<figure><img src="image_16.png" /><figcaption>Fig. 13. (a) Frequency response Functions for increasing levels of excitation from [64] (b) Damper performance curve from [88] plotting amplitude of vibration vs. excitation level (c) Damper performance curve from [233] plotting receptance vs. damper preload over excitation level (d) Damper performance curve from [233] plotting frequency of response peak vs. damper preload over excitation level.</figcaption></figure>

in a numerical environment and reproduce either the blade dynamic response and/or the hysteresis cycles of the damper/blade interface.

The effectiveness of the damper, however, can only be quantified under specific conditions that depend on the test rig architecture. For instance, non-rotating 2B configurations offer only In-Phase and Out-of-phase modes of vibration, while non-rotating and rotating disk setups can be driven to assess any desired Engine Order excitation. Non-rotating 1B configurations test the damper effectiveness with platform kinematics that are very specific and seldom encountered in real working conditions. However, in their most recent version [238,242], they provide the most comprehensive level of model validation, encompassing both blade dynamics and damper contact mechanics. In [220], the contact forces at the blade-platform interface reconstructed from the measurements were directly applied to a finite element model of the blade, thus bypassing the need for a contact model. The resulting blade response was then compared with the measured counterpart.

All of these points are summarised in Table 5. The analysis of the experimental evidence gathered on the test rigs described in this section yielded cross-confirmations and comparable results on several key points, as outlined below.

* **Assessment of damper effectiveness** Blade friction dampers introduce stiffness and damping contributions that depend on the excitation level [272], which in turn triggers varying degrees of slip at the contact interfaces. One common method to evaluate damper performance is to fix the preload (corresponding to a given rotational speed) and adjust the dynamic excitation applied to the bladed array. At low excitation levels, the damper remains fully stuck to the platforms, contributing maximum stiffness but zero damping. As excitation levels increase, slip at the interfaces occurs, leading to increased damping and decreased stiffness contributions. With further increase in excitation, the damper may

begin to slip freely or detach, resulting in a frequency response function resembling that of a free, unbounded bladed system (i.e., without a damper). Results are often presented by displaying the Frequency Response Functions for increasing excitation levels (Fig. 13(a)) and/or by plotting the maximum amplitude and frequency of each curve against the excitation level (Fig. 13(b), (c), (d)). These plots, known as performance curves, aim to summarise damper effectiveness across a wide range of operating conditions. It is noteworthy that the appearance of the damper performance curve may vary depending on the convention used. For example, in Fig. 13(b), where the x-axis is linearly correlated with the excitation level, a fully stuck damper is depicted on the left side and a freely slipping damper on the right. The opposite is true for Fig. 13(c)–(d), where the x-axis represents the preload on the damper relative to the excitation level. Another important distinction lies in the measurement of the blade(s)' dynamic response. In some cases (Fig. 13(b)), the blade(s)' response is directly represented through vibration amplitude, resulting in a continuous increase from full stick to free slip. It is essential to note that the gradient with which the amplitude increases over excitation (i.e. the slope of the curves in Fig. 13(b)) is at its minimum when the damper is operating effectively, as highlighted by the shaded area in Fig. 13(b). An alternative method to represent the blade(s)' response is through receptance (Fig. 13(a) and (c)), i.e., the response normalised by the excitation level. In a linear system, the performance curve would appear as a flat line, as the response would increase linearly with the excitation level. However, in a friction-damped system, the curve exhibits a minimum when the damper operates at maximum efficiency. Similarly, damper effectiveness can be investigated at the mechanics level, independent of the blade dynamics response. In
<figure><img src="image_16.png" /><figcaption>Fig. 13. (a) Frequency response Functions for increasing levels of excitation from [64] (b) Damper performance curve from [88] plotting amplitude of vibration vs. excitation level (c) Damper performance curve from [233] plotting receptance vs. damper preload over excitation level (d) Damper performance curve from [233] plotting frequency of response peak vs. damper preload over excitation level.</figcaption></figure>

in a numerical environment and reproduce either the blade dynamic response and/or the hysteresis cycles of the damper/blade interface.

The effectiveness of the damper, however, can only be quantified under specific conditions that depend on the test rig architecture. For instance, non-rotating 2B configurations offer only In-Phase and Out-of-phase modes of vibration, while non-rotating and rotating disk setups can be driven to assess any desired Engine Order excitation. Non-rotating 1B configurations test the damper effectiveness with platform kinematics that are very specific and seldom encountered in real working conditions. However, in their most recent version [238,242], they provide the most comprehensive level of model validation, encompassing both blade dynamics and damper contact mechanics. In [220], the contact forces at the blade-platform interface reconstructed from the measurements were directly applied to a finite element model of the blade, thus bypassing the need for a contact model. The resulting blade response was then compared with the measured counterpart.

All of these points are summarised in Table 5. The analysis of the experimental evidence gathered on the test rigs described in this section yielded cross-confirmations and comparable results on several key points, as outlined below.

* **Assessment of damper effectiveness** Blade friction dampers introduce stiffness and damping contributions that depend on the excitation level [272], which in turn triggers varying degrees of slip at the contact interfaces. One common method to evaluate damper performance is to fix the preload (corresponding to a given rotational speed) and adjust the dynamic excitation applied to the bladed array. At low excitation levels, the damper remains fully stuck to the platforms, contributing maximum stiffness but zero damping. As excitation levels increase, slip at the interfaces occurs, leading to increased damping and decreased stiffness contributions. With further increase in excitation, the damper may

begin to slip freely or detach, resulting in a frequency response function resembling that of a free, unbounded bladed system (i.e., without a damper). Results are often presented by displaying the Frequency Response Functions for increasing excitation levels (Fig. 13(a)) and/or by plotting the maximum amplitude and frequency of each curve against the excitation level (Fig. 13(b), (c), (d)). These plots, known as performance curves, aim to summarise damper effectiveness across a wide range of operating conditions. It is noteworthy that the appearance of the damper performance curve may vary depending on the convention used. For example, in Fig. 13(b), where the x-axis is linearly correlated with the excitation level, a fully stuck damper is depicted on the left side and a freely slipping damper on the right. The opposite is true for Fig. 13(c)–(d), where the x-axis represents the preload on the damper relative to the excitation level. Another important distinction lies in the measurement of the blade(s)' dynamic response. In some cases (Fig. 13(b)), the blade(s)' response is directly represented through vibration amplitude, resulting in a continuous increase from full stick to free slip. It is essential to note that the gradient with which the amplitude increases over excitation (i.e. the slope of the curves in Fig. 13(b)) is at its minimum when the damper is operating effectively, as highlighted by the shaded area in Fig. 13(b). An alternative method to represent the blade(s)' response is through receptance (Fig. 13(a) and (c)), i.e., the response normalised by the excitation level. In a linear system, the performance curve would appear as a flat line, as the response would increase linearly with the excitation level. However, in a friction-damped system, the curve exhibits a minimum when the damper operates at maximum efficiency. Similarly, damper effectiveness can be investigated at the mechanics level, independent of the blade dynamics response. In
studies such as [212,235,238,246], the contact force on the right side of a damper is measured, leveraging the fact that the damper has a unique contact on the right side. The damper equilibrium is then reconstructed, assuming the damper inertial forces are negligible (Fig. 14(a)). Relative platform-damper kinematics are measured using a differential laser vibrometer at both contact interfaces (Fig. 14(b)), along with relative platform-platform kinematics. For example, as illustrated in [79], to replicate pure blade In-Phase vibration, the relative platform kinematics are vertical (Fig. 14(c)). It can be demonstrated that the total energy dissipated by the damper equals the sum of the areas of the hysteresis cycles at the two contact interfaces (Fig. 14(d)-(e)), which in turn is equal to the platform-platform equivalent hysteresis cycle. This cycle is obtained by plotting the relative platform displacement (directly imposed in these types of test rigs [79,212,235]) against the corresponding component of the contact force at one of the two interfaces (Fig. 14(f)). The shape of the platform-damper hysteresis cycles differs from that shown in Fig. 11 due to the continuous variation of the normal load on the interfaces. Analysing the platform-damper hysteresis cycles allows assessment of the contact state at the different interfaces during the vibration period and provides a direct estimation of the contact parameters $k_t$ and $\mu$.

* **Effect of the damper shape and platform kinematics** The behaviour and effectiveness of a damper are influenced by various factors, including its mass, geometry, and rigid body motion, which, in turn, depends on the kinematics of the platforms, such as the mode of vibration. Some of these test rigs have been used to compare the performance of different types of dampers. A notable example involves solid dampers, characterised as rigid prismatic objects that may be cylindrical, wedge-shaped, or asymmetrical (flat on one side and cylindrical on the other). In [248], the performance of these three types of dampers, positioned at the tip of the blades, was compared under In-Phase and Out-of-Phase modes of vibration. The asymmetrical damper demonstrated superior performance in reducing the response level in both cases, while the cylindrical one consistently showed the least effectiveness (Fig. 15(a)). The reason for the cylinder's lack of performance lies in its kinematics, particularly in the In-Phase mode of vibration, where it involves substantial rotation and minimal slip, as demonstrated in [271] and further illustrated in Fig. 15(b). Wedge-shaped dampers exhibit significant energy dissipation for the Out-of-Phase mode because the damper slides without rotating, as demonstrated in [239] through DIC measurements (Fig. 15(c)) and in [232] through Laser Vibrometer measurements. However, in the In-Phase mode, the wedge damper loses effectiveness due to lift-off of the contact interfaces [232,239]. This is reflected in Frequency Response Functions that exhibit a softening behaviour with jumps. The relationship between damper kinematics and dissipated energy has been further elucidated in the investigation of asymmetric dampers with different platform angles and depth of cut [270]. Further considerations on the variability of experimental results, both due to surface finishing and to solution multiplicity inherent to frictional behaviour are discussed thoroughly in Section 6.2.1.

### 5.2.4. Blade shroud

Table 6 summarises the main features of all the test setups designed to investigate blade shrouds. Three main test rig architectures can be identified in Table 6 and are represented in Fig. 16. These configurations are analogous to those described in Section 5.2.3 with the same names. In detail, non-rotating setups may involve either two blades (2B) coupled through a shroud or one blade (1B) coupled to one or two dummy shrouds, which may be instrumented with load cells to measure contact forces. A non-rotating disk configuration is

also presented in [250], where only two of the blades are loaded, excited, and monitored. In shrouds, the **preload** at the contact is provided by the untwisting due to the centrifugal effect. The same mechanism can be reproduced in rotating setups, while other strategies must be adopted for non-rotating ones. There is not a unique and widespread solution as in test rigs investigating blade roots or dampers. D'Ambrosio et al. [250], in their 1B non-rotating configuration, connected the dummy blade under investigation to a system composed of a blade spring and a steel ball rubbing against a support mounted on a controlled XY table, which is used to charge the contact. The normal and tangential loads are measured by means of a load cell mounted on the fixed block. The advantage of this system is that, since the spring stiffness in the direction of contact is low, it provides a high sensitivity to normal load, thus enabling exploration of a wide range of values. On the other hand, the position and direction of the contact are not representative of realistic working conditions. In 1B configurations, one possible solution is to ensure preload by pushing the dummy shrouds against the blade through deadweights [87,255], or using a screw-driven lift [257]. Other solutions [252] involve a load disk that integrates the dummy shrouds. A moment is applied to the disk, which then rotates and ensures contact with the blade. A similar solution adopted by [259] rotates the blade rather than the support holding the dummy shrouds. These solutions have the advantage of providing a static moment on the blade as in real operating conditions. In [254], the preload is obtained by mounting interference between the two blades and later estimated using strain gauge measurement. In [256], the preload is not provided as the goal of the analysis is to verify the dynamic performance of shrouds undergoing impact as well as friction.

Similar to Section 5.2.3, the most common **excitation** in non-rotating rigs is through a shaker, while in rotating setups, air jets and permanent magnets are used.

The **measured quantities** in this case always include the dynamic response of the blade. Similarly to the previous section, the response is typically presented as a function of frequency for different levels of excitation and preload at the contact. In some specific cases, additional information on the shroud kinematics and on the exchanged contact forces is made available.

A synthesis of noteworthy insights derived from the analysis of the experimental findings is presented below:

* **Comparison with numerical predictions and contact interface detection:** The measured responses are typically compared to numerical predictions, which have become increasingly complex as computational capabilities and model features have improved over the years. Model improvements usually involve the representation of the contact interface. In [249], only resonance frequencies are compared, and the computed counterpart is obtained by imposing a full stick condition on a subset of the nominal contact area. In [230,250,255,258], the complete response is computed, and the importance of a fine discretisation of the contact interface, representative of the effective rather than nominal contact conditions, is emphasised to achieve successful experimental-numerical matching. The influence of the effective contact interface area is further investigated in [257], where changes in contact surfaces were periodically detected over millions of cycles and linked to alterations in the measured hysteresis loops and vibration characteristics.

* **Influence of normal load:** Different works [87,252,254] highlighted the influence of normal load on the shrouds, showing that a decrease in normal load tends to increase damping and limit maximum alternating stresses. However, it was shown in [87] that an excessive reduction in normal load may trigger contact separation and therefore a sharp and sudden increase in blade stress. If the loss of contact is intermittent, on the other hand,
studies such as [212,235,238,246], the contact force on the right side of a damper is measured, leveraging the fact that the damper has a unique contact on the right side. The damper equilibrium is then reconstructed, assuming the damper inertial forces are negligible (Fig. 14(a)). Relative platform-damper kinematics are measured using a differential laser vibrometer at both contact interfaces (Fig. 14(b)), along with relative platform-platform kinematics. For example, as illustrated in [79], to replicate pure blade In-Phase vibration, the relative platform kinematics are vertical (Fig. 14(c)). It can be demonstrated that the total energy dissipated by the damper equals the sum of the areas of the hysteresis cycles at the two contact interfaces (Fig. 14(d)–(e)), which in turn is equal to the platform-platform equivalent hysteresis cycle. This cycle is obtained by plotting the relative platform displacement (directly imposed in these types of test rigs [79,212,235]) against the corresponding component of the contact force at one of the two interfaces (Fig. 14(f)). The shape of the platform-damper hysteresis cycles differs from that shown in Fig. 11 due to the continuous variation of the normal load on the interfaces. Analysing the platform-damper hysteresis cycles allows assessment of the contact state at the different interfaces during the vibration period and provides a direct estimation of the contact parameters *k*<sub>*t*</sub> and *μ*.

* **Effect of the damper shape and platform kinematics** The behaviour and effectiveness of a damper are influenced by various factors, including its mass, geometry, and rigid body motion, which, in turn, depends on the kinematics of the platforms, such as the mode of vibration. Some of these test rigs have been used to compare the performance of different types of dampers. A notable example involves solid dampers, characterised as rigid prismatic objects that may be cylindrical, wedge-shaped, or asymmetrical (flat on one side and cylindrical on the other). In [248], the performance of these three types of dampers, positioned at the tip of the blades, was compared under In-Phase and Out-of-Phase modes of vibration. The asymmetrical damper demonstrated superior performance in reducing the response level in both cases, while the cylindrical one consistently showed the least effectiveness (Fig. 15(a)). The reason for the cylinder’s lack of performance lies in its kinematics, particularly in the In-Phase mode of vibration, where it involves substantial rotation and minimal slip, as demonstrated in [271] and further illustrated in Fig. 15(b). Wedge-shaped dampers exhibit significant energy dissipation for the Out-of-Phase mode because the damper slides without rotating, as demonstrated in [239] through DIC measurements (Fig. 15(c)) and in [232] through Laser Vibrometer measurements. However, in the In-Phase mode, the wedge damper loses effectiveness due to lift-off of the contact interfaces [232,239]. This is reflected in Frequency Response Functions that exhibit a softening behaviour with jumps. The relationship between damper kinematics and dissipated energy has been further elucidated in the investigation of asymmetric dampers with different platform angles and depth of cut [270]. Further considerations on the variability of experimental results, both due to surface finishing and to solution multiplicity inherent to frictional behaviour are discussed thoroughly in Section 6.2.1.

### 5.2.4. Blade shroud

Table 6 summarises the main features of all the test setups designed to investigate blade shrouds. Three main test rig architectures can be identified in Table 6 and are represented in Fig. 16. These configurations are analogous to those described in Section 5.2.3 with the same names. In detail, non-rotating setups may involve either two blades (2B) coupled through a shroud or one blade (1B) coupled to one or two dummy shrouds, which may be instrumented with load cells to measure contact forces. A non-rotating disk configuration is

also presented in [250], where only two of the blades are loaded, excited, and monitored. In shrouds, the **preload** at the contact is provided by the untwisting due to the centrifugal effect. The same mechanism can be reproduced in rotating setups, while other strategies must be adopted for non-rotating ones. There is not a unique and widespread solution as in test rigs investigating blade roots or dampers. D'Ambrosio et al. [250], in their 1B non-rotating configuration, connected the dummy blade under investigation to a system composed of a blade spring and a steel ball rubbing against a support mounted on a controlled XY table, which is used to charge the contact. The normal and tangential loads are measured by means of a load cell mounted on the fixed block. The advantage of this system is that, since the spring stiffness in the direction of contact is low, it provides a high sensitivity to normal load, thus enabling exploration of a wide range of values. On the other hand, the position and direction of the contact are not representative of realistic working conditions. In 1B configurations, one possible solution is to ensure preload by pushing the dummy shrouds against the blade through deadweights [87,255], or using a screw-driven lift [257]. Other solutions [252] involve a load disk that integrates the dummy shrouds. A moment is applied to the disk, which then rotates and ensures contact with the blade. A similar solution adopted by [259] rotates the blade rather than the support holding the dummy shrouds. These solutions have the advantage of providing a static moment on the blade as in real operating conditions. In [254], the preload is obtained by mounting interference between the two blades and later estimated using strain gauge measurement. In [256], the preload is not provided as the goal of the analysis is to verify the dynamic performance of shrouds undergoing impact as well as friction.

Similar to Section 5.2.3, the most common **excitation** in non-rotating rigs is through a shaker, while in rotating setups, air jets and permanent magnets are used.

The **measured quantities** in this case always include the dynamic response of the blade. Similarly to the previous section, the response is typically presented as a function of frequency for different levels of excitation and preload at the contact. In some specific cases, additional information on the shroud kinematics and on the exchanged contact forces is made available.

A synthesis of noteworthy insights derived from the analysis of the experimental findings is presented below:

* **Comparison with numerical predictions and contact interface detection:** The measured responses are typically compared to numerical predictions, which have become increasingly complex as computational capabilities and model features have improved over the years. Model improvements usually involve the representation of the contact interface. In [249], only resonance frequencies are compared, and the computed counterpart is obtained by imposing a full stick condition on a subset of the nominal contact area. In [230,250,255,258], the complete response is computed, and the importance of a fine discretisation of the contact interface, representative of the effective rather than nominal contact conditions, is emphasised to achieve successful experimental-numerical matching. The influence of the effective contact interface area is further investigated in [257], where changes in contact surfaces were periodically detected over millions of cycles and linked to alterations in the measured hysteresis loops and vibration characteristics.

* **Influence of normal load:** Different works [87,252,254] highlighted the influence of normal load on the shrouds, showing that a decrease in normal load tends to increase damping and limit maximum alternating stresses. However, it was shown in [87] that an excessive reduction in normal load may trigger contact separation and therefore a sharp and sudden increase in blade stress. If the loss of contact is intermittent, on the other hand,
<figure><img src="image_17.png" /><figcaption>Fig. 14. (a) Procedure to reconstruct damper equilibrium (b) Procedure to measure/reconstruct damper-platform hysteresis at contact interfaces (c) Procedure to assess the platform-platform equivalent hysteresis for ideal In-Phase motion (d) Measured damper-platform hysteresis at flat-on-flat contact interface for curved-flat damper (e) Measured damper-platform hysteresis at cylinder-on-flat contact interface for curved-flat damper (f) Measured platform-platform equivalent hysteresis for curved-flat damper.</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>Fig. 15. (a) Damper performance curves for asymmetrical, cylindrical and wedge-shaped dampers for In-Phase and Out-of-Phase modes of vibration from [248]. (b) platform-platform hysteresis cycle and damper rotation for asymmetrical and cylindrical dampers for In-Phase platform kinematics from [271]. (c) Wedge-shaped damper kinematics using DIC measurements for In-Phase and Out-of-Phase modes of vibration from [239].</figcaption></figure>

several repeated impacts (i.e., chattering) will introduce additional damping, which will manifest as a response function with an irregular shape [230].

* **Influence of shroud geometry and Engine Order Excitation:** As in the case of friction dampers, different contact angles and relative kinematics at the contact may trigger different contact states as shown in [273] and further exemplified in Fig. 17.

### 5.2.5. Blisk friction ring damper

Designing and commissioning a whole rotating test rig is complex and expensive. There are very few experimental studies on the testing of friction ring dampers in a rotating blisk. The only known experiment on the topic was designed and conducted at the École Centrale de Lyon [112,260], which characteristics are given in Table 7. They tested a high-pressure compressor blisk using a friction ring damper in a rotating test rig. The test rig was developed earlier in a study by [112]
<figure><img src="image_17.png" /><figcaption>Fig. 14. (a) Procedure to reconstruct damper equilibrium (b) Procedure to measure/reconstruct damper-platform hysteresis at contact interfaces (c) Procedure to assess the platform-platform equivalent hysteresis for ideal In-Phase motion (d) Measured damper-platform hysteresis at flat-on-flat contact interface for curved-flat damper (e) Measured damper-platform hysteresis at cylinder-on-flat contact interface for curved-flat damper (f) Measured platform-platform equivalent hysteresis for curved-flat damper.</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>Fig. 15. (a) Damper performance curves for asymmetrical, cylindrical and wedge-shaped dampers for In-Phase and Out-of-Phase modes of vibration from [248]. (b) platform-platform hysteresis cycle and damper rotation for asymmetrical and cylindrical dampers for In-Phase platform kinematics from [271]. (c) Wedge-shaped damper kinematics using DIC measurements for In-Phase and Out-of-Phase modes of vibration from [239].</figcaption></figure>

several repeated impacts (i.e., chattering) will introduce additional damping, which will manifest as a response function with an irregular shape [230].

* **Influence of shroud geometry and Engine Order Excitation:** As in the case of friction dampers, different contact angles and relative kinematics at the contact may trigger different contact states as shown in [273] and further exemplified in Fig. 17.

### 5.2.5. Blisk friction ring damper

Designing and commissioning a whole rotating test rig is complex and expensive. There are very few experimental studies on the testing of friction ring dampers in a rotating blisk. The only known experiment on the topic was designed and conducted at the École Centrale de Lyon [112,260], which characteristics are given in Table 7. They tested a high-pressure compressor blisk using a friction ring damper in a rotating test rig. The test rig was developed earlier in a study by [112]
<table><caption>**Table 5**
Friction Damper Test Rig Configurations vs Usability of Experimental Evidence.</caption><thead><tr><td>Test rig configuration</td><td>Model validation</td><td>Assessment of Damper effectiveness</td></tr></thead><tbody><tr><td>Non-rotating 0B</td><td>Damper mechanics</td><td>Any Engine Order excitation, bending modes only</td></tr><tr><td>Non-rotating 1B</td><td>Damper mechanics<br>Blade dynamics</td><td>Only one condition/mode similar to 90° inter-blade phase angle</td></tr><tr><td>Non-rotating 2B</td><td>Blade dynamics</td><td>In-phase and Out-of-phase modes</td></tr><tr><td>Non-rotating Disk<br>Rotating</td><td>Blade dynamics</td><td>Any Engine Order excitation</td></tr></tbody></table>

<figure><img src="image_19.png" /><figcaption>**Fig. 16.** Test set-ups used to investigate bladed shrouds (a) Non-rotating 1B configuration with two dummy shrouds and (b) rotating configuration from [255]. (c) Non-rotating 1B configuration with one dummy shroud from [257]. (d) Non-Rotating 2B configuration from [254].</figcaption></figure>

<table><caption>**Table 6**
Features of test set-ups focusing on blade shrouds.</caption><thead><tr><td>References</td><td>Set-up</td><td>Preload</td><td>Excitation</td><td>Measured Quantities</td></tr></thead><tbody><tr><td>[249,251,258]</td><td>Rotating Disk</td><td>Centrifugal Effect</td><td>Air Jets</td><td>Dynamic Response</td></tr><tr><td>[87]</td><td>Non-rotating 1B</td><td>Deadweights</td><td>Shaker</td><td>Dynamic Response</td></tr><tr><td>[250]</td><td>Non-Rotating 1B</td><td>Preloaded spring</td><td>Shaker</td><td>Dynamic Response<br>Contact forces<br>Relative Displacements</td></tr><tr><td>[250]</td><td>Non-rotating Disk (2B)</td><td>Loading Screw</td><td>Shaker</td><td>Dynamic Response</td></tr><tr><td>[252]</td><td>Non-rotating 1B</td><td>Load Disk</td><td>Shaker</td><td>Dynamic Response</td></tr><tr><td>[254]</td><td>Non-rotating 2B</td><td>Mounting interference<br>Strain gauge</td><td>Shaker</td><td>Dynamic Response</td></tr><tr><td>[256]</td><td>Non-rotating 2B</td><td>-</td><td>Shaker</td><td>Dynamic Response</td></tr><tr><td>[259]</td><td>Non-Rotating 1B</td><td>Blade rotation</td><td>Shaker</td><td>Dynamic Response<br>Contact forces</td></tr><tr><td>[255]</td><td>Non-rotating 1B</td><td>Deadweights</td><td>Shaker</td><td>Dynamic Response</td></tr><tr><td>[255]</td><td>Rotating Disk</td><td>Centrifugal Effect</td><td>Permanent magnets</td><td>Dynamic Response</td></tr><tr><td>[257]</td><td>Non-Rotating 1B</td><td>Blade rotation</td><td>Shaker</td><td>Dynamic Response<br>Contact forces<br>Relative displacement</td></tr></tbody></table>
<table><caption>**Table 5**
Friction Damper Test Rig Configurations vs Usability of Experimental Evidence.</caption><thead><tr><td>Test rig configuration</td><td>Model validation</td><td>Assessment of Damper effectiveness</td></tr></thead><tbody><tr><td>Non-rotating 0B</td><td>Damper mechanics</td><td>Any Engine Order excitation, bending modes only</td></tr><tr><td>Non-rotating 1B</td><td>Damper mechanics<br>Blade dynamics</td><td>Only one condition/mode similar to 90° inter-blade phase angle</td></tr><tr><td>Non-rotating 2B</td><td>Blade dynamics</td><td>In-phase and Out-of-phase modes</td></tr><tr><td>Non-rotating Disk<br>Rotating</td><td>Blade dynamics</td><td>Any Engine Order excitation</td></tr></tbody></table>

<figure><img src="image_19.png" /><figcaption>**Fig. 16.** Test set-ups used to investigate bladed shrouds (a) Non-rotating 1B configuration with two dummy shrouds and (b) rotating configuration from [255]. (c) Non-rotating 1B configuration with one dummy shroud from [257]. (d) Non-Rotating 2B configuration from [254].</figcaption></figure>

<table><caption>**Table 6**
Features of test set-ups focusing on blade shrouds.</caption><thead><tr><td>References</td><td>Set-up</td><td>Preload</td><td>Excitation</td><td>Measured Quantities</td></tr></thead><tbody><tr><td>[249,251,258]</td><td>Rotating Disk</td><td>Centrifugal Effect</td><td>Air Jets</td><td>Dynamic Response</td></tr><tr><td>[87]</td><td>Non-rotating 1B</td><td>Deadweights</td><td>Shaker</td><td>Dynamic Response</td></tr><tr><td>[250]</td><td>Non-Rotating 1B</td><td>Preloaded spring</td><td>Shaker</td><td>Dynamic Response<br>Contact forces<br>Relative Displacements</td></tr><tr><td>[250]</td><td>Non-rotating Disk (2B)</td><td>Loading Screw</td><td>Shaker</td><td>Dynamic Response</td></tr><tr><td>[252]</td><td>Non-rotating 1B</td><td>Load Disk</td><td>Shaker</td><td>Dynamic Response</td></tr><tr><td>[254]</td><td>Non-rotating 2B</td><td>Mounting interference<br>Strain gauge</td><td>Shaker</td><td>Dynamic Response</td></tr><tr><td>[256]</td><td>Non-rotating 2B</td><td>–</td><td>Shaker</td><td>Dynamic Response</td></tr><tr><td>[259]</td><td>Non-Rotating 1B</td><td>Blade rotation</td><td>Shaker</td><td>Dynamic Response<br>Contact forces</td></tr><tr><td>[255]</td><td>Non-rotating 1B</td><td>Deadweights</td><td>Shaker</td><td>Dynamic Response</td></tr><tr><td>[255]</td><td>Rotating Disk</td><td>Centrifugal Effect</td><td>Permanent magnets</td><td>Dynamic Response</td></tr><tr><td>[257]</td><td>Non-Rotating 1B</td><td>Blade rotation</td><td>Shaker</td><td>Dynamic Response<br>Contact forces<br>Relative displacement</td></tr></tbody></table>
<figure><img src="image_20.png" /><figcaption>**Fig. 17.** (a) Measured vs. simulated nonlinear response of shrouded blade for different preload levels from [87]. (b) Nonlinear response of shrouded blade exhibiting jump phenomenon from [250]. (c) Variability of the nonlinear frequency response and corresponding values of the static component of one of the contact forces from [259]. (d) Hysteresis loops at the shroud-dummy contact recorded for increasing degrees of wear and corresponding frequency response from [257].</figcaption></figure>

<figure><img src="image_21.png" /><figcaption>**Fig. 18.** Experimental set up of a rotating blisk with a friction ring damper [260].</figcaption></figure>

<figure><img src="image_22.png" /><figcaption>**Fig. 19.** Experimental testing results of a rotating blisk with a friction ring damper [260].</figcaption></figure>

to investigate friction damping blade attachment. Fig. 18 shows the experimental setup used in the study. The rig is powered by an electric motor that can rotate at a maximum speed of 5000 rpm. Fig. 18(b) illustrates the use of 24 piezoelectric plates, split into 8 groups, as an excitation system to facilitate precise control for exciting targeted modes. The vibration measurement system comprises 12 semiconductor

strain gauges, with 8 attached to the suction side of the blades and the remaining 4 on the disk. The frequency-strain and damping-strain curves at different rotating speeds were obtained from experimental forced responses using a nonlinear modal identification technique. The test results are used to validate the nonlinear numerical simulation results. The comparison between experimental and numerical results
<figure><img src="image_20.png" /><figcaption>Fig. 17. (a) Measured vs. simulated nonlinear response of shrouded blade for different preload levels from [87]. (b) Nonlinear response of shrouded blade exhibiting jump phenomenon from [250]. (c) Variability of the nonlinear frequency response and corresponding values of the static component of one of the contact forces from [259]. (d) Hysteresis loops at the shroud-dummy contact recorded for increasing degrees of wear and corresponding frequency response from [257].</figcaption></figure>

<figure><img src="image_21.png" /><figcaption>Fig. 18. Experimental set up of a rotating blisk with a friction ring damper [260].</figcaption></figure>

<figure><img src="image_22.png" /><figcaption>Fig. 19. Experimental testing results of a rotating blisk with a friction ring damper [260].</figcaption></figure>

to investigate friction damping blade attachment. Fig. 18 shows the experimental setup used in the study. The rig is powered by an electric motor that can rotate at a maximum speed of 5000 rpm. Fig. 18(b) illustrates the use of 24 piezoelectric plates, split into 8 groups, as an excitation system to facilitate precise control for exciting targeted modes. The vibration measurement system comprises 12 semiconductor

strain gauges, with 8 attached to the suction side of the blades and the remaining 4 on the disk. The frequency-strain and damping-strain curves at different rotating speeds were obtained from experimental forced responses using a nonlinear modal identification technique. The test results are used to validate the nonlinear numerical simulation results. The comparison between experimental and numerical results
<figure><img src="image_23.png" /><figcaption>Fig. 20. Different geometries of UPDs: (a) Wedge-shaped and cylindrical underplatform damper from [274] (b) Modified wedge-shaped underplatform damper from [275]. (c) Semi-cylindrical underplatform damper from [276]. (d) Asymmetrical underplatform damper from [272]. (e) Conical underplatform damper from [277]. (f) Topology optimised underplatform damper from [278].</figcaption></figure>

is illustrated in Fig. 19. It can be observed that the damping ratio and resonant frequency obtained from the numerical simulation are consistent with the experimental results at the same order of magnitude. The damping ratio trend obtained from the numerical simulation at a friction coefficient of 0.5 corresponds well with the curve derived from the experimental findings at lower strain levels. However, when the strain level exceeds 50, the difference between the numerical and experimental results becomes noticeable. The results in [112] also exhibit similar outcomes. These observations indicate that the current numerical technique is relatively dependable for the prediction of nonlinear friction damping.

# 6. Design and optimisation of friction dampers

In the last two sections, the numerical and experimental techniques for the simulation and testing of dynamical systems with friction interfaces were reviewed. This section focuses on advances in terms of the design and optimisation of friction dampers in turbomachinery. The primary objective of design and optimisation is to enhance the performance and robustness of friction damping under various operational conditions, considering diverse sources of uncertainty. This involves modifying structural features such as mass, shape, interface, or material properties.

## 6.1. Structural design and optimisation

### 6.1.1. Underplatform damper (UPD)

The initial investigations, conducted in [34,279,280], aimed to explore the impact of non-geometric parameters (including friction coefficients, excitation level, and contact normal force) on resonance peak frequency and response levels in turbine blades. These studies underscored the significance of such parameters in the damping performance of wedge-shaped friction dampers. A similar sensitivity study was also performed by Krack et al. [36] considering uncertainties in the friction coefficient, the excitation level and the linear damping. Panning et al. [61,229,274] were the first to numerically investigate the influence of contact geometry of both cylindrical and wedge-shaped friction dampers on damping effectiveness by parametrically varying both the geometry of blade-platform and dampers. This confirms the experimental findings detailed in Section 5.2.3 and demonstrates how effective numerical models can be in the design phase, i.e. avoiding costly experimental campaigns. The geometry of the wedge-shaped and

cylindrical damper is shown in Fig. 20(a). Zucca et al. [276] were among the first to model the semi-cylindrical friction damper (shown in Fig. 20(c)) which had been in use in the turbomachinery industry since the 1980s. This shape, as pointed out in the study, ensures full contact between the damper and the blade-platform over the flat damper side when the relative platform angle changes during blade vibration (i.e. self-adjusting). Their numerical investigation made it possible to link the damper kinematics (i.e. lack/presence of rotation) to its stiffening and damping capabilities. This investigation was further developed by Gastaldi et al. [210,272,281], who parametrised the semi-cylindrical damper shape in terms of its angles and depth of cut (see Fig. 20(d)) and proposed pre-optimisation maps which forecast, without the need for numerical iterative computations, the combination of design parameters that will lead to undesirable behaviours such as rotation and jamming for In-Phase and Out-of-Phase modes. More recent works focus on choosing, among the pre-optimised damper configurations, the best fit for a specific blade [38,40,282]. In [283], the coupling between a blade and a semi-cylindrical damper is optimised by considering a design space defined by ten geometric parameters. Among these parameters, three are allocated for the damper. Two surrogate models are introduced and compared, both capable of significantly reducing the computational time needed for a regular design of experiments (DOE) factorial sampling by several orders of magnitude.

Other options to avoid damper rotation have been explored. As shown in Fig. 20(e), Denimal et al. [277] proposed a conical underplatform damper to avoid the rolling phenomenon of traditional wedged-shaped dampers for In-Phase modes through geometric compliance, leading to a better damping performance in both In-Phase and Out-of-Phase mode when compared to classical wedged and circular dampers. In the context of high-fidelity optimisation, Yuan et al. [275] introduced a surrogate-based optimisation approach to enhance the damping performance of a modified wedge-shaped underplatform damper. The geometrical parameterisation of the damper is shown in Fig. 20(b). The study was intended to optimise the shape of the wedge-shaped friction dampers to achieve minimal resonance frequency shift and maximum friction damping. Recently, Denimal et al. performed a further topology optimisation of an underplatform damper [278] where the internal structure of the wedged damper is optimised under stress constraints imposed on the contact interface. One of the topologically optimal dampers is shown in Fig. 20(f). In addition to the design of the damper geometry itself, a simplified and parametric beam model with only one turbine blade and damper was then developed to study
<figure><img src="image_23.png" /><figcaption>**Fig. 20.** Different geometries of UPDs: (a) Wedge-shaped and cylindrical underplatform damper from [274] (b) Modified wedge-shaped underplatform damper from [275]. (c) Semi-cylindrical underplatform damper from [276]. (d) Asymmetrical underplatform damper from [272]. (e) Conical underplatform damper from [277]. (f) Topology optimised underplatform damper from [278].</figcaption></figure>

is illustrated in Fig. 19. It can be observed that the damping ratio and resonant frequency obtained from the numerical simulation are consistent with the experimental results at the same order of magnitude. The damping ratio trend obtained from the numerical simulation at a friction coefficient of 0.5 corresponds well with the curve derived from the experimental findings at lower strain levels. However, when the strain level exceeds 50, the difference between the numerical and experimental results becomes noticeable. The results in [112] also exhibit similar outcomes. These observations indicate that the current numerical technique is relatively dependable for the prediction of nonlinear friction damping.

# 6. Design and optimisation of friction dampers

In the last two sections, the numerical and experimental techniques for the simulation and testing of dynamical systems with friction interfaces were reviewed. This section focuses on advances in terms of the design and optimisation of friction dampers in turbomachinery. The primary objective of design and optimisation is to enhance the performance and robustness of friction damping under various operational conditions, considering diverse sources of uncertainty. This involves modifying structural features such as mass, shape, interface, or material properties.

## 6.1. Structural design and optimisation

### 6.1.1. Underplatform damper (UPD)

The initial investigations, conducted in [34,279,280], aimed to explore the impact of non-geometric parameters (including friction coefficients, excitation level, and contact normal force) on resonance peak frequency and response levels in turbine blades. These studies underscored the significance of such parameters in the damping performance of wedge-shaped friction dampers. A similar sensitivity study was also performed by Krack et al. [36] considering uncertainties in the friction coefficient, the excitation level and the linear damping. Panning et al. [61,229,274] were the first to numerically investigate the influence of contact geometry of both cylindrical and wedge-shaped friction dampers on damping effectiveness by parametrically varying both the geometry of blade-platform and dampers. This confirms the experimental findings detailed in Section 5.2.3 and demonstrates how effective numerical models can be in the design phase, i.e. avoiding costly experimental campaigns. The geometry of the wedge-shaped and

cylindrical damper is shown in Fig. 20(a). Zucca et al. [276] were among the first to model the semi-cylindrical friction damper (shown in Fig. 20(c)) which had been in use in the turbomachinery industry since the 1980s. This shape, as pointed out in the study, ensures full contact between the damper and the blade-platform over the flat damper side when the relative platform angle changes during blade vibration (i.e. self-adjusting). Their numerical investigation made it possible to link the damper kinematics (i.e. lack/presence of rotation) to its stiffening and damping capabilities. This investigation was further developed by Gastaldi et al. [210,272,281], who parametrised the semi-cylindrical damper shape in terms of its angles and depth of cut (see Fig. 20(d)) and proposed pre-optimisation maps which forecast, without the need for numerical iterative computations, the combination of design parameters that will lead to undesirable behaviours such as rotation and jamming for In-Phase and Out-of-Phase modes. More recent works focus on choosing, among the pre-optimised damper configurations, the best fit for a specific blade [38,40,282]. In [283], the coupling between a blade and a semi-cylindrical damper is optimised by considering a design space defined by ten geometric parameters. Among these parameters, three are allocated for the damper. Two surrogate models are introduced and compared, both capable of significantly reducing the computational time needed for a regular design of experiments (DOE) factorial sampling by several orders of magnitude.

Other options to avoid damper rotation have been explored. As shown in Fig. 20(e), Denimal et al. [277] proposed a conical underplatform damper to avoid the rolling phenomenon of traditional wedged-shaped dampers for In-Phase modes through geometric compliance, leading to a better damping performance in both In-Phase and Out-of-Phase mode when compared to classical wedged and circular dampers. In the context of high-fidelity optimisation, Yuan et al. [275] introduced a surrogate-based optimisation approach to enhance the damping performance of a modified wedge-shaped underplatform damper. The geometrical parameterisation of the damper is shown in Fig. 20(b). The study was intended to optimise the shape of the wedge-shaped friction dampers to achieve minimal resonance frequency shift and maximum friction damping. Recently, Denimal et al. performed a further topology optimisation of an underplatform damper [278] where the internal structure of the wedged damper is optimised under stress constraints imposed on the contact interface. One of the topologically optimal dampers is shown in Fig. 20(f). In addition to the design of the damper geometry itself, a simplified and parametric beam model with only one turbine blade and damper was then developed to study
<figure><img src="image_24.png" /><figcaption>Fig. 21. Different friction ring damper models: (a) Ring damper represented by Lumped parameters from [35]. (b) Ring damper represented by 3D beam from [131]. (c) Open ring damper with rectangular cross section from [133]. (d) Ring damper with V shape cross section from [134]. (e) Ring damper with varying top and bottom surface geometry from [135,285]. (f) Multiple sectors composed tuned ring damper where each sector includes a casing and a slender oscillator that rubs against its casing from [286].</figcaption></figure>

<table><caption>**Table 7**
Features of test set-ups focusing on ring dampers.</caption><thead><tr><th>References</th><th>Set-up</th><th>Preload</th><th>Excitation</th><th>Measured Quantities</th></tr></thead><tbody><tr><td>[260]</td><td>Rotating</td><td>Centrifugal effect</td><td>Piezo actuator</td><td>Dynamic response</td></tr></tbody></table>

the influence of platform position and angle of generic friction damper on the nonlinear modal properties [284]. The study indicated that the optimal configuration of the frictional damper can be achieved when the contact angle is around 20° to 30°. The vertical position of the platform to place the damper is highly mode-dependent. Quaegbeur et al. [189] optimised the intentional mistuned pattern of UPDs to reduce both the 1st flexural and torsional modes. An optimal mistuning pattern was identified considering aerodynamic constraints and the negative random mistuning effects on the vibration response. However, the identified mistuned UPD is mode-dependent and may be sensitive to the veering region in lower nodal diameters.

### 6.1.2. Friction ring damper

The initial studies for friction ring dampers were performed using lumped parameter models, as shown in Fig. 21(a) [35,287]. The impact of both the ring damper mass and the rotational speed on the nonlinear forced frequency response [35] and damped nonlinear normal modes [287] were revealed. The studies indicate that these parameters significantly influence damping properties, as they can alter the centrifugal force applied to the friction interface, resulting in a change in friction limits and stick-slip behaviour on the interface. Results demonstrate that when the damper mass increases or when the rotational speed increases, the amplitude of the resonant peaks changes indicating the existence of an optimal damper mass that should be carefully tuned. A full FEM of the blisk with a simplified ring was then considered in [131], modelling the ring damper using 3D-beam elements as shown in Fig. 21(b). The authors emphasised the significant role of ring thickness in friction damping, concluding that there is an optimal ring thickness for maximising friction damping, which is consistent with earlier investigations in [35]. A full high-fidelity FEM including both the blisk and the ring damper was developed in [132] as a case study to validate a novel reduced order modelling method. In the studies [132,288], the impact of a non-dimensional parameter $\rho$,

which is a function of the friction coefficient, normal contact pressure, and forcing level, was evaluated in the context of a nonlinear dynamic response. The sensitivity of the resonance peak and friction damping to the non-dimensional parameter $\rho$ was investigated over a wide range, showing that the friction damping efficiency is proportionally correlated to the non-dimensional parameter, highlighting that it is crucial to control the normal contact load for the optimal design of friction ring dampers. In terms of design and optimisation, the shape of friction ring dampers has been the main focus as it directly impacts the normal contact pressure. Using a similar blisk FEM shown in [132], a parametric study was carried by Tang and Epureanu [134] to assess the damping effectiveness of a V-shaped friction ring damper, whose geometry is shown in Fig. 21(d). To enhance the robustness of the friction damper, a comprehensive sensitivity analysis was conducted in [134] to investigate the influence of geometric parameters, including the angle, length, and thickness of the wedge shape, on the nonlinear dynamic response of the blisk. The location of the ring damper was also studied over a wide variation of $\rho$ to find the optimal placement. In the recent studies [135,285], the top and bottom surface geometries of the friction ring damper were numerically optimised using damped nonlinear modal analysis. The parameterisation of the surface geometry and some examples are shown in Fig. 21(e). It was found that the variation of the contact interface geometry significantly impacts the normal pressure distribution and initial contact conditions, leading to a large variation in the resonant frequency and damping ratio. It was demonstrated that, by taking an optimal geometry, the vibration response at resonance can be reduced by about 50%.

The concept of tuned vibration absorbers can be integrated into the design of friction damper to enhance the damping effects. The idea of the concept is to allow for substantial energy transfer between the blisk and the ring damper by matching the natural frequencies of the ring damper with a specific resonance frequency of the disk. This tuning can greatly amplify friction damping. The theoretical work of this concept was first proposed in [133]. As shown in Fig. 21(c), an open ring damper with a rectangular cross-section was studied for this concept. The concept was further extended in [286] where a pioneering design for a fully tuned friction ring damper was proposed. The design of the ring damper is shown in Fig. 21(f). It is composed of a casing used to hold the system in place and a slender oscillator
<figure><img src="image_24.png" /><figcaption>Fig. 21. Different friction ring damper models: (a) Ring damper represented by Lumped parameters from [35]. (b) Ring damper represented by 3D beam from [131]. (c) Open ring damper with rectangular cross section from [133]. (d) Ring damper with V shape cross section from [134]. (e) Ring damper with varying top and bottom surface geometry from [135,285]. (f) Multiple sectors composed tuned ring damper where each sector includes a casing and a slender oscillator that rubs against its casing from [286].</figcaption></figure>

<table><caption>**Table 7**
Features of test set-ups focusing on ring dampers.</caption><thead><tr><td>References</td><td>Set-up</td><td>Preload</td><td>Excitation</td><td>Measured Quantities</td></tr></thead><tbody><tr><td>[260]</td><td>Rotating</td><td>Centrifugal effect</td><td>Piezo actuator</td><td>Dynamic response</td></tr></tbody></table>

the influence of platform position and angle of generic friction damper on the nonlinear modal properties [284]. The study indicated that the optimal configuration of the frictional damper can be achieved when the contact angle is around 20° to 30°. The vertical position of the platform to place the damper is highly mode-dependent. Quaegbeur et al. [189] optimised the intentional mistuned pattern of UPDs to reduce both the 1st flexural and torsional modes. An optimal mistuning pattern was identified considering aerodynamic constraints and the negative random mistuning effects on the vibration response. However, the identified mistuned UPD is mode-dependent and may be sensitive to the veering region in lower nodal diameters.

### 6.1.2. Friction ring damper

The initial studies for friction ring dampers were performed using lumped parameter models, as shown in Fig. 21(a) [35,287]. The impact of both the ring damper mass and the rotational speed on the nonlinear forced frequency response [35] and damped nonlinear normal modes [287] were revealed. The studies indicate that these parameters significantly influence damping properties, as they can alter the centrifugal force applied to the friction interface, resulting in a change in friction limits and stick-slip behaviour on the interface. Results demonstrate that when the damper mass increases or when the rotational speed increases, the amplitude of the resonant peaks changes indicating the existence of an optimal damper mass that should be carefully tuned. A full FEM of the blisk with a simplified ring was then considered in [131], modelling the ring damper using 3D-beam elements as shown in Fig. 21(b). The authors emphasised the significant role of ring thickness in friction damping, concluding that there is an optimal ring thickness for maximising friction damping, which is consistent with earlier investigations in [35]. A full high-fidelity FEM including both the blisk and the ring damper was developed in [132] as a case study to validate a novel reduced order modelling method. In the studies [132,288], the impact of a non-dimensional parameter $\rho$,

which is a function of the friction coefficient, normal contact pressure, and forcing level, was evaluated in the context of a nonlinear dynamic response. The sensitivity of the resonance peak and friction damping to the non-dimensional parameter $\rho$ was investigated over a wide range, showing that the friction damping efficiency is proportionally correlated to the non-dimensional parameter, highlighting that it is crucial to control the normal contact load for the optimal design of friction ring dampers. In terms of design and optimisation, the shape of friction ring dampers has been the main focus as it directly impacts the normal contact pressure. Using a similar blisk FEM shown in [132], a parametric study was carried by Tang and Epureanu [134] to assess the damping effectiveness of a V-shaped friction ring damper, whose geometry is shown in Fig. 21(d). To enhance the robustness of the friction damper, a comprehensive sensitivity analysis was conducted in [134] to investigate the influence of geometric parameters, including the angle, length, and thickness of the wedge shape, on the nonlinear dynamic response of the blisk. The location of the ring damper was also studied over a wide variation of $\rho$ to find the optimal placement. In the recent studies [135,285], the top and bottom surface geometries of the friction ring damper were numerically optimised using damped nonlinear modal analysis. The parameterisation of the surface geometry and some examples are shown in Fig. 21(e). It was found that the variation of the contact interface geometry significantly impacts the normal pressure distribution and initial contact conditions, leading to a large variation in the resonant frequency and damping ratio. It was demonstrated that, by taking an optimal geometry, the vibration response at resonance can be reduced by about 50%.

The concept of tuned vibration absorbers can be integrated into the design of friction damper to enhance the damping effects. The idea of the concept is to allow for substantial energy transfer between the blisk and the ring damper by matching the natural frequencies of the ring damper with a specific resonance frequency of the disk. This tuning can greatly amplify friction damping. The theoretical work of this concept was first proposed in [133]. As shown in Fig. 21(c), an open ring damper with a rectangular cross-section was studied for this concept. The concept was further extended in [286] where a pioneering design for a fully tuned friction ring damper was proposed. The design of the ring damper is shown in Fig. 21(f). It is composed of a casing used to hold the system in place and a slender oscillator
that rubs against its casing. It enables the internal friction interface to be loaded with a targeted constant normal load, leading to a substantial improvement in damping properties. It was numerically studied using a high-fidelity FE model based on the geometry of the NASA rotor67 [286], demonstrating good effectiveness of this innovative design across various parameters, with the most significant reduction observed in the presence of substantial geometric mistuning, attributed to the absence of one of its sectors. This study also highlighted that a change in the topology or architecture of the damper can drastically improve the damping properties of friction ring dampers. It also shows some robustness even if the tuning is not perfect. By having a mistuned damper, any blisk mode can be targeted and wide frequency ranges can be targeted with the ring damper, even blade-dominated modes that are usually not targetable with those dampers. Similarly to the study in [189] applied to UPD, it appears that mistuned friction dampers have generally positive effects on friction damping and its robustness regardless of the type of damper. Despite numerous numerical investigations on the design of friction ring dampers, to the best of the authors' knowledge, no work with rigorous optimisation studies has been conducted for friction ring dampers, and this may be the topic of future research.

### 6.1.3. Shroud and snubber damping

There are very few studies on the design and optimisation of shroud and snubber damping. One of them was conducted by Wang et al. [273] using normalised slip load and the orientation of the interface as two critical design parameters. The other main parameter to control the friction damping of the shroud is the normal load on the friction interface as indicated in experimental studies [87,252,254]. There will be an optimal normal load to maximise the friction damping while avoiding loss of contact.

In summary, the previous design studies of friction dampers focused mainly on the shape and surface geometries of the damper to improve the damping performance. For underplatform dampers, different geometries were explored to avoid the rolling motion of the damper during the in-phase mode. For friction ring dampers, in addition to the exploration of different shapes, the mesoscale surface geometry was considered as a design parameter, showing a significant impact on the damping performance due to its influence on normal pressure distribution. Innovative designs such as tuned friction mass dampers were explored to increase relative motions between the blisk and the damper. However, experimental verification and rigorous optimisation studies are lacking for friction ring damper designs. For both ring and underplatform dampers, the intentional mistuning design of the friction damper was explored, showing improvement in friction damping.

## 6.2. Robust design and uncertainty quantification

Another crucial aspect in the design of friction dampers is ensuring performance robustness under various sources of uncertainties. One of the main uncertainties comes from the modelling of the friction interface. In this subsection, we will first present the variability of experimental measurements observed during the contact interface characterisation and friction damping device testing. This will be followed by a review of related numerical studies on robust design and uncertainty quantification.

### 6.2.1. Experimental observation of the variation

Experimental measurements often encounter variability and/or lack of repeatability, meaning that the same nominal experimental conditions may produce different measured reference quantities. While some variability is inherent in all experimental observations, an additional factor arises when considering contact friction: the non-uniqueness of the solution. As shown in Section 3, for turbine applications, the contact friction model is usually parametrised using a few variables that can be obtained experimentally using fretting test rigs [84,203,290] as shown

in Section 5.2. However, current contact models are not yet capable of capturing the complexity of frictional phenomena. This results in a large observed variability of the contact parameters. Fig. 22 gives an example of the variation of measured coefficient of friction and contact stiffness from the 1D friction test rig presented in [84], showing that both contact parameters follow a Gaussian distribution in Test 5 with a standard deviation of 0.23% and 3.77% respectively. It can also be observed that there is a larger variation of the mean values of coefficient of friction and contact stiffness among the four tests, highlighting that it is not possible to represent the contact behaviour using a unique set of contact parameters.

In addition to the variation of contact parameters themselves, the initial contact conditions of the friction damping device may lead to a large variation in the damping performance. If a portion of the contact interface in friction damping devices remains permanently stuck during vibration (i.e., microslip of the contact interface), the dynamic behaviour and the friction damping contribution depend on the initial conditions, leading to different static force equilibria. This phenomenon was initially theorised numerically [227] and later experimentally observed [242,246,269]. In [242,246], the same experimental nominal conditions (preload, excitation level) resulted in different frequency response functions on the blade for underplatform and midspan dampers, respectively. In both cases, the observed variability was linked to differences in the static component of the measured contact forces.

Furthermore, flat-on-flat contact interfaces may further amplify variability in damper performance because the nominal contact area may differ from the actual one due to surface imperfections such as roughness and waviness. While this effect is present in all friction damping solutions (e.g., roots, shrouds), it is particularly pronounced in the case of friction dampers. This is because different damper kinematics, such as increased rotation, may be triggered by different support conditions. As shown in Figs. 20 and 21, flat-on-flat contact interfaces are often used for most friction damper designs. In [60], it was observed that numerical predictions of the dynamic response of blades with wedge dampers, assuming a uniform pressure distribution on the grid of contact points on the nominal contact area, failed to match experimental observations. This discrepancy encompassed both the reduction in response amplitude and the difference in frequency (i.e., overall stiffness and damping contribution). As shown in Fig. 23(a), the experimental-numerical agreement significantly improved when the actual pressure distribution was estimated using a Fuji Prescale pressure film and was replicated in numerical simulations. Similar findings were obtained when investigating the damper mechanics of asymmetric underplatform dampers [289]. In that case, a practical solution to minimise damper rotation, ensure an adequate level of damping, and reduce variability was to remove the central portion of the damper's flat surface to ensure contact at the outer edges, as illustrated in Fig. 23(b). A recent study [243] analysed the dynamic response of an industrial high-pressure turbine-bladed disk with wedge dampers measured from engine tests. The study revealed that accurately modelling realistic contact bedding patterns, derived from observations of worn components, significantly improved the agreement between frequency response predictions and engine data, as illustrated in Fig. 23(c). Furthermore, an important implication is that numerical predictions are not unique but rather accompanied by an uncertainty band attributable to contact sensitivity. The studies in [231,291] also compare the dynamic response from experimental testing and numerical models, concluding that the discrepancy obtained comes mainly from the insufficient modelling of the friction interface. Delaune et al. [292] performed a study to investigate the effects of uncertainties associated with fretting wear, showing that it can significantly change the variation of the vibration response.

In summary, as observed in Section 6.2.1, the main source of discrepancy between the experimental and numerical simulation results comes from friction interface modelling. Firstly, current contact models
that rubs against its casing. It enables the internal friction interface to be loaded with a targeted constant normal load, leading to a substantial improvement in damping properties. It was numerically studied using a high-fidelity FE model based on the geometry of the NASA rotor67 [286], demonstrating good effectiveness of this innovative design across various parameters, with the most significant reduction observed in the presence of substantial geometric mistuning, attributed to the absence of one of its sectors. This study also highlighted that a change in the topology or architecture of the damper can drastically improve the damping properties of friction ring dampers. It also shows some robustness even if the tuning is not perfect. By having a mistuned damper, any blisk mode can be targeted and wide frequency ranges can be targeted with the ring damper, even blade-dominated modes that are usually not targetable with those dampers. Similarly to the study in [189] applied to UPD, it appears that mistuned friction dampers have generally positive effects on friction damping and its robustness regardless of the type of damper. Despite numerous numerical investigations on the design of friction ring dampers, to the best of the authors' knowledge, no work with rigorous optimisation studies has been conducted for friction ring dampers, and this may be the topic of future research.

### 6.1.3. Shroud and snubber damping

There are very few studies on the design and optimisation of shroud and snubber damping. One of them was conducted by Wang et al. [273] using normalised slip load and the orientation of the interface as two critical design parameters. The other main parameter to control the friction damping of the shroud is the normal load on the friction interface as indicated in experimental studies [87,252,254]. There will be an optimal normal load to maximise the friction damping while avoiding loss of contact.

In summary, the previous design studies of friction dampers focused mainly on the shape and surface geometries of the damper to improve the damping performance. For underplatform dampers, different geometries were explored to avoid the rolling motion of the damper during the in-phase mode. For friction ring dampers, in addition to the exploration of different shapes, the mesoscale surface geometry was considered as a design parameter, showing a significant impact on the damping performance due to its influence on normal pressure distribution. Innovative designs such as tuned friction mass dampers were explored to increase relative motions between the blisk and the damper. However, experimental verification and rigorous optimisation studies are lacking for friction ring damper designs. For both ring and underplatform dampers, the intentional mistuning design of the friction damper was explored, showing improvement in friction damping.

## 6.2. Robust design and uncertainty quantification

Another crucial aspect in the design of friction dampers is ensuring performance robustness under various sources of uncertainties. One of the main uncertainties comes from the modelling of the friction interface. In this subsection, we will first present the variability of experimental measurements observed during the contact interface characterisation and friction damping device testing. This will be followed by a review of related numerical studies on robust design and uncertainty quantification.

### 6.2.1. Experimental observation of the variation

Experimental measurements often encounter variability and/or lack of repeatability, meaning that the same nominal experimental conditions may produce different measured reference quantities. While some variability is inherent in all experimental observations, an additional factor arises when considering contact friction: the non-uniqueness of the solution. As shown in Section 3, for turbine applications, the contact friction model is usually parametrised using a few variables that can be obtained experimentally using fretting test rigs [84,203,290] as shown

in Section 5.2. However, current contact models are not yet capable of capturing the complexity of frictional phenomena. This results in a large observed variability of the contact parameters. Fig. 22 gives an example of the variation of measured coefficient of friction and contact stiffness from the 1D friction test rig presented in [84], showing that both contact parameters follow a Gaussian distribution in Test 5 with a standard deviation of 0.23% and 3.77% respectively. It can also be observed that there is a larger variation of the mean values of coefficient of friction and contact stiffness among the four tests, highlighting that it is not possible to represent the contact behaviour using a unique set of contact parameters.

In addition to the variation of contact parameters themselves, the initial contact conditions of the friction damping device may lead to a large variation in the damping performance. If a portion of the contact interface in friction damping devices remains permanently stuck during vibration (i.e., microslip of the contact interface), the dynamic behaviour and the friction damping contribution depend on the initial conditions, leading to different static force equilibria. This phenomenon was initially theorised numerically [227] and later experimentally observed [242,246,269]. In [242,246], the same experimental nominal conditions (preload, excitation level) resulted in different frequency response functions on the blade for underplatform and midspan dampers, respectively. In both cases, the observed variability was linked to differences in the static component of the measured contact forces.

Furthermore, flat-on-flat contact interfaces may further amplify variability in damper performance because the nominal contact area may differ from the actual one due to surface imperfections such as roughness and waviness. While this effect is present in all friction damping solutions (e.g., roots, shrouds), it is particularly pronounced in the case of friction dampers. This is because different damper kinematics, such as increased rotation, may be triggered by different support conditions. As shown in Figs. 20 and 21, flat-on-flat contact interfaces are often used for most friction damper designs. In [60], it was observed that numerical predictions of the dynamic response of blades with wedge dampers, assuming a uniform pressure distribution on the grid of contact points on the nominal contact area, failed to match experimental observations. This discrepancy encompassed both the reduction in response amplitude and the difference in frequency (i.e., overall stiffness and damping contribution). As shown in Fig. 23(a), the experimental-numerical agreement significantly improved when the actual pressure distribution was estimated using a Fuji Prescale pressure film and was replicated in numerical simulations. Similar findings were obtained when investigating the damper mechanics of asymmetric underplatform dampers [289]. In that case, a practical solution to minimise damper rotation, ensure an adequate level of damping, and reduce variability was to remove the central portion of the damper's flat surface to ensure contact at the outer edges, as illustrated in Fig. 23(b). A recent study [243] analysed the dynamic response of an industrial high-pressure turbine-bladed disk with wedge dampers measured from engine tests. The study revealed that accurately modelling realistic contact bedding patterns, derived from observations of worn components, significantly improved the agreement between frequency response predictions and engine data, as illustrated in Fig. 23(c). Furthermore, an important implication is that numerical predictions are not unique but rather accompanied by an uncertainty band attributable to contact sensitivity. The studies in [231,291] also compare the dynamic response from experimental testing and numerical models, concluding that the discrepancy obtained comes mainly from the insufficient modelling of the friction interface. Delaune et al. [292] performed a study to investigate the effects of uncertainties associated with fretting wear, showing that it can significantly change the variation of the vibration response.

In summary, as observed in Section 6.2.1, the main source of discrepancy between the experimental and numerical simulation results comes from friction interface modelling. Firstly, current contact models
<figure><img src="image_25.png" /><figcaption>Fig. 22. (a,d) Evolution in time of μ and kᵢ; (b,e) Statistical distribution of steady state μ and kᵢ (values after 200 min); (c,f) Boxplots of steady state μ and kᵢ; Tests 1 and 2 were conducted at a relative displacement of 14μm, and tests 4 and 5 at 22μm [275].</figcaption></figure>

<figure><img src="image_26.png" /><figcaption>Fig. 23. (a) Damper performance curve for Out-of-Phase mode of vibration: measured vs simulated results using different hypotheses on contact pressure distribution [236] (b) (Left) Measured In-Phase platform-platform hysteresis of an asymmetrical underplatform damper with two different flat surface finishings; (Right) In black, profilometer results for damper flat surface and corresponding platform insert and, in blue, schematic representation of re-finishing process for damper flat surface. Adapted from [289]. (c) Simulated frequency response function for bladed disk with wedge dampers under different contact point distribution hypotheses and comparison with experimental results. Adapted from [243].</figcaption></figure>
<figure><img src="image_25.png" /><figcaption>**Fig. 22.** (a,d) Evolution in time of $\mu$ and $k_t$; (b,e) Statistical distribution of steady state $\mu$ and $k_t$ (values after 200 min); (c,f) Boxplots of steady state $\mu$ and $k_t$; Tests 1 and 2 were conducted at a relative displacement of 14µm, and tests 4 and 5 at 22µm [275].</figcaption></figure>

<figure><img src="image_26.png" /><figcaption>**Fig. 23.** (a) Damper performance curve for Out-of-Phase mode of vibration: measured vs simulated results using different hypotheses on contact pressure distribution [236] (b) (Left) Measured In-Phase platform-platform hysteresis of an asymmetrical underplatform damper with two different flat surface finishings; (Right) In black, profilometer results for damper flat surface and corresponding platform insert and, in blue, schematic representation of re-finishing process for damper flat surface. Adapted from [289]. (c) Simulated frequency response function for bladed disk with wedge dampers under different contact point distribution hypotheses and comparison with experimental results. Adapted from [243].</figcaption></figure>
are not yet able to fully capture the contact friction behaviour from ex-
periments, which leads to contact parameter variability. Furthermore,
the inevitable deviation of surface geometry from the nominal condi-
tion, due to manufacturing and finishing tolerances can significantly
impact contact pressure distribution leading to a very different dynamic
response. Finally, the inability to control the initial position of friction
dampers which is a result of the centrifugal loads arising from disk
rotation, can change the contact pressure distribution greatly and thus
influence the dynamic response. In addition, uncertainties may also
arise from environmental conditions during operation such as the vari-
ation of temperature which may change the friction coefficient [293],
and the evolution of material and contact properties over the structure's
lifespan, for example, due to the progression of wear [85,294] and the
influence of creep [295].

6.2.2. *Robust and reliability based design studies*

Considering the various sources of uncertainties highlighted in the
experimental observation, it is imperative to identify and address the
primary uncertainties in the system to enhance numerical predictions,
as demonstrated in previous studies [82]. Petrov [296], using a sensi-
tivity approach and reconstructing the probability density functions of
forced frequency response, demonstrated the substantial impact of un-
certainties in the friction coefficient on the nonlinear dynamic response
of turbine blades. The impact of contact parameters on the variation
of the dynamical response was further confirmed by Krack et al. [297]
through a reliability analysis of a blisk with shroud coupling. To further
understand the uncertainties related to the contact parameters, an
advanced multi-scale modelling approach was developed in [294] con-
sidering surface roughness, waviness and fretting wear of the contact
interface. However, the related robust design and optimisation of the
approach have not yet been carried out due to the high computational
expense.

More recently, Yuan et al. [275] performed a robust optimisation of the geometry of the wedge-shaped underplatform damper considering manufacturing variability. The performance robustness of the damper is defined as the sum of the variance of two simulated parameters, namely resonance amplitude and resonance frequencies. A new optimisation approach taking into account the non-uniqueness of friction forces was proposed for the computation of dynamic response boundaries for turbine bladed disks with contacts [298,299] and midspan dampers [300]. The computation of frequency response variability was further enhanced using a nonlinear-mode-based approach [301].

Sun et al. [284] conducted a stochastic analysis of nonlinear modal
properties, considering variations in the contact angle and the relative
position of the damper to the platform. Monte Carlo simulations to-
gether with Sobol analysis were used to evaluate the global sensitivity
of each geometrical parameter. Yuan et al. [275] propagated the un-
certainties from friction interface parameters in the dynamical response
of bladed disk systems from an underplatform damper. The stochastic
dynamic response obtained through numerical uncertainty propagation
closely aligns with the experimental results of the nonlinear dynamic
response. The key findings of their investigation are depicted in Fig. 24.
It is worth noting that the phase instead of frequency is used in the
study to facilitate the construction of data-driven models for numerical
propagation. The majority of the experimental data points lie within
the 90% confidence interval identified from the stochastic model con-
sidering the uncertainties from contact parameters. It illustrates that
the stochastic dynamic response can improve the predictivity of the
physical model by taking into account the main sources of uncertainty.

In terms of friction ring dampers, the robustness of their shape
design was studied using a 3D full-scale compressor in [285]. The
variation in the manufacturing process was taken into account and
uncertainties were propagated using Kriging-based surrogate models.
The uncertainties associated with friction dampers can also give rise to
mistuning effects, resulting in substantial variations in their impact on
the dynamic response [189]. This depends on the potential deliberately

introduced mistuning pattern. Yuan et al. [86] proposed a methodology
to quantify the influence of mesoscale interface geometry variation
on the nonlinear dynamic response, showing that the edge radii and
surface bumpiness can have significant effects on the nonlinear modal
properties of fan blade systems. In [302,303], the boundary of the
steady-state response was determined using two different numerical
approaches for wedged friction dampers. A criterion that determines
the periodic response boundaries according to the limit tangential force
was proposed.

The different studies presented previously have demonstrated the
significant impact of friction contact uncertainties on the nonlinear
dynamic response for aero-engine structures with friction interfaces,
highlighting the importance of taking into account the uncertainties
in the design stage. Most of the current uncertainty and robust de-
sign studies are related to underplatform dampers using relatively
simplified models. There are very few works focusing on the robust
design of the blade root damping and friction ring dampers due to
the high computational cost of manipulating the variation of high-
fidelity models and associated nonlinear simulations. Classical Monte
Carlo simulation methods are often used for uncertainty propagation
but they become very expensive for high-fidelity models. To over-
come this limitation, Butlin et al. [304,305] proposed a new approach
based on the principle of Maximum Entropy for the prediction of the
response variability of friction-damped turbine blades. The proposed
approach can make the prediction two orders of magnitude faster than
the classical Monte Carlo simulation. It was also validated against
the experimental database through a representative turbine blade test
rig. More efforts are needed in the development of such methods as
well as data-driven modelling techniques to improve the efficiency
of the uncertainty propagation of such nonlinear large-scale models.
The other challenge for uncertainty quantification is that there is not
sufficient experimental data available on the characterisation of friction
interfaces at different scales. More experimental testing will be needed
to create the database for supporting uncertainty quantification and
robust design studies.

7. New trending topics

7.1. *New friction damper designs*

New concepts of friction dampers in turbomachinery have been
emerging in recent years to enhance and control friction damping
for multi-operational conditions and multi-modes in a passive, active
or semi-active manner. Coupling the traditional dry friction dampers
with electrical units, novel materials and energy transfer concepts have
been explored to control the normal pressure or relative tangential dis-
placement on the friction interface. This section introduces innovative
concepts for the next-generation design of friction dampers.

7.1.1. *Shunted piezoelectric damping*

The piezoelectric material can be actively or passively coupled with
friction dampers to improve damping performance. The key concept
of shunted piezoelectric damping is to distribute piezoelectric mate-
rial on the surface of the friction damper [306,307] to dissipate the
strain energy stored in the friction damper or blisk itself through
a shunted impedance. A similar idea was also investigated in [308]
but without the friction ring damper. This damping technique can be
used to suppress the vibration for both disk-dominated modes and
blade-dominated modes [309].

Programmable shunted circuit parameters, combined with a reliable
dry friction damping structure, are a very robust damping technique
for an aero-engine operating at variable working conditions where the
normal force deviates from the design value. The effective working
range can be extended up to 2.6 times [307] by matching piezo-
electric damping and dry friction regarding the working state and
excitation level. Using programmable synthetic impedance [310] or
are not yet able to fully capture the contact friction behaviour from ex-
periments, which leads to contact parameter variability. Furthermore,
the inevitable deviation of surface geometry from the nominal condi-
tion, due to manufacturing and finishing tolerances can significantly
impact contact pressure distribution leading to a very different dynamic
response. Finally, the inability to control the initial position of friction
dampers which is a result of the centrifugal loads arising from disk
rotation, can change the contact pressure distribution greatly and thus
influence the dynamic response. In addition, uncertainties may also
arise from environmental conditions during operation such as the vari-
ation of temperature which may change the friction coefficient [293],
and the evolution of material and contact properties over the structure's
lifespan, for example, due to the progression of wear [85,294] and the
influence of creep [295].

6.2.2. *Robust and reliability based design studies*

Considering the various sources of uncertainties highlighted in the
experimental observation, it is imperative to identify and address the
primary uncertainties in the system to enhance numerical predictions,
as demonstrated in previous studies [82]. Petrov [296], using a sensi-
tivity approach and reconstructing the probability density functions of
forced frequency response, demonstrated the substantial impact of un-
certainties in the friction coefficient on the nonlinear dynamic response
of turbine blades. The impact of contact parameters on the variation
of the dynamical response was further confirmed by Krack et al. [297]
through a reliability analysis of a blisk with shroud coupling. To further
understand the uncertainties related to the contact parameters, an
advanced multi-scale modelling approach was developed in [294] con-
sidering surface roughness, waviness and fretting wear of the contact
interface. However, the related robust design and optimisation of the
approach have not yet been carried out due to the high computational
expense.

More recently, Yuan et al. [275] performed a robust optimisation of the geometry of the wedge-shaped underplatform damper considering manufacturing variability. The performance robustness of the damper is defined as the sum of the variance of two simulated parameters, namely resonance amplitude and resonance frequencies. A new optimisation approach taking into account the non-uniqueness of friction forces was proposed for the computation of dynamic response boundaries for turbine bladed disks with contacts [298,299] and midspan dampers [300]. The computation of frequency response variability was further enhanced using a nonlinear-mode-based approach [301].

Sun et al. [284] conducted a stochastic analysis of nonlinear modal
properties, considering variations in the contact angle and the relative
position of the damper to the platform. Monte Carlo simulations to-
gether with Sobol analysis were used to evaluate the global sensitivity
of each geometrical parameter. Yuan et al. [275] propagated the un-
certainties from friction interface parameters in the dynamical response
of bladed disk systems from an underplatform damper. The stochastic
dynamic response obtained through numerical uncertainty propagation
closely aligns with the experimental results of the nonlinear dynamic
response. The key findings of their investigation are depicted in Fig. 24.
It is worth noting that the phase instead of frequency is used in the
study to facilitate the construction of data-driven models for numerical
propagation. The majority of the experimental data points lie within
the 90% confidence interval identified from the stochastic model con-
sidering the uncertainties from contact parameters. It illustrates that
the stochastic dynamic response can improve the predictivity of the
physical model by taking into account the main sources of uncertainty.

In terms of friction ring dampers, the robustness of their shape
design was studied using a 3D full-scale compressor in [285]. The
variation in the manufacturing process was taken into account and
uncertainties were propagated using Kriging-based surrogate models.
The uncertainties associated with friction dampers can also give rise to
mistuning effects, resulting in substantial variations in their impact on
the dynamic response [189]. This depends on the potential deliberately

introduced mistuning pattern. Yuan et al. [86] proposed a methodology
to quantify the influence of mesoscale interface geometry variation
on the nonlinear dynamic response, showing that the edge radii and
surface bumpiness can have significant effects on the nonlinear modal
properties of fan blade systems. In [302,303], the boundary of the
steady-state response was determined using two different numerical
approaches for wedged friction dampers. A criterion that determines
the periodic response boundaries according to the limit tangential force
was proposed.

The different studies presented previously have demonstrated the
significant impact of friction contact uncertainties on the nonlinear
dynamic response for aero-engine structures with friction interfaces,
highlighting the importance of taking into account the uncertainties
in the design stage. Most of the current uncertainty and robust de-
sign studies are related to underplatform dampers using relatively
simplified models. There are very few works focusing on the robust
design of the blade root damping and friction ring dampers due to
the high computational cost of manipulating the variation of high-
fidelity models and associated nonlinear simulations. Classical Monte
Carlo simulation methods are often used for uncertainty propagation
but they become very expensive for high-fidelity models. To over-
come this limitation, Butlin et al. [304,305] proposed a new approach
based on the principle of Maximum Entropy for the prediction of the
response variability of friction-damped turbine blades. The proposed
approach can make the prediction two orders of magnitude faster than
the classical Monte Carlo simulation. It was also validated against
the experimental database through a representative turbine blade test
rig. More efforts are needed in the development of such methods as
well as data-driven modelling techniques to improve the efficiency
of the uncertainty propagation of such nonlinear large-scale models.
The other challenge for uncertainty quantification is that there is not
sufficient experimental data available on the characterisation of friction
interfaces at different scales. More experimental testing will be needed
to create the database for supporting uncertainty quantification and
robust design studies.

7. New trending topics

7.1. *New friction damper designs*

New concepts of friction dampers in turbomachinery have been
emerging in recent years to enhance and control friction damping
for multi-operational conditions and multi-modes in a passive, active
or semi-active manner. Coupling the traditional dry friction dampers
with electrical units, novel materials and energy transfer concepts have
been explored to control the normal pressure or relative tangential dis-
placement on the friction interface. This section introduces innovative
concepts for the next-generation design of friction dampers.

7.1.1. *Shunted piezoelectric damping*

The piezoelectric material can be actively or passively coupled with
friction dampers to improve damping performance. The key concept
of shunted piezoelectric damping is to distribute piezoelectric mate-
rial on the surface of the friction damper [306,307] to dissipate the
strain energy stored in the friction damper or blisk itself through
a shunted impedance. A similar idea was also investigated in [308]
but without the friction ring damper. This damping technique can be
used to suppress the vibration for both disk-dominated modes and
blade-dominated modes [309].

Programmable shunted circuit parameters, combined with a reliable
dry friction damping structure, are a very robust damping technique
for an aero-engine operating at variable working conditions where the
normal force deviates from the design value. The effective working
range can be extended up to 2.6 times [307] by matching piezo-
electric damping and dry friction regarding the working state and
excitation level. Using programmable synthetic impedance [310] or
<figure><img src="image_27.png" /><figcaption>Fig. 24. (a) Randomly generated FRFs of the turbine blades with underplatform damper at the excitation level of 17 N; (b,c) Comparison of stochastic dynamic phase response and experimental data at the excitation level of 0.96 N (linear case) and 17 N (nonlinear case) [275].</figcaption></figure>

<figure><img src="image_28.png" /><figcaption>Fig. 25. (a) Illustration of piezoelectric ring damper [307]; (b) Conceptual design of contact normal load distribution using piezoelectric actuation [313].</figcaption></figure>

the self-adaptive shunted circuit (e.g. the synchronised switch damping circuit [311]), also leads to vibration mitigation robust to variable working conditions. Fig. 25(a) depicts an illustration of a piezoelectric ring damper, showcasing how damping is imparted through both the friction damper and the piezoelectric damping mechanism. Furthermore, the proposed piezoelectric network can be used to mitigate the amplification effects of the mistuned blisk by distributing piezoelectric patches in an optimal topology [312]. It will facilitate the creation of electric energy pathways between sectors, promoting the dissipation of vibrational energy.

### 7.1.2. Normal pressure-controlled friction damper

The effectiveness of friction damping is primarily governed by the critical friction threshold between two surfaces, which is predominantly influenced by the normal load and coefficient of friction. Recent research efforts have been made to adjust the local normal loading of the friction damper by inputting a control voltage to the distributed piezoelectric materials based on the inverse piezoelectric effect. Wu et al. [314] explored this concept to mitigate the steady-state vibration of the structure by controlling the second harmonic time-varying normal force and revealed a 20% increase in damping compared with the optimal constant normal force. To remove the additional mass brought by the actuators that limits their application prospects in the aerospace field, macro-fibre-composites can be applied to control the normal force of the friction dampers as studied in [315]. Recently, Lasen et al. [313,316] introduced a proof-of-concept experimental campaign aiming for a turbine underplatform damper. Its conceptual design is shown in Fig. 25(b). The pressure distribution due to the normal load (noted NL in Fig. 25(b)) can be altered using a series of piezoelectric actuators leading to different hysteretic behaviour and frequency responses. This conceptual design of normal load control has

been applied to a 1D friction test rig in [313] demonstrating that the stiffness, energy dissipation, and friction limit can be altered through the manipulation of actuation pattern and/or input voltage. Such a control mechanism allows for customising the interface behaviour to achieve the optimal damping performance of the assembled structure. The principal challenge of piezo-based damping technologies is the installation and maintenance of piezoelectric materials.

## 7.2. New modelling and simulation techniques

### 7.2.1. Full data-driven models for contact modelling and simulation

Recent developments in numerical modelling and simulation have emphasised the adoption of data-driven techniques for the modelling and simulation of contact laws. These techniques avoid the expensive trial-and-error process of tuning the contact model parameters. However, using these approaches presents some challenges, such as requiring a large data set and dealing with the difficulty in obtaining friction information. In [317], artificial neural networks (ANN) are used on experimental data for the reconstruction of the normal contact force from the displacement and velocity. In the context of control applications, Dong et al. [318] suggested a method to address friction modelling by incorporating a deep Gaussian process for compensation. In [319], ANN is used to predict the friction force based on the normal load, relative velocity and material properties derived from different elementary experimental benchmarks. Finally, in [320], ANN is used to reduce the computational cost and replace the AFT procedure in the computation of the nonlinear dynamic response.

### 7.2.2. Multi-physics simulations

Recent endeavours have focused on incorporating and coupling with other physical phenomena to enhance predictions and integrate
<figure><img src="image_27.png" /><figcaption>Fig. 24. (a) Randomly generated FRFs of the turbine blades with underplatform damper at the excitation level of 17 N; (b,c) Comparison of stochastic dynamic phase response and experimental data at the excitation level of 0.96 N (linear case) and 17 N (nonlinear case) [275].</figcaption></figure>

<figure><img src="image_28.png" /><figcaption>Fig. 25. (a) Illustration of piezoelectric ring damper [307]; (b) Conceptual design of contact normal load distribution using piezoelectric actuation [313].</figcaption></figure>

the self-adaptive shunted circuit (e.g. the synchronised switch damping circuit [311]), also leads to vibration mitigation robust to variable working conditions. Fig. 25(a) depicts an illustration of a piezoelectric ring damper, showcasing how damping is imparted through both the friction damper and the piezoelectric damping mechanism. Furthermore, the proposed piezoelectric network can be used to mitigate the amplification effects of the mistuned blisk by distributing piezoelectric patches in an optimal topology [312]. It will facilitate the creation of electric energy pathways between sectors, promoting the dissipation of vibrational energy.

### 7.1.2. Normal pressure-controlled friction damper

The effectiveness of friction damping is primarily governed by the critical friction threshold between two surfaces, which is predominantly influenced by the normal load and coefficient of friction. Recent research efforts have been made to adjust the local normal loading of the friction damper by inputting a control voltage to the distributed piezoelectric materials based on the inverse piezoelectric effect. Wu et al. [314] explored this concept to mitigate the steady-state vibration of the structure by controlling the second harmonic time-varying normal force and revealed a 20% increase in damping compared with the optimal constant normal force. To remove the additional mass brought by the actuators that limits their application prospects in the aerospace field, macro-fibre-composites can be applied to control the normal force of the friction dampers as studied in [315]. Recently, Lasen et al. [313,316] introduced a proof-of-concept experimental campaign aiming for a turbine underplatform damper. Its conceptual design is shown in Fig. 25(b). The pressure distribution due to the normal load (noted NL in Fig. 25(b)) can be altered using a series of piezoelectric actuators leading to different hysteretic behaviour and frequency responses. This conceptual design of normal load control has

been applied to a 1D friction test rig in [313] demonstrating that the stiffness, energy dissipation, and friction limit can be altered through the manipulation of actuation pattern and/or input voltage. Such a control mechanism allows for customising the interface behaviour to achieve the optimal damping performance of the assembled structure. The principal challenge of piezo-based damping technologies is the installation and maintenance of piezoelectric materials.

## 7.2. New modelling and simulation techniques

### 7.2.1. Full data-driven models for contact modelling and simulation

Recent developments in numerical modelling and simulation have emphasised the adoption of data-driven techniques for the modelling and simulation of contact laws. These techniques avoid the expensive trial-and-error process of tuning the contact model parameters. However, using these approaches presents some challenges, such as requiring a large data set and dealing with the difficulty in obtaining friction information. In [317], artificial neural networks (ANN) are used on experimental data for the reconstruction of the normal contact force from the displacement and velocity. In the context of control applications, Dong et al. [318] suggested a method to address friction modelling by incorporating a deep Gaussian process for compensation. In [319], ANN is used to predict the friction force based on the normal load, relative velocity and material properties derived from different elementary experimental benchmarks. Finally, in [320], ANN is used to reduce the computational cost and replace the AFT procedure in the computation of the nonlinear dynamic response.

### 7.2.2. Multi-physics simulations

Recent endeavours have focused on incorporating and coupling with other physical phenomena to enhance predictions and integrate
additional design technologies. As an illustration of piezoelectric coupling, Wu et al. [321] introduced the concept of the Nonlinear Modal Electromechanical Coupling Factor (NMEMCF). The paper outlines a numerical strategy for incorporating piezoelectric coupling into nonlinear dynamic computations using the HBM. This approach was validated through testing on a beam structure with a piezo patch, demonstrating its effectiveness in achieving high damping levels. Properly considering piezoelectric coupling is the initial stage of the numerical design of piezoelectric damping, as discussed in Section 7.1.1. In [322], thermal coupling is considered and a numerical strategy to integrate it into the simulations is presented. In the proposed modelling, the friction coefficient depends on the temperature, an unknown variable within the problem, alongside the displacements, and the former is different for each contact point. In the context of aerodynamic coupling, in [323], a fully-coupled Fluid-Structure Interaction solver able to deal with nonlinear effects of both structural and fluid domains is presented. It is based on the HBM and a serial coupling strategy where the structural and fluid problems are solved alternatively.

### 7.3. New experimental techniques

In earlier studies, the investigation of contact interfaces was carried out independently from the study of friction-damped turbomachinery components. However, recent advancements have led to the integration of these two approaches. Consequently, two types of experimental evidence, relating to the same test case, such as dynamic response and contact behaviour (e.g., hysteresis or relative contact kinematics), are now available. This integration enables a more comprehensive validation process and facilitates pinpointing deficiencies in the model with greater precision.

Another recent trend involves heightened focus on the actual contact area, which may deviate from the nominal one derived solely from the geometry of the contact interface. Initially, this was accomplished by scanning the contact surfaces and modelling the particular case under examination [236,289]. Subsequently, common wear patterns were analysed [243] to inform numerical models, enabling the generation of not just one specific response but rather a statistical distribution of responses. Recent studies used ultrasound [207] and total internal reflection [208] methods to actively monitor both the true contact area and the actual contact conditions in real time. In addition, in [257] the periodic recording of the effect of wear on the dynamic response evolution has been observed, while simultaneously monitoring the evolution of the contact interface and the dynamic response.

To improve the efficiency of experimental testing, nonlinear modal testing has been actively explored to capture only the amplitude-dependent resonance response of the system instead of the whole FRF at different force amplitudes. Phase lock control methods based on the extended periodic motion concept were developed to obtain the amplitude-dependent modal properties and were used to validate a turbine blade component test with frictional contacts [324]. Recently, velocity feedback-based nonlinear modal testing was proposed to remove the requirement of a controller [325]. In addition, control-based-continuation [326,327] and response-controlled testing methods [328, 329] are promising alternatives which can also be applied to predict autonomous dynamical responses such as limit cycle oscillations due to friction interfaces. Recently, an experimental test rig for friction saturated limit cycle oscillations was developed for the validation of the numerical prediction methods of turbine blades where an adjustable velocity feedback loop was instrumented [330].

## 8. Conclusions

This review offers an in-depth look at recent progress in friction damping for turbomachinery. The discussion encompasses numerical modelling, simulation, experimental testing, design methodologies, and optimisation studies, providing insights into cutting-edge techniques

and key discoveries. By identifying gaps and suggesting future research directions, this review serves as a valuable resource. Potential research avenues for each topic are given next.

* **Modelling and simulation.** Numerical contact and friction models require enhancement to incorporate various modelling scales and additional physics affecting contact dynamics, which necessitates maintaining computational efficiency and improving current simulation techniques. An ongoing challenge lies in calibrating contact model parameters, often based on limited experimental data and valid only under specific operating conditions. Future endeavours should strive for a systematic methodology for parameter calibration. Furthermore, numerical resolution methods could aim for greater realism with fewer model reduction approximations and more intricate finite element models. Addressing complex phenomena such as mistuning, multi-physics effects, and uncertainties promises more accurate predictions. This progress can be facilitated by computational solution improvements that rely on parallel computing, such as the use of a parallel HBM [331] or parallel model refinement [332]. Both methods offer significant potential for computational speedups and could provide even greater benefits when applied to more realistic models. Another strategy involves data-driven modelling techniques, which emphasise the need for extensive experimental data while avoiding reliance on traditional modelling efforts to account for uncertain multi-physics behaviour. Hybrid analytical and data-driven modelling techniques represent a promising research area. These approaches require further detailed investigation to fully leverage the potential benefits of recent rapid advancements in artificial intelligence algorithms and computing power.

* **Experimental investigation.** Some of the possible topics of future experimental investigation include a thorough campaign on the effects of wear and surface finishing on contact conditions and dynamic response, the contact characterisation of new coatings and materials, including additive manufactured ones and measurement techniques targeting real time contact condition monitoring. Novel experimental techniques to monitor local contact conditions, and improve the efficiency of the testing are needed to investigate the effects of the variation of multi-scale surface geometries, and fretting wear. More experimental test rigs are needed for friction ring dampers to validate numerical modelling of different designs. Additionally, given the increasing interest in the digital twin concept and the expansion of online monitoring capabilities, more full-scale engine test rigs should be envisioned. This would ensure realistic working conditions, and data collection that accounts for statistical variability, enabling robust model validation and providing large datasets for data-driven modelling.

* **Design and optimisation.** The design of friction dampers can still be improved by taking into account the interface geometries at various scales, long-term fretting wear effects and other physics couplings. This can help to expand the design space and increase the reliability of the dampers. Novel designs of friction dampers can be explored by controlling the normal load distribution on friction interfaces to further optimise damping performance across different operational conditions. This can be achieved by leveraging advances in electric controlling systems, material development, and new manufacturing techniques. The damping robustness of structures should be improved further by considering the uncertainties associated with initial preloading position, contact parameter characterisation, and surface geometries. Robust and statistically relevant strategies must be developed to characterise and propagate uncertainties into the nonlinear dynamic response of structures with multiple contact
additional design technologies. As an illustration of piezoelectric coupling, Wu et al. [321] introduced the concept of the Nonlinear Modal Electromechanical Coupling Factor (NMEMCF). The paper outlines a numerical strategy for incorporating piezoelectric coupling into nonlinear dynamic computations using the HBM. This approach was validated through testing on a beam structure with a piezo patch, demonstrating its effectiveness in achieving high damping levels. Properly considering piezoelectric coupling is the initial stage of the numerical design of piezoelectric damping, as discussed in Section 7.1.1. In [322], thermal coupling is considered and a numerical strategy to integrate it into the simulations is presented. In the proposed modelling, the friction coefficient depends on the temperature, an unknown variable within the problem, alongside the displacements, and the former is different for each contact point. In the context of aerodynamic coupling, in [323], a fully-coupled Fluid-Structure Interaction solver able to deal with nonlinear effects of both structural and fluid domains is presented. It is based on the HBM and a serial coupling strategy where the structural and fluid problems are solved alternatively.

### 7.3. New experimental techniques

In earlier studies, the investigation of contact interfaces was carried out independently from the study of friction-damped turbomachinery components. However, recent advancements have led to the integration of these two approaches. Consequently, two types of experimental evidence, relating to the same test case, such as dynamic response and contact behaviour (e.g., hysteresis or relative contact kinematics), are now available. This integration enables a more comprehensive validation process and facilitates pinpointing deficiencies in the model with greater precision.

Another recent trend involves heightened focus on the actual contact area, which may deviate from the nominal one derived solely from the geometry of the contact interface. Initially, this was accomplished by scanning the contact surfaces and modelling the particular case under examination [236,289]. Subsequently, common wear patterns were analysed [243] to inform numerical models, enabling the generation of not just one specific response but rather a statistical distribution of responses. Recent studies used ultrasound [207] and total internal reflection [208] methods to actively monitor both the true contact area and the actual contact conditions in real time. In addition, in [257] the periodic recording of the effect of wear on the dynamic response evolution has been observed, while simultaneously monitoring the evolution of the contact interface and the dynamic response.

To improve the efficiency of experimental testing, nonlinear modal testing has been actively explored to capture only the amplitude-dependent resonance response of the system instead of the whole FRF at different force amplitudes. Phase lock control methods based on the extended periodic motion concept were developed to obtain the amplitude-dependent modal properties and were used to validate a turbine blade component test with frictional contacts [324]. Recently, velocity feedback-based nonlinear modal testing was proposed to remove the requirement of a controller [325]. In addition, control-based-continuation [326,327] and response-controlled testing methods [328, 329] are promising alternatives which can also be applied to predict autonomous dynamical responses such as limit cycle oscillations due to friction interfaces. Recently, an experimental test rig for friction saturated limit cycle oscillations was developed for the validation of the numerical prediction methods of turbine blades where an adjustable velocity feedback loop was instrumented [330].

## 8. Conclusions

This review offers an in-depth look at recent progress in friction damping for turbomachinery. The discussion encompasses numerical modelling, simulation, experimental testing, design methodologies, and optimisation studies, providing insights into cutting-edge techniques

and key discoveries. By identifying gaps and suggesting future research directions, this review serves as a valuable resource. Potential research avenues for each topic are given next.

* **Modelling and simulation.** Numerical contact and friction models require enhancement to incorporate various modelling scales and additional physics affecting contact dynamics, which necessitates maintaining computational efficiency and improving current simulation techniques. An ongoing challenge lies in calibrating contact model parameters, often based on limited experimental data and valid only under specific operating conditions. Future endeavours should strive for a systematic methodology for parameter calibration. Furthermore, numerical resolution methods could aim for greater realism with fewer model reduction approximations and more intricate finite element models. Addressing complex phenomena such as mistuning, multi-physics effects, and uncertainties promises more accurate predictions. This progress can be facilitated by computational solution improvements that rely on parallel computing, such as the use of a parallel HBM [331] or parallel model refinement [332]. Both methods offer significant potential for computational speedups and could provide even greater benefits when applied to more realistic models. Another strategy involves data-driven modelling techniques, which emphasise the need for extensive experimental data while avoiding reliance on traditional modelling efforts to account for uncertain multi-physics behaviour. Hybrid analytical and data-driven modelling techniques represent a promising research area. These approaches require further detailed investigation to fully leverage the potential benefits of recent rapid advancements in artificial intelligence algorithms and computing power.

* **Experimental investigation.** Some of the possible topics of future experimental investigation include a thorough campaign on the effects of wear and surface finishing on contact conditions and dynamic response, the contact characterisation of new coatings and materials, including additive manufactured ones and measurement techniques targeting real time contact condition monitoring. Novel experimental techniques to monitor local contact conditions, and improve the efficiency of the testing are needed to investigate the effects of the variation of multi-scale surface geometries, and fretting wear. More experimental test rigs are needed for friction ring dampers to validate numerical modelling of different designs. Additionally, given the increasing interest in the digital twin concept and the expansion of online monitoring capabilities, more full-scale engine test rigs should be envisioned. This would ensure realistic working conditions, and data collection that accounts for statistical variability, enabling robust model validation and providing large datasets for data-driven modelling.

* **Design and optimisation.** The design of friction dampers can still be improved by taking into account the interface geometries at various scales, long-term fretting wear effects and other physics couplings. This can help to expand the design space and increase the reliability of the dampers. Novel designs of friction dampers can be explored by controlling the normal load distribution on friction interfaces to further optimise damping performance across different operational conditions. This can be achieved by leveraging advances in electric controlling systems, material development, and new manufacturing techniques. The damping robustness of structures should be improved further by considering the uncertainties associated with initial preloading position, contact parameter characterisation, and surface geometries. Robust and statistically relevant strategies must be developed to characterise and propagate uncertainties into the nonlinear dynamic response of structures with multiple contact
interfaces. The future research direction for designing and optimising friction damping should consider integrating novel manufacturing capabilities and mechanically or tribologically tailorable materials. These manufacturing and material technologies will expand the design possibilities and provide more practical options across different scales. Additionally, this will necessitate the development of more efficient and innovative robust topological optimisation methods, which will involve combining the rapid advancements in artificial intelligence technologies.

### CRediT authorship contribution statement

**Jie Yuan:** Writing – review & editing, Writing – original draft, Investigation, Conceptualization. **Chiara Gastaldi:** Writing – review & editing, Writing – original draft, Investigation, Conceptualization. **Enora Denimal Goy:** Writing – review & editing, Writing – original draft, Investigation, Conceptualization. **Benjamin Chouvion:** Writing – review & editing, Writing – original draft, Investigation, Conceptualization.

### Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

### Data availability

No data was used for the research described in the article.

### Acknowledgements

J.Y. and E.D.G. acknowledge the support of Royal Society of Edinburgh Saltire Facilitation Workshop Award (No. 1865). J.Y. also acknowledges the Research Grant from the Royal Academy of Engineering/Leverhulme Trust Research Fellowship (LTRF2223-19-150).

### References

[1] D. Ewins, Control of vibration and resonance in aero engines and rotating machinery–An overview, Int. J. Press. Vessels Pip. 87 (9) (2010) 504–510.

[2] E. Seinturier, Forced response computation for bladed disks industrial practices and advanced methods, in: Lecture Series-Von Karman Institute for Fluid Dynamics, vol. 2, 2007, p. 5.

[3] M.P. Castanier, C. Pierre, Modeling and analysis of mistuned bladed disk vibration: Current status and emerging directions, J. Propuls. Power 22 (2) (2006) 384–396.

[4] J. Yuan, F. Scarpa, G. Allegri, B. Titurus, S. Patsias, R. Rajasekaran, Efficient computational techniques for mistuning analysis of bladed discs: A review, Mech. Syst. Signal Process. 87 (2017) 71–90.

[5] B. Lee, Self-sustained shock oscillations on airfoils at transonic speeds, Prog. Aerosp. Sci. 37 (2) (2001) 147–196.

[6] F. Zhao, J. Nipkau, M. Vahdati, Influence of acoustic reflections on flutter stability of an embedded blade row, Proc. Inst. Mech. Eng. A 230 (1) (2016) 29–43.

[7] F. Zhao, N. Smith, M. Vahdati, A simple model for identifying the flutter bite of fan blades, J. Turbomach. 139 (7) (2017) 071003.

[8] M. Vahdati, G. Simpson, M. Imregun, Mechanisms for wide-chord fan blade flutter, J. Turbomach. 133 (4) (2011) 041029.

[9] A.V. Srinivasan, Flutter and resonant vibration characteristics of engine blades, J. Eng. Gas Turb. Power 119 (4) (1997) 742–775.

[10] L. Fathyunes, M.A. Mohtadi-Bonab, A review on the corrosion and fatigue failure of gas turbines, Metals 13 (4) (2023) 701.

[11] N. Mukhopadhyay, S. Ghosh Chowdhury, G. Das, I. Chattoraj, S. Das, D. Bhattacharya, An investigation of the failure of low pressure steam turbine blades, in: Failure Analysis Case Studies II, Elsevier, 2001, pp. 211–223.

[12] L.E. Goodman, J.H. Klumpp, Analysis of slip damping with reference to turbine-blade vibration, J. Appl. Mech. 23 (3) (1956) 421–429.

[13] K. Cross, W. Lull, R. Newman, a.r. Cavanagh, Potential of graded coatings in vibration damping, J. Aircr. 10 (11) (2004) 689–691.

[14] S. Patsias, C. Saxton, M. Shipton, Hard damping coatings: An experimental procedure for extraction of damping characteristics and modulus of elasticity, Mater. Sci. Eng. A 370 (1) (2004) 412–416.

[15] O.T. Easterday, An Experimental Characterization of Damping Properties of Thermal Barrier Coatings at Elevated Temperatures (Ph.D. thesis), Air Force Institute of Technology, 2011.

[16] Y. Chen, J. Zhai, Q. Han, Vibration and damping analysis of the bladed disk with damping hard coating on blades, Aerosp. Sci. Technol. 58 (2016) 248–257.

[17] L. Rouleau, O. De Smet, J.-F. Deü, Vibration prediction of rotating composite fan blades comprising viscoelastic damping treatments, J. Sound Vib. (2022) 117135.

[18] B. Manayil Santhosh, M. Tufekci, L. Salles, F. Scarpa, J. Yuan, Flutter mitigation of turbofan blades using viscoelastic patches, in: Proceedings of ISMA 2022-International Conference on Noise and Vibration Engineering and USD 2022-International Conference on Uncertainty in Structural Dynamics, 2022.

[19] J. Yuan, G. Allegri, F. Scarpa, Buffeting mitigation using carbon nanotube composites: A feasibility study, Proc. Inst. Mech. Eng. G 227 (9) (2013) 1425–1440.

[20] S. Bograd, P. Reuss, A. Schmidt, L. Gaul, M. Mayer, Modeling the dynamics of mechanical joints, Mech. Syst. Signal Process. 25 (8) (2011) 2801–2826.

[21] M. Krack, L. Salles, F. Thouverez, Vibration prediction of bladed disks coupled by friction joints, Arch. Comput. Methods Eng. 24 (2016) 589–636.

[22] R.P. Kroon, Turbine-blade vibration due to partial admission, J. Appl. Mech. 7 (4) (1940) A161–A165.

[23] J.F. Shannon, Vibration Problems in Gas Turbines, Centrifugal and Axial Flow Compressors, Reports and memoranda, H.M. Stationery Office, 1948.

[24] J.R. Schnittger, The stress problem of vibrating compressor blades, J. Appl. Mech. 22 (1) (1955) 57–64.

[25] X. Guo, J. Zeng, H. Ma, C. Zhao, L. Qu, B. Wen, Dynamic characteristics of a shrouded blade with impact and friction, Front. Mech. Eng. 15 (2) (2020) 209–226.

[26] E. Ferhatoglu, S. Zucca, D. Botto, J. Auciello, L. Arcangeli, Nonlinear vibration analysis of turbine bladed disks with Midspan dampers, J. Eng. Gas Turb. Power 144 (4) (2022).

[27] A. Balakirev, B. Bolotov, A.K. Golovkin, M.S. Nikhamkin, N. Sazhenkov, L. Voronov, I.B. Konev, Experimental evaluation of the efficiency of gas turbine engine parts damping with dry friction dampers using laser vibrometer, in: Proc. of the 29th Congress of the International Council of the Aeronautical Sciences. ICAS 2014, 2014.

[28] E.K. Armstrong, Recent blade vibration techniques, J. Eng. Power 89 (3) (1967) 437–444.

[29] J. Man, X. Bian, W. Zeng, W. Yan, D. Qiao, Experimental investigation of the dynamic response of a flat blade with dual dry friction dampers, Energies 16 (21) (2023) 7401.

[30] J. Man, B. Xue, X. Bian, W. Yan, D. Qiao, W. Zeng, Experimental and numerical investigations on the dynamic response of blades with dual friction dampers, Aerospace 10 (12) (2023) 977.

[31] M.A. Niemotka, J.C. Ziegert, Optimal design of split ring dampers for gas turbine engines, J. Eng. Gas Turb. Power 117 (3) (1995) 569–575.

[32] S. Baek, B. Epureanu, Reduced-order modeling of bladed disks with friction ring dampers, J. Vib. Acoust. 139 (6) (2017) 061011.

[33] J.H. Griffin, Friction damping of resonant stresses in gas turbine engine airfoils, J. Eng. Power 102 (2) (1980) 329–333.

[34] K.Y. Sanliturk, D.J. Ewins, R. Elliott, J.S. Green, Friction damper optimization: Simulation of rainbow tests, J. Eng. Gas Turb. Power 123 (4) (1999) 930–939.

[35] D. Laxalde, F. Thouverez, J.-J. Sinou, J.-P. Lombard, Qualitative analysis of forced response of blisks with friction ring dampers, Eur. J. Mech. A Solids 26 (4) (2007) 676–687.

[36] M. Krack, S. Tatzko, L. Panning-von Scheidt, J. Wallaschek, Reliability optimization of friction-damped systems using nonlinear modes, J. Sound Vib. 333 (13) (2014) 2699–2712.

[37] C. Joannin, B. Chouvion, F. Thouverez, J.-P. Ousty, M. Mbaye, A nonlinear component mode synthesis method for the computation of steady-state vibration in non-conservative systems, Mech. Syst. Signal Process. 83 (2017) 75–92.

[38] C. Gastaldi, M.M. Gola, Platform-centered reduction: A process capturing the essentials for blade-damper coupled optimization, J. Eng. Gas Turb. Power 143 (8) (2021).

[39] M.M. Gola, A general geometrical theory of turbine blade underplatform asymmetric dampers, Mech. Syst. Signal Process. 191 (2023) 110167.

[40] C. Gastaldi, M.M. Gola, Convergence-free mapping of non-linear damper-blade performance, Mech. Syst. Signal Process. 208 (2024) 111062.

[41] S. Quaegbeur, B. Chouvion, F. Thouverez, Nonlinear dynamic analysis of three-dimensional bladed-disks with frictional contact interfaces based on cyclic reduction strategies, Int. J. Solids Struct. 236–237 (2022) 111277.

[42] K. Johnson, Contact Mechanics, Cambridge University Press, UK, 1985.

[43] P. Wriggers, T.A. Laursen, Computational Contact Mechanics, vol. 2, Springer, 2006.

[44] E. M'Ewen, XLI, stresses in elastic cylinders in contact along a generatrix (including the effect of tangential friction), Lond. Edinb. Dublin Philos. Mag. J. Sci. 40 (303) (1949) 454–459.

[45] G.M. Hamilton, L.E. Goodman, The stress field created by a circular sliding contact, J. Appl. Mech. 33 (2) (1966) 371–376.
interfaces. The future research direction for designing and optimising friction damping should consider integrating novel manufacturing capabilities and mechanically or tribologically tailorable materials. These manufacturing and material technologies will expand the design possibilities and provide more practical options across different scales. Additionally, this will necessitate the development of more efficient and innovative robust topological optimisation methods, which will involve combining the rapid advancements in artificial intelligence technologies.

### CRediT authorship contribution statement

**Jie Yuan:** Writing – review & editing, Writing – original draft, Investigation, Conceptualization. **Chiara Gastaldi:** Writing – review & editing, Writing – original draft, Investigation, Conceptualization. **Enora Denimal Goy:** Writing – review & editing, Writing – original draft, Investigation, Conceptualization. **Benjamin Chouvion:** Writing – review & editing, Writing – original draft, Investigation, Conceptualization.

### Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

### Data availability

No data was used for the research described in the article.

### Acknowledgements

J.Y. and E.D.G. acknowledge the support of Royal Society of Edinburgh Saltire Facilitation Workshop Award (No. 1865). J.Y. also acknowledges the Research Grant from the Royal Academy of Engineering/Leverhulme Trust Research Fellowship (LTRF2223-19-150).

### References

[1] D. Ewins, Control of vibration and resonance in aero engines and rotating machinery–An overview, Int. J. Press. Vessels Pip. 87 (9) (2010) 504–510.

[2] E. Seinturier, Forced response computation for bladed disks industrial practices and advanced methods, in: Lecture Series-Von Karman Institute for Fluid Dynamics, vol. 2, 2007, p. 5.

[3] M.P. Castanier, C. Pierre, Modeling and analysis of mistuned bladed disk vibration: Current status and emerging directions, J. Propuls. Power 22 (2) (2006) 384–396.

[4] J. Yuan, F. Scarpa, G. Allegri, B. Titurus, S. Patsias, R. Rajasekaran, Efficient computational techniques for mistuning analysis of bladed discs: A review, Mech. Syst. Signal Process. 87 (2017) 71–90.

[5] B. Lee, Self-sustained shock oscillations on airfoils at transonic speeds, Prog. Aerosp. Sci. 37 (2) (2001) 147–196.

[6] F. Zhao, J. Nipkau, M. Vahdati, Influence of acoustic reflections on flutter stability of an embedded blade row, Proc. Inst. Mech. Eng. A 230 (1) (2016) 29–43.

[7] F. Zhao, N. Smith, M. Vahdati, A simple model for identifying the flutter bite of fan blades, J. Turbomach. 139 (7) (2017) 071003.

[8] M. Vahdati, G. Simpson, M. Imregun, Mechanisms for wide-chord fan blade flutter, J. Turbomach. 133 (4) (2011) 041029.

[9] A.V. Srinivasan, Flutter and resonant vibration characteristics of engine blades, J. Eng. Gas Turb. Power 119 (4) (1997) 742–775.

[10] L. Fathyunes, M.A. Mohtadi-Bonab, A review on the corrosion and fatigue failure of gas turbines, Metals 13 (4) (2023) 701.

[11] N. Mukhopadhyay, S. Ghosh Chowdhury, G. Das, I. Chattoraj, S. Das, D. Bhattacharya, An investigation of the failure of low pressure steam turbine blades, in: Failure Analysis Case Studies II, Elsevier, 2001, pp. 211–223.

[12] L.E. Goodman, J.H. Klumpp, Analysis of slip damping with reference to turbine-blade vibration, J. Appl. Mech. 23 (3) (1956) 421–429.

[13] K. Cross, W. Lull, R. Newman, a.r. Cavanagh, Potential of graded coatings in vibration damping, J. Aircr. 10 (11) (2004) 689–691.

[14] S. Patsias, C. Saxton, M. Shipton, Hard damping coatings: An experimental procedure for extraction of damping characteristics and modulus of elasticity, Mater. Sci. Eng. A 370 (1) (2004) 412–416.

[15] O.T. Easterday, An Experimental Characterization of Damping Properties of Thermal Barrier Coatings at Elevated Temperatures (Ph.D. thesis), Air Force Institute of Technology, 2011.

[16] Y. Chen, J. Zhai, Q. Han, Vibration and damping analysis of the bladed disk with damping hard coating on blades, Aerosp. Sci. Technol. 58 (2016) 248–257.

[17] L. Rouleau, O. De Smet, J.-F. Deü, Vibration prediction of rotating composite fan blades comprising viscoelastic damping treatments, J. Sound Vib. (2022) 117135.

[18] B. Manayil Santhosh, M. Tufekci, L. Salles, F. Scarpa, J. Yuan, Flutter mitigation of turbofan blades using viscoelastic patches, in: Proceedings of ISMA 2022-International Conference on Noise and Vibration Engineering and USD 2022-International Conference on Uncertainty in Structural Dynamics, 2022.

[19] J. Yuan, G. Allegri, F. Scarpa, Buffeting mitigation using carbon nanotube composites: A feasibility study, Proc. Inst. Mech. Eng. G 227 (9) (2013) 1425–1440.

[20] S. Bograd, P. Reuss, A. Schmidt, L. Gaul, M. Mayer, Modeling the dynamics of mechanical joints, Mech. Syst. Signal Process. 25 (8) (2011) 2801–2826.

[21] M. Krack, L. Salles, F. Thouverez, Vibration prediction of bladed disks coupled by friction joints, Arch. Comput. Methods Eng. 24 (2016) 589–636.

[22] R.P. Kroon, Turbine-blade vibration due to partial admission, J. Appl. Mech. 7 (4) (1940) A161–A165.

[23] J.F. Shannon, Vibration Problems in Gas Turbines, Centrifugal and Axial Flow Compressors, Reports and memoranda, H.M. Stationery Office, 1948.

[24] J.R. Schnittger, The stress problem of vibrating compressor blades, J. Appl. Mech. 22 (1) (1955) 57–64.

[25] X. Guo, J. Zeng, H. Ma, C. Zhao, L. Qu, B. Wen, Dynamic characteristics of a shrouded blade with impact and friction, Front. Mech. Eng. 15 (2) (2020) 209–226.

[26] E. Ferhatoglu, S. Zucca, D. Botto, J. Auciello, L. Arcangeli, Nonlinear vibration analysis of turbine bladed disks with Midspan dampers, J. Eng. Gas Turb. Power 144 (4) (2022).

[27] A. Balakirev, B. Bolotov, A.K. Golovkin, M.S. Nikhamkin, N. Sazhenkov, L. Voronov, I.B. Konev, Experimental evaluation of the efficiency of gas turbine engine parts damping with dry friction dampers using laser vibrometer, in: Proc. of the 29th Congress of the International Council of the Aeronautical Sciences. ICAS 2014, 2014.

[28] E.K. Armstrong, Recent blade vibration techniques, J. Eng. Power 89 (3) (1967) 437–444.

[29] J. Man, X. Bian, W. Zeng, W. Yan, D. Qiao, Experimental investigation of the dynamic response of a flat blade with dual dry friction dampers, Energies 16 (21) (2023) 7401.

[30] J. Man, B. Xue, X. Bian, W. Yan, D. Qiao, W. Zeng, Experimental and numerical investigations on the dynamic response of blades with dual friction dampers, Aerospace 10 (12) (2023) 977.

[31] M.A. Niemotka, J.C. Ziegert, Optimal design of split ring dampers for gas turbine engines, J. Eng. Gas Turb. Power 117 (3) (1995) 569–575.

[32] S. Baek, B. Epureanu, Reduced-order modeling of bladed disks with friction ring dampers, J. Vib. Acoust. 139 (6) (2017) 061011.

[33] J.H. Griffin, Friction damping of resonant stresses in gas turbine engine airfoils, J. Eng. Power 102 (2) (1980) 329–333.

[34] K.Y. Sanliturk, D.J. Ewins, R. Elliott, J.S. Green, Friction damper optimization: Simulation of rainbow tests, J. Eng. Gas Turb. Power 123 (4) (1999) 930–939.

[35] D. Laxalde, F. Thouverez, J.-J. Sinou, J.-P. Lombard, Qualitative analysis of forced response of blisks with friction ring dampers, Eur. J. Mech. A Solids 26 (4) (2007) 676–687.

[36] M. Krack, S. Tatzko, L. Panning-von Scheidt, J. Wallaschek, Reliability optimization of friction-damped systems using nonlinear modes, J. Sound Vib. 333 (13) (2014) 2699–2712.

[37] C. Joannin, B. Chouvion, F. Thouverez, J.-P. Ousty, M. Mbaye, A nonlinear component mode synthesis method for the computation of steady-state vibration in non-conservative systems, Mech. Syst. Signal Process. 83 (2017) 75–92.

[38] C. Gastaldi, M.M. Gola, Platform-centered reduction: A process capturing the essentials for blade-damper coupled optimization, J. Eng. Gas Turb. Power 143 (8) (2021).

[39] M.M. Gola, A general geometrical theory of turbine blade underplatform asymmetric dampers, Mech. Syst. Signal Process. 191 (2023) 110167.

[40] C. Gastaldi, M.M. Gola, Convergence-free mapping of non-linear damper-blade performance, Mech. Syst. Signal Process. 208 (2024) 111062.

[41] S. Quaegbeur, B. Chouvion, F. Thouverez, Nonlinear dynamic analysis of three-dimensional bladed-disks with frictional contact interfaces based on cyclic reduction strategies, Int. J. Solids Struct. 236–237 (2022) 111277.

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