right) \int _0^L {\rho _\mathrm{b} A_\mathrm{b} \phi _{1m} U_m \text {d}x} \\ \end{array}} \right) } } \nonumber \\&\quad +\sum _{i=1}^{N_\mathrm{b} } {\sum _{i=1}^{N_{\mathrm{mod}} } {\left( {\begin{array}{l} 2\dot{\theta }\sin \vartheta _i \text {cos}\beta \int _0^L {\rho _\mathrm{b} A_\mathrm{b} \phi _{2m} \dot{V}_m \text {d}x} \\ +\left( {\dot{\theta }^{2}\cos \vartheta _i +\ddot{\theta }\sin \vartheta _i } \right) \text {cos}\beta \int _0^L {\rho _\mathrm{b} A_\mathrm{b} \phi _{2m} V_m \text {d}x} \\ \end{array}} \right) } } \nonumber \\&\quad +\sum _{i=1}^{N_\mathrm{b} } \left( \left( {\begin{array}{l} 2\dot{\theta }_{z\mathrm{d}} \dot{\theta }\sin \vartheta _i +\theta _{Z\mathrm{d}} \dot{\theta }^{2}\cos \vartheta _i +\theta _{Z\mathrm{d}} \ddot{\theta }\sin \vartheta _i \\ +\dot{\theta }^{2}\sin \vartheta _i -\ddot{\theta }\cos \vartheta _i \\ \end{array}} \right) \right. \nonumber \\&\quad \left. \int _0^L {\rho _\mathrm{b} A_\mathrm{b} \left( {\text {R}_\mathrm{d} +x} \right) \text {d}x} \right) .\end{aligned}$$

(50)

• (4) $f_{{\mathrm{nonlinear}},Z}$ is a nonlinear force applied on the disk in Z direction.

$$\begin{aligned} f_{\mathrm{nonlinear,}Z}= & {} \sum _{i=1}^{N_\mathrm{b} } \left( \left( {\begin{array}{l} -2\dot{\theta }_{Y\mathrm{d}} \dot{\theta }\sin \vartheta _i -\theta _{Y\mathrm{d}} \ddot{\theta }\sin \vartheta _i -\theta _{Y\mathrm{d}} \dot{\theta }^{2}\cos \vartheta _i \\ -2\dot{\theta }_{X\mathrm{d}} \dot{\theta }\cos \vartheta _i -\theta _{X\mathrm{d}} \ddot{\theta }\cos \vartheta _i +\theta _{X\mathrm{d}} \dot{\theta }^{2}\sin \vartheta _i \\ \end{array}} \right) \right. \nonumber \\&\left. \quad \int _0^L {\rho _\mathrm{b} A_\mathrm{b} \left( {\text {R}_\mathrm{d} +x} \right) \text {d}x} \right) . \end{aligned}$$

(51)

• (5) $M_{\mathrm{nonlinear},X}$ is a nonlinear bending moment applied on the disk in $\theta _{\mathrm{X}}$ direction.

  $$\begin{aligned}&M_{\mathrm{nonlinear},\,X} =-J_p \dot{\theta }_{Y\mathrm{d}} \dot{\theta }_{Z\mathrm{d}} \nonumber \\&+\sum _{i=1}^{N_\mathrm{b} } \left( \left( {\begin{array}{l} -2\dot{\theta }_{Z\mathrm{d}} \dot{\theta }\hbox {cos}\vartheta _i -\theta _{Z\mathrm{d}} \ddot{\theta }\cos \vartheta _i \\ +\theta _{Z\mathrm{d}} \dot{\theta }^{2}\sin \vartheta _i -\ddot{\theta }\sin \vartheta _i -\dot{\theta }^{2}\cos \vartheta _i \\ \end{array}} \right) \right. \nonumber \\&\quad \left. \sin \beta \cos \beta \int _0^L {\rho _\mathrm{b} I_\mathrm{b} \hbox {d}x} \right) \nonumber \\&+\sum _{i=1}^{N_\mathrm{b} } \sum _{i=1}^{N_{\mathrm{mod}} }\nonumber \\&\times \left( {\begin{array}{l} \displaystyle -2\dot{\theta }\hbox {cos}\vartheta _i \sin \beta \int _0^L {\rho _\mathrm{b} I_\mathrm{b} \phi _{\mathrm{3m}} \dot{\psi }_\mathrm{m} \hbox {d}x} \\ +\left( {-\ddot{\theta }\cos \vartheta _i +\dot{\theta }^{2}\sin \vartheta _i } \right) \sin \beta \int _0^L {\rho _\mathrm{b} I_\mathrm{b} \phi _{\mathrm{3m}} \psi _\mathrm{m} \hbox {d}x} \\ \end{array}} \right) \end{aligned}$$

  (52)

• (6) $M_{\mathrm{nonlinear},Y}$ is a nonlinear bending moment applied on the disk in $\theta _{\mathrm{Y}}$ direction.

  $$\begin{aligned}&M_{\mathrm{nonlinear},\;Y} =J_p \left( {\ddot{\theta }\theta _{X\mathrm{d}} +\ddot{\theta }_{Z\mathrm{d}} \theta _{X\mathrm{d}} +\dot{\theta }_{Z\mathrm{d}} \dot{\theta }_{X\mathrm{d}} } \right) \nonumber \\&+\sum _{i=1}^{N_\mathrm{b} } \left( \left( {\begin{array}{l} -2\dot{\theta }_{Z\mathrm{d}} \dot{\theta }\sin \vartheta _i -\theta _{Z\mathrm{d}} \ddot{\theta }\sin \vartheta _i \\ -\theta _{Z\mathrm{d}} \dot{\theta }^{2}\cos \vartheta _i +\ddot{\theta }\cos \vartheta _i -\dot{\theta }^{2}\sin \vartheta _i \\ \end{array}} \right) \right. \nonumber \\&\quad \left. \sin \beta \cos \beta \int _0^L {\rho _\mathrm{b} I_\mathrm{b} \hbox {d}x} \right) \nonumber \\&+\sum _{i=1}^{N_\mathrm{b} } \sum _{i=1}^{N_{\mathrm{mod}} }\nonumber \\&\times \left( {\begin{array}{l} \displaystyle -2\dot{\theta }\hbox {sin}\vartheta _i \sin \beta \int _0^L {\rho _\mathrm{b} I_\mathrm{b} \phi _{\mathrm{3m}} \dot{\psi }_\mathrm{m} \hbox {d}x} \\ +\left( {-\ddot{\theta }\sin \vartheta _i -\dot{\theta }^{2}\cos \vartheta _i } \right) \sin \beta \int _0^L {\rho _\mathrm{b} I_\mathrm{b} \phi _{\mathrm{3m}} \psi _\mathrm{m} \hbox {d}x} \\ \end{array}} \right) \nonumber \\ .\end{aligned}$$

  (53)

• (7) $M_{\mathrm{nonlinear},Z}$ is a nonlinear torque applied on the disk in $\theta \hbox {z}$ direction.

  $$\begin{aligned}&M_{\mathrm{nonlinear},\;Z} =em_\mathrm{d} \sin \left( {\theta +\theta _{Z\mathrm{d}} } \right) \ddot{X}_\mathrm{d} \nonumber \\&\quad -em_\mathrm{d} \cos \left( {\theta +\theta _{Z\mathrm{d}} } \right) \ddot{Y}_\mathrm{d} -e^{2}m_\mathrm{d} \ddot{\theta }-J_\mathrm{p} \ddot{\theta } \nonumber \\&+J_p \left( {\dot{\theta }_{X\mathrm{d}} \dot{\theta }_{Y\mathrm{d}} +\theta _{X\mathrm{d}} \ddot{\theta }_{Y\mathrm{d}} } \right) \nonumber \\&\quad -\sum _{i=1}^{N_\mathrm{b} } {\left( {\ddot{\theta }\rho _\mathrm{b} \int _0^L {\left( {A_\mathrm{b} \left( {R_\mathrm{d} +x} \right) ^{2}+I_\mathrm{b} \cos ^{2}\beta } \right) } \hbox {d}x} \right) }.\nonumber \\ \end{aligned}$$

  (54)