Influence of Fir-Tree Tenon Blade Parameters on Blade Dynamic Characteristics

In this paper, based on the theory of elastic structure analysis, considering the rotational softening and centrifugal stifening of the blade, the dynamic model of the fr-tree rotating blade and the macroslip hysteresis model of the dry friction on the contact surface of fr-tree tenon are established. Based on elastic mechanics, the pressure distribution model of fr-tree tenon is innovatively established and the relationship between ft clearance and contact pressure has been derived. After the blade is discretized by the fnite element method, the dynamic equation of the blade is solved iteratively by the Newmark-β numerical integration method. According to the dynamic equation of the blade, the infuence of the change of the tenon angle on the forced response and contact stress of the blade is derived. Finally, some novel conclusions are obtained by analyzing the infuence of pressure distribution, tenon angle, friction coefcient, rotational speed, and aerodynamic excitation on blade vibration characteristics. It provides a scheme and relevant basis for the design and development of tenon blades with the most damping efect at diferent speeds in industrial practice.


Introduction
A blade is an important component of turbomachinery such as aeroengine and turbine. Te quality and structure of a blade will directly afect the performance, reliability, and life of turbomachinery. According to statistics, engine failure caused by blade fatigue fracture accounts for more than half of the causes of engine failure. Te fatigue fracture of the blade is sudden, which causes most of the faults of the blade to bring huge economic losses and even casualties. Terefore, it is necessary to prevent the fatigue failure of the blade in industrial practice. Because one of the main causes of blade fatigue failure is the alternating stress in the blade, in recent decades, many scholars and manufacturers in high performance impeller machinery are very concerned about the vibration reduction of the blade to prevent fatigue cracks and even fractures [1][2][3][4]. In order to reduce the alternating stress in the blade and reduce the high cycle fatigue damage of the blade, it is a simple and efective way to reduce the vibration level of the blade by using the friction energy dissipation between the contact surfaces of the tenon joint structure. Terefore, the design of a tenon structure with good vibration damping to maximize its efectiveness in operation and the establishment of an accurate and efcient dynamics model are essential tasks in the design and study of tenon blades [5,6].
Zucca et al. [7] proposed a method for calculating the friction force on the contact surface during blade vibration. Considering the local contact characteristics, the dynamic equation of the dovetail tenon blade was established. Chen et al. [8] established the contact slip mechanical model considering the distribution characteristics of contact pressure and applied it to the dovetail tenon root. Te dynamic equation of the blade was obtained by the fnite element method. Finally, the harmonic balance method was used to solve the dynamic equation, and the accuracy and universality of the model were verifed. Based on the fnite element theory, Li et al. [9] established the fnite element model of a fexible dovetail blade by using the space beam element. Trough the change of friction and dry friction energy dissipation ratio in the process of blade vibration, the vibration reduction mechanism of the blade with dovetail root was expounded. Li et al. [10] established a macroslip friction model at the dovetail tenon root. On this basis, the linearization of nonlinear friction was realized, and the vibration response of the blade was calculated by the time domain method. Chen et al. [11] used 2D and 3D friction contact models to numerically study the friction damping efect on the root contact surface of the dovetail blade. Te hysteresis curves of the tenon root under diferent modeling parameters were analyzed, and the efects of physical parameters such as geometric parameters, friction coefcient, contact stifness, and tangential and normal load on friction damping efect were studied. Wang et al. [12] established the dynamic model of dovetail tenon blade, using spring and viscous damper to describe the normal pressure on the contact surface. Te efects of contact stifness, contact damping, and rotational speed on the dynamic characteristics of the blade are analyzed. She et al. [13] developed a continuum model of a wheel-dovetail beam and investigated the coupled vibration mechanism of the system by employing macroslip dry friction theory. Ma et al. [14] established the fnite element model of dovetail blade-disc and analyzed the infuence of rotational speed and invasion depth on blade vibration response and root contact behavior under blade tip rubbing conditions. Lemoine et al. [15] proposed a calculation method combining the HBM and the DLFT and analyzed the dynamic response of the dovetail tenon blade under diferent friction coefcients and excitation levels. Te change of contact force and contact behavior of the middle node of the contact surface was studied. Asia et al. [16] studied the root-damping efect of the dovetail tenon structure. Te efects of installation angle and centrifugal load on damping ratio and natural frequency were discussed, and the efects of loading times and specimen geometry on tangential stifness were analyzed. Huels et al. [17] established the fnite element model of the tenonconnected turbine blade and analyzed the infuence of the basic geometric design parameters of the blade on the nonlinear vibration response of the blade. Li et al. [18] established the dynamic model of dovetail blade based on the fnite element method and used the dry friction macroslip model to simulate the contact surface friction force. Te nonlinear dynamics of dovetail blade was studied. Li et al. [19] established the fnite element model of dovetail blade and used the one-dimensional dry friction model to capture the mechanical characteristics of the contact interface. Te infuence of tenon structure parameters on the vibration characteristics of the blade was studied.
With the in-depth study of tenon-joint friction damper, scholars have found that although the dovetail tenon-joint friction damper has a signifcant indigenous efect on the vibration reduction of the blade, it also has the following shortcomings: (1) as only one pair of contact surfaces are in play in the dovetail construction, the stresses on the individual contact surfaces are higher. (2) In order to ensure the strength of the tenon joint, the circumferential dimensions and weight of the tenon root are relatively large. (3) Te dovetail tenon structure can only play the role of friction damping when the rotational speed is low, and the damping efect is lost when the rotational speed is high. However, the fr-tree tenon structure not only overcomes the disadvantages of the dovetail construction but also allows for a larger gap between the tenon and the nonsupporting surface of the tenon groove, into which cold air can be passed for cooling, while more blades can be ftted due to the small circumferential size of the tenon. As a result, many scholars have turned their attention to the fr-tree tenon structure. Petrov and Ewins [20] studied the forced vibration response of a frtree tenon structure. A surface friction contact element was introduced in the contact area of the tenon, and the relationship between the friction damping at the root of the blade and the response amplitude of the blade was analyzed. Meguid et al. [21] carried out two-dimensional and threedimensional fnite element analysis on the fr-tree tenon root and pointed out that the three-dimensional calculation could accurately obtain the stress distribution in the blade root rim area, and discussed the infuence of design size on the blade root rim strength. Lavella and Botto [22] convert the centrifugal load of the blade to the tooth surface of the tenon and tenon groove equally when calculating the stresses in the blade and the wheel disc, while cutting down the root and the tooth groove for separate analysis when calculating the contact stresses in the tenon and tooth area, loading the centrifugal force of the blade to the upper surface of the root equally. Koscso and Petrov [23] investigated nonlinear contact interactions at the leaf roots through high-fdelity modeling of fr-tree tenon-and-groove bladed disc assemblies and nonlinear contact modeling of bladed disc contact blocks, analyzing the efects of the number, location, and distribution of nonlinear contact elements and fnding that small diferences in root geometry lead to signifcant changes in the nonlinear forcing response. Ferhatoglu and Zucca [24] proposed a numerical method for determining the variation range periodic response boundaries for all fr-tree turbine blades with frictional damping and a large number of contacts. Te system's damping loss factor was used as the objective function to minimize while satisfying the harmonic balance method (HBM), resulting in a system of nonlinear algebraic equations. Zhao et al. [25] established a twodimensional friction model considering the variation of normal force with excitation at the tenon-and-groove contact surface, investigated the efect of variable normal vibration on the blade response, and analyzed the efect of detuning of blade parameters such as the tangential stifness of the friction surface and structural stifness on the coupled vibration. Shukla and Harsha [26] used the fnite element method to develop a mathematical model of a fr-tree blade, on the basis of which the blade was analyzed and compared with experiments to verify the correctness of the model and the reliability of the response results. Shukla and Harsha [27] established the fnite element model of the fr-tree tenon blade and studied the infuence of diferent size cracks on the vibration characteristics of the blade by adjusting the size of the root crack. Te correctness of the fnite element model was verifed by comparing the analysis results with the experiment. Zucca et al. [28] established a fnite element model of a fr-tree tenon blade with a pin damper. After the matrix was simplifed by CB-CMS, the blade response was calculated based on the harmonic balance method. Te efects of three diferent types of dampers, cylindrical, asymmetric, and wedge-shaped on the vibration characteristics of blades were analyzed. Trough experimental comparison, it was found that the performance of asymmetric dampers was the best. Quaegebeur et al. [29] considered the centrifugal efect and the friction and separation of the tenon joint structure and established the fnite element model of the fr-tree tenon blade with platform damper. Te fnite element model was reduced by combining the nonlinear identifcation of the interaction nodes with the linear component mode. Te dynamic Lagrangian frequency time algorithm was used to analyze the vibration characteristics of the blade. Petrov [30] established the dry friction model of the fr-tree tenon structure through the surface friction contact element. Te dynamic equation of the blade was established by the fnite element method. Krack et al. [31] established the fr-tree tenon blade model with shroud and fange damper by considering microslip and aeroelastic efects. After model reduction, the blade's forced vibration and self-excited vibration were calculated and analyzed by the higher harmonic balance method. Koscso and Petrov [32] established the fnite element model of a fr-tree tenon blade with shroud and studied the vibration characteristics of an anisotropic mistuned blade by the harmonic balance method. Chen et al. [33] established a large fnite element model of the fr-tree tenon structure. A large number of faceto-face elements were used to simulate and capture the microslip phenomenon and action on the tenon contact surface. Te efects of the modal damping factor and blade modal shape on the vibration response were studied and analyzed.
In recent years, the research on the tenon joint structure has almost always been to improve the contact model or use the high degree of freedom model to improve the accuracy. However, the infuence of the contact pressure change of the tenon structure on the vibration characteristics of the blade was not considered. In previous studies, the pressure on the contact surface of the fr-tree tenon was evenly distributed to three pairs of surfaces, without considering the diversity of pressure ratio caused by assembly and other problems in industrial practice. Moreover, the intermittent excitation was almost used in the dynamic analysis of blades, without considering the infuence of actual aerodynamic excitation on the dynamic characteristics of blades.
In this paper, based on the theory of elastic structure analysis, considering the bending, shearing, rotating softening, and centrifugal stifening of rotating blades, the dynamic model of the fr-tree rotating blades is established. Based on the macroslip hysteresis model of the Coulomb friction theorem, the dry friction mechanical model is established on the contact surface of the fr-tree tenon. Under certain conditions, the mathematical expression that should be satisfed between the positive pressure and ftting clearance on the contact surface of the fr-tree tenon is innovatively proposed. According to the principle of force balance and the geometric relationship of the blade, diferent distribution modes of positive pressure on the contact surface of the tenon root are obtained. Te efects of pressure distribution, tenon angle, friction coefcient, rotation speed, and aerodynamic excitation on blade vibration characteristics are studied. It provides a scheme and related theoretical basis for the design and development of tenon blades with the most damping efect at diferent speeds in industrial practice.

Blade Dynamics Model
As a key component in rotating machinery such as gas turbines, a blade should have sufcient stifness and the ability to resist external excitation and suppress vibration. Te fr-tree tenon structure is widely used because of its high strength, small size, and lightweight. Considering the complexity of the fr-tree tenon blade and the non-negligible height-span ratio, the fnite element model of the shear coupled bending fr-tree tenon blade in a rotating state is established based on the theory of elastic structure analysis. Te fnite element model of the blade is shown in Figure 1.Here, oxyz is the local coordinate system of the blade. Te parameters of the fnite element model of the blade are shown in Table 1.
Based on the established fnite element model, the dynamic model of the blade is established using elastic mechanics and energy theory. Te energy Q e of each element of the rotating blade under harmonic excitation can be expressed by the following equation: where U e b is the strain energy caused by the bending deformation of the element, U e s is the strain energy caused by the shear deformation of the element, U e c is the centrifugal strain energy of the element, T e b is the kinetic energy caused by the bending deformation of the element, and T e s is the kinetic energy caused by the shear deformation of the element.

Shock and Vibration
In the local coordinate system oxyz, each of the potential and kinetic energies in equation (1) can be expressed separately in the following equation: where ω e b and ω e s are the displacements caused by the bending deformation and shear deformation of the blade, respectively.
Considering the general expressions of element strain energy and kinetic energy and combining them with equation (2), element bending stifness matrix k e b , element shear stifness matrix k e s , element centrifugal stifness matrix k e c , and element mass matrices m e b and m e s can be obtained. Te whole mass matrix M and stifness matrix K of the blade system are obtained by converting the element matrices in the local coordinate system to the whole coordinate system.
Te size of the damping determines the nature of the vibration of the system. In this paper, alpha-beta damping, also known as Rayleigh damping, is used as the damping of the system in combination with several damping forms commonly used in engineering. Its form can be expressed by the following equation: where M and K are the overall mass and stifness matrices of the system, while λ 1 and λ 2 are the mass and stifness damping coefcients, respectively. Te values of λ 1 and λ 2 can be determined by determining the two-order modal damping ratios and the inherent frequencies, which are expressed as follows: In equation (4), ω 1 and ω 2 are the frst two inherent frequencies of the system and ξ 1 and ξ 2 are the modal damping ratios corresponding to the frst two inherent frequencies, respectively.
According to the mass matrix, stifness matrix, and damping matrix of the blade, the fnite element dynamic equation of the blade is established. Te expression is as follows: where Q is the external excitation acting on the blade and F is the damping force generated by the tenon damper. Figure 1: Te fnite element model of the blade.

Dry Friction Model.
When the fr-tree tenon blade is working, the tenon root of the blade will be closely bonded with the wheel disc due to the centrifugal force so that the blade will be positioned on the wheel disc rotor. Tere is not only positive pressure due to centrifugal force but also dry friction due to blade vibration on the contact surface between the tenon and wheel disc. Te dry friction of the contact surface can consume the energy of the blade vibration to reduce the blade vibration. However, the dry friction forces acting on the blade root surface are very complex and in order to facilitate the description of the nonlinear dry friction forces, the following assumptions are made in this paper: (1) It is assumed that the blade tenon root never separates from the wheel disc during operation (2) It is assumed that the positive pressure on the contact surface between the tenon root and the wheel disc remains constant throughout the vibration of the blade (3) It is assumed that the contact angle of the tenon root remains constant during the vibration of the blade and that the blade only moves horizontally Using artifcial springs to simulate the boundary conditions of the model is a common method to deal with boundary problems [34,35]. Under the abovementioned assumptions, the dry friction model is established as shown in Figure 2. Te model simplifes the blade root to an ideal dry friction damper with massless and fexible. Te specifc parameters of the dry friction model are shown in Table 2, where i � 1, 2, 3; j � 1, 2.
Since the tenon root only moves horizontally y (t), equation (6) can be obtained according to the geometric relationship.
Te contact interface friction depends on the relationship between the spring force k ij (d ij − w ij ) and the maximum static friction force μ ij N ij . When the spring force is less than the maximum static friction force, the contact surfaces will stick to each other and the friction force is equal to the spring force. When the spring force is greater than the maximum static friction force, the contact surface will slip between each other and the friction force is equal to the maximum static friction force. Te expression is as follows: According to the equilibrium condition of force and ignoring the infuence of friction on the contact surface, the positive pressure on each contact surface can be expressed as follows: where N i is the component of the positive pressure on the ith pair of contact surfaces in the radial direction. N i and the centrifugal force N satisfy the following equation: According to the centrifugal force calculation formula and the blade parameters, it is easy to obtain that the centrifugal force N of the blade is as follows: Te friction f ij on the contact surface of the tenon is projected in the horizontal direction to obtain the component of the dry friction in the horizontal direction. Te expression is as follows:

Pressure Ratio
Model. Te small change of the ftting clearance between the tenon root and the tenon groove in the fr-tree tenon joint structure will cause a great change in the positive pressure on the three pairs of contact surfaces and then afect the nonlinear friction force on the contact surface of the tenon joint structure. Terefore, it is necessary to establish the contact model of the relationship between positive pressure and ftting clearance to realize the adjustable distribution of pressure. In order to describe the Shock and Vibration relationship between positive pressure and ftting clearance, this paper makes the following assumptions: (1) It is assumed that the positive pressure on the three pairs of contact surfaces remains unchanged during the vibration of the blade (2) It is assumed that the three pairs of contact surfaces of the blade are in contact and there is a constant contact pressure during the vibration process (3) It is assumed that the deformation process between the tenon root and the tenon groove interface is an elastic deformation process during the vibration of the blade Te fr-tree tenon contact model is shown in Figure 3, where δ 1 , δ 2 , and δ 3 are the ftting clearances between the tenon root and tenon groove when the blade is static.
Assuming that the frst pair of contact surfaces is the main bearing surface during vibration, the contact pressure N i on the three pairs of contact surfaces can be expressed by the following equation: where ∆ is the amount of deformation of the main bearing surface under contact pressure N 1 , ∆x 2 , and ∆x 3 are the gap between the second pair of contact surfaces and the gap between the third pair of contact surfaces, respectively, when the main bearing surfaces frst come into contact. k is the contact stifness of the tenon structure. In this paper, the frst item of the overall stifness matrix is taken as the contact stifness of the tenon structure. Te pressure on the three pairs of contact surfaces of the fr-tree tenon structure is N 1 , N 2 , and N 3 , respectively. By solving equation (12), when the contact pressure on the main bearing surface is N 1 , the deformation ∆ of the main bearing surface and the respective gaps ∆x 2 and ∆x 3 of the other two pairs of contact surfaces can be obtained. Terefore, when the ftting clearance between the main bearing surface is δ, the ftting clearance between the three pairs of contact surfaces of the fr-tree tenon joint structure can be expressed in the following equation:

Newmark-β Numerical Integration
Method. Newmark-β numerical integration method is a common method for solving diferential equations. Since the Newmark-β numerical integration method has the advantages of high accuracy, fast calculation, and high stability, it is widely used in the numerical calculation of the dynamic response of structures. In order to obtain the nonlinear dynamic response of the fr-tree tenon blade, the Newmarkβ numerical integration method is used to solve the dynamic equation of the blade in this paper.
According to the abovementioned, the dynamic equation of the fr-tree tenon blade is as follows: It is assumed that the displacement and velocity of the blade at the moment t i are X i and _ X i . According to the blade dynamics equation, the acceleration of the blade is as follows: Displacement of x on the tangent of the i-th pair of contact surfaces j mm α i Te angle between the i-th pair of contact surfaces 1 and the horizontal direction°β According to the displacement X i , velocity _ X i , and acceleration € X i of the blade at the moment t i , the vibration displacement X i+1 of the blade at the moment t i+1 can be obtained as follows: where It is assumed that the acceleration of the blade changes linearly in ∆t � t i+1 − t i . Te acceleration € X i+1 and velocity _ X i+1 of the blade at the moment t i+1 can be expressed as follows: where the values of parameters α and β are 0.5 and 0.25. Te Newmark-β numerical integration method is unconditionally stable under this condition.

Model Validation.
In order to verify the correctness of the dry friction model and the blade dynamics model developed in this paper, the results of the amplitude-frequency curves are compared with those of [6]. As the blade in the reference is a dovetail blade, the forces applied to the dovetail tenon root in the reference are all applied to the 3rd pair of contact surfaces of the fr-tree tenon root in this paper to simulate the forces applied to the dovetail tenon root in the reference. Secondly, the geometric and material parameters of the blade in this paper are adjusted to match those of the blade in the reference, and exactly the same external excitation Q � 500 sin (wt) is applied at the same location as the reference blade. Finally, taking the blade speed of 200 rad/s and 300 rad/s, respectively, the amplitude-frequency curve of the blade is made and compared with the reference. Te specifc implementation scheme of the fnite element model verifcation is shown in Figure 4, and the comparison results are shown in Figure 5. In Figure 5, the solid and dashed lines show the amplitude-frequency curves of the dovetail blade in the reference at 200 rad/s and 300 rad/s, respectively, while the dots and stars show the amplitude-frequency curves of the blade in this paper at 200 rad/s and 300 rad/s, respectively. As can be seen from the data in the graph, when the speed is 200 rad/s, the absolute error in amplitude is the greatest at ω � 1740 with a value of −2.92 × 10 −5 mm, and the relative error is the greatest at ω � 220 with a value of 2.55%. When the rotation speed is 300 rad/s, the absolute error in amplitude is the greatest at ω � 1630 with a value of 2.64 × 10 −5 mm, and the relative error is the greatest at ω � 200 with a value of 2.20%. As both the absolute and relative errors are small and within allowable limits, the dry friction model and kinetic model developed in this paper can be proved to be correct.

Results and Discussion
In order to further explore the vibration characteristics of the fr-tree tenon blade, this section analyzes the forced vibration response of the blade under diferent pressure distributions, tenon angle, friction coefcient, rotation speed, and aerodynamic excitation. Firstly, the geometrical and material parameters of the blade system are determined and the specifc values are shown in Table 3. Secondly, it is assumed that the friction coefcient on each contact surface of the tenon and tenon groove is the same, and the shear stifness of the three pairs of contact surfaces is k ij � 8 × 10 6 N/m. Finally, the harmonic excitation P(t) � A sin(ωt + φ) acting on the middle of the blade is used to simulate the external excitation of the blade, where the excitation amplitude A � 500 N, and ω and φ represent the excitation angular frequency and phase, respectively.

Te Infuence of Pressure Distribution.
It is known that diferent contact modes of the fr-tree tenon will lead to signifcant changes in nonlinear forced response. Considering that diferent ftting clearances will greatly change the distribution mode of pressure on the root of the fr-tree blade, so as to change the boundary conditions of the blade, and then afect the vibration characteristics of the blade. In  order to intuitively show the pressure distribution mode of the fr-tree tenon structure, the pressure distribution cloud diagram of the fr-tree tenon structure is made. Te results are shown in Figure 6.
In order to further study the infuence law of pressure distribution on blade vibration characteristics, the friction coefcient of the blade tenon root contact surface is 0.1. Te amplitude-frequency response curves of the blade tip under diferent pressure distributions are made to observe the infuence of pressure distribution, and the results are shown in Figures 7 and 8. As can be seen from the fgure, both the positions of the main bearing surface and the pressure distribution have a signifcant infuence on the vibration characteristics of the blade. When the rotational speed is low, the fr-tree tenon joint structure has a better vibration reduction efect in the case of only one pair of contact surfaces acting than in the case of three pairs of contact surfaces acting together, and the peak clipping phenomenon of the amplitude-frequency curve is more obvious when a pair of contact surfaces acting. When a pair of contact surfaces acts, the position of the bearing surface does not afect the vibration characteristics of the blade. When the three pairs of contact surfaces work together, the frst pair of contact surfaces as the main bearing surface will make the tenon structure have a better damping efect. With the increase in speed, when a pair of contact surfaces of the tenon joint structure is used, the vibration reduction efect of the tenon joint structure is the worst, and the peak clipping phenomenon of the blade amplitude-frequency curve almost disappears, which is due to the fact that when the pressure is applied to a pair of contact surfaces, the pressure on a single contact surface is too large, resulting in the sticking between the tenon root and the tenon groove more easily, thus losing the original vibration reduction efect. Under the joint action of three pairs of contact surfaces of the tenon joint structure, the frst pair of contact surfaces as the main bearing surface will make the tenon structure have a better vibration reduction efect, and the greater the pressure of the main bearing surface, the better the vibration reduction efect of the tenon joint structure, and the peak clipping phenomenon of the amplitude-frequency curve is more obvious. Te resonance frequency of the blade under three pairs of contact surfaces is larger than that under a single contact surface. When the rotational speed is high, the vibration reduction efect of the tenon joint structure with three pairs of contact surfaces is still better than that of the tenon joint structure with a single pair of contact surfaces, and the resonance frequency of the blade is higher. However, in the case of the tenon joint structure with three pairs of contact surfaces, the position of the main bearing surface and the pressure distribution between the three pairs of contact surfaces have almost no infuence on the vibration characteristics of the blade. Terefore, when the pressures on the three pairs of contact surfaces are the same, not only is the vibration amplitude of the blade lower, but the stresses in the tenon joint structure are also minimal.
Because the tenon structure relies on friction energy consumption to reduce the vibration amplitude of the blade, the magnitude of friction energy consumption directly refects the damping efect of the tenon structure. When the rotational speed is 300 rad/s, the hysteresis curves on the contact surface of the tenon under diferent pressure distributions are made. Te results are shown in Figure 9. It can be seen from the fgure that the area enclosed by the hysteresis curve is the energy consumed by friction. When the rotational speed is low, the total friction force of the contact surface under the two pressure distribution modes is the same, but the tenon displacement of the single-contact surface is greater than that of the three contact surfaces. Terefore, the tenon structure with single contact surface Shock and Vibration consumes more energy, and the vibration amplitude of the blade is smaller. However, with the increase of rotational speed, the positive pressure of the contact surface of the tenon is too large when the single-pair contact surface acts, which makes the contact surface more prone to stick. However, with the increase of rotational speed, the positive pressure of the contact surface of the tenon is too large when the single contact surface acts, which makes the contact surface more prone to stick. Terefore, the vibration reduction efect of the tenon structure with the single-contact surface is much worse than that of the tenon structure with multiple contact surfaces.
In summary, the fr-tree tenon joint structure can exert better vibration reduction performance only at higher speeds. When the frst pair of the contact surface is the main bearing surface, the vibration reduction efect of the fr-tree tenon joint structure is the best, and the higher the contact pressure of the main bearing surface, the better the vibration reduction efect of the tenon structure. Terefore, under the premise of ensuring strength, the main bearing surface should be subjected to a greater proportion of pressure as far as possible. When the rotational speed is further improved, the average force of the three pairs of contact surfaces of the tenon structure can not only have the best damping efect but also bear the minimum stress.

Te Infuence of Tenon Angle.
Te tenon angle is one of the main parameters of the tenon structure. According to the principle of force balance, it can be seen that the change of tenon angle directly afects the normal pressure on the contact surface, but at the same time, it also afects the friction force on the contact surface in the direction of blade vibration. In order to investigate the infuence of tenon angle variation on the vibration characteristics of rotating blades, the relationship between tenon angle and normal pressure on the contact surface and the relationship between tenon angle and dry friction is derived from theory. When the blade is rotating, the vibration displacement y(t) in the transverse direction of the blade should satisfy the kinetic equation (14).
when the tenon angle changes, the blade mass M, the stifness K, the damping matrix C, and the external excitation F(t) do not change as the blade speed and external excitation remain unchanged. It is assumed that the components of the positive pressure of the three pairs of contact surfaces in the ydirection are N 1 , N 2 , and N 3 , respectively. Taking the frst pair of contact surfaces as an example, according to the balance principle of force and the geometric relationship of structure, the positive pressure N 11 and N 12 of the contact surface can be expressed by equation (15).
By projecting the friction f 11 and f 12 on the contact surfaces of the tenon in the y-direction, the component f 1 of the dry friction in the y-direction can be obtained. f 1 can be represented by equation (16).

(21)
In order to balance the component of the positive pressure on the contact surface in the x-direction, it is necessary to make the tenon angle α 1 � β 2 � η. Te equation (21) can be expressed as follows: It can be found that when the tenon angles α 1 and β 2 are changed at the same time, the friction force of the contact surface in the y-direction is not afected. Te change of the tenon angle will not afect the vibration diferential equation of the blade, so changing the tenon angle of the tenon structure does not afect the response results of the blade. However, the change of tenon angles α 1 and β 2 will directly afect the positive pressure of the contact surface, thus affecting the service life of the blade.
When the rotational speed is 300 rad/s and 800 rad/s, the amplitude-frequency curve of the blade is shown in Figure 10 under diferent tenon angles. It can be found that the change of the tenon angle does not afect the amplitude of the blade, and the result is also consistent with the previous prediction. When the tenon angle is α 1 � β 1 � η, according to equation (15), the positive pressure of the contact surface is as follows: According to equation (23), it can be found that as the tenon angle increases, the contact pressure will gradually increase. On the premise of ensuring the efective joint of the tenon structure, reducing the tenon angle as much as possible can improve the service life of the blade.

Te Infuence of Friction Coefcient.
Te vibration amplitude of the blade can be reduced by the friction energy dissipation of the tenon structure. Te change of friction coefcient will directly afect the size of dry friction, thus afecting the vibration characteristics of the blade. Terefore, it is necessary to study the infuence of friction coefcient on blade vibration. Te blade speed is controlled at 200 rad/s and 300 rad/s, respectively, and the amplitude-frequency curves of the blade under diferent friction coefcients are made to analyze the infuence of friction coefcient on the vibration reduction characteristics of the blade. Te results are shown in Figure 11. It can be seen from the fgure that the friction coefcient of the tenon contact surface has a very signifcant indigenous infuence on the vibration characteristics of the blade. When the rotational speed is constant, the resonance amplitude of the blade increases with the increase of the friction coefcient of the tenon contact surface and the resonance frequency of the blade decreases. When the friction coefcient is small, the peak clipping phenomenon of the response can be obviously observed in the amplitude-frequency curve of the blade, but with the increase of the friction coefcient, the peak clipping phenomenon will become weaker and weaker until it disappears. Te reason for this phenomenon is that when the friction coefcient is small, the contact surface between the tenon and the tenon groove slips due to the small critical friction force, resulting in the peak clipping phenomenon in the amplitude-frequency curve. However, with the increase of friction coefcient, the critical friction force is also increasing, which makes the sliding between the tenon and the tenon groove more and more difcult. When the contact surface between the tenon and the tenon groove is viscous, the peak clipping phenomenon in the amplitude-frequency curve of the blade disappears. In the case of the same friction coefcient, the increase of rotational speed will increase the resonant amplitude of the blade and weaken the peak clipping phenomenon. Terefore, the tenon joint structure will play a better vibration reduction efect when the friction coefcient is low.
In order to further study the infuence of friction coefcient on the vibration reduction performance of the tenon joint structure, the excitation frequency is 1800 Hz near the frst natural frequency of the blade, and the blade speed is 300 rad/s. Te time domain response diagram, frequency spectrum diagram, and the hysteresis curve of dry friction with the displacement of the blade root with a friction coefcient of 0.1, 0.2, and 0.3 are respectively made. Te results are shown in Figure 12. It can be seen from the fgure that the larger the friction coefcient is, the smaller the displacement amplitude of the blade root is. Moreover, when the friction coefcients are 0.1 and 0.2, there are obvious harmonic phenomena in the time domain diagram of the blade root. At the same time, in the frequency domain diagram of the blade root, in addition to the main frequency, there are obvious triple and quintuple frequencies. However, when the friction coefcient is 0.3, the harmonic phenomenon of the blade root disappears, and the displacement of the blade root shows a strict harmonic change over time. At the same time, there is only a principal frequency component in the frequency domain diagram, and there is no obvious triple and quintuple frequency. Te dry friction of the tenon root increases with the increase of friction coefcient, but the slip distance of the contact surface decreases with the increase of friction coefcient. When the friction coefcients are 0.1 and 0.2, the dry friction and displacement show a piecewise linear overall nonlinear relationship, but when the friction coefcient is 0.3, the dry friction and displacement show a strictly linear relationship. Finally, the energy consumed by the dry friction decreases with the increase of the friction coefcient, and when the friction coefcient is 0.3, the dry friction no longer does the power consumption energy. From the abovementioned analysis, it can be found that the friction energy is the cause of the hard nonlinearity of the blade amplitude-frequency curve, and it is also the cause of the harmonic phenomenon in the blade root time domain diagram and the triple frequency and fve times frequency in the spectrum diagram. It is because the energy dissipation of dry friction decreases with the increase of friction coefcient, which leads to the increase of the resonant amplitude with the increase of friction coefcient.
In summary, the increase of the friction coefcient will weaken the vibration reduction performance of the fr-tree tenon structure, making the blade more prone to fatigue damage, but the increase of friction coefcient has little efect on the contact pressure. Terefore, the stability and service life of the blade can be improved by minimizing the roughness of the contact surface under economic conditions.

Te Infuence of Rotational Speed.
Te rotational speed of the blade directly afects the size of the centrifugal force and thus changes the dry friction of the tenon root. Due to the centrifugal stifening efect of the blade, the increase of rotational speed will change the stifness of the blade. Terefore, it is necessary to study the vibration characteristics of blades at diferent speeds. Taking the friction coefcient of the tenon root contact surface of the blade as 0.1, the amplitude-frequency response curve of the blade under diferent rotational speeds is made to observe the infuence of rotational speed. Te results are shown in Figure 13. It can be seen from the fgure that when the blade speed is not high, the amplitude-frequency response curve of the blade shows a certain peak clipping phenomenon. Te resonance frequency of the blade increases with the increase of rotational speed, and the resonance peak increases with the increase of rotational speed, and the resonance peak is signifcantly lower than that without a peak clipping phenomenon. When the rotational speed of the blade is not less than 600 rad/s, there is no obvious peak clipping phenomenon in the amplitude-frequency response curve of the blade at this time. However, with the increase of the rotational speed, the resonant frequency of the blade gradually increases, and the resonant peak gradually decreases. It can also be seen from the fgure that when the blade speed is low, the blade exhibits a certain soft nonlinear phenomenon at a low frequency, and the blade amplitude is larger than that at a high frequency.
In order to further study the infuence of rotational speed on the vibration characteristics of the blade, the excitation frequency is taken as the frequency near the frst-order natural frequency of the blade, and the time domain response diagram, frequency spectrum diagram, and the hysteresis curve of dry friction with displacement change at rotational speeds of 200 rad/s, 300 rad/s, and 600 rad/s are respectively made. Te results are shown in Figure 14. It can be seen from the fgure that the larger the rotational speed is, the smaller the displacement amplitude of the blade root is. When the rotational speed is 200 rad/s and 300 rad/s, the nonlinear dry friction at the blade root causes an obvious harmonic phenomenon in the time domain diagram. At the same time, in the blade root frequency domain diagram, there are three and fve frequencies in addition to the main frequency. However, when the rotational speed is 600 rad/s, the dry friction of the blade root changes linearly with the displacement, the time domain diagram of the blade root is harmonic, and the response amplitude is greatly reduced. At the same time, there is only the main frequency component in the frequency domain diagram. Secondly, the dry friction of the tenon root increases with the increase of rotational speed, but the slip distance between the contact surfaces of the tenon joint structure decreases with the increase of rotational speed. Finally, the energy consumed by the dry friction increases frst. It then decreases with the increase of the rotational speed, and when the rotational speed is 600 rad/s, the dry friction no longer does the power consumption energy. From the abovementioned analysis, it can be found that for the blade with the same parameters, when the rotational speed increases, the resonant amplitude value of the blade frst increases and then decreases. Te reason for the increase is that the increase of the blade energy plays a major role, and the reason for the decrease is that the centrifugal stifening efect of the blade plays a major role. Nonlinear phenomena become weaker and weaker with the increase of rotational speed until they disappear. Te vibration reduction efect of the tenon structure increases frst and then decreases with the increase of rotational speed. Tere is an optimal rotational speed for the blade with specifc parameters, which makes the tenon structure have the best vibration reduction efect, and the blade amplitude decreases the most.
Te blade rotation speed has a great infuence on the positive pressure of the contact surface. According to the equations (8)- (10), it can be seen that the positive pressure of the contact surface of the tenon root is directly related to the centrifugal force of the blade, and the centrifugal force of the blade is proportional to the square of the rotational speed. Terefore, with the increase of the rotational speed of the blade, the contact pressure of the tenon root will increase signifcantly.
In summary, the increase of rotational speed will make the forced response of the blade increase frst and then decrease, while the contact pressure will gradually increase. Terefore, at low rotational speed, the forced response of the blade and the stress of the tenon structure are small, and the blade is not prone to fatigue failure. Tere is an optimal rotational speed that makes the tenon structure have the best vibration reduction efect.

Te Infuence of Aerodynamic Excitation.
Te blade in the process of working due to the engine structure will be subject to the action of aerodynamic excitation. In order to study the vibration characteristics of the blade under aerodynamic excitation, the number of stator blades in front of rotor blades is N � 5. When the rotating speed of the blade is Ω, the aerodynamic excitation Q (t) can be expressed as follows: where A 0 is the aerodynamic excitation amplitude. Because the aerodynamic excitation amplitudes of diferent engines such as aeroengine and ground gas turbine and diferent positions of the same engine are quite diferent, it is necessary to study the vibration characteristics of blades under diferent excitation amplitudes. Te amplitude-frequency response curves of the blade under diferent aerodynamic excitation amplitudes are made, and the results are shown in Figure 15. It can be seen from the fgure that when the amplitude of the aerodynamic excitation is not high, the vibration peak of the blade is low, and the amplitude-frequency response curve does not appear peak clipping phenomenon. With the increase of aerodynamic excitation amplitude, the resonance frequency and resonance peak of the blade increase, and the nonlinear region of the amplitude-frequency response curve becomes wider and the peak clipping phenomenon becomes more obvious. Te reason for this phenomenon is that as the excitation amplitude increases, the vibration amplitude of the blade also increases, resulting in an increase in the slip distance of the tenon root and an increase in friction energy consumption. At the same time, the increase of excitation amplitude increases the rotational speed when the tenon root is viscous, which increases the resonance frequency of the blade and makes the peak clipping phenomenon more obvious. However, when the aerodynamic excitation amplitude is greater than 400 N and the speed is between 350 rad/s and 400 rad/s, the vibration amplitude of the blade is almost unchanged with the increase of the aerodynamic excitation amplitude. From an energy point of view, it can be seen that the fr-tree tenon structure has good vibration reduction ability, which can resist the infuence of aerodynamic excitation amplitude and stabilize the vibration amplitude of the blade in a certain range. When the excitation amplitude is constant, the response amplitude of the blade increases frst and then decreases with the increase of the rotational speed, which is the same as the changing trend of the blade amplitude with the rotational speed under harmonic excitation. It proves the correctness of using harmonic excitation to analyze blade vibration characteristics.
In summary, the increase of the aerodynamic excitation amplitude will cause the blade resonance peak to increase, making the nonlinear region of the amplitude-frequency response curve wider and the peak clipping phenomenon more obvious. When the speed is between 350 rad/s and 400 rad/s, the vibration amplitude of the blade is the least afected by the aerodynamic excitation. Although the excitation amplitude increased signifcantly, the amplitude of the blade can be stabilized in a certain range. Using intermittent excitation to simulate the force of the blade can not only correctly analyze the nonlinear vibration response of the blade, but also make the excitation frequency and the blade rotation speed independent of each other. In this way, the vibration characteristics of blades at diferent speeds can be analyzed more comprehensively, which has a better guiding value for the general design of blades.

Conclusion
In this paper, the fr-tree tenon blade is taken as the research object. After the blade is discretized by the fnite element method, the dynamic model of the rotating blade is established by using the elastic structure analysis theory, the minimum potential energy principle, and numerical integration. According to the ideal dry friction theory, a dry friction model is established to simulate the nonlinear friction of the tenon joint structure, so as to determine the boundary conditions of the blade. Te Newmark method is used to solve the dynamic equation of the blade. Te infuence of pressure distribution, tenon angle, friction coefcient, blade rotation speed, and aerodynamic excitation on blade vibration characteristics is analyzed and the following conclusions are obtained: (1) Te position of the main bearing surface of the tenon structure has a great infuence on the vibration characteristics of the blade. In the case of the same rotational speed, whether it is single-sided contact or multisided contact; when the frst pair of contact surfaces is the main bearing surface, the vibration reduction efect of the tenon structure is the best and the vibration amplitude of the blade is the lowest.
(2) Te contact mode and pressure ratio of the tenon structure have a great infuence on the vibration characteristics of the blade. Te damping efect of single-surface contact is better when the rotational speed is low. Te damping efect of multisurface contact is better when the rotational speed is high,  Amplitude of blade A (mm) Figure 15: Te amplitude-frequency curve of the blade at diferent aerodynamic excitation amplitudes. and the pressure ratio of the main bearing surface should be increased. However, when the rotational speed is too high, the average distribution pressure can make the vibration amplitude of the blade the lowest and the contact stress minimum.
(3) Te friction coefcient and blade rotation speed have a great infuence on the vibration amplitude of the blade, while the change of the tenon angle will not directly afect the vibration amplitude of the blade. Te contact pressure of the tenon root is greatly afected by the tenon angle and the blade rotation speed.
(4) Te increase of aerodynamic excitation will lead to the increase of blade vibration amplitude and make the nonlinear region of the amplitudefrequency response curve wider. Tere is a certain range of rotational speed, which makes the blade least afected by aerodynamic excitation, and the vibration amplitude of the blade is stable in a certain range.

HBM:
Harmonic balance method DLFT: Dynamic Lagrangian frequency-time CB-CMS: Craig-Bampton component mode synthesis Symbols Ω: Te rotational speed of the system R: Blade disc radius L: Length of the blade b: Width of the blade h: Height of the blade n: Number of the blade division elements l � L/n: Length of the element P i : Any point in the i-th element x i : Distance from P i to the frst node of the i-th element E: Young's modulus of the blade υ: Poisson's ratio of the blade I: Moment of inertia of the blade section ρ: Te density of the blade κ: Shear correction factor w: Lateral displacement due to blade deformation F c : Te centrifugal force of the blade F y : Lateral component of centrifugal force c: Shear strain due to shear force θ: Te rotation angle of the blade section φ: Neutral Mass matrix of the blade system K: Stifness matrix of the blade system ω 1 , ω 2 : Te frst two inherent frequencies of the system ξ 1 , ξ 2 : Te modal damping ratios Q: Te external excitation F: Te damping force N: Centrifugal force on the blade N ij : Positive pressure generated by N on the i-th pair of contact surfaces j y (t): Displacement of blade roots in the horizontal direction d ij : Displacement of x on the tangent of the i-th pair of contact surfaces j α i : Te angle between the i-th pair of contact surfaces 1 and the horizontal direction β i : Te angle between the i-th pair of contact surfaces 2 and the horizontal direction A ij : Te surface area of the i-th pair of contact surfaces j w ij : Displacement of the friction dampers on the i-th pair of contact surfaces j k ij : Coefcient of shear stifness on the i-th pair of contact surfaces j μ ij : Coefcient of friction on the i-th pair of contact surfaces j f ij : Friction on the i-th pair of contact surfaces j ∆: Deformation of the main bearing surface ∆x 2 : Te gap between the second pair of contact surfaces ∆x 3 : Te gap between the third pair of contact surfaces k: Te contact stifness of the tenon δ 1 , δ 2 , δ 3 : Te ftting clearances between the tenon root and tenon groove δ: Te ftting clearance between the main bearing surface ω, φ: Te excitation angular frequency and phase.

Data Availability
All data generated or analyzed during this study are included within the article, but the processed data required to reproduce these fndings cannot be shared at this time due to legal or ethical reasons.