1. Introduction
The rotor blades of aircraft engines and general gas turbines are subjected to alternating loads during operation, which frequently experience flow-induced vibrations that have implications for the fatigue life of the structure [
1,
2,
3]. Reducing vibration stress in rotor blades is an issue in the design of aero-gas turbines. One efficacious strategy employed is the integration of dry friction damping mechanisms into blade designs [
4,
5,
6,
7,
8]. Common dry friction damping structures include underplatform damper (UPD) [
9], shroud [
10], and frictional ring [
11], etc. When vibration occurs in the blades, the damping devices convert vibrational energy into heat via frictional interaction between contacting surfaces [
12].
Situated within the interstitial space beneath the blade platform, the UPD makes contact with the platforms of adjacent blades. In consideration of their distinctive structural features and design principles, UPDs adopt a variety of geometric profiles that play a pivotal role in determining their damping effectiveness. Ferhatoglu et al. [
13] found that transitioning the cross-sectional profile of a wedge damper from an isosceles right triangle towards a flatter shape results in a marked diminishment in the amplitude of oscillatory responses. Denimal et al. [
14] observed that when adjacent blades vibrate in phase, the conical dampers demonstrate superior vibration suppression over cylindrical dampers, with a more stable contact status compared to wedge dampers. Furthermore, the damping characteristics of a conical damper can be altered by adjusting the cone angle. To avoid this rolling phenomenon of asymmetric damper, Gastaldi et al. [
15,
16,
17] prescribed a range of geometric parameters and design limitations. It is noteworthy that common wedge and cylindrical UPDs are susceptible to rolling motion, which compromises damping efficiency. Additionally, Panning et al. [
18] innovatively amalgamated the beneficial aspects of wedge and cylindrical dampers in designing an asymmetric damper. Their research showed that during the vibrational process, the locus of the friction force on one of the interface surfaces shifts in accordance with the phase disparity between the adjacent blades.
Several researchers have employed model order reduction techniques to model bladed disks with UPDs. In such instances, the UPD can be treated as a rigid body; only the reduction of blade and disk is needed. Cigeroglu et al. [
19] decomposed the displacement of the model into the linear combination of its normal modes. Salas et al. [
20] used Craig–Bampton Cyclic, Craig–Bampton Multisubstructuring, and subset nominal mode to conduct the reduction of the order model (ROM) of the blade disk assembly. Mehrdad et al. [
21] combined the Craig–Bampton Method with the Loaded Interface Method [
22] to develop an ROM specifically for mistuned bladed disk structure. Gola et al. [
23] modeled the UPD as a point mass moving in a platform, thereby enabling the calculation of its damping characteristic. Rani et al. [
24] represented the blade with UPD using Bond Graph formalism, and subsequently analyzed the model through numerical integration techniques.
In addition to theoretical studies, researchers have carried out extensive experimental research on the vibration-damping capabilities of UPDs. Pesaresi et al. [
25,
26] introduced a pioneering experimental methodology for UPDs, in which a wedge-shaped UPD was tensioned upwards with steel wire to simulate centrifugal load. Laser displacement sensors were employed to measure the displacement of the blade tip. Their analysis delved into the blade’s reaction under varying excitation forces. When adjacent blades were vibrating in phase, the resonance frequency exhibited a significant shift with increasing force input, and the frequency response functions (FRFs) showed softening behavior and indicated a reduced damping effect. Their subsequent experiment used digital image correlation (DIC) and high-speed imaging techniques to observe the rotation of dampers during in-phase vibration.
Zhang et al. [
27] executed vibration experiments using dampers of different cross-sectional geometries and inserted pressure-sensitive paper between the contact surfaces to monitor the contact state. They found that a smaller effective contact area results in a greater resonance response. Ferhatoglu et al. [
28] designed an experimental setup incorporating two dampers and a blade, tensioned using steel wires to simulate centrifugal load. The system included four force sensors and a laser displacement sensor to measure the friction force on the contact surface and the displacement of the blade tip. They performed a series of repetitive experiments under similar conditions to gather distinct FRFs, aiming to explore the occurrence of multiple responses due to the non-uniqueness of friction forces. Using two previously suggested methodologies for the boundaries of FRFs, they compared these predictions with experimental outcomes [
13,
29]. Their experiment not only verified the accuracy of the method of predicting the response boundary but also confirmed that the uncertainty of dynamic response is attributed to the non-uniqueness of tangential forces.
It should be noted that all the previously discussed experimental configurations did not account for dynamic responses influenced by rotational effects such as stress stiffening and rotational softening. Hoffmann et al. [
30] addressed this deficiency by designing a specialized rotating test rig. In their setup, blades were excited using permanent magnets situated underneath, and the resulting measurement data were transmitted from the rotating assembly to the stationary data-acquisition system through the use of slip rings. The test observed a 92% reduction in resonance amplitude, providing experimental evidence of the effectiveness of the dampers.
Building upon the foundational principles of dry friction energy dissipation theory, the research group associated with the author [
31,
32,
33] previously developed a novel approach for computing the damping ratio characteristic curve of a blade with UPDs and a corresponding method for optimizing the damper mass using the damping ratio characteristic curve. Based on this theory, a non-rotating dummy blade damping characteristic testing system was designed. Through this experimental apparatus, the investigation conducted vibration response experiments on prototype blade samples featuring triangular prism-shaped UPDs, employing both the externally applied excitation method and the damping-free vibration testing methodology. This allowed for obtaining the critical damping ratio’s dependence on structural vibration stress, thereby substantiating the validity and applicability of the design and analytical methods. Furthermore, the study delved into the impact of several key structural parameters of UPDs, including inherent inertial loads, length of the shank, and contact areas on the damping ratio. Notably, the experimental scope was confined to examining the first bending mode of the blade.
4. Test Results
4.1. Modal Testing
Modal experimentation was conducted on blade specimens featuring various damper configurations and inertial loads. This involved striking the structure at various measurement points with a force hammer, capturing response signals at accelerometer positions, and analyzing the frequency response functions between different locations. The experiment primarily focused on the first bending mode, determining the first natural frequency of the blade by averaging multiple measurements. The experimental results are summarized in
Table 4. A comparison between the experimental results and modal frequency calculations reveals consistency between simulation and experimentation. It was noted that the installation of underneath dampers resulted in a slight increase in the first modal frequency compared to the one without dampers, with minimal impact from varying simulated inertial loads.
4.2. Sweep Frequency Vibration Test
The UPD was installed on inclined contact surfaces with different centrifugal loads. The experimental and simulation results of the damping ratio characteristic curve are summarized in
Figure 11.
Upon increasing the centrifugal force increased from 30 N to 60 N, an increase in the peak damping ratio was observed. However, upon further increasing the centrifugal force from 60 N to 120 N, no appreciable change in the damping ratio was detected. This phenomenon can be attributed to the fact that, at relatively low levels of normal contact force, changes in normal contact force cause alterations in the tangential contact stiffness. Consequently, this sensitivity leads to the actual tangential contact stiffness being lower than the predicted value when considering a centrifugal load of 30 N. The difference in contact stiffness caused a rightward bias in the damping curve relative to the predicted value. As the centrifugal load increases to a certain level, variations in tangential contact stiffness become less pronounced. With the increase of centrifugal loads, the critical vibration stress increases, while the peak damping ratio remains relatively constant, effectively reflecting this pattern in the simulation results.
The experimental and simulation values for different centrifugal loads on vertical contact surfaces are summarized in
Figure 12, showing that the peak dry friction damping ratio is around 1.5%. Overall, adding dampers significantly enhances the damping effect, achieving the goal of reducing vibration stress. Moreover, as the simulated centrifugal load changes, only the critical vibration stress varies, with the peak damping ratio remaining approximately constant. This consistency aligns with the damping characteristics revealed by simulation analysis.
It should be noticed that the experimental results have an obvious difference from the simulation results, which may be caused by the following results:
The inaccuracy of contact stiffness, especially for the experiments with an exciting force of 30 N.
Our simulation method only considers the tangential displacement of one direction, and the displacement in the other direction can increase the relative displacement of the UPD and blade, which can make the UPD provide more damping.
Only the damping of the contact surface of the complete blade specimen and UPD is considered in the simulation results; however, the contact surface of the bladeless specimen and UPD may provide damping as well, which may cause the maximum experimental damping ratio of the incline contact surface to be higher than the simulation results.
4.3. Free Vibration Test
For the damping-free vibration test with 120 N centrifugal load, with a damper installed on the inclined contact surface, the acceleration sensor measurements pertaining to the first-order bending mode of the damping-free vibration are shown in
Figure 13a. Upon the release of the applied load, the amplitude of the blade specimen diminished due to the combined effects of material damping and the damper.
Noting that during the process of calculating the damping ratio from the acceleration or strain time-domain curves, the peak data points may not perfectly align with the exponential decay function (Equation (
2)), due to limitations imposed by the sampling frequency of the sensing equipment. To mitigate this potential source of error, the envelope curve of the time-domain response was plotted. A zoomed-in view is shown in
Figure 13b. These circled peaks serve as the response points for subsequent damping ratio computations.
Damping-free vibration tests on the blades without dampers were conducted. An acceleration time-domain signal was acquired. The damping ratio is shown in
Figure 14. The average value of the damping ratio is about 0.33% , a value demarcated by a dashed line superimposed on the plot. Therefore, such a damping ratio is attributed to the structural damping ratio of the blade. By subtracting this damping ratio (0.33%) from the experimental data, one can quantitatively isolate and ascertain the damping effect imparted by the UPDs.
Owing to the complexity of directly measuring physical parameters of UPD, such as tangential stiffness and friction coefficient, parameters that exert a significant influence on the damping ratio, experimental estimates for these parameters were obtained through a process of curve fitting and inversion. The objective of this approach is to derive a reliable damping ratio characteristic curve for the damper. For example, applying a 120 N centrifugal load with the damper installed on an incline contact surface, the fitted damping ratio characteristic curve is shown in
Figure 15. The experimental damping ratio characteristic curve of the damper aligns well with the simulation results.
The experimental results for the UPD installed on an inclined contact surface and loaded by varying centrifugal loads are summarized in
Figure 16. Observations indicate that with a 30 N inertia load on the damper, the peak damping ratio reaches 1.8%. As the centrifugal load increases further, the peak damping ratio stabilizes at approximately 2.3%, mirroring observations from the frequency sweep test. Beyond a certain inertia load, the tangential contact stiffness shows no significant change. At this stage, only the critical vibration stress increases with the inertia load, while the peak damping ratio remains largely unchanged. This behavior is consistent with simulation results.
The experimental results of the damper, installed on the vertical contact surface under varied inertia loads, are summarized in
Figure 17. After installation, the damper provides a peak dry friction damping ratio of approximately 1.5%. With the damper set at a 30° angle, as shown in
Figure 2, the normal contact force on the vertical contact surface was equal to that on the inclined contact surface for the same centrifugal load. Overall, installing the UPD significantly enhances damping effects, effectively reducing vibration stress.
To study the damping characteristics of the UPD, experiments begin with applying normal pressure to the damper on the contact surface, followed by employing a frequency sweep excitation method. This method obtained the FRFs at monitoring points before and after damper installation, evaluating the vibration reduction effect. Each experiment provided a single point on the primary damping ratio characteristic curve of UPD under a specific vibration stress. Obtaining a complete damping ratio curve requires adjusting the excitation force and conducting multiple experiments. This process is time-consuming and inefficient and introduces damping due to the shaker, potentially affecting test outcomes. Conversely, a single damping-free vibration test efficiently yields a damping ratio characteristic curve across a broad range of vibration stress.
4.4. The Influence of Contact Area and Root Extension Length
To investigate the effect of the contact area of the UPD on frictional energy dissipation, experimental specimens featuring various contact areas were designed. The key dimensional parameters are listed in
Table 5. These specimens, subjected to a 120 N inertial load on inclined contact surfaces, had their resulting test data shown in
Figure 18.
Observations reveal that UPDs with varying contact areas on inclined surfaces generally exhibit similar damping ratio characteristics. However, when the contact area is too small (about one-third of the total area), the damping ratio shows significant variations with changing vibration stress, indicating instability in the damper’s damping ratio.
To explore the effect of the root extension length or the modal amplitude of the contact surface on damping characteristics, blade specimens with different shank dimensions were adapted. The dimensions for the shank of each blade specimen are listed in
Table 6. Using the same shroud damper with a 120 N inertial load on inclined contact surfaces, the test results are presented in
Figure 19.
Comparative analysis clearly shows that the increase of the root extension length enhances the damping ratio characteristics of the damper, resulting in a higher peak damping ratio.
Obvious differences can be found between the results of the free vibration and sweep frequency experiments. This may be related to the following reasons:
The initial displacement of the free vibration experiment is not exactly the first bending mode, so the process of vibration is a mix of many modes.
During the sweep frequency experiment, both the blade and shaker were vibrating.
The exciting force is hard to maintain constant during the vibration.
The process of the free vibration experiment is quite short and the wear of the contact surface is ignorable, but the sweep frequency experiments take a long time, so the wear may change the contact parameter.
5. Conclusions
This study conducted resonant vibration tests and damping-free vibration tests on a blade simulation component equipped with UPDs. The analysis of the experiment results yields the following conclusions:
(1) The modal frequency of the first-order bending mode derived from the tests closely aligns with the simulated modal frequency results. Following the installation of UPD devices, the blade modal frequency exhibits a slight increase, with the simulated centrifugal load exerting minimal influence on the frequency.
(2) In the resonant vibration tests conducted without UPD devices, the total damping—comprising material damping of the blade simulation component, blade aerodynamic damping, and interface damping—is estimated to be approximately 1%. However, in the damping-free vibration tests, the damping ratio attributed to the test setup itself is approximately 0.33%. The primary difference between the two can be attributed to the material damping contributed by the resonator connection device (resonator head and force rod) during the resonant vibration tests, as well as the internal structure of the resonator that contributes to damping effects. The peak damping ratio achieved by the inclined contact surface UPD device fluctuates in response to changes in the tangential contact stiffness. Overall, the UPD device markedly enhances the damping effect, thereby significantly reducing vibration stress.
(3) Adjusting the inertial load of the damper primarily affects the critical vibration stress, having minimal impact on the peak damping ratio. The damping ratio characteristics derived from simulation analysis correspond to this pattern. Sensitivity tests on the blade root length reveal that with an increase in the blade root length, the damping effect of the UPD significantly intensifies.
(4) It is crucial to note that although the aforementioned results and patterns were derived under non-rotational conditions, they retain significant implications for the design of practical UPDs. The experimental results presented in this study not only validate the analytical methods developed in previous research but also ensure that the systematic conclusions can be effectively applied and verified in the design of aircraft engines.