A Survey of Aero-Engine Blade Modeling and Dynamic Characteristics Analyses

Abstract

The rotating blade is a key component of an aero-engine, and its vibration characteristics have an important impact on the performance of the engine and are vital for condition monitoring. This paper reviews the research progress of blade dynamics, including three main aspects: modeling of blades, solution methods, and vibration characteristics. Firstly, three popular structural dynamics models for blades are reviewed, namely lumped-mass model, finite element model, and semi-analytical model. Then, the solution methods for the blade dynamics are comprehensively described. The advantages and limitations of these methods are summarized. In the third part, this review summarizes the properties of the modal and vibration responses of aero-engine blades and discusses the typical forms and mechanisms of blade vibration. Finally, the deficiencies and limitations in the current research on blade modeling and vibration analysis are summarized, and the directions for future efforts are pointed out. The purpose of this review is to provide meaningful insights to researchers and engineers in the field of aero-engine blade modeling and dynamic characteristics analysis.

1. Introduction

Rotating blades are a very critical component in the aerospace and machinery industries. They feature complex vibration characteristics due to complex loads such as centrifugal force, aerodynamic forces, and torque. With the progress of computer and simulation technology, numerical simulation can complete structural vibration analysis at a lower cost. It has shown great advantages in the study of the vibration characteristics of aero-engine blades and has received more and more attention from scholars and engineers.

In the early design stages, real aero-engine blades with complex geometrical features are often simplified to simple geometrical models due to the unavailability of detailed information about the blades or limitations in computational power. This simple geometric model depends on only important parameters such as the aspect ratio [1], pre-twist angle, mounting angle, and disk radius. Researchers use simplified blade models such as lumped-mass models [1] or equivalent beam and plate models. These models are computationally simple and suitable for rapid evaluation in the preliminary design phase, but they do not meet the growing demand for accuracy. With the development of electronic computers, numerical simulation technology for blade vibration characterization developed rapidly in the 20th century [2,3], which is based on finite discrete theory. Researchers began to establish relatively complex models, such as the two-dimensional shell finite element model and the three-dimensional solid finite element model [4,5,6,7,8]. Mechanics theory and numerical computation technology have made great developments in the research of the aeroelastic analysis of aero-engine blades since the 1980s. At the same time, the models and theories that consider the coupling effect of multiple physical fields have also been improved [9,10]. Nowadays, researchers can use the combination of computational fluid dynamics (CFD) and finite element analysis (FEA) to more accurately simulate and predict the mechanical behavior of blades under different working conditions.

After obtaining the dynamical equations, solving the system of equations and revealing the law of motion of the system are the core goals of dynamics research [11]. The presence of nonlinearities makes the superposition characteristic of linear systems no longer valid. Solving the nonlinear dynamical equations becomes extremely difficult, and this problem has continued throughout the centuries. The problem of improving the accuracy of the solution method has become more important. Many scholars have carried out a lot of research work for blade vibration and achieved fruitful research results.

The methods of solving the dynamic equation model include the analytical method, numerical method, and semi-analytical method. Analytical methods such as the Laplace transform and the Fourier transform directly derive the analytical solutions of systems through mathematical analysis, but the application of such methods is limited by the specific form of the equations and the complexity of the system. For most nonlinear systems in the real world, numerical methods are usually more practical and effective, so the Runge–Kutta method, Newmark, etc., become the main tools for solving. The perturbation method, symplectic integral, harmonic balance method, and other semi-analytical methods are a class of methods between the analytical method and the numerical method, which can deal with some specific nonlinear problems and provide higher accuracy and efficiency than the numerical method.

With the development of modeling and solution techniques, the dynamic characteristics analysis of aero-engine blades has made significant progress in recent years, expanding from traditional finite element analysis and modal analysis to the composite material application, multi-body dynamics model, and multi-physical field coupling analysis. These advances not only enhance the accuracy of predicting the structural response and vibration characteristics of blades, but also provide new methods and theoretical support for optimizing design, improving operational stability, and reducing the risk of accidents. With the application of data-driven technologies, researchers are exploring machine learning and other methods for a deeper understanding of dynamic properties.

Although the study of aero-engine blade vibration has been included in many studies on the vibration analysis of structural systems, the current literature lacks a comprehensive overview of the complete process of aero-engine blades from modeling to vibration characterization. Therefore, this paper provides a comprehensive and in-depth discussion of the mathematical modeling methods, vibration analysis methods, and vibration characteristics of aero-engine blades.

The rest of this paper is organized as follows. Section 2 discusses the mathematical models for modeling aero-engine blades, including the lumped-mass model, the finite element (FE) model, and semi-analytical model. Section 3 details the solution methods for the vibration system based on the development of numerical integration methods, including analytical methods, early numerical solution methods, semi-analytical methods, and computer-aided numerical methods. Section 4 introduces the blade vibration characteristics. This section mainly introduces the forms, causes, and phenomena of the blade vibration, the inherent characteristics of the blade, and the dynamic response characteristics. Section 5 summarizes the shortcomings of the current review and gives an outlook on future research directions.

2. Mathematical Modeling of Blades

The study of blade vibration characteristics needs to be based on accurate modeling. By modeling the blades’ mathematical model, the function design and analysis of the blade can be achieved. The mathematical model of a blade is a mathematical expression reflecting the solid shape of the blade, which can be an independent equation or a set of parametric equations. In this section, the common blade mathematical models are summarized, and the common models are lumped-mass models, FE models [4,5,6,7], and semi-analytical models. The theory of classical and generalized formulas is also summarized.

2.1. Lumped-Mass Model

The lumped-mass method (LMM) is a simplified modeling method. A blade is simplified into several mass blocks interconnected by springs. The lumped-mass model is mainly used for the vibration characterization of rotor systems considering faults such as unbalance, cracks, and friction, and can provide some interesting and valuable qualitative results. As shown in Figure 1, the dynamics model of the blade–rotor-bearing system supported by a single disk sliding bearing, the mass of the rotor is centralized at the disk and bearing, where $O_1$ and $O_2$ are the form centers of the bearing and disk, respectively, and their masses are $m_1$ and $m_2$, respectively, and the eccentricity of the disk is $e$, $l$ is the length of the rotor shaft, and the disk is fixed in the center of the rotor shaft to simplify the blades into a cantilever beam structure [12]. In lumped-mass modeling, the Lagrange equation is often used to establish the dynamical equations of the system. The Lagrange equation is as follows

$${\displaystyle \frac{d}{dt}}({\displaystyle \frac{𝜕L}{𝜕\dot{Q}}})-{\displaystyle \frac{𝜕L}{𝜕Q}}=-{\displaystyle \frac{𝜕D}{𝜕\dot{Q}}}+F$$

(1)

where $L=T-U$ is the Lagrange function of the system, $T$ and $U$ are the kinetic and potential energies of the system, respectively, $D$ is the dissipation function of the system, $F$ is the generalized force of the system, and $Q$ is the generalized coordinate of the system.

According to the theory of rotor dynamics, the kinetic energy of the blades is

$$T_{\mathrm{B}}=\sum _{i=1}^n{\displaystyle \frac{1}{2}}{\displaystyle \underset{0}{\overset{l}{\int }}\rho A{\dot{R}}_{pi}^T{\dot{R}}_{pi}\mathrm{d}x}$$

(2)

where $R_{pl}$ is the displacement of any point $p$ on the ith blade in a fixed coordinate system after the bending deformation.

The strain energy of the blade due to bending is

$$U_{\mathrm{B}}=\sum _{i=1}^n{\displaystyle \underset{0}{\overset{l}{\int }}EI{(\frac{{𝜕}^{2}u}{𝜕x^2})}^2\mathrm{d}x}$$

(3)

The axial contraction potential energy due to transverse deformation is

$$\begin{array}{c}{U}_A={\displaystyle \sum _{i=1}^n}{\displaystyle \frac{1}{2}}{(\mathsf{\Omega }+\dot{\phi })}^2\left\{{\displaystyle \frac{1}{2}}{\int }_0^{\prime }\rho A(l^2-x^2){\left({\displaystyle \frac{𝜕u}{𝜕x}}\right)}^{2}dx+R_{\mathrm{d}}{\int }_0^{\prime }\rho A(l-x){\left({\displaystyle \frac{𝜕u}{𝜕x}}\right)}^{2}dx\right\}\\ ={\displaystyle \frac{n}{2}}{(\mathsf{\Omega }+\dot{\phi })}^2\left\{{\displaystyle \frac{1}{2}}{\int }_0^l\rho A(l^2-x^2){({\displaystyle \frac{𝜕u}{𝜕x}})}^2\mathrm{d}x+R_{\mathrm{d}}{\int }_0^l\rho A(l-x){({\displaystyle \frac{𝜕u}{𝜕x}})}^2\mathrm{d}x\right\}\end{array}$$

(4)

where $R_{\mathrm{d}}$ is the radius of the disk, the root of the blade is fixed on the disk, and $x$ is the axial position of the blade. $\phi$ is the torsion angle of the disk, and $\Omega$ is the rotational speed. A schematic of the lumped-mass model is shown in Figure 2. Ma et al. [13,14] developed a new rotor–blade system dynamics model, where the rotating flexible blade is represented by the Timoshenko beam. The shaft and rigid disk are described by multiple lumped-mass points. These points are connected by massless springs with lateral and torsional stiffness. According to the actual situation, the mathematical model is based on the assumption of isotropy; the contact problems of the blades, disks, and shafts are neglected; the shafts and disks are discretely represented by lumped-mass points; and the bearings are described by a linear stiffness model and simplified by a spring-damper model. The feasibility of the lumped-mass model was verified by comparing the results with the intrinsic frequency and vibration response of the FE model. Zhu et al. [15] performed dynamics modeling on a flexible rotor supported by ball bearings with rubber damping rings. The rotor was constructed using Timoshenko beam units, while the support and bearing outer ring were modeled by the LMM. The theoretical and experimental analyses were carried out by the rotor-bearing test rig, and the error rate between the theoretical and experimental studies was less than 10%. Chen et al. [16] used the LMM in a two-bladed system with a sub-stage damper. The two blades were simplified to 2 lumped mass and 2 corresponding stiffness. Meanwhile, the nonlinear forces are represented by two equivalent coefficients, equivalent damping, and equivalent stiffness. A new lumped-mass model of the system was introduced in a vane disk with a friction ring damper by Laxalde et al. [17]. The model is composed of the basic sector of the impeller disk and its associated friction ring elements. Due to the high contact load resulting from the engine rotation, it is assumed that the ring and blisk remain in contact at all times. In practical applications, the LMM can effectively simplify the analysis process of complex systems and improve the computational efficiency, but it cannot simulate the blade with complex geometric configurations.

2.2. FE Model

The FE method (FEM) divides the continuous blade structure into FEs and builds a mathematical model for each element; the whole blade is finally described by assembling the models of all elements. The FEM provides highly accurate vibration and structural performance analysis due to its ability to accurately model complex geometries and a wide range of physical phenomena. However, it also requires more computational resources and time for simulation and analysis [18].

The aero-engine blade is a complex three-dimensional structure that is usually modeled and analyzed using FE analysis. Generally, beam elements, shell elements, or solid elements can be used to establish the finite element model of an aero-engine blade. These three different element types have different computational accuracies, modeling difficulties, and solving efficiencies.

2.2.1. Beam Element

The blade is the most vibration-sensitive component in an aero-engine, with both rigid rotational motion and elastic deformation characteristics. The blade can be modeled by a one-dimensional beam element with a choice of rotationally flexible or non-rotating beams. The effects of the buckling, pre-twist angle, shear deformation, rotational inertia, and Coriolis effect can be taken into account in the modeling process. Researchers have modeled rotating blades based on different beam theories, including Euler–Bernoulli, Rayleigh, and Timoshenko beams. It is usually difficult to obtain closed-form solutions to the beam equations. Therefore, various approximations have been proposed to study the dynamic properties of rotating beams.

Euler–Bernoulli beam theory is a classical beam theory. Since the Euler–Bernoulli beam ignores the effect of the shear strain, the calculated deformation of the beam is smaller than the actual. Therefore, it is suitable for slender beams, as shown in Figure 3a. The central axis of the beam when it is not deformed is taken as the X-axis. Assuming that the beam has a symmetric plane across the X-axis, the direction in the symmetric plane perpendicular to the X-axis is taken as the Y-axis. When the beam displays flexural vibration in the symmetric plane, there is only a transverse displacement of the beam’s central axis along the X-axis w(x, t), which is the deflection of the beam. The premise of Euler–Bernoulli beam theory analysis is the plane cross-section assumption; that is, the plane remains flat in the bending process of the beam. Assuming that the displacements of any point on the cross-section of the beam in the three directions x, y, and z are u, v, and w, then it follows from Figure 3b that

$$u=-z{\displaystyle \frac{𝜕w(x,t)}{𝜕x}}, v=0,w=w(x,t)$$

(5)

The strain and stress components associated with these displacement fields are

$${\epsilon }_{xx}={\displaystyle \frac{𝜕u}{𝜕x}}=-z{\displaystyle \frac{{𝜕}^{2}w}{𝜕x^2}}, {\epsilon }_{yy}={\epsilon }_{zz}={\epsilon }_{xy}={\epsilon }_{yz}={\epsilon }_{zx}=0$$

(6)

$${\sigma }_{xx}=-Ez{\displaystyle \frac{{𝜕}^{2}w}{𝜕x^2}}, {\sigma }_{yy}={\sigma }_{zz}={\sigma }_{xy}={\sigma }_{yz}={\sigma }_{zx}=0$$

(7)

where ${\epsilon }_{xx}$, ${\epsilon }_{yy}$, and ${\epsilon }_{zz}$ are the normal strains; ${\epsilon }_{xy}$, ${\epsilon }_{yz}$, and ${\epsilon }_{zx}$ are the shear strains; ${\sigma }_{xx}$, ${\sigma }_{yy}$, and ${\sigma }_{zz}$ are the normal stresses; and ${\sigma }_{xy}$, ${\sigma }_{yz}$, and ${\sigma }_{zx}$ are the shear stresses. The strain energy of a Euler–Bernoulli beam is

$$\begin{array}{ll}\mathrm{V}& ={\displaystyle \frac{1}{2}}{{\displaystyle \int }}_V({\sigma }_{xx}{\epsilon }_{xx}+{\sigma }_{yy}{\epsilon }_{yy}+{\sigma }_{zz}{\epsilon }_{zz}+{\sigma }_{xy}{\epsilon }_{xy}+{\sigma }_{yz}{\epsilon }_{yz}+{\sigma }_{zx}{\epsilon }_{zx})dV\\ & ={\displaystyle \frac{1}{2}}{\int }_0^{l}EI{\left({\displaystyle \frac{{\partial }^{2}w}{\partial x^2}}\right)}^{2}dx\end{array}$$

(8)

where E is the elastic modulus and I is the cross-sectional moment of inertia to the Y-axis. The kinetic energy is expressed as

$$T={\displaystyle \frac{1}{2}}{\displaystyle {\int }_0^l{{\displaystyle \int }}_A\rho {\left(\frac{𝜕w}{𝜕t}\right)}^{2}dA\mathrm{d}x}$$

(9)

where ρ denotes the density of the beam and A denotes the cross-sectional area of the beam. The differential equation for the transverse vibration of a slender beam can be obtained as

$$EI{\displaystyle \frac{{𝜕}^{4}w}{𝜕x^4}}+\rho A{\displaystyle \frac{{𝜕}^{2}w}{𝜕t^2}}=f(x,t)$$

(10)

The deformation of the beam calculated by the Timoshenko beam is close to the deformation of the real beam because it takes into account the effect of the shear strain. Therefore, the Timoshenko beam is suitable for short beams. In Timusinko beam theory, the displacement of any point on the surface of the beam element can be obtained by the following equation as

$$\begin{array}{l}{u}_x(x,y,z)=u(x)+z{\theta }_y(x)-y{\theta }_z(x)\\ u_y(x,y,z)=\nu (x)-z{\theta }_x(x)\\ u_z(x,y,z)=w(x)+y{\theta }_x(x)\end{array}$$

(11)

where $u_x(x,y,z), u_y(x,y,z), u_z(x,y,z)$ denote the deformation of a point on the surface of the element along the X-, Y-, and Z-axes, $u(x)$, $v(x)$, $w(x)$ represents the displacement of the mapped point on the neutral axis, ${\theta }_x(x)$, ${\theta }_y(x)$, ${\theta }_z(x)$ denotes the angle of rotation of the mapped point on the neutral axis, and $(x,y,z)$ are the coordinates of the point on the surface of the beam element.

Under the assumption of small strain theory, the relation between the displacement of any point on the surface of the beam element and the normal section strain of the neutral axis where the point is located can be obtained by calculating the partial derivation $x$, $y$, $x$ from the Equation (11), and as follows

$$\begin{array}{l}{\epsilon }_x(x,y,z)={\displaystyle \frac{𝜕u(x)}{𝜕x}}+z{\displaystyle \frac{𝜕{\theta }_y(x)}{𝜕x}}-y{\displaystyle \frac{𝜕{\theta }_z(x)}{𝜕x}}\\ {\gamma }_{xz}(x,y)={\displaystyle \frac{𝜕u_x}{𝜕z}}+{\displaystyle \frac{𝜕u_z}{𝜕x}}={\displaystyle \frac{𝜕w(x)}{𝜕x}}+y{\displaystyle \frac{𝜕{\theta }_x(x)}{𝜕x}}+{\theta }_y(x)\\ {\gamma }_{xy}(x,z)={\displaystyle \frac{𝜕u_x}{𝜕y}}+{\displaystyle \frac{𝜕u_y}{𝜕x}}={\displaystyle \frac{𝜕\nu (x)}{𝜕x}}-z{\displaystyle \frac{𝜕{\theta }_x(x)}{𝜕x}}-{\theta }_z(x)\end{array}$$

(12)

For a beam element of length $L^e$ subjected to concentrated forces and moments at its endpoints, the overall degrees of freedom of the beam can be expressed in terms of a total of fourteen degrees of freedom using two end nodes (1, located at $\mathrm{x}=0$; 2, located at $\mathrm{x}=L^e$) and an intermediate node (r, located at $\mathrm{x}=L^e/2$), as shown in Figure 4. The degrees of freedom on the neutral axis of the beam element can be interpolated using the Lagrangian interpolation form function as

$$\begin{array}{l}u(x)={\displaystyle \sum _{i=1,2}}{L}_i^{(1)}(x)u_i\\ \nu (x)={\displaystyle \sum _{i=1,2}}{L}_i^{(1)}(x)v_i-{\displaystyle \sum _{j=1,\mathrm{r},2}}{N}_j^{(3)}(x){\theta }_{zj}\\ w(x)={\displaystyle \sum _{i=1,2}}{L}_i^{(1)}(x)w_i+{\displaystyle \sum _{j=1,\mathrm{r},2}}{N}_j^{(3)}(x){\theta }_{yj}\\ {\theta }_x(x)={\displaystyle \sum _{i=1,2}}{L}_j^{(1)}(x){\theta }_{xi}\\ {\theta }_y(x)={\displaystyle \sum _{j=1,\mathrm{r},2}}{L}_j^{(2)}(x){\theta }_{yj}\\ {\theta }_z(x)={\displaystyle \sum _{j=1,\mathrm{r},2}}{L}_j^{(2)}(x){\theta }_{zj}\end{array}$$

(13)

where the specific forms of the interpolated shape functions of each order can be found in reference [20]. Equation (13) can also be written in the form of a matrix with the following expression:

$$\mathit{u}(x)=\mathit{N}(x){\mathit{u}}^e$$

(14)

where $\mathit{N}(x)$ is the matrix form of the displacement interpolation shape function. Simultaneous derivation of both sides of the Equation (14) gives the expression between the theoretical cross-section strain of the element and the overall degree of freedom of the beam element as follows

$$\mathit{e}(\mathit{u})=\mathit{B}(x){\mathit{u}}^e$$

(15)

where $\mathit{B}(x)$ is the strain shape function matrix, which is derived from the displacement interpolation shape function matrix $\mathit{N}(x)$.

Based on classical beam theory, many researchers have studied the vibration of thin- or thick-walled beams. Rehfield [22] simplified the composite rotor blade as a thin-walled beam to investigate its design analysis methodology. Subsequently, Chandiramani et al. [23] simulated the rotating pre-twisted blade using a laminated composite, hollow (single-celled), uniform box beam, for which the analysis of the free and forced vibration was carried out. The model considers that the shear strain strains are not uniform through the wall thickness and general orthotropic beam wall theory. Jafari-Talookolaei et al. [24] developed a rotationally delaminated composite beam model with a general lay-up. The model investigated the effects of shear deformation, rotational inertia, material coupling (bending–tension, bending–twist, and tension–twist coupling), and the Poisson effect. By choosing the Legendre polynomials and using their properties, the kinetic and potential energies, the constraints and then functional are made. Smith et al. [25] proposed a structural model for blade airfoil beams. The effects of transverse shear deformation, two-dimensional in-plane elasticity, and torsion-related out-of-plane buckling were investigated. A common beam model is shown in Figure 5 as a cantilever beam structure attached to a rotating rigid hub of radius r. P* is the position of point P before the deformation.

In addition, some researchers have attempted to study the modeling of smart materials and complex cross-section beam elements. Cesnik [26] proposed a hybrid beam model that divides the problem into a general linear two-dimensional problem in the cross-section and a nonlinear global one-dimensional analysis. Ebrahimi et al. [27] modeled a rotating biconical variable cross-section functional gradient (FG) beam, made of porous material. Ren et al. [28] modeled a rotating composite thin-walled beam with shape memory alloy fiber actuators.

Failure of engine blades is in many cases caused by high circumferential fatigue due to vibrational stresses. Designers are increasingly concerned about the impact, caused by nonlinear mechanical models and friction effects to avoid such failures. The designers comprehensively investigated the vibration behavior of the blade by introducing factors such as a pinch angle, springs, and dampers. Griffin [29] developed a one-dimensional friction model and a one-dimensional sliding model including springs and dampers. This model assumes that the damper is characterized by two parameters: the force at which it slips and its stiffness. Yang et al. [30] developed a one-dimensional friction model. Afterwards, the friction model was extended to two dimensions considering the variation in the normal force.

The beam model, as a one-dimensional model, can provide more accurate dynamic characteristics for most rotating mechanisms. These studies provide an important reference and theoretical basis for understanding and analyzing the vibration characteristics of thin- or thick-walled beams.

2.2.2. Shell Element

Most industries have been considering the application of advanced blade configurations to improve aerodynamic performance and reduce vibration loads and noise in recent years. Such advanced configurations include a large curvature and twist for turbomachinery blades. In addition, rotorcraft blades employ blade sweep angle variations and unique tip shapes such as swept tips and dihedral. Accurate analyses of such structures require more accurate two-dimensional models than simple beam models.

The plate/shell model allows for more detailed consideration of the specific shape and size of the blade, as well as the constraints and external loading conditions to which the blade will be subjected to in actual use.

The two most commonly used plate and shell theories are based on Kirchhoff and Reissner–Mindlin assumptions, respectively. Classical theory is based on the Kirchhoff assumption, which ignores the effect of shear deformation and is often applied to the analysis of thin-plate structures. Medium-thickness theory (first-order shear deformation theory) is based on the Reissner–Mindlin assumption, which takes into account the effect of shear deformation and is suitable for the structural analysis of medium-thickness shell. There are three typical types of Mindlin–Reissner theory: (1) Reissner theory; (2) Mindlin theory; and (3) higher-order shear deformation theory. The general (thick-) shell theory is often used for thick-shell structures and cases requiring high-precision analysis, such as composites and shells with complex geometries. Orthogonal thin-shell bending theory is commonly used for thin-shell structures of composites and laminated materials.

According to Kirchhoff’s assumption, the in-plane displacements $u$ and $v$ at any point on the plate are

$$\begin{array}{l}{\mathit{\gamma }}_{xz}=0\Rightarrow \mathit{u}=-z{\displaystyle \frac{𝜕w}{𝜕x}} \\ {\mathit{\gamma }}_{yz}=0\Rightarrow \mathit{v}=-z{\displaystyle \frac{𝜕w}{𝜕y}}\end{array}$$

(16)

where $z$ is the coordinate direction perpendicular to the plate surface. Because the out-of-plane stress component of the thin plate is much smaller than the in-plane stress component, the physical equations of the plane stress problem are used (note the basic assumptions of the plane stress problem) in the theory of thin plates and that is

$$\begin{array}{l}{\sigma }_x={\displaystyle \frac{E}{1-{\nu }^2}}({\displaystyle \frac{𝜕u}{𝜕x}}+\nu {\displaystyle \frac{𝜕v}{𝜕y}})=-{\displaystyle \frac{Ez}{1-{\nu }^2}}({\displaystyle \frac{{𝜕}^{2}w}{𝜕x^2}}+\nu {\displaystyle \frac{{𝜕}^{2}w}{𝜕y^2}})\\ {\sigma }_y={\displaystyle \frac{E}{1-{\nu }^2}}(\nu {\displaystyle \frac{𝜕u}{𝜕x}}+{\displaystyle \frac{𝜕v}{𝜕y}})=-{\displaystyle \frac{Ez}{1-{\nu }^2}}(\nu {\displaystyle \frac{{𝜕}^{2}w}{𝜕x^2}}+{\displaystyle \frac{{𝜕}^{2}w}{𝜕y^2}})\\ \tau ={\displaystyle \frac{E}{2(1+\nu )}}({\displaystyle \frac{𝜕u}{𝜕y}}+{\displaystyle \frac{𝜕v}{𝜕x}})=-{\displaystyle \frac{Ez}{1+\nu }}{\displaystyle \frac{{𝜕}^{2}w}{𝜕x𝜕y}}\end{array}$$

(17)

From the definitions of the bending moment and torque, the relationship between the bending moment and torque, curvature, and torsion can be obtained by integrating along the plate thickness (Simpson/Gaussian integration). Thus, the relationship between the generalized stress $M$ and the generalized strain $\kappa$ of the thin plate is

$$M=D\kappa$$

(18)

where

$$\mathit{M}=\left[\begin{array}{c}{M}_x\\ M_y\\ M_{xy}\end{array}\right]=\left[\begin{array}{c}{\int }_{-h/2}^{h/2}{\sigma }_{x}z\mathrm{d}z\\ {\int }_{-h/2}^{h/2}{\sigma }_{y}z\mathrm{d}z\\ {\int }_{-h/2}^{h/2}\tau z\mathrm{d}z\end{array}\right]$$

(19)

$$\mathit{D}={\displaystyle \frac{E{h}^3}{12(1-{\nu }^2)}}\left[\begin{array}{ccc}1& \nu & 0\\ \nu & 1& 0\\ 0& 0& \left(1-\nu \right)/2\end{array}\right]$$

(20)

$${\kappa }^{\mathrm{T}}=\left[\begin{array}{ccc}-{\displaystyle \frac{{𝜕}^{2}w}{𝜕x^2}}& -{\displaystyle \frac{{𝜕}^{2}w}{𝜕y^2}}& -2{\displaystyle \frac{{𝜕}^{2}w}{𝜕x𝜕y}}\end{array}\right]=\left[\begin{array}{ccc}{\kappa }_x& {\kappa }_y& 2{\kappa }_{xy}\end{array}\right]$$

(21)

The researchers modeled the blades based on multiple shell theories and studied the structural characteristics of the blades. The researchers considered factors such as the pre-mounting angle, pre-twist angle, assembly gap [31], etc. Jie et al. [32] simplified the blade into a cantilever conical shell model, which is based on first-order shear deformation theory. The effects of the pre-mounting angle and pre-twist angle were investigated. Xie et al. [33] developed an improved analytical model that has rotating twisted shrouded blades with stagger angles. Sun et al. [34] investigated the effect of the blade’s rotational speed and system damping using a shell model with pre-twist angles. This simple and practical modeling approach is commonly used for shell models, as shown in Figure 6. Han et al. [35] used a typical cross-section model for a blade with assembly gaps and cubic structural nonlinearities. The main resonance characteristics were investigated under transverse displacement excitation and aerodynamic loading. Sinha et al. [36] derived new governing equations utilizing orthogonal thin-shell bending theory. On this basis, Chen et al. [37] investigated the effect of the system parameters on the modal characteristics. The development of the above-mentioned shell models provides support for describing the stresses and deformations of the blade during rotation. It provides a theoretical basis for further structural design and performance optimization.

It should be noted that the two-dimensional model is a simplification of the actual three-dimensional structure, and there may be a certain degree of error in carrying out the analysis. Therefore, it is necessary to consider different factors in the actual engineering design to ensure the accuracy and reliability of the model.

2.2.3. Solid Element

Compared with two-dimensional plate/shell elements, the advantages of three-dimensional solid elements include applicability to various types of structures with non-uniform properties and complex geometries; better numerical stability and convergence when dealing with structures with large thicknesses; ability to more accurately simulate complex loading conditions such as multi-physics field coupling; and suitability for detailed localized analyses. Therefore, three-dimensional solid finite elements are widely used in analyzing and designing aero-engine blades [38].

Common three-dimensional solid elements can be of two types: hexahedral and tetrahedral. Tetrahedral meshing is simple, but not very accurate and has a large number of meshes. Hexahedral cell delineation takes a lot of time and requires high meshing experience, but the number of meshes is small, which saves computation time and has high accuracy. In practice, it can be selected according to different engineering applications.

As shown in Figure 7, the straight hexahedral element has edge lengths 2a, 2b, and 2c. the straight hexahedral element has 8 principal nodes, ‘1–8’, 3 displacement components per node, and 24 degrees of freedom, and each displacement function can include the following 8 terms.

$$u(x,y,z)=a_1+a_{2}x+a_{3}y+a_{4}z+a_{5}xy+a_{6}yz+a_{7}zx+a_{8}xyz$$

(22)

The other two displacement functions also take the same basis, and similar displacement functions can be written. Similar to the planar rectangular element, the shape function of each node can be obtained by Lagrange interpolation.

$$\begin{array}{c}{N}_1={\displaystyle \frac{1}{8abc}}(a+x)(b-y)(c-z), N_2={\displaystyle \frac{1}{8abc}}(a+x)(b+y)(c-z)\\ N_3={\displaystyle \frac{1}{8abc}}(a+x)(b-x)(c-z), N_4={\displaystyle \frac{1}{8abc}}(a-x)(b-y)(c-z)\\ N_5={\displaystyle \frac{1}{8abc}}(a-x)(b-y)(c+z), N_6={\displaystyle \frac{1}{8abc}}(a+x)(b+y)(c+z)\\ N_7={\displaystyle \frac{1}{8abc}}(a-x)(b+y)(c+z), N_8={\displaystyle \frac{1}{8abc}}(a-x)(b-y)(c+z)\end{array}$$

(23)

If $\epsilon ={\displaystyle \frac{x}{a}},\eta ={\displaystyle \frac{y}{b}},\zeta ={\displaystyle \frac{z}{c}}$, then Equation (23) can be expressed in dimensionless natural coordinates as

$$\begin{array}{c}{N}_1\left(\mathit{\xi },\mathit{\eta },\mathit{\zeta }\right)={\displaystyle \frac{1}{8}}\left(1+\mathit{\xi }\right)\left(1-\mathit{\eta }\right)\left(1-\mathit{\xi }\right)\\ N_2\left(\mathit{\xi },\mathit{\eta },\mathit{\zeta }\right)={\displaystyle \frac{1}{8}}\left(1+\mathit{\xi }\right)\left(1+\mathit{\eta }\right)\left(1-\mathit{\xi }\right)\\ N_3\left(\mathit{\xi },\mathit{\eta },\mathit{\zeta }\right)={\displaystyle \frac{1}{8}}\left(1-\mathit{\xi }\right)\left(1+\mathit{\eta }\right)\left(1-\mathit{\xi }\right)\\ N_4\left(\mathit{\xi },\mathit{\eta },\mathit{\zeta }\right)={\displaystyle \frac{1}{8}}\left(1-\mathit{\xi }\right)\left(1-\mathit{\eta }\right)\left(1-\mathit{\xi }\right)\\ N_5\left(\mathit{\xi },\mathit{\eta },\mathit{\zeta }\right)={\displaystyle \frac{1}{8}}\left(1+\mathit{\xi }\right)\left(1-\mathit{\eta }\right)\left(1+\mathit{\xi }\right)\\ N_6\left(\mathit{\xi },\mathit{\eta },\mathit{\zeta }\right)={\displaystyle \frac{1}{8}}\left(1+\mathit{\xi }\right)\left(1+\mathit{\eta }\right)\left(1+\mathit{\xi }\right)\\ N_7\left(\mathit{\xi },\mathit{\eta },\mathit{\zeta }\right)={\displaystyle \frac{1}{8}}\left(1-\mathit{\xi }\right)\left(1+\mathit{\eta }\right)\left(1+\mathit{\xi }\right)\\ N_8\left(\mathit{\xi },\mathit{\eta },\mathit{\zeta }\right)={\displaystyle \frac{1}{8}}\left(1-\mathit{\xi }\right)\left(1-\mathit{\eta }\right)\left(1+\mathit{\xi }\right)\end{array}$$

(24)

or written in generic form as

$$N_i(\xi ,\eta ,\zeta )={\displaystyle \frac{1}{8}}(1+{\xi }_i\xi )(1+{\eta }_i\eta )(1+{\zeta }_i\zeta )\hspace{1em}(i=1,2,\cdots ,8)$$

(25)

• General three-dimensional models

The design of engine blades can be divided into two categories: forward design and reverse design. Most of the engine blades belong to commercial blades, and there are a lack of key initial design data, especially the more important airfoil data are unknown. It has become a mainstream trend to use digital inverse modeling technology to establish the blade computational model required for research [39]. Various methods have been applied to blade three-dimensional modeling and analysis in recent years. They cover vibration testing, scale-down modeling, parametric modeling, point cloud processing, and finite element analysis.

Bayoumy et al. [40] derived a continuum-based three-dimensional FE model of an elastic wind turbine blade using the absolute nodal coordinate formulation (ANCF). An effective model consisting of six thin-plate elements was proposed for this blade with an inhomogeneous and twisted nature. The validity of the model is demonstrated by comparing the results computed with the ANSYS code. Yeo et al. [41] systematically compare the one-dimensional and three-dimensional analyses of advanced geometrical blades. The intrinsic frequencies of the two models were calculated at different rotor speeds and the differences were quantified [42]. The results show that the one-dimensional model can capture the free vibration characteristics of various advanced geometry beams and blades fairly well when there is no coupling arising from the geometry or material. This holds for at least the six lowest frequency modes when the beam length is greater than 10 times the chord. When coupling arises, discrepancies between one-dimensional and three-dimensional analyses occur, especially for high-frequency modes. Kee et al. [43] carried out an analysis of the dynamics of rotating blades using an eighteen-node solid element. One-dimensional beams and three-dimensional solid models were compared. Yuan [44] introduced a novel reduced-order model technique to describe the dynamic behavior of turbofan aero-engine blades. The model provides excellent accuracy with reduced computational time compared to a high-fidelity FE model. Lee et al. [45] used the kriging method to construct a surrogate model to determine the optimal aerodynamic improvements for a three-dimensional single-stage turbine. This optimization method significantly reduces the design variables and makes the optimization process more efficient. The optimized rotor shape greatly increases the loads that can be carried by the blades. Noever-Castelos et al. [46] introduced a fully parametric blade modeling method. The method consists of a parametric definition based on a spline of all the design and material parameters, which enables fast and easy parametric analysis. The validity of the model was verified after comparison with measured blades. Liu et al. [47] used modified cracked hexahedral finite elements to model three-dimensional cracked blades and other cracked structures. Figure 8 shows a single-blade model after meshing, commonly used in the analysis of three-dimensional models. NASA rotor 37 [48] is an open-blade geometry. The blade is designed to have an air pressure ratio of 2.05, a rotor speed of 454 m/s, and a rotor exhibition ratio of 1.19. It has multiple circular arc blade shapes with a tip hardness of 1.3 and an aspect ratio of 1:19. Kojtych et al. [49] introduced a blade model generation code based on open NASA blades, called OpenMCAD, for the subsequent utilization of open-blade models. These works are important for improving the design accuracy and performance optimization of blades, as well as exploring the application of machine learning in blade optimization.

2.

Three-dimensional model considering friction behavior

Friction affects the dynamic behavior of blades and system stability in blade vibration analysis. Different friction models can simulate realistic contact forces [51]. To study the friction behavior of the blade, Guo et al. [52] established a new dynamic model for the elastic support of the blade based on the Timoshenko beam theory and gradually improved it to a three-dimensional model. In the study of the damping characteristics of flexible blades, different friction models are used to consider the contact force between the damper and the blade. Li et al. [53] used a one-dimensional friction model and a three-dimensional friction model at the root of the blade and the damper under the flat table, respectively. Liu et al. [54] proposed a friction contact stiffness model to describe the friction forces at different rough interfaces and under different positive pressures. Yu [55] found that when the rotor is in a sudden state of unbalance and inertial asymmetry, the friction impact will be the main factor determining the dynamic behavior of the rotor. The states of motion that may occur under certain conditions include intermittent friction and synchronized full-ring friction. Wu et al. [51] established a three-dimensional model of a multi-bladed system containing under-platform dampers (UPDs) and discretized the model using the FEM.

Researchers are gradually realizing that there is a large difference between the single mechanical relationship and the actual loads on the blade, and the analysis results often cannot effectively serve the structural design. Therefore, to make the blade response analysis closer to reality, the simple unidirectional fluid–solid coupling model and bidirectional fluid–solid coupling model began to be applied to blade frequency response analysis and vibration research.

3.

Three-dimensional model considering fluid action

To accurately simulate the effect of aerodynamic loads on the vibration characteristics of blades and improve the accuracy and reliability of the analysis, the fluid–solid coupling model was studied. Filsinger et al. [56] proposed an analytical method for unidirectional coupled CFD-FEM for determining the dynamic loading of turbine blades. Mao et al. [57] developed a unidirectional fluid–solid coupling model for blades and calculated the equivalent stress distribution of the blade under constant aerodynamic forces. In unidirectional fluid–solid coupling applications, researchers assume that the blade is in a small deformation state under the fluid acting on the blade. This assumption is based on constant flow. In actual operation, the flow inside a pressurized gas turbine is often greatly affected by structural deformation. This calculation method will bring a large error in the calculation results. Subsequently, researchers began to explore bidirectional fluid–solid coupling numerical algorithms, to solve the error brought by the unidirectional coupling assumption.

Aerodynamic intelligent modeling based on machine learning methods has been applied in assessing blade vibration stability. Zhang et al. [58] used the eXtreme Gradient Boosting (XGBoost) algorithm in machine learning to build a three-dimensional non-stationary aerodynamic reduced-order model. The prediction results of the machine learning model were compared with those of CFD, to evaluate and validate the reliability of the intelligent model of blade aerodynamic loads. The results show that the prediction results based on the machine learning model are in good agreement with the CFD results, and the computational efficiency is significantly improved. Janeliukstis et al. [59] used continuous wavelet transform to estimate the modal parameters of the blade flapping bending mode based on the acceleration response. The proposed modal parameter extraction method extracts a large number of observations to enable statistical decision-making and machine learning capabilities. Yang et al. [60] used a fuzzy neural network PID to control the input parameters and simulated the control of the vibration loads on the rotor hub.

The research on numerical algorithms for the bidirectional fluid–solid coupling of blades has developed rapidly [61] and has been more widely used. This is due to the continuous improvement in the research system of computational structural mechanics, computational fluid dynamics, and other disciplines. Cai et al. [62] analyzed the internal and external flows, temperatures, and thermal stresses of the turbine blade by using the coupled heat-flow–solid method based on the original and improved models of the turbine blade established by finite elements. The results show that the coupled heat-flow–solid method can better predict the ablation and thermal stress damage of the turbine blade. Yu et al. [63] used a three-dimensional time-domain fluid–solid coupled method to study the aerodynamic stability of blades. Li et al. [64] established a three-dimensional shell model of the blade based on Solidworks and studied the stress and deformation of the blade in the flow field based on the fluid–solid coupling method. Internal resonance as an important dynamic behavior in multi-degree-of-freedom nonlinear vibration systems may also occur during blade vibration. Wang et al. [65] constructed a three-degree-of-freedom model to simulate blade and airflow interaction.

2.3. Semi-Analytical Model

The semi-analytical method is mainly based on the blade dynamics model established by the beam theory and the plate/shell theory, which can well balance the calculation accuracy and efficiency. In this model, the coupling effect between the blade and blade, and the coupling between the blade and wheel disk are mainly simulated by a massless spring. Most of the semi-analytical methods are based on the Euler–Bernoulli beam theory and Timoshenko beam theory, which have been introduced in the previous section. Reboul et al. [66] proposed a simplified semi-analytical model to predict the noise emitted by helicopter blades. They focused on the load noise due to blade vortex interaction. Li et al. [67] compared the FEM with the semi-analytical method. Two dynamic models of a rotating shell system considering the effect of the frictional impact were developed using continuous beam theory and the FE theory. The reliability of the semi-analytical model is demonstrated by the fact that the two models yield the same blade intrinsic frequencies and mode shapes. Bontempo et al. [68] proposed a semi-analytical model that can deal with hubs of arbitrary shapes and rotor load distributions. The method strongly couples the flow induced by the rotor and the hub. The efficiency and feasibility of the method are demonstrated by comparing it with an actuator disk model based on computational fluid dynamics techniques. Gozum et al. [69] developed a novel semi-analytical model for the dynamic analysis of plates with discrete and/or continuous inhomogeneities. Xi et al. [70] developed a semi-analytical model for the aerodynamic damping of horizontal-axis wind turbines. The model can accurately predict the dynamic response of horizontal-axis wind turbines under wind–seismic combined load excitation. Guo et al. [71] proposed a semi-analytical model for a flexible variable-section disk blade system with elastic support, as shown in Figure 9. The proposed semi-analytical model considered the transverse vibration of the disk as well as the longitudinal and transverse vibration of the blade. In addition, the validity of the proposed model is verified by comparing the intrinsic frequencies, model shapes, and vibration responses obtained from the proposed model, FE model, and experiments. The results show that the proposed semi-analytical model has a high accuracy and computational efficiency, which can provide an alternative solution for predicting the vibration characteristics of variable cross-section disk blades and overcome the problem of the high computational cost of the FE model.

2.4. Summary of Rotating Blade Mechanics Models

According to the previous research work of researchers, the research of the vibration problem is based on classical theory, using different assumptions or constraints to solve the vibration problem under different working conditions. Most of the blade vibration analysis models can be categorized as lumped-mass models, semi-analytical models, and FE models. FE models can be categorized into independent beam models, plate and shell models, and solid models according to their geometry and dimension. The models for aero-engine blades are summarized in

Table 1. Summarization of blade models.

Model Type		Literature Sources	Applications	Advantages	Limitations
Lumped-mass model		[13,15,16,17]	Lumped-mass model is typically applied to simplified blade dynamic response prediction and preliminary design stages.	Simple principle, high computational efficiency.	Difficult to accurately determine mechanical parameters, can only simulate low-order modes, low modeling accuracy.
FE model	One-dimensional models (beam models)	[22,23,24,25,26,27,28,73]	Suitable for simplified modeling of blades for use at the stage of preliminary design and monolithic analysis, usually for low-frequency vibration studies.	The calculations are efficient, suitable for preliminary design and quick analyses, and helpful for the initial understanding of the overall structure.	It does not accurately capture the complex three-dimensional geometry and higher-order vibration modes of the blade and may not be accurate enough for fine vibration analyses when the requirements are high.
	Two-dimensional model (plate/shell models)	[29,30,32,33,34,35,36,37]	More accurate simulation of bending and torsional vibration of blades for mid-frequency vibration studies.	It is suitable for more detailed vibration analysis and provides accurate simulation results for blades with regular geometries.	For some complex blades, such as twisted or multilayered structures, there may still be limitations in the geometry that do not allow for full consideration of three-dimensional effects.
	Three-dimensional models (solid models)	[40,41,43,44,45,46,47,50,51,52,53,54,55,58,59,64,65]	Suitable for accurate modeling of blades, able to take into account more geometrical, fluid action, and material nonlinear effects.	High accuracy of calculation results, able to take into account complex geometries and anisotropic materials, and suitable for complex structures and high-frequency vibration analysis.	High computational complexity, requires more computational resources, and is not suitable for rapid design and preliminary analysis.
Semi-analytical model		[66,67,68,69,70,71]	Semi-analytical models are suitable for dealing with nonlinear effects such as large amplitude vibrations, material nonlinearities, and multiscale problems.	Offering a balance between precision and computational speed.	High difficulty in model development, limited scope of application, high dependence on model accuracy, difficulty in determining parameters.

. The lumped-mass model has high computational efficiency and is usually used to understand the mechanism, but it cannot fully simulate the complex geometry of the blade. Semi-analytical methods are widely used because of their high accuracy and computational efficiency, but they also have difficulties in modeling complex blade structures. FE models have advantages in dealing with complex blade geometries, loading conditions, material properties, and boundary conditions. However, the computational efficiency of finite element models decreases with increasing degrees of freedom, which has prompted some researchers to use hybrid modeling approaches to improve the computational efficiency [18,72].

In summary, the choice of model depends on the specific needs of the study and the complexity of the problem. In the preliminary design and rapid analysis stages, it may be more practical to use beam and plate/shell elements in the FE model. Solid elements, on the other hand, are more appropriate for vibration analyses that require higher accuracy, especially when nonlinearities and fluid–structure interactions are considered. With the development of computers, machine learning will continue to play an important role in future research and lead to more breakthroughs and applications in this field.

3. Numerical Solution Methods

Linear systems are idealized approximations under specific assumptions, but the problem of the vibration analysis of blades is inherently nonlinear. The methods for solving nonlinear dynamical systems have gone through four periods: analytical, early numerical, semi-analytical, and computer-aided numerical methods [74]. Figure 10 shows the development history of numerical integration methods for nonlinear systems and the process of computer-aided numerical methods.

At the beginning of the study of blade vibration analysis, researchers mostly used direct solution of differential equations, Laplace transform, and Fourier transform. However, nonlinear equations cannot be solved due to their complexity, and closed analytical solutions do not even exist.

With the development of early numerical solution methods, commonly used numerical methods for blade analysis are Euler’s method and the Runge–Kutta method, Newmark-beta method, finite difference method [75], etc. Gerolymos et al. [76] used Euler’s equations to numerically integrate the blade surface formulation. Zhang et al. [77] used the Runge–Kutta method for numerical integration to determine the validity of the developed novel nonlinear vibration equations. Gu et al. [78] used the Runge–Kutta method to study the nonlinear vibration at two critical speeds nonlinear vibrations at 1:1 internal resonance. Robinson et al. [79] used the obtained modal shapes to build a system of ordinary differential equations (ODEs) and solved the ODEs using the Newmark-beta algorithm. Subrahmanyam et al. [80] solved the problem of calculating the intrinsic frequency and vibration mode of rotating blades by a modified finite difference method based on second-order center difference. Li et al. [81] solved the integral differential equations to obtain the intrinsic frequency and vibration mode by using the finite difference method. The accuracy and robustness of these methods have been validated by a large number of studies and have been integrated into many commercial finite element software such as ANSYS 2024 R1, ABAQUS 6.14, COMSOL 6.2, etc.

Subsequently, Poincaré proposed an asymptotic analysis method using small parameter expansions as an approximation (semi-analytical methods) for solving such complex dynamical system problems. In addition, common semi-analytic methods, such as the harmonic balance method, the symplectic integral method (for Hamiltonian systems), and the fractional-order calculus method, have been applied to solve blade nonlinear equation problems. Kan et al. [82] developed an approximate symplectic integration in a Hamiltonian framework to simulate the vibration of each blade in a system of misaligned vane disks. Semi-analytical methods require a large number of formulae derivation in the use of the process, often only the first few orders of the calculation, and the strong nonlinear problem-solving effect is not good. With the development of computer technology, new vigor has been injected into the solution of nonlinear dynamical systems.

The computer-aided numerical method essentially discretizes complex problems in spatial coordinate systems into algebraic equations (statics problems) or systems of ordinary differential equations (dynamics problems) by finite difference methods or FEM. The researchers applied this method to solve the nonlinear dynamics equations of a blade and showed good results. For solving the dynamics problem of a blade, it is mainly divided into global and local methods.

The global method is suitable for solving nonlinear dynamical systems with periodic characteristics, including the time-domain collocation method, the harmonic balance method, and the high-dimensional harmonic balance method, which are widely used in solving nonlinear numerical equations for blades. When the solutions of nonlinear dynamical systems are periodic, the traditional solution methods are the perturbation method and the harmonic balance method. Bakhle et al. [83] used the harmonic balance method to solve the problem of experimental blades experiencing flutter during wind tunnel testing at 85% and 75% of the rotational speed. The aeroelasticity code used was compared with the experimental data and the results were in good agreement with the experimental data, but the solution underestimated the pressure ratio by 3%. The above method is limited by complex symbolic derivations and weak nonlinearities, which are not favorable for high-precision calculations. The local method is to solve the problem by discretizing the time domain into a finite number of subdomains and then solving the problem. Each subdomain can be solved by various global methods, including numerical integration methods such as variational iterative integration and the improved Runge–Kutta method. Ferri et al. [84] developed three reduced-order models for eliminating the numerical stiffness problem using the singular perturbation method. The results show that the developed reduced-order models are consistent with physically driven approximation models [85]. The accuracy of each approximate model was checked by comparing it with the original full-order model for a range of system and excitation parameters. Tian et al. [86] calculated and corrected the intrinsic frequencies of the structural parameters by the matrix perturbation method. The results were then compared with finite element calculations as well as measured results. The results proved the reliability, validity, and practicability of the method. Nešić et al. [87] proposed a method for determining the steady-state amplitude–frequency response by the incremental harmonic balance method and the continuum technique. The obtained periodic solutions in the strongly nonlinear case were verified by the Newmark numerical integration method. The reliability of the incremental harmonic balance method for structural analysis based on the theory of nonlocal strain gradients in strongly nonlinear systems is well demonstrated.

The methods for solving nonlinear equations in blade vibration analysis have distinct advantages and disadvantages. Analytical methods, like Laplace and Fourier transforms, are limited by the complexity of nonlinear equations. Numerical methods (e.g., Runge–Kutta, Newmark-beta) offer accuracy and robustness but can be computationally intensive. Semi-analytical methods, such as the harmonic balance method, simplify derivations but struggle with strong nonlinearity. Computer-aided methods, including FEM and global/local methods, provide precise solutions but may involve complex symbolic derivations and can be limited by computational resources.

4. Dynamic Characteristics

4.1. Vibration Mechanisms and Phenomena

Blade vibration forms include bending, torsion, axial or longitudinal, coupled bending–torsion, resonance, local vibration, and chattering. These forms of vibration are an important part of the structural dynamics of the blade in engineering practice. All of them have a critical impact on the design, performance, and life of the blade.

There is a fundamental difference between the action of torsional and bending modes under vacuum conditions. Bending vibration causes the blade to induce alternating bending stresses, with resonant fatigue being the main cause of damage. Fatigue induced by low-order torsional vibration can also damage the blade. It has been shown that the bending intrinsic frequency of a blade is related to the length of the blade. Due to the complex coupling of these modes, multiple stable solutions may result. This provides additional possibilities and challenges for exploring material properties and applications.

Different types of excitation force will lead to different forms of blade vibration in the work of the blade. The blade is subjected to very complex excitation forces during operation. The main categories are mechanical and pneumatic excitation. Mechanical excitation belongs to non-direct excitation, such as the vibration of the compressor disk or hub. This vibration is transmitted to the blade through contact or interaction between the blade and other mechanical parts, causing blade bending, torsion, or compound vibration. Pneumatic excitation is due to the gas force acting directly on the blade, the excitation force is larger, and the harm to the blade is heavier. According to the nature of excitation, pneumatic excitation is divided into wake excitation, rotating stall, and chattering. The chattering frequency is determined by the blade’s own geometry and material properties and thus is also known as self-excited vibration. The nature and magnitude of these excitation forces can have a significant impact on the dynamic response and structural stability of the blade.

Other factors related to the blade vibration form are nonlinear effects, such as friction, collisions, and material nonlinearities; resonance effects, such as matching the resonance frequency; material and structural problems, such as material failures and structural design issues; environmental conditions, including temperature and humidity variations; and manufacturing and assembly problems such as poor manufacturing and assembly errors. The causes of blade vibration are usually a complex combination of multiple factors.

The phenomena of blade vibration refer to the specific manifestations of vibration, including amplitude, frequency, mode shape, etc., as well as complex dynamical behaviors such as resonance phenomena, internal resonance, periodic motions, multiplicative periodic motions, and chaotic motions [52,88]. With external aerodynamic loads, the blade has very complex dynamics such as flip-flop bifurcation, period doubling, and period shortening sequences. Aerodynamic loads lead to richer blade dynamics, including quasi-periodic motions and possible chaotic motions. These phenomena are largely affected by the causes and forms of blade vibration. These nonlinear dynamical phenomena can lead to instability and fracture of the blade, which can damage the whole engine structure. Therefore, it is important to avoid or control these nonlinear dynamics that may occur during blade operation.

In addition, the difference in materials also causes a change in the characteristics of the vibration. The introduction of the FG graphene plate gives the rotating pre-twisted composite cantilever plate the characteristics of a stiff spring [89]. The nonlinear characteristics exhibited by the curve results become apparent at specific frequency ratios and rotational speeds. Meanwhile, the FG graphene plate-reinforced rotating pre-twisted composite cantilever plate may exhibit chaotic vibration in a specific parameter range, and the nonlinear vibration amplitude of the graphene sheet-reinforced cantilever plate varies from model to model. Miller et al. [90] have used bird strike tests to verify the impact resistance of composite materials based on engineering data. The final results show that due to the improvement in the ply lay-up configuration, the addition of new materials, and test settings of composite materials, the bird strike results of the blades have changed from serious damage to almost no visual damage.

In the field of aero-engines, it is crucial to study the impact of blade vibration on safety and reliability. Considering parameters such as the blade’s material components, rotational disturbance speed, and airflow velocity, the type of excitation force and vibration form of the blade must be considered comprehensively. Safe and reliable operation of the blade can be ensured by taking appropriate measures to reduce the impact of the excitation force on the blade.

4.2. Modal Characteristics

There is a lot of research on the modal characteristics of structures with different material shapes, such as rotating composite laminate blades, swept-tip paddles, tilt-symmetric laminate combination beams, and other mechanical structures. The intrinsic frequency and modal shapes of rotating blades can be affected by several factors. Among them, the rotational speed, the characteristic parameters of the blade, the coupling effect between the blade and other factors, and the nonlinear characteristics are the main factors affecting the intrinsic frequency and modal vibration pattern. Li et al. [91] investigated various factors affecting the intrinsic frequency and modes of engine blades. Figure 11a–d show the stress distribution of the blade subjected to rotational speed, aerodynamic load, thermodynamic and thermal field, the coupling of the centrifugal force field and thermal field, and then the modal analysis that was carried out. The blade vibration mode is minimally affected by the centrifugal force field, aerodynamic force field, temperature field, and thermal force field. Both the centrifugal force field and the temperature field cause significant changes in the intrinsic vibration frequency of the blade under operating conditions. The centrifugal force field increases the vibration frequency of the blade, while the temperature field decreases the frequency. Kou et al. [92] found that the intrinsic characteristics of the blade change under different loads, considering the effect of the centrifugal and steady-state aerodynamic loads.

As the rotational speed changes, the intrinsic frequency of the system changes due to factors such as Coriolis and centrifugal forces. This may lead to frequency shifts and frequency crossover phenomena. Chen et al. [37] showed that the rotational speed affects the intrinsic frequency and vibration mode of rotating composite laminate blades. Figure 12 shows the third- and fourth-order modes of the blade at different rotational speeds. The frequency and vibration mode increase with the increase in the pre-twist angle, stagger angle, and hub radius, but decrease with the increase in the thickness ratio. Kee et al. [43] explored the intrinsic frequency of swept-tip propellers. The results show that the frequencies of all modes increase with the increasing rotational speed and are particularly affected by the centrifugal reinforcement effect.

In addition, the parameters of the blade also affect the intrinsic frequency of the rotating blade. Researchers have investigated the effect of many parameters on the vibration characteristics such as the pre-twist, stagger angle, hub radius, porosity, composite material properties, temperature, rotational speed, external excitation, and thickness ratio. The interaction of many factors also affects the vibration characteristics of rotating blades. Yoo et al. [73] studied the bending vibration of a rotating combined beam flap with an inclined symmetric layer. As the angular velocity increases, the intrinsic frequency of this combined beam flap increases. The slope of the position of the intrinsic frequency increases as the hub radius ratio increases. Chandiramani et al. [23] detailed the effect of pre-twist on the intrinsic frequency at a low angular velocity or large layer angle conditions. Pre-twist, the ply angle affects the intrinsic frequency, but the effect on the third coupling intrinsic frequency diminishes with the increasing rotational speed. Meanwhile, the increase in pre-twist produces a combined effect of softening and stiffening on the same eigenfrequency. Yang et al. [93] analyzed a three-dimensional model with a 15° inclined blade model to describe the different vibration patterns of a turbine blade. Studies have shown that the vibration stresses of the blade at the resonant speed are much higher than those at other speeds. Inala et al. [94] showed that many factors such as power-law exponent values, width taper, etc., also affect the intrinsic frequency. Ebrahimi [27] studied the effect of porosity on FGM materials and found it is related to the power-law exponent. The width taper ratio also affects the vibration frequency of the beam.

The blade–disk coupling structure plays a very important role in an aero-engine, and the modal vibration pattern will be more complex due to the coupling effect. Wang et al. [95] found that the rotational speed has a great influence on the intrinsic frequency and stiffness effect of the coupled structure. Li et al. [96] found that the centrifugal load has a significant effect on the vibration characteristics of the lobe–disk structure. Elhami et al. [97] coupled the vibration and pressure variations of the rotor and analyzed the characteristics of the structure under non-fluid–solid coupling. The applied pressure was found to have a small effect on the modal but a direct effect on the amplitude. Under the action of applied moments on the rotor surface, the applied pressure has a significant effect on the twisting vibration. In addition to the blade–disk coupling structure, the shrouded blade has also been studied extensively. Shrouded blades will also consider the influence of the contact state of the leaf shrouded, the friction of the leaf shrouded, and other factors compared to the general blade. Zhou et al. [98] found that the intrinsic frequency and modal vibration patterns of the whole round-shrouded blade are quite different from those of the individual blades. The vibration modes and stress distribution are essentially unchanged at different operating speeds but are strongly influenced by temperature. Zhao et al. [99] found the presence of annular bending vibration, axial bending vibration, and combined vibration of different orders in the blade.

Blade vibration becomes more complex under the effect of nonlinear factors. Kee et al. [43] analyzed and compared the variation in modal frequencies of each order under linear and nonlinear conditions. Under nonlinear conditions, the magnitude of change in the modal frequencies of each order is small, but the position of the modal shape may change abruptly. The effect of rotation on specific modes is different from linear analysis in shallow-shelled blades and shows complex effects on deep-shelled blades as well. The dynamic properties of shell blades are significantly affected by the blade curvature and geometrical nonlinearities. Wang et al. [100] pointed out that the first two intrinsic frequencies of the system have different trends to the power-law exponent. As a result of the nonlinear modal interaction between the first two modes, the system shows a complex frequency response relationship. The frequency response curves show multivalued characteristics. There are multiple stable branches, showing obvious nonlinear stiff spring characteristics and a wide frequency range of the two-mode resonance.

The modal characteristics of blades are affected by many factors, and the study of these factors is of great significance for guiding blade design optimization, fault diagnosis, and improving the safety of the aircraft.

4.3. Vibration Response

The vibration state of the system caused by the blade under the action of excitation is called the response, and the steady-state response is usually analyzed by the amplitude– and phase-frequency characteristics of the system. Resonance occurs when the excitation frequency is close to the intrinsic frequency of a linear system [101], resulting in fatigue damage to the blades. The nonlinear vibration system will exhibit the phenomena of jumping, frequency doubling, and internal resonance under excitation. This section summarizes and analyzes the vibration-influencing factors of the blade, as well as three different common vibration characteristics, which are the internal resonance, chattering, and asynchronous resonance phenomena. It is found that the dynamic response of the system is affected by different factors, such as the speed, aerodynamic force, friction coefficient, asymmetric configuration [102], and so on.

4.3.1. Influencing Factors on Blade Vibration Response

The vibration response characteristics of the blade are not only affected by the speed but also by the aerodynamic force and turbine load. Li et al. [96] found the effect of the centrifugal load on the amplitude of the system. When the mode of vibration is dominated by internal mass vibration, centrifugal loading reduces the vibration frequency. When the vibration mode is dominated by external mass vibration, its frequency increases with the centrifugal loading. Pan et al. [103] studied the resonance of blades of a vanless runoff turbine. It was found that there is a V-shaped relationship between the blade amplitude and turbine load at a low excitation order. At a high expansion ratio, the amplitude is positively correlated with the load. Xie et al. [104] studied the coupled vibration of the overall structure of a double rotor blade and found that the rotational speed affected the vibration amplitude of the blade. With the increase in the rotational speed, the vibration amplitude decreased. The lateral external force influences the vibration of the whole disk blade, and the amplitude increases with the increase in the accompanying force.

Friction affects the resonant frequency and amplitude of the blade, for example, a different surface roughness, normal load, contact area, friction coefficient, geometrical deformation, and friction direction changes will change the dynamic response characteristics of the blade [105]. For the dynamic response vibration characteristics analysis of shrouded blades, Liu [54] showed that the resonance frequency and amplitude of shielded blades with a different surface roughness are related to the initial normal load and contact area. Li et al. [106] studied the effects of the blade speed, contact interface friction coefficient, and external excitation amplitude on the natural characteristics of blades by using the ideal dry friction model and FEM. As shown in Figure 13, it can be seen that the amplitude–frequency curve is also different at different speeds and the rotational speed will affect the nonlinear phenomenon and the resonance peak. Wang et al. [107] considered large geometrical deformations, positive pressure on the contact surface of the shoulder table, and the angle between the friction surface and the vibration direction of the blade in the blade study. They found that the change in the direction of the friction affects the resonance characteristics of the system and the amplitude–frequency response curve varies considerably. Kojtych et al. [49] performed a typical analysis of blade-tip/casing contact related to the partial blade rotors (35, 68, and 74A2) supplied by NASA, especially the frictional interactions. The results show that the dynamics of blade-tip/casing contact are very complicated. The lack of a wear-resistant coating promotes vibration behavior involving higher-frequency modes with potentially low vibration amplitudes. When the angular speeds are slightly higher than the angular speeds at the intersection of the first bending mode and engine order 4, each blade exhibits a very obvious vibration behavior.

The frequency and amplitude of blade vibration will not only be affected by the rotational speed, but also by the airflow velocity, excitation amplitude, and shape of the damping blade [108]. It is also necessary to investigate the nonlinear characteristics of the blade vibration response. Wang et al. [100] pointed out that the translational velocity, excitation amplitude, damping, and in-plane tension had a significant effect on the nonlinear dynamical response of the translational FGM plate, especially as the quantitative effects of the different power-law indices were different. Wu et al. [109] mainly discussed the influencing factors of blade nonlinear vibration characteristics, including the rotor torsion amplitude, steady rotational speed, incoming gas flow rate, blade shape parameters, etc. Zhang et al. [110] investigated the complex nonlinear vibration and internal resonance of a pre-twisted rotating cantilevered rectangular plate induced by cross-section variation and rotational speed change under aerodynamic forces. The system was found to exhibit hardening and softening nonlinear characteristics, multivalued and jumping phenomena under cross-section change, and rotational speed perturbation.

Different materials also have different effects on the dynamic response of the system [111]. Hao et al. [112] thoroughly discussed the effect of different asymmetric double-layer low-temperature silicon carbide beams on the nonlinear vibration behavior of the system. Chang et al. [113] studied a cantilever plate model of a functional gradient material and found internal resonance phenomena excited by higher-order modes during rotation. Aerodynamic forces cause the resonance to occur earlier, and an increase in the component parameters of FGM leads to a decrease in the peak value and a deepening of the hardening of the spring characteristics. Functional gradient materials have little effect on internal resonance transfer but do have an effect on mode complexity.

4.3.2. Resonance Behavior

The analysis of resonance phenomena can reveal a significant enhancement in the blade’s vibration response when the external excitation frequency is close to the blade’s intrinsic frequency, which plays a key role in avoiding structural fatigue, damage, and failure. Wu et al. [114] showed that the ratio of the first- and second-order modal frequencies of the structure was made close to 1:3 by the modal design. The experimental results show that the internal resonance behavior between the first and second modes can be stimulated by the first and second mode frequencies. At the same time, there are many kinds of combined resonance behaviors in the structure. Ding et al. [115] considered the effect of the temperature and volume fraction on the stability of the response at a 1:2 resonance based on FG plates. In this case, the temperature rise and volume fraction modes interact, and the presence and instability behavior of the pure modes change over different temperature ranges.

Internal resonance is a phenomenon that occurs during the vibration of a nonlinear system since the intrinsic frequencies of each order of modes are close or equal, resulting in the transfer of energy between multiple modes. It can be regarded as a specific response of a dynamical system to an external excitation. Internal resonances can be affected by factors such as precession, small differences in the rotational speed, damping coefficients [116], and gas pressure amplitudes. Failure to avoid or control these phenomena can lead to instability or fracture of the blades, which can damage the entire engine structure. Wang et al. [65] showed that under certain conditions, saddle nodes and Hopf bifurcations may occur in the response of rotating blades under the influence of upstream incoming flow. Reducing the vibration frequency helps to suppress structural vibration and there is an energy transfer phenomenon. Zhang et al. [117] found that when the main resonance occurs at 2:1, factors such as pre-deflection, small differences in the rotational speed, damping coefficients, and gas pressure have a significant effect on the frequency response of rotating blades under thermal gradients. Luo et al. [118] carried out a saddle-point bifurcation characterization study of an asymmetric Duffing system containing a constant excitation. The results show that increasing the system damping or reducing the amplitude of the simple excitation help to suppress the multiple solutions and complex vibration jumps in the main resonance response of the system.

4.3.3. Flutter Phenomenon

The flutter problem also occurs during blade vibration. The flutter problem is an aeroelastic instability phenomenon that couples aerodynamic, elastic, and inertial forces, which can cause blade fatigue or breakage quickly. The flutter phenomenon is affected by shock waves, blade top clearance, modes, and acoustic waves. Sanders et al. [119] successfully reproduced the stall flutter of a transonic fan with a low-aspect ratio through numerical simulation and analyzed that the shock wave is the main factor for energy transfer between the blade and flow field. Although flow separation exists in this working condition, the separation area is small and only plays a secondary role in energy transfer. The simulation results of Isomura et al. [120] also show that the vibration of the shock wave leads to the occurrence of flutter. Huang et al. [121] investigated the effect of tip clearance on aerodynamic damping with the help of plane turbine cascade experiments and found that a large gap (5%) would reduce the aerodynamic damping, while a small gap (1.25%~2.5%) would increase the aerodynamic damping. Vahdati et al. [122] focused on the effect of the vibration modes on fan flutter. Jin et al. [123] studied the torsional flutter problem of three-dimensional turbine blades. The angle of attack and back pressure cause the flutter phenomenon, and the intensity of the flutter can be investigated by studying the characteristic curves of the effect of the angle of attack and back pressure on the blade amplitude.

4.3.4. Non-Synchronous Resonance

With the continuous increase in the aero-engine compressor load and the decrease in blade damping, an aero-engine faces the problem of non-synchronous vibration, which further leads to blade failure. Non-synchronous resonance is a vibration phenomenon of the blade, which is generated due to the interaction between the moving blade and the surrounding fluid. According to the existing studies, the influencing factors of non-synchronous vibration are broad, as follows: assembly clearance, torsional geometrical nonlinearity [124,125], rotor shaft transverse vibration displacement, blade-free stream velocity, frequency ratio, channel vortex formed by tip leakage flow and main flow stream, aerodynamic excitation force, blade modal aerodynamic damping ratio, etc. Han et al. [35] focused on assembly clearance, torsional geometrical nonlinearity, and rotor shaft transverse vibration displacement on system vibration. It was found that reducing the rotor shaft vibration amplitude attenuates the amplitude of the main resonance response, while the assembly clearance is the main cause of the constraint hysteresis of the system. Increasing the blade-free flow velocity increases the vibration amplitude and the increase in the frequency ratio leads to a wider vibration response lag.

Overall, these studies have provided important information and methods for understanding and analyzing the vibration characteristics of turbine blades. However, there is still a need for more research into the dynamic behavior and stability of these systems.

5. Discussion and Prospect

Currently, there are some shortcomings in the simulation studies of engine blade vibration. Firstly, the current multi-physics field coupling analysis is usually based on the constant flow assumption, which simplifies or ignores the non-constant heat flow under actual working conditions. Secondly, there is a relative lack of simulation studies on new materials and composites in terms of blade vibration characteristics and fatigue life. In addition, in-depth experimental validation and data-driven studies are needed to improve the accuracy of the simulation models. Research on vibration analysis and complex nonlinear vibration under high-temperature and high-speed operating conditions is also relatively limited. In practical engineering applications, an in-depth study of these aspects will help to more comprehensively and accurately understand and simulate the dynamic characteristics of engine blade vibration, providing more reliable theoretical support for improving engine performance and flight safety. For future aero-engine blade vibration analyses, the following points are proposed:

• Enhancing the research of multi-physical field coupling problems. Structural vibration is affected by a variety of factors, including solid mechanics, aerodynamics, heat, and other physical coupling [9,10]. In different coupling situations, the vibration characteristics of the blade have large differences [61]. Future research can enhance the multi-physics field coupled simulation, especially considering the non-constant heat flow under real operating conditions, to improve the realism and accuracy of the simulation.
• More attention needs to be paid to the application of the newest materials for blades. The application of new materials opens up new possibilities in blade design. With a higher strength and lighter weight, they can change the vibration characteristics of blades and improve engine performance. Experiments based on engineering data [90] show that the addition of new materials improves the impact resistance of blades, which provides a strong reference for future research directions. Future research will therefore focus on material properties, component design, and test methods. This understanding will be critical in determining the suitability of these materials for practical use.
• Research should explore data-driven simulation methods. High-dimensional machine learning methods should be used as well as measured data to optimize the simulation model, reduce the computational costs, and improve the computational efficiency and accuracy. Many methods in machine learning are suitable for solving high-dimensional, constrained [126], and multi-objective optimization problems and the key to solving the nonlinear dynamics of blades is currently to reduce the computational complexity. More nonlinear factors, multi-physical field coupling, and other factors can be considered using data-driven models to improve the realism and reliability of the simulation.
• Structural parameters with nonlinear characteristics, such as damping, deflection, and external forces, have a significant effect on the vibration response of a structure. Future research could focus on optimizing these parameters to improve the performance and stability of the structure. In addition, active or semi-active control strategies can be considered to suppress or reduce nonlinear vibrations.