Research on the effect of fretting wear on the vibration characteristics of the tenon joint bladed disk

The tenon joint structure is a typical structure of aero-engine bladed disk. Under external excitation, micro-slip inevitably occurs between the tenon and the tenon groove of the blade, which will cause fretting wear on the contact surface. The resonance frequency and margin of the bladed disk structure will change after fretting wear occurs. In this paper, a finite element subprogram is compiled based on the basic principle of Archard wear, which realizes the generation of surface wear morphology and multi-step iterative update of the double-tooth fir-tree joint model under fretting load. The subroutine is used to quantitatively analyze the wear depth distribution of the tenon joint surface under fretting loads. On this basis, the resonance margin and the vibration response characteristics of the bladed disk are quantitatively studied after fretting wear. The research results show that the natural vibration frequency of the bladed disk structure decreases at different wear times. This caused a decrease in the resonance margin of more than 80% of the measuring points. In addition, the resonance response amplitudes all showed an upward trend with the increase of wear depth.


Introduction
The fretting wear will have a certain influence on the dynamic response of the mortise-joint bladed disk structure, and the change of the response will also affect the fretting wear process on the contact surface.The fretting process involves many factors.In the literature [1], 180 laws about wear are summarized.Fouvry et al. [2] proposed a model based on energy dissipation to quantify the amount of wear directly.In the process of quantifying the amount of wear, the most commonly used model is the Archard model [3].
The amount of structural fretting wear can also be obtained by a variety of numerical methods, and the standard numerical method is the finite element method [4,5].Although the finite element method can be used to calculate the wear more accurately, it is necessary to establish a fine enough grid to ensure that the calculation of the normal contact force is accurate enough, and the grid update is required so the calculation efficiency is low.To improve the efficiency, P˜odra et al. [6] proposed an intelligent and efficient algorithm based on Winkler's method to predict and analyze the wear of the contact surface.In addition to the traditional finite element method, Gallego et al. [7] and Sfantos et al. [8] used the semianalytical and boundary element methods to predict and analyze the contact surface wear, respectively.Salles et al. [9] assumed that the wear was in a periodic steady state and used the finite element method to dynamically predict the wear process.
The damping characteristics of the structure will change due to the wear of the contact structure, and the dynamic response will also change.Laxalde et al. [10] used the method of coupling the amplitude of the forced response and the wear to predict and analyze the wear characteristics of the engine blade contact structure based on a high-fidelity finite element model.The research of Salles et al. [11,12] reveals a phenomenon: even the fretting wear with a depth of a few microns will greatly influence the vibration response of the structure.Lemoine et al. [13] conducted a detailed analysis and calculation on the tenon joint structure's contact wear and vibration characteristics based on the numerical theoretical analysis and experimental.Armand [14] et al. expounded the coupling relationship between fretting wear and nonlinear vibration characteristics between the underground damper and the blade flange at different time scales.Gallego [15] et al. proposed a quasi-static multi-scale method and performed coupling analysis with the finite element method, which improved the computational efficiency.
However, the existing research mainly focuses on qualitative analysis and lacks the quantitative analysis of the influence of fretting wear on the vibration characteristics of tenon joint bladed disk structure.In this paper, the surface contact characteristics of the mortise and tenon joints are first analyzed, and a finite element subroutine based on the fretting contact algorithm is compiled to realize the generation of the surface wear profile of the double-tooth fir-tree tenon joint under fretting loads.Finally, the influence of different fretting wear depths on the bladed disk structure's natural vibration characteristics and resonance margin are obtained.

Analysis of contact characteristics
In the literature [16], the contact characteristics of the mortise joint surface were analyzed in detail, and the finite element simulation results were verified by combining the fretting wear test.Using the model in literature [16], the finite element simulation calculation of the normal and shear stress distribution on the tenon joint surface of the upper and lower tenon teeth is carried out.The calculation results are shown in figure 1-5.The calculation results in figure 1-5 show that the contact surface of the tenon joint bladed disk is in a complex stress state.For the double-tooth fir tree tenon, the average normal and shear stress level of the upper tenon tooth are higher than those of the lower tenon tooth, and the two sides (left and right) and the bottom (near the center side) of the tenon tooth contact surface are higher than the center and upper areas.It can be inferred that the area above the center of the upper tenon tooth contact surface produces greater relative sliding during vibration, leading to relatively severe fretting wear.

Calculation principle of fretting wear depth
Based on the basic assumptions of fretting wear [17] and fretting contact mechanics calculations [18], the Archard equation [3,19,20] can be applied to the local surface of the contact area in the process of calculating the fretting wear depth.The basic equation is as follows: , where K is the dimensionless wear coefficient, H is the material hardness (MPa), P is the normal pressure, V is the total fretting wear volume, and S is the cumulative displacement.To simulate the evolution of the fretting wear morphology of the contact surface, it is necessary to define the local wear depth and the corresponding horizontal position coordinate x at each contact node of the finite element model.Therefore, for an infinitely small surface contact area dA, the corresponding sliding distance is dS, and equation ( 1) can be expressed as: , where dd VS on the left side of equation ( 2) can be defined as the local wear rate r W , which can be measured and calculated by wear tests.To further simplify equation ( 2), dividing both sides by dA gives: , where dd PA represents the local contact pressure () px at this position, dd VA is the local wear depth to be solved.Here, d = d d h V A is defined, and the prediction equation for the fretting wear depth increment can be obtained: Here, k* is used instead of K/H to represent the local wear coefficient.Equation (4) shows that the wear depth increment at a given node is positively related to the wear coefficient, local contact pressure and local slip increment.Since there is no accurate evaluation method for k* at present, the acceptable method in engineering uses the average method to estimate the wear coefficient: , where W is the width of the wear area, b is the width of the test piece, hm is the average depth of the wear surface, Nt is the number of fretting wear cycles, and  is fretting amplitude.The research content of this chapter does not involve the influence and change law of the wear coefficient.In the process of numerical analysis, it is set as a constant value to participate in the numerical simulation calculation.
From equations ( 3) and ( 4) we can get: It can be seen from the above equation ( 6) that the parameters proportional to the wear depth are pressure distribution, dimensionless wear coefficient, and sliding distance, and the parameter inversely proportional to the wear depth is material hardness.Therefore, the wear depth of the contact surface can be controlled by changing the material parameters and load parameters to meet the engineering needs.
A subroutine based on incremental wear accumulation calculation was developed in the commercial finite element software ABAQUS.The subroutine uses the Fortran language as the assembly language, and the calculation of the wear depth is completed through multiple iterations between the subroutine and the main program.The algorithm and flow of the main program (gray area) and subroutine (blue area) are shown in figure 6.In the main program, the finite element modeling of the contact structure, the application of loads and boundary conditions, the generation of .inpfiles, and an initial static calculation are completed.Contact information (pressure and sliding distance distribution) on the contact surface can be obtained.After completing the initial calculation of the main program, the user-defined displacement constraint is used to call the subroutine to calculate the wear depth.After entering the subroutine, first call the contact node information calculated in the main program to judge whether it is an initial cycle and perform an initial wear depth calculation to eliminate the influence of the initial singular value.Then, the mesh is adaptively updated.In the process of subroutine programming, define the number of fretting cycles as N, the number of mesh updates (KMESH) as n, and define n=4 in each incremental step.When the calculation reaches the set number of cycles, the cycle is ended, and the calculation is completed.At the same time, the calculation result of the last subroutine is imported into the main program.Then, the .inpfile is updated to obtain the distribution of the wear depth of the final node on the contact surface.

Calculation example of fretting wear depth
The finite element model of the basic fretting structure is shown in figure 7. The gray elements in the figure is the motion unit, which participates in the periodic movement along the z direction during the iterative calculation of the subroutine.And a constant load along the y direction is applied on its upper surface to ensure that the structure is always in contact during the fretting process.The green element is a fixed unit, and a fixed constraint is imposed on the lower surface.The contact surface of the upper and lower parts of the structure adopts an adaptive grid and is marked with a red wireframe.During the fretting wear process, the adaptive mesh participates in the iterative calculation of the wear depth in the subroutine and updates the mesh in each load step.After the calculation is completed, the fretting wear morphology can be obtained.During the calculation of fretting wear, only the green cell area is considered to be worn.In the calculation process, the sliding amplitude is defined as 0.0025mm, the positive pressure is 200N, the friction coefficient is 0.5, and the wear coefficient is 8.2e-7.The upper surface of the lower mass block is the surface to be worn, and the calculation results are shown in figure 8.   8.The three-dimensional wear morphology exhibits similar shapes under different load steps, and they all appear in the form of "bathtub" surfaces.And with the progress of fretting wear, the wear depth accumulates, and the surface depth gradually deepens.

Influence of Fretting Wear on System Vibration Frequency
Based on the analysis in the second chapter, this section conducts modal analysis on the tenon joint bladed disk sector structure and only considers the modal of the blade.To obtain the vibration characteristics at different speeds, this section calculates the modals and the corresponding contact surface wear depth at speeds of 10000rpm, 15000rpm, and 18000rpm, respectively.The calculation results are shown in table 1-4 • By comparing the table horizontally, it can be seen that as the speed increases, the fretting wear depth of the contact surface gradually increases.According to the longitudinal comparison table, as the wear process progresses, the wear depth on the tenon-tooth contact surface gradually increases.Moreover, there are differences in the wear conditions of the upper and lower tenon teeth, and the wear depth of the upper tenon teeth is always greater than that of the lower tenon teeth.
• Fretting wear will cause the system's natural frequency to drop, and the average drop percentage value is shown in Table 5.It can be seen that when the speed is 15000rpm, the average value of the natural frequency drop percentage is the largest among the four calculated moments.
Through longitudinal comparison, it can be seen that with the progress of fretting wear, the amplitude of natural frequency decrease caused by fretting wear gradually increases.• As the speed increases, there is an obvious upward trend in fretting wear, as shown in figure 10.
By comparing figure 10 with table 5, it can be seen that although the fretting wear depth is the largest at 18000rpm, the degree of frequency drop caused by this is smaller than that corresponding to 15000rpm.Resonance analysis requires the determination of the excitation order.According to the aero-engine design manual and relevant design experience, the excitation order k in this paper is 1, 2, 3, 4, 6, 8.Among them, the selection of excitation orders as 1, 2, and 3 usually needs to be considered in engineering calculations.The calculated Campbell diagram is shown in figure 11, where (a) is in the unworn state, and (b), (c), (d), and (e) correspond to wear times of 100s, 200s, 600s, and 1500s, respectively.It can be seen that when the structure fretting wear occurs, the resonance margin at different speeds will change significantly.The number of calculation points for the resonance margin is 360, and there are 240 calculation points among them where the resonance margin decreases.Table 15 shows the ratio of the number of points where the resonance margin decreases under different excitation orders.It can be seen that fretting wear will lead to a decrease in the resonance margin of the bladed disk structure, which dramatically increases the safety hazard of the bladed disk structure resonance.According to the requirements of the aeroengine design manual, the frequency resonance margin of the bladed disk rotor system should not be less than 20%, and the resonance margin of parts such as blades and disks should not be less than 10%.For the bladed disk structure without fretting wear, in calculating the resonance margin at three speeds, only 5 points have a resonance margin less than 20%, and all other points have a resonance margin greater than 20%.After the fretting wear occurs, the resonance margin of 40 points is less than 20% at three speeds, and the resonance margin of 25 points is less than 10%.The minimum value (0.1%) of the resonance margin is located at the second-order resonance under the triple frequency excitation at 15000rpm, which is in a perilous state.Therefore, it is necessary to focus on the influence of fretting wear on the resonance characteristics in the design process of the tenon joint bladed disk of the aero-engine.

Conclusion
According to the quantitative analysis of the influence of fretting wear on the vibration characteristics, the following conclusions can be obtained.
• Fretting wear will significantly impact the natural vibration characteristics of the bladed disk structure.At the same speed, the natural frequency of double-tooth mortise-jointed bladed disk structure decreases gradually with the progress of fretting wear.At the same wear moment, the fretting wear depth increases with increased rotational speed.When the speed is 15000rpm, the natural frequency drop caused by wear is the largest.• The change of the natural frequency directly affects the resonance margin of the tenon joint bladed disk.This paper calculates the resonance margin of 360 points, and the resonance margin of 240 calculation points decreases after wear.The calculation points with resonance margin less than 20% increased from 5 before wear to 40 after wear, and 25 of them had resonance margin less than 10%.

1 . 2 . 3 .
(a) Normal stress (b)Distribution along the contact nodes Figure Normal stress of upper tenon tooth.(a) Normal stress (b)Distribution along the contact nodes Figure Normal stress of lower tenon tooth.(a) Shear stress (b)Distribution along the contact nodes Figure Shear stress of upper tenon tooth.

4 .
(a) Shear stress (b)Distribution along the contact nodes Figure Shear stress of lower tenon tooth.

5 .
(a) Upper tenon tooth (b) Lower tenon tooth Figure The transient displacement response of tenon and groove.

Figure 6 .
Figure 6.Algorithm and process of fretting wear depth calculation.In the main program, the finite element modeling of the contact structure, the application of loads and boundary conditions, the generation of .inpfiles, and an initial static calculation are completed.Contact information (pressure and sliding distance distribution) on the contact surface can be obtained.After completing the initial calculation of the main program, the user-defined displacement constraint is used to call the subroutine to calculate the wear depth.After entering the subroutine, first call the contact

Figure 7 .
Figure 7.The finite element model of the basic fretting structure.During the calculation of fretting wear, only the green cell area is considered to be worn.In the calculation process, the sliding amplitude is defined as 0.0025mm, the positive pressure is 200N, the friction coefficient is 0.5, and the wear coefficient is 8.2e-7.The upper surface of the lower mass block is the surface to be worn, and the calculation results are shown in figure8.

8 .
(a) Centerline position of the contact area (b) Node wear depth at the centerline position Figure Wear depth variation of the nodes on the centerline of the contact surface.

Figure 9 .
Fretting wear morphology.The three-dimensional wear morphology of the basic fretting structure is shown in figure 9(a), (b) and (c) under different load steps.The topography of the 3D surface after fretting wear corresponds to the 2D image in figure

Figure 10 .
Figure 10.Fretting wear depth changes with rotating speed.

3. 2 .
Effect of fretting wear on resonance margin Section 3.1 analyzes the change in the natural vibration frequency of the structure caused by fretting wear.The change of the natural vibration characteristics will directly affect the resonance margin, and the change of the resonance margin can increase the risk of engine resonance.It causes great hidden danger to the structural safety of aero-engine and flight safety.Therefore, attention must be paid to the change of resonance characteristics caused by fretting wear.This section completes the calculation and analysis of the resonance margin after fretting wear.

Figure 11 .
no fretting wear (b)fretting wear for 100 seconds The 17th Asian International Conference on Fluid Machinery (AICFM 17 2023) Journal of Physics: Conference Series 2707 Campbell plots before and after fretting wear.

Table 4 .
Modal and fretting wear (t=1500s).The following conclusions can be drawn by analyzing the calculation results in the above tables.

Table 5 .
Percentage of the natural frequency drop.

Table 15 .
Percentage of resonance margin reduction.