**1.2. A quick critical review of contact parameter estimation**

All contact models require knowledge or information of contact-friction parameters to provide meaningful predictions.

Realistic models, based on the integration over the whole contact surface of mechanical principles applied at the asperity level, for example [9], are the only kind which allow for a predictive contact parameter estimation. In other words, these models can be calibrated using information concerning the geometry, roughness distribution, material properties, etc. Unfortunately, at least in many applications such as the turbomachinery field, this level of refinement has not yet been achieved and heuristic models are preferred.

Heuristic models are instead based on phenomenological friction laws (e.g. Coulomb's friction law), and their calibration is based on fitting to empirical observations.

Taking a state-of-the-art macroslip contact model with normal compliance described in Section 1.1, the parameters to be determined are normal and tangential contact stiffness k<sup>n</sup> and k<sup>t</sup> and friction coefficient μ.

The first, and perhaps, most obvious choice, is the use of single-contact test arrangements capable of providing the hysteresis cycle at a given (constant) normal load. Friction coefficients can be easily determined taking the ratio of the limit value of the tangential friction force during slip and the corresponding normal load [5, 25]. Tangential contact stiffness k<sup>t</sup> [2, 25–27] can be estimated by taking the slope of the hysteresis curve in stick condition. This methodology is effective, as it can explore different temperatures, mean normal loads and frequencies. However, as will be shown in the following sections, it may fail to capture the true contact conditions and kinematics, especially for complex multi-interface contacts such as underplatform dampers.

Other methods are available, especially for the determination of contact stiffness values.


account this effect. Unfortunately, a study performed in [32] suggests that this method gives a poor estimation of the contact stiffnesses.

Other ad-hoc elements built to take into account microslip exist [19–24], however they are typically applied to conforming surfaces, which is somewhat limiting, as the kinematics of the contact, which play a significant role in the non-linear dynamic behavior, are not well represented.

All contact models require knowledge or information of contact-friction parameters to pro-

Realistic models, based on the integration over the whole contact surface of mechanical principles applied at the asperity level, for example [9], are the only kind which allow for a predictive contact parameter estimation. In other words, these models can be calibrated using information concerning the geometry, roughness distribution, material properties, etc. Unfortunately, at least in many applications such as the turbomachinery field, this level of

Heuristic models are instead based on phenomenological friction laws (e.g. Coulomb's fric-

Taking a state-of-the-art macroslip contact model with normal compliance described in Section 1.1, the parameters to be determined are normal and tangential contact stiffness k<sup>n</sup>

The first, and perhaps, most obvious choice, is the use of single-contact test arrangements capable of providing the hysteresis cycle at a given (constant) normal load. Friction coefficients can be easily determined taking the ratio of the limit value of the tangential friction force during slip

mated by taking the slope of the hysteresis curve in stick condition. This methodology is effective, as it can explore different temperatures, mean normal loads and frequencies. However, as will be shown in the following sections, it may fail to capture the true contact conditions and kinematics, especially for complex multi-interface contacts such as underplatform dampers.

Other methods are available, especially for the determination of contact stiffness values.

areas (i.e. Hertz theory), thus of limited interest in turbomachinery applications, while the

**2.** Another possibility is to mimic the single-contact tests using non-linear FE analysis [29]: two contacting bodies are modeled using a very fine FE mesh and each node is assigned a Coulomb-like slip criterion. Stiffness values are evaluated from computed force-deformation curves. Results were found to be 6–11% higher with respect to measured counterparts, possibly because of *"the neglected surface roughness as well as adhesive contact and viscous-*

**3.** In 2002, the "residual stiffness" method was proposed [31]. It is based on the observation that typical reduction techniques (e.g. CB-CMS [4]), used to reduce the size of FE models, may neglect the small local deformations. A "correction factor" is introduced to take into

[2, 25–27] can be esti-

) is available only for circular or elliptical contact

**1.2. A quick critical review of contact parameter estimation**

refinement has not yet been achieved and heuristic models are preferred.

tion law), and their calibration is based on fitting to empirical observations.

and the corresponding normal load [5, 25]. Tangential contact stiffness k<sup>t</sup>

normal compliance (kn) is available for cylinder-on-flat contacts [28].

vide meaningful predictions.

100 Contact and Fracture Mechanics

and friction coefficient μ.

**1.** A complete analytical solution (kn and k<sup>t</sup>

*elastic solid behavior*" [30].

and k<sup>t</sup>


One common point to approaches 2, 3 and 5 is that they use, as a benchmark, the solution offered by the full model in the FE software environment. This implies relying predictions performed with the Penalty or the Augmented Lagrangian Method to enforce the impenetrability condition and neglecting the possible influence of surface roughness.

Using, as a benchmark, numerical evidence is certainly quite convenient, as it does not require experiments and it is generally quite "complete" (the user can interrogate the software and retrieve displacements, stresses at any point of the mesh). However, it is based on the strong assumption that sees the full FE model as representative of the true contact conditions.

Approach 5 is usually regarded as the fastest and most effective, as it guarantees that the simulated target evidence matches the reference one. However, this local adjustment of parameters does not truly add knowledge to the field. In fact, it has a strong ad-hoc character, and must be repeated for every new system the designer comes across.

Another possible "drawback" of approach 5 is the possible under-determinacy of the contact parameter problem. As shown in [37], there exist multiple combinations of contact parameters capable of satisfying a given FRF. Therefore, if the number of contact parameters to be determined is larger than the (observed or computed) target features to be matched during tuning (or the influence of contact parameters is weak on the available target features), multiple solutions are possible. Two sets of contact parameters which produce equivalent responses at a given excitation level, may give rise to radically different solutions if the excitation level changes (see **Figure 1**, for example, with a curved-flat damper between a set of blades). This may not be a critical issue if a large number of target evidence can be produced (e.g. if the reference evidence is obtained numerically) and/or if the contact parameters to be determined are limited. However, it becomes a strong limitation if curved-flat underplatform dampers, characterized by a complex kinematics and multiple contact interfaces, are considered.

For all these reasons, it is here believed that the contact parameter estimation problem should be tackled using dedicated experimental evidence which focuses on the damper-blade interface. An increased attention to the damper kinematics has been demonstrated by other notable researchers in the field: in detail in [8] laser measurements have been employed to record damper rotation, while in [38] Digital Image Correlation has been used to investigate contact displacements.

the sole indicator of the damper performance. The FRF is certainly an important design indicator, however by itself, it is not capable of offering enough information on the damper working conditions. Furthermore, if FRFs are the only experimental evidence available it is likely that, as pointed out in Section 1.2, the contact parameter problem will remain underdetermined.

Modeling Friction for Turbomachinery Applications: Tuning Techniques and Adequacy…

http://dx.doi.org/10.5772/intechopen.72676

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For all these reasons, in 2009, the AERMEC lab proposed a novel kind of test rig (see **Figure 3a**) which focuses directly on the underplatform damper. No blades are present (i.e. it is not a resonant rig). On the other hand, two dummy platforms are used to connect the system to the

• The left platform is connected to two piezoelectric actuators inserted into a purposely designed mechanical structure. This system allows imposing any user-defined in-plane displacements simulating the so-called In-Phase (IP, vertical) and Out-of-Phase (OOP, hor-

• The right platform is connected to two uniaxial force sensors by means of a tripod structure to the purpose of measuring the forces transmitted between the two platforms through the

The damper is pulled by a deadweight simulating the centrifugal force, CF. The main purpose of the rig is to relate contact forces to the displacements that produce them (see also **Figure 3c**). For this reason, a differential laser head is employed to measure the platforms relative displacement (a necessary precaution owing to the lack of closed loop control of the piezoelectric actuators), the damper radial displacement and rotation angle and the damper-platform relative displacement at the contact. A scheme representing the laser positioning to obtain the tangential relative motion at the contacts and the damper rotation is shown in **Figure 3d**.

The key features of the test rig described above remain unchanged since its first version [39], however several subsequent improvements have been performed (see **Figure 4** for a graphical representation). In detail, the tripod and the structure hosting the force sensors have been redesigned to increase the overall stiffness of the rig [18]. This had a positive impact over the frequency operating range which increased from [≈5–80] Hz to [≈5–160 Hz]. In [40] each platform has been redesigned into two parts: a "fixed" part connected to the rest of the test rig (the left platform to the actuators, the right one to the force sensors) and a second part, termed here "insert" in contact with the damper. This configuration has several advantages: (i) the "insert" can be substituted to test different platform angles, (ii) the contact is localized along the damper axis by means of 4 mm wide protrusions present on both platform inserts which

Lastly, the new platform inserts and dampers have been machined with cube-like protrusions oriented with one of the faces perpendicular to the contact line. Each contact line (left and right) is equipped with two cubes (one on the damper and one on the corresponding platform).

ensure high contact pressures even with moderate deadweights on the damper.

izontal) relative motion between the blades platforms or combinations of the two.

input motion generation and to the force measuring mechanism.

**2.1. The Piezo Damper Rig**

damper.

**2.2. The test rig evolution**

**Figure 2.** (a) Example of tuning of contact parameters in the full stick regime: two sets of contact parameters (one from Section 3 and one with a simplified assumption) leading to the "same" FRF. (b) Resulting FRFs (excitation level out of the tuning range) produced by the two sets of contact parameters from **Figure 1a**. CF: centrifugal force on the damper, FE: external excitation on blades.

## **1.3. Goals of the chapter**

The main purpose of this chapter is to present the latest advances made by the AERMEC lab to improve the fidelity of damper modeling and to rigorously assess processes needed for reliable predictions/estimation of contact parameters (see **Figure 2**).

In detail, Section 2 briefly describes the Piezo Damper Rig (see **Figure 3a**), first presented in [39], and recounts its latest improvements.

Section 3 with reference to Section 1.1, defines a numerical damper model (also represented in **Figure 3b**) and justifies all modeling choices.

Section 3.2 uses the experimental evidence gathered on the above-mentioned rig to estimate all contact parameters necessary to represent a curved-flat damper between a set of platforms (conforming and nonconforming surfaces both).

In Section 4, the adequacy of the chosen contact model is discussed on the basis of an experimental campaign on numerous damper samples. Furthermore, the role of rotation of nonconforming contacts, a topic which has never been addressed in this context to the author's knowledge, will be explored.

The chapter conclusion (Section 5) includes a series of warning and recommendation for the damper designer/tester.
