# Friction damping for turbomachinery: A comprehensive review of modelling, design strategies, and testing capabilities

Jie Yuan, Chiara Gastaldi, Enora Denimal, Benjamin Chouvion

► To cite this version:

Jie Yuan, Chiara Gastaldi, Enora Denimal, Benjamin Chouvion. Friction damping for turbomachinery: A comprehensive review of modelling, design strategies, and testing capabilities. Progress in Aerospace Sciences, 2024, 147, pp.101018. 10.1016/j.paerosci.2024.101018. hal-04617385

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Submitted on 19 Jun 2024

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<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Friction damping for turbomachinery: A comprehensive review of modelling, design strategies, and testing capabilities

Jie Yuan, Chiara Gastaldi, Enora Denimal, Benjamin Chouvion

► To cite this version:

Jie Yuan, Chiara Gastaldi, Enora Denimal, Benjamin Chouvion. Friction damping for turbomachinery: A comprehensive review of modelling, design strategies, and testing capabilities. Progress in Aerospace Sciences, 2024, 147, pp.101018. 10.1016/j.paerosci.2024.101018. hal-04617385

HAL Id: hal-04617385

https://inria.hal.science/hal-04617385v1

Submitted on 19 Jun 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
<figure><img src="image_2.png" /></figure>

# Friction damping for turbomachinery: A comprehensive review of modelling, design strategies, and testing capabilities

Jie Yuan<sup>a,*</sup>, Chiara Gastaldi<sup>b</sup>, Enora Denimal Goy<sup>c</sup>, Benjamin Chouvion<sup>d</sup>

<figure><img src="image_3.png" /></figure>

## ARTICLE INFO

**Keywords:**

Friction dampers

Aero-engine

Nonlinear vibration

Vibration control

Contact modelling

Reduced order modelling

Structural design

Uncertainty quantification

Experimental characterisation

Friction bench design

## ABSTRACT

This paper presents a comprehensive review of recent advancements in modelling approaches, design strategies, and testing techniques applied to friction damping in turbomachinery. It critically evaluates experimental testing, design processes, and optimisation studies, along with the latest developments in numerical modelling techniques. The review begins with an overview of vibration mitigation methods and the historical development of friction dampers for bladed disk systems. Subsequent sections explore research efforts aimed at enhancing numerical and simulation modelling capabilities, encompassing contact friction models, reduced-order modelling methods, and numerical solvers suitable for real-world applications and industrial high-fidelity models. The paper also delves into available testing rigs for experimental validation and characterisation of various friction damper types, as well as the literature on uncertainty quantification in friction damping. It concludes by highlighting recent trends in novel concepts, modelling techniques, and testing technologies shaping the design of next-generation friction dampers.

## 1. Introduction

### 1.1. Vibration problems in turbomachinery

Bladed disks in turbomachinery represent about 30% of the overall weight and are critical components for the structural and aerodynamic efficiency of the whole system. They work under severe operational environments and loadings making them undergo extreme static and dynamic stresses. The former is predominantly attributed to thermal loads, static fluid pressures, and rotation-induced centrifugal loads, whereas the latter arises from mechanical vibrations caused by two fundamental mechanisms:

* Synchronous vibrations: they originate from blade rotation within a pressure field that is non-uniform circumferentially, induced by interactions between blade rows and non-uniform inflow conditions (non-uniformity in the intake, obstruction to the flow, etc.). From the rotating bladed disk point of view, the pressure field takes the shape of a wave moving at the rotor speed. Consequently, the structure experiences dynamic loading at frequencies that are multiples of the rotational frequency, leading to synchronous excitation. When the excitation frequency of the

pressure field coincides with a natural frequency of the structure, leading to resonance, the forced vibration response can become significantly amplified. Further explanation about synchronous vibrations can be found in [1,2]. The synchronous vibration becomes more complex when mistuning due to manufacturing tolerance and defects is present. It will lead to mode localisation phenomena and to the modal energy being confined to a certain set of blades. The vibration can be triggered by multiple harmonics of the rotating frequency possibly, resulting in much larger vibration amplitudes compared to the tuned counterpart. Readers are invited to refer to [3,4] for a more detailed review of this topic. In addition, out of balance of the rotor is also a common source for dynamic loading that alternates at a frequency equal to the speed of the rotation [1].

* Non-synchronous vibrations: they appear at frequencies that are not proportional to the rotational speed, but that still correspond to one of the structure's modes of vibration. In this context, non-synchronous vibrations group unsteady flow-driven vibration (vortex shedding, buffering, rotating stall, etc.) [5], flutter [6] and acoustic resonance [7,8].

<footnote><sup>a</sup> Department of Aeronautics and Astronautics Engineering, University of Southampton, Southampton, SO17 1BJ, UK</footnote>

<footnote><sup>b</sup> Department of Mechanical & Aerospace Engineering, Politecnico Di Torino, Corso Duca degli Abruzzi 24, Turin 10129, Italy</footnote>

<footnote><sup>c</sup> Inria, CMAP, Ecole Polytechnique, IPP, Palaiseau, France</footnote>

<footnote><sup>d</sup> Centre de Recherche de l'Ecole de l'Air (CREA), Ecole de l'air et de l'espace, 13661 Salon-de-Provence, France</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail addresses: j.yuan@soton.ac.uk (J. Yuan), chiara.gastaldi@polito.it (C. Gastaldi), enora.denimal-goy@inria.fr (E. Denimal Goy), benjamin.chouvion@ecole-air.fr (B. Chouvion).</footnote>
<figure><img src="image_2.png" /></figure>

# Friction damping for turbomachinery: A comprehensive review of modelling, design strategies, and testing capabilities

Jie Yuan<sup>a,*</sup>, Chiara Gastaldi<sup>b</sup>, Enora Denimal Goy<sup>c</sup>, Benjamin Chouvion<sup>d</sup>

<figure><img src="image_3.png" /></figure>

## ARTICLE INFO

**Keywords:**

Friction dampers

Aero-engine

Nonlinear vibration

Vibration control

Contact modelling

Reduced order modelling

Structural design

Uncertainty quantification

Experimental characterisation

Friction bench design

## ABSTRACT

This paper presents a comprehensive review of recent advancements in modelling approaches, design strategies, and testing techniques applied to friction damping in turbomachinery. It critically evaluates experimental testing, design processes, and optimisation studies, along with the latest developments in numerical modelling techniques. The review begins with an overview of vibration mitigation methods and the historical development of friction dampers for bladed disk systems. Subsequent sections explore research efforts aimed at enhancing numerical and simulation modelling capabilities, encompassing contact friction models, reduced-order modelling methods, and numerical solvers suitable for real-world applications and industrial high-fidelity models. The paper also delves into available testing rigs for experimental validation and characterisation of various friction damper types, as well as the literature on uncertainty quantification in friction damping. It concludes by highlighting recent trends in novel concepts, modelling techniques, and testing technologies shaping the design of next-generation friction dampers.

## 1. Introduction

### 1.1. Vibration problems in turbomachinery

Bladed disks in turbomachinery represent about 30% of the overall weight and are critical components for the structural and aerodynamic efficiency of the whole system. They work under severe operational environments and loadings making them undergo extreme static and dynamic stresses. The former is predominantly attributed to thermal loads, static fluid pressures, and rotation-induced centrifugal loads, whereas the latter arises from mechanical vibrations caused by two fundamental mechanisms:

* Synchronous vibrations: they originate from blade rotation within a pressure field that is non-uniform circumferentially, induced by interactions between blade rows and non-uniform inflow conditions (non-uniformity in the intake, obstruction to the flow, etc.). From the rotating bladed disk point of view, the pressure field takes the shape of a wave moving at the rotor speed. Consequently, the structure experiences dynamic loading at frequencies that are multiples of the rotational frequency, leading to synchronous excitation. When the excitation frequency of the

pressure field coincides with a natural frequency of the structure, leading to resonance, the forced vibration response can become significantly amplified. Further explanation about synchronous vibrations can be found in [1,2]. The synchronous vibration becomes more complex when mistuning due to manufacturing tolerance and defects is present. It will lead to mode localisation phenomena and to the modal energy being confined to a certain set of blades. The vibration can be triggered by multiple harmonics of the rotating frequency possibly, resulting in much larger vibration amplitudes compared to the tuned counterpart. Readers are invited to refer to [3,4] for a more detailed review of this topic. In addition, out of balance of the rotor is also a common source for dynamic loading that alternates at a frequency equal to the speed of the rotation [1].

* Non-synchronous vibrations: they appear at frequencies that are not proportional to the rotational speed, but that still correspond to one of the structure's modes of vibration. In this context, non-synchronous vibrations group unsteady flow-driven vibration (vortex shedding, buffering, rotating stall, etc.) [5], flutter [6] and acoustic resonance [7,8].

<footnote><sup>a</sup> Department of Aeronautics and Astronautics Engineering, University of Southampton, Southampton, SO17 1BJ, UK</footnote>

<footnote><sup>b</sup> Department of Mechanical & Aerospace Engineering, Politecnico Di Torino, Corso Duca degli Abruzzi 24, Turin 10129, Italy</footnote>

<footnote><sup>c</sup> Inria, CMAP, Ecole Polytechnique, IPP, Palaiseau, France</footnote>

<footnote><sup>d</sup> Centre de Recherche de l'Ecole de l'Air (CREA), Ecole de l'air et de l'espace, 13661 Salon-de-Provence, France</footnote>

<footnote>* Corresponding author.
E-mail addresses: j.yuan@soton.ac.uk (J. Yuan), chiara.gastaldi@polito.it (C. Gastaldi), enora.denimal-goy@inria.fr (E. Denimal Goy), benjamin.chouvion@ecole-air.fr (B. Chouvion).</footnote>
The persistence of high dynamic stresses, contingent on the extent of static stress, can result in high cycle fatigue (HCF), which is one of the most common causes of failure in gas [9,10] and steam [11] turbines.

## 1.2. Vibration mitigation methods

The most effective way to reduce vibrations is to prevent any potential excitations of forced resonances, aeroelastic instabilities, and mechanical interactions. During the design stage, adjusting design parameters to change the natural frequencies can help to some extent. However, mitigating the onset of resonance regimes caused by synchronous forced responses is challenging, given the extensive modal density in bladed disks and the broad spectrum of aerodynamic excitation forces. Moreover, the incidence of flutter is on the rise as a result of the growing trend towards slender blades with increased aspect ratios, aimed at curbing fuel consumption and reducing weight. Still, to preserve structural integrity, it is essential to ensure that the vibration response remains within acceptable limits by introducing effective damping in the system that reduces the sharpness of the resonance and lowers the peak values of alternating stresses to safe levels. It is worth mentioning here that active damping solutions are not covered by the paper.

Setting aside aerodynamic damping, which is beneficial in most cases but may become negative (i.e. exciting source) in the case of flutter, as already observed in 1956 by Goodman and Klumpp [12], the remaining sources of damping are connected to hysteresis dissipation within the material and to dry friction between the different components. In terms of material dissipation, coating the surface of the fan blade system with, for example, ceramic materials [13–16], or viscoelastic material [17–19] can effectively enhance the damping performance although it can also impact the aerodynamic efficiency. Due to the frequent inadequacy of various material and mechanical damping alternatives in enduring the challenging conditions within a gas turbine, friction damping emerges as the prevailing damping technique in aero-engines. Despite potential challenges such as fretting wear effects on the friction interface, which may raise maintenance concerns, it remains its prominence due to its resilience under high temperatures and complex loading conditions. Noteworthy is its capacity to provide substantially higher damping levels compared to material damping, with a potential increase ranging from 10 to 100 times, as emphasised in [20].

Friction damping has demonstrated its effectiveness in numerous industrial applications and is now widespread, particularly in turbomachinery where challenging operating conditions limit the feasibility of alternative damping sources. While introducing multiple contact interfaces within the system may be beneficial as it increases the damping sources, it also adds complexity in terms of modelling, simulation, and prediction of the dynamic behaviour. This is primarily due to the nonlinear nature of friction damping, the presence of intricate phenomena such as fretting wear, and the challenge of accurately modelling the friction. It represents numerous open questions that have been at the centre of many research studies over the past decades.

## 1.3. Contribution and organisation of the paper

This paper aims to review the current state of the art of friction damping in terms of modelling, design and testing. Although the applications are focused on turbomachinery, the methodologies and comments remain applicable to other fields where friction damping occurs. The recent review paper [21] provides an extensive review of numerical methods to predict vibration in bladed disks with friction joints. The current paper differs in that it mainly addresses the challenges of practical engineering aspects of friction dampers. Particularly, it includes a comprehensive evaluation of experimental capabilities and outcomes obtained from experimental tests, as well as a detailed section on design and optimisation strategies for friction damping, which have never been covered previously. The different topics addressed in the paper are summarised in Fig. 1 and include:

* A brief historical and technical description of the introduction of friction damping in turbine blade design (Section 2).

* A presentation of prominent numerical modelling methods for modelling contact interfaces of structural components, exploring recent advancements such as the incorporation of multi-scale analysis (Section 3).

* A section on numerical simulation techniques including general ideas on how to solve the nonlinear dynamic equations, reduced-order modelling methods and recent concepts such as nonlinear normal modes (Section 4).

* A comprehensive section on experimental works, including testing capabilities and challenges, model parameter estimation and model validation (Section 5).

* A comprehensive section on design and optimisation methods related to friction damping, aimed to provide new robust and efficient designs (Section 6).

* A section on current trends and future challenges in the study of friction damping (Section 7).

# 2. A brief history of the integration of friction damping in turbine blade design

The importance of friction damping in reducing the incidence of turbine failures was first mentioned in 1940, when Kroon [22] attempted to measure the material damping contribution as a function of stress, temperature and excitation frequency of blades, and stated that: "Besides internal damping in the materials, there is a certain amount of damping caused by microscopic rubbing and impacting along the fits between rotor and blades and in the shroud connections". In 1946, Shannon [23], compared the order of magnitude of friction damping at the blade root with the internal material damping. While the different sources of damping depend on many factors, e.g. design of the blade root, preload, temperature, stress on the blade, etc, the comparison is still striking. The damping connected to friction was estimated to be 5 times greater than the one connected to internal material damping [24]. It should be noted that this estimate is related to a joint, i.e. blade root, which was not designed for the purpose of dissipating energy through friction. This gain can be enhanced when friction damping efficiency is targeted [20]. Yet, the contribution to the overall structural damping was significant. Current technologies exploit this mechanism by introducing friction contact interfaces in bladed disks to increase damping efficiency.

Fig. 2 depicts the progression of diverse friction dampers implemented on turbine blades, utilising images sourced from significant patents. The illustration serves a dual purpose. Firstly, it provides an estimation of the timeline for the introduction of various damping techniques. Even though the filing dates of these patents often do not align with the actual implementation of such devices in turbine designs, they still provide valuable and informative insights into the evolution of these technologies within the industry. Secondly, it offers a visual portrayal, though not exhaustive, of the varying adaptations of each damping mechanism. Complementary, Fig. 3 illustrates different damping solutions in real systems.

Although blade root attachments and shrouds were observed to contribute to structural damping [22], neither of them were purposely designed as friction damping devices. Blade root attachments responded to the technological need to connect the blades to the rotor in an era in which producing a blisk, i.e. an integrated bladed rotor, was not feasible. It also allowed the same blade casting to be re-used on rotors with a different number of blades. Blade roots come into different configurations, the most common being the “dovetail”, the “fir-tree” and the “t-shape”. Some of them are visible in the upper portion of Fig. 2.

Shrouds are integral protuberances found on each blade, interlocking at speed to form a ring that couples the blades and enhances the overall stiffness of the assembly. They were introduced to provide an
The persistence of high dynamic stresses, contingent on the extent of static stress, can result in high cycle fatigue (HCF), which is one of the most common causes of failure in gas [9,10] and steam [11] turbines.

## 1.2. Vibration mitigation methods

The most effective way to reduce vibrations is to prevent any potential excitations of forced resonances, aeroelastic instabilities, and mechanical interactions. During the design stage, adjusting design parameters to change the natural frequencies can help to some extent. However, mitigating the onset of resonance regimes caused by synchronous forced responses is challenging, given the extensive modal density in bladed disks and the broad spectrum of aerodynamic excitation forces. Moreover, the incidence of flutter is on the rise as a result of the growing trend towards slender blades with increased aspect ratios, aimed at curbing fuel consumption and reducing weight. Still, to preserve structural integrity, it is essential to ensure that the vibration response remains within acceptable limits by introducing effective damping in the system that reduces the sharpness of the resonance and lowers the peak values of alternating stresses to safe levels. It is worth mentioning here that active damping solutions are not covered by the paper.

Setting aside aerodynamic damping, which is beneficial in most cases but may become negative (i.e. exciting source) in the case of flutter, as already observed in 1956 by Goodman and Klumpp [12], the remaining sources of damping are connected to hysteresis dissipation within the material and to dry friction between the different components. In terms of material dissipation, coating the surface of the fan blade system with, for example, ceramic materials [13–16], or viscoelastic material [17–19] can effectively enhance the damping performance although it can also impact the aerodynamic efficiency. Due to the frequent inadequacy of various material and mechanical damping alternatives in enduring the challenging conditions within a gas turbine, friction damping emerges as the prevailing damping technique in aero-engines. Despite potential challenges such as fretting wear effects on the friction interface, which may raise maintenance concerns, it remains its prominence due to its resilience under high temperatures and complex loading conditions. Noteworthy is its capacity to provide substantially higher damping levels compared to material damping, with a potential increase ranging from 10 to 100 times, as emphasised in [20].

Friction damping has demonstrated its effectiveness in numerous industrial applications and is now widespread, particularly in turbomachinery where challenging operating conditions limit the feasibility of alternative damping sources. While introducing multiple contact interfaces within the system may be beneficial as it increases the damping sources, it also adds complexity in terms of modelling, simulation, and prediction of the dynamic behaviour. This is primarily due to the nonlinear nature of friction damping, the presence of intricate phenomena such as fretting wear, and the challenge of accurately modelling the friction. It represents numerous open questions that have been at the centre of many research studies over the past decades.

## 1.3. Contribution and organisation of the paper

This paper aims to review the current state of the art of friction damping in terms of modelling, design and testing. Although the applications are focused on turbomachinery, the methodologies and comments remain applicable to other fields where friction damping occurs. The recent review paper [21] provides an extensive review of numerical methods to predict vibration in bladed disks with friction joints. The current paper differs in that it mainly addresses the challenges of practical engineering aspects of friction dampers. Particularly, it includes a comprehensive evaluation of experimental capabilities and outcomes obtained from experimental tests, as well as a detailed section on design and optimisation strategies for friction damping, which have never been covered previously. The different topics addressed in the paper are summarised in Fig. 1 and include:

* A brief historical and technical description of the introduction of friction damping in turbine blade design (Section 2).

* A presentation of prominent numerical modelling methods for modelling contact interfaces of structural components, exploring recent advancements such as the incorporation of multi-scale analysis (Section 3).

* A section on numerical simulation techniques including general ideas on how to solve the nonlinear dynamic equations, reduced-order modelling methods and recent concepts such as nonlinear normal modes (Section 4).

* A comprehensive section on experimental works, including testing capabilities and challenges, model parameter estimation and model validation (Section 5).

* A comprehensive section on design and optimisation methods related to friction damping, aimed to provide new robust and efficient designs (Section 6).

* A section on current trends and future challenges in the study of friction damping (Section 7).

# 2. A brief history of the integration of friction damping in turbine blade design

The importance of friction damping in reducing the incidence of turbine failures was first mentioned in 1940, when Kroon [22] attempted to measure the material damping contribution as a function of stress, temperature and excitation frequency of blades, and stated that: "Besides internal damping in the materials, there is a certain amount of damping caused by microscopic rubbing and impacting along the fits between rotor and blades and in the shroud connections". In 1946, Shannon [23], compared the order of magnitude of friction damping at the blade root with the internal material damping. While the different sources of damping depend on many factors, e.g. design of the blade root, preload, temperature, stress on the blade, etc, the comparison is still striking. The damping connected to friction was estimated to be 5 times greater than the one connected to internal material damping [24]. It should be noted that this estimate is related to a joint, i.e. blade root, which was not designed for the purpose of dissipating energy through friction. This gain can be enhanced when friction damping efficiency is targeted [20]. Yet, the contribution to the overall structural damping was significant. Current technologies exploit this mechanism by introducing friction contact interfaces in bladed disks to increase damping efficiency.

Fig. 2 depicts the progression of diverse friction dampers implemented on turbine blades, utilising images sourced from significant patents. The illustration serves a dual purpose. Firstly, it provides an estimation of the timeline for the introduction of various damping techniques. Even though the filing dates of these patents often do not align with the actual implementation of such devices in turbine designs, they still provide valuable and informative insights into the evolution of these technologies within the industry. Secondly, it offers a visual portrayal, though not exhaustive, of the varying adaptations of each damping mechanism. Complementary, Fig. 3 illustrates different damping solutions in real systems.

Although blade root attachments and shrouds were observed to contribute to structural damping [22], neither of them were purposely designed as friction damping devices. Blade root attachments responded to the technological need to connect the blades to the rotor in an era in which producing a blisk, i.e. an integrated bladed rotor, was not feasible. It also allowed the same blade casting to be re-used on rotors with a different number of blades. Blade roots come into different configurations, the most common being the “dovetail”, the “fir-tree” and the “t-shape”. Some of them are visible in the upper portion of Fig. 2.

Shrouds are integral protuberances found on each blade, interlocking at speed to form a ring that couples the blades and enhances the overall stiffness of the assembly. They were introduced to provide an
<figure><img src="image_4.png" /><figcaption>Fig. 1. Research topics related to the predictive modelling and design of friction dampers in turbomachinery.</figcaption></figure>

elastic support at the mid-span of blades to prevent flutter. Initially, the same function was achieved by “tied blades”, i.e. adjacent blades that were tied together by rods “whose ends were screwed into threaded bushes set in the blade forms” [28]. This technique was later discontinued due to the stress concentrations at the holes in the airfoils, and was replaced by snubbers or shrouds. In current terminology, snubbers usually refer to protrusions, integral to the blade airfoil, found at approximately two-thirds of the airfoil span, while “tip-shrouds” are found at the tip of the blade. The concept of “tied blades” also evolved into that of lacing wires as friction dampers, i.e. loose flexible rods which come into contact with the inner surface of the lacing holes within the turbine blade airfoil. In time, as shown in Fig. 2, other shapes of dampers placed mid-span on the blade airfoil were introduced. Such mid-span dampers may pass through holes, or may rest against protrusions integral to the airfoil. In both cases, the contact is ensured by the centrifugal force. The protrusions of adjacent blades may or may not come into contact. In the former case, a combination of snubber and mid-span damper is achieved. Both lacing wires and mid-span dampers are commonly used on last-stage blades of steam turbines. These blades are typically thin, long and flexible, making them particularly susceptible to substantial centrifugal and aerodynamic loads.

An external damper may also be placed in other positions along the blade. It may be positioned at the airfoil's tip, where it rests against blade covers or shrouds. Alternatively, one of the most common choices, especially in gas turbines, is to place it at the airfoil's root, against two adjacent inner platforms. This last configuration, which is more common with shroud-less shorter blades, has the advantage of not interfering with the aerodynamic flow. These dampers are usually referred to as underplatform dampers. The use of dual friction dampers, as proposed recently [29,30], involves coupling two separate dampers to the blade at different radial positions, offering enhanced damping and stiffening. However, careful consideration is required in determining the radial position to avoid disrupting the aerodynamic flow, akin to mid-span dampers. One proposed solution [30] involves placing one damper near the blade root, effectively replacing one lobe

of the disk sector. While this solution avoids flow interference, it may substantially increase stresses at the blade root-disk interface.

In the 1980s, blisks, an acronym for “Bladed Integrated Disks”, were introduced as a response to the need for increased efficiency and reduced weight in gas turbine engines for aircraft and power generation. While they offer significant advantages in terms of aerodynamic performance and overall efficiency, they also present some challenges. Since the blisk is manufactured as a single piece, no inherent contact surfaces are available to dissipate energy via friction. For this reason, starting in the 1990s, split ring dampers were added to the annular air seal which is commonly used in gas turbine engines to prevent gas flow from one section of the engine to another [31] or, more generally, under the rim [32].

In all the above-mentioned cases, the rotation caused by the centrifugal load is responsible for the contact preload, while the relative motion necessary for friction dissipation is caused by the blades' vibration itself.

## 3. Numerical modelling of friction interfaces

### 3.1. An overview of modelling approaches

Predicting the damping resulting from friction in bladed disks requires modelling different components and their coupling through the contact friction interface, which is associated with many complex phenomena at different scales. When modelling mechanical components such as the bladed disk or dampers for dynamic studies, the emphasis is on deriving the structural mass, stiffness and damping matrices. These can be obtained analytically for lumped-parameter models in early-stage studies [33–35]. More recent analytical and semi-analytical models can be found in [36–40]. These models, characterised by a short simulation time, capture nonlinear dynamic behaviours and/or energy dissipation, and are well-suited for proof of concept. In practical applications, precise modelling of geometries and mechanical properties is essential for an accurate prediction of modal properties. In this
<figure><img src="image_4.png" /><figcaption>Fig. 1. Research topics related to the predictive modelling and design of friction dampers in turbomachinery.</figcaption></figure>

elastic support at the mid-span of blades to prevent flutter. Initially, the same function was achieved by “tied blades”, i.e. adjacent blades that were tied together by rods “whose ends were screwed into threaded bushes set in the blade forms” [28]. This technique was later discontinued due to the stress concentrations at the holes in the airfoils, and was replaced by snubbers or shrouds. In current terminology, snubbers usually refer to protrusions, integral to the blade airfoil, found at approximately two-thirds of the airfoil span, while “tip-shrouds” are found at the tip of the blade. The concept of “tied blades” also evolved into that of lacing wires as friction dampers, i.e. loose flexible rods which come into contact with the inner surface of the lacing holes within the turbine blade airfoil. In time, as shown in Fig. 2, other shapes of dampers placed mid-span on the blade airfoil were introduced. Such mid-span dampers may pass through holes, or may rest against protrusions integral to the airfoil. In both cases, the contact is ensured by the centrifugal force. The protrusions of adjacent blades may or may not come into contact. In the former case, a combination of snubber and mid-span damper is achieved. Both lacing wires and mid-span dampers are commonly used on last-stage blades of steam turbines. These blades are typically thin, long and flexible, making them particularly susceptible to substantial centrifugal and aerodynamic loads.

An external damper may also be placed in other positions along the blade. It may be positioned at the airfoil's tip, where it rests against blade covers or shrouds. Alternatively, one of the most common choices, especially in gas turbines, is to place it at the airfoil's root, against two adjacent inner platforms. This last configuration, which is more common with shroud-less shorter blades, has the advantage of not interfering with the aerodynamic flow. These dampers are usually referred to as underplatform dampers. The use of dual friction dampers, as proposed recently [29,30], involves coupling two separate dampers to the blade at different radial positions, offering enhanced damping and stiffening. However, careful consideration is required in determining the radial position to avoid disrupting the aerodynamic flow, akin to mid-span dampers. One proposed solution [30] involves placing one damper near the blade root, effectively replacing one lobe

of the disk sector. While this solution avoids flow interference, it may substantially increase stresses at the blade root-disk interface.

In the 1980s, blisks, an acronym for “Bladed Integrated Disks”, were introduced as a response to the need for increased efficiency and reduced weight in gas turbine engines for aircraft and power generation. While they offer significant advantages in terms of aerodynamic performance and overall efficiency, they also present some challenges. Since the blisk is manufactured as a single piece, no inherent contact surfaces are available to dissipate energy via friction. For this reason, starting in the 1990s, split ring dampers were added to the annular air seal which is commonly used in gas turbine engines to prevent gas flow from one section of the engine to another [31] or, more generally, under the rim [32].

In all the above-mentioned cases, the rotation caused by the centrifugal load is responsible for the contact preload, while the relative motion necessary for friction dissipation is caused by the blades' vibration itself.

### 3. Numerical modelling of friction interfaces

#### 3.1. An overview of modelling approaches

Predicting the damping resulting from friction in bladed disks requires modelling different components and their coupling through the contact friction interface, which is associated with many complex phenomena at different scales. When modelling mechanical components such as the bladed disk or dampers for dynamic studies, the emphasis is on deriving the structural mass, stiffness and damping matrices. These can be obtained analytically for lumped-parameter models in early-stage studies [33–35]. More recent analytical and semi-analytical models can be found in [36–40]. These models, characterised by a short simulation time, capture nonlinear dynamic behaviours and/or energy dissipation, and are well-suited for proof of concept. In practical applications, precise modelling of geometries and mechanical properties is essential for an accurate prediction of modal properties. In this
<figure><img src="image_5.png" /><figcaption>Fig. 2. Timeline representing the introduction of the different kinds of friction dampers in bladed disks.</figcaption></figure>

context, full Finite Element Models (FEM) are usually employed to compute the structural matrices. With such a modelling approach, high-fidelity models can be produced, but they are computationally heavy on the other hand. When modelling bladed disk structures, preliminary studies usually consider the system to be perfectly cyclically symmetric to reduce computational cost while considering only a single sector. However, adapted strategies must be used to account for mistuning effects, which can have a significant impact on the system dynamics. More details on general reduction techniques and specific approaches dedicated to mistuning are given in Section 4.2.2. As an illustration, a lumped-parameter model of a bladed disk with a ring damper [35] and a high-fidelity FEM of a bladed disk with UPD [41] are displayed respectively in Figs. 4(a), and 4(b). The former has 8 DOF per sector

(2 for the blade, 3 for the disk and 3 for the ring damper) while the latter, with 30840 DOF per sector (14430 for the blade, 14628 for the disk and 1782 for the UPD), highlights the large size of realistic FEM and the need for model reduction, as will be developed in Section 4, to ensure affordable computational time.

Another challenge in modelling systems that include friction dampers lies in the representation of the contact friction interface. In this context, one is interested in modelling the normal loads, i.e. the non-penetration of the solids, and the tangential loads with a friction law, such as the Coulomb law, for example. Both are strongly non-linear leading to numerical difficulties when solving such problems. More advanced models include phenomena taking place at the contact interface, such as fretting wear.
<figure><img src="image_5.png" /><figcaption>Fig. 2. Timeline representing the introduction of the different kinds of friction dampers in bladed disks.</figcaption></figure>

context, full Finite Element Models (FEM) are usually employed to compute the structural matrices. With such a modelling approach, high-fidelity models can be produced, but they are computationally heavy on the other hand. When modelling bladed disk structures, preliminary studies usually consider the system to be perfectly cyclically symmetric to reduce computational cost while considering only a single sector. However, adapted strategies must be used to account for mistuning effects, which can have a significant impact on the system dynamics. More details on general reduction techniques and specific approaches dedicated to mistuning are given in Section 4.2.2. As an illustration, a lumped-parameter model of a bladed disk with a ring damper [35] and a high-fidelity FEM of a bladed disk with UPD [41] are displayed respectively in Figs. 4(a), and 4(b). The former has 8 DOF per sector

(2 for the blade, 3 for the disk and 3 for the ring damper) while the latter, with 30840 DOF per sector (14430 for the blade, 14628 for the disk and 1782 for the UPD), highlights the large size of realistic FEM and the need for model reduction, as will be developed in Section 4, to ensure affordable computational time.

Another challenge in modelling systems that include friction dampers lies in the representation of the contact friction interface. In this context, one is interested in modelling the normal loads, i.e. the non-penetration of the solids, and the tangential loads with a friction law, such as the Coulomb law, for example. Both are strongly non-linear leading to numerical difficulties when solving such problems. More advanced models include phenomena taking place at the contact interface, such as fretting wear.
<figure><img src="image_6.png" /><figcaption>Fig. 3. In black and white, schematic representation of different kinds of damping solutions. Shrouds and roots (Photographs from [25]), Midspan dampers (©2020 Baker Hughes Company - All rights reserved, already published in [26]), Ring dampers [27].</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 4. (a) Lumped parameter model of a friction ring damper [35] and (b) high-fidelity FEM of the sector of a bladed disk with UPD [41].</figcaption></figure>

This section is dedicated to a brief presentation of the main different friction interface models that exist and are used in the numerical modelling of friction damping for turbomachinery. It is worth mentioning here that the numerical modelling of contact interfaces, in general, constitutes a broad research domain, and turbomachinery applications represent only a small section of it. In such applications, the contact is highly loaded and vibrations are of small amplitudes. A commonly employed assumption is to decouple these two problems and analyse the dynamic behaviour around the static solution. For a general

overview of contact modelling, one can refer to [42,43] and to [21] for application to bladed disks.

## 3.2. Analytical contact models

Analytical contact models were initially developed for static problems using the Hertz theory [42], primarily considering simple geometries and without accounting for frictional effects or interactions between surface asperities. For rough surfaces, the statistical approaches
<figure><img src="image_6.png" /><figcaption>Fig. 3. In black and white, schematic representation of different kinds of damping solutions. Shrouds and roots (Photographs from [25]), Midspan dampers (©2020 Baker Hughes Company - All rights reserved, already published in [26]), Ring dampers [27].</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 4. (a) Lumped parameter model of a friction ring damper [35] and (b) high-fidelity FEM of the sector of a bladed disk with UPD [41].</figcaption></figure>

This section is dedicated to a brief presentation of the main different friction interface models that exist and are used in the numerical modelling of friction damping for turbomachinery. It is worth mentioning here that the numerical modelling of contact interfaces, in general, constitutes a broad research domain, and turbomachinery applications represent only a small section of it. In such applications, the contact is highly loaded and vibrations are of small amplitudes. A commonly employed assumption is to decouple these two problems and analyse the dynamic behaviour around the static solution. For a general

overview of contact modelling, one can refer to [42,43] and to [21] for application to bladed disks.

## 3.2. Analytical contact models

Analytical contact models were initially developed for static problems using the Hertz theory [42], primarily considering simple geometries and without accounting for frictional effects or interactions between surface asperities. For rough surfaces, the statistical approaches
based on analytical solutions are not able to track the local contact behaviour. Analytical models were subsequently expanded to account for various specific contact geometries and include frictional effects modelled as a Coulomb law [44,45]. Partial slip was later incorporated in [46,47]. While these methods have the advantage of being exact, they can only be used in very specific cases and are difficult to extend to complex interface geometries and/or more advanced friction laws. A more detailed review of analytical methods for contact friction modelling can be found in [48].

### 3.3. Macro-scale contact models

Finite element-based numerical methods have been developed to address dynamic contact problems across a wide range of structures and contact geometries. These methods gained widespread popularity and efficiency with the advancements and proliferation of computational capacity. Consequently, it became feasible to model the frictional contact interface and simulate its dynamic behaviour [43]. Nevertheless, to accurately capture contact dynamics, a highly refined mesh is essential, particularly in the presence of sharp strain gradients. This significantly increases computational time, rendering the method impractical for real-world applications. Alternatives based on the elastic half-space assumption have been proposed based on the use of Boundary Element Methods [49–51], zero-thickness elements [52] or thin-layer elements [53,54] to mesh only the contact elements. Even though nonlinear relations may be considered to model the friction law, the normal law is usually regularised with a linear-elastic unilateral law [55]. A major drawback of these methods comes from this regularisation, which is often too strong and tends to soften the friction contact effects. For these different reasons, these methods are not adapted for turbomachinery applications, where small vibrations with high contact pressure and change of contact status (impact, stick, slip) are experienced.

The most widespread strategy to model the friction contact interface in turbomachinery applications involves utilising a node-to-node contact formulation to define the interface's contact law between the two structures [33,56–58]. Such approaches discretise the contact surface by pairing a contact point from one solid with a contact point from the other solid. As the two points interact, it is assumed that the displacements remain sufficiently small, and that two points in contact can thus be considered to almost face-to-face during the vibrations. After establishing the pairing, the next step involves defining the contact law between the points to replicate the hysteresis-like behaviour, including both a normal law and a tangential law. These contact laws are usually referred to as contact models, or sometimes as friction elements. They are composed of additional stiffness and damping forces in the contact area. Depending on the case, macro-slip or micro-slip elements can be used [21]. Micro-slip elements give a smooth and progressive transition between the elastic deformation and the sliding contact [59,60]. However, such elements usually require the tuning of many parameters, making them difficult to use in practice. Such elements are mostly used when the normal load is constant over time [21]. Alternatively, the main drawback of using macro-slip elements is that for one contact point, the transition between sliding and sticking conditions is sharp. However, if the contact surface is discretised with many macro-slip elements, or if the hysteresis behaviour is reproduced implicitly with the contact law, then a smooth and progressive transition between the elastic deformation and the sliding contact can be reached for the contact surface [21,61–64].

In terms of contact models, classical options include the LuGre [65], Bouc-Wen [66], Iwan [67], Valanis [68], and Dahl [69] models. A full description and comparison of the different models can be found in [70]. The most commonly used contact element for turbomachinery is based on the Jenkins element, originally developed in [56,71] for a 2D problem. It was extended to 3D in [72] by coupling the movement in the two tangential directions to the contact surface, while considering a constant normal load. This was later expanded [57] to incorporate

variations in the normal load, accounting for possible loss of contact. Recent studies [73,74] focused on applying these advanced full 3D contact models. The latter additionally presents the Jacobian matrix for the contact model, required in many numerical methods for the computation of the dynamic response, as will be explained in Section 4.

In Fig. 5, three hysteresis loops are shown to illustrate the impact of the contact law. The total dissipated energy is represented by the area within the hysteresis loop. The first hysteresis loop corresponds to classic the Coulomb law, where the contact alternates between stick and slip conditions. The second loop represents a Jenkins element, which includes an elastic spring in addition to a Coulomb element. In this case, there is still a sharp transition between the stick and slip states. The third hysteresis loop depicts a typical microslip contact element such as the Valanis model. Here, we can observe a smooth transition between the two states, indicating micro-slip, i.e., partial slip on the contact surface. It is evident that the latter can describe more complex dissipative behaviour at the contact interface, but also requires more parameters to be explained.

The presented approaches are based on models that rely on multiple parameters, necessitating meticulous tuning for accurate predictions, which often requires expensive model updating techniques. The most popular method consists of updating the model parameters until the simulation fits experimental measurements [12,64,75–80], as will be detailed in Section 5. After this updating phase, the predictions can be reliable for the considered configuration [64]. However, this relies on the idea that the FEM with the contact model can accurately represent the real contact conditions, i.e. that there exists a set of model parameters that give a good fit with the experimental data. It is worth mentioning here that such approaches are applicable if the uncertainties, emerging from noise measurement or environmental variations for example, do not influence the system properties. In a context where uncertainties have a substantial impact, as shown in Section 6.2, more advanced updating methods must be employed (sensitivity-based, optimisation, Bayesian approaches, etc.) to consider the statistical aspect of the problem. More details can be found in [81]. Furthermore, conducting the necessary experiments can sometimes be challenging and/or costly. Finally, the updated parameters remain valid only for the considered scenario or structure, as they change quickly with the environment and geometry [82]. This poses a significant constraint during the design stage, particularly when no prior knowledge is available for contact modelling.

### 3.4. Multiscale contact model

More sophisticated modelling techniques have also been proposed to include more advanced phenomena taking place at the contact interface, such as wear [49,83], fretting [50,51,84,85], and the consideration of coating [13–16]. The methods described in the previous section only use a coarse description of the contact interface with a limited number of contact points and do not consider the microscale or mesoscale interface geometries. As examples, surface roughness affects the contact stiffness [43], while the mesoscale geometry and fretting wear significantly influence the distribution of normal pressure [86]. In this context, multiscale approaches have been developed to establish a connection between contact mechanics and dynamics [87,88]. To consider surface with microscale asperities, a compliant model incorporating a statistical distribution of asperities was developed [89]. It can assess the effects of the surface asperities in a statistically averaged sense while being more computationally feasible compared to the FE approach. With such an approach, the surface roughness and its impact on the effective contact surface are modelled and quantified. Thus, relations between the contact load and the real contact pressure or between the contact stiffness and the normal load are obtained. They are typically described by power laws. Such relationships can directly be included as a linear penalty stiffness [90] or a nonlinear penalty model [91] to solve the normal contact problem. Recently, fractal
based on analytical solutions are not able to track the local contact behaviour. Analytical models were subsequently expanded to account for various specific contact geometries and include frictional effects modelled as a Coulomb law [44,45]. Partial slip was later incorporated in [46,47]. While these methods have the advantage of being exact, they can only be used in very specific cases and are difficult to extend to complex interface geometries and/or more advanced friction laws. A more detailed review of analytical methods for contact friction modelling can be found in [48].

### 3.3. Macro-scale contact models

Finite element-based numerical methods have been developed to address dynamic contact problems across a wide range of structures and contact geometries. These methods gained widespread popularity and efficiency with the advancements and proliferation of computational capacity. Consequently, it became feasible to model the frictional contact interface and simulate its dynamic behaviour [43]. Nevertheless, to accurately capture contact dynamics, a highly refined mesh is essential, particularly in the presence of sharp strain gradients. This significantly increases computational time, rendering the method impractical for real-world applications. Alternatives based on the elastic half-space assumption have been proposed based on the use of Boundary Element Methods [49–51], zero-thickness elements [52] or thin-layer elements [53,54] to mesh only the contact elements. Even though nonlinear relations may be considered to model the friction law, the normal law is usually regularised with a linear-elastic unilateral law [55]. A major drawback of these methods comes from this regularisation, which is often too strong and tends to soften the friction contact effects. For these different reasons, these methods are not adapted for turbomachinery applications, where small vibrations with high contact pressure and change of contact status (impact, stick, slip) are experienced.

The most widespread strategy to model the friction contact interface in turbomachinery applications involves utilising a node-to-node contact formulation to define the interface's contact law between the two structures [33,56–58]. Such approaches discretise the contact surface by pairing a contact point from one solid with a contact point from the other solid. As the two points interact, it is assumed that the displacements remain sufficiently small, and that two points in contact can thus be considered to almost face-to-face during the vibrations. After establishing the pairing, the next step involves defining the contact law between the points to replicate the hysteresis-like behaviour, including both a normal law and a tangential law. These contact laws are usually referred to as contact models, or sometimes as friction elements. They are composed of additional stiffness and damping forces in the contact area. Depending on the case, macro-slip or micro-slip elements can be used [21]. Micro-slip elements give a smooth and progressive transition between the elastic deformation and the sliding contact [59,60]. However, such elements usually require the tuning of many parameters, making them difficult to use in practice. Such elements are mostly used when the normal load is constant over time [21]. Alternatively, the main drawback of using macro-slip elements is that for one contact point, the transition between sliding and sticking conditions is sharp. However, if the contact surface is discretised with many macro-slip elements, or if the hysteresis behaviour is reproduced implicitly with the contact law, then a smooth and progressive transition between the elastic deformation and the sliding contact can be reached for the contact surface [21,61–64].

In terms of contact models, classical options include the LuGre [65], Bouc-Wen [66], Iwan [67], Valanis [68], and Dahl [69] models. A full description and comparison of the different models can be found in [70]. The most commonly used contact element for turbomachinery is based on the Jenkins element, originally developed in [56,71] for a 2D problem. It was extended to 3D in [72] by coupling the movement in the two tangential directions to the contact surface, while considering a constant normal load. This was later expanded [57] to incorporate

variations in the normal load, accounting for possible loss of contact. Recent studies [73,74] focused on applying these advanced full 3D contact models. The latter additionally presents the Jacobian matrix for the contact model, required in many numerical methods for the computation of the dynamic response, as will be explained in Section 4.

In Fig. 5, three hysteresis loops are shown to illustrate the impact of the contact law. The total dissipated energy is represented by the area within the hysteresis loop. The first hysteresis loop corresponds to classic the Coulomb law, where the contact alternates between stick and slip conditions. The second loop represents a Jenkins element, which includes an elastic spring in addition to a Coulomb element. In this case, there is still a sharp transition between the stick and slip states. The third hysteresis loop depicts a typical microslip contact element such as the Valanis model. Here, we can observe a smooth transition between the two states, indicating micro-slip, i.e., partial slip on the contact surface. It is evident that the latter can describe more complex dissipative behaviour at the contact interface, but also requires more parameters to be explained.

The presented approaches are based on models that rely on multiple parameters, necessitating meticulous tuning for accurate predictions, which often requires expensive model updating techniques. The most popular method consists of updating the model parameters until the simulation fits experimental measurements [12,64,75–80], as will be detailed in Section 5. After this updating phase, the predictions can be reliable for the considered configuration [64]. However, this relies on the idea that the FEM with the contact model can accurately represent the real contact conditions, i.e. that there exists a set of model parameters that give a good fit with the experimental data. It is worth mentioning here that such approaches are applicable if the uncertainties, emerging from noise measurement or environmental variations for example, do not influence the system properties. In a context where uncertainties have a substantial impact, as shown in Section 6.2, more advanced updating methods must be employed (sensitivity-based, optimisation, Bayesian approaches, etc.) to consider the statistical aspect of the problem. More details can be found in [81]. Furthermore, conducting the necessary experiments can sometimes be challenging and/or costly. Finally, the updated parameters remain valid only for the considered scenario or structure, as they change quickly with the environment and geometry [82]. This poses a significant constraint during the design stage, particularly when no prior knowledge is available for contact modelling.

### 3.4. Multiscale contact model

More sophisticated modelling techniques have also been proposed to include more advanced phenomena taking place at the contact interface, such as wear [49,83], fretting [50,51,84,85], and the consideration of coating [13–16]. The methods described in the previous section only use a coarse description of the contact interface with a limited number of contact points and do not consider the microscale or mesoscale interface geometries. As examples, surface roughness affects the contact stiffness [43], while the mesoscale geometry and fretting wear significantly influence the distribution of normal pressure [86]. In this context, multiscale approaches have been developed to establish a connection between contact mechanics and dynamics [87,88]. To consider surface with microscale asperities, a compliant model incorporating a statistical distribution of asperities was developed [89]. It can assess the effects of the surface asperities in a statistically averaged sense while being more computationally feasible compared to the FE approach. With such an approach, the surface roughness and its impact on the effective contact surface are modelled and quantified. Thus, relations between the contact load and the real contact pressure or between the contact stiffness and the normal load are obtained. They are typically described by power laws. Such relationships can directly be included as a linear penalty stiffness [90] or a nonlinear penalty model [91] to solve the normal contact problem. Recently, fractal
<figure><img src="image_8.png" /><figcaption>Fig. 5. Hysteresis curves for a Coulomb contact model (a), a Jenkins contact model (b) and a Valanis contact model (c) - Taken from [20].</figcaption></figure>

and self-affine geometries have been employed to model more realistic roughness across different length scales [92]. The authors proposed a model to represent surface roughness using fractal geometry with a fractal dimension D. Using this model, the contact stiffness $K_n$ is expressed as a function of the fractal dimension and the normal load P. They demonstrated that the contact stiffness is a function of a power law of the normal load, i.e. $K_n \propto P^\alpha$, where the power $\alpha$ depends on the fractal dimension D. This modelling allows for capturing the evolution of the first natural frequency and damping with respect to the contact pressure for a fan blade dov tail joint test case. Also, semi-analytical methods based on half-space theory were proposed to consider multiscale friction interface geometry that deals with surface roughness [93], mesoscale interface geometry [86,94] and fretting fatigue [85,93]. Semi-analytical methods discretise the contact interface to address the overall numerical challenge by analytically solving numerous elementary problems at smaller scales associated with each discretised element. These solvers combine macro- and micro-scale models. First, an initial quasi-static analysis is performed on the macro-scale model to compute the applied pressure and the shear traction fields [42,95]. Then, a micro-scale contact model is used to calculate the elastic deflections of the surface in the normal and tangential directions. Under the half-space assumption, this can be done with the Boussinesq and Cerruti potentials. The solver is speeded up with the projected conjugate gradient method [96] and a discrete-convolution Fast Fourier Transform (FFT). With such solvers, very refined contact meshes can be considered, allowing for the inclusion of surface roughness and profile in the simulations. These analytical solutions of contact properties are then incorporated into the nonlinear structural dynamics solver. Some studies, for example [86], demonstrate that the mesoscale surface geometry has a much greater impact on the nonlinear dynamic properties than surface roughness. An alternative multiscale framework was recently proposed in [97] for modelling the dynamics of bolted structures through zero-thickness elements to couple the effects of microscale roughness with the dynamic analysis of the system.

## 4. Numerical simulation

Using any of the modelling approaches mentioned in Section 3, the equation governing the dynamics of the whole system, including local friction joints, generally takes the form of the following ordinary differential equation:

$$M\ddot{x}(t) + C\dot{x}(t) + Kx(t) + f_{nl}(x, \dot{x}, t) = f_e(t) \quad (1)$$

where M, C, K are the mass, damping and stiffness matrices of the system, respectively, and x the unknown DOF. $f_e$ and $f_{nl}$ express the external excitation and nonlinear efforts, the latter corresponding to the nonlinear contact forces arising from the friction joints. The aim of this section is to present the general and most common ways to solve Eq. (1). Considering that in a realistic finite element modelling of an industrial structure, the number of DOF may be very large, some kinds of reduction techniques are required to simulate the dynamics in an

acceptable amount of time. Some of the reduction techniques existing in the literature were constructed from specificities offered by spectral methods. This section hence first introduces details on possible solution methodologies, with special attention paid to the harmonic balance method, before illustrating some recent reduction approaches adapted to this kind of problem.

### 4.1. Solving the nonlinear ODE

Two main approaches exist to solve Eq. (1): numerical time integration or methods that directly search for the stationary regime.

Numerical time integration techniques possess a primary advantage in their capability to handle diverse systems and nonlinearities without assumptions regarding the form or characteristics of the solution. These methods involve the resolution of Eq. (1) through a sequential iterative process starting with some a priori chosen initial conditions to simulate the evolution of the dynamics with time. The choice of initial conditions has a major influence on the solution reached by the solver as the scheme will naturally tend to stay within the same basin of attraction. It practically means that the solution to which converges the numerical integration depends on the initial conditions.

Integration methods are broadly categorised into explicit and implicit schemes.<fnref n="1" /> Explicit schemes compute the current system response based on prior steps and often necessitate smaller time steps to ensure numerical stability [99]. For instance, the widely recognised Runge-Kutta algorithm [100] operates as an explicit scheme. In contrast, implicit schemes derive the current response based on both the present and previous steps. While most implicit schemes demonstrate unconditional stability in linear systems, they typically demand increased storage capacity and higher computational efforts compared to explicit solution strategies. There exist several implicit methods, such as for instance the Newmark scheme [101] or the Houbolt method [102]. A general review of numerical time integration can be found in [103] and some useful information for practical implementation in [104]. In [105] a series of time-marching methods tailored to friction-induced nonlinearities with a specific focus on turbomachinery applications was proposed.

If the objective is finding the solution (or at least one particular solution) describing the steady state reached after time, numerical time integration is not the best choice as it may require too great a computational effort for a system with a large number of DOF to calculate all the transient period. This justifies the limited use of these approaches in investigating nonlinear vibrations of structures with friction damping. The direct search for a static equilibrium state can be performed by a Newton-Raphson algorithm solving the equation $Kx+f_{nl}(x) = f_e$. In aero-engine applications, and more specifically in the simulation of structures with dampers, this step corresponds to solving

<footnote n="1">¹ There also exists a combination of both, known as the Implicit-Explicit Method [98].</footnote>
<figure><img src="image_8.png" /><figcaption>Fig. 5. Hysteresis curves for a Coulomb contact model (a), a Jenkins contact model (b) and a Valanis contact model (c) - Taken from [20].</figcaption></figure>

and self-affine geometries have been employed to model more realistic roughness across different length scales [92]. The authors proposed a model to represent surface roughness using fractal geometry with a fractal dimension D. Using this model, the contact stiffness $K_n$ is expressed as a function of the fractal dimension and the normal load P. They demonstrated that the contact stiffness is a function of a power law of the normal load, i.e. $K_n \propto P^\alpha$, where the power $\alpha$ depends on the fractal dimension D. This modelling allows for capturing the evolution of the first natural frequency and damping with respect to the contact pressure for a fan blade dov tail joint test case. Also, semi-analytical methods based on half-space theory were proposed to consider multiscale friction interface geometry that deals with surface roughness [93], mesoscale interface geometry [86,94] and fretting fatigue [85,93]. Semi-analytical methods discretise the contact interface to address the overall numerical challenge by analytically solving numerous elementary problems at smaller scales associated with each discretised element. These solvers combine macro- and micro-scale models. First, an initial quasi-static analysis is performed on the macro-scale model to compute the applied pressure and the shear traction fields [42,95]. Then, a micro-scale contact model is used to calculate the elastic deflections of the surface in the normal and tangential directions. Under the half-space assumption, this can be done with the Boussinesq and Cerruti potentials. The solver is speeded up with the projected conjugate gradient method [96] and a discrete-convolution Fast Fourier Transform (FFT). With such solvers, very refined contact meshes can be considered, allowing for the inclusion of surface roughness and profile in the simulations. These analytical solutions of contact properties are then incorporated into the nonlinear structural dynamics solver. Some studies, for example [86], demonstrate that the mesoscale surface geometry has a much greater impact on the nonlinear dynamic properties than surface roughness. An alternative multiscale framework was recently proposed in [97] for modelling the dynamics of bolted structures through zero-thickness elements to couple the effects of microscale roughness with the dynamic analysis of the system.

## 4. Numerical simulation

Using any of the modelling approaches mentioned in Section 3, the equation governing the dynamics of the whole system, including local friction joints, generally takes the form of the following ordinary differential equation:

$$M\ddot{x}(t) + C\dot{x}(t) + Kx(t) + f_{nl}(x, \dot{x}, t) = f_e(t) \quad (1)$$

where M, C, K are the mass, damping and stiffness matrices of the system, respectively, and x the unknown DOF. $f_e$ and $f_{nl}$ express the external excitation and nonlinear efforts, the latter corresponding to the nonlinear contact forces arising from the friction joints. The aim of this section is to present the general and most common ways to solve Eq. (1). Considering that in a realistic finite element modelling of an industrial structure, the number of DOF may be very large, some kinds of reduction techniques are required to simulate the dynamics in an

acceptable amount of time. Some of the reduction techniques existing in the literature were constructed from specificities offered by spectral methods. This section hence first introduces details on possible solution methodologies, with special attention paid to the harmonic balance method, before illustrating some recent reduction approaches adapted to this kind of problem.

### 4.1. Solving the nonlinear ODE

Two main approaches exist to solve Eq. (1): numerical time integration or methods that directly search for the stationary regime.

Numerical time integration techniques possess a primary advantage in their capability to handle diverse systems and nonlinearities without assumptions regarding the form or characteristics of the solution. These methods involve the resolution of Eq. (1) through a sequential iterative process starting with some a priori chosen initial conditions to simulate the evolution of the dynamics with time. The choice of initial conditions has a major influence on the solution reached by the solver as the scheme will naturally tend to stay within the same basin of attraction. It practically means that the solution to which converges the numerical integration depends on the initial conditions.

Integration methods are broadly categorised into explicit and implicit schemes.<fnref n="1" /> Explicit schemes compute the current system response based on prior steps and often necessitate smaller time steps to ensure numerical stability [99]. For instance, the widely recognised Runge-Kutta algorithm [100] operates as an explicit scheme. In contrast, implicit schemes derive the current response based on both the present and previous steps. While most implicit schemes demonstrate unconditional stability in linear systems, they typically demand increased storage capacity and higher computational efforts compared to explicit solution strategies. There exist several implicit methods, such as for instance the Newmark scheme [101] or the Houbolt method [102]. A general review of numerical time integration can be found in [103] and some useful information for practical implementation in [104]. In [105] a series of time-marching methods tailored to friction-induced nonlinearities with a specific focus on turbomachinery applications was proposed.

If the objective is finding the solution (or at least one particular solution) describing the steady state reached after time, numerical time integration is not the best choice as it may require too great a computational effort for a system with a large number of DOF to calculate all the transient period. This justifies the limited use of these approaches in investigating nonlinear vibrations of structures with friction damping. The direct search for a static equilibrium state can be performed by a Newton-Raphson algorithm solving the equation $Kx+f_{nl}(x) = f_e$. In aero-engine applications, and more specifically in the simulation of structures with dampers, this step corresponds to solving

<footnote n="1">¹ There also exists a combination of both, known as the *Implicit-Explicit Method* [98].</footnote>
for the pre-stressed structural state induced by centrifugal effects that apply normal forces on the different interfaces. In this expression, the possible linear part of the contact forces is implicitly included in the K-matrix. As the dissipation mechanism induced by friction depends on these normal forces, calculation of the pre-stressed state is therefore essential in the correct prediction of the underlying dynamics.

A direct search for a periodic steady state is possible by assuming a priori the periodic form of the solution and looking for only a solution of this kind. The shooting method, for instance, will iterate on the initial conditions and look for the periodicity condition to be satisfied after a single time period integration. Shooting techniques have been used widely in friction related applications, see for instance [106,107]. Another approach is to use a Galerkin procedure known as the Harmonic Balance Method (or HBM) that employs the Fourier basis as both base and weight functions. For this mean, the DOF $x(t)$ are expanded on a Fourier basis and substituted in Eq. (1). The residual error is then orthogonalised (i.e. minimised) with respect to the same basis by projection. This results in the following algebraic system of equations to solve:

$$Z(\omega)\hat{x} + \hat{f}_{nl}(\hat{x}, \omega) - \hat{f}_e = 0 \quad (2)$$

In this equation, $\hat{x}$ are the unknown Fourier coefficients of the DOF, $Z$ is the block diagonal multi-harmonic dynamic stiffness matrix function of the structural matrices (M, C and K) and also function of the frequency $\omega$ that defines the periodic solution, which is generally equal to the forcing frequency. The terms $\hat{f}_{nl}(\hat{x}, \omega)$ and $\hat{f}_e = 0$ are the Fourier coefficients of the nonlinear forces and external forces, respectively, obtained during the projection step of the Galerkin procedure. The numerical evaluation of $\hat{f}_{nl}(\hat{x}, \omega)$ can be challenging. The most common approach is to use the alternating frequency/time (AFT) scheme [108] on regularised contact laws. These are required for convergence, although this inherently involves some discretisation errors and approximation. Another method, also based on frequency/time alternation, is the Dynamic Lagrangian Frequency Time (DLFT) [109]. It relies on a prediction-correction process to determine the contact state and associated forces. Because of this process, the DLFT can accurately verify non-regular contact laws and ensure non-interpenetration between the bodies in contact. A comparison between the DLFT formulation and a friction model based on normal and tangential stiffnesses is given in [110]. A detailed review of various strategies for handling nonlinear forces can be found in [21]. The HBM has been largely employed for its efficiency and robustness in solving for the periodic solution of nonlinear ODE, especially in the application of bladed disks with friction interfaces [35, 111–116].

Since several solutions may co-exist due to the nonlinear effects, the Newton-Raphson algorithm used to solve Eq. (2) may or may not converge towards (only) one of them, depending on the initialisation and accuracy of the descent direction used in the algorithm. To improve the latter, providing the solver with an analytical expression of the Jacobian matrix is particularly useful, see for example in [37] for the detailed expression in the case of a regularised Coulomb law. Numerical path continuation methods will provide convergence towards a particular solution and also make it possible to retrieve unstable ones. One can refer to [117–121] for different approaches and an explanation of their numerical implementation. By varying a parameter defined as the continuation parameter, for instance the rotational speed or the excitation frequency, each of these solutions will form a solution branch. In practical numerical solution procedures, one usually initiates the continuation scheme far from a resonant peak, in a region where nonlinear effects are negligible. Then, following and constructing step by step a solution branch as the continuation parameter varies, it is possible to detect possible bifurcations [122,123], switch to other solution branches [124–126] and assess their stability [127,128] in order to simulate a larger scope of possibly reached steady states.

## 4.2. Reduced order modelling

To conduct quantitative analyses on larger industrial numerical models, it has proved necessary to use model reduction techniques. Among these methods, one encounters classical Galerkin projection approaches including component mode syntheses that could be applied to any kind of structures, and some specific techniques relevant only to systems showing cyclic symmetry properties.

### 4.2.1. General approaches

**Condensation.** The simplest technique to reduce the size of an algebraic system with some nonlinear DOF (for instance Eq. (2)) involves using a condensation method. Particularly useful when the nonlinear effects are localised, this approach leverages the fact that only a small subset of the problem's DOF is affected by the nonlinear terms. The method partitions the unknown vector, $\hat{x}$ in Eq. (2), into two parts and isolates the nonlinear DOF, those directly involved in the term $\hat{f}_{nl}$ from the purely linear DOF. By a simple matrix inversion, it then becomes possible to express the linear DOF in terms of the nonlinear DOF. This form is substituted into the other part of the equation to find a new, significantly reduced-size algebraic equation whose size is equal to the number of nonlinear DOF in the problem. In cases where the nonlinear terms are expressed in terms of relative variables, as in most friction problems in which laws involve a node-to-node contact formulation, it is also possible to condense the relative DOF. These straightforward condensation techniques have, for example, been used in [32,41,129–135] for bladed disk finite element models.

Note that these condensation methods do not involve approximation: they simply entail rewriting equations to streamline the nonlinear solver. Conversely, the Galerkin projection approaches mentioned later on are approximations that involve information loss during the transition to the reduced model.

**Galerkin projection.** The primary idea behind the Galerkin projection reduction method is to seek an approximate solution to the dynamic problem as a linear combination of a reduced-size basis relative to the initial space. This involves expressing $x(t)$ as $x(t) = \Phi q(t)$ in the equation of motion (1) and then projecting the resultant onto the reduced space. The classical reduction on a linear modal basis is part of these Galerkin methods. The form of the resulting equation is similar to the un-reduced equation, allowing for the use of previously described solution-seeking and periodic solution methods to find $q(t)$, and then to backtrack to $x(t)$. The primary challenges lie in choosing the reduction base $\Phi$ and computing the projected nonlinear terms on the new base $\Phi^T \hat{f}_{nl}(\Phi q, \Phi \dot{q})$. Poor handling of the reduced nonlinear efforts assessment may result in overestimation or underestimation of the nonlinear effects, leading to a flawed behaviour of the reduced model. The direct approach to compute these nonlinear forces involves simply projecting the nonlinear term onto the reduction basis, akin to the treatment of linear matrices in the system. This method may require manipulating large quantities when the number of DOF is substantial. Sometimes, a spectral substitution technique can be employed to replace this nonlinear term with a previously calculated nonlinear mode, see [37].

To build a reduced base through linear transformation, linear modes can be associated with their modal derivatives [136], companion modes (or dual modes) [137], or even nonlinear modes (see [138,139] for more details on their interpretation and calculation). It may also be possible to use constraint modes [140], hybrid modes [141], or a single dominant mode supplemented with static stiffness matrix compensation [142]. When adopting nonlinear modes, the change of variables between the bases remains linear, but the amplitude of the additional modes changes with frequency. The modal deformation is then evaluated at each frequency iteration from a backbone curve representing the pre-calculated nonlinear mode. This nonlinear modal synthesis, initially proposed in [143], was later extended to non-conservative
for the pre-stressed structural state induced by centrifugal effects that apply normal forces on the different interfaces. In this expression, the possible linear part of the contact forces is implicitly included in the K-matrix. As the dissipation mechanism induced by friction depends on these normal forces, calculation of the pre-stressed state is therefore essential in the correct prediction of the underlying dynamics.

A direct search for a periodic steady state is possible by assuming a priori the periodic form of the solution and looking for only a solution of this kind. The shooting method, for instance, will iterate on the initial conditions and look for the periodicity condition to be satisfied after a single time period integration. Shooting techniques have been used widely in friction related applications, see for instance [106,107]. Another approach is to use a Galerkin procedure known as the Harmonic Balance Method (or HBM) that employs the Fourier basis as both base and weight functions. For this mean, the DOF $x(t)$ are expanded on a Fourier basis and substituted in Eq. (1). The residual error is then orthogonalised (i.e. minimised) with respect to the same basis by projection. This results in the following algebraic system of equations to solve:

$$Z(\omega)\hat{x} + \hat{f}_{nl}(\hat{x}, \omega) - \hat{f}_e = 0 \quad (2)$$

In this equation, $\hat{x}$ are the unknown Fourier coefficients of the DOF, $Z$ is the block diagonal multi-harmonic dynamic stiffness matrix function of the structural matrices (M, C and K) and also function of the frequency $\omega$ that defines the periodic solution, which is generally equal to the forcing frequency. The terms $\hat{f}_{nl}(\hat{x}, \omega)$ and $\hat{f}_e = 0$ are the Fourier coefficients of the nonlinear forces and external forces, respectively, obtained during the projection step of the Galerkin procedure. The numerical evaluation of $\hat{f}_{nl}(\hat{x}, \omega)$ can be challenging. The most common approach is to use the alternating frequency/time (AFT) scheme [108] on regularised contact laws. These are required for convergence, although this inherently involves some discretisation errors and approximation. Another method, also based on frequency/time alternation, is the Dynamic Lagrangian Frequency Time (DLFT) [109]. It relies on a prediction-correction process to determine the contact state and associated forces. Because of this process, the DLFT can accurately verify non-regular contact laws and ensure non-interpenetration between the bodies in contact. A comparison between the DLFT formulation and a friction model based on normal and tangential stiffnesses is given in [110]. A detailed review of various strategies for handling nonlinear forces can be found in [21]. The HBM has been largely employed for its efficiency and robustness in solving for the periodic solution of nonlinear ODE, especially in the application of bladed disks with friction interfaces [35, 111–116].

Since several solutions may co-exist due to the nonlinear effects, the Newton-Raphson algorithm used to solve Eq. (2) may or may not converge towards (only) one of them, depending on the initialisation and accuracy of the descent direction used in the algorithm. To improve the latter, providing the solver with an analytical expression of the Jacobian matrix is particularly useful, see for example in [37] for the detailed expression in the case of a regularised Coulomb law. Numerical path continuation methods will provide convergence towards a particular solution and also make it possible to retrieve unstable ones. One can refer to [117–121] for different approaches and an explanation of their numerical implementation. By varying a parameter defined as the continuation parameter, for instance the rotational speed or the excitation frequency, each of these solutions will form a solution branch. In practical numerical solution procedures, one usually initiates the continuation scheme far from a resonant peak, in a region where nonlinear effects are negligible. Then, following and constructing step by step a solution branch as the continuation parameter varies, it is possible to detect possible bifurcations [122,123], switch to other solution branches [124–126] and assess their stability [127,128] in order to simulate a larger scope of possibly reached steady states.

## 4.2. Reduced order modelling

To conduct quantitative analyses on larger industrial numerical models, it has proved necessary to use model reduction techniques. Among these methods, one encounters classical Galerkin projection approaches including component mode syntheses that could be applied to any kind of structures, and some specific techniques relevant only to systems showing cyclic symmetry properties.

### 4.2.1. General approaches

**Condensation.** The simplest technique to reduce the size of an algebraic system with some nonlinear DOF (for instance Eq. (2)) involves using a condensation method. Particularly useful when the nonlinear effects are localised, this approach leverages the fact that only a small subset of the problem's DOF is affected by the nonlinear terms. The method partitions the unknown vector, $\hat{x}$ in Eq. (2), into two parts and isolates the nonlinear DOF, those directly involved in the term $\hat{f}_{nl}$ from the purely linear DOF. By a simple matrix inversion, it then becomes possible to express the linear DOF in terms of the nonlinear DOF. This form is substituted into the other part of the equation to find a new, significantly reduced-size algebraic equation whose size is equal to the number of nonlinear DOF in the problem. In cases where the nonlinear terms are expressed in terms of relative variables, as in most friction problems in which laws involve a node-to-node contact formulation, it is also possible to condense the relative DOF. These straightforward condensation techniques have, for example, been used in [32,41,129–135] for bladed disk finite element models.

Note that these condensation methods do not involve approximation: they simply entail rewriting equations to streamline the nonlinear solver. Conversely, the Galerkin projection approaches mentioned later on are approximations that involve information loss during the transition to the reduced model.

**Galerkin projection.** The primary idea behind the Galerkin projection reduction method is to seek an approximate solution to the dynamic problem as a linear combination of a reduced-size basis relative to the initial space. This involves expressing $x(t)$ as $x(t) = \Phi q(t)$ in the equation of motion (1) and then projecting the resultant onto the reduced space. The classical reduction on a linear modal basis is part of these Galerkin methods. The form of the resulting equation is similar to the un-reduced equation, allowing for the use of previously described solution-seeking and periodic solution methods to find $q(t)$, and then to backtrack to $x(t)$. The primary challenges lie in choosing the reduction base $\Phi$ and computing the projected nonlinear terms on the new base $\Phi^T \hat{f}_{nl}(\Phi q, \Phi \dot{q})$. Poor handling of the reduced nonlinear efforts assessment may result in overestimation or underestimation of the nonlinear effects, leading to a flawed behaviour of the reduced model. The direct approach to compute these nonlinear forces involves simply projecting the nonlinear term onto the reduction basis, akin to the treatment of linear matrices in the system. This method may require manipulating large quantities when the number of DOF is substantial. Sometimes, a spectral substitution technique can be employed to replace this nonlinear term with a previously calculated nonlinear mode, see [37].

To build a reduced base through linear transformation, linear modes can be associated with their modal derivatives [136], companion modes (or dual modes) [137], or even nonlinear modes (see [138,139] for more details on their interpretation and calculation). It may also be possible to use constraint modes [140], hybrid modes [141], or a single dominant mode supplemented with static stiffness matrix compensation [142]. When adopting nonlinear modes, the change of variables between the bases remains linear, but the amplitude of the additional modes changes with frequency. The modal deformation is then evaluated at each frequency iteration from a backbone curve representing the pre-calculated nonlinear mode. This nonlinear modal synthesis, initially proposed in [143], was later extended to non-conservative
<figure><img src="image_9.png" /><figcaption>Fig. 6. Nonlinear modes and FRFs for different forcing amplitudes: (a) FEM of a fan system [146] and (b) lumped parameter model of a mistuned structure, adapted from [147].</figcaption></figure>

systems [111] and was further developed by Krack et al. for the calculation of nonlinear modes with multiple harmonics [144]. These two approaches, developed within a framework of synthesis with a single nonlinear mode, were generalised to a multi-mode synthesis without assuming mode orthogonality in [37]. One of the limitations of these methods is that the projected nonlinear forces are substituted without coupling via other modes, preventing the ability to simulate internal resonances. A relatively similar approach introduced “response-dependent nonlinear modes” to enhance the projection basis [145]. It proposes a structural modification using the Dual Modal Space method to reduce the computational effort in evaluating nonlinear effects. The complex nonlinear modes defined in [111] represent a pseudo-periodic solution of the autonomous system (1), that is, with $f_e = 0$, yet allowing for dissipative nonlinearities, contrasting with the nonlinear (real) normal modes [139].

The calculation of nonlinear modes can also be useful to avoid frequency forced responses simulations and save computation time as they can depict the loci of the resonant response for a given energy input without having to simulate an entire solution branch. Recent studies have been dedicated to this concept, showcasing its efficiency in analysing systems of both academic and industrial complexity. For example, nonlinear modal analysis has been applied to simulate the response of turbines with underplatform dampers [36], blisks with ring dampers [135] and fan blade systems with dovetail joints [146]. The relationship between nonlinear modes and forced nonlinear frequency response is illustrated in Fig. 6(a) for a FEM of a fan system with dovetail joints and in Fig. 6(b) for an academic model of a mistuned system with blade root friction. For a given excitation forcing level, the vibration amplitude corresponding to the forced resonance response can be identified through the knowledge of the nonlinear mode.

**Component mode synthesis.** Component Mode Synthesis (CMS) reduced order models are particularly adapted to structures composed of several components (or sub-structures) assembled at interfaces [148,149]. Each sub-structure can be reduced using the above-mentioned techniques and coupling at interfaces is accounted for by adding static modes, associated with the nodes retained within the interface of each sub-structure, to the global reduction basis. The number of dynamic modes retained is chosen based on a conventional frequency criterion. After this reduction, only generalised coordinates of the reduced substructures and the DOF associated with the nodes within interfaces remain. Depending on the boundary conditions applied to the interface while treating a separate sub-structure, CMS approaches can be classified into free [150,151], fixed [152,153] or hybrid interface methods [154,155]. An assessment of these techniques for assembled structures with friction interfaces can be found in [156]. The Craig-Bampton method [153] is known as one of the most popular fixed interface CMS approaches because of its efficiency. Its extension to account for nonlinear effects incoming from frictional interfaces in bladed disks was performed in [129] by adding complex nonlinear normal modes, computed with a fixed interface, to the reduction basis of the blade sub-structure.

**Interface reduction.** CMS approaches present a significant limitation for structures with a high mesh density at the interfaces as they result in a reduced model that maintains an excessive number of physical coordinates linked to sector boundaries. To overcome this issue, given the assumption that the model retains an excess of information, double component modal synthesis approaches have been devised [157] to filter out the dynamics of the interfaces and retain only the necessary information according to the calculation requirements. It is based on the concept of interface mode [158]. For this purpose, after reassembling the complete structure via the boundary nodes (which have been retained so far in the Craig-Bampton procedure) to form the reduced mass and stiffness matrices of the entire system, the sub-blocks corresponding to the boundaries are then extracted from the reduced matrices, and the corresponding modes are computed. The reduction of the interfaces is performed by selecting certain eigenvectors used as a reduction basis for the boundary degrees of freedom. This selection can be based on a frequency criterion, retaining only the modes whose frequencies are of the same order as those of the fixed-interface modes preserved for the creation of the super-elements [153]. These retained modes are sometimes referred to as partial interface modes [159] or branch modes [160]. Combining a nonlinear component mode synthesis and reduction of the cyclic symmetry interface, Joannin et al. [129] simulated efficiently nonlinear forced responses of a realistically meshed bladed disk with friction nonlinearities.

The general technique mentioned above is well suited when the interface forces acting between sub-structures are linear. Other methods were developed especially to reduce frictional interfaces that require a high level of discretisation to capture singularities, see for instance [149,161,162]. More recently, adaptive component mode synthesis [163] has been proposed and validated for fan blade systems through a reformulated equation of motion by introducing an internal penalty variable depending on the online contact condition. In recent years, reduction techniques of increasing complexity [164–168] have been introduced, aiming to exploit the piecewise linearity of friction states (i.e., stick, slip, and lift-off). Despite the reduced size of the system, the computation of nonlinear forces is still performed in full-space, resulting in substantial online computational costs, especially for high-fidelity modelling of joints requiring a large number of contact nodes. To address this drawback, hyper-reduction techniques have emerged [32,168,169], which evaluate nonlinear forces in the reduced space, further decreasing the computational burden.

### 4.2.2. Approaches specific to rotationally cyclic structures

The cyclic symmetry of bladed disks can also be used to reduce the size of the problem to solve. The original method that simplifies the analysis of cyclic symmetric structures primarily stems from the work of Wildheim [170] and Thomas [171], extending the study conducted in [172], where the structures considered were infinitely long linear chains of identical substructures. For more in-depth insights, a reference such as [173] discusses the transformation from complex cyclic components to real cyclic components.
<figure><img src="image_9.png" /><figcaption>Fig. 6. Nonlinear modes and FRFs for different forcing amplitudes: (a) FEM of a fan system [146] and (b) lumped parameter model of a mistuned structure, adapted from [147].</figcaption></figure>

systems [111] and was further developed by Krack et al. for the calculation of nonlinear modes with multiple harmonics [144]. These two approaches, developed within a framework of synthesis with a single nonlinear mode, were generalised to a multi-mode synthesis without assuming mode orthogonality in [37]. One of the limitations of these methods is that the projected nonlinear forces are substituted without coupling via other modes, preventing the ability to simulate internal resonances. A relatively similar approach introduced “response-dependent nonlinear modes” to enhance the projection basis [145]. It proposes a structural modification using the Dual Modal Space method to reduce the computational effort in evaluating nonlinear effects. The complex nonlinear modes defined in [111] represent a pseudo-periodic solution of the autonomous system (1), that is, with $f_e = 0$, yet allowing for dissipative nonlinearities, contrasting with the nonlinear (real) normal modes [139].

The calculation of nonlinear modes can also be useful to avoid frequency forced responses simulations and save computation time as they can depict the loci of the resonant response for a given energy input without having to simulate an entire solution branch. Recent studies have been dedicated to this concept, showcasing its efficiency in analysing systems of both academic and industrial complexity. For example, nonlinear modal analysis has been applied to simulate the response of turbines with underplatform dampers [36], blisks with ring dampers [135] and fan blade systems with dovetail joints [146]. The relationship between nonlinear modes and forced nonlinear frequency response is illustrated in Fig. 6(a) for a FEM of a fan system with dovetail joints and in Fig. 6(b) for an academic model of a mistuned system with blade root friction. For a given excitation forcing level, the vibration amplitude corresponding to the forced resonance response can be identified through the knowledge of the nonlinear mode.

**Component mode synthesis.** Component Mode Synthesis (CMS) reduced order models are particularly adapted to structures composed of several components (or sub-structures) assembled at interfaces [148,149]. Each sub-structure can be reduced using the above-mentioned techniques and coupling at interfaces is accounted for by adding static modes, associated with the nodes retained within the interface of each sub-structure, to the global reduction basis. The number of dynamic modes retained is chosen based on a conventional frequency criterion. After this reduction, only generalised coordinates of the reduced substructures and the DOF associated with the nodes within interfaces remain. Depending on the boundary conditions applied to the interface while treating a separate sub-structure, CMS approaches can be classified into free [150,151], fixed [152,153] or hybrid interface methods [154,155]. An assessment of these techniques for assembled structures with friction interfaces can be found in [156]. The Craig-Bampton method [153] is known as one of the most popular fixed interface CMS approaches because of its efficiency. Its extension to account for nonlinear effects incoming from frictional interfaces in bladed disks was performed in [129] by adding complex nonlinear normal modes, computed with a fixed interface, to the reduction basis of the blade sub-structure.

**Interface reduction.** CMS approaches present a significant limitation for structures with a high mesh density at the interfaces as they result in a reduced model that maintains an excessive number of physical coordinates linked to sector boundaries. To overcome this issue, given the assumption that the model retains an excess of information, double component modal synthesis approaches have been devised [157] to filter out the dynamics of the interfaces and retain only the necessary information according to the calculation requirements. It is based on the concept of interface mode [158]. For this purpose, after reassembling the complete structure via the boundary nodes (which have been retained so far in the Craig-Bampton procedure) to form the reduced mass and stiffness matrices of the entire system, the sub-blocks corresponding to the boundaries are then extracted from the reduced matrices, and the corresponding modes are computed. The reduction of the interfaces is performed by selecting certain eigenvectors used as a reduction basis for the boundary degrees of freedom. This selection can be based on a frequency criterion, retaining only the modes whose frequencies are of the same order as those of the fixed-interface modes preserved for the creation of the super-elements [153]. These retained modes are sometimes referred to as partial interface modes [159] or branch modes [160]. Combining a nonlinear component mode synthesis and reduction of the cyclic symmetry interface, Joannin et al. [129] simulated efficiently nonlinear forced responses of a realistically meshed bladed disk with friction nonlinearities.

The general technique mentioned above is well suited when the interface forces acting between sub-structures are linear. Other methods were developed especially to reduce frictional interfaces that require a high level of discretisation to capture singularities, see for instance [149,161,162]. More recently, adaptive component mode synthesis [163] has been proposed and validated for fan blade systems through a reformulated equation of motion by introducing an internal penalty variable depending on the online contact condition. In recent years, reduction techniques of increasing complexity [164–168] have been introduced, aiming to exploit the piecewise linearity of friction states (i.e., stick, slip, and lift-off). Despite the reduced size of the system, the computation of nonlinear forces is still performed in full-space, resulting in substantial online computational costs, especially for high-fidelity modelling of joints requiring a large number of contact nodes. To address this drawback, hyper-reduction techniques have emerged [32,168,169], which evaluate nonlinear forces in the reduced space, further decreasing the computational burden.

### 4.2.2. Approaches specific to rotationally cyclic structures

The cyclic symmetry of bladed disks can also be used to reduce the size of the problem to solve. The original method that simplifies the analysis of cyclic symmetric structures primarily stems from the work of Wildheim [170] and Thomas [171], extending the study conducted in [172], where the structures considered were infinitely long linear chains of identical substructures. For more in-depth insights, a reference such as [173] discusses the transformation from complex cyclic components to real cyclic components.
The general method for obtaining the equation of motion in a cyclic basis involves applying a change of variables between physical DOF and cyclic components (via a discrete spatial Fourier transform) and relies on the concept of continuity between sectors. With a partitioning of physical DOF into internal and boundary ones, it may be possible to retain only the structural matrices of the reference sector in the equation of motion projected into the cyclic basis. After projection, partitioning, and imposition of boundary conditions, the resulting structural cyclic matrices are block-diagonal (due to the orthogonality of the Fourier basis). By this means, all the physical quantities of a cyclically symmetric structure can be described using a cyclic basis with cyclic components defined over the reference sector and a harmonic index, or spatial wave number, often called nodal diameter. In a linear and perfectly tuned system, the change to cyclic coordinates allows decoupling the system by each nodal diameter.

Nonlinear forces introduce coupling among the different nodal diameters via higher time or space harmonics and very few procedures exist to reduce the nonlinear dynamics of a cyclically symmetric structure. Most studies on this subject, for example [173,174], solve the complete nonlinear system without using the advantages offered by cyclic periodicity. In 2004, Petrov [175] assumed the displacement of a cyclically symmetric structure subjected to wave type excitation to have the same form as the excitation. He then achieved reduction by reducing the complex dynamic stiffness matrix arising from the HBM application. This assumption has been employed in the literature, among other things, for studies on the influence of contact zones [176] or wear [85] on the system's dynamics. In practice, depending on the form of the external excitation, it is possible that not all the cyclic components will be coupled by nonlinear forces. The method developed in [177] theoretically demonstrates which cyclic components will actually be coupled when the structure is subjected to a waveform excitation. Focusing only on the corresponding harmonics then further reduces the size of the system to be solved while removing the restrictive assumption imposed by the solution's form.

Given the highly specific issue posed by blade mistuning in the context of turbomachinery, several reduction approaches that deal with the effect of mistuning have been developed [178–184]. Furthermore, the consideration of both nonlinearities and mistuning within numerical simulations remains a major challenge that no existing technique appears to effectively address. Hence, the development of new methods to tackle this challenge has been identified by Castanier and Pierre [3] as one of the significant endeavours for bladed disk design, and this has been the subject of recent research [37,185–189].

## 5. Experimental testing

Due to the complex nature of dry friction, the dynamic design of friction-damped turbine blades still relies heavily on experimental verification. Numerical models (see Section 3) used during the design phase rely on assumptions/approximations<fnref n="2" /> and thus need tuning against experimental evidence, often gathered on purposely developed test rigs. These test rigs are a vital part of the research in the friction damping field, as they offer experimental evidence in a controlled laboratory environment which is used not only to tune, but also to validate the numerical models later used as design tools. Over the years, different research groups across the globe have designed, produced, calibrated and tried out a large number of rigs. The main purpose of this section is to highlight the key challenges common to these test rigs, review the solutions adopted, and discuss the trends and key developments.

The experimental work related to friction-damped structures subjected to dynamic loading can be classified into three main categories:

1. Test setups that concentrate on the contact mechanics of individual interfaces, with the goal of estimating calibration parameters required for contact models.

2. Test setups designed to replicate the dynamics and/or mechanics of gas turbine components, with the aim of validating numerical models and assessing the effectiveness of specific friction dampers.

3. Test setups that pursue both objectives above, involving the monitoring of structure dynamics while simultaneously observing the contact interfaces.

Section 5.1 focuses on the first kind of test rigs, while Section 5.2 focuses on the second and third kind of test rigs.

### 5.1. Contact interface characterisation

The ability to accurately predict the dynamics of friction-damped structures is hindered by the challenge of integrating suitable models for contact interfaces. As discussed in Section 4, numerical models typically aim at reproducing the hysteresis-like behaviour observed at the contact, by introducing additional stiffness and damping parameters. Such parameters are often formulated in terms of tangential and normal contact stiffness, $k_t$ and $k_n$, and friction coefficient $\mu$. While the normal contact stiffness $k_n$ is typically estimated using analytical and/or numerical formulations [46,190,191], $k_t$ and $\mu$ are estimated starting from hysteresis curves (see Fig. 7) measured in controlled conditions on purposely developed test rigs. Hysteresis measurement and contact mechanics represent extensive research fields. This section focuses on the test rigs developed to target turbomachinery applications, where specific materials and operating conditions are of interest. In detail, this section will focus on the latest efforts of research groups active in the field. Additional notable works, related to other applications (e.g. bolted joints [192–194], brake squeal [195,196], static metal seals [197], torsional fretting [198], low and ultra-low friction systems [199]), share some features with the presented test rigs but are beyond the scope of the present review.

#### 5.1.1. The test rigs

In gas turbine applications, obtaining hysteresis data is crucial for various materials and combinations, spanning a broad spectrum of temperatures, normal load, amplitudes within slipping regimes, and considerations related to the running time and wear-dependent properties. To address these requirements, these test rigs were specifically designed for measuring hysteresis properties within a nominal contact zone or area. The gathered data were subsequently integrated into finite element models appropriately scaling the contact stiffness information. Since the nominal contact area is here used as a “representative unit”, its configuration should be meticulously controlled, incorporating not only general features such as material, presence of coating, etc., but also specific details such as the applied normal load, the surface finish of the contact region, the spectrum of slippage amplitudes, the frequencies of cyclic loading and the contact area nominal extension. This last feature can be numerically evaluated for flat-on-flat interfaces only, while it is theoretically zero for hemispherical-on-flat interfaces and a simple line for cylinder-on-flat interfaces.

All these test setups have a set of shared objectives:

* to apply a normal load $N$ between a pair of specimens;

* to generate tangential loading either via controlled displacement or controlled force on one side of the contact;

* to measure the tangential resisting force $T$ on the opposite side;

* to measure the relative displacement between contact surfaces.

Although the objective remains constant, the design choices vary among the different rigs. An initial comprehensive examination of these test rigs is available in [200]. Table 1 seeks to revise this review with the most recent contributions, emphasising crucial aspects related to

<footnote n="2">² Approximations are also introduced by certain solution techniques, as discussed in Section 4. However, these will not be addressed further in this paper.</footnote>
The general method for obtaining the equation of motion in a cyclic basis involves applying a change of variables between physical DOF and cyclic components (via a discrete spatial Fourier transform) and relies on the concept of continuity between sectors. With a partitioning of physical DOF into internal and boundary ones, it may be possible to retain only the structural matrices of the reference sector in the equation of motion projected into the cyclic basis. After projection, partitioning, and imposition of boundary conditions, the resulting structural cyclic matrices are block-diagonal (due to the orthogonality of the Fourier basis). By this means, all the physical quantities of a cyclically symmetric structure can be described using a cyclic basis with cyclic components defined over the reference sector and a harmonic index, or spatial wave number, often called nodal diameter. In a linear and perfectly tuned system, the change to cyclic coordinates allows decoupling the system by each nodal diameter.

Nonlinear forces introduce coupling among the different nodal diameters via higher time or space harmonics and very few procedures exist to reduce the nonlinear dynamics of a cyclically symmetric structure. Most studies on this subject, for example [173,174], solve the complete nonlinear system without using the advantages offered by cyclic periodicity. In 2004, Petrov [175] assumed the displacement of a cyclically symmetric structure subjected to wave type excitation to have the same form as the excitation. He then achieved reduction by reducing the complex dynamic stiffness matrix arising from the HBM application. This assumption has been employed in the literature, among other things, for studies on the influence of contact zones [176] or wear [85] on the system's dynamics. In practice, depending on the form of the external excitation, it is possible that not all the cyclic components will be coupled by nonlinear forces. The method developed in [177] theoretically demonstrates which cyclic components will actually be coupled when the structure is subjected to a waveform excitation. Focusing only on the corresponding harmonics then further reduces the size of the system to be solved while removing the restrictive assumption imposed by the solution's form.

Given the highly specific issue posed by blade mistuning in the context of turbomachinery, several reduction approaches that deal with the effect of mistuning have been developed [178–184]. Furthermore, the consideration of both nonlinearities and mistuning within numerical simulations remains a major challenge that no existing technique appears to effectively address. Hence, the development of new methods to tackle this challenge has been identified by Castanier and Pierre [3] as one of the significant endeavours for bladed disk design, and this has been the subject of recent research [37,185–189].

## 5. Experimental testing

Due to the complex nature of dry friction, the dynamic design of friction-damped turbine blades still relies heavily on experimental verification. Numerical models (see Section 3) used during the design phase rely on assumptions/approximations<fnref n="2" /> and thus need tuning against experimental evidence, often gathered on purposely developed test rigs. These test rigs are a vital part of the research in the friction damping field, as they offer experimental evidence in a controlled laboratory environment which is used not only to tune, but also to validate the numerical models later used as design tools. Over the years, different research groups across the globe have designed, produced, calibrated and tried out a large number of rigs. The main purpose of this section is to highlight the key challenges common to these test rigs, review the solutions adopted, and discuss the trends and key developments.

The experimental work related to friction-damped structures subjected to dynamic loading can be classified into three main categories:

1. Test setups that concentrate on the contact mechanics of individual interfaces, with the goal of estimating calibration parameters required for contact models.

2. Test setups designed to replicate the dynamics and/or mechanics of gas turbine components, with the aim of validating numerical models and assessing the effectiveness of specific friction dampers.

3. Test setups that pursue both objectives above, involving the monitoring of structure dynamics while simultaneously observing the contact interfaces.

Section 5.1 focuses on the first kind of test rigs, while Section 5.2 focuses on the second and third kind of test rigs.

### 5.1. Contact interface characterisation

The ability to accurately predict the dynamics of friction-damped structures is hindered by the challenge of integrating suitable models for contact interfaces. As discussed in Section 4, numerical models typically aim at reproducing the hysteresis-like behaviour observed at the contact, by introducing additional stiffness and damping parameters. Such parameters are often formulated in terms of tangential and normal contact stiffness, $k_t$ and $k_n$, and friction coefficient $\mu$. While the normal contact stiffness $k_n$ is typically estimated using analytical and/or numerical formulations [46,190,191], $k_t$ and $\mu$ are estimated starting from hysteresis curves (see Fig. 7) measured in controlled conditions on purposely developed test rigs. Hysteresis measurement and contact mechanics represent extensive research fields. This section focuses on the test rigs developed to target turbomachinery applications, where specific materials and operating conditions are of interest. In detail, this section will focus on the latest efforts of research groups active in the field. Additional notable works, related to other applications (e.g. bolted joints [192–194], brake squeal [195,196], static metal seals [197], torsional fretting [198], low and ultra-low friction systems [199]), share some features with the presented test rigs but are beyond the scope of the present review.

#### 5.1.1. The test rigs

In gas turbine applications, obtaining hysteresis data is crucial for various materials and combinations, spanning a broad spectrum of temperatures, normal load, amplitudes within slipping regimes, and considerations related to the running time and wear-dependent properties. To address these requirements, these test rigs were specifically designed for measuring hysteresis properties within a nominal contact zone or area. The gathered data were subsequently integrated into finite element models appropriately scaling the contact stiffness information. Since the nominal contact area is here used as a “representative unit”, its configuration should be meticulously controlled, incorporating not only general features such as material, presence of coating, etc., but also specific details such as the applied normal load, the surface finish of the contact region, the spectrum of slippage amplitudes, the frequencies of cyclic loading and the contact area nominal extension. This last feature can be numerically evaluated for flat-on-flat interfaces only, while it is theoretically zero for hemispherical-on-flat interfaces and a simple line for cylinder-on-flat interfaces.

All these test setups have a set of shared objectives:

* to apply a normal load $N$ between a pair of specimens;

* to generate tangential loading either via controlled displacement or controlled force on one side of the contact;

* to measure the tangential resisting force $T$ on the opposite side;

* to measure the relative displacement between contact surfaces.

Although the objective remains constant, the design choices vary among the different rigs. An initial comprehensive examination of these test rigs is available in [200]. Table 1 seeks to revise this review with the most recent contributions, emphasising crucial aspects related to

<footnote n="2">² Approximations are also introduced by certain solution techniques, as discussed in Section 4. However, these will not be addressed further in this paper.</footnote>
<figure><img src="image_10.png" /><figcaption>Fig. 7. Graphic representation of the link between a measured and simulated hysteresis cycle.</figcaption></figure>

<table><caption>**Table 1**
Main features of the test rigs dedicated to the experimental characterisation of contact interfaces for turbomachinery applications.</caption><thead><tr><th>References</th><th>[201]</th><th>[202]</th><th>[59,203]</th><th>[204]</th><th>[205]</th></tr></thead><tbody><tr><td>Contact configuration</td><td>Flat-on-flat</td><td>Flat-on-flat</td><td>Flat-on-Flat<br>Cylinder-on-Flat</td><td>Flat-on-flat</td><td>Sphere-on-flat</td></tr><tr><td>Nominal<br>contact area</td><td>Square<br>[1–1.4] mm²</td><td>Two rect. patches<br>[25–100] mm²</td><td>Two rect. patches<br>[5–50] mm²</td><td>Two rect. patches<br>[6,18] mm²</td><td>Sphere<br>radius: 6 mm</td></tr><tr><td>Nominal contact<br>pressure/load</td><td>&lt;500 MPa</td><td>[20–200] MPa</td><td>[1–30] MPa</td><td>[1–35] MPa</td><td>&lt;4700 N</td></tr><tr><td>Frequency</td><td>&lt;200 Hz</td><td>&lt;10 Hz</td><td>&lt;300 Hz</td><td>&lt;80 Hz</td><td>&lt;2 kHz</td></tr><tr><td>Sliding amplitude</td><td>&lt; 100 µm</td><td>[20–500] µm</td><td>&lt; 150 µm</td><td>&lt; 50 µm</td><td>&lt;2 mm</td></tr><tr><td>Temperature</td><td>&lt;800 °C</td><td>&lt;900 °C</td><td>&lt;1000 °C</td><td>Room temperature</td><td>Room temperature</td></tr><tr><td>Tangential Excitation</td><td>Shaker</td><td>Hydraulic actuator</td><td>Shaker</td><td>Piezoelectric actuator</td><td>Shaker</td></tr><tr><td>x measurement</td><td>Two LDVs</td><td>DIC</td><td>Two LDVs</td><td>One LDV</td><td>One LDV</td></tr><tr><td>T measurement</td><td>Piezo force<br>transducer</td><td>Strain force<br>transducer</td><td>Piezo force<br>transducer</td><td>Strain force<br>transducer</td><td>Piezo force<br>transducer</td></tr><tr><td>N application</td><td>Pneumatic actuator</td><td>Hydraulic actuator</td><td>Deadweights</td><td>Piezoelectric actuator</td><td>Piezoelectric actuator</td></tr><tr><td>N measurement</td><td>-</td><td>-</td><td>-</td><td>Strain force<br>transducer</td><td>Strain force<br>transducer</td></tr><tr><td>Temperature<br>application/control</td><td>Furnace</td><td>Not disclosed</td><td>Induction with<br>thermocouple feedback</td><td>-</td><td>-</td></tr></tbody></table>

measurement and motion techniques. It should be noted that, over the years, some of the research groups have designed different generations of test rigs. In this instance, the earlier versions (fully described in [200]) have been omitted, as they have been surpassed by their successors.

Effectively producing and measuring hysteresis cycles pose a series of challenges which have been faced in different ways by the different research groups. In the following a few of these challenges are analysed, highlighting the pros and cons of each adopted solution.

**The selection of contact interface geometry** significantly influences contact parameters and the dispersion of estimated results. The sphere-on-flat contact is less susceptible to uncertainty, making it suitable for

exploring test rig functionality and assessing the impact of variables such as normal load variation [205]. However, as the focus shifts to interfaces commonly found in turbomachinery friction dampers, a natural progression to “flat-on-flat” and “cylinder-on-flat” contacts is anticipated. The “flat-on-flat” interface, a prevalent choice, is notably influenced by surface morphology and aligning systems. While many test rigs for flat-on-flat interfaces employ a rigid aligning system, the strategy presented in [203] advocates a “self-aligning” system. In this setup, the specimen with two contact tracks is free to rotate around two axes, unaffected by factors other than the normal load, providing a condition closer to the actual working environment. The choice between a rigid or self-aligning system depends on the test's
<figure><img src="image_10.png" /><figcaption>Fig. 7. Graphic representation of the link between a measured and simulated hysteresis cycle.</figcaption></figure>

<table><caption>**Table 1**
Main features of the test rigs dedicated to the experimental characterisation of contact interfaces for turbomachinery applications.</caption><thead><tr><th>References</th><th>[201]</th><th>[202]</th><th>[59,203]</th><th>[204]</th><th>[205]</th></tr></thead><tbody><tr><td>Contact configuration</td><td>Flat-on-flat</td><td>Flat-on-flat</td><td>Flat-on-Flat<br>Cylinder-on-Flat</td><td>Flat-on-flat</td><td>Sphere-on-flat</td></tr><tr><td>Nominal<br>contact area</td><td>Square<br>[1–1.4] mm²</td><td>Two rect. patches<br>[25–100] mm²</td><td>Two rect. patches<br>[5–50] mm²</td><td>Two rect. patches<br>[6,18] mm²</td><td>Sphere<br>radius: 6 mm</td></tr><tr><td>Nominal contact<br>pressure/load</td><td>&lt;500 MPa</td><td>[20–200] MPa</td><td>[1–30] MPa</td><td>[1–35] MPa</td><td>&lt;4700 N</td></tr><tr><td>Frequency</td><td>&lt;200 Hz</td><td>&lt;10 Hz</td><td>&lt;300 Hz</td><td>&lt;80 Hz</td><td>&lt;2 kHz</td></tr><tr><td>Sliding amplitude</td><td>&lt; 100 µm</td><td>[20–500] µm</td><td>&lt; 150 µm</td><td>&lt; 50 µm</td><td>&lt;2 mm</td></tr><tr><td>Temperature</td><td>&lt;800 °C</td><td>&lt;900 °C</td><td>&lt;1000 °C</td><td>Room temperature</td><td>Room temperature</td></tr><tr><td>Tangential Excitation</td><td>Shaker</td><td>Hydraulic actuator</td><td>Shaker</td><td>Piezoelectric actuator</td><td>Shaker</td></tr><tr><td>x measurement</td><td>Two LDVs</td><td>DIC</td><td>Two LDVs</td><td>One LDV</td><td>One LDV</td></tr><tr><td>T measurement</td><td>Piezo force<br>transducer</td><td>Strain force<br>transducer</td><td>Piezo force<br>transducer</td><td>Strain force<br>transducer</td><td>Piezo force<br>transducer</td></tr><tr><td>N application</td><td>Pneumatic actuator</td><td>Hydraulic actuator</td><td>Deadweights</td><td>Piezoelectric actuator</td><td>Piezoelectric actuator</td></tr><tr><td>N measurement</td><td>-</td><td>-</td><td>-</td><td>Strain force<br>transducer</td><td>Strain force<br>transducer</td></tr><tr><td>Temperature<br>application/control</td><td>Furnace</td><td>Not disclosed</td><td>Induction with<br>thermocouple feedback</td><td>-</td><td>-</td></tr></tbody></table>

measurement and motion techniques. It should be noted that, over the years, some of the research groups have designed different generations of test rigs. In this instance, the earlier versions (fully described in [200]) have been omitted, as they have been surpassed by their successors.

Effectively producing and measuring hysteresis cycles pose a series of challenges which have been faced in different ways by the different research groups. In the following a few of these challenges are analysed, highlighting the pros and cons of each adopted solution.

**The selection of contact interface geometry** significantly influences contact parameters and the dispersion of estimated results. The sphere-on-flat contact is less susceptible to uncertainty, making it suitable for

exploring test rig functionality and assessing the impact of variables such as normal load variation [205]. However, as the focus shifts to interfaces commonly found in turbomachinery friction dampers, a natural progression to “flat-on-flat” and “cylinder-on-flat” contacts is anticipated. The “flat-on-flat” interface, a prevalent choice, is notably influenced by surface morphology and aligning systems. While many test rigs for flat-on-flat interfaces employ a rigid aligning system, the strategy presented in [203] advocates a “self-aligning” system. In this setup, the specimen with two contact tracks is free to rotate around two axes, unaffected by factors other than the normal load, providing a condition closer to the actual working environment. The choice between a rigid or self-aligning system depends on the test's
purpose [206]. For more repeatable tests, a rigid setup is preferred, while a self-aligning system is recommended for testing complex and larger interfaces. An essential consideration highlighted by various research groups is the design of flat-on-flat contact interfaces. Both the mobile and fixed contact surfaces should have the same width along the sliding motion direction. Failure to meet this condition may result in the shorter specimen wearing a groove within the wider one, leading to a hardening effect in the hysteresis [203].

**Tangential excitation** is crucial for inducing relative motion between specimens and, consequently, generating friction force. Closed-loop actuation mechanisms, whether force or displacement-based, are commonly employed. The shaker is the preferred choice for high-frequency measurements; however, careful dynamic design of the test rig structure and its joints must be consistently conducted to ensure that the resonance frequencies of the rig are well above those of the measurements.

**The application of the normal load N** is intricately tied to the test rig's purpose. Stable application of a constant normal load, with built-in measurement capability, can be achieved using actuators or simple deadweights. Rigs operating under a constant normal load, with the goal of attaining high contact pressure, often opt for pneumatic or hydraulic actuators [201,202]. For rigs designed to apply a normal load varying harmonically around a mean value, a piezo actuator is the preferred choice [204,205].

**Accurately measuring the contact force T** (and N in [204,205]) involves not only selecting suitable force sensors but also ensuring a smooth force transfer, while preventing potential compensation or misalignment issues arising from the test rig structure. Piezo-based force transducers are the preferred choice for test rigs aiming to achieve higher frequency measurements. These sensors are affected by the volatility of the electric charge, posing challenges in attaining long-term stable measurements. This is not a limiting factor if the goal of the measurement is related to assessing properties such as contact stiffness or dissipated energy (i.e. area of the cycle), as an incorrect assessment of the mean value of T would not change the result. The correct assessment of the mean value of T becomes essential when considering the presence of multiple static equilibria under the same nominal conditions, as further discussed in Section 6.2.1. Measurement of the normal force N is accomplished through a strain force transducer. This selection is justified by the superior long-term stability of strain force sensors compared to piezo-based force transducers. This stability is crucial for accurately capturing the stable mean force value. It is important to note that the softer structure of the strain force transducer comes with a limitation in suitability for high-frequency measurements. The force-transfer mechanism from the contact to the load cell(s) is another crucial aspect, as discussed previously. Various structural approaches, such as elastic beams, three-claw stabilisers, and tripods, have been employed to ensure high stiffness, perfect alignment, and the transfer of forces without introducing spurious components.

**Measuring the relative displacement at the contact x** is crucial for accurately capturing hysteresis and subsequently estimating the tangential contact stiffness $k_t$. There appears to be a consensus among various research groups favouring techniques such as Laser Doppler Vibrometers (LDV) or Digital Image Correlation (DIC). Both methods ensure an adequate resolution when measuring within a low displacement range, i.e. [0.1–2] µm, typically corresponding to a fully stuck condition, a prerequisite for estimating tangential contact stiffness. The necessity for high-precision measurement techniques is underscored by a comparison between DIC and a linear variable differential transformer in [202]. Additionally, the definition of “relative displacement” itself is noteworthy. In certain test rigs, only the displacement of the moving specimen is measured, assuming the compliance of the test rig connected to the fixed specimen is negligible. While this approach may be suitable for specific test rig architectures, it is not universally applicable, as demonstrated in [202]. Opting for a differential measurement, either with DIC or using two LDVs positioned as close as possible to the contact, appears to be the safest choice.

**The effect of wear** has been observed in all test rigs, to various degrees. In detail, hysteresis properties exhibit a significant dependence on overall slipping time, closely tied to the wear of the contact zones. Some research groups incorporate within their measurement protocol, besides $\mu$ and $k_t$, which can be recorded as a function of the number of cycles, other wear-related parameters, e.g. the possibility of measuring the wear volume history of the same pair of specimens with a negligible perturbation to the wear processes [203].

**The influence of frequency** does not appear to be significant, particularly within the 20–100 Hz range. While higher frequency measurements offer advantages for assessing wear, they also bring about a greater influence of the test rig itself. As the frequency increases, the signals tend to become more disrupted, posing challenges for accurate evaluation.

**Real time contact state detection** is a recent focus of research in the field of single interface testing. Recent works have used ultrasound [207] and total internal reflection [208] to detect in real time the effective contact area and contact state of non-transparent and transparent samples respectively using the test rig detailed in [201].

### 5.1.2. An overview of published results

Several rigs have been designed, constructed, and used over the years to measure hysteresis cycles of friction contacts. A quantitative comparison of results is not always possible due to the large variety of test conditions (contact interface geometry, materials, temperature, etc.), and the fact that the influence of each variable on the contact parameters is not known with precision. Nevertheless, a number of common features and trends have been observed and are summarised below:

* **Hysteresis Shape.** A positive slope of the macroslip portions of the hysteresis cycles depicted in Fig. 7 (unexplained by a Coulomb-type friction model) is often observed. Furthermore, the macroslip portion of the cycles may exhibit fluctuations, which stem from interactions among wear scars, often traceable to a ridge being formed at each end of a multi-cycle test run [198].

* **Friction Coefficient.** This parameter exhibits remarkable stability, as consistent outcomes are observed when testing the same material on various rigs. Results from the Polito-Imperial round robin on steel-on-steel specimens [209] showed variations within a 15% range. Generally, for flat-on-flat interfaces in steel-on-steel and other materials used in turbomachinery, $\mu$ ranges from 0.4 to 0.8 at room temperature, with most values falling within the [0.6–0.7] range [198,201,202,210]. At temperatures exceeding 200 °, $\mu$ decreases to [0.2–0.3] [200,201,211]. No major influence of parameters such as worn area of contact or normal load have been observed.

* **Contact stiffness.** The $k_t$ values are much more scattered, with discrepancies up to 70% between the same experiment on different rigs [209]. Nevertheless, a dependency on the worn area $A_w$, rather than the commonly practiced nominal area, has been evidenced in [209] for flat-on-flat contacts. This relation has been here confirmed using the results published in [204], as evidenced in Fig. 8. A further confirmation of this dependence is given by the fact that $k_t$ was observed to gradually increase during the test due to the expanding worn contact area. Normalised contact stiffness $k_t/A_w$, is notably lower in hysteresis loops with macroslip than in fully stuck hysteresis loops measured at lower displacement amplitudes. This decrease in loops with macroslip is attributed to a reduced time for the ageing of asperity contacts at higher velocities, indicating a velocity dependency [209].

* **Harmonic variation of normal load.** The use of a harmonic normal force results in non-uniform energy dissipation, leading to a rougher worn surface, while under constant normal force, the worn surface tends to flatten. This variation also influences the evolution of contact parameters, which exhibit a continuous
purpose [206]. For more repeatable tests, a rigid setup is preferred, while a self-aligning system is recommended for testing complex and larger interfaces. An essential consideration highlighted by various research groups is the design of flat-on-flat contact interfaces. Both the mobile and fixed contact surfaces should have the same width along the sliding motion direction. Failure to meet this condition may result in the shorter specimen wearing a groove within the wider one, leading to a hardening effect in the hysteresis [203].

**Tangential excitation** is crucial for inducing relative motion between specimens and, consequently, generating friction force. Closed-loop actuation mechanisms, whether force or displacement-based, are commonly employed. The shaker is the preferred choice for high-frequency measurements; however, careful dynamic design of the test rig structure and its joints must be consistently conducted to ensure that the resonance frequencies of the rig are well above those of the measurements.

**The application of the normal load N** is intricately tied to the test rig's purpose. Stable application of a constant normal load, with built-in measurement capability, can be achieved using actuators or simple deadweights. Rigs operating under a constant normal load, with the goal of attaining high contact pressure, often opt for pneumatic or hydraulic actuators [201,202]. For rigs designed to apply a normal load varying harmonically around a mean value, a piezo actuator is the preferred choice [204,205].

**Accurately measuring the contact force T** (and N in [204,205]) involves not only selecting suitable force sensors but also ensuring a smooth force transfer, while preventing potential compensation or misalignment issues arising from the test rig structure. Piezo-based force transducers are the preferred choice for test rigs aiming to achieve higher frequency measurements. These sensors are affected by the volatility of the electric charge, posing challenges in attaining long-term stable measurements. This is not a limiting factor if the goal of the measurement is related to assessing properties such as contact stiffness or dissipated energy (i.e. area of the cycle), as an incorrect assessment of the mean value of T would not change the result. The correct assessment of the mean value of T becomes essential when considering the presence of multiple static equilibria under the same nominal conditions, as further discussed in Section 6.2.1. Measurement of the normal force N is accomplished through a strain force transducer. This selection is justified by the superior long-term stability of strain force sensors compared to piezo-based force transducers. This stability is crucial for accurately capturing the stable mean force value. It is important to note that the softer structure of the strain force transducer comes with a limitation in suitability for high-frequency measurements. The force-transfer mechanism from the contact to the load cell(s) is another crucial aspect, as discussed previously. Various structural approaches, such as elastic beams, three-claw stabilisers, and tripods, have been employed to ensure high stiffness, perfect alignment, and the transfer of forces without introducing spurious components.

**Measuring the relative displacement at the contact x** is crucial for accurately capturing hysteresis and subsequently estimating the tangential contact stiffness $k_t$. There appears to be a consensus among various research groups favouring techniques such as Laser Doppler Vibrometers (LDV) or Digital Image Correlation (DIC). Both methods ensure an adequate resolution when measuring within a low displacement range, i.e. [0.1–2] µm, typically corresponding to a fully stuck condition, a prerequisite for estimating tangential contact stiffness. The necessity for high-precision measurement techniques is underscored by a comparison between DIC and a linear variable differential transformer in [202]. Additionally, the definition of “relative displacement” itself is noteworthy. In certain test rigs, only the displacement of the moving specimen is measured, assuming the compliance of the test rig connected to the fixed specimen is negligible. While this approach may be suitable for specific test rig architectures, it is not universally applicable, as demonstrated in [202]. Opting for a differential measurement, either with DIC or using two LDVs positioned as close as possible to the contact, appears to be the safest choice.

**The effect of wear** has been observed in all test rigs, to various degrees. In detail, hysteresis properties exhibit a significant dependence on overall slipping time, closely tied to the wear of the contact zones. Some research groups incorporate within their measurement protocol, besides $\mu$ and $k_t$, which can be recorded as a function of the number of cycles, other wear-related parameters, e.g. the possibility of measuring the wear volume history of the same pair of specimens with a negligible perturbation to the wear processes [203].

**The influence of frequency** does not appear to be significant, particularly within the 20–100 Hz range. While higher frequency measurements offer advantages for assessing wear, they also bring about a greater influence of the test rig itself. As the frequency increases, the signals tend to become more disrupted, posing challenges for accurate evaluation.

**Real time contact state detection** is a recent focus of research in the field of single interface testing. Recent works have used ultrasound [207] and total internal reflection [208] to detect in real time the effective contact area and contact state of non-transparent and transparent samples respectively using the test rig detailed in [201].

### 5.1.2. An overview of published results

Several rigs have been designed, constructed, and used over the years to measure hysteresis cycles of friction contacts. A quantitative comparison of results is not always possible due to the large variety of test conditions (contact interface geometry, materials, temperature, etc.), and the fact that the influence of each variable on the contact parameters is not known with precision. Nevertheless, a number of common features and trends have been observed and are summarised below:

* **Hysteresis Shape.** A positive slope of the macroslip portions of the hysteresis cycles depicted in Fig. 7 (unexplained by a Coulomb-type friction model) is often observed. Furthermore, the macroslip portion of the cycles may exhibit fluctuations, which stem from interactions among wear scars, often traceable to a ridge being formed at each end of a multi-cycle test run [198].

* **Friction Coefficient.** This parameter exhibits remarkable stability, as consistent outcomes are observed when testing the same material on various rigs. Results from the Polito-Imperial round robin on steel-on-steel specimens [209] showed variations within a 15% range. Generally, for flat-on-flat interfaces in steel-on-steel and other materials used in turbomachinery, $\mu$ ranges from 0.4 to 0.8 at room temperature, with most values falling within the [0.6–0.7] range [198,201,202,210]. At temperatures exceeding 200 °, $\mu$ decreases to [0.2–0.3] [200,201,211]. No major influence of parameters such as worn area of contact or normal load have been observed.

* **Contact stiffness.** The $k_t$ values are much more scattered, with discrepancies up to 70% between the same experiment on different rigs [209]. Nevertheless, a dependency on the worn area $A_w$, rather than the commonly practiced nominal area, has been evidenced in [209] for flat-on-flat contacts. This relation has been here confirmed using the results published in [204], as evidenced in Fig. 8. A further confirmation of this dependence is given by the fact that $k_t$ was observed to gradually increase during the test due to the expanding worn contact area. Normalised contact stiffness $k_t/A_w$, is notably lower in hysteresis loops with macroslip than in fully stuck hysteresis loops measured at lower displacement amplitudes. This decrease in loops with macroslip is attributed to a reduced time for the ageing of asperity contacts at higher velocities, indicating a velocity dependency [209].

* **Harmonic variation of normal load.** The use of a harmonic normal force results in non-uniform energy dissipation, leading to a rougher worn surface, while under constant normal force, the worn surface tends to flatten. This variation also influences the evolution of contact parameters, which exhibit a continuous
<figure><img src="image_11.png" /><figcaption>Fig. 8. Tangential contact stiffness $k_t$ normalised by the worn area $A_w$ versus $N/A_w$.</figcaption></figure>

Source: Adapted from [209] with the addition of results from [204].

<caption>**Table 2**</caption>

Main contributions found in the literature on experimental characterisation of friction damping devices for turbine disks.

<table><thead><tr><td>Friction damping device</td><td>References</td></tr></thead><tbody><tr><td>Blade root</td><td>[9,12,75,76,78,112,213–222]</td></tr><tr><td>Underplatform damper</td><td>[63,88,216,223–232]<br>[62,64,79,212,233–243]</td></tr><tr><td>Mid-span damper</td><td>[244–246]</td></tr><tr><td>Tip damper</td><td>[247,248]</td></tr><tr><td>Blade shroud</td><td>[87,249–259]</td></tr><tr><td>Ring damper</td><td>[260]</td></tr></tbody></table>

evolution due to the fluctuations in interface conformity [205]. While the normal load is typically constant for shrouds and blade roots when the rotational speed of the disk is constant, this investigation is particularly relevant in the case of dampers [212]. Here, the inclined contact surface, combined with the damper kinematics, produces a harmonic variation of both tangential and normal contact forces even at constant disk rotational speed.

## 5.2. Friction damping devices

This section is devoted to test setups designed to replicate, in a controlled environment, the dynamics and/or mechanics of gas turbine components. Test setups that include friction damping elements such as dampers and shrouds are commonly employed to validate numerical models or assess the effectiveness of a particular damping device. Table 2 provides an overview of key references detailing the construction of these test setups and the corresponding outcomes. The references are categorised based on the type of friction damping device investigated by the test rig.

### 5.2.1. Common features and challenges

While the forthcoming sections will highlight variations, these test setups share common objectives:

* to ensure normal preload at the contact;

* to provide excitation and/or enforce appropriate kinematics;

* to measure relevant quantities, either on the blade(s) and/or at the contact.

The execution of these tasks may differ based on the test rig configuration, whether rotating or non-rotating, leading to distinct challenges. Several examples for each task are discussed below.<fnref n="3" />

* **Normal preload at the contact** may be provided by the centrifugal effect in rotating test rigs, as in real working conditions, but often the scale of laboratory testing in terms of disk dimension and rotational speed results in contact pressures which are below those actually experienced in operating conditions. Non rotating test rigs provide the contact preload often using deadweights; in this case the resulting pressure distribution may be different from the one provided by centrifugal load.

* **External excitation** is typically provided in non-rotating test setups utilising a shaker. This choice offers flexibility to explore a broad range of excitation levels from full stick to gross slip. Despite its widespread adoption, the shaker is not free from drawbacks. Since the shaker needs to be mechanically attached to the structure, it is almost impossible to avoid some sort of interaction between them. Depending on a number of factors such as the location and kind of stinger, excitation level etc., the structure under scrutiny may physically push back against the shaker attachment, producing non-fundamental harmonic content in the force measured by the load cell, entailing the need for specific shaker models at the validation stage [261,262]. Another important point to consider when selecting the test rig configuration is the mode shape of the structure. The kinematics in a non-rotating setup, where a complete disk is not represented (e.g., two blades with an interposed damper), are limited to pure In-Phase and Out-of-Phase modes. While this does not constitute a limitation if the purpose of the test rig is validation, it does prevent the direct assessment of the friction damping performance in an extensive set of working conditions. In contrast, classical rotating rigs often employ air jets on blades for excitation. While this may not induce a full-slip situation in all cases, it allows for exploring additional inter-blade phase angles [263]. The preferred choice in newly designed rotating set-ups is piezoelectric actuators, as in [264], which, compared to air-jets ensure a better control on the amplitude of excitation. An additional, less common test setup, comprises a complete disk in a non-rotating configuration,

<footnote n="3">³ This discussion is not exhaustive, and a comprehensive overview is provided in subsequent sections for each specific damping device.</footnote>
<figure><img src="image_11.png" /><figcaption>Fig. 8. Tangential contact stiffness $k_t$ normalised by the worn area $A_w$ versus $N/A_w$.</figcaption></figure>

Source: Adapted from [209] with the addition of results from [204].

<caption>**Table 2**</caption>

Main contributions found in the literature on experimental characterisation of friction damping devices for turbine disks.

<table><thead><tr><td>Friction damping device</td><td>References</td></tr></thead><tbody><tr><td>Blade root</td><td>[9,12,75,76,78,112,213–222]</td></tr><tr><td>Underplatform damper</td><td>[63,88,216,223–232]<br>[62,64,79,212,233–243]</td></tr><tr><td>Mid-span damper</td><td>[244–246]</td></tr><tr><td>Tip damper</td><td>[247,248]</td></tr><tr><td>Blade shroud</td><td>[87,249–259]</td></tr><tr><td>Ring damper</td><td>[260]</td></tr></tbody></table>

evolution due to the fluctuations in interface conformity [205]. While the normal load is typically constant for shrouds and blade roots when the rotational speed of the disk is constant, this investigation is particularly relevant in the case of dampers [212]. Here, the inclined contact surface, combined with the damper kinematics, produces a harmonic variation of both tangential and normal contact forces even at constant disk rotational speed.

## 5.2. Friction damping devices

This section is devoted to test setups designed to replicate, in a controlled environment, the dynamics and/or mechanics of gas turbine components. Test setups that include friction damping elements such as dampers and shrouds are commonly employed to validate numerical models or assess the effectiveness of a particular damping device. Table 2 provides an overview of key references detailing the construction of these test setups and the corresponding outcomes. The references are categorised based on the type of friction damping device investigated by the test rig.

### 5.2.1. Common features and challenges

While the forthcoming sections will highlight variations, these test setups share common objectives:

* to ensure normal preload at the contact;

* to provide excitation and/or enforce appropriate kinematics;

* to measure relevant quantities, either on the blade(s) and/or at the contact.

The execution of these tasks may differ based on the test rig configuration, whether rotating or non-rotating, leading to distinct challenges. Several examples for each task are discussed below.<fnref n="3" />

* **Normal preload at the contact** may be provided by the centrifugal effect in rotating test rigs, as in real working conditions, but often the scale of laboratory testing in terms of disk dimension and rotational speed results in contact pressures which are below those actually experienced in operating conditions. Non rotating test rigs provide the contact preload often using deadweights; in this case the resulting pressure distribution may be different from the one provided by centrifugal load.

* **External excitation** is typically provided in non-rotating test setups utilising a shaker. This choice offers flexibility to explore a broad range of excitation levels from full stick to gross slip. Despite its widespread adoption, the shaker is not free from drawbacks. Since the shaker needs to be mechanically attached to the structure, it is almost impossible to avoid some sort of interaction between them. Depending on a number of factors such as the location and kind of stinger, excitation level etc., the structure under scrutiny may physically push back against the shaker attachment, producing non-fundamental harmonic content in the force measured by the load cell, entailing the need for specific shaker models at the validation stage [261,262]. Another important point to consider when selecting the test rig configuration is the mode shape of the structure. The kinematics in a non-rotating setup, where a complete disk is not represented (e.g., two blades with an interposed damper), are limited to pure In-Phase and Out-of-Phase modes. While this does not constitute a limitation if the purpose of the test rig is validation, it does prevent the direct assessment of the friction damping performance in an extensive set of working conditions. In contrast, classical rotating rigs often employ air jets on blades for excitation. While this may not induce a full-slip situation in all cases, it allows for exploring additional inter-blade phase angles [263]. The preferred choice in newly designed rotating set-ups is piezoelectric actuators, as in [264], which, compared to air-jets ensure a better control on the amplitude of excitation. An additional, less common test setup, comprises a complete disk in a non-rotating configuration,

<footnote n="3">³ This discussion is not exhaustive, and a comprehensive overview is provided in subsequent sections for each specific damping device.</footnote>
<figure><img src="image_12.png" /><figcaption>**Fig. 9.** Test set-ups used to investigate blade root damping. (a) Rotating configuration from [112]. (b) Non-rotating single blade configurations from [78] (preload from loading screw) and [265] (preload from thermal shrinkage). (c) Non-rotating double blade configurations from [213,214].</figcaption></figure>

<table><caption>**Table 3**
Features of test set-ups focusing on blade root damping.</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>[12]</td><td>Non-rotating</td><td>Clamps</td><td>Vibrating table</td><td>Dissipated Energy</td></tr><tr><td>[75]</td><td>Non-rotating</td><td>Spring-loaded wires</td><td>Shaker</td><td>Blade dynamic response<br>η</td></tr><tr><td>[76]</td><td>Non-rotating</td><td>Loading screw</td><td>Vibrating table</td><td>Blade dynamic response<br>η</td></tr><tr><td>[213]</td><td>Non-rotating</td><td>Thermal Shrinkage</td><td>Strikers</td><td>Blade dynamic response<br>δ</td></tr><tr><td>[214]</td><td>Non-rotating</td><td>Hydraulic cylinder</td><td>Impulse Hammer<br>Resonance Electromagnet</td><td>Resonance frequency<br>η, Q</td></tr><tr><td>[112]</td><td>Rotating</td><td>Centrifugal effect</td><td>Piezo actuator</td><td>Dynamic response<br>ζ</td></tr><tr><td>[216]</td><td>Rotating</td><td>Centrifugal effect</td><td>Piezoelectric exciters</td><td>Dynamic response<br>ζ</td></tr><tr><td>[78]</td><td>Non-rotating</td><td>Loading screw</td><td>Shaker Platform</td><td>Dynamic response<br>ζ</td></tr><tr><td>[265]</td><td>Non-rotating</td><td>Thermal Shrinkage</td><td>Shaker</td><td>Dynamic response<br>η</td></tr><tr><td>[217]</td><td>Non-rotating</td><td>Hydraulic cylinder</td><td>Hydraulic cylinder</td><td>Load-Stress curve</td></tr><tr><td>[218]</td><td>Non-rotating</td><td>Hydraulic cylinder</td><td>Impulse Hammer<br>Resonance Electromagnet</td><td>Dynamic response</td></tr><tr><td>[219,220]</td><td>Non-rotating</td><td>Loading screw</td><td>Impulse Hammer<br>Shaker</td><td>Dynamic response</td></tr><tr><td>[80]</td><td>Non-rotating</td><td>Hydraulic cylinder</td><td>Shaker</td><td>Contact parameters<br>η</td></tr><tr><td>[221]</td><td>Non-rotating</td><td>Hydraulic cylinder</td><td>Shaker + Electromagnet</td><td>ζ</td></tr><tr><td>[222]</td><td>Non-rotating</td><td>Hydraulic cylinder</td><td>Shaker + Electromagnet</td><td>ζ</td></tr></tbody></table>

where the excitation is given by electromagnets, generating a travelling wave.

* **Measurement of relevant quantities** includes the dynamic response of the structure and, less frequently, the force-displacement hysteresis at the contact. Dynamic response measurements are easily obtained with accelerometers or laser scanning in non-rotating setups. Rotating rigs, often placed in spin-pits under vacuum, are less accessible, making tip-timing and strain gauge measurements more common. Force-displacement hysteresis measurements are currently feasible only in non-rotating rigs.

### 5.2.2. Blade root

Table 3 summarises the main features of all the test set-ups designed to investigate blade root damping. Three main architectures can be identified and are represented in Fig. 9 and are closely connected with

the strategy used to ensure the preload at the contact. In rotating set-ups [112,216] the load is guaranteed, as in real conditions, by the centrifugal effect. Non-rotating test set-ups can be further differentiated into single or double blade configurations. Statically pulling a single blade does introduce an additional source of damping due to the contact between the blade and the pushing/pulling device. One strategy involves the minimisation of this damping contribution: in single blade test set-ups the preload may be guaranteed by thin pulling wires loaded with springs [75] in an effort to avoid the disruption of the blade root motion during slip, or by loading screws. Such screws are equipped with strain-gauges [76,78] or load cells [219,220] to measure the load. The load is transferred to the test piece via a small diameter bearing to obtain a uniform load distribution. An alternative non-rotating architecture comprises a dummy blade where a root attachment is machined at both ends thus creating a symmetric arrangement [80,214,217,221] or two blades welded at the tip [213]. The symmetry of the architecture, the equivalence of loads and joint kinematics ensures a load application
<figure><img src="image_12.png" /><figcaption>**Fig. 9.** Test set-ups used to investigate blade root damping. (a) Rotating configuration from [112]. (b) Non-rotating single blade configurations from [78] (preload from loading screw) and [265] (preload from thermal shrinkage). (c) Non-rotating double blade configurations from [213,214].</figcaption></figure>

<table><caption>**Table 3**
Features of test set-ups focusing on blade root damping.</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>[12]</td><td>Non-rotating</td><td>Clamps</td><td>Vibrating table</td><td>Dissipated Energy</td></tr><tr><td>[75]</td><td>Non-rotating</td><td>Spring-loaded wires</td><td>Shaker</td><td>Blade dynamic response η</td></tr><tr><td>[76]</td><td>Non-rotating</td><td>Loading screw</td><td>Vibrating table</td><td>Blade dynamic response η</td></tr><tr><td>[213]</td><td>Non-rotating</td><td>Thermal Shrinkage</td><td>Strikers</td><td>Blade dynamic response δ</td></tr><tr><td>[214]</td><td>Non-rotating</td><td>Hydraulic cylinder</td><td>Impulse Hammer<br>Resonance Electromagnet</td><td>Resonance frequency η, Q</td></tr><tr><td>[112]</td><td>Rotating</td><td>Centrifugal effect</td><td>Piezo actuator</td><td>Dynamic response ζ</td></tr><tr><td>[216]</td><td>Rotating</td><td>Centrifugal effect</td><td>Piezoelectric exciters</td><td>Dynamic response ζ</td></tr><tr><td>[78]</td><td>Non-rotating</td><td>Loading screw</td><td>Shaker Platform</td><td>Dynamic response ζ</td></tr><tr><td>[265]</td><td>Non-rotating</td><td>Thermal Shrinkage</td><td>Shaker</td><td>Dynamic response η</td></tr><tr><td>[217]</td><td>Non-rotating</td><td>Hydraulic cylinder</td><td>Hydraulic cylinder</td><td>Load-Stress curve</td></tr><tr><td>[218]</td><td>Non-rotating</td><td>Hydraulic cylinder</td><td>Impulse Hammer<br>Resonance Electromagnet</td><td>Dynamic response</td></tr><tr><td>[219,220]</td><td>Non-rotating</td><td>Loading screw</td><td>Impulse Hammer<br>Shaker</td><td>Dynamic response</td></tr><tr><td>[80]</td><td>Non-rotating</td><td>Hydraulic cylinder</td><td>Shaker</td><td>Contact parameters η</td></tr><tr><td>[221]</td><td>Non-rotating</td><td>Hydraulic cylinder</td><td>Shaker + Electromagnet</td><td>Dynamic response ζ</td></tr><tr><td>[222]</td><td>Non-rotating</td><td>Hydraulic cylinder</td><td>Shaker + Electromagnet</td><td>Dynamic response ζ</td></tr></tbody></table>

where the excitation is given by electromagnets, generating a travelling wave.

* **Measurement of relevant quantities** includes the dynamic response of the structure and, less frequently, the force-displacement hysteresis at the contact. Dynamic response measurements are easily obtained with accelerometers or laser scanning in non-rotating setups. Rotating rigs, often placed in spin-pits under vacuum, are less accessible, making tip-timing and strain gauge measurements more common. Force-displacement hysteresis measurements are currently feasible only in non-rotating rigs.

### 5.2.2. Blade root

Table 3 summarises the main features of all the test set-ups designed to investigate blade root damping. Three main architectures can be identified and are represented in Fig. 9 and are closely connected with

the strategy used to ensure the preload at the contact. In rotating set-ups [112,216] the load is guaranteed, as in real conditions, by the centrifugal effect. Non-rotating test set-ups can be further differentiated into single or double blade configurations. Statically pulling a single blade does introduce an additional source of damping due to the contact between the blade and the pushing/pulling device. One strategy involves the minimisation of this damping contribution: in single blade test set-ups the preload may be guaranteed by thin pulling wires loaded with springs [75] in an effort to avoid the disruption of the blade root motion during slip, or by loading screws. Such screws are equipped with strain-gauges [76,78] or load cells [219,220] to measure the load. The load is transferred to the test piece via a small diameter bearing to obtain a uniform load distribution. An alternative non-rotating architecture comprises a dummy blade where a root attachment is machined at both ends thus creating a symmetric arrangement [80,214,217,221] or two blades welded at the tip [213]. The symmetry of the architecture, the equivalence of loads and joint kinematics ensures a load application
very similar to the real operating conditions. In this case, preload is typically obtained using a hydraulic cylinder. An alternative idea, valid for both single [265] and double-blade [213] arrangements, involves providing an initial mismatch between the contact surfaces at the blade root that is removed by thermal expansion. The preload is then obtained through thermal shrinkage.

Two main kinds of excitation can be found in these test rigs:

* Harmonic/resonance excitation. This excitation is typically provided through a shaker or using a vibrating table. In the second case there is a reduced risk of introducing an additional damping/stiffness contribution through the shaker attachment. Other non-contact methods include electromagnets. In rotating set-ups the preferred choice is piezoelectric actuators.

* Impulse excitation, i.e. hammer test, is typically employed when evaluating free decay response. One limitation of hammer tests is that the maximum achievable level of excitation is typically lower than that obtained with other excitation devices [214]. Depending on the specific structure, the preload at the contact and the direction of excitation, such excitation levels may [266,267] or may not [214] be sufficient to trigger slip of the interfaces. A new excitation system presented in [222], proposes the use of power-to-release and classical electromagnets to detach a shaker and pushrod from the specimen after reaching steady-state high frequency vibration.

All friction joints modify the dynamic response of the system. They are part of/attached to through two contributions [218,221]:

* they introduce an additional stiffness, which is maximum in full stick regime and decreases as the joint starts to slip

* they introduce damping, which is null in full stick regime and increases as the joint starts to slip

Given the position of the blade root joint, the first contribution is less significant and often masked by the stiffening effect of the centrifugal load on the blade. Blade root damping, on the other hand, has been recognised since the 1940s [22,23] and represents the target measured quantity of most of the test set-ups detailed in Table 3. The results were initially given in terms of empirical observation, expressed as logarithmic decrement $\delta$, in order to be easily compared with other sources of damping, such as material damping and aerodynamic damping (or excitation) [24] and/or to include the presence of the root joint in the damping matrix C without the need for modelling the contact. With time, this and other damping measures were introduced and related to relevant design quantities of the blade root such as stress at key locations, centrifugal load, root geometry, temperature, etc. As of today, there is no unified quantity to represent damping. One of the most common measures is the loss factor $\eta$ which is defined as the ratio of energy dissipated in unit volume per radian of oscillation to the maximum strain energy per unit volume. The loss factor $\eta$ may be obtained through the half power method as shown in Fig. 10. In order to clarify the contribution of each work detailed in Table 3, the relation between $\eta$ and the various damping-related quantities is given below:

$$ \eta \approx \frac{\Delta f}{f_n} \approx \frac{\delta}{\pi} \approx 2\zeta \approx \frac{1}{Q} \qquad (3) $$

where $\delta$ is the logarithmic decrement damping, $\zeta$ is the damping ratio and $Q$ is the quality or Q-factor. It should be noted that Eq. (3) is valid within a 5% error for $0 < \eta < 0.3$ [268], while more complex relations, e.g. $\eta = 2\zeta\sqrt{1-\zeta^2}$, hold outside of that range. As will be apparent from Fig. 10, the experimental set-up and its exciting source may guide the way results are presented. As an example, test rigs with harmonic excitation [75,76], typically rely on the half-power method to estimate damping ($\eta$), while those test set-ups which use impulse excitation/vibratory decay tests resort to logarithmic decrement $\delta$ computation [213] or to the Hilbert Transform to estimate $\eta$ [214].

Nevertheless, Eq. (3) may be used to offer an estimate of the quantities which have not been directly measured.

In [75,76], the effect of slip at the root on the dynamic response of blades was quantified through the loss factor $\eta$. A compressor and a fan blade, both equipped with a dovetail root were investigated respectively. In both cases various experiments were conducted by adjusting the centrifugal loads to replicate scenarios ranging from a completely clamped blade (high load, no slip at the root) to situations with decreasing loads that induced different degrees of slip. Similar conclusions which can be summarised below were drawn:

1. Loss factors decrease rapidly with an increase in centrifugal load.

2. Higher levels of excitation result in greater friction damping for low centrifugal loads, but this dependence diminishes with increased centrifugal loads.

3. Loss factors plateau at moderate centrifugal loads, gradually approaching non-friction damping levels, indicating a tendency for minimal slip at the root.

Confirmation of points 1–3 can be found in [214,265], where $\eta$ is plotted against vibration amplitude for varying centrifugal loads. Further support for point 2 is presented in [216] and in [213], where a nearly linear relationship between the logarithmic decrement $\delta$ and vibration amplitude at the blade tip is evidenced. However, recent experiments [221] demonstrate the damping-dynamic displacement/excitation relationship over a broader displacement range. For low excitation levels corresponding to the full stick condition, damping remains approximately constant and aligns with the material's intrinsic damping $\eta_0$, as corroborated in [78,216], which compared $\eta$ and $\zeta$ values for bonded and un-bonded root joints. As the root starts to slip, damping increases to a maximum with rising displacement, consistent with observations in [75,76], but subsequently asymptotically decreases to $\eta_0$. It is noteworthy that this decrease corresponds to joint interfaces in advanced gross-slip, a condition rarely encountered in practical working conditions.

Other relevant measured quantities include the stress on a key location on the disk sector [217] or at the (dummy) blade root section [78], and the friction-dissipated energy [12]. Regardless of the measured quantity, several works try to replicate measured results with a model which includes friction. All contact models, as detailed in Section 5.1, require calibration parameters, i.e. friction coefficient $\mu$ and, when implemented, normal and tangential contact stiffness $k_n$ and $k_t$. Such values are either tuned on the basis of the experimental results [75,76], identified with a purposely developed numerical technique which compares measured frequency and damping with those obtained from a FE model [80], or measured through a separate contact fretting test set-up replicating the material and geometry of a single lobe of a blade root joint [12,78].

### 5.2.3. Blade friction damper

Table 4 summarises the main features of all the test setups designed to investigate blade friction dampers, here defined as separate objects, unlike shrouds or roots, which are integral to the blade. Depending on their position, they may be classified as ‘underplatform’, ‘mid-span’, or ‘tip’ dampers. Nevertheless, the features of the test rigs used to investigate their performance can be applied to all these types of dampers. Test rig architectures can be subdivided into rotating and non-rotating. Non rotating rigs can be further classified depending on the number of blades employed. A total of five main test rig architectures can be identified in Table 4 and are represented in Fig. 12:

* Two Blades (2B): by far the most common non-rotating test rig architecture, comprising two blades with an interposed damper.

* Single Blade (1B): a single blade is in contact with (i) a ground damper, (ii) two dampers, one on each side of the blade. The other side of the dampers is in contact with a stiffer structure called a dummy blade/platform, which may be equipped with force sensors to measure contact forces, as in [238].
very similar to the real operating conditions. In this case, preload is typically obtained using a hydraulic cylinder. An alternative idea, valid for both single [265] and double-blade [213] arrangements, involves providing an initial mismatch between the contact surfaces at the blade root that is removed by thermal expansion. The preload is then obtained through thermal shrinkage.

Two main kinds of excitation can be found in these test rigs:

* Harmonic/resonance excitation. This excitation is typically provided through a shaker or using a vibrating table. In the second case there is a reduced risk of introducing an additional damping/stiffness contribution through the shaker attachment. Other non-contact methods include electromagnets. In rotating set-ups the preferred choice is piezoelectric actuators.

* Impulse excitation, i.e. hammer test, is typically employed when evaluating free decay response. One limitation of hammer tests is that the maximum achievable level of excitation is typically lower than that obtained with other excitation devices [214]. Depending on the specific structure, the preload at the contact and the direction of excitation, such excitation levels may [266,267] or may not [214] be sufficient to trigger slip of the interfaces. A new excitation system presented in [222], proposes the use of power-to-release and classical electromagnets to detach a shaker and pushrod from the specimen after reaching steady-state high frequency vibration.

All friction joints modify the dynamic response of the system. They are part of/attached to through two contributions [218,221]:

* they introduce an additional stiffness, which is maximum in full stick regime and decreases as the joint starts to slip

* they introduce damping, which is null in full stick regime and increases as the joint starts to slip

Given the position of the blade root joint, the first contribution is less significant and often masked by the stiffening effect of the centrifugal load on the blade. Blade root damping, on the other hand, has been recognised since the 1940s [22,23] and represents the target measured quantity of most of the test set-ups detailed in Table 3. The results were initially given in terms of empirical observation, expressed as logarithmic decrement $\delta$, in order to be easily compared with other sources of damping, such as material damping and aerodynamic damping (or excitation) [24] and/or to include the presence of the root joint in the damping matrix C without the need for modelling the contact. With time, this and other damping measures were introduced and related to relevant design quantities of the blade root such as stress at key locations, centrifugal load, root geometry, temperature, etc. As of today, there is no unified quantity to represent damping. One of the most common measures is the loss factor $\eta$ which is defined as the ratio of energy dissipated in unit volume per radian of oscillation to the maximum strain energy per unit volume. The loss factor $\eta$ may be obtained through the half power method as shown in Fig. 10. In order to clarify the contribution of each work detailed in Table 3, the relation between $\eta$ and the various damping-related quantities is given below:

$$ \eta \approx \frac{\Delta f}{f_n} \approx \frac{\delta}{\pi} \approx 2\zeta \approx \frac{1}{Q} \qquad (3) $$

where $\delta$ is the logarithmic decrement damping, $\zeta$ is the damping ratio and $Q$ is the quality or Q-factor. It should be noted that Eq. (3) is valid within a 5% error for $0 < \eta < 0.3$ [268], while more complex relations, e.g. $\eta = 2\zeta\sqrt{1 - \zeta^2}$, hold outside of that range. As will be apparent from Fig. 10, the experimental set-up and its exciting source may guide the way results are presented. As an example, test rigs with harmonic excitation [75,76], typically rely on the half-power method to estimate damping ($\eta$), while those test set-ups which use impulse excitation/vibratory decay tests resort to logarithmic decrement $\delta$ computation [213] or to the Hilbert Transform to estimate $\eta$ [214].

Nevertheless, Eq. (3) may be used to offer an estimate of the quantities which have not been directly measured.

In [75,76], the effect of slip at the root on the dynamic response of blades was quantified through the loss factor $\eta$. A compressor and a fan blade, both equipped with a dovetail root were investigated respectively. In both cases various experiments were conducted by adjusting the centrifugal loads to replicate scenarios ranging from a completely clamped blade (high load, no slip at the root) to situations with decreasing loads that induced different degrees of slip. Similar conclusions which can be summarised below were drawn:

1. Loss factors decrease rapidly with an increase in centrifugal load.

2. Higher levels of excitation result in greater friction damping for low centrifugal loads, but this dependence diminishes with increased centrifugal loads.

3. Loss factors plateau at moderate centrifugal loads, gradually approaching non-friction damping levels, indicating a tendency for minimal slip at the root.

Confirmation of points 1–3 can be found in [214,265], where $\eta$ is plotted against vibration amplitude for varying centrifugal loads. Further support for point 2 is presented in [216] and in [213], where a nearly linear relationship between the logarithmic decrement $\delta$ and vibration amplitude at the blade tip is evidenced. However, recent experiments [221] demonstrate the damping-dynamic displacement/excitation relationship over a broader displacement range. For low excitation levels corresponding to the full stick condition, damping remains approximately constant and aligns with the material’s intrinsic damping $\eta_0$, as corroborated in [78,216], which compared $\eta$ and $\zeta$ values for bonded and un-bonded root joints. As the root starts to slip, damping increases to a maximum with rising displacement, consistent with observations in [75,76], but subsequently asymptotically decreases to $\eta_0$. It is noteworthy that this decrease corresponds to joint interfaces in advanced gross-slip, a condition rarely encountered in practical working conditions.

Other relevant measured quantities include the stress on a key location on the disk sector [217] or at the (dummy) blade root section [78], and the friction-dissipated energy [12]. Regardless of the measured quantity, several works try to replicate measured results with a model which includes friction. All contact models, as detailed in Section 5.1, require calibration parameters, i.e. friction coefficient $\mu$ and, when implemented, normal and tangential contact stiffness $k_n$ and $k_t$. Such values are either tuned on the basis of the experimental results [75,76], identified with a purposely developed numerical technique which compares measured frequency and damping with those obtained from a FE model [80], or measured through a separate contact fretting test set-up replicating the material and geometry of a single lobe of a blade root joint [12,78].

### 5.2.3. Blade friction damper

Table 4 summarises the main features of all the test setups designed to investigate blade friction dampers, here defined as separate objects, unlike shrouds or roots, which are integral to the blade. Depending on their position, they may be classified as ‘underplatform’, ‘mid-span’, or ‘tip’ dampers. Nevertheless, the features of the test rigs used to investigate their performance can be applied to all these types of dampers. Test rig architectures can be subdivided into rotating and non-rotating. Non rotating rigs can be further classified depending on the number of blades employed. A total of five main test rig architectures can be identified in Table 4 and are represented in Fig. 12:

* Two Blades (2B): by far the most common non-rotating test rig architecture, comprising two blades with an interposed damper.

* Single Blade (1B): a single blade is in contact with (i) a ground damper, (ii) two dampers, one on each side of the blade. The other side of the dampers is in contact with a stiffer structure called a dummy blade/platform, which may be equipped with force sensors to measure contact forces, as in [238].
<figure><img src="image_13.png" /><figcaption>**Fig. 10.** (a) Graphical representation of the half-power method to estimate the loss factor $\eta$ (b) Graphical representation of the logarithmic decrement method to estimate parameter $\delta$.</figcaption></figure>

<table><caption>**Table 4**
Features of test set-ups focusing on blade friction damper.</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>[63,88,225,228,229]<br>[64,234,237,248,269]</td><td>Non-Rotating 2B</td><td>Deadweights</td><td>Shaker</td><td>Dynamic response</td></tr><tr><td>[62]</td><td>Non-Rotating 2B</td><td>Deadweights</td><td>Air jets</td><td>Dynamic response</td></tr><tr><td>[247]</td><td>Non-Rotating 2B</td><td>Deadweights</td><td>Electromagnet</td><td>Free decay</td></tr><tr><td>[240,241]</td><td>Non-Rotating 2B</td><td>Compressed Air</td><td>Shaker</td><td>Dynamic response</td></tr><tr><td>[232,236,239]</td><td>Non-Rotating 2B</td><td>Deadweights</td><td>Shaker</td><td>Dynamic response<br>Damper kinematics</td></tr><tr><td>[223]</td><td>Non-Rotating 1B</td><td>Deadweights</td><td>Shaker</td><td>Dynamic response</td></tr><tr><td>[227]</td><td>Non-Rotating 1B</td><td>Deadweights</td><td>Air jets</td><td>Dynamic response</td></tr><tr><td>[220,238,242,246]</td><td>Non-Rotating 1B</td><td>Deadweights</td><td>Shaker</td><td>Dynamic response<br>Contact forces<br>Damper-plat kinematics</td></tr><tr><td>[79,212,235,270,271]</td><td>Non-rotating 0B</td><td>Deadweights</td><td>Piezo actuators</td><td>Contact forces<br>Damper-plat kinematics</td></tr><tr><td>[233]</td><td>Non-Rotating Disk</td><td>Deadweights</td><td>Magnets</td><td>Dynamic response</td></tr><tr><td>[226]</td><td>Rotating</td><td>Centrifugal Effect</td><td>Air jets</td><td>Dynamic response</td></tr><tr><td>[224,231]</td><td>Rotating</td><td>Centrifugal effect</td><td>Magnets</td><td>Dynamic response</td></tr><tr><td>[216,230]</td><td>Rotating</td><td>Centrifugal effect</td><td>Piezo exciters</td><td>Dynamic response</td></tr><tr><td>[243–245]</td><td>Rotating</td><td>Centrifugal effect</td><td>Acc/Deceleration</td><td>Dynamic response</td></tr></tbody></table>

* No Blade (0B): the damper is in contact with two dummy platforms. One platform is connected to a piezo actuator to reproduce the platform kinematics; the other dummy platform is connected to a set of force sensors to measure contact forces.

* Non-rotating Disk.

* Rotating Disk.

While in rotating rigs (RD), the preload is naturally ensured by the centrifugal effect, non-rotating setups (2B, 1B, 0B) face the challenge of ensuring a realistic contact pressure distribution. For solid dampers (i.e., prismatic dampers), the most common method involves using wires pulled by deadweights. This requires machining holes in the dampers, typically located at their centre of mass, to replicate the pull of the centrifugal load. This methodology, despite its widespread use, may not be representative of real working conditions, as further discussed below. In some cases, to achieve optimal contact pressure aligned with operating conditions without increasing deadweights, the damper or platform contact surface is machined to localise the contact. The impact of the effective contact surface on the results will be further analysed below. Wires and deadweights are impractical for strip dampers. In this case, the preload is guaranteed by a purposely developed compressed air system [269], as illustrated in Fig. 12(a), which loads the strip without any contact, mimicking the effect of the centrifugal force. In this case, a measure of the preload and its distribution is achieved using a calibration system based on a pressure-sensitive film.

The excitation depends on the test rig architecture and its goal. In non-rotating setups, with one or more blades present (1B, 2B), a shaker is the most common choice to provide dynamic excitation. In 2B configurations, typically only one blade is excited and transfers energy to the other blade through the damper and the base. In [241], both blades are equipped with a shaker to ensure that both are equally excited and their motion has equal amplitude. Permanent magnets are used to excite rotating [224,231] and non-rotating [233] disk configurations. This technique, like piezo exciters on rotating configurations, ensures the ability to trigger different Engine Order excitations. Non-rotating 0B configurations employ piezo actuators to input the desired kinematics to one of the dummy platforms. It is important to note that the 0B configuration, unlike all others, does not need dynamic excitation, as the purpose of the rig is to investigate damper mechanics. In this sense, non-rotating 0B configurations share more similarities with single-contact test arrangements as described in Section 5.1. The measured quantities are closely aligned with the two primary objectives of these test rigs, as discussed in Section 5:

* to validate numerical models incorporating dry friction dampers,

* to assess the effectiveness of a specific damper configuration.

Both of these aims can be pursued by considering the dynamics of a bladed arrangement and/or the contact mechanics of the damper. Model validation to some extent is feasible with all the test rigs outlined in this section, as it is possible to replicate the test rig configuration
<figure><img src="image_13.png" /><figcaption>**Fig. 10.** (a) Graphical representation of the half-power method to estimate the loss factor $\eta$ (b) Graphical representation of the logarithmic decrement method to estimate parameter $\delta$.</figcaption></figure>

<table><caption>**Table 4**
Features of test set-ups focusing on blade friction damper.</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>[63,88,225,228,229]<br>[64,234,237,248,269]</td><td>Non-Rotating 2B</td><td>Deadweights</td><td>Shaker</td><td>Dynamic response</td></tr><tr><td>[62]</td><td>Non-Rotating 2B</td><td>Deadweights</td><td>Air jets</td><td>Dynamic response</td></tr><tr><td>[247]</td><td>Non-Rotating 2B</td><td>Deadweights</td><td>Electromagnet</td><td>Free decay</td></tr><tr><td>[240,241]</td><td>Non-Rotating 2B</td><td>Compressed Air</td><td>Shaker</td><td>Dynamic response</td></tr><tr><td>[232,236,239]</td><td>Non-Rotating 2B</td><td>Deadweights</td><td>Shaker</td><td>Dynamic response<br>Damper kinematics</td></tr><tr><td>[223]</td><td>Non-Rotating 1B</td><td>Deadweights</td><td>Shaker</td><td>Dynamic response</td></tr><tr><td>[227]</td><td>Non-Rotating 1B</td><td>Deadweights</td><td>Air jets</td><td>Dynamic response</td></tr><tr><td>[220,238,242,246]</td><td>Non-Rotating 1B</td><td>Deadweights</td><td>Shaker</td><td>Dynamic response<br>Contact forces<br>Damper-plat kinematics</td></tr><tr><td>[79,212,235,270,271]</td><td>Non-rotating 0B</td><td>Deadweights</td><td>Piezo actuators</td><td>Contact forces<br>Damper-plat kinematics</td></tr><tr><td>[233]</td><td>Non-Rotating Disk</td><td>Deadweights</td><td>Magnets</td><td>Dynamic response</td></tr><tr><td>[226]</td><td>Rotating</td><td>Centrifugal Effect</td><td>Air jets</td><td>Dynamic response</td></tr><tr><td>[224,231]</td><td>Rotating</td><td>Centrifugal effect</td><td>Magnets</td><td>Dynamic response</td></tr><tr><td>[216,230]</td><td>Rotating</td><td>Centrifugal effect</td><td>Piezo exciters</td><td>Dynamic response</td></tr><tr><td>[243–245]</td><td>Rotating</td><td>Centrifugal effect</td><td>Acc/Deceleration</td><td>Dynamic response</td></tr></tbody></table>

* No Blade (0B): the damper is in contact with two dummy platforms. One platform is connected to a piezo actuator to reproduce the platform kinematics; the other dummy platform is connected to a set of force sensors to measure contact forces.

* Non-rotating Disk.

* Rotating Disk.

While in rotating rigs (RD), the preload is naturally ensured by the centrifugal effect, non-rotating setups (2B, 1B, 0B) face the challenge of ensuring a realistic contact pressure distribution. For solid dampers (i.e., prismatic dampers), the most common method involves using wires pulled by deadweights. This requires machining holes in the dampers, typically located at their centre of mass, to replicate the pull of the centrifugal load. This methodology, despite its widespread use, may not be representative of real working conditions, as further discussed below. In some cases, to achieve optimal contact pressure aligned with operating conditions without increasing deadweights, the damper or platform contact surface is machined to localise the contact. The impact of the effective contact surface on the results will be further analysed below. Wires and deadweights are impractical for strip dampers. In this case, the preload is guaranteed by a purposely developed compressed air system [269], as illustrated in Fig. 12(a), which loads the strip without any contact, mimicking the effect of the centrifugal force. In this case, a measure of the preload and its distribution is achieved using a calibration system based on a pressure-sensitive film.

The excitation depends on the test rig architecture and its goal. In non-rotating setups, with one or more blades present (1B, 2B), a shaker is the most common choice to provide dynamic excitation. In 2B configurations, typically only one blade is excited and transfers energy to the other blade through the damper and the base. In [241], both blades are equipped with a shaker to ensure that both are equally excited and their motion has equal amplitude. Permanent magnets are used to excite rotating [224,231] and non-rotating [233] disk configurations. This technique, like piezo exciters on rotating configurations, ensures the ability to trigger different Engine Order excitations. Non-rotating 0B configurations employ piezo actuators to input the desired kinematics to one of the dummy platforms. It is important to note that the 0B configuration, unlike all others, does not need dynamic excitation, as the purpose of the rig is to investigate damper mechanics. In this sense, non-rotating 0B configurations share more similarities with single-contact test arrangements as described in Section 5.1. The measured quantities are closely aligned with the two primary objectives of these test rigs, as discussed in Section 5:

* to validate numerical models incorporating dry friction dampers,

* to assess the effectiveness of a specific damper configuration.

Both of these aims can be pursued by considering the dynamics of a bladed arrangement and/or the contact mechanics of the damper. Model validation to some extent is feasible with all the test rigs outlined in this section, as it is possible to replicate the test rig configuration
<figure><img src="image_14.png" /><figcaption>Fig. 11. Damping measurement as a function of (a) centrifugal load [76], (b) amplitude of the response [214], (c) rotational speed [216], and (d) amplitude displacement [221].</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>Fig. 12. Test set-ups used to investigate underplatform dampers (a) Non-rotating Blade-Damper-Blade (BDB) configuration with preload from deadweights-wires system pulling a solid damper [64,229] and with preload from compressed air pushing a strip damper [241] (b) Non-rotating Platform-Damper-Blade (PDB) configuration [242]. (c) Non-Rotating full disk configuration. (d) Non-rotating Platform-Damper-Platform configuration [212] (e) Rotating rig [231].</figcaption></figure>
<figure><img src="image_14.png" /><figcaption>Fig. 11. Damping measurement as a function of (a) centrifugal load [76], (b) amplitude of the response [214], (c) rotational speed [216], and (d) amplitude of displacement [221].</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>Fig. 12. Test set-ups used to investigate underplatform dampers (a) Non-rotating Blade–Damper–Blade (BDB) configuration with preload from deadweights-wires system pulling a solid damper [64,229] and with preload from compressed air pushing a strip damper [241] (b) Non-rotating Platform–Damper–Blade (PDB) configuration [242]. (c) Non-Rotating full disk configuration. (d) Non-rotating Platform–Damper–Platform configuration [212] (e) Rotating rig [231].</figcaption></figure>