5 Stability Robustness Analysis
5.1 Ground Resonance Parametric Analysis
The uncertainties introduced in the dynamic system in Eq. (1) are related to blade hinge stiffness and are presented as a function of in-plane lead-lag resonance fre- quency squared. In this work, only uncertainties in the 1st blade are accounted for
118 D. B. P. de Oliveira et al.
the robustness analysis. By assuming𝜔b1 as the nominal natural frequency of the blade, the modified frequency𝜔b1is given by:
𝜔2b1= (1 +𝛿1)𝜔b1 (5) The first robustness analysis is to be done by choosing different combinations ofCb andCxthat gives a specific peak (h) of the real part of the eigenvalues. One admits therefore the same stability level of the aircraft for all set of damping coefficients (Cb,Cx) considered. In order to do so, first the curve that gives combinations of both damping that yield the peakh= −0.5(i.e., the maximum real part of the eigenvalue) has to be found. Afterwards, for each pair of (Cb,Cx), the𝜇-analysis determines the minimal perturbation𝛿1that destabilizes the Helicopter Types 1 and 2, accordingly to Tables2and3, respectively. Note that the analysis was carried out at four different rotor speeds𝛺= [2,4,6,8]Hz. It is important to note from Tables2and3the mini- mal perturbation𝛿1that leads the helicopter unstable remains practically constant at a given rotor speed and for different set of (Cb,Cx). Also, the result does not change when DVA is considered. Moreover, the perturbation𝛿1 evolves as the rotor speed increases, which means the system is less robust in low rotor speeds.
By taking into account Eq. (5), one can note that the system is extremely robust once the perturbations less than−1lead to negative blade natural frequencies. This means the helicopter can lose one of its rotor blade stiffness and still be a stable system with those combinations of damping at these rotating speed. Also, consider- ing both helicopters have the same stability level (h= −0.5) for all set of damping coefficients, the fact of adding the DVA on the system does not change the aircraft robustness. This is concluded since the same perturbations (𝛿1) are obtained for HT1 and HT2 systems. From a practical point of view, this fact means only positive con- tritions of the DVA for the helicopter.
Table 2 Perturbations𝛿1for Helicopter Type 1
Cb(N m s/rad) Cx(N s/m) 𝛺= 2Hz 𝛺= 4Hz 𝛺= 6Hz 𝛺= 8Hz
𝛿1 𝛿1 𝛿1 𝛿1
1290 7688.5636 −1.0586 −1.2821 −1.6038 −2.0627
1668 6261.2820 −1.0586 −1.2827 −1.6039 −2.0627
2047 5515.3003 −1.0586 −1.2829 −1.6040 −2.0628
2426 5049.8542 −1.0586 −1.2830 −1.6040 −2.0628
2805 4741.7484 −1.0586 −1.2831 −1.6040 −2.0628
3184 4518.1112 −1.0586 −1.2832 −1.6040 −2.0628
3563 4344.1412 −1.0586 −1.2832 −1.6040 −2.0628
3942 4217.9519 −1.0586 −1.2832 −1.6041 −2.0628
4321 4119.8877 −1.0586 −1.2832 −1.6041 −2.0628
4700 4025.2589 −1.0586 −1.2833 −1.6041 −2.0628
Dynamics of Helicopters with DVA Under Structural Uncertainties 119
Table 3 Perturbations𝛿1for Helicopter Type 2
Cb(N m s/rad) Cx(N s/m) 𝛺= 2Hz 𝛺= 4Hz 𝛺= 6Hz 𝛺= 8Hz
𝛿1 𝛿1 𝛿1 𝛿1
1131 7031.7050 −1.0586 −1.2828 −1.6040 −2.0628
1533 5126.2450 −1.0586 −1.2835 −1.6041 −2.0628
1935 4218.1842 −1.0585 −1.2837 −1.6042 −2.0629
2337 3733.5303 −1.0585 −1.2838 −1.6042 −2.0629
2740 3386.5867 −1.0585 −1.2839 −1.6042 −2.0629
3142 3167.5168 −1.0585 −1.2839 −1.6042 −2.0629
3544 3016.2318 −1.0585 −1.2840 −1.6042 −2.0629
3946 2859.3988 −1.0585 −1.2840 −1.6042 −2.0629
348 2803.9668 −1.0585 −1.2840 −1.6042 −2.0629
4750 2651.9956 −1.0585 −1.2840 −1.6042 −2.0629
(a) Helicopter Type 1 (b) Helicopter Type 2
Fig. 3 Rotor stability according to rotor speed𝛺and perturbations𝛿1
The previous analysis for both Helicopter Type 1 and Helicopter Type 2 was done considering only the peak of the maximum real part as h= −0.5. However, it is important to assess the different perturbations related to all peak levels (h). As seen before, the perturbations are practically constant along the curve for a specific peak of real part, which means that every combination of fuselage and rotor damping that lies in that curve will give the same perturbation as result. With that in mind, it is possible to analyze the minimum destabilizing perturbations for differenthby choosing one set of damping coefficients. Here, the peaks chosen are −0.6,−0.4,
−0.2,0,0.2,0.4and0.6. In this case, the perturbations were evaluated in the range (i.e.,0≤𝛺≤10Hz) in which the helicopter is on the ground. Figure3shows the minimum perturbations at different rotor speed allowed to guarantee stability for helicopters HT1 and HT2.
120 D. B. P. de Oliveira et al.
(a) Helicopter Type 1 (b) Helicopter Type 2
Fig. 4 The influence of fuselage damping on the rotor stability sensitivity due to blade stiffness asymmetry: a) Helicopter Type 1Cb= 4500N s/m; b) Helicopter Type 2Cb= 1250N s/m
It is worth mentioning that the𝜇-analysis method can only be applied when the system is nominally stable. Therefore, the analysis is not performed in the range 4≤𝛺≤6Hz since, whenh>0, the system is unstable. Also, in order to be able to compare all the results, the analysis was not performed in that region forh<0.
One can see from Fig.3a that in the region0≤𝛺≤3Hz, the perturbations are practically the same for allh, with some small differences. This means that the peak of the real part has almost no influence in the robustness of the system. However, in the range of3≤𝛺≤4Hz, it is possible to see that the perturbations are different between positive and negativehvalues. In fact, the higher the peak (from negative to positiveh), the less robust the system is, since the perturbations are bigger. In the region6≤𝛺≤10Hz the perturbations are exactly the same for all values ofh, meaning that the peak has absolutely no effect in the robustness of the system. The same conclusions can be made for Helicopter Type 2 in Fig.3b. Also, by comparing the results for Helicopter Type 1 and Helicopter Type 2 in the region3≤𝛺≤4Hz, it is possible to see that the perturbations for the helicopter with DVA are greater than the ones for the helicopter without DVA, which means that Helicopter Type 2 is more robust than Helicopter Type 1 in that region.
The robustness analysis made for different values of peak of real part does not assess the influence of fuselage and rotor blade damping in the robustness of the system, since for each value ofha different set ofCb andCxis considered. There- fore, robustness analyses are carried out by assuming isolated variations at fuse- lage damping coefficients (see Fig.4) and at blade damping coefficients (see Fig.5), respectively.
In order to evaluate the influence ofCxin helicopters robustness in Fig.4, the rotor blade damping was considered fixed atCb= 4500N m s/rad for Helicopter Type 1 andCb= 1250N m s/rad for Helicopter Type 2 and the fuselage damping was varied from900N s/m≤Cx≤9000N s/m with increments of900N s/m. For ease to see
Dynamics of Helicopters with DVA Under Structural Uncertainties 121
(a) Helicopter Type 1 (b) Helicopter Type 2
Fig. 5 The influence of blade damping on the rotor stability sensitivity due to blade stiffness asymmetry: a) Helicopter Type 1Cb= 4500N s/m; b) Helicopter Type 2Cb= 1250N s/m
the results, for both cases the fuselage damping were given subscript index in which Cx1= 900N s/m,Cx2= 1800N s/m, and so on untilCx10 = 9000N s/m.
One can see from Fig.4that the perturbations are practically the same in all the range of rotor speed, with the exception being in the region around𝛺= 3Hz. At this region one may conclude that the helicopter robustness is sensitive to the fuselage damping, specially concerning low values ofCxthat yield higher perturbations val- ues. Also, with the exception of that region, both helicopters have the same values for perturbation, meaning that the DVA has no effect on the robustness of the system, which corroborates with the conclusions made with the analysis of the influence of the peak of real part.
Now the influence of rotor blade damping is evaluated in Fig.5. For both heli- copters the fuselage damping is fixed atCx= 9000N s/m and the rotor blade damp- ing is varied from 500N m s/rad≤Cb≤4500N m s/rad. One can see from the results that the perturbations are exactly the same in all the range of rotor speed, which means that the rotor blade damping has no influence on the robustness of both systems. Also, both helicopters have the same values for perturbation, which once again corroborates with the previous conclusions.
6 Conclusion
Electrical or mechanical devices might be attached to the fuselage of helicopters, which may be modeled as spring-mass-damping vibration systems and can interact with the helicopter ground oscillations, which might alter the stability characteristics of the helicopter. The influence of the SMDVS parameters added to the fuselage of a helicopter were under investigation in the present paper. Firstly, the influence of fuselage and rotor blade damping, specially their combination, was evaluated. The
122 D. B. P. de Oliveira et al.
results showed that in order to vanish the instabilities due to ground resonance, the system should have the proper combination of damping. Also, it was possible to see that the addition of the DVA is good for the overall system stability, since by applying it to the fuselage, the helicopter can make use of smaller values for damping, which makes the system more compact and less complex.
Lastly, a stability robustness analysis of the entire system was performed by changing the stiffness properties of one rotor blade for the helicopters with and with- out DVA to assess how the system handles those variations. The results showed that for both helicopters, as the rotor speed increases, the system become more robust.
It was seen that with the proper combination of fuselage and rotor blade damping, the system can be so robust as to lose one of its rotor blade stiffness and still remain stable. Also, the analysis showed that in the region around the natural frequency of the fuselage, the robustness of the system is sensitive to the fuselage damping, since different values of it yield different values of perturbations.
Acknowledgements The authors would like to thank the French and Brazilian funding provided by CNPq (Scholarship IC-CNPQ2015-0139 and Project 460207/2014-8), through INCT-EIE and FAPEMIG.
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