The numerical model

Figure 1 shows the numerical model of the damper/platform system. In it, the damper is modelled as a rigid body with mass and inertia properties. The rigid body assumption is considered valid given the bulkiness of the damper and has been successfully applied to several test cases in the last few years (12, 13, 08). The relative motion imposed to the platforms is harmonic to simulate the motion of the blades at resonance (and that of the test rig).

The contact model used is a standard macroslip element (22). The following section will present the procedure used to tune the contact parameters (six in total): with reference to Figure 1 friction coefficients (μ_R, μ_L), tangential contact stiffness (k_tR, k_tL) and normal contact stiffness (k_nR, k_nL). The flat-on-flat contact is represented using multiple macroslip elements (typically 4, as in the present case): the k_tL and k_nL values refer to the complete interface.

The benchmark case

The chosen experimental benchmark has been obtained by imposing a 5 Hz, w_LP = 30 µm (reduced to 22 µm due to the lack of a closed loop control on the piezo) In-Phase harmonic motion to the left platform. The centrifugal load on the damper is set at 50 N. The frequency is set to 5 Hz in order to have a quasi-static hysteresis measurement (i.e. cleaner signal) for contact parameter estimation. It should be noted that the damper has been tested in the complete frequency range allowed by the test rig ([5–160 Hz]) and the results have been found unchanged, as already shown in (11).

The centrifugal force is set at 50 N. However, the contact pressure is greatly increased by the presence of two 4 mm long tracks on the platforms, described in (08) and partially visible in Figure 1. These tracks have a double function: they standardize the length of contact and increase the contact pressure to 50% of that experienced by a 170 mm long damper with the same cross section mounted on the fourth stage of a turbine for energy production.

Figure 2 plots the comparison between numerical and experimental vertical platform-to-platform hysteresis cycle. The measured cycle is reproduced very satisfactorily (error on equivalent stiffness and damping <1%). Moreover, it should be noted that this piece of experimental evidence was not used during tuning.

Figure 3 to Figure 5 display additional experimental diagrams, relative to the same measurement and experimental conditions: this set of diagrams, described in the following section will be used to estimate the damper/platforms contact parameters.

Figure 5.

(a) Scheme representing the flat surface model. (b) Measured damper equilibrium during the cycle. (c) Measured moment vs. rotation plot.

Figure 3.

Measured T/N force ratio diagram corresponding to the cycle in Figure 2.

Friction coefficients

Friction coefficients at both interfaces are estimated using the time history of the tangential/normal force ratio at the two contacts as described in (13, 08). If the force ratio is constant in time and equal to a maximum then it is assumed to be a friction coefficient (e.g. 3-4D and 5-1U in Figure 3). The measured values for the case in Figure 3 are reported in Table 1.

The components of the right contact force (T_R and N_R) are directly measured, while the left components (T_L and N_L) are reconstructed: the numerical model has shown that inertia effects of the damper are negligible in the investigated frequency range therefore it is possible to reconstruct the equilibrium as shown in Figure 1.

Estimation accuracy

The variability between independent measurements is negligible, therefore the estimation accuracy includes two effects: the uncertainty on the force components (4% on T_R/N_R and 8% on T_L/N_L) derived from the load cells specifications and from subsequent error propagation techniques (13) and the error committed in reading the value on the diagram (≈±0.05 as shown in Figure 3). These two effects are summed is a maximizing indicator (i.e. the actual error will be certainly contained within those limits), shown as an error bar in Figure 8.

Figure 8.

Friction coefficient μ_R acceptable variation to ensure an error on the performance indicators limited to 5% (ΔKR=±5% , ΔKI=±5% ).

The variation of the T/N force ratio during slip is slightly sharper in the case of the right contact (cylinder-on-flat), possibly due to the different interface and the different platform angles. This behaviour is quite repeatable and may indicate that a contact model with constant friction coefficient is not completely adequate to represent reality (i.e. the authors have willingly turned epistemic uncertainty into measurement uncertainty). However, as it will be shown later the effect of this variation produces negligible effects on the output quantities of interest at blade level, therefore these authors still deem the model adequate.

Tangential contact stiffness values

The dummy platform have been recently equipped with cube-like protrusions oriented with one of the faces perpendicular to the contact line (see Figure 4). Each contact line (left and right) is equipped with two cubes (one on the damper and one on the corresponding platform). This allows for the measurement of the tangential relative motion at the contact. The hysteresis at the contact (as shown in Figure 4) is obtained by relating this relative motion to the corresponding tangential force: the slopes of the portion of hysteresis cycle identified as being in stick condition can be used to estimate the tangential contact stiffness values on each side of the damper. The stick condition is identified by looking at the corresponding T/N force ratio in Figure 3 (e.g. stage 1U-2 is identified as stick because T_L/N_L is not constant in time) as shown in (09). Cylinder-on-flat and flat-on-flat contact interfaces have both been tested.

Figure 4.

(a) Functional scheme representing the experimental procedure used to estimate the tangential contact stiffness values. (b) Measured platform-to-damper hysteresis at the left contact.

Flat-on-flat normal contact stiffness

The normal contact stiffness at the flat side k_nL is assumed to be uniformly distributed along the flat interface, dk_nL/dx (see Figure 5, 09). As shown in Figure 5 during the cycle the normal component of the left contact force resultant N_L travels along the flat surface during the cycle. If N_L enters the inner third portion of the flat interface, then the complete surface is in contact. Under this condition, it is possible to write the moment of N_L around O (see Figure 5) as a function of the rotation β:

The normal contact stiffness per unit length is the slope highlighted in Figure 5. It should be noted that the slope has been estimated using a specific portion of the curve in Figure 5: the portion corresponds to instants in the cycle where N_L falls inside the inner third portion of the contact interface. This can be estimated checking the corresponding experimental diagram in Figure 5.

Cylinder-on-flat normal contact stiffness

The normal contact stiffness for the cylinder-on-plane contact has been obtained by taking the initial slope of the corresponding normal displacement-normal force curve. This curve has been obtained from interpolations of experimental data (14), later confirmed by theoretical investigations (04).

Harmonic complex springs: equivalent stiffness and damping

The nonlinear dynamic response is usually computed using Harmonic Balance methods (HBM) where forces and displacements are decomposed in their harmonic components. If the platforms’ imposed displacement is assumed to be mono-harmonic wLP−wRP= w¯(1C) cos(ωt), then it is sufficient to consider the first harmonic of the corresponding friction force VR ≅ V¯R(1C) cos(ωt) + iV¯R(1S) sin(ωt) to correctly represent respectively the real and imaginary part of a complex spring KR + iKi. The real part KR= V¯R(1C)/w¯(1C), in phase with the displacement, represents the elastic stiffness introduced by the damper between the platforms, the imaginary part KI= V¯R(1S)/w¯(1C), out of phase with respect to the input displacement, represents the energy dissipated by the damper (i.e. the area of the hysteresis cycle equals the area of the ellipse A = π⋅ w¯(1C)⋅V¯R(1S)). The recorded difference between the area of the hysteresis cycle and that of the corresponding ellipse is lower than ±0.05%, and it is due only to numerical errors in the integration. In fact, if the motion is mono-harmonic, only the corresponding component of the friction force will produce any dissipation (all others being orthogonal to the displacement), retaining multiple harmonics in the force signal will improve the shape of the resulting cycle but will not modify its area. Therefore, the HBM can be considered a simplified yet adequate representation of the original hysteresis shape (see Figure 6).

Figure 6.

Hysteresis cycle with imposed displacement w¯(1C)=30 μm (black line) and corresponding HBM equivalent ellipse (grey line).

Figure 7 plots the complex springs values against the imposed displacement w¯(1C). Figure 7 plots three hysteresis cycles corresponding to three different points highlighted on the complex springs curves. For w¯(1C) <1μm the damper is fully stuck to the platforms, therefore K1 = 0 and KR is equal to its maximum. For w¯(1C) = 5 μm, KI is at its maximum since 5μm is the lowest displacement that allows the damper to reach gross slip. For higher values of w¯(1C), KI and KR progressively diminish. These complex spring values are used as performance indicators (of damper equivalent stiffness and dissipation) in the following section.

Figure 7.

(a) Complex spring values against imposed displacements. (b) Corresponding platform-to-platform hysteresis cycles for increasing imposed displacements.

Sensitivity

The purpose of this section is to show the acceptable variation of each contact parameter to ensure an error on the performance indicators limited to 5% (ΔKR = ±5%, ΔKI = ±5% ). The benchmark case draws the contact parameter values from the tuning procedure described in Section 3, the resulting spring values are diagrammed in Figure 7. It was here decided to assess the effect of each contact parameter independently, i.e. starting from the benchmark case, numerically vary one at a time, while keeping the others set at their “nominal” value.

This numerical analysis is carried out for platforms’ imposed displacements ranging from w¯(1C) = 0.3  to 60  μm. This range of displacements has been selected because it encompasses a broad range of damper contact states (i.e. from full stick to gross slip at all interfaces). The actual working conditions of the damper will depend on the blade, the mode of vibration and the excitation amplitude, however the authors’ intention was that of offering a comprehensive overview of all damper operating conditions. For each value of imposed displacement the acceptable variation of each contact parameter is compared to its accuracy.

Sensitivity to variation of contact stiffness values

Figure 10 plots the acceptable variation of the tangential contact stiffness at the flat side (ktL). A similar trend holds for ktR (here not shown for brevity): the relevant data is summarized in Table 2 and Table 3. Since the limits vary with the platforms’ imposed displacement the most conservative values are selected.

Figure 10.

Tangential contact stiffness k_tL acceptable variation to ensure an error on the performance indicators limited to 5% (ΔKR=±5% , ΔKI=±5% ).

In Figure 10 the black dashed line corresponds to the value of contact stiffness estimated using the procedure presented in the previous section and is assumed to be the correct one. The corresponding error bars are located at w¯(1C) = 30 μm.

Several caveats can be drawn from the observation of these diagrams:

• in the w¯(1C) = [0.3–5) μm range, only minor variations of contact stiffness values are allowed (<15%). Since the damper is fully stuck, the cycle corresponds to a line whose slope is a linear combination of all contact stiffness values. This slope is equal to K_R, therefore Δk_tL is directly proportional to ΔK_R.

• in the gross slip range w¯(1C) = [5–60] μm K_R is less affected by variations of contact stiffness since the gross-slip phase prevails. Moreover, a positive Δk_tL leads to a decrease of K_R. This happens because higher contact stiffness values decrease the duration of the stick phase (see Figure 10).

• in the w¯(1C) = [0.3–5) μm range, K_I = 0 therefore the error lines cannot be plotted.

• in the range w¯(1C) = 5 μm, K_I is influenced by variations of contact stiffness values because, as shown in Figure 11 the stick and gross-slip stages have a similar duration

  Figure 11.

  Effect of variation of k_tL on platform-to-platform hysteresis cycle area for increasing imposed displacements (a) 5 µm and (b) 30 µm.

  

• in the gross slip range w¯(1C) = [10–60] μm K_I displays a diminishing sensitivity to contact stiffness values variations. This is explained by the fact that only a minimal portion of the overall area is accounted for by the variation of the hysteresis slopes (see Figure 11). In some cases (i.e. k_nR, see Table 3), it was not possible to find the |ΔK_I| = 5% limit in the investigated range.

• as evidenced in Table 2 and Table 3 the accuracy obtained on the tangential contact stiffness values is comparable with the |ΔK_R| and |ΔK_I| 5% limits.

• as evidenced in Table 3, k_nR, being the highest, has a very low influence compared to the others.