60 A. H. Haslam et al. ratio at which this transition from FW modes increasing with speed to BW modes increasing with speed is at D 1. Note that for this counter-rotating system, in all the modes one of the rotors is whirling FW compared to its own spin direction, so it will be possible for more of the modes to be strongly excited by unbalances. The high speed mode shapes are plotted in Fig. 6.4b, and the same three types from the co-rotating case are observed. However, some of the low-speed modes now tend to different types of high-speed mode, when compared to the co-rotating case. The 2 highest frequency modes (modes 7 & 8, compared to modes 6 & 8 before) now tend to “precessional” mode shapes. This is because modes 7 & 8 are now the highest frequency modes where rotor 1 and 2 are whirling FW w.r.t their own direction of rotation respectively. The 2 lowest frequency modes (now modes 1 & 2, compared to modes 1 & 3 before) now tend to “static” mode shapes. This is because modes 1 & 2 are now the lowest frequency modes where rotor 2 and 1 are whirling BW w.r.t their own direction of rotation respectively. These are now both IP. The remaining modes become “flat” modes, but it is no longer the case that IP modes must tend towards IP “flat” modes and similarly for OP modes. For example, modes 3 & 4 start out as OP cylindrical modes @ 0 rpm, but tend towards IP ‘flat” modes at high speeds. There is a greater density of critical speeds at lower speeds than for the co-rotating system, as shown in Fig. 6.5. There are now 3 critical speeds for rotor 1 unbalances compared to only 2 before, as there is now an additional FW mode in the bottom half of the Campbell diagram. Rotor 2 unbalances now excite BW modes so the precessional mode cannot be excited @ 8000 rpm. However, there are still 3 critical speeds as before, because the engine order line crosses 3 BW modes in the bottom half of the Campbell diagram. 6.3.3 Summary It has been shown that the modes at low-speeds of the dual-rotor system can be classified based on the following 3 criteria: • Shape: Whether the mode appears “cylindrical” involving primarily translational motion, or “conical” involving primarily rotational motion • Whirl: Whether there is forwards or backward whirl • Phase: Whether the rotors move in-phase or out-of-phase with each other The first two criteria apply to single-rotor systems as well, but the third criterion is unique to dual-rotor systems. For co-rotating arrangements, the FW modes always increase in frequency with speed as with a single rotor. However, for counter-rotation the BW modes can instead increase with frequency depending on the speed ratio. Although not shown here, the specific rotor configuration will also have an influence. At high speeds, the modes tend towards 3 types of mode shape: • “Static”: Purely conical modes which tend towards very low frequencies. • “Precessional”: Modes where one rotor precesses around the other, and the natural frequency increases linearly with speed. • “Flat”: Purely cylindrical modes which tend towards constant non-zero natural frequencies. There are both in-phase and out-of-phase varieties. The same types of high-speed modes arise in both co- or counter-rotating systems. However, each type of low-speed mode can asymptote towards a different type of high-speed mode, depending on the speed ratio. 6.4 Parameter Study The focus then moved onto understanding how the dynamics of the system vary with certain key parameters. The mode shape and natural frequencies were computed at 0 rpm and 20,000 rpm, as well as the critical speeds. Unlike the rotor properties, there is often more design freedom in the choice of the inter-shaft bearing for rotating machinery, so the stiffness of this bearing kc was chosen as the first parameter to be varied. The relative rotation speed of the two rotors can often vary during operation of turbomachinery, so the speed ratio was chosen as the second parameter.
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