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Nonlinear vibration and dynamic stability analysis of rotor-blade system with nonlinear supports

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Abstract

A dynamic model of a rotor-blade system is established considering the effect of nonlinear supports at both ends. In the proposed model, the shaft is modeled as a rotating beam where the gyroscopic effect is considered, while the shear deformation is ignored. The blades are modeled as Euler–Bernoulli beams where the centrifugal stiffening effect is considered. The equations of motion of the system are derived by Hamilton principle, and then, Coleman and complex transformations are adopted to obtain the reduced-order system. The nonlinear vibration and stability of the system are studied by multiple scales method. The influences of the normal rubbing force, friction coefficient, damping and support stiffness on the response of the rotor-blade system are investigated. The results show that the original hardening type of nonlinearity may be enhanced or transformed into softening type due to the positive or negative nonlinear stiffness terms of the bearing. Compared with the system with higher support stiffness, the damping of the bearing has a more powerful effect on the system stability under lower support stiffness. With the increase in rubbing force and support stiffness, the jump-down frequency, resonant peak and the frequency range in which the system has unstable responses increase.

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Abbreviations

A, \(A'\) :

Cross-sectional area of the shaft and blade

\({{A}}_{1}, {{A}}_{2}\) :

The complex functions of the dimensionless displacements to be solved

c, \( c_{\mathrm{blade}}\), \( c_{\mathrm{bearing}}\) :

Damping coefficients of the shaft, blade and bearing

\({{c}}_{\mathrm{b}{{y}}{1}}, {{c}}_{\mathrm{b}{{z}}{1}}, {{c}}_{\mathrm{b}{{y}}{ 2}}, {{c}}_{\mathrm{b}{{z}}{2}}\) :

The damping coefficients of bearing 1 and bearing 2 along Y and Z directions

\({{D}}_{{0}},{{D}}_{{2}}\) :

Partial derivative with respect to \({{T}}_{0}\) and \({{T}}_{2}\)

\(D_{11},D_{22},D_{33}\) :

Torsional and flexural stiffness

\(e_{y}, e_{z}\) :

Eccentricity with respect to y and z axes

E :

Young’s modulus

\({{E}}_{{{y}}}, {{E}}_{{{z}}}\) :

Misalignment along Y and Z directions

\(f_\mathrm{f}\) , \(f_\mathrm{b}\) :

Forward and backward whirl mode frequencies

\(F_{\mathrm{n}}, F_{\mathrm{t}}\) :

Normal and tangential rubbing forces at the tip of blade

\(F_{\mathrm{nmax}},{{F}}_{\mathrm{nmaxi}}\) :

Maximum normal rubbing force

\(F_{\mathrm{t}{{i}}},{{F}}_{\mathrm{t}{{xi}}},{{F}}_{ \mathrm{t}{{yi}}},{{F}}_{\mathrm{n}i}\) :

Rubbing forces on the ith blade

FFNF, FBNF:

First-order forward and backward natural frequencies

FTNF:

First-order torsional natural frequency

G :

Shear modulus

I :

Cross section inertia moment of the blade

\(I_{1}, I_{2}, I_{3}\) :

Polar and diametral mass moments of inertia

\(I'_{11}\) :

Area moment of inertia of the blade

\({{I}}_{\mathrm{disk}},{{J}}_{\mathrm{disk}}\) :

Diametral and polar mass moment of inertia of the disk

\(I_\mathrm{s}\) :

Cross section inertia moment of the shaft

k :

Linear support stiffness

\(k_{1}, k_{2}, k_{3}\) :

Shaft curvatures

L, l :

Length of the blade and shaft

\({{l}}_{{1}},{{l}}_{{2}}\) :

The distances of the disk to the left and right end

m :

Mass per unit length of the shaft

\(m'\) :

Density of the blade

\({{m}}_{\mathrm{D}},{{J}}_{\mathrm{p}},{{J}}_{\mathrm{d}}\) :

The mass, the polar and diametral mass moment of inertia of bladed disk

\({{m}}_{\mathrm{disk}}\) :

The mass of the disk

\(N_{11}\) :

Longitudinal stiffness

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

The number of blades

\({{O}}_{{1}}\) :

The center of rotating blade

\({{O}}_{{ 2}},{{O}}'_{{ 2}}\) :

The center of static and rubbed casing

\(p^{*}\) :

Dimensionless vibration displacement of the blade in the complex plane

\({\mathrm{ra}}\) :

The ratio of excitation frequency to rotating frequency

\({{r}}_{\mathrm{g}}\) :

The radius of the blade-tip orbit

\({{R}}_{{ 0}}\) :

The radius of the casing

\(R_{\mathrm{d}}\) :

The radius of the disk

SFNF, SBNF:

Second-order forward and backward natural frequencies

t :

Time

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

Contact time

\({{t}}_{\mathrm{p}}\) :

Rotating period

\({{t}}_{0}\) :

Start time of the rubbing

\({{T}},{{T}}_{\mathrm{shaft}},{{T}}_{\mathrm{blades}},{{T}}_{\mathrm{disk}}\) :

Kinetic energy, kinetic energies of shaft, blades and disk

\({{T}}_{{ 0}},{{T}}_{{ 2}}\) :

Components of time on large scale and second-order small scale

u, v, w :

Longitudinal and transverse displacements of the shaft

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

The displacement of the casing

\({{V}},{{V}}_{\mathrm{shaft}},{{V}}_{\mathrm{blades}},{{V}}_{\mathrm{bearing}}\) :

Potential energy, potential energies of shaft, blades and bearings

\({{W}},W_{F_{ni}} \) :

The work, the work done by rubbing force applied on ith blade

\({{x}}_{\mathrm{b}},{{y}}_{\mathrm{b}}\) :

The location of the point along the flapwise and chordwise directions

\({{x}}_{\mathrm{d}}\) :

The location of the disk

\({{z}}^{*}\) :

Dimensionless displacement of the shaft in the complex plane

\(\alpha \) :

strain along the neutral axis of the shaft

\(\alpha _{ni}, \beta _{ni}, c_{1i},c_{2i},c_{3i},c_{4i }(i=1,2)\) :

The coefficients of the mode shape of the shaft to be solved

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

Contact angle

\(\beta _\mathrm{f},\beta _\mathrm{b}\) :

Dimensionless forward and backward whirl mode frequencies

\(\gamma \) :

Stagger angle of the blade

\(\delta \) :

Variational operator

\(\delta ({{x}})\) :

Dirac delta function

\(\varepsilon \) :

Non-dimensional small-scale parameter

\(\zeta \) :

The distance of the point from the blade root

\({{\theta }}_{{{i}}},{{\theta }}_{{{yi}}},{{\theta }}_{{{zi}}}({{i=1,2}})\) :

Angular displacements of the ith part of the shaft

\(\varTheta _\mathrm{f},\varTheta _\mathrm{b}\) :

Forward and backward mode shape coefficients of the blades

\(\varTheta _{ni}(x)\,(i=1,2)\) :

Piecewise nth-order mode shapes of angular displacements in the complex plane

k :

Shear correction factor

\(\lambda _{i},r_{i} (i=1,2)\) :

The coefficients of free vibration differential equation of the shaft to be solved

\(\varLambda \) :

Vibration amplitude of the blade

\(\mu \) :

Friction coefficient

\(\xi ,\eta \) :

Coleman transformation parameters

\(\rho \) :

Density of the shaft

\(\sigma \) :

Detuning parameter

\(\psi ,\theta ,\beta \) :

Euler angles

\({{\psi }}_{{\mathrm{f}}},{{\psi }}_{{\mathrm{b}}}\) :

Forward and backward mode shapes of the rotating blades

\(\omega \) :

The frequency of harmonic motion

\(\omega _{1},\omega _{2},\omega _{3}\) :

Angular velocities of the rotating shaft

\(\varOmega \) :

Rotating speed

\(\vartheta \) :

Duffing term coefficient

\(\vartheta _{i}\) :

The azimuth angle of the ith blade on the disk

\(\phi \) :

Torsional deformation

\(\phi _\mathrm{f},\phi _\mathrm{b}\) :

Forward and backward whirl mode shapes of the shaft

\(\phi _{i}\) :

Angular position of the ith blade

\(\phi _{ni}\) :

Piecewise mode shapes of the shaft at nth-order critical speed

\({{k}}_{\phi }\) :

Torsional stiffness

References

  1. Khadem, S.E., Shahgholi, M., Hosseini, S.A.A.: Primary resonances of a nonlinear in-extensional rotating shaft. Mech. Mach. Theory 45(8), 1067–1081 (2010)

    Article  MATH  Google Scholar 

  2. Khadem, S.E., Shahgholi, M., Hosseini, S.A.A.: Two-mode combination resonances of an in-extensional rotating shaft with large amplitude. Nonlinear Dyn. 65(3), 217–233 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Shahgholi, M., Khadem, S.: Resonances of an in-extensional asymmetrical spinning shaft with speed fluctuations. Meccanica 48(1), 103–120 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chiu, Y.Z., Chen, D.Z.: The coupled vibration in a rotating multi-disk rotor system. Int. J. Mech. Sci. 53(1), 1–10 (2011)

    Article  Google Scholar 

  5. Chiu, Y.Z., Huang, S.C.: The influence on coupling vibration of a rotor system due to a mistuned blade length. Int. J. Mech. Sci. 49(4), 522–532 (2007)

    Article  Google Scholar 

  6. Ma, H., Lu, Y., Wu, Z.Y., et al.: A new dynamic model of rotor-blade systems. J. Sound Vib. 357, 168–194 (2015)

    Article  Google Scholar 

  7. Genta, G., Feng, C., Tonoli, A.: Dynamics behavior of rotating bladed discs: a finite element formulation for the study of second and higher order harmonics. J. Sound Vib. 329(25), 5289–5306 (2010)

    Article  Google Scholar 

  8. Diken, H., Alnefaie, K.: Effect of unbalanced rotor whirl on blade vibrations. J. Sound Vib. 330(14), 3498–3506 (2011)

    Article  Google Scholar 

  9. Wang, L., Cao, D.Q., Huang, W.: Nonlinear coupled dynamics of flexible blade-rotor-bearing systems. Tribol. Int. 43(4), 759–778 (2010)

    Article  Google Scholar 

  10. Santos, I.F., et al.: Contribution to experimental validation of linear and non-linear dynamic models for representing rotor-blade parametric coupled vibrations. J. Sound Vib. 271(3–5), 883–904 (2004)

    Article  Google Scholar 

  11. Najafi, A., Ghazavi, M.R., Jafari, A.A.: Stability and Hamiltonian Hopf bifurcation for a nonlinear symmetric bladed rotor. Nonlinear Dyn. 78(2), 1049–1064 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. Najafi, A., Ghazavi, M.R., Jafari, A.A.: Application of Krein’s theorem and bifurcation theory for stability analysis of a bladed rotor. Meccanica 49(6), 1507–1526 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. Ghazavi, M.R., Najafi, A., Jafari, A.A.: Bifurcation and nonlinear analysis of nonconservative interaction between rotor and blade row. Mech. Mach. Theory 65, 29–45 (2013)

    Article  Google Scholar 

  14. Sanches, L., et al.: Instability zones for isotropic and anisotropic multibladed rotor configurations. Mech. Mach. Theory 46(8), 1054–1065 (2011)

    Article  MATH  Google Scholar 

  15. Parent, M.O., Thouverez, F.: Phenomenological model for stability analysis of bladed rotor-to-stator contacts. Int. Symp. Transp. Phenom. Dyn. Rotating Mach. 10–15, 1–13 (2016)

    Google Scholar 

  16. Hou, L., Chen, Y.S., Fu, Y.Q., Li, Z.G.: Nonlinear response and bifurcation analysis of a Duffing type rotor model under sine maneuver load. Int. J. Non-linear Mech. 78, 133–141 (2016)

    Article  Google Scholar 

  17. Hou, L., Chen, Y.S., Cao, Q.J.: Nonlinear vibration phenomenon of an air-craft rub-impact rotor system due to hovering flight. Commun. Nonlinear Sci. Numer. Simulat. 19, 286–297 (2014)

    Article  MATH  Google Scholar 

  18. Bolotin, V.V.: Nonconservative Problems of the Theory of Elastic Stability. Pergamon Press, New York (1963)

    MATH  Google Scholar 

  19. Kleim, W., Pommer, C., Stoustrup, J.: Stability of rotor systems: a complex modeling approach. ZAMP 49, 644–655 (1998)

    MATH  Google Scholar 

  20. Ehrich, F.F.: Shaft whirl induced by rotor. J. Appl. Mech. 31(2), 279–282 (1964)

    Article  MATH  Google Scholar 

  21. Bucciarelli, L.L.: On the instability of rotating shafts due to internal damping. J. Appl. Mech. 49(2), 425–428 (1982)

    Article  MATH  Google Scholar 

  22. Sorge, F., Cammalleri, M.: On the beneficial effect of rotor suspension anisotropy on viscous-dry hysteretic instability. Meccanica 47(7), 1705–1722 (2012)

    Article  MATH  Google Scholar 

  23. Samantaray, A.K., Mukherjee, A., Bhattacharyya, R.: Some studies on rotors with polynomial type non-linear external and internal damping. Int. J. Non-linear Mech. 41(9), 1007–1015 (2006)

    Article  MATH  Google Scholar 

  24. Genta, G.: On the stability of rotating blade arrays. J. Sound Vib. 273(4–5), 805–836 (2004)

    Article  Google Scholar 

  25. Bab, S., Khadem, S.E., Abbsi, A., Shahgholi, M.: Dynamic stability and nonlinear vibration analysis of a rotor system with flexible/rigid blades. Mech. Mach. Theory 105, 633–653 (2016)

    Article  Google Scholar 

  26. Petrov, E.: Multiharmonic analysis of nonlinear whole engine dynamics with bladed disc-casing rubbing contacts. In: ASME Turbo Expo 2012: Turbine Technical Conference and Exposition. American Society of Mechanical Engineers

  27. Petrov, E.: Analysis of bifurcations in multiharmonic analysis of nonlinear forced vibrations of gas-turbine engine structures with friction and gaps. In: ASME Turbo Expo 2015: Turbine Technical Conference and Exposition. American Society of Mechanical Engineers

  28. Sinha, S.K.: Non-linear dynamic response of a rotating radial Timoshenko beam with periodic pulse loading at the free-end. Int. J. Nonlinear Mech. 40, 113–149 (2015)

    Article  MATH  Google Scholar 

  29. Sinha, S.K.: Combined torsional-bending-axial dynamics of a twisted rotating cantilever Timoshenko beam with contact-impact loads at the free end. J. Appl. Mech. 74, 505–522 (2007)

    Article  MATH  Google Scholar 

  30. Turner, K., Adams, M., Dunn, M.: Simulation of engine blade tip-rub induced vibration. In: ASME Turbo Expo: Power for Land, Sea and Air, pp. 391–396 (2005)

  31. Turner, K., Dunn, M., Padova, C.: Airfoil deflection characteristics during rub events. J. Turbomach. 134, 112–121 (2012)

    Article  Google Scholar 

  32. Kou, H.J., Yuan, H.Q.: Rub-induced non-linear vibrations of a rotating large deflection plate. Int. J. Nonlinear Mech. 58, 283–294 (2014)

    Article  Google Scholar 

  33. Ma, H., Wang, D., Tai, X.Y., Wen, B.C.: Vibration response analysis of blade-disk dovetail structure under blade tip rubbing condition. J. Vib. Control 23, 252–271 (2017)

    Article  Google Scholar 

  34. Wang, C., Zhang, D.Y., Ma, Y.H., Liang, Z.C., Hong, J.: Dynamic behavior of aero-engine rotor with fusing design suffering blade off. Chin. J. Aeronaut. 30, 918–931 (2017)

    Article  Google Scholar 

  35. Wang, C., Zhang, D.Y., Ma, Y.H., Liang, Z.C., Hong, J.: Theoretical and experimental investigation on the sudden unbalance and rub-impact in rotor system caused by blade off. Mech. Syst. Signal Process. 76–77, 111–135 (2016)

    Article  Google Scholar 

  36. Yu, P.C., Zhang, D.Y., Ma, Y.H., Hong, J.: Dynamic modeling and vibration characteristics analysis of the aero-engine dual-rotor system with fan blade out. Mech. Syst. Signal Process. 106, 158–175 (2018)

    Article  Google Scholar 

  37. Ma, H., Yin, F.L., Wu, Z.Y., Tai, L.X.Y.: Nonlinear vibration response analysis of a rotor-blade system with blade-tip rubbing. Nonlinear Dyn. 84, 1225–1258 (2016)

    Article  MathSciNet  Google Scholar 

  38. Ma, H., Tai, X.Y., Han, Q.K., Wu, Z.Y., Wang, D., Wen, B.C.: A revised model for rubbing between rotating blade and elastic casing. J. Sound Vib. 337, 301–320 (2015)

    Article  Google Scholar 

  39. Yuan, H.Q.: Analysis Method of Rotor Dynamics. Metallurgical Industry Press, Beijing (2017)

    Google Scholar 

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Author information

Authors and Affiliations

  1. School of Mechanical Engineering and Automation, Northeastern University, Shenyang, 110819, Liaoning, People’s Republic of China

    Bingqiang Li, Hui Ma, Xi Yu, Jin Zeng, Xumin Guo & Bangchun Wen

  2. Key Laboratory of Vibration and Control of Aero-Propulsion System Ministry of Education, Northeastern University, Shenyang, 110819, Liaoning, People’s Republic of China

    Hui Ma

Authors
  1. Bingqiang Li
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  2. Hui Ma
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  3. Xi Yu
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  4. Jin Zeng
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Correspondence to Hui Ma.

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Funding

This project is supported by the National Natural Science Foundation (Grant No. 11772089), the Fundamental Research Funds for the Central Universities (Grant Nos. N170308028) and Program for the Innovative Talents of Higher Learning Institutions of Liaoning (LR2017035).

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Appendices

Appendix A: The first- and third-order equations extracted from multiple scales method

$$\begin{aligned}&O(\varepsilon ): \nonumber \\&\quad D_0^2 z_1^{*} +m_{\mathrm{r}} {\delta }\left( {x^{*}-x_{\mathrm{d}}^{*} } \right) D_0^2 z_1^{*} +\hbox {j}\Omega I_1^{*} D_0 {z''}_1^{*} +\hbox {j}\Omega ^{*}J_{\mathrm{r}} \delta \left( {x^{*}-x_\mathrm{d}^{*} } \right) D_0 {z}_1{''^{*}}\nonumber \\&\qquad -I_2^{*} D_0^2 {z''}_1^{*} -I_{\mathrm{r}} D_0^{2} {z''}_1^{*} {\delta }\left( x^{*}-x_{\mathrm{d}}^{*} \right) +z_1^{{IV}^{*}} \nonumber \\&\qquad +kz_1^{*} {\delta }\left( {x^{*}} \right) +kz_1^{*} {\delta }\left( {x^{*}-1} \right) +m_{\mathrm {r}2} {\delta }\left( {x^{*}-x_{\mathrm{d}}^{*} } \right) \cos \gamma \int _{0}^{1} {D_0^{2} p_{1}^{*} } \hbox {d}\varsigma ^{*}=0 \end{aligned}$$
(A.1)
$$\begin{aligned}&D_{0}^{2} p_{1}^{*} +{I'}_{11}^{*}\frac{\partial ^{4}p_{1}^{*}}{\partial \varsigma ^{*}{4}}+\Omega ^{{*}^{2}}\sin ^{2}\gamma p_{1}^{*} +\left( {R_{\mathrm{d}}^{{*}} +\varsigma ^{*}} \right) \Omega ^{{*}^{2}}\frac{\partial p_{1}^{*} }{\partial \varsigma ^{*}}-\frac{\Omega ^{{*}^{2}}}{2}\left( {\left( {R_{\mathrm{d}}^{{*}} +1} \right) ^{2}-\left( {R_{\mathrm{d}}^{{*}} +\varsigma ^{*}} \right) ^{2}} \right) \frac{\partial ^{2}p_{1}^{*} }{\partial \varsigma ^{*}{2}} \nonumber \\&\quad +2\hbox {j}\Omega ^{*}D_{0} p_{1}^{*} -l^{*}D_{0}^{2} z_{1}^{*} {\delta }\left( {x^{*}-x_{\mathrm{d}}^{{*}} } \right) \cos \gamma =0 \end{aligned}$$
(A.2)
$$\begin{aligned}&O(\varepsilon ^{3}):\nonumber \\&\quad D_0^2 z_3^{*} +m_{\mathrm{r}} {\delta }\left( {x^{*}-x_{\mathrm{d}}^{{*}} } \right) D_0^2 z_3^{*} +\hbox {j}\Omega I_1^{*} D_0 {z''}_3^{*} +\hbox {j}\Omega ^{*}J_{\mathrm{r}} {\delta }\left( {x^{*}-x_{\mathrm{d}}^{{*}} } \right) D_0 {z''}_3^{*}\nonumber \\&\qquad -I_2^{*} D_0^2 {z''}_3^{*} -I_{\mathrm{r}} D_0^2 {z''}_3^{*} {\delta }\left( {x^{*}-x_{\mathrm{d}}^{{*}} } \right) +z_3^{{IV}^{*}} \nonumber \\&\qquad +kz_{3}^{*} {\delta }\left( {x^{*}} \right) +kz_{3}^{*} {\delta }\left( {x^{*}-1} \right) =-2D_2 D_0 z_1^{*} -2m_{\mathrm{r}} {\delta }\left( {x^{*}-x_{\mathrm{d}}^{{*}} } \right) D_2 D_0 z_1^{*} \nonumber \\&\qquad -\hbox {j}\Omega I_1^{*} D_2 {z''}_1^{*} -\hbox {j}\Omega ^{*}J_{\mathrm{r}} {\delta }\left( {x^{*}-x_{\mathrm{d}}^{{*}} } \right) D_0 {z''}_1^{*} \nonumber \\&\qquad +2I_{2}^{*} D_{2} D_{0} {z''}_{1}^{*} +2I_{\mathrm{r}} D_{2} D_{0} {z''}_{1}^{*} {\delta }\left( x^{*}-x_{\mathrm{d}}^{{*}} \right) -c^{{*}} D_{0} z_{1}^{*} -c_{\mathrm {bearing}}^{{*}} D_{0} z_{1}^{*} {\delta }\left( {x^{*}}\right) \nonumber \\&\qquad -c_{\mathrm {bearing}}^{{*}} D_0 z_1^{*} {\delta }\left( {x^{*}-1} \right) -\vartheta z_{1}^{*} \left| {z_{1}^{*}} \right| ^{2}\left( {{\delta }\left( {x^{*}} \right) +{\delta }\left( {x^{*}-1} \right) } \right) \nonumber \\&\qquad +\left( {m_{\mathrm {disk}}^{{*}} \left( {\hbox {j}e_z +e_y } \right) \varOmega ^{{*}^{2}}e^{\mathrm {j}\varOmega ^{*}T_0 }-\sum _{i=1}^{N_{\mathrm{b}} } {F_{\mathrm {n}i} } (1+\hbox {j}\mu )e^{\mathrm {j}\phi _i }} \right) {\delta }\left( {x^{*}-x_{\mathrm{d}}^{{*}} } \right) -m_{\mathrm {r}2} {\delta }\left( {x^{*}-x_{\mathrm{d}}^{{*}} } \right) \cos \gamma \nonumber \\&\qquad \int _0^{1} {2D_{2} D_{0} p_{1}^{*}} \hbox {d}\varsigma ^{*} \nonumber \\&\qquad -\frac{{z'}_1^{*}}{2}\int _0^{{x^{*}}} {\left( {2D_0 {z'}_1^{*}D_0 {\dot{\bar{z}}}_1{'^{*}}+D_0^{2} {z'}_1 ^{*}{{\bar{z}}_1{'^{*}}}+D_0^{2} {\bar{z}}_1{'^{*}}{z'}_1 ^{*}} \right) \hbox {d}x^{*}} \nonumber \\&\qquad -\frac{{z''}_1 ^{*}}{2}\int _1^{{x^{*}}} {\int _0^{{x^{*}}} {\left( {2D_0 {z'}_1 ^{*}D_0 {\dot{\bar{z}}}_1{'^{*}}+D_0^2 {z'}_1 ^{*}{\bar{z'}}_1^{*}+D_0^2 {\bar{z}}_1{'^{*}}{z'}_1 ^{*}} \right) \hbox {d}x^{*}\hbox {d}x^{*}} } \nonumber \\&\qquad -m_{\mathrm{r}} {\delta }\left( {x^{*}-x_{\mathrm{d}}^{{*}} } \right) \left( \int _0^{x_{\mathrm{d}}^{\mathrm {*}}} {\left( {2D_0 {z'}_1^{*}D_0 {\dot{\bar{z}'}}_1^{*}+D_0^2 {z'}_1^{*}{\bar{z}}_1{'^{*}}+D_0^2 {\bar{z'}}_1^{*}{{z}_1}{'^{*}}} \right) \hbox {d}x^{*}} \right. \nonumber \\&\left. \qquad + \frac{{{z''}_1}^{*}}{2}\int _1^{{x_{\mathrm{d}}^{{*}} }} {\int _0^{x_{\mathrm{d}}^{{*}} } {\left( {2D_0 {z'}_1^{*}D_0 {\dot{\bar{z}}}_1{'^{*}}+ D_0^2 {z}_1{'^{*}}{\bar{z'}}_1 ^{*}+D_0^2 {\bar{z}}_1{'^{*}}{z}_1{'^{*}}} \right) \hbox {d}x^{*}\hbox {d}x^{*}} } \right) \nonumber \\&\qquad -2{z}_1{''^{*}}{z'}_1^{*}{\bar{z}}{'''^{*}}_1 -\frac{1}{2}{z}_1{''^{*}}{{\bar{z}}_1}{'^{*}}{z}_1{'''^{*}}-\frac{3}{2}{\bar{z}}_1{''^{*}}{{z'}_1}^{*}{z}_1{'''^{*}}-{{\bar{z}}_1}{''^{*}}{{{z''}_1}^{*}}^{2}-\frac{1}{2}{z_1 ^{\left( {IV} \right) }}^{*}{{\bar{z}}_1}{'^{*}}{{z'}_1}^{*} -\frac{1}{2} {\bar{z}_{1}^{(IV)*}} {{{z'}_1}^{*}}^{2} \end{aligned}$$
(A.3)
$$\begin{aligned}&D_0^{2} p_{3}^{*} +{I}{'^{*}}_{11}\frac{\partial ^{4}p_3^{*} }{\partial \varsigma ^{*}{4}}+{\Omega ^{*}}^{2}\sin ^{2}\gamma p_3^{*} +2\hbox {j}\Omega ^{*}D_0 p_3^{*} +\left( {R_{\mathrm{d}}^{{*}} +\varsigma ^{*}} \right) {\Omega ^{*}}^{2}\frac{\partial p_{3}^{*} }{\partial \varsigma ^{*}}\nonumber \\&\quad -\frac{{\Omega ^{*}}^{2}}{2}\left( {\left( {R_{\mathrm{d}}^{{*}} +1} \right) ^{2}-\left( {R_{\mathrm{d}}^{{*}} +\varsigma ^{*}} \right) ^{2}} \right) \frac{\partial ^{2}p_3^{*} }{\partial \varsigma ^{*2}} \nonumber \\&\quad +D_0^2 z_3^{*} {\delta }\left( {x^{*}-x_\mathrm{d}^{*} } \right) \cos \gamma =-2D_2 D_0 p_1^{*} -c_{\mathrm {blade}}^{{*}} D_0 p_1^{*} -2\hbox {j}\Omega ^{*}D_2 p_1^{*} -2D_2 D_0 z_1^{*} {\delta }\left( {x^{*}-x_{\mathrm{d}}^{{*}} } \right) \cos \gamma \nonumber \\&\quad -\hbox {j}c_{\mathrm {blade}}^{{*}} \Omega ^{*}p_1^{*} -\frac{2}{N_{\mathrm{b}} }\mu \cos \gamma {\delta }\left( {\varsigma ^{*}-1} \right) \sum _{i=1}^{N_{\mathrm{b}} } {F_{\mathrm {n}i}^{*} r^{*}} e^{-\hbox {j}\phi _i } \end{aligned}$$
(A.4)

Appendix B: The coefficients of \(\hbox {e}^{\mathrm{j}{\beta _{f}}{T}_{0}}\) and \(\hbox {e}^{\mathrm{j}{\beta _{b}}{T}_{0}}\)

$$\begin{aligned}&{H_{z,f}}\left( {{x^*},{T_2}} \right) = \left[ {2{\mathrm{j}}{{{{\mathrm{d}}^2}{\phi _f}({x^*})} \over {{\mathrm{d}}{x^*}^2}}{\beta _\mathrm{f}}I_2^*} \right. - 2{\mathrm{j}}{\beta _f}{\phi _f}({x^*}) - 2{\mathrm{j}}{m_{\mathrm{{r}}}}{{ \delta }}\left( {{x^*} - x_{\mathrm{{d}}}^{\mathrm{{*}}}} \right) {\beta _f}{\phi _f}({x^*}) - {\mathrm{j}}{\Omega ^*}I_1^*{{{{\mathrm{d}}^2}{\phi _f}({x^*})} \over {{\mathrm{d}}{x^*}^2}} \nonumber \\&\quad - 2{\mathrm{j}}{m_{{\mathrm{r}}2}}\cos \gamma {\beta _f}{{ \delta }}\left( {{x^*} - x_{\mathrm{{d}}}^{\mathrm{{*}}}} \right) {\varTheta _f}\int _0^1 {{\psi _\mathrm{f}}} ({\varsigma ^*}){\mathrm{d}}{\varsigma ^*} - {\mathrm{j}}{\Omega ^*}{J_{\mathrm{{r}}}}{{ \delta }}\left( {{x^*} - x_{\mathrm{{d}}}^{\mathrm{{*}}}} \right) {{{{\mathrm{d}}^2}{\phi _f}({x^*})} \over {{\mathrm{d}}{x^*}^2}} \nonumber \\&\quad \left. { + 2{I_{\mathrm{{r}}}}{\beta _\mathrm{f}}{{ \delta }}\left( {{x^*} - x_{\mathrm{{d}}}^{\mathrm{{*}}}} \right) {{{{\mathrm{d}}^2}{\phi _f}({x^*})} \over {{\mathrm{d}}{x^*}^2}}} \right] {{{\mathrm{d}}{A_1}({T_2})} \over {\hbox {d}{T_2}}} + \left[ {{{{{\left( {{\beta _\mathrm{f}} - {\beta _\mathrm{b}}} \right) }^2}} \over 2}{{{\mathrm{d}}{\phi _b}({x^*})} \over {{\mathrm{d}}{x^*}}}} \right. \left( {\int _0^{{x^*}} {{{{\mathrm{d}}{\phi _f}({x^*})} \over {{\mathrm{d}}{x^*}}}{{{\mathrm{d}}{\phi _b}({x^*})} \over {{\mathrm{d}}{x^*}}}{\mathrm{d}}{x^*}} } \right) \nonumber \\ \end{aligned}$$
(B.1)
$$\begin{aligned}&\quad + {{{{\left( {{\beta _\mathrm{f}} - {\beta _\mathrm{b}}} \right) }^2}} \over 2}{{{\mathrm{d}}{\phi ^2}_b({x^*})} \over {{\mathrm{d}}{x^*}^2}}\left( {\int _1^{{x^*}} {\int _0^{{x^*}} {{{{\mathrm{d}}{\phi _f}({x^*})} \over {{\mathrm{d}}{x^*}}}{{{\mathrm{d}}{\phi _b}({x^*})} \over {{\mathrm{d}}{x^*}}}{\mathrm{d}}{x^*}{\mathrm{d}}{x^*}} } } \right) \\&\quad + {{{m_{\mathrm{{r}}}}{{ \delta }}\left( {{x^*} - x_{\mathrm{{d}}}^{\mathrm{{*}}}} \right) {{\left( {{\beta _\mathrm{f}} - {\beta _\mathrm{b}}} \right) }^2}} \over 2}{{{\mathrm{d}}{\phi _b}({x^*})} \over {{\mathrm{d}}{x^*}}}\left( {\int _0^{x_{\mathrm{{d}}}^{\mathrm{{*}}}} {{{{\mathrm{d}}{\phi _f}({x^*})} \over {{\mathrm{d}}{x^*}}}{{{\mathrm{d}}{\phi _b}({x^*})} \over {{\mathrm{d}}{x^*}}}{\mathrm{d}}{x^*}} } \right) \\&\quad + {{{m_{\mathrm{{r}}}}{{ \delta }}\left( {{x^*} - x_{\mathrm{{d}}}^{\mathrm{{*}}}} \right) {{\left( {{\beta _\mathrm{f}} - {\beta _\mathrm{b}}} \right) }^2}} \over 2}{{{\mathrm{d}}{\phi ^2}_b({x^*})} \over {{\mathrm{d}}{x^*}^2}}\left( {\int _1^{x_{\mathrm{{d}}}^{\mathrm{{*}}}} {\int _0^{x_{\mathrm{{d}}}^{\mathrm{{*}}}} {{{{\mathrm{d}}{\phi _f}({x^*})} \over {{\mathrm{d}}{x^*}}}{{{\mathrm{d}}{\phi _b}({x^*})} \over {{\mathrm{d}}{x^*}}}{\mathrm{d}}{x^*}{\mathrm{d}}{x^*}} } } \right) \\&\quad - {5 \over 2}{{{{\mathrm{d}}^2}{\phi _b}({x^*})} \over {{\mathrm{d}}{x^*}^2}}{{{\mathrm{d}}{\phi _b}({x^*})} \over {{\mathrm{d}}{x^*}}}{{{{\mathrm{d}}^3}{\phi _f}({x^*})} \over {{\mathrm{d}}{x^*}^3}} - {7 \over 2}{{{{\mathrm{d}}^2}{\phi _b}({x^*})} \over {{\mathrm{d}}{x^*}^2}}{{{\mathrm{d}}{\phi _f}({x^*})} \over {{\mathrm{d}}{x^*}}}{{{{\mathrm{d}}^3}{\phi _b}({x^*})} \over {{\mathrm{d}}{x^*}^3}} \\&\quad - 2{{{{\mathrm{d}}^2}{\phi _f}({x^*})} \over {{\mathrm{d}}{x^*}^2}}{{{\mathrm{d}}{\phi _b}({x^*})} \over {{\mathrm{d}}{x^*}}}{{{{\mathrm{d}}^3}{\phi _b}({x^*})} \over {{\mathrm{d}}{x^*}^3}} - {1 \over 2}{{{{\mathrm{d}}^4}{\phi _b}({x^*})} \over {{\mathrm{d}}{x^*}^4}}{{{\mathrm{d}}{\phi _b}({x^*})} \over {{\mathrm{d}}{x^*}}}{{{\mathrm{d}}{\phi _f}({x^*})} \over {{\mathrm{d}}{x^*}}} \\&\quad - 2{\left( {{{{{\mathrm{d}}^2}{\phi _b}({x^*})} \over {{\mathrm{d}}{x^*}^2}}} \right) ^2}{{{{\mathrm{d}}^2}{\phi _f}({x^*})} \over {{\mathrm{d}}{x^*}^2}}\left. { - {1 \over 2}{{{{\mathrm{d}}^4}{\phi _f}({x^*})} \over {{\mathrm{d}}{x^*}^4}}{{\left( {{{{\mathrm{d}}{\phi _b}({x^*})} \over {{\mathrm{d}}{x^*}}}} \right) }^2}} \right] {A_1}({T_2}){A_2}({T_2}){{{{\bar{A}}}}_2}({T_2}) \\&\quad + \left[ { - 4{{{{\mathrm{d}}^2}{\phi _f}({x^*})} \over {{\mathrm{d}}{x^*}^2}}{{{\mathrm{d}}{\phi _f}({x^*})} \over {{\mathrm{d}}{x^*}}}{{{{\mathrm{d}}^3}{\phi _f}({x^*})} \over {{\mathrm{d}}{x^*}^3}} - {\vartheta ^*}{{\left( {{\phi _f}({x^*})} \right) }^3}\left( {{{ \delta }}\left( {{x^*}} \right) + {{ \delta }}\left( {{x^*} - 1} \right) } \right) } \right. \\&\quad \left. { - {{\left( {{{{{\mathrm{d}}^2}{\phi _f}({x^*})} \over {{\mathrm{d}}{x^*}^2}}} \right) }^3} - {1 \over 2}{{{{\mathrm{d}}^4}{\phi _f}({x^*})} \over {{\mathrm{d}}{x^*}^4}}{{\left( {{{{\mathrm{d}}{\phi _f}({x^*})} \over {{\mathrm{d}}{x^*}}}} \right) }^2}} \right] A_1^2({T_2}){{{{\bar{A}}}}_1}({T_2}) \\&\quad - {\mathrm{j}}{c^*}{\phi _f}({x^*}){\beta _\mathrm{f}}{A_1}({T_2}) + m_{{\mathrm{disk}}}^{\mathrm{{*}}}{\Omega ^*}^2\left( {{\mathrm{j}}{e_z} + {e_y}} \right) {{ \delta }}\left( {{x^*} - x_{\mathrm{{d}}}^{\mathrm{{*}}}} \right) {e^{{\mathrm{j}}\sigma {T_2}}} - {{ \delta }}\left( {{x^*} - x_{\mathrm{{d}}}^{\mathrm{{*}}}} \right) \sum \limits _{i = 1}^{{N_{\mathrm{{b}}}}} {F_{{\mathrm{n}}i}^*} (1 + {\mathrm{j}}\mu ){e^{{\mathrm{j}}\sigma {T_2} + {\mathrm{j}}{\vartheta _i}}} \\&\quad - {\mathrm{j}}c_{{\mathrm{bearing}}}^{\mathrm{{*}}}{\phi _f}({x^*}){\beta _\mathrm{f}}{A_1}({T_2}){{ \delta }}\left( {{x^*}} \right) - {\mathrm{j}}c_{{\mathrm{bearing}}}^{\mathrm{{*}}}{\phi _f}({x^*}){\beta _f}{A_1}({T_2}){{ \delta }}\left( {{x^*} - 1} \right) = 0 \end{aligned}$$
$$\begin{aligned}&{H_{z,b}}\left( {{x^*},{T_2}} \right) = \left[ {2{\mathrm{j}}{{{{\mathrm{d}}^2}{\phi _b}({x^*})} \over {{\mathrm{d}}{x^*}^2}}{\beta _\mathrm{b}}I_2^*} \right. - 2{\mathrm{j}}{\beta _b}{\phi _b}({x^*}) - 2{\mathrm{j}}{m_{\mathrm{r}}}{{ \delta }}\left( {{x^*} - x_{\mathrm{d}}^{*}} \right) {\beta _\mathrm{b}}{\phi _b}({x^*}) - {\mathrm{j}}{\Omega ^*}I_1^*{{{{\mathrm{d}}^2}{\phi _b}({x^*})} \over {{\mathrm{d}}{x^*}^2}} \nonumber \\&\quad - 2{\mathrm{j}}{m_{{\mathrm{r}}2}}\cos \gamma {\beta _b}{{ \delta }}\left( {{x^*} - x_{\mathrm{d}}^{*}} \right) {\varTheta _b}\int _0^1 {{\psi _\mathrm{b}}} ({\varsigma ^*}){\mathrm{d}}{\varsigma ^*} - {\mathrm{j}}{\Omega ^*}{J_{\mathrm{r}}}{{ \delta }}\left( {{x^*} - x_{\mathrm{d}}^{*}} \right) {{{{\mathrm{d}}^2}{\phi _b}({x^*})} \over {{\mathrm{d}}{x^*}^2}} \nonumber \\&\quad \left. { + 2{I_{\mathrm{r}}}{\beta _\mathrm{b}}{{ \delta }}\left( {{x^*} - x_{\mathrm{d}}^{*}} \right) {{{{\mathrm{d}}^2}{\phi _b}({x^*})} \over {{\mathrm{d}}{x^*}^2}}} \right] {{{\mathrm{d}}{A_2}({T_2})} \over {\hbox {d}{T_2}}} + \left[ {{{{{\left( {{\beta _\mathrm{f}} - {\beta _\mathrm{b}}} \right) }^2}} \over 2}{{{\mathrm{d}}{\phi _f}({x^*})} \over {{\mathrm{d}}{x^*}}}} \right. \left( {\int _0^{{x^*}} {{{{\mathrm{d}}{\phi _f}({x^*})} \over {{\mathrm{d}}{x^*}}}{{{\mathrm{d}}{\phi _b}({x^*})} \over {{\mathrm{d}}{x^*}}}{\mathrm{d}}{x^*}} } \right) \nonumber \\&\quad + {{{{\left( {{\beta _\mathrm{f}} - {\beta _\mathrm{b}}} \right) }^2}} \over 2}{{{\mathrm{d}}{\phi ^2}_f({x^*})} \over {{\mathrm{d}}{x^*}^2}}\left( {\int _1^{{x^*}} {\int _0^{{x^*}} {{{{\mathrm{d}}{\phi _f}({x^*})} \over {{\mathrm{d}}{x^*}}}{{{\mathrm{d}}{\phi _b}({x^*})} \over {{\mathrm{d}}{x^*}}}{\mathrm{d}}{x^*}{\mathrm{d}}{x^*}} } } \right) \nonumber \\&\quad + {{{m_{\mathrm{r}}}{{ \delta }}\left( {{x^*} - x_{\mathrm{d}}^{*}} \right) {{\left( {{\beta _\mathrm{f}} - {\beta _\mathrm{b}}} \right) }^2}} \over 2}{{{\mathrm{d}}{\phi _f}({x^*})} \over {{\mathrm{d}}{x^*}}}\left( {\int _0^{x_{\mathrm{d}}^{*}} {{{{\mathrm{d}}{\phi _f}({x^*})} \over {{\mathrm{d}}{x^*}}}{{{\mathrm{d}}{\phi _b}({x^*})} \over {{\mathrm{d}}{x^*}}}{\mathrm{d}}{x^*}} } \right) \nonumber \\&\quad + {{{m_{\mathrm{r}}}{{ \delta }}\left( {{x^*} - x_{\mathrm{d}}^{*}} \right) {{\left( {{\beta _\mathrm{f}} - {\beta _\mathrm{b}}} \right) }^2}} \over 2}{{{\mathrm{d}}{\phi ^2}_f({x^*})} \over {{\mathrm{d}}{x^*}^2}}\left( {\int _1^{x_{\mathrm{d}}^{*}} {\int _0^{x_{\mathrm{d}}^{*}} {{{{\mathrm{d}}{\phi _f}({x^*})} \over {{\mathrm{d}}{x^*}}}{{{\mathrm{d}}{\phi _b}({x^*})} \over {{\mathrm{d}}{x^*}}}{\mathrm{d}}{x^*}{\mathrm{d}}{x^*}} } } \right) \nonumber \\&\quad - {5 \over 2}{{{{\mathrm{d}}^2}{\phi _f}({x^*})} \over {{\mathrm{d}}{x^*}^2}}{{{\mathrm{d}}{\phi _f}({x^*})} \over {{\mathrm{d}}{x^*}}}{{{{\mathrm{d}}^3}{\phi _b}({x^*})} \over {{\mathrm{d}}{x^*}^3}} - {7 \over 2}{{{{\mathrm{d}}^2}{\phi _f}({x^*})} \over {{\mathrm{d}}{x^*}^2}}{{{\mathrm{d}}{\phi _b}({x^*})} \over {{\mathrm{d}}{x^*}}}{{{{\mathrm{d}}^3}{\phi _f}({x^*})} \over {{\mathrm{d}}{x^*}^3}} \nonumber \\&\quad - 2{{{{\mathrm{d}}^2}{\phi _b}({x^*})} \over {{\mathrm{d}}{x^*}^2}}{{{\mathrm{d}}{\phi _f}({x^*})} \over {{\mathrm{d}}{x^*}}}{{{{\mathrm{d}}^3}{\phi _f}({x^*})} \over {{\mathrm{d}}{x^*}^3}} - {1 \over 2}{{{{\mathrm{d}}^4}{\phi _f}({x^*})} \over {{\mathrm{d}}{x^*}^4}}{{{\mathrm{d}}{\phi _f}({x^*})} \over {{\mathrm{d}}{x^*}}}{{{\mathrm{d}}{\phi _b}({x^*})} \over {{\mathrm{d}}{x^*}}} \nonumber \\&\quad - 2{\left( {{{{{\mathrm{d}}^2}{\phi _f}({x^*})} \over {{\mathrm{d}}{x^*}^2}}} \right) ^2}{{{{\mathrm{d}}^2}{\phi _b}({x^*})} \over {{\mathrm{d}}{x^*}^2}}\left. { - {1 \over 2}{{{{\mathrm{d}}^4}{\phi _b}({x^*})} \over {{\mathrm{d}}{x^*}^4}}{{\left( {{{{\mathrm{d}}{\phi _f}({x^*})} \over {{\mathrm{d}}{x^*}}}} \right) }^2}} \right] {A_1}({T_2}){A_2}({T_2}){{{\bar{A}}}_1}({T_2}) \nonumber \\&\quad + \left[ { - 4{{{{\mathrm{d}}^2}{\phi _f}({x^*})} \over {{\mathrm{d}}{x^*}^2}}{{{\mathrm{d}}{\phi _f}({x^*})} \over {{\mathrm{d}}{x^*}}}{{{{\mathrm{d}}^3}{\phi _f}({x^*})} \over {{\mathrm{d}}{x^*}^3}} - {\vartheta ^*}{{\left( {{\phi _f}({x^*})} \right) }^3}\left( {{{ \delta }}\left( {{x^*}} \right) + {{\delta }}\left( {{x^*} - 1} \right) } \right) } \right. \nonumber \\&\quad \left. { - {{\left( {{{{{\mathrm{d}}^2}{\phi _f}({x^*})} \over {{\mathrm{d}}{x^*}^2}}} \right) }^3} - {1 \over 2}{{{{\mathrm{d}}^4}{\phi _f}({x^*})} \over {{\mathrm{d}}{x^*}^4}}{{\left( {{{{\mathrm{d}}{\phi _f}({x^*})} \over {{\mathrm{d}}{x^*}}}} \right) }^2}} \right] A_2^2({T_2}){{\bar{A}}_2}({T_2}) \nonumber \\&\quad - {\mathrm{j}}{c^*}{\phi _b}({x^*}){\beta _\mathrm{b}}{A_2}({T_2}) - {\mathrm{j}}c_{{\mathrm{bearing}}}^{*}{\phi _b}(0){\beta _b}{A_2}({T_2}) - {\mathrm{j}}c_{{\mathrm{bearing}}}^{*}{\phi _b}(1){\beta _\mathrm{b}}{A_2}({T_2}) = 0 \end{aligned}$$
(B.2)
$$\begin{aligned}&H_{p,f} \left( {\varsigma ^{*},T_2 } \right) =-2\hbox {j}\Theta _f \beta _\mathrm{f} \psi _\mathrm{f} \left( {\varsigma ^{*}} \right) \frac{\hbox {d}A_1 (T_2 )}{\hbox {d}T_2 }-\hbox {j}c_{\mathrm {blade}}^{{*}} \Theta _f \beta _\mathrm{f} \psi _\mathrm{f} \left( {\varsigma ^{*}} \right) A_1 (T_2 )-2\hbox {j}\Omega ^{*}\Theta _f \psi _\mathrm{f} \left( {\varsigma ^{*}} \right) \frac{\hbox {d}A_1 (T_2 )}{\hbox {d}T_2 } \nonumber \\&\quad -\hbox {j}c_{\mathrm {blade}}^{{*}} \Theta _f \Omega ^{*}\psi _\mathrm{f} \left( {\varsigma ^{*}} \right) A_1 (T_2 )-2\hbox {j}\beta _\mathrm{f} \phi _f (x_\mathrm{d} )\frac{\hbox {d}A_1 (T_2 )}{\hbox {d}T_2 }\cos \gamma -\frac{2}{N_{\mathrm{b}} }\mu \cos \gamma \sum _{i=1}^{N_\mathrm{b} } {F_{\mathrm {n}i}^{*} } r^{*}e^{\mathrm {j}\sigma T_2 +\hbox {j}^\vartheta _{i}} \end{aligned}$$
(B.3)
$$\begin{aligned}&H_{p,b} \left( {\varsigma ^{*},T_2 } \right) =-2\hbox {j}\Theta _b \beta _\mathrm{b} \psi _\mathrm{b} \left( {\varsigma ^{*}} \right) \frac{\hbox {d}A_2 (T_2 )}{\hbox {d}T_2 }-\hbox {j}c_{\mathrm {blade}}^{{*}} \Theta _b \beta _\mathrm{b} \psi _\mathrm{b} \left( {\varsigma ^{*}} \right) A_2 (T_2 )-2\hbox {j}\Omega ^{*}\Theta _b \psi _\mathrm{b} \left( {\varsigma ^{*}} \right) \frac{\hbox {d}A_2 (T_2 )}{\hbox {d}T_2 } \nonumber \\&\quad -\hbox {j}c_{\mathrm {blade}}^{{*}} \Theta _b \Omega ^{*}\psi _\mathrm{b} \left( {\varsigma ^{*}} \right) A_2 (T_2 )-2\hbox {j}\beta _\mathrm{b} \phi _b (x_\mathrm{d} )\frac{\hbox {d}A_2 (T_2 )}{\hbox {d}T_2 }\cos \gamma \end{aligned}$$
(B.4)

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Li, B., Ma, H., Yu, X. et al. Nonlinear vibration and dynamic stability analysis of rotor-blade system with nonlinear supports. Arch Appl Mech 89, 1375–1402 (2019). https://doi.org/10.1007/s00419-019-01509-0

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