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

Abstract

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

1 Introduction

The clearance between the blade and casing is a key parameter of turbine machinery, which influences the gas leakage. For example, a large clearance will lead to a reduction of the compression efficiency in turbine machinery. However, reducing the clearance, which can improve the compression efficiency, will increase the risk of the blade-casing rubbing. Blade vibration caused by blade-tip rubbing and rubbing fault diagnosis are attracting increasing attention [1–4]. Only considering the blade vibration, the rubbing between the blade/bladed-disk and the casing has been investigated, such as Legrand et al. [5, 6], Lesaffre et al. [7], Sinha [8, 9], Kou and Yuan [10], Yuan and Kou [11], Almeida et al. [12], Batailly et al. [13] and Ma et al. [14, 15].

For the blade installed on a flexible rotor system, the rotor whirl has a great influence on the blade-casing rubbing [16]. Many researchers worked on rotor-blade systems and investigated the dynamic characteristics due to the blade-casing rubbing [17–27]. On the basis of a lumped mass model (LMM), Kascak et al. [17] investigated the responses of a rotor-bearing system with smearing or abradable rubbing. Padovan et al. [18] established an LMM of a rotor system and analyzed the single- and multiple-blade rubbings responses considering the influence of unbalance magnitude, blade/rotor stiffness, system damping and rubbing interface friction characteristics. Based on a finite element (FE) model which considers the rubbing between the rigid blade and rigid casing, Lawrence et al. [19] simulated the interactions between the blade and casing. Sinha [20] developed a dynamic model for a bladed rotor system supported by multiple bearings and discussed the transient response of the rotor due to blade-tip rubbing during both the acceleration and deceleration processes. Based on a nonsymmetric bladed rotor system where blades are simulated by pre-twisted thin shallow shells, Sinha [21] analyzed the rubbing load under the blade missing. His simulation results show that the sudden rubbing load can increase by an order of magnitude over the unbalance force. Lesaffre et al. [22] established a flexible bladed rotor model in the rotating frame and observed an unstable phenomenon around the critical speed of the stator even under frictionless sliding. Based on a coupling model of flexible bladed rotor and flexible casing, Parent et al. [23] analyzed the blade-casing rubbing phenomena. Based on Ref. [23], Parent et al. [24] analyzed the effects of 3D contact formulation on both rubbing detection and the system stability due to the blade-casing rubbing. Based on contact dynamics, Ma et al. [25] established a rotor-blade-casing FE model using ANSYS software and analyzed complicated vibration responses caused by the blade-casing rubbing. Petrov [26] proposed a multi-harmonic analysis method to simulate whole-engine vibration due to the blade-casing rubbing, and he also demonstrated the high accuracy and computational efficiency of the proposed methods using a set of test cases and an example of analysis of a realistic gas turbine structure. Thinery et al. [27] studied the dynamic behaviors of a misaligned Kaplan turbine with blade-to-stator contacts. In their model, the rotor is modeled using the FE method with beam elements, while the rigid blades are adopted to deal with the contact between the rotor and casing.

Fig. 1

a Neglecting the swing of the disk in Ref. [34], b considering the swing of the disk in this study

Experimental tests have been used to study the blade-casing rubbing [14, 28–33]. By simplifying the blade as a rotating uniform beam and the casing as an arc structure, Ahrens et al. [28] measured contact forces and the contact duration by experiment. Considering the effects of elastic casing, Ma et al. [14] established a test rig of blade-casing rubbing and analyzed the normal rubbing force under different casing materials and rotational speeds. Padova et al. [29, 30] established an in-ground spin-pit facility (SPF) whose maximum rotational speed can reach 20000 rev/min and studied the metal-to-metal contact due to sudden penetrations with different penetration depths. Chen et al. [31] carried out rubbing experiments with different rubbing positions by using a rotor experiment rig of aero-engine and analyzed the relation of rubbing feature and rubbing position by the cepstrum. Adopting wavelet analysis to deal with the measured blade-casing rubbing data, Lim and Leong [32] and Abdelrhman et al. [33] detected the changes of rotor dynamics caused by blade-casing rubbing.

From the above literature reviews, it can be observed that most researchers focused on the rubbing between cantilever blades (beam or plate) and casings. The studies considering both the rotor whirl and the flexibility of the blades are very limited. Based on our previous research works [14, 34], the focuses of this paper include:

(1) An improved rotor-blade model is developed. In the improved model, the effects of the swing of the disk are considered (see Fig. 1b); however, they are not considered in Ref. [34] (see Fig. 1a). The proposed model can improve the calculation accuracy, especially under the flexible shaft condition. In addition, the stagger angle of the blade can also be considered in the improved model.

In our previous work [34], the shaft is modeled using lumped-parameter model (lumped mass points); however, it is difficult to accurately determine the masses of these discrete points in this model. In this study, the shaft is modeled using an FE method, which adopts element matrices of mass, stiffness and damping to assemble the whole matrices’ of mass, stiffness and damping, and it is convenient to implement for the modeling of the shaft. Furthermore, the mode shapes of rotor-blade systems can be also obtained based on the proposed model, which is also another improvement to the model in Ref. [34].

Fig. 2

Schematic of rotor-blade systems

Fig. 3

a Coordinate systems of the disk, b local coordinate systems of the blade, c Timoshenko beam in local coordinate system of the blade

(2) The rubbings between both the single and four blades and casing are simulated based on a normal rubbing force model presented in our previous work [14]. In Ref. [14], the simulated normal rubbing force is determined using the model in which the blade is presented using a cantilever beam model to represent the blade, and the casing is described using a two DOF model; the simulated normal rubbing force is validated using experimental results. In this study, the rotor whirl is considered, and the bending vibrations of the blades, the lateral and torsional vibrations of the shaft and casing vibration are discussed. In addition, the simulated results are also validated using the measured results in a test rig. Some new coupling vibration phenomena of the rotor-blade-casing system are also evaluated using the experiment results.

The paper is organized as follows. After this introduction, an improved dynamic model of rotor-blade systems is developed using the Hamilton’s principle in conjunction with assumed modes method in Sect. 2.1. The proposed model is validated by comparing the natural frequencies obtained from FE method and literature results in Sect. 2.2. A dynamic model of the rotor-blade system with blade-casing rubbing is presented in Sect. 3. In Sect. 4, simulated and measured vibration responses of the system are compared and some typical fault features for single- and four-blade rubbings are summarized. Finally, the conclusions are drawn in Sect. 5.

2 An improved dynamic model of rotor-blade systems

2.1 Dynamic model of rotor-blade systems

Considering the coupling effects of lateral and torsional vibrations of the rotor and longitudinal and bending/flexural vibrations of the blade, a schematic of a rotor-blade system is shown in Fig. 2. The rotor is composed of a shaft and rigid disk. The cantilever Timoshenko beam is used to simulate flexible blade, attached to the rigid disk. In Fig. 2, OXYZ is the global coordinate, and ${ox}^{\mathrm{d}}y^{\mathrm{d}}z^{\mathrm{d}}$ is the disk body coordinate. In addition, ${ox}^{\mathrm{r}}y^{\mathrm{r}}z^{\mathrm{r}}$ and $ox^{\mathrm{b}}y^{\mathrm{b}}z^{\mathrm{b}}$ represent the rotational coordinate and local coordinate systems of the blade, respectively. Symbols u, v and w represent the deformations in longitudinal, bending/flexural and swing directions of the blade, and $\varphi$ represents the cross-sectional rotation of the blade in local coordinate system ${ox}^{\mathrm{b}}y^{\mathrm{b}}z^{\mathrm{b}}$ (see Fig. 3).

The mathematical model of rotor-blade system is simplified according to the following assumptions.

• (1) Isotropic material is adopted, and the constitutive relationship satisfies Hooke’s law;

• (2) The contact problems of the blade, disk and shaft are neglected;

• (3) The disk is considered to be rigid, i.e., its flexibility is neglected, and it is described using a lumped mass point;

• (4) The shaft is described using FE method;

• (5) The blades are represented using uniform cantilever beams;

• (6) The bearing is simplified by a linear spring-damping model.

The position vector of any point Q on the blade can be written in the global coordinate system as:

$$\begin{aligned} \varvec{r}_Q =\left[ {{\begin{array}{c} {X_\mathrm{d} } \\ {Y_\mathrm{d} } \\ {Z_\mathrm{d} } \\ \end{array} }} \right] +\varvec{A}_4 \varvec{A}_3 \varvec{A}_2 \varvec{A}_1 \left[ {{\begin{array}{c} {R_\mathrm{d} +x+u-y\varphi } \\ {v+y} \\ w \\ \end{array} }} \right] ,\quad \end{aligned}$$

(1)

where four rotational transformation matrices ${\varvec{A}}_{1}, {\varvec{A}}_{2}, {\varvec{A}}_{3}$ and ${\varvec{A}}_{4}$ are given as follows:

$$\begin{aligned} \varvec{A}_1= & {} \left[ {{\begin{array}{lll} 1&{}\quad 0&{}\quad 0 \\ 0&{}\quad {\cos \beta }&{}\quad {-\sin \beta } \\ 0&{}\quad {\sin \beta }&{}\quad {\cos \beta } \\ \end{array} }} \right] , \end{aligned}$$

(2)

$$\begin{aligned} \varvec{A}_2= & {} \left[ {{\begin{array}{lll} {\cos \left( {\vartheta _i +\theta _{Z\mathrm{d}} } \right) }&{}\quad {-\sin \left( {\vartheta _i +\theta _{Z\mathrm{d}} } \right) }&{}\quad 0 \\ {\sin \left( {\vartheta _i +\theta _{Z\mathrm{d}} } \right) }&{}\quad {\cos \left( {\vartheta _i +\theta _{Z\mathrm{d}} } \right) }&{}\quad 0 \\ 0&{}\quad 0&{}\quad 1 \\ \end{array} }} \right] , \end{aligned}$$

(3)

$$\begin{aligned} \varvec{A}_3= & {} \left[ {{\begin{array}{lll} 1&{}\quad 0&{}\quad 0 \\ 0&{}\quad {\cos \theta _{X\mathrm{d}} }&{}\quad {-\sin \theta _{X\mathrm{d}} } \\ 0&{}\quad {\sin \theta _{X\mathrm{d}} }&{}\quad {\cos \theta _{X\mathrm{d}} } \\ \end{array} }} \right] \nonumber \\= & {} \left[ {{\begin{array}{lll} 1&{}\quad 0&{}\quad 0 \\ 0&{}\quad 1&{}\quad {-\theta _{X\mathrm{d}} } \\ 0&{}\quad {\theta _{X\mathrm{d}} }&{}\quad 1 \\ \end{array} }} \right] , \end{aligned}$$

(4)

$$\begin{aligned} \varvec{A}_4= & {} \left[ {{\begin{array}{lll} {\cos \theta _{Y\mathrm{d}} }&{}\quad 0&{}\quad {\sin \theta _{Y\mathrm{d}} } \\ 0&{}\quad 1&{}\quad 0 \\ {-\sin \theta _{Y\mathrm{d}} }&{}\quad 0&{}\quad {\cos \theta _{Y\mathrm{d}} } \\ \end{array} }} \right] \nonumber \\= & {} \left[ {{\begin{array}{lll} 1&{}\quad 0&{}\quad {\theta _{Y\mathrm{d}} } \\ 0&{}\quad 1&{}\quad 0 \\ {-\theta _{Y\mathrm{d}} }&{}\quad 0&{}\quad 1 \\ \end{array} }} \right] . \end{aligned}$$

(5)

In Eqs. (2)–(5), $\beta$ is the stagger angle of the blade; $\vartheta _i =\theta (t)+\left( {i-1} \right) \frac{2{\pi }}{N_\mathrm{b} }$, where $\theta (t)$ is the angular displacement of the disk; $\left( {i-1} \right) \frac{2{\pi }}{N_\mathrm{b} }$describes the position of the ith blade in the blade group; $\theta _{Zd}$ is a shaft torsional angle at disk hub; and $N_\mathrm{b}$ is the number of the blade. It is worth noting that the motion in the swing direction of the blade is neglected, i.e. $w=0$.

Substituting Eqs. (2)–(5) into Eq. (1) and ignoring high-order terms, ${\varvec{r}}_{Q}$ can then be expressed as:

$$\begin{aligned} \varvec{r}_Q =\left[ {{\begin{array}{l} {\left( {\begin{array}{l} X_\mathrm{d} +y\theta _{Y\mathrm{d}} \sin \beta -\left( {\left( {y+v} \right) \cos \beta +\left( {R_\mathrm{d} +x} \right) \theta _{Z\mathrm{d}} } \right) \sin \vartheta _i \\ +\left( {R_\mathrm{d} +x+u-y\varphi -y\theta _{Z\mathrm{d}} \cos \beta } \right) \text {cos}\vartheta _i \\ \end{array}} \right) } \\ {\left( {\begin{array}{l} Y_\mathrm{d} -y\theta _{X\mathrm{d}} \sin \beta +\left( {R_\mathrm{d} +x+u-y\varphi -y\theta _{Z\mathrm{d}} \cos \beta } \right) \text {sin}\vartheta _i \\ +\left( {\left( {y+v} \right) \cos \beta +\left( {R_\mathrm{d} +x} \right) \theta _{Z\mathrm{d}} } \right) \cos \vartheta _i \\ \end{array}} \right) } \\ {\left( {\begin{array}{l} Z_\mathrm{d} +\left( {y+v} \right) \sin \beta +\left( {y\theta _{Y\mathrm{d}} \cos \beta +\left( {R_\mathrm{d} +x} \right) \theta _{X\mathrm{d}} } \right) \text {sin}\vartheta _i \\ +\left( {y\theta _{X\mathrm{d}} \cos \beta -\left( {R_\mathrm{d} +x} \right) \theta _{Y\mathrm{d}} } \right) \text {sin}\vartheta _i \\ \end{array}} \right) } \\ \end{array} }} \right] . \end{aligned}$$

(6)

The expressions of total kinetic energy $T_{\mathrm{total}}$ and total potential energy $V_{\mathrm{total}}$ for the SDB system are as follows:

$$\begin{aligned} T_{\mathrm{total}}= & {} \sum _{i=1}^{N_\mathrm{b} } {T_{\mathrm{blade}} } +T_{\mathrm{shaft}} +T_{\mathrm{disk}}, \nonumber \\ V_{\mathrm{total}}= & {} \sum _{i=1}^{N_\mathrm{b} } {V_{\mathrm{blade}} } +V_{\mathrm{shaft}} +V_{\mathrm{bearing}} \end{aligned}$$

(7)

where $T_{\mathrm{blade}}, T_{\mathrm{shaft}}$ and $T_{\mathrm{disk}}$ are the kinetic energy of the blade, shaft and disk, and $V_{\mathrm{blade}}, V_{\mathrm{shaft}}$ and $V_{\mathrm{disk}}$ are the potential energy of the blade, shaft and disk, respectively. More details about these energy expressions can refer to [34] and the specific matrices and vectors formulas can be found in “Appendix”.

The equations of motion of the rotor system and blade system are assembled together to form the global matrices of the rotor-blade system. The schematic of the matrix assembling is shown in Fig. 4 where $N_{\mathrm{dof}}$ is the number of DOFs for the ith blade, and $N_{N\mathrm{s}}$ is the number of DOFs for the rotor.

Equations of motion of the rotor-blade system can then be written as follows:

$$\begin{aligned}&\varvec{M}_{\mathrm{RB}} \ddot{\varvec{q}}_{\mathrm{RB}} +(\varvec{C}_{\mathrm{RB}} +\varvec{G}_{\mathrm{RB}} )\dot{\varvec{q}}_{\mathrm{RB}} +\varvec{K}_{\mathrm{RB}} \varvec{q}_{\mathrm{RB}} \nonumber \\&\quad =\varvec{F}_{\mathrm{nonlinear}} +\varvec{F}_{\mathrm{rub}}, \end{aligned}$$

(8)

where ${\varvec{M}}_{\mathrm{RB}}$ is the mass matrix of the system; ${\varvec{C}}_{\mathrm{RB}}$ is the viscous damping matrix of the system, which is simulated by the Rayleigh damping; ${\varvec{G}}_{\mathrm{RB}}$ is the other damping matrix of the system except for viscous damping matrix, which includes the Coriolis force matrices of the blades, damping matrix of bearings, and gyroscopic matrices of the shaft and rigid disk; ${\varvec{K}}_{\mathrm{RB}}$ is the stiffness matrix of the system; ${\varvec{q}}_{\mathrm{RB}}$ is the generalized displacement vector; and ${\varvec{F}}_{\mathrm{nonlinear}}$ and ${\varvec{F}}_{\mathrm{rub}}$ are the nonlinear coupling force vector and rubbing force vector, respectively.

Fig. 4

Schematic diagram of assembled matrices for rotor-blade systems: a mass matrix ${\varvec{M}}_{\mathrm{RB}}$, b damping matrix ${\varvec{G}}_{\mathrm{RB}}$, c stiffness matrix ${\varvec{K}}_{\mathrm{RB}}$

In the equation, generalized displacement vector of rotor-blade systems ${\varvec{q}}_{\mathrm{RB}}$ can be expressed as:

$$\begin{aligned} \varvec{q}_{\mathrm{RB}} =\left[ {{\begin{array}{ll} {\varvec{q}_\mathrm{b} }&{}\quad {\varvec{q}_\mathrm{s} } \\ \end{array} }} \right] ^{\mathrm{T}}, \end{aligned}$$

(9)

where ${\varvec{q}}_{\mathrm{b}}$ and ${\varvec{q}}_{\mathrm{s}}$ are the displacement vectors of the blade and shaft, respectively.

The nonlinear force vector ${\varvec{F}}_{\mathrm{nonlinear}}$ can be expressed as:

$$\begin{aligned} \varvec{F}_{\mathrm{nonlinear}} =[\varvec{F}_{\mathrm{nonlinear,\,b}} \;\;\;\varvec{F}_{\mathrm{nonlinear,\,s}} ]^{\mathrm{T}}, \end{aligned}$$

(10)

where ${\varvec{F}}_\mathrm{nonlinear,b}$ is the nonlinear force vector of the blade (see “Appendix 5”). ${\varvec{F}}_\mathrm{nonlinear,s}$ is the nonlinear force vector of the shaft and its expression can be given as follows:

$$\begin{aligned} \varvec{F}_{\mathrm{nonlinear,s}}= & {} [0\;\cdots f_{\mathrm{nonlinear},X} \;f_{\mathrm{nonlinear},Y} \;f_{\mathrm{nonlinear},Z} \nonumber \\&M_{\mathrm{nonlinear},X} \;M_{\mathrm{nonlinear},Y}\nonumber \\&M_{\mathrm{nonlinear},Z} \;\cdots \;0]^{\mathrm{T}}, \end{aligned}$$

(11)

where $f_{\mathrm{nonlinear},X},$ $f_{\mathrm{nonlinear},Y}$, $f_{\mathrm{nonlinear},Z}$, $M_{\mathrm{nonlinear},X}, M_{\mathrm{nonlinear},Y}$ and $M_{\mathrm{nonlinear},Z}$ are nonlinear forces and moments applied at the disk position (see “Appendix 5”).

Rayleigh damping matrix ${\varvec{C}}_{\mathrm{RB}}$ can be expressed as:

$$\begin{aligned}&\varvec{C}_{\mathrm{RB}} =\zeta \varvec{M}_{\mathrm{RB}} +\eta \varvec{K}_{\mathrm{RB}}, \end{aligned}$$

(12)

$$\begin{aligned}&\left\{ {{\begin{array}{l} \zeta =\frac{{4\pi }f_\mathrm{n1} f_\mathrm{n2} (\xi _1 f_{\mathrm{n}2} -\xi _2 f_\mathrm{n1} )}{(f_{\mathrm{n}2}^2 -f_{\mathrm{n}1}^2 )} \\ \eta =\frac{\xi _2 f_\mathrm{n2} -\xi _1 f_\mathrm{n1} }{{\pi }(f_{\mathrm{n}2}^2 -f_\mathrm{n1}^2 )} \\ \end{array} }} \right. , \end{aligned}$$

(13)

where $f_\mathrm{n1}$ and $f_{\mathrm{n}2}$ represent the first and second natural frequency (Hz) of the rotor-blade system, respectively, and $\xi _1$ and $\xi _2$ (in this paper, $\xi _1 =\xi _2 =0.02)$ are the corresponding modal damping ratios.

2.2 Model verification based on natural characteristics

In this section, a flexible rotor-blade model is used to verify the proposed model, and the detailed rotor-blade physical dimensions are shown in Fig. 5 and Table 1. The detailed elements used to describe the shaft, disk, blades and bearings can be found in Ref. [34]. The natural frequencies are calculated using both methods [34], including an FE modeling method and an analytical method.

The results are shown in Table 2. The natural frequencies determined using FE and analytical methods in [34] are given in the second and third columns. The fifth and sixth columns show the percentage differences between the analytical model in Ref. [34] and the proposed model relative to the results of the FE model, respectively. The results in Table 2 show that the proposed model has a higher accuracy than that of the analytical method in Ref. [34], especially, for the vibration modes related to the disk swing. For example, the percentage differences of natural frequencies related to the disk swing $(f_{\mathrm{n4}}$ and $f_{\mathrm{n5}})$ reduce from 3.9340 % for the analytic model in [34] to 0.1042 % for the proposed model. Moreover, the mode shapes obtained from the proposed model also show a good agreement with those obtained from the FE model, as shown in Fig. 6.

Fig. 5

Physical dimensions of a rotor-blade system: a physical dimensions of the shaft, b physical dimensions of the disk and blade

Table 1 Model parameters of the rotor-blade system

Table 2 Natural frequencies comparison of the rotor-blade system

Besides the comparison with the FE model, the model is also validated by comparing the natural frequencies obtained from the proposed model with those from Yang’s method in Ref. [35], as shown in Table 3. The results obtained from the proposed method are in good agreement with those obtained from Yang’s method, and the maximum percentage difference of the natural frequency is 4.7919 %. The trends of some natural frequencies (see Table 3) are the same as that in Ref. [35]. Some natural frequencies do not change because the coupling effect between the disk and blade is not considered for the proposed model.

Fig. 6

Comparison of mode shapes: a $f_{\mathrm{n1}},$ b $f_{\mathrm{n2}},$ c $f_{\mathrm{n4}}$, d $f_{\mathrm{n7}}$, e $f_{\mathrm{n13}}$, f $f_{\mathrm{n14}}$

Table 3 Natural frequencies comparison of a rotor-blade system considering stagger angles

3 A dynamic model of rotor-blade systems with blade-casing rubbing

Rubbing between the blade-tip and casing can happen due to the rotor whirl and blade elongation. A schematic of blade-casing rubbing forces is shown in Fig. 7 where $F_\mathrm{n}^i$ and $F_\mathrm{t}^i$ are the normal and tangential rubbing forces applied on the ith rubbing blade, respectively. For the shaft, $F_\mathrm{n}^i$ can be translated and equivalent to a force $F_{\mathrm{nr}}^i$, and $F_\mathrm{t}^i$ can be equivalent to a force $F_{\mathrm{tr}}^i$ and a torque $M_{\mathrm{tr}}^i$, as shown in the left figure of Fig. 7. For the rubbing blade, the direction of $F_\mathrm{n}^i$ along the blade pointing toward the disk center, and the tangential rubbing force $F_\mathrm{t}^i$ in the local coordinate system of the blade can be decomposed into two forces $F_{\mathrm{t}z}^i$ and $F_{\mathrm{t}y}^i$, as shown in the right figure of Fig. 7.

Considering the influence of the casing vibration on the blade-tip rubbing and simplifying the casing as an LMP with two DOFs, the equations of motion of the rotor-blade-casing system can be written as follows:

$$\begin{aligned} \left\{ {\begin{array}{l} \varvec{M}_{\mathrm{RB}} \ddot{\varvec{q}}_{\mathrm{RB}} +(\varvec{C}_{\mathrm{RB}} +\varvec{G}_{\mathrm{RB}} )\dot{\varvec{q}}_{\mathrm{RB}} +\varvec{K}_{\mathrm{RB}} \varvec{q}_{\mathrm{RB}} =\varvec{F}_{\mathrm{RB}} \\ \varvec{M}_\mathrm{c} \ddot{\varvec{q}}_\mathrm{c} +\varvec{D}_\mathrm{c} \dot{\varvec{q}}_\mathrm{c} +\varvec{K}_\mathrm{c} \varvec{q}_\mathrm{c} =\varvec{F}_\mathrm{c} \\ \end{array}} \right. ,\quad \nonumber \\ \end{aligned}$$

(14)

where $\varvec{M}_\mathrm{c}, \varvec{D}_\mathrm{c}$ and $\varvec{K}_\mathrm{c}$ are mass, damping and stiffness matrices of the casing, respectively; ${\varvec{q}}_{\mathrm{c}}$ and ${\varvec{F}}_{\mathrm{c}}$ are the generalized coordinate vector and rubbing force vector of the casing, respectively. In this equation, the external force $\varvec{F}_{\mathrm{RB}} =\varvec{F}_{\mathrm{nonlinear}} +\varvec{F}_{\mathrm{rub}}$, where ${\varvec{F}}_{\mathrm{nonlinear}}$ (see “Appendix 5”) and ${\varvec{F}}_{\mathrm{rub}}$ are the nonlinear force vector and rubbing force vector of the rotor-blade system, respectively.

Fig. 7

Schematic of blade-casing rubbing forces

The expression of ${\varvec{F}}_{\mathrm{rub}}$ is

$$\begin{aligned} \varvec{F}_{\mathrm{rub}} =\left[ {\underbrace{\varvec{F}_{\mathrm{rub,\,b}} }_{\mathrm{Blade}}\;\;\;\underbrace{0\;\;\;\varvec{F}_{\mathrm{rub,\,d}} \;\;\;0}_{\mathrm{Rotor}}} \right] ^{\mathrm{T}}. \end{aligned}$$

(15)

The rubbing force vector ${\varvec{F}}_{\mathrm{rub,b}}$ of the blades can be written as $\varvec{F}_{\mathrm{rub},\text {b}} =\left[ {\cdots \;\varvec{F}_{\mathrm{rub,\,b}}^i \;\cdots } \right]$ where $\varvec{F}_{\mathrm{rub,\,b}}^i$ denotes the rubbing force vector of the ith rubbing blade and it can be expressed as

$$\begin{aligned} \varvec{F}_{\mathrm{rub,b}}^i =\left[ {\begin{array}{c} -F_\mathrm{n}^i \left. {\phi _{1m} } \right| _{x=L} \\ \left. {-F_\mathrm{t}^i \cos \beta \phi _{2m} } \right| _{x=L} \\ 0 \\ \end{array}} \right] ^{\mathrm{T}}. \end{aligned}$$

(16)

The rubbing force vector applied on the disk ${\varvec{F}}_{\mathrm{rub,d}}$ is

$$\begin{aligned} \varvec{F}_{\mathrm{rub,d}} =\sum _{i=1}^{N_\mathrm{b} } \left[ {\begin{array}{c} F_\mathrm{t}^i \text {sin}\vartheta _i -F_\mathrm{n}^i \cos \vartheta _i \\ -F_\mathrm{t}^i \cos \vartheta _i -F_\mathrm{n}^i \text {sin}\vartheta _i \\ 0 \\ 0 \\ 0 \\ -\left( {R_\mathrm{d} +L} \right) F_\mathrm{t}^i \\ \end{array}} \right] ^{\mathrm{T}}. \end{aligned}$$

(17)

The expressions of $\varvec{M}_\mathrm{c}, \varvec{D}_\mathrm{c}, \varvec{K}_\mathrm{c}, {\varvec{q}}_{\mathrm{c}}$ and ${\varvec{F}}_{\mathrm{c}}$ are given as follows:

$$\begin{aligned}&\varvec{M}_\mathrm{c} =\left[ {{\begin{array}{ll} {m_\mathrm{c} }&{}\quad 0 \\ 0&{}\quad {m_\mathrm{c} } \\ \end{array} }} \right] , \quad \varvec{K}_\mathrm{c} =\left[ {{\begin{array}{ll} {k_{\mathrm{c}X} }&{}\quad 0 \\ 0&{}\quad {k_{\mathrm{c}Y} } \\ \end{array} }} \right] ,\nonumber \\&\quad \varvec{D}_\mathrm{c} =\left[ {{\begin{array}{ll} {c_{\mathrm{c}X} }&{}\quad 0 \\ 0&{}\quad {c_{\mathrm{c}Y} } \\ \end{array} }} \right] +\zeta \varvec{M}_c+\eta \varvec{K}_c, \end{aligned}$$

(18)

$$\begin{aligned}&\varvec{q}_\mathrm{c} =\left[ {{\begin{array}{ll} {X_\mathrm{c} }&{}\quad {Y_\mathrm{c} } \\ \end{array} }} \right] ^{\mathrm{T}},\nonumber \\&\quad \varvec{F}_\mathrm{c} = \sum _{i=1}^{N_\mathrm{b} } \left[ \begin{array}{l} -F_\mathrm{t}^i \text {sin}\vartheta _i +F_\mathrm{n}^i \cos \vartheta _i\\ F_\mathrm{t}^i \cos \vartheta _i +F_\mathrm{n}^i \text {sin}\vartheta _i \end{array} \right] , \end{aligned}$$

(19)

where $m_{\mathrm{c}}$ is the casing mass; $c_{\mathrm{c}X}$ and $c_{\mathrm{c}Y}$ are damping of the casing; $k_{\mathrm{c}X}$ and $k_{\mathrm{c}Y}$ are stiffness of the casing; $X_{\mathrm{c}}$ and $Y_{\mathrm{c}}$ are displacements of the casing. The subscripts X and Y denote X and Y directions, respectively.

Normal rubbing force $F_{\mathrm{n}}$ can be expressed as [14]:

$$\begin{aligned} F_\mathrm{n} =-\left( {L\varGamma _1 k_\mathrm{c} \frac{-5\left( {\alpha \varGamma _1 -2\frac{\delta }{L}} \right) +\sqrt{5}\alpha \sqrt{5\varGamma _1 \left( {\varGamma _1 +\frac{4}{\alpha }\frac{\delta }{L}} \right) +12\mu ^{2}\frac{\delta }{L}}}{20\varGamma _1 -\frac{10}{\alpha }\frac{\delta }{L}{+}6\alpha \mu ^{2}}} \right) ,\nonumber \\ \end{aligned}$$

(20)

where $\delta$ is the penetration depth; $k_{\mathrm{c}}$ is the equivalent stiffness of the casing, here, $k_{\mathrm{c}}=k_{\mathrm{c}X};\mu$ is the friction coefficient; $\varGamma _1 =\frac{\varGamma _0 }{k_\mathrm{c}}; \varGamma _0 =E_\mathrm{b} I_\mathrm{b} \frac{3}{L^{3}}+\rho _\mathrm{b} A_\mathrm{b} \dot{\theta }^{2}\left( {\frac{81}{280}L+\frac{3}{8}R_\mathrm{d} } \right) ; \alpha =\frac{R_\mathrm{d} +L}{L}$. Minus sign denotes the direction of normal rubbing force is from blade-tip to the center of rotor.

The penetration depth $\delta$, which is related to the radial elongation of the blade, is caused by centrifugal loads and relative geometric position between the blade-tip and casing (see Fig. 8). In the published literature, the effects of the casing distortion are usually considered in the calculation of the blade-casing relative positions [13, 26]. It is worth noting that in our study, the effects of the casing distortion are not considered because only a small arc-shaped casing (see Fig. 8) is used to simulate the local blade-casing rubbing in the test rig. The disk and the casing are concentric in the original position (see Fig. 8a), and $g_{0}$ is the gap between the concentric bladed disk and casing, $g_0 =R_\mathrm{c} -\left( {R_\mathrm{d} +L} \right) \ge 0$, where $R_{\mathrm{c}}$ is the radius of the casing. However, due to assembling misalignment, the clearance between rotor and stator can be asymmetrical. Hence, the blade tip may penetrate the casing due to the rotor whirl motion and centrifugal force (see Fig. 8b).

Fig. 8

Schematic of blade-casing rubbing

The expression of penetration depth between the ith blade and casing is obtained as [26]:

$$\begin{aligned} \delta ^{i}(t)=\varvec{u}_i^\mathrm{T} \varvec{n}_i -g_0, \end{aligned}$$

(21)

where ${\varvec{n}}_{\mathrm{i}}$ is unit normal vector to the contact surface.

$$\begin{aligned} \varvec{n}_i =\left[ {\cos \vartheta _i \;\;\sin \vartheta _i } \right] ^{\mathrm{T}}, \end{aligned}$$

and ${\varvec{u}}_{\mathrm{i}}$ is a vector of the ith blade-casing relative motion in the global coordinate,

$$\begin{aligned} \varvec{u}_i =\varvec{u}_\mathrm{b}^i -\varvec{u}_\mathrm{c} -\varvec{e}_\mathrm{c}, \end{aligned}$$

where ${\varvec{e}}_{\mathrm{c}}$ is the eccentricity vector related to the misalignment of the rotor and casing centers; $\varvec{e}_\mathrm{c} =\left[ {{\begin{array}{ll} {e_X }&{}\quad {e_Y } \\ \end{array} }} \right] ^{\mathrm{T}}; \varvec{u}_{\mathrm{b}}^i$ and ${\varvec{u}}_{\mathrm{c}}$ are the displacement vectors of the ith blade-tip and casing in the global coordinate, respectively.

$$\begin{aligned} \varvec{u}_\mathrm{b}^i= & {} \left[ {{\begin{array}{l} {X_\mathrm{d} } \\ {\begin{array}{l} Y_\mathrm{d} \\ Z_d \\ \end{array}} \\ \end{array} }} \right] +\varvec{A}_4 \varvec{A}_3 \varvec{A}_2 \varvec{A}_1 \left[ {{\begin{array}{l} {u_i } \\ v_i \\ 0 \\ \end{array} }} \right] ,\nonumber \\ \varvec{u}_\mathrm{c}= & {} \varvec{q}_\mathrm{c} =\left[ {{\begin{array}{ll} {X_\mathrm{c} }&{}\quad {Y_\mathrm{c} } \\ \end{array} }} \right] ^{\mathrm{T}},\nonumber \\ \vartheta _i= & {} \theta (t)+\left( {i-1} \right) \frac{2{\pi }}{N_\mathrm{b} }. \end{aligned}$$

(22)

Tangential force $F_{\mathrm{t}}$ is generated from the friction between the blade-tip and casing, of which the direction is opposite to slip direction on the contact surface. Hence, tangential force can be written as:

$$\begin{aligned} F_\mathrm{t} =\mu F_\mathrm{n}. \end{aligned}$$

(23)

Because of the effects of the blade-tip rubbing, the equations of motion of the rotor-blade-casing system are nonlinear. In this study, Newmark integral method is adopted to calculate the system vibration response. The detailed simulation flowchart is shown in Fig. 9.

Fig. 9

Flowchart of blade-casing rubbing simulation

4 Numerical studies and experimental verification

The physical dimensions of a rotor-blade system in the test rig are the same as those in Ref. [34]. For the rotor-blade system, the bearing stiffness in horizontal and vertical directions are set as $k_{bX1}=k_{bY1}=k_{bX2}=k_{bY2}=1.5\times 10^{7}$ N/m. The bearing stiffness in axial direction are $k_{bZ1}=k_{bZ2}=4\times 10^{6}$ N/m. The bearing damping in three directions are $c_{bX1}=c_{bY1}=c_{bX2}=c_{bY2}=c_{bZ1}=c_{bZ2}=1000$ Ns/m. The stagger angle of the blade is $\beta =0^{\circ }$. It is worth noting that the torsional DOF of the right-most node (node 11) is restrained while that of the left-most node (node 1) is free, which is another revision to the system in Ref. [34]. This is because that the right end of the shaft is connected with the driving motor by a coupling in this study. Other parameters are the same as those in Ref. [34]. The simulated and measured natural frequencies of the rotor-blade system are shown in Table 4. This results show that the proposed model has a higher accuracy than the model in Ref. [34]. Percentage differences of the natural frequencies $f_{\mathrm{n5}}$ and $f_{\mathrm{n6}}$ decrease from 4.3973 % in Ref. [34] to 3.1661 % in this study (see Table 4).

In order to evaluate the rubbing-induced casing vibration, the natural frequencies of the casing system without and with blade-casing contact are also measured using the same test rig [34]. The measured natural frequencies are listed in Table 5.

Table 4 Natural frequencies of the rotor-blade system at zero rotational speed

Table 5 Natural frequencies of the casing system without and with blade-casing contact

Table 6 Parameters for simulation and experiment under five cases

Assuming that both the directions of the rotor whirl of rotation are counterclockwise, and the whirl velocity and the rotational speed are the same, in this section, two kinds of rubbing forms: single- and four-blade rubbings will be simulated. In addition, the experimental results are also used to validate the simulated results. The simulation parameters are set as follows:

Eccentricity between the geometrical center of the rotor-blade system and its mass center is 1 mm. The casing mass is $m_{{c}}=5$ kg and casing stiffness in horizontal and vertical directions are $k_{{cX}}=3\times 10^{6}$ N/m and $k_{{cY}}=7\times 10^{7}$ N/m, i.e., the natural frequencies of the casing in X and Y directions are 123.3 and 595.5 Hz, respectively. The casing damping in horizontal and vertical directions are $c_{{\mathrm{c}X}}= c_{{\mathrm{c}Y}}=2000$ Ns/m. The radius of the casing is $R_{{\mathrm{c}}}=224$ mm, the vector of the eccentricity is $\varvec{e}_\mathrm{c} =[{e_X }\quad {e_Y }]^{\mathrm{T}}, e_X =g_0 +\delta _0 \;(g_0 =2\;\text {mm}), e_Y =0$ where $\delta _{0}$ is the initial penetration depth. The friction coefficient between the blade and casing is ${\mu }=0.2$.

Five cases are selected to compare the simulated results with experimental results: healthy condition (case 1), single-blade rubbing condition (cases 2 and 3), and four-blade rubbing condition (cases 4 and 5), as are listed in Table 6. It should be noted that one slightly longer blade is artificially assembled under the single-blade rubbing condition, and four blades with the same dimension is assembled under the four-blade rubbing condition.

4.1 No rubbing condition

The comparison of simulated and measured vibration responses are shown in Fig. 10. The simulated lateral vibration of the rotor in X direction only contains the rotational frequency component $(f_\mathrm{r})$. However, the measured result shows multiple frequency components $({nf}_{\mathrm{r}})$ and a frequency component between $2f_\mathrm{r}$ and $3f_\mathrm{r}$, which may be excited by the rotor misalignment and bearing nonlinearity. There are also some errors about the vibration amplitude due to the effects of the assumed bearing stiffness, unbalance, and system damping.

Fig. 10

Vibration responses of the rotor-blade system without rubbing: a simulated results, b experimental results. Note The figures from left to right are time-domain waveform, amplitude spectrum and rotor orbit, respectively

4.2 Single-blade rubbing condition

Assuming that the rubbing between only single blade (blade 1) and casing appears, a comparison of simulated and experimental results under case 2 is shown in Figs. 11 and 12, respectively. For the simulated results, rotor vibration (node 9) in lateral (X) and torsional $(\theta _{\mathrm{Z}})$ directions, casing vibration in X direction, blade bending vibrations in local coordinate of the blades, and normal rubbing forces, are used to analyze the fault features. For the experimental results, only the rotor lateral vibration, casing vibration, and normal and tangential rubbing forces are obtained due to some limitations in experimental equipments. Based on the results in Figs. 11 and 12, some dynamic phenomena can be observed as follows:

• (1) Simulated results show that amplitude amplification phenomena occur when the multiple frequency components coincide with the natural frequencies of the system. The amplification phenomena can be observed at $8f_\mathrm{r}$ near the conical natural frequencies ($f_{\mathrm{n2}}$ and $f_{\mathrm{n3}}$ in Table 4), $8f_\mathrm{r}$ near the casing natural frequency, $10f_\mathrm{r}$ near the torsional natural frequency ($f_{\mathrm{n4}}$ in Table 4), and $22f_\mathrm{r}$ near the bending natural frequencies of the blades ($f_{\mathrm{n7}}, f_{\mathrm{n8}}, f_{\mathrm{n9}}$ and $f_{\mathrm{n10}}$ in Table 4), as shown in Fig. 11. The enlargement degrees for the torsional vibration of the rotor, casing vibration and blade bending vibration are more significant than that of the lateral vibration of the rotor.

• (2) The measured results also show the similar amplification phenomena when the multiple frequency components coincide with the conical natural frequencies (see Fig. 12b). The casing vibration shows that the amplification phenomena appear at $7f_\mathrm{r}$ which is related to the casing natural frequency ($f_{\mathrm{nc1}}=124.1$ Hz, see Table 5). In addition, other higher multiple frequency components at $58f_\mathrm{r}$ and $70f_\mathrm{r}$ which coincide with the casing natural frequencies $f_{\mathrm{ncr3}}$ (or $f_{\mathrm{nc5}})$ and $f_{\mathrm{nc6}}$, respectively, also show large amplitudes. Based on the ratio of the maximum tangential rubbing force and corresponding normal rubbing force in a rotational period, the dynamic friction coefficient is about 0.166, as shown in Fig. 12f, g.

• (3) The measured rubbing forces and casing vibration have a significant difference from the simulated results. Main reasons for these errors are that (1) the casing is overly simplified. Actually, the casing is a complicated assembly, and the simulation can only consider the first natural frequency; (2) the assumed system damping, bearing stiffness, amount of unbalance have some differences with those in real system; and (3) accurately controlling the penetration depth is difficult due to the micro-feeding errors of ballscrew driving, which may also lead to some errors between the simulated and measured rubbing forces.

Fig. 11

Simulated results with four-blade rubbing (case 2): a rotor displacement waveform in X direction, b amplitude spectrum of the rotor in X direction, c rotor orbit, d acceleration waveform of casing in X direction, e amplitude spectrum of casing acceleration in X direction, f normal rubbing force, g torsional displacement waveform of the rotor, h torsional amplitude spectrum of the rotor, i bending displacement of the rubbing blade (blade 1), j amplitude spectrum of bending displacement of the rubbing blade (blade 1)

Fig. 12

Experimental results under case 2: a time-domain waveform of the rotor in X direction, b amplitude spectrum of the rotor, c rotor orbit, d time-domain waveform of the casing in X direction, e amplitude spectrum of the casing, f normal rubbing force, g tangential rubbing force

Fig. 13

Simulated results with four-blade rubbing (case 3): a rotor displacement waveform in X direction, b amplitude spectrum of the rotor in X direction, c rotor orbit, d acceleration waveform of casing in X direction, e amplitude spectrum of casing acceleration, f normal rubbing force, g torsional displacement waveform of the rotor, h torsional amplitude spectrum of the rotor, i bending displacement of the rubbing blade (blade 1), j amplitude spectrum of bending displacement of the rubbing blade (blade 1)

Fig. 14

Experimental results under case 3: a time-domain waveform of the rotor in X direction, b amplitude spectrum of the rotor, c rotor orbit, d time-domain waveform of the casing in X direction, e amplitude spectrum of the casing, f normal rubbing force, g tangential rubbing force

Fig. 15

Simulated results with four-blade rubbing (case 4): a rotor displacement waveform in X direction, b amplitude spectrum of the rotor in X direction, c rotor orbit, d acceleration waveform of casing in X direction, e amplitude spectrum of casing acceleration, f normal rubbing force, g torsional displacement waveform of the rotor, h torsional amplitude spectrum of the rotor, i bending displacements of the blades, j amplitude spectrum of bending displacements of the blade 1

Fig. 16

Experimental results under case 4: a time-domain waveform of the rotor in X direction, b amplitude spectrum of the rotor, c rotor orbit, d time-domain waveform of the casing in X direction, e amplitude spectrum of the casing, f normal rubbing force, g tangential rubbing force. Note Numbers 1, 4, 3 and 2 denote the rubbing time for blades 1, 4, 3 and 2

Simulated and measured responses of the system under case 3 are shown in Figs. 13 and 14. Compared with those under case 2, some new vibration features are summarized as follows:

• (1) Under the higher rotational speeds, the rotor lateral vibration increase due to the increase of the rubbing level which can be justified by the increased normal rubbing force. The dynamic friction coefficient slightly decreases from 0.166 under case 2 (986.4 rev/min) to 0.158 under case 3 (1478.4 rev/min).

• (2) Some frequencies, at which amplification phenomena are observed, may lightly change due to the effects of rubbing nonlinearity on the system natural frequencies. For example, $48f_\mathrm{r}$ under case 3 are slightly larger than $70f_\mathrm{r}$ under case 2.

4.3 Four-blade rubbing condition

Simulated and measured results under case 4 are shown in Figs. 15 and 16, which show the following dynamic phenomena.

• (1) For the simulated results, amplitude amplification phenomena can also be observed, for example, $8f_\mathrm{r}$ related to the conical natural frequencies (see Fig. 15b) and casing natural frequency (see Fig. 15e), $9f_\mathrm{r}$ related to the torsional natural frequency (see Fig. 15h), and $22f_\mathrm{r}$ related to the bending natural frequencies of the blades (see Fig. 15j). For the four-blade rubbing, bigger amplitudes can also be observed at the blade passing frequency (BPF, $4f_\mathrm{r)}$ and its multiple frequencies, such as BPF, 2BPF, 3BPF, 4BPF, 5BPF and 6BPF, i.e., $4f_\mathrm{r}, 8f_\mathrm{r}, 12f_\mathrm{r}, 16f_\mathrm{r}, 20f_\mathrm{r}$ and $24f_\mathrm{r}$ (see Fig. 15b, e, h).

• (2) For the measured results, both the lateral vibration of the rotor and casing vibration show the amplitude amplification phenomena which agree well with the simulated results. The similar experimental results were also reported in Ref. [32].

  The measured normal and tangential rubbing forces under the four-blade and single-blade rubbing conditions have some differences. There are some reasons for these: (1) In order to generate rubbing for the four blades, some blades were grinded, and the lengths of these blades were reduced. (2) The natural frequencies of the mechanical structure, e.g., the triaxial force sensor, are excited by the contact forces, which lead to an amplification and a phase distortion of the force signals [28]. For the four-blade rubbing, the effect of the transient vibration of the force sensor on the measured results increases due to the reducing rubbing period. (3) The control of the penetration depths can also lead to some errors. These reasons also cause large errors in the dynamic friction coefficients for the four blades. So the dynamic friction coefficients are not suggested to be measured under four-blade rubbing conditions, and dynamic friction coefficients under this condition will not be compared in the following analysis.

• (3) Simulated and measured spectra show that the amplitude amplification phenomena often appear at the BPF and its multiple frequencies. These features are especially obvious for the amplitude spectrum of the measured casing vibration, which indicates that the rubbing-induced casing vibration can appear at the high-order natural frequencies (see Fig. 16e).

• (4) Simulated and measured normal rubbing forces show that the rubbing level of blade 1 is most severe, and those of blades 2 and 3 are moderate, and that of blade 4 is the least severe. A detailed explanation can be found in Fig. 17, which shows that the penetration depth is maximal for blade 1, followed by those for blade 2 and blade 4, and minimal for blade 3. In addition, it is also possible that the rubbing between the blade 3 and casing may not appear due to the effects of rotor whirl under small penetration depths.

The simulated results under case 5 are shown in Fig. 18, the similar amplitude amplification phenomena and frequency distributed features related to BPF can be observed. In addition, the rubbing between the blade 3 and casing does not appear. This also verifies the above analysis (see Fig. 17). By increasing the penetration from $50 \upmu \text {m}$ to $80 \upmu \text {m}$, the simulated results are shown in Fig. 19, which shows that the four-blade rubbing appears under relatively large penetration depths. Fig. 20 shows the measured results of four-blade rubbing under case 5. The collision levels under case 5 are slightly different from the above analysis under case 4. The penetration depths cannot accurately agree with the simulated conditions. This may lead to the difference between the simulated and measured results. In addition, the blade-casing abrasions can also lead to the change of the clearance between the blade and casing, which also affects the measured results to some extent.

Fig. 17

A schematic of blade-casing rubbing explanation

Fig. 18

Simulated results with four-blade rubbing (case 5): a rotor displacement waveform in X direction, b amplitude spectrum of the rotor in X direction, c rotor orbit, d acceleration waveform of casing in X direction, e amplitude spectrum of casing acceleration, f normal rubbing force, g torsional displacement waveform of the rotor, h torsional amplitude spectrum of the rotor, i bending displacements of the blades, j amplitude spectrum of bending displacements of the blade 1

Fig. 19

Simulated results with four-blade rubbing under $\delta _{0}=80 \upmu \text {m}$: a rotor displacement waveform in X direction, b amplitude spectrum of the rotor in X direction, c rotor orbit, d acceleration waveform of casing in X direction, e amplitude spectrum of casing acceleration, f normal rubbing forces, g torsional displacement waveform of the rotor, h torsional amplitude spectrum of the rotor, i bending displacements of the blades, j amplitude spectrum of bending displacements of the blade 1

Fig. 20

Experimental results under case 5: a time-domain waveform of the rotor in X direction, b amplitude spectrum of the rotor, c rotor orbit, d time-domain waveform of the casing in X direction, e amplitude spectrum of the casing, f normal rubbing force, g tangential rubbing force. Note Numbers 1, 4, 3 and 2 denote the rubbing time for blades 1, 4, 3 and 2

Table 7 Fault features summary for single- and four-blade rubbings

4.4 Summary of rubbing fault features

Typical fault features for single- and four-blade rubbings are listed in Table 7. For the single-blade rubbing, simulated results show that the amplitude amplification phenomena can be observed when the multiple frequency components of rotational frequency ($f_\mathrm{r})$ coincide with the conical natural frequencies, torsional natural frequency of the rotor-blade system, casing natural frequency and bending natural frequencies of the blades. For the four-blade rubbing, simulated results show that BPF and its multiple frequency components ($n\text {BPF}, n=1, 2, 3,{\ldots }$) with larger amplitudes can be viewed as a distinguished feature besides the amplitude amplification phenomena.

For single- and four-blade rubbings, measured results also show the similar amplitude amplification phenomena. Moreover, BPF and its multiple frequency components are also obvious for the four-blade rubbing. However, it should be noted that the amplitude spectra of the casing are complicated because the adopted casing model is simple and it cannot completely simulate the dynamic characteristics of the actual casing structure.

5 Conclusions

An improved rotor-blade dynamic model is developed, and the model has some significant advantages compared with our previous model in [34]. In addition, the proposed model is also compared with finite element model [34], analytical models [34, 35], and experiment results [34]. Then single- and four-blade rubbings are simulated by simplifying the casing as a lumped mass point with two degrees of freedom (DOFs). Finally, the simulated results are also validated using the experimental results. Main conclusions are summarized as follows:

• (1) Compared with the model in Ref. [34], the proposed model considers the effects of the swing of the rigid disk and stagger angle of the blade. Natural frequencies of a flexible rotor-blade system obtained from the proposed model indicate that natural frequencies related to the disk swing are closer to those obtained from finite element (FE) model. For the case under study, the maximum percentage differences reduce from 3.934 % based on the analytic model [34] to 0.1042 % based on the proposed model.

  In addition, the shaft is discretized using FE method rather than the lumped mass method, which makes the shaft modeling more general. Furthermore, the mode shapes of the rotor-blade systems can be also obtained from the proposed model, which is also a new development relative to the analytical model in Ref. [34].

• (2) Simulated rubbing responses show a good agreement with the experimental results. For the single-blade rubbing, amplitude amplification phenomena appear when the multiple frequency of the rotational frequency $(f_\mathrm{r})$ coincides with the conical and torsional natural frequencies of the rotor-blade system, the bending natural frequencies of the blades, and the natural frequencies of the casing. For the four-blade rubbing, the blade passing frequency (BPF, $4f_\mathrm{r})$ and its multiple frequency components are also obvious besides these frequencies related to amplitude amplification phenomena.

• (3) Rubbing levels of the blades are related to the rotor whirl. The most severe rubbing appears for the blade which locates at the right end of the whirl orbit, i.e., is nearest to the casing. The least severe rubbing appears for the blade located at the left end of the whirl orbit, which is the opposite side for the most severe rubbing blade. The rubbing levels related to other two blades are moderate.

In this study, the casing is simulated as a two DOFs system and only the whole displacement of the casing is considered, and only the rotor vibration displacements are measured. In the future studies, more complicated casing models such as elastic ring models will be introduced in the system model, and the effects of stagger angles of blades on the rubbing will be considered, and the shaft torsional vibration and blade dynamic stress will also be measured.

Appendices

Appendix 1: Vectors and matrices related to the blades

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

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

  (24)

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

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

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

  (25)

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

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

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

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

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

  (26)

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

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

  and the surplus elements are all zero.

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

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

  (27)

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

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

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

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

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

and the surplus elements are all zero.

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

1.1 Mass matrix

The mass coupling matrix of rotor-blade systems is

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

(28)

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

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

  (29)

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

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

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

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

  (30)

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

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

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

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

  (31)

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

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

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

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

  (32)

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

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

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

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

  (33)

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

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

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

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

  (34)

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

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

1.2 Damping matrix

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

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

(35)

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

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

(36)

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

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

1.3 Stiffness matrix

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

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

(37)

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

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

(38)

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

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

The stiffness coupling matrix of rotor-blade systems is

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

(39)

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

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

(40)

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

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

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

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

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

(41)

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

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

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

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

  (42)

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

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

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

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

  (43)

where

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

Appendix 4: Element matrices of the rotational shaft

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

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

(44)

where

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

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

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

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

(45)

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

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

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

(46)

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

Appendix 5: Nonlinear forces of the rotor-blade system

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

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

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

  (48)

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

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

(49)

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

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

(50)

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

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

(51)

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

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

  (52)

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

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

  (53)

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

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

  (54)