**2.3. Measurement protocol**

**Diagram Goal Quantities Measurement** 

and derived contact forces. (d) Laser positioning to obtain relevant kinematical quantities.

t

**Figure 3.** (a) Piezo Damper Rig scheme and relevant quantities. (b) Piezo Damper Rig numerical model. (c) Measured

t

TL, NL, application point

This example is carried out in case of IP motion, a similar diagram can be obtained in case of OOP motion by plotting the

horizontal force component HR against the corresponding horizontal platform relative displacement uLP − uRP.

Estimate friction coefficient Identify contact states

Estimate tangential contact

Estimate tangential contact

Estimate normal contact stiffness (conf. contact)

Validation, check position of

Damper rotation is here obtained as described in, that is, with reference to **Figure 3d**, *β<sup>D</sup>* = (*wD*

left contact force

stiffness

stiffness

T/N force ratios (**Figure 5a**)

Hysteresis at nonconforming contact

Hysteresis at conforming contact (**Figure 5b**)

104 Contact and Fracture Mechanics

Moment vs. Rotation diagram (**Figure 5c**)

Platform-to-platform hysteresis cycle (**Figure 5d**)<sup>2</sup>

Contact forces diagram

**Table 1.** Essentials in damper diagrams.

(**Figure 6a**)

1

2

**technique**

RD-tRP Laser vibrometer 0.08 μm (~0.2%)

LD-tLP Laser vibrometer 0.08 μm (~0.2%)

TR/N<sup>R</sup> Derived 3–5% TL/NL Derived 6–10%

T<sup>R</sup> Load cells 2%

TL Derived 2.5%

β<sup>D</sup> Derived1 5% M = NL \* x Derived 5–7%

VR Load cells 2%

TR, N<sup>R</sup> 2%

Validation wLP-wRP Laser vibrometer 0.08 μm (~0.2%)

**Uncertainty**

2.5%, <1 mm

*<sup>A</sup>*<sup>0</sup> − *wD AR* )/¯ *A*<sup>0</sup> *AR*. Each experimental nominal condition is defined by: damper configuration (i.e. shape, platform angles, etc.), centrifugal load on the damper, excitation frequency, amplitude and direction of motion.

The analysis of the damper performance under each nominal experimental condition is operated through the cross-comparison of a series of quantities (whose graphical representation can be found in **Figure 3b**–**d**) organized into diagrams (summarized in **Table 1**, shown in **Figures 5** and **6** and further commented in Section 3.2). Both contact forces and damper/ platform kinematics are taken into account for the purpose of uncovering the cross-relations existing between them and to estimate contact parameters. It should be noted that some of these quantities are directly measured (e.g. tangential and normal forces at the nonconforming contact TR and NR and all damper displacements), while other quantities are derived (e.g. tangential and normal forces at the conforming contact TL and NL are obtained through the damper equilibrium by neglecting inertia forces at frequencies where this is correct, as shown

**Figure 4.** Piezo Damper Rig evolution.

Only in-plane motion is addressed here (typical of blades bending modes, where dampers are most effective), however a more general 3D version of the same model is available for more

Modeling Friction for Turbomachinery Applications: Tuning Techniques and Adequacy…

{*FC*} (1)

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

, {*U*} <sup>=</sup> {*uD*, *wD*, *<sup>β</sup>D*, *uLP*, *wLP*, *uRP*, *wRP*}, {*FE*} <sup>=</sup> {0, <sup>−</sup>*CF*, <sup>0</sup>,

, *NLnC*} and [*T*] is a transformation matrix. In detail,

*uL*, *k wL*, [*k* ′ *<sup>R</sup>*]) 107

(2)

complex cases. The general equilibrium equation to be solved at this stage is:

*uR* − *k* ′

*uR* sin<sup>2</sup> *<sup>α</sup>* <sup>+</sup> *<sup>k</sup>* ′

where with reference to **Figure 3b**, [*M*] <sup>=</sup> *diag*(*mD*, *mD*, *mD*, *mLP*, *mLP*, *mRP*, *mRP*), [*K*] <sup>=</sup> *diag*(0, <sup>0</sup>, <sup>0</sup>, *<sup>k</sup>*

a larger system be considered, multi-Harmonic Balance Method can be applied [35].

*wR*)sin*α*cos*α*

*wR* cos<sup>2</sup> *<sup>α</sup>*]

vector {*FC*} is the output of the contact elements which are fed by the correct relative displace-

In this chapter, Direct Time Integration [40] is used to avoid approximations, however should

The reader will notice that the damper is modeled as a rigid body, a quite reasonable assump-

The contact elements here applied are state-of-the-art in the gross slip regime [11], which is the focus of this chapter's investigation. The nonconforming contact (cylinder-on-flat) is modeled using one element, while the conforming contact requires at least two contact elements (four in **Figure 3b**). Increasing the number of contact elements will smoothen the hysteresis shape but not change significantly the damper behavior. The position of the contact points is typically set at equal intervals along the flat interface using the two edges as limits (i.e. start-

In principle, friction is a material property, therefore all interfaces, both conforming and nonconforming, should share the same contact parameter values. Friction is indeed a material property at microscopical level, therefore if a reliable and validated "realistic" model was available one could start from material properties and surface characteristics, and integration over the contact area would do the rest. However, since the selected contact elements are of

In detail previous experience has shown that the geometry of the contact surface (line vs. area contact), contact surface kinematics and normal load play a significant role [37]. The influence of normal load will be addressed in Section 4, while, in order to take into account the influence

> knR ≠ knL ktR ≠ ktL μR ≠ μ<sup>L</sup>

and μ values.

[*M*]{*U*¨} + [*K*]{*U*} = {*Fe*} + [*T*]

*wR* sin<sup>2</sup> *<sup>α</sup>* (*<sup>k</sup>* ′

, *NR* , *TL*<sup>1</sup> , *NL*<sup>1</sup> , …,*TLnC*

*wR*)sin*α*cos*α k* ′

with [*<sup>k</sup>* ′

*k uL* ∙ *uvol*, *k*

*<sup>R</sup>*] <sup>=</sup> [ *k* ′

ments at the contact.

ing and ending points).

**3.1. Definition of the unknowns**

the "heuristic" kind, other factors influence kn, k<sup>t</sup>

of the contact areas different geometries and kinematics, it holds:

for a total of six unknowns (also represented in **Figure 3b**).

(*k* ′ *uR* − *k* ′

*wL* <sup>∙</sup> *wvol*, <sup>0</sup>, <sup>0</sup>}, {*FC*} <sup>=</sup> {*TR*

*uR* cos<sup>2</sup> *<sup>α</sup>* <sup>+</sup> *<sup>k</sup>* ′

tion given the bulkiness of the damper.

**Figure 5.** Measured vs. simulated. (a) T/N force ratio diagram. (b) Platform-to-damper hysteresis cycle at the flat-on-flat contact. (c) Moment vs. rotation diagram. (d) Platform-to-platform hysteresis cycle. Measured: dotted line, simulated: solid line. IP case, with CF = 4.65 kg.

in **Figure 3c**). Each quantity is equipped with a proper level of uncertainty. Measurement uncertainty, minimized through a purposely developed protocol, ensures significant trustworthy results (error up to 7%).
