**1.1. A quick critical review of contact modeling in the turbomachinery field**

They are caused by the large response levels at resonance. Since turbine blades do not benefit significantly from material hysteresis and aerodynamic damping, the only option is to add external sources of damping, for example, in the form of dry friction devices [2, 3] such as the underplatform damper. Underplatform dampers, available in several shapes (cylindrical, curved-flat and wedge-like), are small metallic objects placed on the underside of two adjacent blades. As shown in **Figure 1a**, the centrifugal force (CF) provides the necessary precompression and the resonant-induced blade vibration triggers the damper-platform relative motion and therefore friction dissipation. Dampers are extensively used in turbine designs because they are easy to manufacture, install and substitute, while relatively inexpensive.

Whenever a damper is added to the bladed system, its dynamic response is modified into two

• the blades' resonant frequencies increase since the damper acts as an additional constraint

• the blades' response diminishes for the combined effect of the stiffness introduced by the

An additional complication is posed by the nonlinearity introduced by friction: it is well known that the non-linear dynamic response of bladed systems (both in frequency and maximum amplitude) is tightly coupled to the motion of the damper and its contact states

Accounting for the presence of friction is not an easy task. The presence of friction-induced nonlinearities makes solving the equilibrium equations a challenging task, therefore standard FE codes are not suited to the purpose: a complex hierarchy of techniques has been devel-

oped, a thorough review can be found in [4]. Furthermore, modeling friction entails:

**1.** finding a reliable model for the force-displacement relation at the contact interface and

**Figure 1.** (a) Sketch representing curved-flat underplatform dampers mounted on a turbine disk. (b) Example of standard

macroslip contact element used to represent conforming and nonconforming contacts.

(with a given stiffness) between the platforms and

damper (which acts as a constraint) and of friction damping.

fundamental ways:

98 Contact and Fracture Mechanics

(stick-slip-separation).

**2.** a proper way to estimate its parameters.

In the technical literature, the problem of modeling periodical contact forces at friction contacts is still ongoing [5] and has been addressed by several authors, leading to different contact models and techniques. Some authors adopt a Dynamic Lagrangian method to solve on the contact patch [6, 7], that is, the contact constraints are taken into account in their non-regularized form without additional compliance. Other authors, for example, [4, 8] apply a contact element to each meshed node belonging to the contact area, introducing normal and tangential stiffnesses and a Coulomb friction law. This last method is preferred here, as its calibration parameters (kn, k<sup>t</sup> and μ), however difficult to determine, represent a physical measurable property.<sup>1</sup>

The contact elements typically used in turbomachinery belong to the "spring-slider" family, a class of displacement-dependent contact models which neglect features like viscous forces along the normal direction and friction's velocity-dependence. These features, while relevant in other fields, are not typically considered in turbomachinery applications. These models belong to the larger family of heuristic models, as opposed to microscale "realistic" models where asperities and surface roughness are modeled using stochastic distributions [9].

These interactions can be geometrically divided in the normal and the tangential directions. A unilateral contact law is often considered in the normal direction (with or without normal contact stiffness) and frictional law for the tangential contact. The spring-slider elements have undergone an evolution, starting from 1D tangential motion without normal compliance [2] up to a fully coupled 3D motion [10], passing through a 1D element with normal compliance (2D motion) [11]. This last element has been adopted by many authors because of its simplicity and versatility. In fact, it can be applied to represent 1D in-plane relative motion (a quite common occurrence if the first bending modes of the blades are considered), or, with a simple upgrade [12], to give a simplified representation of 2D in-plane motion.<sup>2</sup>

Modeling conforming (i.e. flat-on-flat) or nonconforming (e.g. cylinder-on-flat) surfaces requires a different strategy. Nevertheless, the same standard macroslip contact element presented in [11] can be applied (as it is done in this Chapter, see also **Figure 1b**).

Conforming contact surfaces are typically discretized into contact points (or nodes in FE terms) and each one is assigned a standard macroslip element, either with uncoupled 2D inplane motion [8, 13, 14] or with a coupled one [15]. This choice allows to account for the presence of "microslip", first theorized by Cattaneo in 1938 [16], and later explored by Mindlin [17]. Modeling microslip is particularly relevant in those cases where high normal loads prevent actual slipping of the complete interface: in that case the gradual loss of stiffness that forecomes gross slip and the consequent dissipation does have an impact on the system response, while it becomes negligible if the gross slip regime is reached [18].

Nonconforming contacts are, in most cases, represented using one of the standard macroslip contact elements described above. Recently, a novel contact element, fit to take into account microslip as well as the nonlinearity in the normal direction typical of nonconforming contacts, has been proposed [18].

<sup>1</sup> Furthermore, Herzog et al. [7] have shown that Dynamic Lagrangians may incur in convergence problems for penalty parameters lower than 107 N/m, thus highlighting a possible limitation of their use in case of "softer" contact interfaces. 2 Where the 2D tangential motion is albeit considered as the combination of two uncoupled 1D motions.

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.

account this effect. Unfortunately, a study performed in [32] suggests that this method

Modeling Friction for Turbomachinery Applications: Tuning Techniques and Adequacy…

rounded edges pressed against an infinite half-plane. However, its results were found to

**5.** The last (and perhaps the most popular) method is based on tuning against experimental [35] or numerically obtained [36] Frequency Response Functions (FRFs). Contact stiffness values are tuned until the experimental (or full FE model) evidence and that obtained from the reduced model with contact elements match. This operation is performed using evidence in full stick condition, so that all contact stiffness are "active" and accounted for.

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 impenetra-

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

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

bility condition and neglecting the possible influence of surface roughness.

must be repeated for every new system the designer comes across.

displacements.

of a 3D flat indenter with

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gives a poor estimation of the contact stiffnesses.

be overestimating observed compliances [34].

**4.** In 2009, Allara [33] proposed a model to determine kn and k<sup>t</sup>
