**4.2. Remark on rolling nonconforming contacts**

The level of uncertainty is estimated taking into account two contributions: measurement uncertainty and the uncertainty introduced by data processing techniques (e.g. reading error). Variability will be investigated at different levels in order to answer the following questions: 1. If the same damper is tested more than once under the same nominal conditions, do the estimated contact parameters change? 2. Are contact parameters dependent on the damper

**Figure 7.** (a) TR/NR during slip as a function of NR. (b) ktR values as a function of the mean value of NR during the

**Figure 6.** (a) Measured (dotted) vs. simulated (solid) contact forces diagram. (b) Representative scheme of the distribution

The answer to point 3 is investigated using three pre-optimized damper configurations [40], that is, curved-flat dampers not affected by lift-off/rolling, jamming or partial detachment.

The user-controlled working conditions investigated during this chapter are limited to a variation of centrifugal load on the damper (i.e. normal load at the contact). Other factors such as temperature, length/area of contact may affect the contacts. These dependences can and should be mapped in order to build a "database" and avoid testing each new

of different damper samples working under very similar working conditions?

3. How different are contact parameters

working conditions (e.g. normal contact pressure)?<sup>4</sup>

corresponding stick stage. Three different damper samples are represented.

of normal contact springs. (c) Derived position of NL and resulting q(x) at stage 2.

110 Contact and Fracture Mechanics

component. This chapter should be intended as a first attempt in this direction.

4

The case of rolling contact is of scarce interest for curved-flat dampers, as it was demonstrated that large rotations (damper in lift-off) lead to a sharp decrease in dissipation capabilities [40]. However, purely cylindrical dampers are widely used and thus require a separate investigation.

The procedure to evaluate ktR described in Section 3.2 cannot be operated if the damper is rotation is large (~10 times higher than that observed in **Figure 5c**). In fact, in that case, the reading "tRD-tRP" would give a false indication. As shown in **Figure 8a**, the laser, which is initially tracking point A ends up tracking point A\*. However, the physical point initially corresponding to A is now A′, not A\*. This apparently minor difference, at micrometer level, impairs the effectiveness of this technique.

Fortunately, an alternative procedure based on the equivalent slopes of the platform-to-platform hysteresis cycle can be successfully carried out, both for cylindrical dampers and for curved-flat dampers [37, 39]. It is interesting to notice that the resulting ktR values are 3.5–4 times lower than those obtained for non-rolling cylinder-on-flat contacts, all other parameters

<sup>5</sup> This is to be expected, as the cylinder-on-flat surface has a line contact, and an increase in normal load will lead to a number of asperities coming into contact which is proportionately much smaller than that obtained for a rectangular contact area (flat-on-flat contact).

**Figure 8.** (a) Error committed by laser in case of large rolling motion mixed with sliding. (b) Behavior of the contact model in case of pure rolling motion.

(material, length of contact, radius, normal load, etc.) being equal (i.e. 8.5 N/μm vs. 30 N/μm for a 8 mm long contact). These predictions have been successfully validated both at damper [37] and at FRF level [34]. A possible explanation for this repeatable difference resides in the kinematics of the contact. In case of pure tangential translation (~0 rotation) the contact point coincides with the same physical point (same asperities) throughout the period of vibration. If larger rotations are at play, the "physical point" in contact keeps changing during rolling motion (as shown in **Figure 8b** contact point CR initially coincident with physical point DR, moves to CR′ and DR, unloaded, moves to DR′). This periodic unloading of contact regions may contribute to lower compenetration of the asperities, and therefore a lower ktR. As shown in **Figure 8b**, the heuristic contact model applied here does not model this effect, therefore a case-specific calibration of ktR is to be expected. In fact, during pure rolling motion, the contact model remains linked to the same nodes (physical points) PR and DR~DR′.

#### **4.3. Conforming contacts**

Friction coefficient (μL) and contact stiffness values (ktL and dknL/dx) of the flat-on-flat interface display a higher variability. Results for the same damper are very repeatable, but change from damper to damper.

On the other hand, Dampers B and C sport wear traces limited to the edges of the flat contact surfaces (see **Figure 9c**), therefore contact pressures are maximum at the two edges and much lower in the inner portion of the contact patch. As a result, friction coefficients increase (probably due to localized very high contact pressures) and the uniform distribution of contact springs assumption does not hold anymore. In fact, platform-to-damper hysteresis cycles (see **Figure 9b**), repeatedly display a non-unique slope during the stick stage. A minor gradual loss of stiffness could be explained by simple microslip, but this sharp two-slopes curve is

**Figure 9.** (a). ktL as a function of NL for Damper A. (b) Typical platform-to-damper flat-on-flat hysteresis cycle for damper C (similar to Damper B). (c) Contact surfaces of Damper A and C. (d) Representative scheme of Damper C's contact

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Hysteresis cycles similar to those obtained for Dampers B and C can indeed be obtained in the simulation if one accepts to distribute the contact stiffness values in a non-uniform manner (see **Figure 9d**). The average height of Damper C's asperities is not the same throughout the entire nominal contact surface, rather it has two maxima at the edges. Therefore, for a given normal load, the equivalent stiffness at the edges is bound to be higher than that in the inner portion of the nominal contact surface. Therefore, if one wishes to represent the surface behavior with a limited number of equivalent macroslip elements the only option is to assign a different value to the different elements, depending on their position, as shown in **Figure 9d**. This strategy has been adopted to produce the very satisfactory match in **Figure 9b** (see dashed line). The procedure will have to be performed again if the mean value of normal load varies. Once again this "local fitting" (stiffness values vary with normal load and with

simply not compatible with the uniform distribution of contact springs assumption.

surfaces with non-uniform ktL values.

Dampers B and C have repeatedly higher μL ≈ 0.57 with respect to Damper A, μL ≈ 0.45 (for all investigated normal loads) and the uncertainty levels (7% uncertainty and a 0.05 reading error) do not justify this marked and repeatable difference. Since the loading condition and kinematics of the flat interfaces are similar in all investigated cases, the cause of this difference may reside in the contact surfaces conditions.

In fact, Damper A, which has been tested for a higher number of cycles (it is completely "run-in") has a continuous wear trace (see **Figure 9c**). Unsurprisingly, it is easy to estimate ktL (see **Figure 5b**). The same holds for dknL/dx (see **Figure 5c**) values. In other words, the uniform distribution of contact springs postulated in Section 3, is verified. Although both ktL and knL values are positively correlated to normal loads (see **Figure 9a**). The adopted heuristic model does not take into account an increasing number of asperities coming into contact with increasing contact pressures. However, the normal load dependence can be easily mapped.

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(material, length of contact, radius, normal load, etc.) being equal (i.e. 8.5 N/μm vs. 30 N/μm for a 8 mm long contact). These predictions have been successfully validated both at damper [37] and at FRF level [34]. A possible explanation for this repeatable difference resides in the kinematics of the contact. In case of pure tangential translation (~0 rotation) the contact point coincides with the same physical point (same asperities) throughout the period of vibration. If larger rotations are at play, the "physical point" in contact keeps changing during rolling motion (as shown in **Figure 8b** contact point CR initially coincident with physical point DR, moves to CR′ and DR, unloaded, moves to DR′). This periodic unloading of contact regions may contribute to lower compenetration of the asperities, and therefore a lower ktR. As shown in **Figure 8b**, the heuristic contact model applied here does not model this effect, therefore a case-specific calibration of ktR is to be expected. In fact, during pure rolling motion, the contact

**Figure 8.** (a) Error committed by laser in case of large rolling motion mixed with sliding. (b) Behavior of the contact

Friction coefficient (μL) and contact stiffness values (ktL and dknL/dx) of the flat-on-flat interface display a higher variability. Results for the same damper are very repeatable, but change

Dampers B and C have repeatedly higher μL ≈ 0.57 with respect to Damper A, μL ≈ 0.45 (for all investigated normal loads) and the uncertainty levels (7% uncertainty and a 0.05 reading error) do not justify this marked and repeatable difference. Since the loading condition and kinematics of the flat interfaces are similar in all investigated cases, the cause of this difference

In fact, Damper A, which has been tested for a higher number of cycles (it is completely "run-in") has a continuous wear trace (see **Figure 9c**). Unsurprisingly, it is easy to estimate ktL (see **Figure 5b**). The same holds for dknL/dx (see **Figure 5c**) values. In other words, the uniform distribution of contact springs postulated in Section 3, is verified. Although both ktL and knL values are positively correlated to normal loads (see **Figure 9a**). The adopted heuristic model does not take into account an increasing number of asperities coming into contact with increasing contact pressures. However, the normal load dependence can be easily mapped.

model remains linked to the same nodes (physical points) PR and DR~DR′.

**4.3. Conforming contacts**

model in case of pure rolling motion.

112 Contact and Fracture Mechanics

from damper to damper.

may reside in the contact surfaces conditions.

**Figure 9.** (a). ktL as a function of NL for Damper A. (b) Typical platform-to-damper flat-on-flat hysteresis cycle for damper C (similar to Damper B). (c) Contact surfaces of Damper A and C. (d) Representative scheme of Damper C's contact surfaces with non-uniform ktL values.

On the other hand, Dampers B and C sport wear traces limited to the edges of the flat contact surfaces (see **Figure 9c**), therefore contact pressures are maximum at the two edges and much lower in the inner portion of the contact patch. As a result, friction coefficients increase (probably due to localized very high contact pressures) and the uniform distribution of contact springs assumption does not hold anymore. In fact, platform-to-damper hysteresis cycles (see **Figure 9b**), repeatedly display a non-unique slope during the stick stage. A minor gradual loss of stiffness could be explained by simple microslip, but this sharp two-slopes curve is simply not compatible with the uniform distribution of contact springs assumption.

Hysteresis cycles similar to those obtained for Dampers B and C can indeed be obtained in the simulation if one accepts to distribute the contact stiffness values in a non-uniform manner (see **Figure 9d**). The average height of Damper C's asperities is not the same throughout the entire nominal contact surface, rather it has two maxima at the edges. Therefore, for a given normal load, the equivalent stiffness at the edges is bound to be higher than that in the inner portion of the nominal contact surface. Therefore, if one wishes to represent the surface behavior with a limited number of equivalent macroslip elements the only option is to assign a different value to the different elements, depending on their position, as shown in **Figure 9d**. This strategy has been adopted to produce the very satisfactory match in **Figure 9b** (see dashed line). The procedure will have to be performed again if the mean value of normal load varies. Once again this "local fitting" (stiffness values vary with normal load and with position now) denounces the inadequacy of the contact model. The inadequacy of the model forces the user to tune the contact stiffness values with increasing values of CF and, in some cases, with the contact point position.

a given flat-on-flat contact surface, nor how long it will take for that surface to evolve towards a uniform distribution of contacts, this necessity for "adjustments" of contact parameters values translates into higher uncertainty levels. In other words, the state-of-the-art contact model used in this chapter is only partially adequate to represent all the complex phenomena

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On the other hand, other recalibrations (such as that needed for increasing normal loads at the flat-on-flat contact) or for very large rolling motions still signal that the heuristic model is not 100% adequate. Still, these dependences can be easily mapped and therefore do not add

One main outcome of this careful investigation, apart from the best fit values of the contact parameters (and the methodology used to obtain them), is an increased awareness of the limits and capabilities of heuristic contact models. The logical next step, the author is now working on, is the assessment of the influence that the uncertainty on contact parameters has

observed. This adds its contribution to uncertainty.

to the uncertainty.

**Nomenclature**

at the blade response level.

Variables, matrices and vectors

CF Centrifugal force {F} Generic force vector

μ Friction coefficient

[T] Transformation matrix

{U} Vector of displacements

θ Platform angle

R Damper radius

Additional subscripts C Contact D Damper E External L, R Left and right P Platforms

M Moment produced by left contact force

T, N Tangential and normal contact forces

u, w Horizontal and vertical displacements

nc Number of contact points used to represent the flat-on-flat contact

t, n Aligned along the normal and tangential direction, respectively

t, n Tangential and normal displacements at the contact

[M], [K] Mass and stiffness matrices

β Rotation

k Stiffness
