Research on Vibration Response of Shrouded Blades Based on Contact State

The shrouded is widely used in vibration reduction of turbine blades. In this study, the damping characteristics of the shrouded blade in gas turbine were studied. Firstly, a finite element model of the shrouded blade was established, and the full set of blades was reduced to a single sector based on the principle of cyclic symmetry. The contact state of the contact surface of the blade at different interference and speeds was calculated based on the displacement extraction method. Secondly, a microslip friction contact model was established based on the contact state of the contact surface. The nonlinear vibration response of shrouded blades was calculated based on the harmonic balance method. The results indicate that the shrouded has excellent vibration reduction performance. The interference and rotational speed of the blade will affect the damping efficiency.


Introduction
The turbine rotor blades in rotating machinery endure multiple loads during operation, including centrifugal forces, temperature load, and pressure load.Consequently, its operational condition is harsh and susceptible to resonance, which will result in high-cycle fatigue issues [1,2].The use of shrouded blades effectively addresses this issue by offering supplementary damping and dissipating structural vibration energy.This helps attain the objective of enhancing the structure's life and reliability [3].However, a nonlinear contact boundary exists between the working surfaces of shrouded blades, rendering structural design a challenging task.As a result, scholars have conducted comprehensive research to delve into the nonlinear factors introduced by these contact surfaces.Yang [4] formulated a macroslip model to describe the interactions between the contact surfaces, calculating the equivalent stiffness and damping by considering the relative motion of the tangential and normal direction of contact surfaces.Liu [5] developed an improved hybrid time-frequency domain solution method for nonlinear friction and used harmonic balance method to solve the vibration response of shrouded blades.However, estimating the stiffness of the contact surface in vibration response calculation is always a challenge.Shen [6] used displacement extraction method to calculate the contact stress of the shroud contact surface and evaluate the static strength according to the convergent displacement of contact surface.Gao [7] calculated the modal characteristics of the blades and combined them with the energy principle to calculate the damping ratio of the shrouded blades.In order to obtain the nonlinear vibration response of shrouded blades under different contact conditions, this paper calculates the contact state of the shroud contact surface for different interference and speed conditions according to the displacement extraction method.Based on the contact state, a microslip friction model was introduced to calculate the nonlinear vibration response of shrouded blades.

Calculation Model
The shrouded blades utilize pre-twisting during assembly to generate an initial normal load on the shrouded contact.Figure 1(a) illustrates the results of the overall blade assembly, while figure 1(b) displays the individual blade sector model.The section of the shrouded that makes contact with adjacent blades after assembly is referred to as the "shrouded contact surface."In this turbine, there are a total of 72 blades, all of which display cyclic symmetry with respect to their structural arrangement.Therefore, the analysis of the blade contact is based on a cyclic symmetry model [8].To facilitate the application of periodic boundary conditions to the blades, specific modifications were made to the shrouded, as depicted in figure 1(c).A portion of the blade was removed and rotated to achieve periodic symmetric boundary conditions, allowing for the complete periodic processing of individual blade sectors.To emphasize the damping effect of the shrouded, the frictional contact between the blade root and the blade disk has been neglected, effectively excluding the disk structure from the analysis.The material of the blades is superalloy, with a density of

Contact State of Shrouded Blades
In the process of calculating the contact behavior of the shrouded blade, it becomes essential to refine the mesh on the contact surface, as visually demonstrated in figure 3, to achieve optimal results.However, an intriguing observation emerged during these calculations.As the mesh size within the finite element model continues to decrease, the contact stress on the contact surface gradually fails to converge.In typical finite element problems, reducing the mesh size typically leads to enhanced calculation accuracy, and the results often converge.However, contact-related issues present a distinct behavior.When performing contact analyses using nonlinear static methods, the computed contact stress results are dependent on both the mesh quality of the contact surface and the selected contact algorithm.In contrast, the displacement calculations in analogous scenarios demonstrate a higher level of robustness.In this study, the displacement extraction method, as detailed in reference [6], is adopted.The core idea is as follows: conventional contact calculations are first conducted.When the displacement results reach convergence, the contact state between the contact surfaces is canceled, and the displacement distribution of the contact boundary is extracted and reintroduced into the calculation as a displacement boundary condition until the stress converges.Subsequently, the mesh size at convergence is employed for contact analysis to calculate the contact stress of the contact boundary.
After a series of iterative calculations, the optimal mesh size was determined, guided by the displacement extraction method.Ultimately, a mesh size of 2mm for the body and 0.5mm for the contact surface was found to be the most appropriate.This mesh configuration serves as the foundation for calculating contact stress and, subsequently, solving for the corresponding contact surface load.To aid in this analysis, a local coordinate system is established on the contact surface, as shown in figure 4.This allows for the determination of the magnitude of the load acting on the contact surface, particularly the normal load in n Z direction.After obtaining the magnitude of the normal load on the contact surface, the calculation of the contact stiffness can be carried out.For the contact form of the shrouded, equation ( 1) can be used to solve for the normal contact stiffness: where 0 N is the normal load and  is the normal relative displacement of the contact surface.Due to the interference fit of the shrouded,  is the interference amount.The displacement extraction method can obtain the average displacement  change of the contact surface under corresponding normal load, which is caused by the twisting and the contact of the shrouded blade.Therefore, the displacement caused by contact is  =  − .
The normal contact stiffness in the initial state of the shrouded assembly can be obtained.For general spherical and flat contact [9], the tangential and normal contact stiffness satisfies: The contact surface of the shrouded belongs to the plane-plane contact form, so the value  is slightly small.Here, 0.4 is chosen.Through the above calculation, the tangential and normal stiffness of the contact surface can be obtained.
When the interference of the shrouded contact surface or the blade speed changes, it will have an impact on the calculation of contact stiffness.Firstly, the interference of the blade  was calculated using three different values, as shown in table 1. , the equivalent stress of the full structure was shown in figure 5(a).The contact state and contact stress distribution of the shrouded contact surface are shown in figure 5(b).The primary sliding region on the surface is notably concentrated in the upper left corner, while the majority of other areas remain in a quasi-contact state, with the potential to transition into a sliding state.This transition is favorable for enhancing the damping effect on the blade.Additionally, the concentrated distribution of contact stress is also situated in the upper left corner of the blade's contact surface, offering certain guidance for subsequent vibration response analysis.From the relationship between rotational speed, normal load, and contact stiffness, it can be seen that as the working speed of the blade continues to increase, the interference between the blade contact surfaces continues to increase, resulting in an increase in normal load and initial contact stiffness.

Contact Model
The interaction between the two shrouds involves elastic surfaces, resulting in a complex relative motion between them.The characteristics of this contact are significantly influenced by the normal load.As the normal load on the contact surface changes, various states such as stick, slip, and separation may occur.For friction contact, two primary models are employed: the macroslip model and the microslip model.The macroslip model assumes that under external forces, the stress and strain fields within the contact area of two surfaces are identical, making it possible to describe the overall motion of the entire contact surface using a single-point representation.In contrast, the microslip model describes a multi-point contact surface, considering it as a series or parallel connection of multiple macro elements.In comparison to the macroslip model, the microslip model offers a more precise depiction of the behavior of the surface, ranging from full stick to partial sliding and eventual overall sliding.This enhanced accuracy makes the microslip model particularly well-suited for addressing a broader range of practical engineering challenges.
A typical type of microslip model [10] was selected in this article to represent the contact state of the contact surface.The initial loading curve of the given friction force is: where u is the tangential displacement, F is the tangential load,  is the friction coefficient, and 0 N is the initial normal load. is the stiffness factor, specifically t gross KK  = , and gross K is the average contact stiffness during the sliding process, as detailed in [11].00 The excitation force F is a harmonic variation.To establish a stable-state hysteresis curve for friction displacement within a vibration cycle, considering unloading and reloading processes is essential.In accordance with the Masing Rule [12], the hysteresis loop path generated during periodic loading and unloading of materials mirrors the form of monotonic loading.However, it is magnified twofold and subsequently mapped to generate a closed symmetric hysteresis loop curve.This hypothesis is mathematically represented as follows: where n u is the unloading displacement, and it is a function of F .It is obtained by subtracting twice the function M from the critical displacement 0 u , and 0 F denotes the corresponding frictional force when the critical sliding condition is reached.The function M describes the relationship between force and displacement under monotonic loading.The reload function, r u , is defined as shown in equation ( 5): By substituting equations (3) into both equations ( 4) and ( 5), we can establish the relationship between the friction force and displacement occurring at the contact surface during microslip.When the sliding exceeds the critical state, the sticking process transitions into an overall sliding regime, where friction becomes the product of the friction coefficient and the normal load.Consequently, the forcedisplacement relationship curve during an entire loading cycle can be obtained.This leads us to the expression for the nonlinear friction force in the respective state: where u is the relative motion amplitude of the contact surface, as the form of simple harmonic motion is assumed.Therefore, the relative displacement of the contact surface is cos( ) uu  = , and in addition, is the phase angle of the stick/sliding state transition.

Solution Method
The point where the maximum vibration stress of the blade often occurs in the first order bending mode.
For the analyzed blade, a first order bending mode with 20 nodal diameter is selected.Therefore, the vibration equation of the system can be expressed as: where u is the vibration displacement, M is the mass matrix, K is the stiffness matrix, C is the damping matrix, and e F is the excitation force, F is the nonlinear friction force.For shrouds contact that maintain stable contact without interface separation characteristics, the influence of higher-order harmonic components is relatively small.Therefore, the nonlinear friction force is equivalent in the form of a first harmonic: where Ft is the approximate friction force, 0 1 1 ,, a a b are the three coefficients in the Fourier expansion equation.Since the relative displacement of the contact surface has a simple harmonic form, the vibration displacement can be expressed as the Fourier expansion form that retains the fundamental frequency.Substitute the displacement expression into equation (7) to obtain the equivalent stiffness and damping of the blade represented by the first-order harmonic balance method: ( , )cos( ) After calculating the equivalent damping and stiffness, the corresponding values can be added to the matrix 27 element to conduct harmonic response analysis and obtain the final results.
The value of u in the frictional contact model determines the area of the hysteresis curve and also determines the damping effect.However, it serves as both an output parameter of the friction model and a result parameter of the vibration response.Therefore, it needs to be iteratively solved.Therefore, a vibration response parameterization solution program has been developed: finite element modeling of the structure and harmonic response solution of linear systems have been achieved, and results such as element displacement at the contact surface of the blade can be output.Given the relative displacement u and solution accuracy of the blade contact surface.Solve the equivalent stiffness and damping of the corresponding sliding state structure based on the friction model.Add matrix27 elements to add equivalent damping and equivalent stiffness to the finite element model, and perform iterative calculations again.The iteration calculation ends when the error of the calculation results before and after the tangential relative displacement of the contact surface is less than the given error.

Vibration Response Analysis of Shrouded Blades
Establish a finite element model of shrouded blades, and use SOLID186 elements for mesh generation.The total number of elements in a single sector is 12563, and the stiffness and damping between shrouds are simulated using matrix27 elements.

Influence of Normal Load
The interference caused by centrifugal force is reflected by the relationship between rotational speed, normal load, and initial equivalent contact stiffness.Five equivalent normal loads at different rotational speeds were selected here, as shown in table 2, and the vibration response was calculated.The vibration response results of the blades are shown in figure 6  Drawing from the information presented in figure 6(a), the resonance frequency and resonance amplitude curve under different normal loads can be obtained, as illustrated in figure 6(b).Notably, this depiction reveals two key observations: Firstly, as the normal load changes, the resonance amplitude of the system exhibits a pattern of initial reduction followed by an increase.This phenomenon validates the existence of an optimal normal load that minimizes the resonance amplitude.Secondly, the resonance frequency with the rise in normal load, ultimately stabilizing within a specific range.This behavior can be attributed to the fact that when the normal load reaches a sufficiently high level, the contact surface is in a fully stick state, rendering the system predominantly linear.Consequently, the resonance frequency remains relatively stable.

Influence of Excitation Force
The variation in the magnitude of the excitation force also exerts an influence on the contact state of the contact surface.To explore this relationship, a constant normal load is maintained on the contact surface, with the variable being the excitation force.The resulting amplitude frequency response curves are depicted in figure 7(a) for excitation forces of 1N, 5N, and 15N, respectively.As the normal load remains constant and the excitation force continues to increase, several noteworthy effects emerge.Firstly, the resonance amplitude of the system gradually rises, and the resonance frequency gradually decreases, leading to a phenomenon known as stiffness softening.This phenomenon can be attributed to two key factors.On one hand, the increase in excitation force corresponds to an augmentation in the inherent viscous damping of the material.On the other hand, the excitation force leads to an increase in the relative displacement of the shroud contact surface.This results in a greater proportion of slip during the vibration cycle, ultimately reducing the provided equivalent stiffness.It's evident that as the excitation force increases, the shroud exhibits an enhanced damping effect over a broader range of normal loads.In other words, the shroud effectively imparts damping at larger vibration amplitudes.Additionally, under the same normal load, a smaller excitation force corresponds to a higher resonance frequency.The resonance frequency gradually increases with the rise in normal load, ultimately stabilizing at a certain value.

Conclusion
This paper calculates the contact state of the contact surface under the pre-twisted condition of the shrouded blades.The nonlinear dynamic response characteristics of shrouded were analyzed, and the effects of interference and excitation force on the vibration characteristics of the system were studied.Some results were obtained as follows: 1) Conventional contact result in the calculation relies on the quality of the mesh on the contact surface and the specific contact algorithm being used.In contrast, displacement results are generally more stable and dependable.The displacement extraction method was utilized to calculate the distribution of contact stress and the normal load on the contact surface.
2) A greater initial interference results in a higher normal load, and as the rotational speed increases, both the normal load and initial contact stiffness also increase.An optimal normal load exists that provides the most effective damping effect for the shrouded blade.
3) The increase of the excitation force results stiffness softening, when the rotational speed remains constant.Additionally, with higher excitation force, the shrouds exhibit an improved ability to provide effective damping across a broader range of normal loads.

3 7950
kg m , an elastic modulus at normal temperature of 189GPa , and a Poisson's ratio of 0.3 .

Figure 1 .
Figure 1.(a) Overall model (b) Single sector model (c) Periodic processing.The boundary conditions of the model are established in accordance with the actual installation configuration of the blades.As depicted in figure 2, fixed constraints are applied at the root position A of the blade.Cyclic symmetric boundary conditions are employed on the cut surfaces B of the shrouds, while contact pairs are defined on the contact surface C of the shrouds, enabling a nonlinear static analysis.This analysis facilitates the determination of the contact state of the shrouded blade contact surface in its initial installation configuration.

Figure 3 .
Figure 3. Fine mesh of contact surface.

Figure 4 .
Figure 4. Local coordinate system of contact surface.

Figure 5 .
Figure 5. (a) Equivalent stress distribution of the full model (b) Contact state and equivalent stress distribution of the contact surface. (a).

Figure 6 .
Figure 6.(a) Amplitude-frequency curves under different normal load (b) Resonance amplitude/frequency with different normal load.

Figure 7 .
Figure 7. a) Amplitude frequency curves under different excitations b) Resonance amplitude/frequency with different normal load.By subjecting the system to excitation forces of 1N, 5N, and 15N, we can derive the relationship curves between amplitude/frequency and various normal loads, akin to figure6(b).These results are depicted in figure7(b).It's evident that as the excitation force increases, the shroud exhibits an enhanced damping effect over a broader range of normal loads.In other words, the shroud effectively imparts damping at larger vibration amplitudes.Additionally, under the same normal load, a smaller excitation force corresponds to a higher resonance frequency.The resonance frequency gradually increases with the rise in normal load, ultimately stabilizing at a certain value.

Table 1 .
Normal load and contact stiffness under different  .

Table 2 .
Normal load and contact stiffness at different rotational speeds. )