**3. Numerical model and contact parameters estimation**

The diagrams summarized in **Table 1** and shown in **Figures 5** and **6a** are represented together with the corresponding simulated counterpart, obtained using the numerical model shown in **Figure 3b**. 3 The numerical model represents the damper inside the test rig, however the same numerical routine can be incorporated into a code which substitutes the test rig presence with that of a FE bladed system [35].

<sup>3</sup> The springs representing the tripod and the mechanical structure hosting the piezo actuators have been experimentally measured as described in [39].

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

$$\text{[M]} [\text{\color{red}{Ul}}] + \text{[K]} [\text{\color{red}{Ul}}] = \text{ \color{red}{\{F\_c\}} + \text{[T]} \{F\_c\}} \tag{1}$$

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> uL*, *k wL*, [*k* ′ *<sup>R</sup>*]) with [*<sup>k</sup>* ′ *<sup>R</sup>*] <sup>=</sup> [ *k* ′ *uR* cos<sup>2</sup> *<sup>α</sup>* <sup>+</sup> *<sup>k</sup>* ′ *wR* sin<sup>2</sup> *<sup>α</sup>* (*<sup>k</sup>* ′ *uR* − *k* ′ *wR*)sin*α*cos*α* (*k* ′ *uR* − *k* ′ *wR*)sin*α*cos*α k* ′ *uR* sin<sup>2</sup> *<sup>α</sup>* <sup>+</sup> *<sup>k</sup>* ′ *wR* cos<sup>2</sup> *<sup>α</sup>*] , {*U*} <sup>=</sup> {*uD*, *wD*, *<sup>β</sup>D*, *uLP*, *wLP*, *uRP*, *wRP*}, {*FE*} <sup>=</sup> {0, <sup>−</sup>*CF*, <sup>0</sup>, *k uL* ∙ *uvol*, *k wL* <sup>∙</sup> *wvol*, <sup>0</sup>, <sup>0</sup>}, {*FC*} <sup>=</sup> {*TR* , *NR* , *TL*<sup>1</sup> , *NL*<sup>1</sup> , …,*TLnC* , *NLnC*} and [*T*] is a transformation matrix. In detail, vector {*FC*} is the output of the contact elements which are fed by the correct relative displacements at the contact.

In this chapter, Direct Time Integration [40] is used to avoid approximations, however should a larger system be considered, multi-Harmonic Balance Method can be applied [35].

The reader will notice that the damper is modeled as a rigid body, a quite reasonable assumption given the bulkiness of the damper.

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. starting and ending points).

## **3.1. Definition of the unknowns**

in **Figure 3c**). Each quantity is equipped with a proper level of uncertainty. Measurement uncertainty, minimized through a purposely developed protocol, ensures significant trust-

**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:

The diagrams summarized in **Table 1** and shown in **Figures 5** and **6a** are represented together with the corresponding simulated counterpart, obtained using the numerical model shown in

numerical routine can be incorporated into a code which substitutes the test rig presence with

The springs representing the tripod and the mechanical structure hosting the piezo actuators have been experimentally

The numerical model represents the damper inside the test rig, however the same

**3. Numerical model and contact parameters estimation**

worthy results (error up to 7%).

solid line. IP case, with CF = 4.65 kg.

106 Contact and Fracture Mechanics

that of a FE bladed system [35].

**Figure 3b**.

3

3

measured as described in [39].

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 the "heuristic" kind, other factors influence kn, k<sup>t</sup> and μ values.

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

$$\begin{array}{ll} \mathbf{k}\_{\text{nK}} \neq \mathbf{k}\_{\text{nL}} \\ \mathbf{k}\_{\text{tK}} \neq \mathbf{k}\_{\text{tL}} \\ \mu\_{\text{g}} \neq \mu\_{\text{L}} \end{array} \tag{2}$$

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

Contact parameters of the flat-on-flat interface are typically distributed uniformly among the contact points, for example, considering the normal contact spring:

$$\mathbf{k}\_{\rm hL} = \frac{\mathbf{k}\_{\rm nL}}{\mathbf{n}\_{\rm c}} \tag{3}$$

The distribution of the normal contact stiffness at the flat-on-flat interface per unit length dknL/dx is obtained by linking the damper inclination (i.e. rotation angle βD at a given instant in time) to the position of the left contact force resultant NL. In other words, it is postulated that forces (i.e. moments) and displacements (i.e. rotation) are linearly linked. The technique also relies on two assumptions: (1) the normal contact stiffness is uniformly distributed along the flat interface (see Eq. 3 and **Figure 6b**); (2) the force per unit length q(x) related to the normal component of the left contact force resultant NL has a linear distribution. In detail, with reference to **Figure 6b**, let us define a reference system x, parallel to the contact with its origin in O, the mid-point of the flat interface. As shown in **Figure 6a**, the normal component of the left contact force resultant NL travels along the flat surface during the cycle. If, however NL enters the inner third portion of the flat interface (see **Figure 6c**) the complete surface is in

Modeling Friction for Turbomachinery Applications: Tuning Techniques and Adequacy…

dx *L*3 \_\_

This relation is graphically represented in **Figure 5c**, where the shaded are corresponds to the

All experimental evidence shown and commented in this chapter has been represented together with its simulated counterpart for validation purposes. Some of the features of the diagrams (i.e. T/N levels during slip stages and slopes of the hysteresis cycles during stick stages) are meant to be similar because they are used as a calibration key in the contact parameter estimation process. Other features such as force trajectories and left contact force resultant application point (**Figure 6a**); the platform-to-platform hysteresis cycle (e.g. **Figure 5d**); the transition between contact states and the time instant at which they take place (e.g. **Figure 5a**); are not part of the calibration process. Therefore, the goodness of fit of these observed and simulated signals is a further proof of the soundness of the model and of the correctness of the

portion of the cycle during which NL enters the inner third portion of the flat interface.

**4. Contact parameters variability and contact model adequacy**

**Table 3** and will be further commented on in the following subsections.

The purpose of this section is to detail the level of uncertainty of each of the contact parameters estimated in Section 3.2 and to investigate their variability. Results are summarized in

**Contact parameter μR ktR (N/μm) μL ktL (N/μm) dknL/dx (N/m2**

Uniform flat contact uncertainty bands [0.6–0.75] [25–35] 0.45–0.5 [20–30] [0.8–1.2] × 1010 Irregular flat contact uncertainty bands [0.6–0.75] [25–35] 0.45–0.6 [20–100] [0.4–1.2] × 1010

Normal load dependence No No Yes Yes Yes

**Table 3.** Uncertainty bands and normal load dependence of contact parameters obtained at CF = 4.65 kg.

<sup>12</sup> β<sup>D</sup> (4)

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

109

**)**

contact. Under this condition, the following holds:

<sup>M</sup> <sup>=</sup> NL <sup>∙</sup> <sup>x</sup> <sup>=</sup> <sup>d</sup> <sup>k</sup>\_\_\_\_nL

**3.3. Remark on experimental evidence and validation**

contact parameters used to calibrate it.

The validity of this assumption will be further assessed in Section 4.

#### **3.2. Step-by-step contact parameter estimation procedure**

Contact parameters are estimated starting from the experimental evidence, organized into diagrams as summarized in **Table 1**. In detail, for a given experimental nominal condition:



**Table 2.** Experimental contact states identification strategy.

The distribution of the normal contact stiffness at the flat-on-flat interface per unit length dknL/dx is obtained by linking the damper inclination (i.e. rotation angle βD at a given instant in time) to the position of the left contact force resultant NL. In other words, it is postulated that forces (i.e. moments) and displacements (i.e. rotation) are linearly linked. The technique also relies on two assumptions: (1) the normal contact stiffness is uniformly distributed along the flat interface (see Eq. 3 and **Figure 6b**); (2) the force per unit length q(x) related to the normal component of the left contact force resultant NL has a linear distribution. In detail, with reference to **Figure 6b**, let us define a reference system x, parallel to the contact with its origin in O, the mid-point of the flat interface. As shown in **Figure 6a**, the normal component of the left contact force resultant NL travels along the flat surface during the cycle. If, however NL enters the inner third portion of the flat interface (see **Figure 6c**) the complete surface is in contact. Under this condition, the following holds:

$$\mathbf{M} = \mathbf{N}\_{\mathrm{L}} \cdot \mathbf{x} = \frac{\mathrm{d} \, \mathrm{k}\_{\mathrm{nl}}}{\mathrm{d} \mathbf{x}} \, \frac{\mathrm{L}^3}{12} \beta\_{\mathrm{D}} \tag{4}$$

This relation is graphically represented in **Figure 5c**, where the shaded are corresponds to the portion of the cycle during which NL enters the inner third portion of the flat interface.

#### **3.3. Remark on experimental evidence and validation**

Contact parameters of the flat-on-flat interface are typically distributed uniformly among

Contact parameters are estimated starting from the experimental evidence, organized into diagrams as summarized in **Table 1**. In detail, for a given experimental nominal condition:

**Step 1.** Reference points on the diagrams in **Figure 5** have been marked by a symbol and a number: they are useful to guide the analysis of the cycle through cross-comparison.

**Step 2.** The cross-comparison of the T/N diagram (**Figure 5a**) and of the observed relative

displacement at the contact interfaces can be used to make a hypothesis on the contact states experienced by the damper during one period of vibration. Namely, if, during a given portion of the cycle, the T/N force ratio is constant and the relative displacement at the contact is non-negligible, then that interface is assumed to be in slip condition. On the other hand, if the T/N force ratio is varying and the corresponding displacement at the contact is negligible, the damper is likely to be stuck to the platform. This allows the user to assign each stage of a cycle a given contact state (see **Table 2**). **Step 3.** Based on the results of **Table 2**, friction coefficients at the right (cylinder-on-flat) and

left (flat-on-flat) interface, μR and μL, are estimated using the time history of the cor-

(referring to the stages identified as being in stick condition) can be used to estimate

 **(μm) Contact state T/N slope ΔtL (μm) Contact state**

, where τ is the time variable.

**Step 4.** Based on the results of **Table 2**, the slopes of the platform-to-damper hysteresis cycles

**Step 5.** The normal contact stiffness at the cylinder-on-flat interface knR is estimated using Brandlein's formula [28]. To this purpose only material properties and length of contact are needed. The new inserts equipped with contact "tracks" described in Section

nC

(3)

the contact points, for example, considering the normal contact spring:

The validity of this assumption will be further assessed in Section 4.

respondent T/N force ratio during slip (see **Figure 5a**).

**Right interface Left interface**

*t RD* (*τ*3) − *t RP* (*τ*3) ) <sup>−</sup> ( *t RD* (*τ*2) − *t RP* (*τ*2) )

1U–2 High ~1 Stick High 1.8 Stick 2–3 High <0.2 Stick Medium 7.8 Partial slip 3–4D Low 21.3 Slip Low 13.1 Slip 4D–5 High −0.6 Stick High −1.7 Stick 5–1U Low −21.7 Slip Low −21.03 Slip

tangential contact stiffness values (ktR and ktL).

2.2 ensure a controlled length of contact.

**1**

**Stage T/N slope Δt<sup>R</sup>**

ΔtR of stage, for example 2–3, is defined as (

**Table 2.** Experimental contact states identification strategy.

1

**3.2. Step-by-step contact parameter estimation procedure**

knLi <sup>=</sup> <sup>k</sup>\_\_\_nL

108 Contact and Fracture Mechanics

All experimental evidence shown and commented in this chapter has been represented together with its simulated counterpart for validation purposes. Some of the features of the diagrams (i.e. T/N levels during slip stages and slopes of the hysteresis cycles during stick stages) are meant to be similar because they are used as a calibration key in the contact parameter estimation process. Other features such as force trajectories and left contact force resultant application point (**Figure 6a**); the platform-to-platform hysteresis cycle (e.g. **Figure 5d**); the transition between contact states and the time instant at which they take place (e.g. **Figure 5a**); are not part of the calibration process. Therefore, the goodness of fit of these observed and simulated signals is a further proof of the soundness of the model and of the correctness of the contact parameters used to calibrate it.
