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.
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Abbreviations
- \({\varvec{C}}_{\mathrm{RB}}\) :
-
Viscous damping matrix of the rotor-blade system
- \(c_{\mathrm{b}X}, c_{\mathrm{b}Y}, c_{\mathrm{b}Z}\) :
-
Bearing damping in X, Y and Z directions
- \(c_{\mathrm{c}X}, c_{\mathrm{c}Y}\) :
-
Damping of the casing in X and Y directions
- E :
-
Young’s modulus of blade
- \({\varvec{e}}_{\mathrm{c}}\) :
-
Vector of the eccentricity of the static equilibrium positions of the rotor and casing center line
- \({\varvec{F}}_{\mathrm{c}}\) :
-
Rubbing force vector of the casing
- \({\varvec{F}}_{\mathrm{nonlinear},}, {\varvec{F}}_{\mathrm{rub}}\) :
-
Nonlinear force and rubbing force vectors of the rotor-blade system
- \({\varvec{F}}_{\mathrm{nonlinear,b}},{\varvec{F}}_{\mathrm{nonlinear,s}}\) :
-
Nonlinear force vectors of the blade and shaft
- \(F_{\mathrm{n}}, F_{\mathrm{t}}\) :
-
Normal and tangential rubbing forces
- \(f_{\mathrm{n}i}\) :
-
The ith natural frequency of the rotor-blade system (Hz)
- \(f_\mathrm{r}\) :
-
Rotational frequency
- \(G_{\mathrm{b}}\) :
-
Shear modulus of the blade
- \({\varvec{G}}_{\mathrm{b}},{\varvec{G}}_{\mathrm{d}}\) :
-
Coriolis matrices of the blade and disk
- \({\varvec{G}}_{\mathrm{c}1},\ldots ,{\varvec{G}}_{\mathrm{c}6}\) :
-
Coupling terms of damping matrix
- \({\varvec{G}}^{\mathrm{e}}\) :
-
Gyroscopic matrix of the Timoshenko beam element
- \({\varvec{G}}_{\mathrm{RB}}\) :
-
A matrix including the Coriolis force matrices of the blades, damping matrix of bearings, and gyroscopic matrices of the shaft and rigid disk
- \(g_{0}\) :
-
Gap between concentric rotor-blade and casing
- \(J_{d}, J_{p}\) :
-
Diametric and polar mass moments of inertia of the disk
- \({\varvec{K}}_{\mathrm{b}1}, {\varvec{K}}_{\mathrm{b}2}\) :
-
Stiffness matrices of the left and right bearings
- \(k_{\mathrm{b}X}, k_{\mathrm{b}Y}, k_{\mathrm{b}Z}\) :
-
Stiffness of the bearing in X, Y and Z directions
- \({\varvec{K}}_{\mathrm{c}}\) :
-
Stiffness matrix of the casing
- \(k_{\mathrm{c}}\) :
-
Equivalent stiffness of the casing
- \(k_{\mathrm{c}X}, k_{\mathrm{c}Y}\) :
-
Stiffness of the casing in X and Y directions
- \({\varvec{K}}^{\mathbf{e}}\) :
-
Stiffness matrix of the Timoshenko beam element
- \({\varvec{K}}_{\varvec{s}}\) :
-
The stiffness matrix of the shaft
- L :
-
Length of the blade
- \({\varvec{M}}_{\mathrm{b}}, {\varvec{M}}_{\mathrm{RB }}\) :
-
Mass matrices of the blade and rotor-blade system
- \({\varvec{M}}_{\mathrm{c}}, {\varvec{M}}_{\mathrm{d}}\) :
-
Mass matrices of the casing and disk
- \({\varvec{M}}_{\mathrm{c}1}, {\varvec{M}}_{\mathrm{c}2}, {\varvec{M}}_{\mathrm{c}3}\) :
-
Coupling terms of mass matrices
- \({\varvec{M}}^{\mathbf{e}}\) :
-
mass matrix of the Timoshenko beam element
- \({\varvec{M}}_{\mathbf{s}}\) :
-
Mass matrix of the shaft
- \(M_{\mathrm{rub},X}, M_{\mathrm{rub},Y}, M_{\mathrm{rub},Z}\) :
-
Bending moments at the disk location in \(\theta \) \(_{X}\) and \(\theta \) \(_{Y}\) directions and the torque in \(\theta \) \(_{Z}\) direction
- \(m_{\mathrm{d}}\) :
-
Mass of the disk
- \(N_{\mathrm{b}}\) :
-
Blade number
- \(N_{\mathrm{dof}}\) :
-
Number of DOFs for ith blade
- \(N_{\mathrm{mod}}\) :
-
Number of modal truncation
- \(N_{N\mathrm{s}}\) :
-
Number of DOFs for the shaft
- \({\varvec{n}}_{i}\) :
-
Unit normal vector to the contact surface for the ith blade
- \({\varvec{q}}_{\mathrm{b}}, {\varvec{q}}_{\mathrm{c}},{\varvec{q}}_{\mathrm{d}}, {\varvec{q}}_{\mathrm{r}}, {\varvec{q}}_{\mathrm{s}},{\varvec{q}}_{\mathrm{RB},}\) :
-
Generalized coordinate vectors of the blade, casing, disk, rotor, shaft and rotor-blade system
- \(R_{\mathrm{c}}, R_{\mathrm{d}}\) :
-
Radius of the casing and disk
- \(U_{m}(t), V_{m}(t), \psi _{m}(t)\) :
-
Canonical coordinates
- \(\mathbf{u}_\mathrm{b}^i \) :
-
Displacement vector of the ith blade in the global coordinate system
- \({\varvec{u}}_{\mathrm{c}}\) :
-
Displacement vector of casing
- \({\varvec{u}}_{i}\) :
-
Displacement vector of the ith blade-casing relative motion in the global coordinate system
- \(u, v,w, \varphi \) :
-
Longitudinal deformation, lateral deformation, swing deformation and cross-sectional rotation in blade local coordinate system
- \(X_{\mathrm{d}}, Y_{\mathrm{d}}, Z_{\mathrm{d}}\) :
-
Displacements of the disk in X, Y and Z directions in the global coordinate system
- \(\beta \) :
-
Stagger angle of the blade
- \(\delta \) :
-
The penetration depth
- \(\delta _{0}\) :
-
Initial penetration depth
- \(\delta {u},{\delta {v}}, {\delta \varphi }\) :
-
Independence variables of variational operation
- \(\theta (t)\) :
-
The angular displacement of the disk
- \(\theta _{X\mathrm{d}}, \theta _{Y\mathrm{d}}\) and \(\theta _{Z\mathrm{d}} \) :
-
Swing angle of the disk in X and Y directions and torsional angle of the shaft
- \(\dot{\theta }\) :
-
Angular velocity
- \(\kappa \) :
-
Shear correction factor of the blade
- \(\mu \) :
-
Friction coefficient
- \(\xi _1, \xi _2\) :
-
Modal damping ratios (in this paper, \(\xi _1 =\xi _2 =0.02)\)
- \(\upsilon \) :
-
Poisson’s ratio
- \(\rho _{\mathrm{b}}\) :
-
Material density of the blade
- \(\phi _{1m} (x), \phi _{2m} (x), \phi _{3m} (x)\) :
-
Modal shape functions
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Acknowledgments
This project is supported by Program for the Joint Funds of the National Natural Science Foundation and the Civil Aviation Administration of China (Grant No. U1433109), the Fundamental Research Funds for the Central Universities (Grant Nos. N130403006 and N140301001) and the National Basic Research Program of China (Grant No. 2011CB706504) for providing financial support for this work. We also thank the anonymous reviewers for their valuable comments and Professor Ziqiang Lang and Dr. Yuzhu Guo from the University of Sheffield for the proof reading of the final version of the paper and offering suggestions on the improvement of presentation.
Conflict of interest
The authors declare that there is no conflict of interests regarding the publication of this article.
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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:
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
-
(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:
1.2 Damping matrix
The damping coupling matrix of rotor-blade systems at the disk location is
where \({\varvec{G}}_{\mathrm{c}i} =\mathbf 0 (i=1,2{\ldots }5)\), and the expression of \(\varvec{G}_{\mathrm{c6}} \) is
where the elements of \(\varvec{G}_{\mathrm{c6}}^i \) are given as follows:
1.3 Stiffness matrix
The stiffness coupling matrix of rotor-blade systems related to the acceleration at the disk location is
where \({\varvec{K}}_{\mathrm{ac}i}=\varvec{0 }(i=1,2{\ldots }5)\), and the expression of \({\varvec{K}}_{\mathrm{ac6}}\) is
The elements of \(\varvec{K}_{\mathrm{ac6}}^i \) are given as follows:
The stiffness coupling matrix of rotor-blade systems is
where \({\varvec{K}}_{\mathrm{c}i}=\mathbf 0 \,(i=1,2{\ldots }5)\), and the expression of \({\varvec{K}}_{\mathrm{c6}}\) is
The elements of \(\varvec{K}_{\mathrm{c6}}^i \) are given as follows:
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.
where, \(\tilde{M}_{{XX}} =\sum _{i=1}^{N_\mathrm{b} } {\int _0^L {\rho _\mathrm{b} } } A_\mathrm{b} \text {d}x, \tilde{M}_{{YY}} =\sum _{i=1}^{N_\mathrm{b} } {\int _0^L {\rho _\mathrm{b} } } A_\mathrm{b} \text {d}x, \tilde{M}_{{ZZ}} =\sum _{i=1}^{N_\mathrm{b} } {\int _0^L {\rho _\mathrm{b} } } A_\mathrm{b} \text {d}x\),
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(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
Appendix 4: Element matrices of the rotational shaft
The element consistent mass matrix of Timoshenko beam \({\varvec{M}}^{\mathrm{e}}\) is
where
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
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
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
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(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.
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(3)
\(f_{{\mathrm{nonlinear}},Y}\) is a nonlinear force applied on the disk in Y direction.
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(4)
\(f_{{\mathrm{nonlinear}},Z}\) is a nonlinear force applied on the disk in Z direction.
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(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)
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(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)
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Ma, H., Yin, F., Wu, Z. et al. Nonlinear vibration response analysis of a rotor-blade system with blade-tip rubbing. Nonlinear Dyn 84, 1225–1258 (2016). https://doi.org/10.1007/s11071-015-2564-5
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DOI: https://doi.org/10.1007/s11071-015-2564-5