Parametric Study of Geometric Variations

In this study, 28 V-shaped dampers are created by varying either the length of contact area from 3 to 6 mm, or the opening angle of the V-groove from ^π₃ to ^5π₆. The thickness of the dampers are adjusted accordingly to maintain the volume constraint. Several examples of the V-shaped dampers are shown in Fig.5.5. Nonlinear forced responses are extracted at the tip of the blades for each of the 28 blisk-damper models. A traveling wave excitation is

(a) l= 3, θ =^π₆ (b) l= 4.5, θ =^π₃ (c) l= 6, θ = ^π₄

Figure 5.5: Examples of V-shaped dampers with various lengths of contact area and open-ing angles of the groove

used as the external periodic forcing function, which can be expressed as follows

f=

N

X

j=1

Fe^{−i( j−1)EO2πN}, (5.9)

0 10 20 30 40 50

0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75

ρ

u

Figure 5.6: Forced responses of a UM validation blisk in contact with a typical V-shaped damper under a traveling wave excitation

As an example, Figure5.6shows the forced responses of a blisk-damper systems. The damper geometry is shown in Fig.5.5b, which has a length of contact area of 4.5 mm, and an opening angle of ^π₃. To comprehensively evaluate the relative importance among the pre-stress magnitude N₀, forcing magnitude F, and the coefficient of friction µ, the ratio ρ = ^µN_F⁰ is used in this study. Also, to account for the variations of damper effectiveness due to external forcing, the forced response magnitude x is normalized with respect to the

external forcing magnitude F. Thus, the normalized response magnitude u= _F^x is used in this study. The maximum response amplitude within the range of excitation frequency is extracted at each ρ value, ranging from 0 to 40. The effectiveness of a damper is often evaluated by the reduction ratio of forced responses R from the maximum value, typically when the damper is full sliding or completely stuck with the blisk, to the minimum re-sponse where the friction damping is most effective. Moreover, from a design perspective, a damper is often required to be effective over a wide range of frictional coefficients, or under different forcing conditions. This requirement is reflected in Fig.5.6as to maintain the reduction ratio above certain threshold over a wide range of ρ values, referred hereafter as the effective range of ρ. In this study, the two metrics, reduction ratio R and the effective range of ρ, are used to evaluate damper performance.

0 10 20 30 40

0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75

ρ

u

Figure 5.7: Forced responses of a UM validation blisk in contact with 28 V-shaped dampers of different variations in geometric parameters

Figure 5.7 shows the nonlinear responses of 28 blisk-damper systems. Surprisingly, the forced responses of all 28 models are almost identical to each other, with the same reduction ratio and the same effective range of ρ. This observation indicates that small variations of damper geometry has little or no effect on the damper effectiveness. Thus, it is of interest to explore the aspects of damper geometry that control damper effectiveness.

A cone-shaped damper is created as shown in Fig.5.8a, which possess half of the contact area compared with a V-shaped damper (Fig. 5.8b). It is expected that under the same forcing condition, the cone-shaped damper should reach at least half of the reduction ratio that a V-shaped damper provides.

However, Figure 5.9a shows that the cone-shaped damper has a reduction ratio of

(a) A cone-shaped damper (b) A V-shaped damper

Figure 5.8: A cone-shaped damper that has half contact area of a V-shaped damper

Sliding Sticking Frequency

Frequency (Hz) Frequency (Hz) Split (Hz) Cone-Shaped Damper 399.5890 399.6969 0.1079

V-Shaped Damper 399.0624 399.6754 0.6130

Table 5.1: Natural frequencies of a UM Validation Blisk in contact with a cone-shaped and a V-shaped damper

19.46%. Compared to the reduction ratio of 46.95% provided by the V-shaped damper (Fig. 5.9b), the cone-shaped damper fails to meet the expectation of reaching half of the effectiveness that the V-shaped damper provides. Moreover, the effective range of ρ values that the cone-shaped damper provides is 0.45, if the threshold of reduction ratio is set as 10%. This range is much narrower than the range of 24 that the V-shaped damper provides with the same threshold.

With the same forcing condition, and half of the contact area of the V-shaped damper, it is suspected that the ineffectiveness of the cone-shaped damper is due to the joint that the V-shaped damper possesses. The main contribution of the joint is additional bending stiffness along the tangential direction. Thus, the natural frequencies of the blisk-damper systems under sliding and sticking conditions are computed. Presented in Tab.5.1are the natural frequencies that fall into the range of excitation frequency. Note that while the sticking frequency remains the same for both cases, the sliding frequency has changed by approximately 0.5 Hz. This change leads to a larger difference between sliding and sticking frequencies, hereafter referred to as the frequency split, in the case of a V-shaped damper. It is believed that with a narrow frequency split, a small perturbation in excitation

0 0.5 1 1.5 2 0.65

0.7 0.75 0.8

u

Cone−Shaped Damper

ρ

(a) A cone-shaped damper

0 20 40 60 80

0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75

ρ

u

V−Shaped Damper

(b) A V-shaped damper

Figure 5.9: Forced responses of a cone-shaped damper in comparison with the responses of a V-shaped damper that has twice the contact area

frequency can result in a sudden change of contact status of the entire contact surface from a sliding to a sticking condition. In contrast, a large frequency split provides more tolerance in perturbation in excitation frequency, and allows more micro-slip to occur. To prove this point, a double cone-shaped damper is created as shown in Fig.5.10, and its Young’s modulus is tuned to increase its bending stiffness so that its frequency split matches that of the V-shaped damper, as shown in Tab.5.2.

Figure 5.10: A double cone-shaped damper

Forced response analyses reveal that the double cone-shaped damper and the V-shaped

Sliding Sticking Frequency Frequency (Hz) Frequency (Hz) Split (Hz)

Double Cone- 399.3131 399.9290 0.6159

Shaped Damper

V-Shaped Damper 399.0624 399.6754 0.6130

Table 5.2: Natural frequencies of a UM Validation Blisk in contact with a double cone-shaped and a V-cone-shaped damper

0 20 40 60 80

(a) A double cone-shaped damper

0 20 40 60 80

Figure 5.11: Forced responses of a double cone-shaped damper in comparison with the responses of a V-shaped damper that has twice the contact area

damper has similar reduction ratio of forced response amplitudes, as shown in Fig. 5.11.

This observation is consistent with the statement that under the same forcing condition, the frequency split that damper provides determines its effectiveness. This is further confirmed by revisiting the frequency split provided by the 28 V-shaped damper models. It is revealed that all the 28 models have nearly identical frequency split of approximately 0.58 to 0.62 Hz.