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Steady-state coupling vibration analysis of shaft–disk–blade system with blade crack

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Abstract

Rotating shaft–disk–blade (RSDB) system is one of the most important parts of turbomachinery, such as aero-engine, gas turbine and power plant. The coupling vibration of RSDB system with blade crack is vital for the blade health monitoring and crack detection of rotating blade. This study aims at addressing the dynamic modeling and steady-state coupling vibration mechanism of RSDB system with blade crack. First and foremost, on the basis of the stress state at crack section, an improved analytical breathing crack model (modified stress-based breathing crack model, MSBCM) for rotating blade is proposed. The validity of the proposed breathing crack model is verified by comparing the results obtained by MSBCM, finite element contact crack model and conventional analytical crack models. The comparative results suggest that MSBCM is of high fidelity and behaves best among the analytical crack models. Subsequently, a comprehensive dynamic model of the coupling vibration for RSDB system with blade crack is formulated on the basis of continuum beam theory and Lagrange equation. The shaft bending, shaft torsion, blade bending and blade radial deformation are comprehensively considered in this model. The validity of the proposed dynamic model is verified through comparison with finite element simulation and experimentation results. By introducing the proposed MSBCM, the dynamic coupling vibration model of the RSDB system with blade crack is formulated. At last, the steady-state coupling vibration mechanism of two typical structures for RSDB system is comprehensively investigated. It is suggested that the shaft torsional vibration is much more sensitive to blade crack than the shaft bending vibration be, which indicates that the vibration features of shaft torsional vibration may offer indicators for the presence of blade crack.

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Abbreviations

\(\gamma \) :

Relative crack location

\(\lambda \) :

Relative crack depth

\(\nu _\mathrm{b}\) :

Poisson ratio of blade

\(\omega \) :

Shaft rotating speed

\(\psi _\mathrm{s}\) :

Shaft torsional vibration displacement

\(\varphi _{K_j}\) :

Phase of excitation force

\(A_\mathrm{b}\) :

Blade cross-section area

\(a_\mathrm{c}\) :

Crack depth

\(A_\mathrm{s}\) :

Shaft cross-section area

\(E_\mathrm{b}\) :

Elastic modulus of blade

\(E_\mathrm{d}\) :

Elastic modulus of disk

\(E_\mathrm{s}\) :

Elastic modulus of shaft

\(f(y_0)\) :

Breathing function of the SBCM

\(F_0\) :

Load amplitude

\(f_\mathrm{c}(x)\) :

Centrifugal force of rotating blade

\(f_{v_{i}}(x,t)\) :

Excitation force loaded on the \(i\mathrm{th}\) blade

\(g(y_0)\) :

Breathing function of the MSBCM

\(h_\mathrm{b}\) :

Blade height

\(h_\mathrm{d}\) :

Disk thickness

\(I_\mathrm{b}\) :

Polar potential moment of inertial of blade cross section

\(I_\mathrm{b}\) :

Polar potential moment of inertial of shaft cross section

\(K_\mathrm{I}\) :

Stress intensity factor at crack tip

\(L_\mathrm{b}\) :

Blade length

\(l_\mathrm{c}\) :

Crack location

\(L_\mathrm{s}\) :

Shaft length

\(N_\mathrm{b}\) :

Number of blades installed on rotating disk

\(P_0\) :

DC component of excitation force

\(P_{K_j}\) :

Aerodynamic force amplitude

\(Q({\lambda })\) :

Geometrical correction factor

\(R_\mathrm{d}\) :

Disk radius

u, v, w :

Blade vibration displacements in span-wise, lateral and chordwise directions

\(x_\mathrm{d}\), \(y_\mathrm{d}\), \(z_\mathrm{d}\) :

Shaft displacements at disk

\(x_\mathrm{s}\), \(y_\mathrm{s}\) :

Shaft bending vibration displacements in lateral directions

\(y_\mathrm{c}\) :

Height of crack tip

\({\sigma _{{\mathrm{bu}}}}\) :

Unit bending stress at the crack section

\({\sigma _{{\mathrm{c}}}}\) :

Centrifugal stress at the crack section

\({\theta _{{\mathrm{s}}x}}\), \({\theta _{{\mathrm{s}}y}}\) :

Shaft rotational displacement along with axes OX and OY

\({{\varvec{{D}}}_{\text {b}}}\) :

Damping coefficients matrix of blade

\({{\varvec{{D}}}_{\text {RB}}}\) :

Damping coefficients matrix of RSDB system

\({{\varvec{{F}}}_{\text {b}}}\) :

External force vector of blade

\({{\varvec{{F}}}_{\text {RB}}}\) :

External force vector of RSDB system

\({{\varvec{{G}}}_{\text {b}}}\) :

Gyroscopic matrix of blade

\({{\varvec{{G}}}_{\text {RB}}}\) :

Gyroscopic matrix of RSDB system

\({{\varvec{{K}}}^\mathrm{\Omega }_{\text {RB}}}\) :

Spin softening stiffness matrix of RSDB system

\({{\varvec{{K}}}^\mathrm{e}_{\text {RB}}}\) :

Structure stiffness matrix of RSDB system

\({{\varvec{{K}}}_{\text {acc}}}\) :

Stiffness matrix caused by acceleration of blade

\({{\varvec{{K}}}_{\text {b}}}\) :

Stiffness matrix of blade

\({{\varvec{{K}}}_{\text {cr}}}\) :

Stiffness alteration caused by fully open crack

\({{\varvec{{K}}}_{\text {e}}}\) :

Structural stiffness matrix of blade

\({{\varvec{{K}}}_{\text {so}}}\) :

Spin softening matrix of blade

\({{\varvec{{K}}}_{\text {st}}}\) :

Centrifugal stiffening matrix of blade

\({{\varvec{{M}}}_{\text {b}}}\) :

Mass matrix of blade

\({{\varvec{{M}}}_{\text {RB}}}\) :

Mass matrix of RSDB system

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Acknowledgements

This work is sponsored by the National Major Project of China (No. 2017-V-0009) and the Natural Science Foundation of China (No. 51705397). The first author acknowledges the host and support from the Structural Dynamics and Acoustic Systems Laboratory at the University of Massachusetts Lowell. The authors acknowledge the Laboratory of Science and Technology on Integrated Logistics Support at National University of Defense Technology for providing the blade vibration measurement test rig.

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Authors and Affiliations

  1. The State Key Laboratory for Manufacturing Systems Engineering, Xi’an Jiaotong University, Xi’an, 710049, Shaanxi, People’s Republic of China

    Lai-Hao Yang, Shu-Ming Wu, Xue-Feng Chen & Ru-Qiang Yan

  2. The Structural Dynamics and Acoustic Systems Laboratory, University of Massachusetts Lowell, Lowell, MA, 01854, USA

    Lai-Hao Yang & Zhu Mao

  3. Department of Engineering Science, University of Oxford, Parks Road, Oxford, OX1 3PJ, UK

    Shu-Ming Wu

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  1. Lai-Hao Yang
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  2. Zhu Mao
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  3. Shu-Ming Wu
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Contributions

Lai-Hao Yang contributed to theory modeling, numerical simulation, experimental verification, writing—original draft. Zhu Mao contributed to conceptualization, supervision, writing—review and editing. Shu-Ming Wu contributed to conceptualization, numerical simulation, experimental verification. Xue-Feng Chen contributed to conceptualization, supervision, funding acquisition, project administration. Ru-Qiang Yan contributed to supervision, writing—review and editing.

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Correspondence to Lai-Hao Yang.

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Matrices and vectors related to RSDB system

Matrices and vectors related to RSDB system

  1. (1)

    \({{\mathbf {M}}_{\text {s}}}\) is the lumped mass matrix related to the shaft translation and bending and can be shown as follows

    $$\begin{aligned} {\mathbf{M}}_{\mathrm{s}}^{}{\mathrm{= }}\left[ {\begin{array}{*{20}{c}} {{\mathbf{M}}_{{\mathrm{s1}}}^{{\mathrm{shaft}}}}&{}{\mathbf{0}}\\ {\mathbf{0}}&{}{{\mathbf{M}}_{{\mathrm{s2}}}^{{\mathrm{shaft}}}} \end{array}} \right] + \left[ {\begin{array}{*{20}{c}} {{\mathbf{M}}_{{\mathrm{s1}}}^{{\mathrm{disc}}}}&{}{\mathbf{0}}\\ {\mathbf{0}}&{}{{\mathbf{M}}_{{\mathrm{s2}}}^{{\mathrm{disc}}}} \end{array}} \right] + \left[ {\begin{array}{*{20}{c}} {{\mathbf{M}}_{{\mathrm{s1}}}^{{\mathrm{blade}}}}&{}{\mathbf{0}}\\ {\mathbf{0}}&{}{{\mathbf{M}}_{{\mathrm{s2}}}^{{\mathrm{blade}}}} \end{array}} \right] \end{aligned}$$
    (A.1)
    $$\begin{aligned}&\left\{ \begin{array}{l} {\mathbf{M}}_{{\mathrm{s1}}}^{{\mathrm{shaft}}}{\mathrm{= }}\displaystyle \int _0^{{L_{\mathrm{s}}}} {{\rho _{\mathrm{s}}}{A_{\mathrm{s}}}\left( {{{\mathbf{X}}^{\mathrm{T}}}{\mathbf{X}}} \right) } {\mathrm{d}}z + \displaystyle \int _0^{{L_{\mathrm{s}}}} {{J_{{\mathrm{ds}}}}\left( {{{{\mathbf{X'}}}^{\mathrm{T}}}{\mathbf{X'}}} \right) } {\mathrm{d}}z\\ {\mathbf{M}}_{{\mathrm{s2}}}^{{\mathrm{shaft}}}{\mathrm{= }}\displaystyle \int _0^{{L_{\mathrm{s}}}} {{\rho _{\mathrm{s}}}{A_{\mathrm{s}}}\left( {{{\mathbf{Y}}^{\mathrm{T}}}{\mathbf{Y}}} \right) } {\mathrm{d}}z + \displaystyle \int _0^{{L_{\mathrm{s}}}} {{J_{{\mathrm{ds}}}}\left( {{{{\mathbf{Y'}}}^{\mathrm{T}}}{\mathbf{Y'}}} \right) } {\mathrm{d}}z \end{array} \right. \end{aligned}$$
    (A.2a)
    $$\begin{aligned}&\left\{ \begin{array}{l} {\mathbf{M}}_{{\mathrm{s1}}}^{{\mathrm{disc}}} = {m_{\mathrm{d}}}{{\mathbf{X}}^{\mathrm{T}}}\left( {{z_{\mathrm{d}}}} \right) {\mathbf{X}}\left( {{z_{\mathrm{d}}}} \right) {\mathrm{+ }}{J_{{\mathrm{dd}}}}{{{\mathbf{X'}}}^{\mathrm{T}}}\left( {{z_{\mathrm{d}}}} \right) {\mathbf{X'}}\left( {{z_{\mathrm{d}}}} \right) \\ {\mathbf{M}}_{{\mathrm{s2}}}^{{\mathrm{disc}}} = {m_{\mathrm{d}}}{{\mathbf{Y}}^{\mathrm{T}}}\left( {{z_{\mathrm{d}}}} \right) {\mathbf{Y}}\left( {{z_{\mathrm{d}}}} \right) {\mathrm{+ }}{J_{{\mathrm{dd}}}}{{{\mathbf{Y'}}}^{\mathrm{T}}}\left( {{z_{\mathrm{d}}}} \right) {\mathbf{Y'}}\left( {{z_{\mathrm{d}}}} \right) \end{array} \right. \end{aligned}$$
    (A.2b)
    $$\begin{aligned}&\left\{ \begin{array}{l} {\mathbf{M}}_{{\mathrm{s1}}}^{{\mathrm{blade}}} = {N_{\mathrm{b}}}{\rho _{\mathrm{b}}}{A_{\mathrm{b}}}\displaystyle \int _{\mathrm{0}}^{{L_{\mathrm{b}}}} {\left[ {{{\mathbf{X}}^{\mathrm{T}}}\left( {{z_{\mathrm{d}}}} \right) {\mathbf{X}}\left( {{z_{\mathrm{d}}}} \right) + \dfrac{1}{2}{{\left( {{R_{\mathrm{d}}} + x} \right) }^2}{{{\mathbf{X'}}}^{\mathrm{T}}}\left( {{z_{\mathrm{d}}}} \right) {\mathbf{X'}}\left( {{z_{\mathrm{d}}}} \right) } \right] {\mathrm{d}}x\;} \\ {\mathbf{M}}_{{\mathrm{s2}}}^{{\mathrm{blade}}} = {N_{\mathrm{b}}}{\rho _{\mathrm{b}}}{A_{\mathrm{b}}}\displaystyle \int _{\mathrm{0}}^{{L_{\mathrm{b}}}} {\left[ {{{\mathbf{Y}}^{\mathrm{T}}}\left( {{z_{\mathrm{d}}}} \right) {\mathbf{Y}}\left( {{z_{\mathrm{d}}}} \right) + \dfrac{1}{2}{{\left( {{R_{\mathrm{d}}} + x} \right) }^2}{{{\mathbf{Y'}}}^{\mathrm{T}}}\left( {{z_{\mathrm{d}}}} \right) {\mathbf{Y'}}\left( {{z_{\mathrm{d}}}} \right) } \right] {\mathrm{d}}x} \end{array} \right. \end{aligned}$$
    (A.2c)
  2. (2)

    \({{\mathbf {M}}_{\uptheta }}\) is the lumped mass matrix related to shaft torsional vibration and can be shown as follows

    $$\begin{aligned}&{{\varvec{M}}_{\varvec{\theta }}} = {\varvec{M}}_{\varvec{\theta }}^{{\mathrm{shaft}}} + {\mathbf{M}}_{\varvec{\theta }}^{{\mathrm{disc}}} + {\varvec{M}}_{\varvec{\theta }}^{{\mathrm{blade}}} \end{aligned}$$
    (A.3)
    $$\begin{aligned}&\left\{ \begin{array}{l} {\varvec{M}}_{\varvec{\theta }}^{{\mathrm{shaft}}}{\mathrm{= }}\displaystyle \int _0^{{L_{\mathrm{s}}}} {{J_{{\mathrm{ps}}}}\left( {{{\varvec{\Psi }}^{\mathrm{T}}}{\varvec{\Psi }}} \right) } {\mathrm{d}}z\\ {\varvec{M}}_{\varvec{\theta }}^{{\mathrm{disc}}}={J_{{\mathrm{pd}}}}{{\varvec{\Psi }}^{\mathrm{T}}}\left( {{z_{\mathrm{d}}}} \right) {\varvec{\Psi }}\left( {{z_{\mathrm{d}}}} \right) + {m_{\mathrm{d}}}{e^2}{{\varvec{\Psi }}^{\mathrm{T}}}\left( {{z_{\mathrm{d}}}} \right) {\varvec{\Psi }}\left( {{z_{\mathrm{d}}}} \right) \\ {\varvec{M}}_{\varvec{\theta }}^{{\mathrm{blade}}}={N_{\mathrm{b}}}{\rho _{\mathrm{b}}}{A_{\mathrm{b}}}\displaystyle \int _{\mathrm{0}}^{{L_{\mathrm{b}}}} {\left[ {{{\left( {R_{\mathrm{d}} + x} \right) }^2}{{\mathbf{\Psi }}^{\mathrm{T}}}\left( {{z_{\mathrm{d}}}} \right) {\mathbf{\Psi }}\left( {{z_{\mathrm{d}}}} \right) } \right] {\mathrm{d}}x} \end{array} \right. \end{aligned}$$
    (A.4)
  3. (3)

    \({{\varvec{M}}_{{{\mathrm{B}}_i}}} \) is the mass matrix related to blade bending and can be shown as follows

    $$\begin{aligned}&{{\varvec{M}}_{{{\mathrm{B}}_i}}} = \left[ {\begin{array}{*{20}{c}} {{\varvec{M}}_{{{\mathrm{B}}_i}}^u}&{}{}\\ {}&{}{{\varvec{M}}_{{{\mathrm{B}}_i}}^v} \end{array}} \right] \end{aligned}$$
    (A.5)
    $$\begin{aligned}&\left\{ {\begin{array}{*{20}{l}} {{\varvec{M}}_{{{\mathrm{B}}_i}}^u = {\rho _{\mathrm{b}}}{A_{\mathrm{b}}}\displaystyle \int _{\mathrm{0}}^{{L_{\mathrm{b}}}} {\left[ {{{\mathbf{U}}^{\mathrm{T}}}{\mathbf{U}}} \right] {\mathrm{d}}x} }\\ {{\varvec{M}}_{{{\mathrm{B}}_i}}^v = {\rho _{\mathrm{b}}}{A_{\mathrm{b}}}\displaystyle \int _{\mathrm{0}}^{{L_{\mathrm{b}}}} {\left[ {{{\mathbf{V}}^{\mathrm{T}}}{\mathbf{V}}} \right] {\mathrm{d}}x} } \end{array}} \right. \end{aligned}$$
    (A.6)
  4. (4)

    \({{\mathbf{G}}_{{{\mathrm{s}}}}} \) is the lumped gyroscopic matrix related the shaft rotation and can be shown as follows:

    $$\begin{aligned} {\mathbf{G}}_{\mathrm{s}}^{}{\mathrm{= }}\left[ {\begin{array}{*{20}{c}} {\mathbf{0}}&{}{{\mathbf{G}}_{{\mathrm{s1}}}^{{\mathrm{shaft}}}}\\ {{\mathbf{G}}_{{\mathrm{s2}}}^{{\mathrm{shaft}}}}&{}{\mathbf{0}} \end{array}} \right] + \left[ {\begin{array}{*{20}{c}} {\mathbf{0}}&{}{{\mathbf{G}}_{{\mathrm{s1}}}^{{\mathrm{disc}}}}\\ {{\mathbf{G}}_{{\mathrm{s2}}}^{{\mathrm{disc}}}}&{}{\mathbf{0}} \end{array}} \right] + \left[ {\begin{array}{*{20}{c}} {\mathbf{0}}&{}{{\mathbf{G}}_{{\mathrm{s1}}}^{{\mathrm{blade}}}}\\ {{\mathbf{G}}_{{\mathrm{s2}}}^{{\mathrm{blade}}}}&{}{\mathbf{0}} \end{array}} \right] \end{aligned}$$
    (A.7)
    $$\begin{aligned}&\left\{ \begin{array}{l} {\mathbf{G}}_{{\mathrm{s1}}}^{{\mathrm{shaft}}}=\Omega \displaystyle \int _0^{{L_{\mathrm{s}}}} {{J_{{\mathrm{ps}}}}{{{\mathbf{Y'}}}^{\mathrm{T}}}{\mathbf{X'}}} {\mathrm{d}}z\\ {\mathbf{G}}_{{\mathrm{s2}}}^{{\mathrm{shaft}}} = - \Omega \displaystyle \int _0^{{L_{\mathrm{s}}}} {{J_{{\mathrm{ps}}}}{{{\mathbf{X'}}}^{\mathrm{T}}}{\mathbf{Y'}}} {\mathrm{d}}z \end{array} \right. \end{aligned}$$
    (A.8a)
    $$\begin{aligned}&\left\{ \begin{array}{l} {\mathbf{G}}_{{\mathrm{s1}}}^{{\mathrm{disc}}} = \Omega {J_{{\mathrm{pd}}}}{{{\mathbf{Y'}}}^{\mathrm{T}}}\left( {{z_{\mathrm{d}}}} \right) {\mathbf{X'}}\left( {{z_{\mathrm{d}}}} \right) \\ {\mathbf{G}}_{{\mathrm{s2}}}^{{\mathrm{disc}}} = - \Omega {J_{{\mathrm{pd}}}}{{{\mathbf{Y'}}}^{\mathrm{T}}}\left( {{z_{\mathrm{d}}}} \right) {\mathbf{X'}}\left( {{z_{\mathrm{d}}}} \right) \end{array} \right. \end{aligned}$$
    (A.8b)
    $$\begin{aligned}&\left\{ \begin{array}{l} {\mathbf{G}}_{{\mathrm{s1}}}^{{\mathrm{blade}}} = \Omega {N_{\mathrm{b}}}{\rho _{\mathrm{b}}}{A_{\mathrm{b}}}\displaystyle \int _{\mathrm{0}}^{{L_{\mathrm{b}}}} {\left[ {{{\left( {R_{\mathrm{d}} + x} \right) }^2}{{{\mathbf{Y'}}}^{\mathrm{T}}}\left( {{z_{\mathrm{d}}}} \right) {\mathbf{X'}}\left( {{z_{\mathrm{d}}}} \right) } \right] {\mathrm{d}}x} \\ {\mathbf{G}}_{{\mathrm{s2}}}^{{\mathrm{blade}}} = - \Omega {N_{\mathrm{b}}}{\rho _{\mathrm{b}}}{A_{\mathrm{b}}}\displaystyle \int _{\mathrm{0}}^{{L_{\mathrm{b}}}} {\left[ {{{\left( {R_{\mathrm{d}} + x} \right) }^2}{{{\mathbf{X'}}}^{\mathrm{T}}}\left( {{z_{\mathrm{d}}}} \right) {\mathbf{Y'}}\left( {{z_{\mathrm{d}}}} \right) } \right] {\mathrm{d}}x} \end{array} \right. \end{aligned}$$
    (A.8c)
  5. (5)

    \({{\mathbf{G}}_{{{\mathrm{B}}_{i}}}} \) is the lumped gyroscopic matrix related the shaft rotation and can be shown as follows

    $$\begin{aligned}&{\mathbf{G}}_{{{\mathrm{B}}_i}}^{}= \left[ {\begin{array}{*{20}{c}} {}&{}{{\mathbf{G}}_{{{\mathrm{B}}_i}}^{vu}}\\ {{\mathbf{G}}_{{{\mathrm{B}}_i}}^{uv}}&{}{} \end{array}} \right] \end{aligned}$$
    (A.9)
    $$\begin{aligned}&\left\{ \begin{array}{l} {\mathbf{G}}_{\mathrm{s}}^{vu} = - 2\Omega {\rho _{\mathrm{b}}}{A_{\mathrm{b}}}\displaystyle \int _{\mathrm{0}}^{{L_{\mathrm{b}}}} {\left[ {\cos \left( \beta \right) {{\mathbf{U}}^{\mathrm{T}}}{\mathbf{V}}} \right] {\mathrm{d}}x} \\ {\mathbf{G}}_{\mathrm{s}}^{uv} = 2\Omega {\rho _{\mathrm{b}}}{A_{\mathrm{b}}}\displaystyle \int _{\mathrm{0}}^{{L_{\mathrm{b}}}} {\left[ {\cos \left( \beta \right) {{\mathbf{V}}^{\mathrm{T}}}{\mathbf{U}}} \right] {\mathrm{d}}x} \end{array} \right. \end{aligned}$$
    (A.10)
  6. (6)

    \({{\mathbf{K}}_{{{\mathrm{S}}}}} \) is the structure stiffness matrix related to shaft bending deformation and can be shown as follows

    $$\begin{aligned}&{{\mathbf{K}}_{\mathrm{S}}} = \left[ {\begin{array}{*{20}{c}} {{\mathbf{K}}_{{\mathrm{S1}}}^{\mathrm{e}}}&{}{}\\ {}&{}{{\mathbf{K}}_{{\mathrm{S2}}}^{\mathrm{e}}} \end{array}} \right] \end{aligned}$$
    (A.11)
    $$\begin{aligned}&\left\{ \begin{array}{l} {\mathbf{K}}_{{\mathrm{S1}}}^{\mathrm{e}} =\displaystyle \int _0^{{L_{\mathrm{s}}}} {{E_{\mathrm{s}}}{I_{\mathrm{s}}}\left( {{{{\mathbf{X''}}}^{\mathrm{T}}}{\mathbf{X''}}} \right) } {\mathrm{d}}z\\ {\mathbf{K}}_{{\mathrm{S2}}}^{\mathrm{e}} = \displaystyle \int _0^{{L_{\mathrm{s}}}} {{E_{\mathrm{s}}}{I_{\mathrm{s}}}\left( {{{{\mathbf{Y''}}}^{\mathrm{T}}}{\mathbf{Y''}}} \right) } {\mathrm{d}}z \end{array} \right. \end{aligned}$$
    (A.12)
  7. (7)

    \({\mathbf{K}}_{\mathrm{\uptheta }}^{\mathrm{e}}\) is the structure stiffness matrix related to shaft torsional deformation and can be shown as follows

    $$\begin{aligned} {{\mathbf{K}}_{\mathrm{\theta }}^{\mathrm{e}} = \int _0^{{L_{\mathrm{s}}}} {{G_{\mathrm{s}}}{J_{\mathrm{s}}}\left( {{{{\varvec{\Psi '}}}^{\mathrm{T}}}{\varvec{\Psi '}}} \right) } {\mathrm{d}}z} \end{aligned}$$
    (A.13)

    (8) \({{\mathbf{K}}_{{{\mathrm{S}}}}} \) is the structure stiffness matrix related to blade bending deformation and can be shown as follows

    $$\begin{aligned}&{{\mathbf {K}}_{{{\text {B}}_i}}} = \left[ {\begin{array}{*{20}{c}} {{\mathbf {K}}_{{\text {u}}i}^{\text {e}}}&{}{} \\ {}&{}{{\mathbf {K}}_{{\text {v}}i}^{\text {e}}} \end{array}} \right] \end{aligned}$$
    (A.14)
    $$\begin{aligned}&\left\{ \begin{array}{l} {\mathbf{K}}_{{\mathrm{u}}i}^{\mathrm{e}} = {E_{\mathrm{b}}}{A_{\mathrm{b}}}\displaystyle \int _{\mathrm{0}}^{{L_{\mathrm{b}}}} {\left[ {{{{\mathbf{U'}}}^{\mathrm{T}}}{\mathbf{U'}}} \right] {\mathrm{d}}x} \\ {\mathbf{K}}_{{\mathrm{v}}i}^{\mathrm{e}} = {E_{\mathrm{b}}}{I_{\mathrm{b}}}\displaystyle \int _{\mathrm{0}}^{{L_{\mathrm{b}}}} {\left[ {{{{\mathbf{V''}}}^{\mathrm{T}}}{\mathbf{V''}}} \right] {\mathrm{d}}x} + \int _{\mathrm{0}}^{{L_{\mathrm{b}}}} {{f_{\mathrm{c}}}\left( x \right) \left[ {{{{\mathbf{V'}}}^{\mathrm{T}}}{\mathbf{V'}}} \right] {\mathrm{d}}x} \end{array} \right. \end{aligned}$$
    (A.15)
  8. (9)

    \({\mathbf{K}}_{\mathrm{s}}^\Omega \) is the spin softening stiffness matrix caused by disk and blade spinning, which affects the shaft bending vibration and can be shown as follows

    $$\begin{aligned} {\mathbf{K}}_{\mathrm{s}}^\Omega = {\mathbf{K}}_{\mathrm{s}}^{\Omega {\mathrm{disc}}} + {\mathbf{K}}_{\mathrm{s}}^{\Omega {\mathrm{blade}}} = \left[ {\begin{array}{*{20}{c}} {{\mathbf{K}}_{{\mathrm{s1}}}^{\Omega {\mathrm{disc}}}}&{}{}\\ {}&{}{{\mathbf{K}}_{{\mathrm{s2}}}^{\Omega {\mathrm{disc}}}} \end{array}} \right] + \left[ {\begin{array}{*{20}{c}} {{\mathbf{K}}_{{\mathrm{s1}}}^{\Omega {\mathrm{blade}}}}&{}{}\\ {}&{}{{\mathbf{K}}_{{\mathrm{s2}}}^{\Omega {\mathrm{blade}}}} \end{array}} \right] \nonumber \\ \end{aligned}$$
    (A.16)
    $$\begin{aligned}&\left\{ \begin{array}{l} {\mathbf{K}}_{{\mathrm{s}}1}^{\Omega {\mathrm{disc}}} = {\Omega ^2}{J_{{\mathrm{dd}}}}{{{\mathbf{X'}}}^{\mathrm{T}}}\left( {{z_{\mathrm{d}}}} \right) {\mathbf{X'}}\left( {{z_{\mathrm{d}}}} \right) \\ {\mathbf{K}}_{{\mathrm{s2}}}^{\Omega {\mathrm{disc}}} = {\Omega ^2}{J_{{\mathrm{dd}}}}{{{\mathbf{Y'}}}^{\mathrm{T}}}\left( {{z_{\mathrm{d}}}} \right) {\mathbf{Y'}}\left( {{z_{\mathrm{d}}}} \right) \end{array} \right. \end{aligned}$$
    (A.17a)
    $$\begin{aligned}&\left\{ \begin{array}{l} {\mathbf{K}}_{{\mathrm{s1}}}^{\Omega {\mathrm{blade}}} = \dfrac{1}{2}{N_{\mathrm{b}}}{\Omega ^2}{\rho _{\mathrm{b}}}{A_{\mathrm{b}}}\displaystyle \int _{\mathrm{0}}^{{L_{\mathrm{b}}}} {\left[ {{{\left( {R_{\mathrm{d}} + x} \right) }^2}{{{\mathbf{X'}}}^{\mathrm{T}}}\left( {{z_{\mathrm{d}}}} \right) {\mathbf{X'}}\left( {{z_{\mathrm{d}}}} \right) } \right] {\mathrm{d}}x} \\ {\mathbf{K}}_{{\mathrm{s2}}}^{\Omega {\mathrm{blade}}} = \dfrac{1}{2}{N_{\mathrm{b}}}{\Omega ^2}{\rho _{\mathrm{b}}}{A_{\mathrm{b}}}\displaystyle \int _{\mathrm{0}}^{{L_{\mathrm{b}}}} {\left[ {{{\left( {R_{\mathrm{d}} + x} \right) }^2}{{{\mathbf{Y'}}}^{\mathrm{T}}}\left( {{z_{\mathrm{d}}}} \right) {\mathbf{Y'}}\left( {{z_{\mathrm{d}}}} \right) } \right] {\mathrm{d}}x} \end{array} \right. \end{aligned}$$
    (A.17b)
  9. (10)

    \({\mathbf{K}}_{\mathrm{s}}^\Omega \) is the spin softening stiffness matrix caused by disk and blade spinning, which affects the shaft torsional vibration and can be shown as follows

    $$\begin{aligned}&{\mathbf{K}}_{\mathrm{\theta }}^\Omega = {\mathbf{K}}{_{\mathrm{\theta }}^{\Omega {{\mathrm{disc}}}}}+ {\mathbf{K}}{_{\mathrm{\theta }}^{\Omega {{\mathrm{blade}}}}} \end{aligned}$$
    (A.18)
    $$\begin{aligned}&\left\{ \begin{array}{l} {\mathbf{K}}{_{\mathrm{\uptheta }}^{\Omega {{\mathrm{disc}}}}}= {J_{{\mathrm{pd}}}}{{\mathbf{\Psi }}^{\mathrm{T}}}\left( {{z_{\mathrm{d}}}} \right) {\mathbf{\Psi }}\left( {{z_{\mathrm{d}}}} \right) \\ {\mathbf{K}}{_{\mathrm{\uptheta }}^{\Omega {{\mathrm{blade}}}}} = {N_{\mathrm{b}}}{\rho _{\mathrm{b}}}{A_{\mathrm{b}}}\displaystyle \int _{\mathrm{0}}^{{L_{\mathrm{b}}}} {\left[ {{\Omega ^2}{{\left( {R_{\mathrm{d}} + x} \right) }^2}{{\mathbf{\Psi }}^{\mathrm{T}}}\left( {{z_{\mathrm{d}}}} \right) {\varvec{\Psi }}\left( {{z_{\mathrm{d}}}} \right) } \right] {\mathrm{d}}x} \end{array} \right. \end{aligned}$$
    (A.19)
  10. (11)

    \({\mathbf{K}}_{{{\mathrm{B}}_i}}^\Omega \) is the spin softening stiffness matrix caused by blade spinning and can be shown as follows

    $$\begin{aligned}&{\mathbf{K}}_{{{\mathrm{B}}_i}}^\Omega = \left[ {\begin{array}{*{20}{c}} {{\mathbf{K}}_{{\mathrm{u}}i}^\Omega }&{}{}\\ {}&{}{{\mathbf{K}}_{{\mathrm{v}}i}^\Omega } \end{array}} \right] \end{aligned}$$
    (A.20)
    $$\begin{aligned}&\left\{ {\begin{array}{*{20}{l}} {{\mathbf{K}}_{{\mathrm{u}}i}^\Omega = {\Omega ^2}{\rho _{\mathrm{b}}}{A_{\mathrm{b}}}\displaystyle \int _{\mathrm{0}}^{{L_{\mathrm{b}}}} {\left[ {{{\mathbf{U}}^{\mathrm{T}}}{\mathbf{U}}} \right] {\mathrm{d}}x} }\\ {{\mathbf{K}}_{{\mathrm{v}}i}^\Omega = {\Omega ^2}{\rho _{\mathrm{b}}}{A_{\mathrm{b}}}\displaystyle \int _{\mathrm{0}}^{{L_{\mathrm{b}}}} {\left[ {{{\cos }^2}\left( \beta \right) {{\mathbf{V}}^{\mathrm{T}}}{\mathbf{V}}} \right] {\mathrm{d}}x} } \end{array}} \right. \end{aligned}$$
    (A.21)
  11. (12)

    \({\mathbf{M}}_{{\mathrm{\theta B}}}\) is the static-coupling mass matrix between shaft and blade bending deformation and can be shown as follows

    $$\begin{aligned}&{{\varvec{M}}_{{\mathrm{\theta B}}}}{{ = }}\left[ {\begin{array}{*{20}{c}} {{\varvec{M}}_{{\mathrm{\theta B}}}^u}&{{\varvec{M}}_{{\mathrm{\uptheta B}}}^v} \end{array}} \right] \end{aligned}$$
    (A.22)
    $$\begin{aligned}&\left\{ {\begin{array}{*{20}{l}} {{\varvec{M}}_{{\mathrm{\theta B}}}^u = {\mathbf{0}}}\\[10pt] {{\varvec{M}}_{{\mathrm{\theta B}}}^v = {\rho _{\mathrm{b}}}{A_{\mathrm{b}}}\displaystyle \int _{\mathrm{0}}^{{L_{\mathrm{b}}}} {\left[ {\cos \left( \beta \right) \left( {R_{\mathrm{d}} + x} \right) {{\mathbf{V}}^{\mathrm{T}}}{\mathbf{\Psi }}\left( {{z_{\mathrm{d}}}} \right) } \right] {\mathrm{d}}x} } \end{array}} \right. \end{aligned}$$
    (A.23)
  12. (13)

    \({\mathbf{G}}_{{\mathrm{\theta B}}}\) is the static-coupling gyroscopic matrix between shaft and blade bending deformation and can be shown as follows

    $$\begin{aligned}&{\mathbf{G}}_{{\mathrm{\theta B}}}^{}{{ = }}\left[ {\begin{array}{*{20}{c}} {{\mathbf{G}}_{{\mathrm{\theta B}}}^u}&{{\mathbf{G}}_{{\mathrm{\theta B}}}^v} \end{array}} \right] \end{aligned}$$
    (A.24)
    $$\begin{aligned}&\left\{ \begin{array}{l} {\mathbf{G}}_{{\mathrm{\theta B}}}^u = {\mathrm{2}}\Omega {\rho _{\mathrm{b}}}{A_{\mathrm{b}}}\displaystyle \int _{\mathrm{0}}^{{L_{\mathrm{b}}}} {\left[ {\left( {R_{\mathrm{d}} + x} \right) {{\varvec{\Psi }}^{\mathrm{T}}}\left( {{z_{\mathrm{d}}}} \right) {\mathbf{U}}} \right] {\mathrm{d}}x} \\ {\mathbf{G}}_{{\mathrm{\theta B}}}^v = {\mathbf{0}} \end{array} \right. \end{aligned}$$
    (A.25)
  13. (14)

    \({\mathbf{K}}_{{\mathrm{\theta B}}}^\Omega \) is the static-coupling stiffness matrix between shaft and blade bending deformation and can be shown as follows

    $$\begin{aligned}&{\mathbf{K}}_{{\mathrm{\theta B}}}^\Omega {{ = }}\left[ {\begin{array}{*{20}{c}} {{\mathbf{K}}_{{\mathrm{\theta B}}}^{\Omega u} }&{{\mathbf{K}}_{{\mathrm{\theta B}}}^{\Omega v} } \end{array}} \right] \end{aligned}$$
    (A.26)
    $$\begin{aligned}&\left\{ \begin{array}{l} {\mathbf{K}}_{{\mathrm{\theta B}}}^{\Omega u} = {\mathbf{0}}\\ {\mathbf{K}}_{{\mathrm{\theta B}}}^{\Omega v} ={\rho _{\mathrm{b}}}{A_{\mathrm{b}}}\displaystyle \int _{\mathrm{0}}^{{L_{\mathrm{b}}}} {\left[ {\cos \left( \beta \right) {\Omega ^2}\left( {R_{\mathrm{d}} + x} \right) {{\mathbf{V}}^{\mathrm{T}}}{\mathbf{\Psi }}\left( {{z_{\mathrm{d}}}} \right) } \right] {\mathrm{d}}x}\nonumber \end{array} \right. \\ \end{aligned}$$
    (A.27)
  14. (15)

    The dynamic coupling terms shown in Fig. 7 can be expressed with following forms:

    $$\begin{aligned}&{{\varvec{M}}_{{\mathrm{S\theta }}}}={\left[ {\begin{array}{*{20}{c}} {{\varvec{M}}_{{\mathrm{S\theta }}}^x}&{{\varvec{M}}_{{\mathrm{S\theta }}}^y} \end{array}} \right] ^{\mathrm{T}}} = \cos \left( {\Omega t} \right) {{\varvec{M}}_{{\mathrm{S\theta }}}^c} + \sin \left( {\Omega t} \right) {{\varvec{M}}_{{\mathrm{S\theta }}}^s} \end{aligned}$$
    (A.28a)
    $$\begin{aligned}&{{\varvec{M}}_{{\mathrm{SB}}}}=\left[ {\begin{array}{*{20}{c}} {{\varvec{M}}_{{\mathrm{SB}}}^{ux}}&{}{{\varvec{M}}_{{\mathrm{SB}}}^{vx}}\\ {{\varvec{M}}_{{\mathrm{SB}}}^{uy}}&{}{{\varvec{M}}_{{\mathrm{SB}}}^{vy}} \end{array}} \right] = \cos \left( {{\theta _i}} \right) {\mathbf {M}}_{{\mathrm{SB}}}^c + {\mathrm{sin}}\left( {{\theta _i}} \right) {\mathbf {M}}_{{\mathrm{SB}}}^s \end{aligned}$$
    (A.28b)
    $$\begin{aligned}&{{\mathbf{G}}_{{\mathrm{SB}}}}=\left[ {\begin{array}{*{20}{c}} {{\mathbf{G}}_{{\mathrm{SB}}}^{ux}}&{}{{\mathbf{G}}_{{\mathrm{SB}}}^{vx}}\\ {{\mathbf{G}}_{{\mathrm{SB}}}^{uy}}&{}{{\mathbf{G}}_{{\mathrm{SB}}}^{vy}} \end{array}} \right] =2\Omega \cos \left( {{\theta _i}} \right) {\mathbf {G}}_{{\mathrm{SB}}}^c+2\Omega {\mathrm{sin}}\left( {{\theta _i}} \right) {\mathbf {G}}_{{\mathrm{SB}}}^s \end{aligned}$$
    (A.28c)
    $$\begin{aligned}&{{\mathbf{G}}_{{\mathrm{BS}}}}=\left[ {\begin{array}{*{20}{c}} {{\mathbf{G}}_{{\mathrm{BS}}}^{ux}}&{}{{\mathbf{G}}_{{\mathrm{BS}}}^{vx}}\\ {{\mathbf{G}}_{{\mathrm{BS}}}^{uy}}&{}{{\mathbf{G}}_{{\mathrm{BS}}}^{vy}} \end{array}} \right] =2\Omega \cos \left( {{\theta _i}} \right) {\mathbf {G}}_{{\mathrm{BS}}}^c+2\Omega {\mathrm{sin}}\left( {{\theta _i}} \right) {\mathbf {G}}_{{\mathrm{BS}}}^s \end{aligned}$$
    (A.28d)
    $$\begin{aligned}&{{\mathbf{K}}_{{\mathrm{SB}}}}=\left[ {\begin{array}{*{20}{c}} {{\mathbf{K}}_{{\mathrm{SB}}}^{ux}}&{}{{\mathbf{K}}_{{\mathrm{SB}}}^{vx}}\\ {{\mathbf{K}}_{{\mathrm{SB}}}^{uy}}&{}{{\mathbf{K}}_{{\mathrm{SB}}}^{vy}} \end{array}} \right] =\Omega ^2 \cos \left( {{\theta _i}} \right) {\mathbf {K}}_{{\mathrm{SB}}}^c+\Omega ^2 {\mathrm{sin}}\left( {{\theta _i}} \right) {\mathbf {K}}_{{\mathrm{SB}}}^s \end{aligned}$$
    (A.28e)
    $$\begin{aligned}&{{\mathbf{K}}_{{\mathrm{BS}}}}=\left[ {\begin{array}{*{20}{c}} {{\mathbf{K}}_{{\mathrm{BS}}}^{ux}}&{}{{\mathbf{K}}_{{\mathrm{BS}}}^{vx}}\\ {{\mathbf{K}}_{{\mathrm{BS}}}^{uy}}&{}{{\mathbf{K}}_{{\mathrm{BS}}}^{vy}} \end{array}} \right] =\Omega ^2 \cos \left( {{\theta _i}} \right) {\mathbf {K}}_{{\mathrm{BS}}}^c+\Omega ^2 {\mathrm{sin}}\left( {{\theta _i}} \right) {\mathbf {K}}_{{\mathrm{BS}}}^s \end{aligned}$$
    (A.28f)

    where, the superscript ‘\(^\mathrm{s}\)’ and ‘\(^\mathrm{c}\)’ denote that the coefficients are corresponding to the “cos” and “sin” part, respectively. Considering the relationship \(X_i(z)=Y_i(z)\), those coefficients can be expressed as

    $$\begin{aligned}&{{\varvec{M}}_{{\mathrm{S\uptheta }}}^c} = {\left[ {\begin{array}{*{20}{c}} {\mathbf{0}}&{ - {\mathbf{C}}_{{\mathrm{S\theta }}}^1} \end{array}} \right] ^{\mathrm{T}}},&{{\varvec{M}}_{{\mathrm{S\theta }}}^s}&= {\left[ {\begin{array}{*{20}{c}} {{\mathbf{C}}_{{\mathrm{S\theta }}}^1}&{\mathbf{0}} \end{array}} \right] ^{\mathrm{T}}}\end{aligned}$$
    (A.29a)
    $$\begin{aligned}&{\mathbf {M}}_{{\mathrm{SB}}}^c = \left[ {\begin{array}{*{20}{c}} {{\mathbf{C}}_{{\mathrm{SB}}}^1}&{}{ - {\mathbf{C}}_{{\mathrm{SB}}}^2}\\ {\mathbf{0}}&{}{{\mathbf{C}}_{{\mathrm{SB}}}^3} \end{array}} \right] ,&{\mathbf {M}}_{{\mathrm{SB}}}^s&= \left[ {\begin{array}{*{20}{c}} {\mathbf{0}}&{}{ - {\mathbf{C}}_{{\mathrm{SB}}}^3}\\ {{\mathbf{C}}_{{\mathrm{SB}}}^1}&{}{ - {\mathbf{C}}_{{\mathrm{SB}}}^2} \end{array}} \right] \end{aligned}$$
    (A.29b)
    $$\begin{aligned}&{\mathbf {G}}_{{\mathrm{SB}}}^c = \left[ {\begin{array}{*{20}{c}} {\mathbf{0}}&{}{ - {\mathbf{C}}_{{\mathrm{SB}}}^3}\\ {{\mathbf{C}}_{{\mathrm{SB}}}^1}&{}{\mathbf{0}} \end{array}} \right] ,&{\mathbf {G}}_{{\mathrm{SB}}}^s&= \left[ {\begin{array}{*{20}{c}} { - {\mathbf{C}}_{{\mathrm{SB}}}^1}&{}{\mathbf{0}}\\ {\mathbf{0}}&{}{ - {\mathbf{C}}_{{\mathrm{SB}}}^3} \end{array}} \right] \end{aligned}$$
    (A.29c)
    $$\begin{aligned}&{\mathbf {G}}_{{\mathrm{BS}}}^c =\left[ {\begin{array}{*{20}{c}} {\mathbf{0}}&{}{\mathbf{0}}\\ {\mathbf{0}}&{}{ - {\mathbf{C}}_{{\mathrm{SB}}}^2} \end{array}} \right] ,&{\mathbf {G}}_{{\mathrm{BS}}}^s&=\left[ {\begin{array}{*{20}{c}} {\mathbf{0}}&{}{\mathbf{0}}\\ {{\mathbf{C}}_{{\mathrm{SB}}}^2}&{}{\mathbf{0}} \end{array}} \right] \end{aligned}$$
    (A.29d)
    $$\begin{aligned}&{\mathbf {K}}_{{\mathrm{SB}}}^c =\left[ {\begin{array}{*{20}{c}} { - {\mathbf{C}}_{{\mathrm{SB}}}^1}&{}{\mathbf{0}}\\ {\mathbf{0}}&{}{ - {\mathbf{C}}_{{\mathrm{SB}}}^3} \end{array}} \right] ,&{\mathbf {K}}_{{\mathrm{SB}}}^s&= \left[ {\begin{array}{*{20}{c}} {\mathbf{0}}&{}{{\mathbf{C}}_{{\mathrm{SB}}}^3}\\ { - {\mathbf{C}}_{{\mathrm{SB}}}^1}&{}{\mathbf{0}} \end{array}} \right] \end{aligned}$$
    (A.29e)
    $$\begin{aligned}&{\mathbf {K}}_{{\mathrm{BS}}}^c =\left[ {\begin{array}{*{20}{c}} {\mathbf{0}}&{}{\mathbf{0}}\\ {{\mathbf{C}}_{{\mathrm{SB}}}^2}&{}{\mathbf{0}} \end{array}} \right] ,{\mathrm{}}&{\mathbf {K}}_{{\mathrm{SB}}}^s{\mathrm{}}&=\left[ {\begin{array}{*{20}{c}} {\mathbf{0}}&{}{\mathbf{0}}\\ {\mathbf{0}}&{}{{\mathbf{C}}_{{\mathrm{SB}}}^2} \end{array}} \right] \end{aligned}$$
    (A.29f)

    with

    $$\begin{aligned} {{\mathbf{C}}_{{\mathrm{S\theta }}}^{1} = {m_{\mathrm{e}}}e{\varvec{\Phi }}\left( {{z_{\mathrm{d}}}} \right) {\mathbf{Y}}\left( {{z_{\mathrm{d}}}} \right) } \end{aligned}$$
    (A.30)

    and

    $$\begin{aligned}&{\mathbf {C}}_{{\mathrm{SB}}}^{1} = {\rho _{\mathrm{b}}}{A_{\mathrm{b}}}\displaystyle \int _{\mathrm{0}}^{{L_{\mathrm{b}}}} {\left[ {{{\mathbf {U}}^{\mathrm{T}}}{\mathbf {X}}\left( {{z_{\mathrm{d}}}} \right) } \right] {\mathrm{d}}x} \end{aligned}$$
    (A.31a)
    $$\begin{aligned}&{\mathbf {C}}_{{\mathrm{SB}}}^{2} = {\rho _{\mathrm{b}}}{A_{\mathrm{b}}}\displaystyle \int _{\mathrm{0}}^{{L_{\mathrm{b}}}} {\left[ { \sin \left( \beta \right) \left( {{R_{\mathrm{d}}} + x} \right) {{\mathbf {V}}^{\mathrm{T}}}{\mathbf {X'}}\left( {{z_{\mathrm{d}}}} \right) } \right] {\mathrm{d}}x} \end{aligned}$$
    (A.31b)
    $$\begin{aligned}&{\mathbf {C}}_{{\mathrm{SB}}}^{3} = {\rho _{\mathrm{b}}}{A_{\mathrm{b}}}\displaystyle \int _{\mathrm{0}}^{{L_{\mathrm{b}}}} {\left[ {\cos \left( \beta \right) {{\mathbf {V}}^{\mathrm{T}}}{\mathbf {Y}}\left( {{z_{\mathrm{d}}}} \right) } \right] {\mathrm{d}}x} \end{aligned}$$
    (A.31c)

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Yang, LH., Mao, Z., Wu, SM. et al. Steady-state coupling vibration analysis of shaft–disk–blade system with blade crack. Nonlinear Dyn 105, 61–98 (2021). https://doi.org/10.1007/s11071-021-06645-3

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