6 Analysis of the Dynamic Response of Coupled Coaxial Rotors 63 0 Ampl. Mode 1 18 20 22 24 26 0 Mode 2 18 20 22 24 26 ω(Hz) 0 Ampl. Mode 3 34 36 38 40 42 0 Mode 4 34 36 38 40 42 w (Hz) 0 Ampl. Mode 5 52 54 56 58 60 0 Mode 6 52 54 56 58 60 w (Hz) −4 −2 0 2 4 0 h Ampl. Mode 7 92 94 96 98 100 −4 −2 0 2 4 0 h Mode 8 92 94 96 98 100 w (Hz) (a) 0 Ampl. Mode 1 2 4 6 8 10 0 Mode 2 0 10 20 w (Hz) 0 Ampl. Mode 3 10 15 20 0 Mode 4 20 30 40 w (Hz) 0 Ampl. Mode 5 20 30 40 0 Mode 6 0 200 400 600 w (Hz) −4 −2 0 2 4 0 h Ampl. Mode 7 0 1,000 2,000 3,000 −4 −2 0 2 4 0 h Mode 8 1,000 2,000 3,000 w (Hz) (b) Fig. 6.9 Variation of the mode shapes of the dual-rotor system with speed ratio : rotor 1L( ), R( ); rotor 2 L ( ), R( ); natural frequency ( ). (a) Mode shapes @ 0 rpm. (b) Mode shapes @ 20,000 rpm For all the critical speeds which do vary with kc, the sensitivity to stiffness eventually decreases. This is because the elastic forces start to dominate when the stiffness is very large, so that the natural frequency ! pkc. The critical speeds will therefore tend towards a similar power law. 6.4.2 Rotor Speed Ratio The relative speed of the inner rotor was now varied by changing the speed ratio . The inter-shaft bearing stiffness was reset to its initial value kc D1MNm 1. Note that all modes have been numbered based on their frequencies at D2so that they agree with Figs. 6.2 and 6.3 for the baseline co-rotating case. 6.4.2.1 Natural Frequencies and Mode Shapes The natural frequencies have been plotted in Fig. 6.9, in the same way as for the bearing stiffness. The mode shapes and natural frequencies are entirely unaffected by the value of for low rotation speeds as shown in Fig. 6.9a, because gyroscopic loads are not significant in this regime. However, the high speed behaviour is significantly affected as shown in Fig. 6.9b, and all of the modes have different behaviour for positive and negative values of . Mode 1 is always a “static” mode as discussed in Sect. 6.3.2, but rotor 1 has a larger displacement for < 0 and rotor 2 for > 0. The frequency of this mode decay towards 0 Hz more slowly when < 0 because the gyroscopic moments of the 2 rotors partially cancel out. Mode 8 is always a “precessional” mode as discussed in Sect. 6.3.2, but rotor 1 precesses for >1 and rotor 2 precesses for <1. This is because changes which rotor has the highest angular momentum and therefore the highest precession frequency. The frequency increases rapidly with when >1, because scales the rotor 2 precession frequency. The type of asymptotic behaviour of the other modes changes with the sign of . Modes 6 & 7 switch between different “flat” modes with lower frequencies and “precessional” modes with considerably higher frequencies. Modes 2 & 3 similarly switch between “static” modes and “flat” modes. These changes arise because the whirl direction of each mode is affected by the value ; however it must remain the case that the 2 highest modes where each rotor whirls FW must tend towards “precessional” modes; and the 2 lowest modes where each rotor whirls BW must tend towards “static” modes, as discussed in Sect. 6.3.2. Modes 4 & 5 remain “flat” modes, because these modes are never the highest FW or BW modes, but they do swap between the IP and OP varieties.
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