To determine the best achievable Q as a function of the mistuning, we studied a simple model with 2 resonant masses (see elog 150). Now we have a model with 3 resonant masses (presented in elog 217, 219, T1900846).
As a reminder, in order to obtain the same transfer function with 2 masses model and the 3 masses model, we need to multiply the mass of the unique BRD by 2 in the model with 2 masses (see figure 1).
Here we assumed the Q of the standalone BRD is 110 and perfect tuning. The resulting Q of the two peaks is around 220. This is in line with the rule of thumb that states "for perfect tuning, the resultant Qs of the bounce and roll modes are approximately 2x the Q of the damper" (Norna's email). However, with the 3 masses model, the resonance of the BS corresponds to the "dip" in the transfer function. At this frequency, the Q value is exactly the Q of the damper. Therefore, there is no factor 2 for the BS mode.
%% Recalling here some results:
- m
_{B} = 10 g → k_{B} = 110 N/m,
- m
_{R} = 7.3 g → k_{R} = 171 N/m
Mass ratio is therefore:
- μ
_{B} = 10g/27.78 kg = 3.6e-4
- μ
_{R}= 7.3g/12.90 kg = 5.7e-4
And we measured:
- Q
_{B} ~ 100-110
- Q
_{R} ~ 60-80
%%
In T1500271, Brett derives the best achievable Q of a damper for a given mass ratio:
**Q = 1/(2*w1/w2*sqrt(3*mu(ii)/(8*(1+mu(ii))^3)));** % from equation 12 in T1500271
Comparing these values to our blade v5 measurements for bounce and roll modes, we can see that the actual Qs for the standalone BRDs are a factor 2.5 higher than this ideal case (see figure 2). We therefore updated the study of elog 150 with the new 3 resonant masses model and the Qs of the standalone BRDs equal to 2.5 times the ideal Qs. The results are summarized in graphs 3 and 4. We can see that:
- The Q of the BS decreases with the increase of the mass ratio. Therefore, the Q of the bounce mode will be higher than the roll mode.
- The Q of the BS increases with the detuning. 1% mistuning allows to maintain the Qs below 200.
- The Q of the BRDs decreases with the increase of the mass ratio.
- The Q of the BRDs decreases with the detuning. At perfect tuning, the Q of the bounce mode is about 225.
So we could aim for a tuning of the BRDs within 1% of the BS resonant frequencies. |