A review on dynamic characteristics of blade–casing rubbing

Abstract

Blade–casing interaction/rubbing can occur under the small blade-tip clearance, which may lead to the rubbing damage, excessive wear of the abradable coating, and efficiency loss caused by the increasing tip leakage flow. Moreover, the rubbing process involves complicated dynamic phenomena, such as friction-induced wear and heat, coupling vibration of the blade and elastic casing, which has recently received a great concern. This paper provides a review on the dynamic characteristics of blade, rotor, and casing as well as the mechanism of coating wear when the rubbing between the blade and bare or coating casing occurs. Firstly, the modeling methods for blade–casing rubbing are categorized into three types in accordance with different casing types, namely models for the rubbing between the blade and bare casing, models for the rubbing between the blade and coating casing without considering wear, and models for the rubbing between the blade and coating casing considering wear. Then, the simulated blade–casing dynamic characteristics due to rubbing are described for both bare and coating casings. After that, the experimental results of the blade–casing rubbing are reviewed and summarized for the bare and coating casings, respectively. Finally, the open problems for blade–casing rubbings are stated, and some recommendations for future research are also pointed out.

1 Introduction

The radial distance between the rotating blade and casing is referred to as blade-tip clearance in aeroengine (see Fig. 1), which is one of the important parameters affecting tip leakage flow. To improve the performance of the machine of this kind, the modern design for such improvements is minimizing the blade–casing clearance [1]. However, with the decreasing clearance, the probability of blade–casing rubbing increases. This has long been regarded as a significant contributor to rub-induced engine failure. The blade–casing rubbing may lead to complicated vibration of the overall unit, reduction in system performance, and shorter lives of blade and casing. In 1973, the National Transportation Safety Board (NTSB) reported a case that one of the engine fan assemblies was disintegrated during the flight because of the interaction between the fan blade tip and the fan casing [2]. In June 23, 2014, F-35A engine fired and leaded to a fleetwide grounding because the excessive rubbing between the turbine blades and the cowling occurred [3]. As the problems happened one after another, the blade–casing rubbing has attracted extensive attentions [1].

Fig. 1

A schematic diagram of blade–casing clearance

For the vibration responses and wear mechanism caused by blade–casing rubbing, many researchers carried out a lot of studies on rotor-to-stator/blade-to-blade/blade-to-casing interactions. A list of key articles [4–11] have addressed this issue, and fruitful results were published. Muszynska [8] presented a literature review on rub-related phenomena in rotating machinery, such as wear and heat caused by friction, repetitive periodic impacts, stiffening of the shaft, and coupling effects varying with the normal contact force. Ahmad [9] summarized the rotor–casing rubbing phenomenon and introduced some key parameters which influence the rubbing, such as dynamic stiffness, external damping, Coulomb friction, rotor acceleration, asymmetry of support structure, thermal effects due to friction and disk flexibility. Jiang et al. [10] reviewed the literature on the rotor–stator rubbing in the past half century from view of dynamics and control, and classified the existing modeling on the rotor–stator rubbing into two categories: the local rubbing models and the system rubbing models. Moreover, the authors also discussed the progress in the study of synchronous rubbing responses, sub-harmonic and super-harmonic rubbing responses, quasi-periodic partial rubbing, dry whirl/whip, coexistence of multi-stable rubbing responses as well as bifurcation and chaos of rubbing responses and also summarized the ideas and results of passive and active suppression of rubbing in the literature. Jacquet-Richardet et al. [11] summarized the main works related to the rotor–stator rubbing and blade–casing rubbing, introduced existing numerical models and test rigs, and emphasized the complicated contact phenomena.

Muszynska indicated that the rubbing often occurs at seals, seldom, but more dangerous, a blade rubs against the stator or vane [8]. Jiang et al. [10] gave the following statement: “It is known that rubbing often occurs at seals and the blade–casing rubbing occurs much less frequently, but the latter is more dangerous than the former due to higher line velocity at blade tip and larger impact energy, which also has a great influence on the rotor dynamics. Blade–casing rubbing (see Fig. 2) is a more complicated impact process: (a) Collision occurs between the flexible body and elastic body with a relatively large stiffness or two flexible bodies (the casing can be regarded as a thin-walled shell structure); here, the impact force mainly depends on the whole deformation of flexible body rather than the local deformation; (b) the blade–casing rubbing is an oblique collision when there is a certain angle between the blade tip and the line velocity, which easily causes the bending–torsion coupling deformation of the blade; (c) the impact is not transient during the rubbing process due to the obvious movement of the blade-tip location.”

Fig. 2

Schematic of blade–casing rubbing [10]

The above literatures show that the present theoretical researches mainly focus on the simulation of the rubbing-induced dynamic characteristics and rotor–stator interaction mechanism. However, literatures on blade–casing rubbing are fairly limited, and the associated rubbing and wear phenomena are still not fully understood. This paper aims to summarize the blade–casing rubbing models and rubbing-induced complicated dynamic characteristics for bare casing (the casing without coating), coating casing without considering wear and coating casing considering wear, and proposes the future researches on the blade–casing rubbing.

This paper consists of five sections. After this introduction, in Sect. 2, modeling methods for blade–casing rubbings are summarized. Sections 3 and 4 review the simulated and measured blade–casing dynamic characteristics due to rubbing, respectively. Finally, conclusions and outlooks are given in Sect. 5. Figure 3 summarizes the involved contents of this review.

Fig. 3

Contents of this review

2 Modeling methods for blade–casing rubbings

Blade–casing rubbing is a complicated nonlinear contact process, and the rubbing mechanisms are multiphysical including impact, friction, heating, and wear [11, 12]. In addition, the rubbing can lead to the strong coupling between the global dynamics of the system and local phenomena that involve material and tribological characteristics (temperature, wear, etc.) [11]. How to accurately describe the impact, friction, and wear during rubbing through mathematical models has attracted a huge amount of interest already. As in the earlier study, the rubbings between the blade and bare casing are mainly focused on. However, in recent 10 years, the researches on the rubbings between the blade and coating casing are paid more attention. The modeling methods for blade–casing rubbings are classified according to three types of casings (bare casings, coating casings without considering wear, and coating casings considering wear, i.e., casings with abradable coating).

2.1 Models for the rubbing between blade and bare casing

According to different assumptions on the impact and friction, the modeling methods for the bare casing can be divided into the following six types:

• 1) Based on quasi-static assumptions, three normal rubbing force models considering collision energy conservation (NRFMCCEC) are developed [13–15]. These models can consider the physical dimensions of the blades such as the length and cross-sectional inertia moment of the blade and friction coefficient between the blade and casing [13–15]. Moreover, the effects of the rotational speed and radius of the disk are introduced into the rubbing models in [14, 15], and the casing stiffness is introduced into the rubbing model in [15].

• 2) The blade tip may melt and deposit a smear layer on the colder metal surface when a metal blade scrapes a metal casing, the tangential and normal rubbing forces under this condition are similar to the forces in a squeeze film bearing. Based on this analogy, smearing rubbing model (SRM) is proposed [16].

• 3) According to the assumptions that the normal rubbing force is or is not proportional to the penetration depth $\delta$, the linear or nonlinear spring models (LSM or NLSM) are defined [17–24], which are widely used to simulate the traditional rotor–stator rubbing.

• 4) For some typical rubbing forms, such as point or partial rubbing, the normal rubbing force is similar to the periodic pulse loading [15]. Aiming at this specific rubbing condition, many types of pulse forces, such as the half-sine wave pulse, rectangular pulse, and sawtooth pulse, are adopted to simulate the local blade–casing rubbing [25–30].

• 5) Considering the added constraint due to the blade–casing contact, constraint mechanical model for rubbing (CMMFR) is proposed [31].

• 6) Based on analytical or FE methods, blade–casing rubbing can be simulated as contact problems with impact, friction, and initial gaps [32–47]. This simulation method is called “rubbing model based on contact dynamics (RMBOCD).”

2.1.1 Normal rubbing force model considering collision energy conservation (NRFMCCEC)

Assuming the bare casing is rigid, Padovan et al. [13] modeled the blade as a cantilever beam and derived the relationship between the normal rubbing force $F_{\mathrm{n}}$ and the penetration depth $\delta$ under the single-blade rubbing condition (see Fig. 2).

$$\begin{aligned} F_\mathrm{n} =\frac{\pi ^{2}}{4}\frac{EI}{L^{2}}\frac{\frac{\pi }{2}\sqrt{\frac{\delta }{L}}}{\mu +\frac{\pi }{2}\sqrt{\frac{\delta }{L}}}, \end{aligned}$$

(1)

where $E, I, L, \delta$, and $\mu$ are Young’s modulus, the cross-sectional inertia moment of blade, the length of blade, penetration depth, and the friction coefficient, respectively.

Fig. 4

Blade-tip rubbing schematic under elastic casing condition [15]

Fig. 5

Normal rubbing force comparison of three models: a the effects of penetration depth, b the effects of rotational speed

Considering the centrifugal stiffening effects caused by the rotation of the blade, Jiang et al. [14] deduced a revised normal rubbing force between the rotating blade and rigid casing based on Padovan’s model. The revised expression is given as follows:

$$\begin{aligned} F_\mathrm{n}= & {} 2.5\frac{EI}{L^{2}}\frac{1.549\sqrt{\frac{\delta }{L}}}{\mu +1.549\sqrt{\frac{\delta }{L}}}\nonumber \\&+\,\frac{11}{56}\rho AL\varOmega ^{2}\left( {\frac{5}{22}L+\frac{35}{22}R_\mathrm{d}} \right) \frac{1.549\sqrt{\frac{\delta }{L}}}{\mu +1.549\sqrt{\frac{\delta }{L}}},\nonumber \\ \end{aligned}$$

(2)

where $\rho$ is the material density, A is the cross-sectional area of blade, $\varOmega$ is the angular speed, $R_{\mathrm{d}}$ is the radius of the disk, and the other parameters are the same as those in Eq. (1).

Fig. 6

Smearing blade-tip rubbing model [16]

Taking the effects of the deformation of elastic casing into account (see Fig. 4) and assuming the blade as a variable cross-sectional cantilever beam, Ma et al. [15] derived an improved normal rubbing force model based on Jiang’s model [14] and verified the improved model through the experiment. The expression of their model is given as follows:

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

(3)

where

$$\begin{aligned} \varGamma _0= & {} EI_0 \left( \frac{3}{L^{3}}+A_1\right) +\rho A_0 \left( \frac{81}{280}L+\frac{3}{8}R_\mathrm{d}+A_2\right) , \\ A_1= & {} -\frac{3\tau _\mathrm{b}}{4L^{3}}-\frac{9\tau _\mathrm{h}}{4L^{3}}+\frac{9\tau _\mathrm{b} \tau _\mathrm{h}}{10L^{3}}-\frac{9}{20L^{3}}\tau _\mathrm{b} \tau _\mathrm{h}^{2}\\&-\frac{3}{20L^{3}}\tau _\mathrm{h}^{3}+\frac{3}{35L^{3}}\tau _\mathrm{b} \tau _\mathrm{h}^{3},\\ A_2= & {} -\frac{369}{2240}L\tau _\mathrm{b} -\frac{57}{280}R_\mathrm{d} \tau _\mathrm{b} -\frac{369}{2240}L\tau _\mathrm{h}\\&-\frac{57}{280}R_\mathrm{d} \tau _\mathrm{h} +\frac{59}{560}L\tau _\mathrm{b} \tau _\mathrm{h} +\frac{141}{1120}R_\mathrm{d} \tau _\mathrm{b} \tau _\mathrm{h} .\\ \end{aligned}$$

Here, $\tau _\mathrm{b}$ and $\tau _\mathrm{h}$ are the taper ratios in the width and height directions of the blade; $\varGamma _1 =\frac{\varGamma _0}{k_\mathrm{c}}$; $\alpha =\frac{R_\mathrm{d} +L}{L}$.

Normal rubbing forces which are produced by the three models are compared in Fig. 5. The figure shows that all the three models have the same increasing trend with the increase in penetration depths (see Fig. 5a), and Jing’s model and Ma’s model have the same increasing trend with the increase in rotational speeds. There are some errors between Jiang’s model and Ma’s model. This is because the different deflection curves of cantilever beams are adopted, and the work caused by the tangential rubbing force is added in Ma’s model. Another reason is that some higher-order terms are neglected to obtain the analytical expression in Ma’s model. The comparison between Ma’s model and experiment is also carried out in [15], and the calculation errors of simulation and experiment are also evaluated.

For the above three models, the quasi-static assumptions are adopted during modeling, i.e., the kinetic energy of the blade and casing is ignored. This will lead to that a smaller simulated quasi-state rubbing force than the actual dynamic rubbing force. It should be noted that the analytical expression for the dynamic rubbing force is difficult to obtain due to the solution process. Under this condition, a method of polynomial fitting may be suitable. There are still some challenges in determining the local blade-tip rubbing force for the real blades with complicated sections.

Fig. 7

Sketch of the gap definition [17]

2.1.2 Smearing rubbing model (SRM)

The blade tip may melt and deposit a smear layer on the colder metal surface during a metal blade scraping a metal casing (see Fig. 6). Under this condition, the blade-tip tangential force can be evaluated by the blade moving velocity u in a viscous medium, and the normal force by the relative radial velocities between the blade tip and the substrate. Analogizing the tangential and normal rubbing forces to the forces in a squeeze film bearing, Kascak et al. [16] developed a smearing rubbing model, which assumes that b / a (the ratio of blade width to blade thickness) is large. The normal and tangential rubbing forces ($F_{\mathrm{n}}$ and $F_\mathrm{t})$ are

$$\begin{aligned} \left\{ {\begin{array}{ll} \left\{ {{\begin{array}{ll} F_\mathrm{n} =cvb\left( {\frac{a}{h}} \right) ^{3}&{} F_\mathrm{n} \le F_\mathrm{s} \\ F_\mathrm{n} =k_\mathrm{c} (r-C)&{} F_\mathrm{n} >F_\mathrm{s} \\ \end{array}} } \right. \\ F_\mathrm{t} =\frac{cuab}{h} &{} \\ \end{array}}, \right. \end{aligned}$$

(4)

where c is the viscosity of the molten metal, $F_\mathrm{s}$ is the force supported by the substrate, $k_\mathrm{c}$ is the casing stiffness, r is the blade-tip radial displacement, C is the radial clearance between the blade tip and casing, and other parameters are shown in Fig. 6.

There are some difficulties in applications of SRM: 1) The thickness of the molten metal is hard to quantify although it is closely related to the rubbing process; 2) the viscosity of molten metal is also hard to evaluate; 3) how to determine the force supported by the substrate $F_\mathrm{s}$ is not provided in [16], and only the condition of $F_\mathrm{n} \le F_\mathrm{s}$ is adopted in [16]. Alternatively, some empirical values determined by the experimental data from steady in-house rub tests are given in [16], such as $h=2\,\upmu \hbox {m}$ and $c=10$–$50\,\hbox {MNs/m}^{2}$.

2.1.3 Linear or nonlinear spring models (LSM or NLSM)

A rubbing model is widely used in rotor–stator rubbing field, which assumes that the normal rubbing force is proportional to the penetration depth. Many researchers [17–24] have also adopted this model to simulate the blade–casing rubbing because it is simple but effective under some conditions.

By determining the gap function (see Fig. 7), Parent et al. [17] adopted a linear spring model to simulate the blade–casing rubbing.

$$\begin{aligned} \left\{ {\begin{array}{lll} F_{\mathrm{n}j} =k_\mathrm{c} g_j \\ g_j =L+R_\mathrm{d} +g_0 +u(\alpha _\mathrm{i},t)-\sqrt{{B}'_j (1)^{2}+{B}'_j (2)^{2}} \\ \end{array}} \right. , \end{aligned}$$

(5)

where the subscript j denotes the jth blade. Based on the same calculation method of the normal rubbing force [17], Parent et al. [18] developed a three-dimensional (3D) contact detection law (see Fig. 8), which combines 3D model kinematics with the 3D local geometry of the contact area. Assuming that the blades are rigid and adopting linear spring model to describe normal rubbing force, Lawrence et al. [19] and Thiery et al. [20] analyzed the vibration response of the rotor under the blade–casing rubbing. Based on the assumption of elastic blades, Zhao et al. [21] described the normal blade–casing rubbing force using a linear spring model.

Fig. 8

3D contact detection in Ref. [18]

Considering the different vibration displacements of the blade tip along the whole blade width (see Fig. 9), Yuan and Kou [22] adopted multiple linear springs to simulate the normal rubbing force $F_\mathrm{n}(y)$.

$$\begin{aligned} F_\mathrm{n}(y)=\left\{ {{\begin{array}{llll} k_\mathrm{c} ({u}'-e) &{} {u}'>e \\ 0&{} {u}'\le e \\ \end{array}} } \right. ,\;{u}'=u(a,y)\nonumber \\ \qquad \qquad \quad \qquad \qquad \qquad +\,x_\mathrm{D} \cos \varOmega t+z_\mathrm{D} \sin \varOmega t, \end{aligned}$$

(6)

where u(a, y) is the x-direction displacement, $x_\mathrm{D}$ and $z_\mathrm{D}$ are displacement excitations in x- and z-directions, $k_\mathrm{c}$ is the contact stiffness, e is the initial clearance between the plate and casing, $\varOmega$ is the angular speed (rad/s). In order to simulate the rubbing between single-blade/multiple-blade and casing, Petrov [23, 24] developed contact interface elements based on a linear spring model similar to Eq. (6) and a nonlinear spring model given by a table of values with high-order polynomial dependency.

Fig. 9

Blade–casing rubbing schematic [22]: a contact model under thermal shock, b local contact model

2.1.4 Pulse loading model (PLM)

For some typical rubbing forms, such as point or partial rubbing, the normal rubbing force is similar to a periodic pulse loading [15]. For this specific rubbing form, many researchers simulated the local blade–casing rubbing by applying pulse forces on the blade tip [25–30], which is called pulse loading model in this paper. Sinha [25, 26] proposed several mathematical expressions for the PLM, such as half-sine wave pulse, rectangular pulse, and sawtooth pulse. Turner et al. [27, 28] adopted a similar half-sine wave pulse to simulate the blade–casing rubbing (see Fig. 10a). In these models, the unit pulse load can be expressed as follows:

$$\begin{aligned}&\varPhi (t)=\frac{4}{\varDelta t}\left( {1-\frac{t}{\varDelta t}} \right) t,\nonumber \\&\quad \varDelta t=\frac{2\cos ^{-1}\left( {\frac{R_\mathrm{c}^2 +(G_\mathrm{r} +\delta )^{2}-R_\mathrm{g}^2}{2R_\mathrm{c} (G_\mathrm{r} +\delta )}} \right) }{\varOmega }, \end{aligned}$$

(7)

Fig. 10

a Pulse load definition, b rubbing duration determination [28]

where the rubbing duration $\varDelta t$ can be determined by the penetration depth $\delta$ and angular speed $\varOmega$ (see Fig. 10b). The ultimate normal blade-tip rubbing force can be expressed as the product of a space-varying component F(x) and a time-varying component $\varPhi (t)$.

$$\begin{aligned} {{\mathbf {P}}}(t)=\varPhi (t)\cdot F(x)=\frac{4}{\varDelta t}\left( {1-\frac{t}{\varDelta t}} \right) t\sum _{i=1}^n {F_i} , \end{aligned}$$

(8)

where $F_{i}$ is the ith node load vector and n is the loaded node number. Kou and Yuan [29] adopted two kinds of functions: sine pulse function and continuous sine function to simulate the blade-tip rubbing. Ma et al. [30] determined the maximum normal rubbing force between the blade and elastic casing using Eq. (3).

2.1.5 Constraint mechanical model for rubbing (CMMFR)

Not only rubbing forces but also constraints will be generated during blade–casing rubbing. The heavier the rubbing is, the deeper the blade penetrates into the casing and the stronger the constraint will be. The constraints vanish while the blade separates from the casing, and contact and separation occur alternately during the rubbing process. Considering the constraint caused by rubbing, Ma et al. [31] proposed a mechanical model (see Fig. 11), of which the expression is

$$\begin{aligned} \left\{ {\begin{array}{llll} M\ddot{z}+D\dot{z}+Kz=mr\varOmega ^{2}\mathrm{e}^{(\hbox {j}\varOmega t+\varphi )} &{} |z|<c \\ M\ddot{z}+(D+{D}')\dot{z}+(K+{K}')z\\ \qquad =mr\varOmega ^{2}\mathrm{e}^{(\mathrm{j}\varOmega t+\varphi )}&{} |z|>c \\ \end{array}} \right. , \end{aligned}$$

(9)

where $z=x+\hbox {j}y, x$ and y are the displacements in x- and y-directions; M, K, and D are the mass, the stiffness, and damping of the system before rubbing; $K+K$’ and $D+D$’ are the stiffness and damping of the system after rubbing.

Fig. 11

Constraint mechanical model for rubbing [31]

Fig. 12

a Bladed disk schematic, b the ith predicted clearance between blade i and casing [35]

2.1.6 Rubbing model based on contact dynamics (RMBOCD)

The rubbing between the blade and casing can be viewed as contact problems with gap, and the impact and friction due to rubbing can be accurately described by contact dynamics. In recent years, FE theory with unilateral contact and friction conditions has been widely used to simulate the blade–casing rubbing [32–47]. Some researchers adopted lumped mass [32], generalized spring [45], beam [33–38], and solid models [39–42] to simulate the blade, employed curved beam [36–38, 43], thin-walled cylindrical shell [45–47] to simulate the casing, so as to reproduce the rubbing process between the blade and casing.

Fig. 13

a 2D FE models of the blade–casing system [36], b 2D FE models of the shaft-disk-blade-casing system [38]

Fig. 14

a 3D FE models of the disk-blade-casing coupling system [40], b impeller–casing rubbing schematic [41]

Some researchers adopted analytical method to model the rotor–blade systems, and simulated the blade–casing rubbing using Lagrange multiplier method [32] and penalty method [33, 34]. Assuming the casing as an elastic ring and ignoring the effect of the friction, Lesaffre et al. [32] developed a dynamic model of a bladed rotor system using Lagrange equation and regarded the blade–casing dynamic contact problem as a static contact problem in some rotational speed ranges. Assuming the rotating blades as cantilever beams with uniform section and determining the radial rubbing load from the rigid casing using the penalty method, Sinha [33, 34] developed a dynamic model of a bladed flexible rotor–bearing system. His work suggests that during the hard-rubbing against the casing, the normal rubbing load is similar to Hertzian contact [33].

Simplifying the blade–casing rubbing as a 2D in-plane contact problem, the blade–casing rubbing is described using Lagrange multiplier method in [35–38]. Ignoring the friction effect and applying impenetrability condition by Lagrange multiplier method, Legrand et al. [35] developed a two-dimensional (2D) model of outer casing and the bladed disk (see Fig. 12a) and studied the modal interaction phenomenon between the bladed disk and casing (see Fig. 12b). By simplifying the blade and casing as a straight beam and a curved beam, respectively, Legrand et al. [36], Batailly et al. [37], and Salvat et al. [38] adopted an explicit time integration method, which combines the Lagrange multiplier method with Coulomb friction law, to deal with blade–casing interaction (see Fig. 13). Based on an FE model of a coupling system of the bladed disk and flexible casing, Almeida et al. [39] simulated the blade–casing interaction using an explicit time marching scheme with the Lagrange multiplier method, and described the friction and wear using Coulomb’s friction law and Archard’s laws, respectively.

Recently, 3D contact schemes have also been used to simulate the blade–casing rubbing [40–43]. Legrand et al. [40] developed a full 3D contact algorithm to simulate the blade–casing rubbing (see Fig. 14a), which combines with a smoothing procedure including bicubic B-spline patches and a Lagrange multiplier. Using a similar method, Batailly et al. [41, 42] studied the rubbing between the blade and oval casing in a centrifugal compressor (see Fig. 14b). By changing the motion equation with contact constraint into a linear complementarity problem form and solving it in a time-discretized period by using a set of algebraic equations, Meingast et al. [43] proposed a method to obtain periodic solutions for blade–casing systems with blade-tip rubbing.

Some contact elements in commercial FE softwares such as NASTRAN [19], ANSYS [44], Samcef [45], and LS-DYNA [46, 47] are also used to simulate blade–casing rubbing. Using ANSYS software, Ma et al. [44] established an FE model of a shaft–disk–casing coupling system where point-to-point contact elements were adopted to simulate the blade–casing rubbing, and the augmented Lagrangian method and the Coulomb friction model were used to handle contact constraint conditions and the friction between the blade and casing. Arnoult et al. [45] proposed two methods (method 1: a geometrical approach and method 2: a simplified contact element using a penalty function) to simulate the contact phenomena using an implicit FE code, which have been included in the FE code Samcef. Based on LS-DYNA software, Garza [46] and Arzina [47] simulated the blade–casing rubbing using contact elements and analyzed the complicated vibration responses caused by the blade–casing rubbing.

Fig. 15

a Rubbing between the blade and coating casing, b blade damage and coating wear [1]

2.2 Models for the rubbing between blade and coating casing without considering wear

Taking the structural stiffness of both the flexible blade and casing, and local stiffness of coating painted both the blade tip and casing, and energy loss during rubbing into consideration, Cao et al. [48] proposed a hysteretic contact force model (HCFM) to describe the mechanisms of rubbing between the rotor blade and casing of an aeroengine. In their model, two important parameters: contact stiffness and nonlinear hysteretic damping are involved, where the contact stiffness is composed of two parts, i.e., the structure stiffness of rotor components (blades and casing), and local contact stiffness of soft or hard coating, while the nonlinear hysteretic damping is adopted to describe the energy loss during rubbing. The general expression of HCFM is

$$\begin{aligned} F=\frac{k_\mathrm{c} k_\mathrm{b} k_\mathrm{h} \varDelta _\mathrm{h}^{0.5}}{k_\mathrm{c} k_\mathrm{b} +(k_\mathrm{c} k_\mathrm{h} +k_\mathrm{b} k_\mathrm{h})\varDelta _\mathrm{h}^{0.5}} \varDelta \left[ {1+\frac{3(1-e^{2})\dot{\varDelta } }{4{\mathop {\varDelta }\limits ^{\cdot -}}}}\right] , \end{aligned}$$

(10)

where $k_\mathrm{c}, k_\mathrm{b}, k_\mathrm{h}, e, \varDelta$ and ${\mathop {\varDelta }\limits ^{\cdot -}}$ are structural stiffness of casing, structural stiffness of blade, Hertz contact stiffness, coefficient of restitution, total elastic deformation of the system, initial embedding velocity, respectively. The expression of $\varDelta _\mathrm{h}$ is

$$\begin{aligned} \varDelta _\mathrm{h}=\left[ {\frac{1}{6}\frac{4^{1/3}f(\varDelta )}{k_\mathrm{h}} +\frac{1}{6}\frac{k_\mathrm{s}^2 4^{2/3}}{k_\mathrm{h} f(\varDelta )}-\frac{1}{3}\frac{k_\mathrm{s}}{k_\mathrm{h}}}\right] ^{2}, \end{aligned}$$

(11)

where $k_\mathrm{s}$ is the structural stiffness of rotor system, and the expression of $f(\varDelta )$ is

$$\begin{aligned} f(\varDelta )=\left\{ {k_\mathrm{s} \left[ {27\varDelta k_\mathrm{h}^2 -2k_\mathrm{s}^2 +3\sqrt{3}\sqrt{\varDelta (27\varDelta k_\mathrm{h}^2 -4k_\mathrm{s}^2)}k_\mathrm{h}} \right] } \right\} ^{1/3}.\nonumber \\ \end{aligned}$$

(12)

2.3 Models for the rubbing between blade and coating casing considering wear

In order to avoid catastrophic failures due to the blade tip penetrating into a bare metal casing and adjust the blade–casing operating clearances, abradable coating has been widely applied (see Fig. 15a). However, some obvious blade damage and coating wear may occur during the rubbing between the blade and coating casing from experimental observations [1] (see Fig. 15b).

Fig. 16

Analogy between abradable coating wear and metal machining: a Ref. [16], b Ref. [49]

In order to elucidate the blade-tip rubbing and coating wear mechanisms, it is very essential to develop some mathematical models to reproduce the rubbing and wear processes. So far, five kinds of modeling methods for the coating casing are widely used [16, 49–63].

• 1) When the blade scrapes the coating, small coating particles are removed after each rubbing, which is similar to metal machining. Based on this idea, some models are proposed, where these models are called RMBAMM (rubbing model by analogizing metal machining) [16, 49, 50].

• 2) Considering the linear wear of the coating, a new modeling method, called RMBOLWL (rubbing model based on linear wear law), is developed [51, 52].

• 3) Based on a piecewise linear plastic constitutive law with which the current abradable liner profile can be obtained using a time-stepping method, a macroscopic model of material removal is proposed [53–58]. This model is called RMBOPCL (rubbing model based on plastic constitutive law), where three main parameters: Young’s modulus, the plastic modulus, and the yield limit are needed.

• 4) The RMBOAL (rubbing model based on Archard’s Laws) [59, 60] describes the rubbing between the blade and coating casing using an explicitly time marching scheme with the Lagrange multipliers method, and formulates friction and coating wear using Coulomb’s and Archard’s laws.

• 5) Based on explicit dynamic FE methods or commercial FE software such as LS-DYNA, the blade–casing rubbing and coating wear can also be simulated using elastic–plastic material models of the blade, coating, and casing [61–63].

2.3.1 Rubbing model by analogizing metal machining (RMBAMM)

When the blade scrapes the abradable coating, small coating particles are removed after each rubbing, which is similar to metal machining (see Fig. 16a). In view of this idea, Kascak et al. [16] developed an abradable rubbing model, which can be expressed as:

$$\begin{aligned} \left\{ {\begin{array}{l} F_\mathrm{t} =(r-C)Ub \\ F_\mathrm{n} =k_\mathrm{c} (r-C) \\ \end{array}} \right. , \end{aligned}$$

(13)

where U is the energy per volume of material removed per blade, and $k_\mathrm{c}$ is the casing stiffness. Assuming the process of blade-tip scratching the abradable coating as a milling processes (see Fig. 16b), Salvat et al. [49] qualitatively analyzed the removal process of abradable coating on the basis of delay differential equations. Based on the cluster treatment of characteristic roots (CTCR), Olgac et al. [50] evaluated the system stability caused by the blade–casing rubbing by analogizing the rubbing dynamics with machine tool chatter.

2.3.2 Rubbing model based on linear wear law (RMBOLWL)

Williams [51] presented a new model to simulate blade–casing rubbing which includes a detailed interaction model considering the linear wear (see Fig. 17). In his model, the free incursion depth $e_\mathrm{free}$ is a key parameter, whose expression is

$$\begin{aligned} e_{\mathrm{free}} =e_{\mathrm{wear}} +\varDelta e_{\mathrm{wear}} (P_\mathrm{r})\cdot \varDelta t+e_{\mathrm{blade}} (P_\mathrm{r}), \end{aligned}$$

(14)

where $e_\mathrm{wear}$ is the cumulative linear wear loss at the initial time step, $\varDelta e_{\mathrm{wear}}$ is the increased linear penetration depth, $e_\mathrm{blade}$ is the blade radial compression, and $P_\mathrm{r}$ is the radial load. Based on Williams’s model [51], Ronchi et al. [52] discussed the effects of blade mode shape on the rubbing severity.

Fig. 17

Schematics of liner wear model and radial load balance [51]

2.3.3 Rubbing model based on plastic constitutive law (RMBOPCL)

Based on a piecewise linear plastic constitutive law, Legrand et al. [53] and Batailly et al. [54–58] developed a blade–casing rubbing model considering the effects of the material removal (see Fig. 18). In their model, three main parameters: Young’s modulus, the plastic modulus, and the yield limit are used to handle the coating wear, and the real-time wear profile of the coating can then be obtained using a time-stepping method.

Fig. 18

Schematic of the interaction between the blade and casing with abradable coating [53]

2.3.4 Rubbing model based on Archard’s Law (RMBOAL)

Almeida et al. [59] simulated the rubbing between the blade and coating casing using a simplified FE model of a rotating bladed disk and a flexible casing. In their model, the contact algorithm adopts an explicitly time marching scheme with the Lagrange multipliers method, and friction and coating wear are formulated using Coulomb’s and Archard’s laws [60].

2.3.5 Rubbing model based on explicit dynamics algorithm (RMBOEDA)

Based on an FE model with contact elements, Beaupain et al. [61] analyzed the wear degree of the abradable coating, the normal contact force, and the vibration response of the blade during the blade–casing rubbing interaction. Neglecting the effects of the friction damping between the blade and casing, and the influences of air pressure and gravity, Chai et al. [62] simulated the rubbing between the blades and casing with abradable coating by using LS-DYNA software. Wei et al. [63] simulated the blade–casing rubbing for many kinds of accelerated speed (see Fig. 19) without considering the effects of aerodynamic force in the process of speed acceleration.

Fig. 19

a One scratch in the inlet duct, b two scratches stacking in the inlet duct, c the picture of the actual fault [63]

2.4 Summary and discussion on rubbing models

The mathematical models for simulating the rubbing between the blade and bare casing or coating casing regardless of wear are summarized in Table 1. Some observations from Table 1 are discussed as follows:

• (1) Different rubbing models all have their respective assumptions and application range. For NRFMCCEC, Padovan’s model and Jiang’s model can simulate the elastic interaction between the blade and rigid casing in the ground gas turbine [13, 14]; Ma’s model may be applied for the elastic casing in aeroengine [15]; however, its expression is derived by quasi-state assumption, and the dynamic collision process and the effects of coating material need to be considered in the future. For HCFM, structural stiffness of both the flexible blade and casing, and the local stiffness of coating painted on both the blade tip and casing, and energy loss during rubbing are considered [48], which enables this model to simulate the blade–casing rubbing more accurately in aeroengine. However, this model may meet with some difficulties in accurately estimating the values of model parameters, such as the coefficient of restitution and embedding velocity.

• (2) SRM can be fit for serious friction-dominated rubbing where the blade tip melts due to friction-induced high temperature. LSM can be accurate under some special conditions, such as smaller penetration depths for the elastic casing. PLM can easily deal with local rubbing condition for complicated real blade, avoiding the complicated rubbing detection and nonlinear iterative solution. However, the magnitude of normal rubbing force needs to be evaluated by other model, such as NRFMCCEC and LSM; the contact time for PLM also needs to be estimated; moreover, PLM can only simulate the line vibration and cannot simulate the nonlinear vibration caused by rubbing. It is difficult to determine the added stiffness and damping for CMMFR because these parameters may be time-varying with the changing rubbing level.

• (3) RMBOCD is relatively closer to the actual rubbing process than other models because it only needs to know the geometries, material properties, and kinematic parameters of contact bodies and does not need to introduce too many parameters. However, the model needs a complicated nonlinear iterative solution which is extremely time-consuming and less efficient compared with other models.

Table 1 Mathematical models of rubbing between the blade and bare casing or coating casing regardless of wear

The mathematical models for simulating the rubbing between the blade and casing with the abradable coating are summarized in Table 2. Some observations from Table 2 are discussed as follows:

Generally, RMBAMM can be used to simulate the blade milling the abradable coating, which is fit for non-separated blade–coating interaction. The researches [49, 50] of this aspect mainly focus on the coating removal-induced instability, and improvements need to be carried out, such as considering the effects of the friction and the vibration of the flexible casing. RMBOLWL shows a good agreement with the experiment results, and the model may be acceptable for the blade displacement in the range of elastic deformation [51]. RMBOPCL is also well coincided with the test results [53] while the disadvantage of this model is not easy for accurately determining the material parameters of abradable coating, such as Young’s modulus, the plastic modulus, and the yield limit. RMBOAL can only deal with the elastodynamic contact with friction and wear [59], and is unfit for plastic contact and wear conditions. RMBOEDA is very close to the real situations [62], however, the poor computation efficiency and the bad simulation convergence could be its main weakness.

Table 2 Mathematical models of rubbing between the blade and casing with abradable coating

3 Simulation studies of blade–casing dynamic characteristics

Using above blade–casing rubbing models, complicated dynamic characteristics of blades, bladed disks, and rotor–blade systems are studied. The summary is then classified into two parts corresponding to the bare and coating casings. For the bare casing, the following contents are focused on.

• 1) Assume the blade is rigid, single-blade- or multiple-blade-induced vibration responses of the rotor systems are studied by transferring the rubbing load to the shaft [13, 16, 20, 33]. The effects of some key parameters such as unbalance, contact stiffness, rotational speed and friction on the system stability, nodal diameter interaction, rotor transient responses, and rotor whirl are evaluated.

• 2) For the blade or blade–disk systems, vibration characteristics of the blades are studied, such as model interaction phenomenon [36], rubbing-induced instability [32], nonlinear vibration behaviors [22, 29], and amplitude amplification phenomenon [30].

For the coating casing, studies are focused on the following topics:

The relationship between the instability lobes and the bending and torsion modal responses [49], and the coupling mechanisms of the critical rotating velocities and blade eigenvalues [53], are studied. The couplings of the coating material properties, casing deformation, and blade physical dimension were studied in [54]. The effects of aerodynamic loading on the blade vibration and coating wear are discussed in [57, 58].

3.1 Simulation studies of dynamic characteristics for bare casing

3.1.1 Vibration response based on NRFMCCEC, SRM, LSM or NLSM, PLM, and CMMFR

Based on NRFMCCEC [Eq. (1)], the effects of the friction coefficient, imbalance, and blade stiffness on the vibration responses of the rotor and blade such as backward whirl and blade stress were discussed in [13]; based on NRFMCCEC, the differences between the quasi-state rubbing force [Eq. (3)] and simulated dynamic rubbing force were evaluated in [15].

Based on the smearing rubbing model [Eq. (4) for SRM] and abradable rubbing model [Eq. (13) for RMBAMM], the effects of the blade–casing rubbing on the rotor whirl are discussed in [16].

On the basis of LSM for 2D [17] and 3D contact detection [18], the blade–casing rubbing phenomena were simulated using a coupling model of flexible bladed rotor and flexible casing. For the 2D contact detection, a combination is generated with one-nodal-diameter (1ND) interaction related to the whole model and a 2ND interaction involving the local models of the blades and casing [17]; the results based on 3D contact detection show that the inclination of the casing inner surface has a strong influence on the rubbing detection, and the 3D formulation affects the system stability [18]. Based on the similar LSM, the effects of rotational speeds, contact stiffness, damping, and eccentricity on the dynamic responses of rotor–blade systems with blade-tip rubbing are studied [20]; the influences of friction coefficient, contact stiffness, and thermal transient on the vibration response of the plate are analyzed [22]. Based on LSM and NLSM models, Petrov [23] pointed out that the odd harmonics can be excited due to blade–casing rubbing; nonlinear vibration responses obtained from LSM agree well with those from NLSM under the low rotational speed range. However, for both models, there exists a large difference under the high rotational speed range.

Based on PLM, the effects of the friction, amplitude of the rubbing force, and application time of the rubbing load on the system vibration responses were analyzed in [25, 29]; the impacts of Coriolis forces due to axial–lateral coupling and twist angles of blades on the rubbing-induced vibration were discussed in [26]; the errors between the simulation using PLM and experiment were evaluated in [27, 28]; based on a 3D FE model of a bladed disk structure, the effects of the rubbing, which is described using PLM, on the contact behaviors of dovetail interface were studied in [30].

The rubbing between the rotor–blade and casing in aeroengines is simulated using CMMFR in [31]. The results showed that the responses with rubbing constraint are closely related to the rubbing position and the mode shape of the rotor, and the rubbing can lead to quasi-periodic motions [31].

3.1.2 Vibration response based on RMBOCD

Lesaffre et al. [32] established a model of a flexible bladed rotor with rubbing and pointed out that the steady balanced static contact states of the system are the function of the rotational speed. An unstable phenomenon around the critical speed of the stator can be observed even under frictionless sliding condition. Another paper published by Lesaffre et al. [64] indicates that the radial rotating loads applied on an elastic ring can lead to divergence instability and post-critical coupling modes; in addition, the beam rubbing the ring due to lateral motion can also excite coupling modes and cause locus-veering phenomenon. Based on a dynamic model of a bladed rotor system supported by multiple bearings, Sinha [33] pointed that a local blade-tip rubbing is nonlinear in nature under hard-rubbing because the normal rubbing force is similar to Hertzian contact force. On the basis of a non-symmetrical bladed rotor system where the blades are simulated by pre-twisted thin shallow shells, Sinha [34] indicated that the sudden rubbing load can increase by an order of magnitude over the unbalance force.

Model interaction phenomenon between the blade and casing has also been discussed, and a dangerous case of traveling wave speed coincidence is stated as $\omega _\mathrm{c} =n_\mathrm{d} \varOmega -\omega _\mathrm{bd}$, where $\omega _\mathrm{bd}$ and $\omega _\mathrm{c}$ are the natural frequencies of $n_\mathrm{d}$-nodal models of the bladed disk and casing, and $\varOmega$ is the rotational speed of the disk [35]. A modal coincidence when the vibrations of blade and casing have a k-nodal-diameter traveling wave feature for the axis-symmetry structure is studied in [36]. In order to improve the computational efficiency during rubbing simulation, the work [36] is extended by developing reduced-order models on the basis of two different component mode synthesis methods [37]. A further improvement for the model [37] considering the effects of the rotor whirl was performed in [38]. Effects of coating wear are evaluated, and some spectrum features are shown based on a simplified finite element model of a rotating bladed disk and a flexible casing [39].

Legrand et al. [40] pointed out that many critical velocities should be noticed, and the critical areas for stress levels are situated on the trailing and leading edges mainly near the blade root. The rubbing between the centrifugal compressor blade tip and rigid casing was firstly studied in [41, 42]. These studies show that rubbing phenomena in centrifugal compressors are similar to those in axial compressors [41], and super-harmonics appear due to nonlinear nature of unilateral contact, and many critical interaction velocities cannot be evaluated by the traditional linear model [42].

Based on commercial FE software ANSYS, Ma et al. [44] analyzed complicated vibration responses caused by blade–casing rubbing under two cases and displayed different vibration behaviors under two excitation conditions. Using LS-DYNA software, Garza [46] studied the rubbing between the blade and casing by simulation and experiment, and pointed out that (1) the blade first rubs the casing at about 75 % chord; (2) the radial load on the blade tip decreases with the rubbing progresses; (3) the blade-tip stress oscillates, which shows a modal vibration or chattering type of response during the rubbing; (4) vibration responses for the simulation and experiment during the rubbing are very similar; however, there are some big errors in predicating the vibration level. Adopting LS-DYNA as well, Arzina [47] displayed that torsional vibration can be observed in the radial direction between the leading edge and trailing edge, and the nonlinear vibration may lead to resonant characteristic of the blade at non-resonant frequencies.

3.2 Simulation studies of dynamic characteristics for coating casing

Based on RMBAMM, Salvat et al. [49] evaluated potentially dangerous rubbing regions and pointed that instability lobes are related to both the bending and torsion modal responses; Olgac et al. [50] provided a selection criterion for rub-strip stiffness and operational speed to guarantee a stable rubbing.

Adopting RMBOPCL, Williams [51] showed that the short-duration instability may appear when several conditions happen simultaneously, and the radial vibration in the fundamental modes has a violent effect on the lateral vibration response; Ronchi et al. [52] indicated that when the excitation is applied on just one tip node, the blade response is always stable for forward mode shapes, while it can turn to instability for backward modes.

On the basis of RMBOPCL, Legrand et al. [53] pointed out that (1) the critical rotating velocities are related to the blade eigenvalues; (2) the blade vibration levels affect the mechanical characteristic of the abradable coating; in turn, the coating may cause more severe blade vibrations; (3) the tangential rubbing force slightly softens the rubbing conditions and leads to the appearance of some unexpected blade vibrations. Batailly et al. [54] indicated that the simulated wear profiles are very similar to the measured ones, and the sudden jumps in the measured vibration amplitude may be related to the desynchronization of the blade on its first flexural mode, and the rubbing phenomenon is closely related to three factors: blade physical dimension, coating material properties, and a global casing deformation. Batailly and Legrand [56] observed low vibration amplitudes caused by a non-synchronization of the blade vibration with deformed casing, and underlined the effects of the initial clearance on the blade vibration response near a critical rotating frequency. The effects of aerodynamic loading on the blade–casing rubbing were also considered in [57, 58]. Batailly et al. [57] pointed that unexplained experimental vibratory behaviors are related to the vacuum conditions of the experiment; Batailly et al. [58] first emphasized that iterated profile advantageously leads to a significant drop of the maximum static stress with the blade.

Based on RMBOAL, Almeida et al. [59] pointed out that (1) the friction can cause system unstable when a sideband of the excitation frequency coincides with a one-nodal-diameter mode of the bladed disk; (2) wear can lead to a vibration reduction, when the abradable material is removed during a wear process; (3) the number of coating wear lobes is related to the ratio between the vibration frequency of the blades and the rotational speed.

Table 3 Typical rubbing features for bare casings

Based on LS-DYNA software, Chai et al. [62] indicated that the rubbing between the wide chord swept fan blade tip and the coating casing can lead to the local buckling of the blade-tip leading edge, and acceleration loads can cause considerable flexural and torsional deformation of the blade; Wei et al. [63] showed that when the starting acceleration speed, blade-tip clearance and axial length of the case coating are insufficient at the same time, the blades can touch and scrape the casing, and blades can make plastic deformation, both of the coating and air inlet may be damaged.

Table 4 Typical rubbing features for coating casings

3.3 Summary and discussion on simulated dynamic features

For the numerical simulations, some typical rubbing features for the bare and coating casings are summarized in Tables 3 and 4. These features are discussed as follows:

For the numerical simulation on the rubbing-induced vibration, earlier studies were mainly based on lumped-mass models of rotor systems, and adopted numerical methods, and focused on the rubbing-induced system vibration by applying the blade-tip rubbing loads on the rotor [13, 16]. With the development of the research, many researchers developed both blade models [22, 25, 26, 29, 39, 64] and rotor–blade models [17, 18, 32–34] based on beam or plate assumptions using the semi-analytical methods. In this research field, Sinha [33] firstly developed a rotor–blade model using Euler–Bernoulli beam, and some subsequent in-depth studies based on this reference, such as Sinha [25, 26, 34], Lesaffre et al. [32, 64], Parent et al. [17, 18], Kou et al. [29], and Yuan et al. [22], are carried out. Semi-analytical method has a good computation efficient and can qualitatively analyze the rubbing mechanism; however, this method meets with many difficulties in dealing with complicated blade structures and coupling of multiple connected components.

In the practical engineering applications, FE method is widely used to simulate real bladed disk structure from simple 2D blade–casing system models [36–38] to complicated 3D models [23, 24, 40–42, 46, 47]. Considering the effects of the rotor whirl, FE models of shaft–disk–blade–casing systems were developed in [38, 44] based on beam and shell elements. Generally, there exists an abradable coating on the casing, and many new published references [49–54, 56–62] mainly focus on the rubbing between the blade and abradable coating. During the numerical study by FE method, some key technical problems are solved, such as algorithms for contact treatment, component mode synthesis methods for obtaining reduced-order dynamic model, wear simulation for abradable coating, and convergence and stability in solving nonlinear equations due to the blade–coating rubbing. A group from McGill University, such as Legrand et al. [40, 53], Batailly et al. [37, 41, 42, 54–58], and Salvat et al. [38, 49], made great contributions in these aspects. These contributions are summarized as follows:

• 1) For 3D FE model, a functional smoothing technique of the contact surface by introducing bi-cubic B-splines is successfully used to accurately determine the exact distances between the blade and casing [40].

• 2) The Craig–Martinez method and Craig–Bampton method are compared, and the application ranges of both are pointed out, namely the former is well fit for 2D FE model, and the later for 3D FE model [40].

• 3) A one-dimensional two-node bar element based on a nonlinear plastic constitutive law is developed to reproduce the coating removal. The wear model involving three parameters: Young’s modulus E, the plastic modulus K, and the elastic limit $\sigma _\mathrm{Y}$ shows a good agreement with the experiment [54].

• 4) Explicit central difference method is proposed to solve the numerical equations, and the time step $\varDelta \, t$ and the reduction parameter are firstly evaluated to ensure the accuracy of the simulation [38].

4 Experimental studies of blade–casing dynamic characteristics

For the bare casing, early rubbing experiments were carried out based on some test rigs. In these setups, the blades and casings are, respectively, simplified as pendulum arms and flat plates in [65, 66]; rotor–blade systems of which the blade is simplified as the rectangle beam were operated in a lower rotational speed range (1000–2000 rev/min) [15, 67–69]; more real rotor–blade systems with real blades were operated in a higher rotational speed range (20,000–58,500 rev/min) [70–74].

For the casing with abradable coating, most studies focused on the coating wear mechanisms such as adhesive transfer, cutting, and melting. In these studies, simplified blades are used [75–80, 85–88]. In order to achieve a higher blade-tip velocity, two other simplified test rigs are, respectively, adopted in [89–91] and [92, 93] in which there is a significant difference between the adopted blade and real blade. Some relatively real test rigs are used in [81–84], which are capable of considering the coupling effects of the real blade vibration and coating wear, and can reproduce the blade–coating rubbing to a large extent.

4.1 Experimental studies of dynamic characteristics for bare casing

Simplifying the blade as a single pendulum and the casing as a plate and considering the related thermal and mechanical phenomena, Kennedy [65] performed both experimental and analytical researches on the temperatures and deformations under the effects of rubbing between a steel blade tip and a flat copper plate. Wang et al. [66] carried out an experimental study on the rubbing between a rotating blade (a pendulum arm) and a static casing (a plate) and analyzed the influences of the penetration depth and sliding speed on the rubbing force. By simplifying the blade as a rotating uniform beam and the casing as an arc structure adjusted between $0^\circ$ and $90^\circ$ in the blade motion direction, Jiang [67] and Ahrens et al. [68] measured contact forces and the contact duration by experiment. Considering the effects of elastic casing, Ma et al. [15] established a blade–casing test rig and analyzed the normal rubbing force under different casing materials at lower speeds (1000, 1500, and 2000 rev/min). Abdelrhman et al. [69] established an experimental test rig of a multistage rotor system, which consists of three rows of rotor blade each with 8, 11, and 13 blades, and extracted the rubbing fault features using casing vibration signals at the rotational speed of 1200 rev/min.

In the previous experimental researches, blade–casing rubbing experiments were carried out using a simple blade structure (uniform beam) under lower rotational speeds [15, 65–69], which has a big difference with the real systems in both speed and blade structure. In order to make up for these deficiencies, Gas Turbine Laboratory in the Ohio State University established an in-ground spin-pit facility (SPF) whose maximum speed can reach 20,000 rev/min. It can reproduce the blade–casing rubbing at operating speeds of the engine [70, 71]. Padova et al. introduced the SPF in detail in their paper [70]. In their experiment, a $90^\circ$ sector casing is forced to contact a single-blade tip under different rubbing levels; the rubbing force is measured by three triaxial force sensors (Kistler Type 9047B); rubbing-induced vibrations are monitored by five accelerometers; the blade deformation, friction heat, and contact times can, respectively, be measured by strain gauge, thermocouple, and an electric circuit. In another paper, Padova et al. [71] studied the metal-to-metal contact due to sudden penetrations with different incursion depths from 13 to $762\,\upmu \hbox {m}$. Using the same test rig in [70, 71] but a segmented shroud, Langenbrunner et al. [72] measured the vibration response of a metal and a ceramic matrix composite (CMC) turbine blades under a controlled rubbing condition, and compared the rubbing features of different blade materials and tip designs. Using an aeroengine rotor experimental rig with casings, Chen [73, 74] carried out the blade–casing rubbing experiments which consider both the single-point rubbing and the partial rubbing, and extracted the rubbing faults’ characteristics by the casing vibration acceleration signals.

4.2 Experimental studies of dynamic characteristics for casing with abradable coating

Many researches on the wear mechanisms, such as adhesive transfer, cutting, and melting, underlying in the process of blades rubbing abradable coating have been carried out by metallographic observation on the abradable coating and blade [75–80]. These experiments on coating wear are mainly evaluated from the perspective of material science, which is out of the interest of this review. This review focuses on the published literatures on the dynamic characteristics of the blade and casing due to the blade–coating rubbing and coating wear. Based on the same test rig [70, 71], Padova et al. [81] carried out the experiments on the compressor blade rubbing bare and coating casings at engine speed, and analyzed the rubbing due to progressive and sudden penetrations of blade. Millecamps et al. [82] carried out experiments on bladed disk/casing rubbing and discussed the effect of the thermomechanical coupling on the blade vibration response. Reiss [83] established a test rig to simulate rubbing between the blade and coating, characterized the wear of the abradable coating using lobes, and measure rubbing-induced heat by infrared camera. His results show that under divergence, the number of these lobes corresponds to the ratio of response frequency to speed frequency. Almeida et al. [84] carried out experiments on the rubbing between a centrifugal compressor impeller and its casing with abradable coating at constant speed.

Stringer et al. [85], Fois et al. [86], and Fois et al. [87] established a high-speed test rig, analyzed the wear characteristics using image analysis of the blade, stroboscopic imaging of the abrasion and SEM (scanning electron microscope) pictures of the blade tip. By bonding cubic boron nitride (cBN) grits and coating Cr(Al)N to the blade tips, Watson et al. [88] studied the wear and failure mechanisms during the blade-tip rubbing coating.

Baiz et al. [89] developed a test rig to investigate contact forces, coating wear, and blade dynamics under the blade–seal rubbing, which is equipped with a piezoelectric load cell (KISTLER 9031A), a high-speed camera, a strain gauge (TML FLA2), and three laser displacement sensors (two KEYENCE LC2400 sensors and one LKG82 sensor). Based on the test rig in [89], Mandard et al. [90] proposed an experiment method to estimate the blade–casing rubbing force by indirect measurements. The tangential force was evaluated by the measured blade flexural motion, an analytical model of the blade, and a model inversion technique; the normal force was estimated by using force and acceleration measurements combining with an accelerometric compensation equation. Adopting the continuous wavelet transform (CWT) and fast Fourier transform (FFT) simultaneously, Mandard et al. [91] estimated the blade friction-caused vibration modes and studied the interactions between the blade vibration and abradable coating removal.

Based on a laboratory test rig, Sutter et al. [92] investigated the interaction phenomena between a titanium alloy (TA6V) and an abradable material (M601) to analogize blade–casing rubbing in an aircraft engine. Adopting the test rig in [92], Cuny et al. [93] investigated the blade–casing rubbing phenomena which were simulated by high-speed orthogonal cutting, and presented a correction technique on the basis of modal analysis principles to reduce the effect of dynamic behavior of the sensor on the measured contact force.

4.3 Summary and discussion on measured dynamic features

Measured typical rubbing characteristics for the bare and coating casings are listed in Tables 5 and 6, respectively. These typical test rigs and rubbing-induced vibration features are discussed as follows:

For the experimental researches on the rubbing-induced vibration, rubbing, and wear mechanisms, early studies on the blade–casing rubbing are relatively simple, where the blade is often simplified as a single pendulum and the casing as a plate [65, 66]. These simplified experiments have some differences from the realistic blade–casing rubbing, such as the blade structure, rotational speed, and casing shape. Some improved test rigs are established in [15, 67, 68] which can simulate the rotation of the blade and measure the rubbing force; however, the blade structure (uniform beam) is still simple, and the operational speed is much lower than the actual operational speed in turbine machine or aircraft engine. A special test rig is established in ONERA (the French Aerospace Lab) to simulate the blade–coating interaction [89–91], and some responses, such as rubbing forces, wear profiles, bending displacement, and strain of the blade, incursion depth, are measured at blade-tip tangential speed $V_\mathrm{t}=19\,\hbox {m/s}$ [89, 90] and $V_\mathrm{t} =92\,\hbox {m/s}$ [91]. In order to obtain more realistic results, the test rig may be improved by increasing the blade-tip tangential speed and adopting the concave contact geometry [73, 74].

A research group from Gas Turbine Laboratory at the Ohio State University, especially Padova et al. [70, 71, 81] and Garza [46], developed an in-ground spin-pit facility (SPF) which adopts the realistic blade structure, can reach the maximum rotational speed 20,000 rev/min, and can simulate compressor blade rubbing bare and coating casings at engine speed. This test rig can measure rubbing force, blade dynamic stress, friction heat, casing acceleration, and rubbing times. To our knowledge, this test rig can carry out a relatively all-sided reproduction for the blade–casing rubbing in axial compressor. A real blade structure is also applied for a test rig in [82] to emphasize the blade failure mechanism due to the coupling among thermomechanics, wear, and dynamic. Besides the blade rubbing in axial compressor, the first impeller–casing rubbing experiment in centrifugal compressor is also reported in [84], which adopts a realistic low-pressure centrifugal compress impeller with 10 main blades and 10 secondary blades and an actual casing fixed near the leading edge inside a rigid vacuum tank. Blade–casing rubbing is also simulated in a real turbojet aeroengine, whose maximum rotational speed is up to 58,500 rev/min [73, 74].

Table 5 Measured typical rubbing features for bare casing

Some test rigs are established to carry out the researches mainly focused on the wear mechanism [85–87, 92, 93]. A high-speed test rig is used to study wear profiles of the blade and coating and two kinds of wear mechanisms (cutting and adhesive transfer) at $100\,\hbox {m/s}<V_\mathrm{t} \le 200\,\hbox {m/s}$ [85, 86], and the relationship between the abrasion and interaction force for abradable materials is studied at $V_\mathrm{t} =100\,\hbox {m/s}$ [87], and effects of blade surface treatments in tip–shroud abradable contacts are researched at $V_\mathrm{t} =200\,\hbox {m/s}$ [88]. A facility is developed to measure the contact force and reveal wear mechanisms related to abradable materials at $10\,\hbox {m/s}<V_\mathrm{t} < 107\,\hbox {m/s}$ [92] and $60\,\hbox {m/s}<V_\mathrm{t} < 270\,\hbox {m/s}$ [93], respectively. In these experiments [92, 93], the sample depositing abradable material is launched to rub a cutting tool simulating the blade. In these researches [85–88, 92, 93], the small rigid blade is adopted to make an accurate control on the penetration depth and speed.

Some typical vibration characteristics due to the blade–casing rubbing can be summarized as follows:

• 1) Under the rigid casing conditions, the circumferential load increases significantly with the increasing penetration; however, the maximum axial load increases much less, which reveals the nonlinear characteristic of the rubbing [15, 68, 70, 71, 81]. Under small casing stiffness, the normal contact force has the linear characteristic with penetration depth [15].

• 2) Blade–casing rubbing-induced vibration response and the number of wear lobes are related to the excitation force [82, 84]. The number of abrasion lobes is connected with the first blade flexural mode [82]; vibration amplification can be observed when the rotational speed harmonics coincide with the casing or impeller modes [84]; the number of wear lobes is related to the ratio of the excited mode to the rotational speed [84].

• 3) Casing vibration during the rubbing is coupled with the blade vibration, so it is feasible to determine the rubbing level and extract rubbing features by casing vibrations [69, 73, 74]. The blade passage frequency can be viewed as a distinguished feature for multiple-blade rubbing [69, 73, 74], and some obvious frequency components of the rotational frequency and its multiple frequencies can be observed in the cepstrum [74], and higher closed frequency components can also be observed in multistage rotor systems [69].

Table 6 Measured typical rubbing features for coating casing

5 Conclusions and outlooks

In the previous sections, we have reviewed the published results on the blade–casing rubbing dynamics. In fact, the literatures in this filed are numerous and various. A summary on all the literatures is infeasible, and omission of some references is unavoidable. The review is composed of three main parts, i.e., modeling methods of the rubbing between rotating blade and casing, simulation studies, and experimental studies of dynamic characteristics under rubbing between rotating blade and casing. In each of three parts, the summaries are performed according to two types of casing: bare casing and casing with abradable coating. Some discussions and summaries for three parts are provided in the end of each section. Some further studies in this field are summarized as follows:

Fig. 20

Morphology of the fracture in [97]: a macro-morphology, b SEM

5.1 Theory research aspects

• 1) Adopting different rubbing models, vibration response of the blade [23] and rubbing force [15] has a small difference under lower rotational speeds; however, the difference under higher rotational speeds is significant. Because of the fact that the blade vibration and blade–casing interaction force will have a significant impact on the coating removal, it is essential to develop an improved blade–casing rubbing model considering the effects of coating material parameters, such as plastic modulus and limit strength, dynamic loads including kinetic energies of the blade and elastic casing for aircraft engine, on the basis of the existing rubbing models.

• 2) For 3D FE models of the blade adopted in published references [40–43, 53, 54, 56–58], only centrifugal stiffening due to rotation is involved; however, other rotating effects, such as rotating softening due to the large deformation of the blade and Coriolis force due to the dynamic coupling of axial motion and flexural deflection of the blade, are not included in the reduced FE models. Sinha [26] pointed out that Coriolis force has a great influence on the rubbing-caused vibration. The combined action of the centrifugal stiffening, rotating softening, and Coriolis force will affect the blade elongation and vibration, which has a certain impact on the contact detection and penetration depth evaluation.

• 3) For the flexible casing without coating, the bladed disk–casing modal contact phenomenon is studied in [35–38], and the single blade–casing rubbing considering the coating wear effects is focused on the rigid casing in [53, 54, 56–58]. Considering the effects of a flexible casing with abradable coating, how to analyze the rubbing and wear mechanisms due to the modal contact between bladed–disk and elastic casing will be a next interesting issue. By analogizing the blade rubbing the coating to the cutting tool cutting the workpiece [16, 49, 50, 92], the blade–coating rubbing needs to be further studied by the finite element method, which can use automatic remeshing/rezoning in the coating and chip to replace those distorted elements with ones of better shape [94]. Moreover, for the bladed–disk structure, although blade mistuning due to imperfections in bladed–disk manufacture, assembling, and material scatters can drastically affect the forced vibration responses [95], however, to our knowledge, the effects of blade mistuning on the blade–casing rubbing and coating wear have not been involved.

• 4) Considering the effects of the rotor whirl, the rubbing-induced blade and rotor vibrations have been evaluated by many researchers, such as Parent et al. [17, 18], Lesaffre et al. [32], Sinha [33, 34], Ma et al. [44], Salvat et al. [38]. The vibration coupling of shaft, blade, and casing is considered in [38, 44]; however, the shaft in the former [38] is simplified as the springs which cannot consider the gyroscopic effect of the rotor and complicated coupling modes of the shaft and bladed disk; in the latter [44], the complicated model contact phenomena of the shaft-disk-blade and casing are not involved. Taking the coupling effects of rotor whirl involving multi-modes [96] and the vibrations of the bladed disk and casing into account, study on the effects of shaft–disk–blade–casing coupling vibration on the rubbing and coating wear mechanisms is also an important research direction. The effects of blade–casing rubbing on the dynamic loads of bearings should also be evaluated because the rubbing will generate time-varying impact load which can reduce bearing life. Moreover, evaluations of the blade vibration and coating wear by using the casing vibration are also an interesting topic.

• 5) Generally, blade is subjected to complicated loading conditions, such as aerodynamic loading, temperature field, unbalance and rubbing-induced vibration loads. Assuming that the aerodynamic loading on the blade is sinusoidal in time, the rubbing simulation is carried out in [57, 58]. Most researches, except for [57, 58], on aerodynamic performance and rubbing-caused vibration are carried out independently. Under the combined action of the aerodynamic loading, periodic rubbing load, and temperature field, the rubbing-induced vibration will be strongly nonlinear, which is also an important research issue. The existing literatures indicated that the rubbing mainly excites lower-order mode, such as the first bending mode [53, 54, 58, 70, 71, 81, 82, 91] and the first torsional mode [71, 81]. However, a new reference [97] reported that a blade loss-angle failure (Fig. 20) due to the second torsional resonance exited by blade–casing rubbing, which provides a new research content. That is, the rubbing-induced high-order mode vibration and fatigue damage mechanism should be focused on, which was also stated in [74] as follows: “However, the high-frequency segment contains more important information about aeroengine rubbing faults.”

Table 7 Research teams summarization

5.2 Experimental research aspects

• 1) Published experimental researches mostly adopted small rigid blades [85–88, 92, 93]; however, the practical blade is mostly flexible, especially for low-pressure compressor. Considering the effects of flexible blade and elastic casing vibrations, rubbing and wear experiments for “truer” blade conditions including the high blade-tip tangential velocity (about 500 m/s for low-pressure compressor [91, 93]), operating temperature $300^{\circ }\hbox {C}$ [93], and blade oblique angles will be an important improvement. Moreover, coating wear involves multiple subjects, such as materials, mechanical engineering and mechanics, and evaluation on rubbing-induced vibration law, and wear mechanism is also an important imperative based on the combination of research conceptions of different disciplines and combination of the macroscopic and microscopic ways.

• 2) During the test, the effects of sensors on the measured results need to be corrected, for example the effects of the additional sensor masses on the blade vibration characteristics and amplitude and phase distortion of the force signal provided by force sensor due to intermittent contact forces (impulse type) [68, 93].

5.3 Experiment–simulation combination aspects

Some key model parameters which are vital to the simulations of the blade–casing rubbing and coating wear need to be measured by experiment, such as the elastic casing stiffness [15], modal damping ratios, and rubbing forces. Some new experiment–simulation combination methods are desired to evaluate the plastic parameters of the coating materials. For example, by an FE simulation and experiment verification, Peyraut et al. [98] indicated a bilinear plastic law can accurately simulate the mechanical behavior of coating material.

Finally, a personal view on researches in these fields is given as follows: During the carding and summarization of the literatures, we deeply realize that the development in these research fields is promoted by the practical demand, and some research teams which are carrying out collaborative study with companies are listed in Table 7. In author’s personal opinion, the outstanding contributions on the blade–casing rubbing by the researchers in France are largely attributed to the practical needs and efficient university–enterprise cooperation modes.